Heft 219 Ferdinand Beck 
 
Generation of Spatially Correlated 
Synthetic Rainfall Time Series in  
High Temporal Resolution -  
A Data Driven Approach 
 
 
 
 
 
Generation of Spatially Correlated Synthetic Rainfall Time 
Series in High Temporal Resolution 
A Data Driven Approach 
 
 
 
 
 
 
 
Von der Fakultät Bau- und Umweltingenieurwissenschaften der 
Universität Stuttgart zur Erlangung der Würde eines 
Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung  
 
 
 
Vorgelegt von 
Ferdinand Beck 
aus Backnang, Deutschland 
 
 
 
 
Hauptberichter: Prof. Dr. rer.nat. Dr.-Ing. András Bárdossy 
Mitberichter: Prof. Dr.-Ing. Uwe Haberlandt 
 
 
 
Tag der mündlichen Prüfung: 12. November 2012 
 
 
 
 
 
 
 
 
Institut für Wasser- und Umweltsystemmodellierung 
der Universität Stuttgart 
2013 
 
 
 
 
 
 
 
Heft 219 Generation of Spatially 
Correlated Synthetic Rainfall 
Time Series in High Temporal 
Resolution - A Data Driven 
Approach 
 
 
 
 von 
Dr.-Ing. 
Ferdinand Beck 
 
 
 
 
 
 
 
 
 
 
 
Eigenverlag des Instituts für Wasser- und Umweltsystemmodellierung 
der Universität Stuttgart 
D93 Generation of Spatially Correlated Synthetic Rainfall Time Series in 
High Temporal Resolution - A Data Driven Approach 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über 
http://www.d-nb.de abrufbar 
 
 
Beck, Ferdinand 
Generation of Spatially Correlated Synthetic Rainfall Time Series in High Temporal 
Resolution - A Data Driven Approach von Ferdinand Beck. Institut für 
Wasser- und Umweltsystemmodellierung, Universität Stuttgart. - Stuttgart: 
Institut für Wasser- und Umweltsystemmodellierung, 2013 
 
(Mitteilungen Institut für Wasser- und Umweltsystemmodellierung, Universität 
Stuttgart: H. 219) 
Zugl.: Stuttgart, Univ., Diss., 2013 
ISBN 978-3-942036-23-8 
 
NE: Institut für Wasser- und Umweltsystemmodellierung <Stuttgart>: Mitteilungen 
 
 
Gegen Vervielfältigung und Übersetzung bestehen keine Einwände, es wird lediglich 
um Quellenangabe gebeten. 
 
 
 
Herausgegeben 2013 vom Eigenverlag des Instituts für Wasser- und Umwelt-
systemmodellierung 
 
Druck: Document Center S. Kästl, Ostfildern
Danksagung
Mein herzlichster Dank gilt Herrn Professor Bárdossy; nicht nur für die Möglichkeit an
seinem Lehrstuhl mitzuarbeiten, sondern vor allem für die vielen Ideen und sein großes hy-
drologisches und statistisches Wissen, an dem er mich über die Jahre hat teilhaben lassen.
Er hat mir die Faszination der Statistik näher gebracht, was meine Sicht auf die Welt mit
prägen wird.
Genauso möchte ich Herrn Professor Haberlandt herzlich dafür danken, dass er den Mit-
bericht übernommen hat; für seine sehr freundliche, hilfsbereite und unkomplizierte Be-
treuung.
Vielen Dank an Herrn Professor Helmig für seine umgängliche Art als Leiter der Prü-
fungskommission und die sehr angenehme Atmosphäre während der Prüfung.
In meinen Jahren am Institut für Wasserbau habe ich das Arbeitsklima dort immer als sehr
freundschaftlich und fröhlich empfunden. Ich bin jeden Tag gerne dort hingegangen und
dafür möchte ich den Kollegen in der Abteilung für Hydrologie und Geohydrologie herzlich
danken, besonders den Mitstatistikern Dirk und Philipp für interessante und fordernde
Diskussionen und, ganz wichtig, Thomas dafür, dass er mir “richtig” programmieren beige-
bracht hat.
Vielen Dank meinen Eltern, dafür, dass es sie gibt, dass es mich gibt und dass sie mir
meine schulische Ausbildung und mein Studium ermöglicht haben. Nicht zuletzt danke
ich meinen Brüdern und meinen Freunden, für die Zerstreuung, den Spaß und manchmal
auch das offene Ohr, wenn es doch mal zu viel wurde mit der Statistik.

Contents
List of Figures V
List of Tables IX
List of Abbreviations XI
Notation XII
Zusammenfassung XV
1. Introduction 1
1.1. Context and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Characterization of the Hydrological Conditions in Baden-Württemberg 5
2.1. Context in the Global Atmospheric Circulation . . . . . . . . . . . . . . . . . . 5
2.2. Regional Conditions in Baden-Württemberg . . . . . . . . . . . . . . . . . . . 10
3. Acquisition of Precipitation Data 12
3.1. Point Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1. Rain Gauge after Hellmann . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.2. Disdrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.3. Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.3.1. Examples of Recording and Data Transmission Errors . . . . 14
3.1.3.2. Some Aspects of Error Identification . . . . . . . . . . . . . . 16
3.2. Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1. Radar Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2. Satellite Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3. The Data Set of Precipitation Measurements . . . . . . . . . . . . . . . . . . . . 23
3.3.1. Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4. Statistical Properties of Precipitation 27
4.1. Spatial and Temporal Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2. Intermittency and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3. Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5. Modeling of Precipitation 33
5.1. Purpose of Precipitation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2. Spatial Interpolation of Precipitation Data . . . . . . . . . . . . . . . . . . . . . 35
II Contents
5.3. Markov Chain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.4. Point Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.5. Alternating Renewal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.6. Scaling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.6.1. Autoregressive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6. Generation of Point Rainfall 55
6.1. The Synthetic Rainfall Time Series Generator NiedSim . . . . . . . . . . . . . . 55
6.1.1. Necessary Adaptations to NiedSim . . . . . . . . . . . . . . . . . . . . 56
6.2. Statistical Parameters of Hourly Precipitation Time Series . . . . . . . . . . . . 58
6.3. Methods for the Statistical Description of Distributions . . . . . . . . . . . . . 59
6.3.1. Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.3.2. Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3.3. Pareto Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3.4. Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3.5. Comparison of Theoretical Distributions . . . . . . . . . . . . . . . . . 63
6.3.6. Maximum Likelihood Method for the Fit of Distribution Functions . . 64
6.4. Methods for the Description of Multivariate Dependencies . . . . . . . . . . . 64
6.4.1. Covariance and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.4.2. Spearman’s Rank Correlation . . . . . . . . . . . . . . . . . . . . . . . . 65
6.4.3. Copulas as a Measure of Dependence . . . . . . . . . . . . . . . . . . . 66
6.4.3.1. Empirical Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4.3.2. Theoretical Copulas . . . . . . . . . . . . . . . . . . . . . . . . 68
6.4.3.3. The Gaussian Copula . . . . . . . . . . . . . . . . . . . . . . . 69
6.5. Analysis of Observed Precipitation Data . . . . . . . . . . . . . . . . . . . . . . 70
6.5.1. Multivariate Dependence in Observed Data . . . . . . . . . . . . . . . 70
6.5.1.1. Gaussian Copula Dependence used for Rainfall Generation . 75
6.5.2. Observed Marginal Distributions . . . . . . . . . . . . . . . . . . . . . . 75
6.5.2.1. Marginal Distributions used for Rainfall Generation . . . . . 81
6.6. The Generation Algorithm for 1h Precipitation Time Series . . . . . . . . . . . 83
6.6.1. Parameter Estimation for the Initial Time Series . . . . . . . . . . . . . 83
6.6.2. Rainfall Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.6.2.1. Statistical Parameters in the Objective Function . . . . . . . . 86
6.7. Comparison of Simulated and Observed Rainfall Time Series . . . . . . . . . . 88
6.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7. Spatial Generation 94
7.1. A High Dimensional Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2. Generation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3. Observed Multivariate Dependencies in Daily Precipitation Data . . . . . . . 96
7.4. Marginal Distributions of Daily Precipitation Data . . . . . . . . . . . . . . . . 99
7.5. Representation of Spatial Dependencies . . . . . . . . . . . . . . . . . . . . . . 101
7.5.1. Observed Interdependencies of Statistical Parameters at Different
Measurement Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.5.1.1. Analysis of Anisotropy . . . . . . . . . . . . . . . . . . . . . . 104
Contents III
7.5.1.2. Spatial Rank Correlations Used for the Generation of Simul-
taneous Daily Time Series . . . . . . . . . . . . . . . . . . . . 107
7.5.2. Spatial Correlation According to Atmospheric Circulation Patterns . . 109
7.5.2.1. Definition of the Circulation Pattern Classification System . . 109
7.5.2.2. CP Dependent Precipitation Statistics . . . . . . . . . . . . . . 111
7.5.2.3. Analysis of Spatial Correlation . . . . . . . . . . . . . . . . . . 111
7.5.2.4. CP Dependent Correlations Used For the Generation of Si-
multaneous Time Series . . . . . . . . . . . . . . . . . . . . . . 121
7.6. Generation Algorithm for Simultaneous 1h Precipitation Time Series . . . . . 122
7.6.1. Parameter Estimation for the Initial Daily Time Series . . . . . . . . . . 122
7.6.2. Generation of Simultaneous Daily Precipitation Time Series . . . . . . 124
7.6.3. Disaggregation to hourly values . . . . . . . . . . . . . . . . . . . . . . 126
7.7. Comparison of Simulated and Observed Simultaneous Time Series . . . . . . 127
7.7.1. Comparison of Observed and Simulated Distributions . . . . . . . . . 128
7.7.2. Representation of the Spatial Dependence . . . . . . . . . . . . . . . . . 138
7.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8. Some Aspects of Climate Change 146
8.1. Trend Signals in Observed Precipitation Time Series . . . . . . . . . . . . . . . 148
8.1.1. Regional Trend in Extreme Precipitation . . . . . . . . . . . . . . . . . . 149
8.1.2. Change in Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.2. Global and Regional Circulation Models . . . . . . . . . . . . . . . . . . . . . . 157
8.2.1. Emission Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.2.2. Model Errors in Global and Regional Circulation Models . . . . . . . . 159
8.2.3. Correction and Downscaling of GCM Precipitatioin Data . . . . . . . . 160
8.2.3.1. Direct Correction of GCM Precipitation Data . . . . . . . . . 160
8.2.3.2. Indirect Correction by Additional Information . . . . . . . . . 163
8.3. Prediction of Future Precipitation by GCM Defined Circulation Patterns . . . 164
8.3.1. Changes in CP-Frequencies According to GCM and Reanalysis Data . 167
8.3.2. Temperature Sensitivity of CP-related Precipitation . . . . . . . . . . . 171
8.3.3. Expected Trend in 1h Precipitation Amounts due to Changes in Atmo-
spheric Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.4.1. Reliability of Estimated Trend Signals . . . . . . . . . . . . . . . . . . . 180
8.4.2. Potential for Climate Change Adaptations in Rainfall Generation . . . 181
8.4.3. Future Predictions as an Ensembles . . . . . . . . . . . . . . . . . . . . 182
9. Conclusions and Outlook 184
Bibliography 187
A. Generation of Point Rainfall 197
A.1. Marginal Distributions of All 292 Stations . . . . . . . . . . . . . . . . . . . . . 197
A.1.1. Monthly Precipitation Sum . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.1.2. Average Precipitation in Wet Hours . . . . . . . . . . . . . . . . . . . . 199
IV Contents
A.1.3. Standard Deviation of Precipitation Amounts in Wet Hours . . . . . . 201
A.1.4. Hourly Rainfall Probability . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.2. Drawing Conditioned Values from a Multivariate Normal Distribution . . . . 205
B. Spatial Generation 206
B.1. Observed Multivariate Dependencies in Daily Precipitation Data . . . . . . . 206
B.2. Marginal Distribution in Daily Resolution . . . . . . . . . . . . . . . . . . . . . 209
B.2.1. Average Precipitation in Wet Hours . . . . . . . . . . . . . . . . . . . . 209
B.2.2. Standard Deviation of Precipitation Amounts in Wet Hours . . . . . . 211
B.2.3. Daily Rainfall Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 213
B.3. Representation of Spatial Dependencies . . . . . . . . . . . . . . . . . . . . . . 215
List of Figures
1. Beispiel empirischer Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX
2. Empirische Copula Monatssumme/Regenwahrscheinlichkeit . . . . . . . . . XIX
3. Jährliche Serie Holzgerlingen . . . . . . . . . . . . . . . . . . . . . . . . . . . . XX
4. Isobarenkarte, CP2 und CP3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXII
5. Räumliche Korrelation CP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII
6. Räumliche Korrelation CP3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIV
2.1. Average number of rain days, montly sum Freiburg . . . . . . . . . . . . . . . 8
2.2. Average number of rain days, montly sum Freiburg . . . . . . . . . . . . . . . 8
2.3. Elevation and Yearly Precipitation Baden-Württemberg . . . . . . . . . . . . . 9
3.1. Example of an error in a precipitation time series, probably occurred during
data transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2. Example of an error in a precipitation time series, probably due to unreported
calibration and maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3. Example of an error in a precipitation time series, probably due to unreported
calibration and maintenance (continued) . . . . . . . . . . . . . . . . . . . . . . 17
3.4. Hourly and daily precipitation stations . . . . . . . . . . . . . . . . . . . . . . 24
3.5. Subset of 137 high resolution rain gauges with continuous measurements
from 1997 to 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6. Subset of 30 high resolution rain gauges with continuous measurements from
1991 to 2003 and low number of missing values. . . . . . . . . . . . . . . . . . 26
4.1. Observed precipitation data: correlation vs. distance . . . . . . . . . . . . . . . 27
5.1. Schematic for Point Process Rectangular Pulse Model . . . . . . . . . . . . . . 43
6.1. Principle of the permutation scheme in NiedSim . . . . . . . . . . . . . . . . . 56
6.2. Empirical histogram Neckarsulm . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3. Effect of wrong fit - Log-Normal Distribution . . . . . . . . . . . . . . . . . . . 63
6.4. Gauss copula density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.5. Monthwise empirical bivariate copula, monthly sum - 1h rainfall probability 71
6.6. Monthwise empirical bivariate copula, monthly sum - average 1h rainfall
amount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.7. Monthwise empirical bivariate copula, monthly sum - standard deviation of
1h rainfall amount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.8. Monthwise empirical bivariate copula, rainfall probability - average of 1h
rainfall amounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
VI List of Figures
6.9. Monthwise empirical bivariate copula, monthly sum - standard deviation of
1h rainfall amount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.10. Monthwise empirical bivariate copula, average - standard deviation of 1h
rainfall amount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.11. Marginal distribution of monthly precipitation sum, Holzgerlingen . . . . . . 77
6.12. Marginal distribution of average precipitation of wet hours, Holzgerlingen . 78
6.13. Log-Normal and Lomax Distribution moof nthly precipitation sum, Geislin-
gen Waldhof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.14. Marginal distribution of standard deviation of wet hours, Holzgerlingen . . . 80
6.15. Marginal distribution of hourly rainfall frequency, Holzgerlingen . . . . . . . 81
6.16. Simulated monthly sum, Holzgerlingen . . . . . . . . . . . . . . . . . . . . . . 88
6.17. Annual series Holzgerlingen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.18. Annual series - 1h January an July, Holzgerlingen . . . . . . . . . . . . . . . . 91
6.19. Wet spells, Holzgerlingen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.20. 100 longest wet spells, Holzgerlingen . . . . . . . . . . . . . . . . . . . . . . . 92
6.21. Dry spells, Holzgerlingen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.22. 100 longest dry spells, Holzgerlingen . . . . . . . . . . . . . . . . . . . . . . . . 93
7.1. Marginal distribution of average wet day precipitation, Schönaich . . . . . . . 99
7.2. Marginal distribution of standard deviation of wet day precipitation, Schönaich 99
7.3. Marginal distribution of rain day frequency, Schönaich . . . . . . . . . . . . . 100
7.4. Absolute frequency of the distances between 575 daily precipitation stations . 101
7.5. Correlation vs. distance – rain day frequency . . . . . . . . . . . . . . . . . . . 102
7.6. Correlation vs. distance – average vs. standard deviation of wet day precipi-
tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.7. Sketch of the sector division for rank correlation analysis . . . . . . . . . . . . 105
7.8. Correlation vs. distance in different orientated sectors . . . . . . . . . . . . . 106
7.9. Principle of the fuzzy rule based CP classification . . . . . . . . . . . . . . . . 110
7.10. Average pressure anomalies for CPs . . . . . . . . . . . . . . . . . . . . . . . . 112
7.11. Direction dependent correlations – CP2 . . . . . . . . . . . . . . . . . . . . . . 114
7.12. Direction dependent correlations – CP3 . . . . . . . . . . . . . . . . . . . . . . 115
7.13. Direction dependent correlations – CP15 . . . . . . . . . . . . . . . . . . . . . . 116
7.14. Standard deviation of correlation values CP2 . . . . . . . . . . . . . . . . . . . 118
7.15. Standard deviation of correlation values CP3 . . . . . . . . . . . . . . . . . . . 119
7.16. Standard deviation of correlation values CP15 . . . . . . . . . . . . . . . . . . 120
7.17. Principle of the Kriging interpolation of CP dependent correlations . . . . . . 121
7.18. Matrix of the parameter correlations in the standard normal transformed
space for the generation of simultaneous time series . . . . . . . . . . . . . . . 123
7.19. Overview of the target location for the generation of simultaneous time series 128
7.20. Annual Series – multi generation Holzgerlingen 1 . . . . . . . . . . . . . . . . 131
7.21. Annual Series – multi generation Holzgerlingen 1 . . . . . . . . . . . . . . . . 132
7.22. Annual Series – multi-generation, Holzgerlingen January and July . . . . . . 133
7.23. Wet spells, multi-generation Holzgerlingen . . . . . . . . . . . . . . . . . . . . 134
7.24. 100 longest wet spells, multi-generation Holzgerlingen . . . . . . . . . . . . . 135
7.25. Dry spells, multi-generation Holzgerlingen . . . . . . . . . . . . . . . . . . . . 136
List of Figures VII
7.26. 100 longest dry spells, multi-generation Holzgerlingen . . . . . . . . . . . . . 137
8.1. Regional trend in extreme precipitation, Baden-Württemberg 1991 to 2003 . . 150
8.2. Regional trend in extreme precipitation, Bavaria 1990 to 2006 . . . . . . . . . . 152
8.3. Average scaling 1985-1980 to 1981-2003 . . . . . . . . . . . . . . . . . . . . . . 154
8.4. Principle of the change in scaling exponent . . . . . . . . . . . . . . . . . . . . 155
8.5. Map of change in scale exponent . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.6. Scenarios for Green House Gas (GHG) emissions from 2000 to 2100 (Nakicen-
ovic and Swart, 2000) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.7. Example of QQ-Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.8. Principle of the objective function for the CP definition in the simulated an-
nealing scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.9. MSLP of CPs leading to high precipitation intensites . . . . . . . . . . . . . . . 166
8.10. Exceedance frequency of the 95% and 99% quantile for two seasons. Quantiles
and frequencies are calculated for all hours with H ≥ 0.1mm . . . . . . . . . . 167
8.11. Absolute frequency of anticyclonic CPs (CP2, CP3, CP5 and CP8) according
to NCEP/NCAR reanalysis during two seasons . . . . . . . . . . . . . . . . . 168
8.12. Absolute frequency of cyclonic CPs (CP7, CP10, CP11 and CP12) according to
NCEP/NCAR reanalysis during two seasons . . . . . . . . . . . . . . . . . . . 168
8.13. Trends in frequency of anticyclonic CPs . . . . . . . . . . . . . . . . . . . . . . 169
8.14. Trends in frequency of cyclonic CPs . . . . . . . . . . . . . . . . . . . . . . . . 170
8.15. MSLP of CPs leading to high precipitation intensities - ECHAM5 . . . . . . . 171
8.16. Precipitation frequency and extreme value frequency vs. temperature – CP2 . 172
8.17. Precipitation frequency and extreme value frequency vs. temperature – CP7 . 173
8.18. Precipitation frequency and extreme value frequency vs. temperature – CP11 174
8.19. Expected statistics 1961 to 2060 – ECHAM5 . . . . . . . . . . . . . . . . . . . . 176
8.20. Linear trend in precipitation statistics –ECHAM5 . . . . . . . . . . . . . . . . . 178
8.21. Linear trend in precipitation statistics –ECHAM5 May to August . . . . . . . 179
A.1. Histogram and Weibull PDF - monthly precipitation sum . . . . . . . . . . . . 197
A.2. Linearized Weibull fit - monthly precipitation sum . . . . . . . . . . . . . . . . 198
A.3. Histogram and Log-Normal PDF - average wet hour precipitation . . . . . . . 199
A.4. Linearized Log-Normal fit - average wet hour precipitation . . . . . . . . . . . 200
A.5. Histogram and Log-Normal PDF - standard deviation of wet hour precipitation201
A.6. Linearized Log-Normal fit - standard deviation of wet hour precipitation . . . 202
A.7. Histogram and Beta PDF - hourly rainfall probability . . . . . . . . . . . . . . 203
A.8. Beta CDF - hourly rainfall probability . . . . . . . . . . . . . . . . . . . . . . . 204
B.1. Monthwise empirical bivariate copula, monthly sum - rainfall probability;
daily values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
B.2. Monthwise empirical bivariate copula, monthly sum - average wet day pre-
cipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
B.3. Monthwise empirical bivariate copula, monthly sum - standard deviation of
wet day precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
VIII List of Figures
B.4. Monthwise empirical bivariate copula, rainfall frequency - average wet day
precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.5. Monthwise empirical bivariate copula, rainfall frequency - standard deviation
of wet day precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
B.6. Monthwise empirical bivariate copula, average - standard deviation of wet
day precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
B.7. histogram and Log-Normal PDF - average wet day precipitation . . . . . . . . 209
B.8. linearized Log-Normal fit -average wet day precipitation . . . . . . . . . . . . 210
B.9. histogram and Log-Normal PDF - standard deviation of wet day precipitation 211
B.10. linearized Log-Normal fit - standard deviation of wet day precipitation . . . . 212
B.11. histogram and Beta PDF - daily rainfall frequency . . . . . . . . . . . . . . . . 213
B.12. Beta CDF - daily rainfall frequency . . . . . . . . . . . . . . . . . . . . . . . . . 214
B.13. Correlation vs. distance – monthly sum . . . . . . . . . . . . . . . . . . . . . . 215
B.14. Correlation vs. distance – average wet day precipitation . . . . . . . . . . . . . 215
B.15. Correlation vs. distance – standard deviation of wet day precipitation . . . . 216
B.16. Correlation vs. distance – monthly sum vs. rain day frequency . . . . . . . . . 216
B.17. Correlation vs. distance – monthly sum vs. average precipitation . . . . . . . 217
B.18. Correlation vs. distance – monthly sum vs. standard deviation of wet day
precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.19. Correlation vs. distance – rain day frequency vs. average wet day precipitation 218
B.20. Correlation vs. distance – rain day frequency vs. average wet day precipitation 218
List of Tables
6.1. Rank correlation and symmetry monthwise . . . . . . . . . . . . . . . . . . . . 72
6.2. Correlation hourly precipitation statistics . . . . . . . . . . . . . . . . . . . . . 76
6.3. Distribution functions used in the parameter estimation for the rainfall gen-
eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4. Basic statistics simulation for Holzgerlingen . . . . . . . . . . . . . . . . . . . . 89
7.1. Rank correlation and symmetry monthwise - daily values . . . . . . . . . . . . 97
7.2. Correlations in Standard Normal transformed daily precipitation statistics . . 98
7.3. Distribution functions used in the parameter estimation for the generation of
simultaneous daily rainfall time series . . . . . . . . . . . . . . . . . . . . . . . 101
7.4. Parameters of the regression functions according to Eq. (7.5) for the estimation
of parameter rank correlation over distance - same parameter at both stations 107
7.5. Parameters of the regression functions according to Eq. (7.5) for the estima-
tion of parameter rank correlation over distance - combination of different
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.6. Basic daily precipitation statistic for all anticyclonic, dry CPs . . . . . . . . . . 113
7.7. Basic daily precipitation statistic for all moderate humid CPs . . . . . . . . . . 113
7.8. Basic daily precipitation statistic for all cyclonic humid CPs . . . . . . . . . . . 113
7.9. Coordinates - simultaneous simulation . . . . . . . . . . . . . . . . . . . . . . . 127
7.10. Basic hourly statistics Holzgerlingen - multi site generation . . . . . . . . . . . 129
7.11. Basic daily statistics Holzgerlingen - multi site generation . . . . . . . . . . . . 129
7.12. Daily correlation – simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.13. Daily correlation – target values . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.14. Daily correlation – simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.15. Daily correlation – target values . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.16. hourly correlation - simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.17. hourly correlation w. time shift - simulations . . . . . . . . . . . . . . . . . . . 140
7.18. hourly correlation w. time shift - observed . . . . . . . . . . . . . . . . . . . . . 140
7.19. hourly correlation with time shift - simulations with high distance . . . . . . . 140
7.20. CP-dependent correlation, hourly values . . . . . . . . . . . . . . . . . . . . . 141
7.21. CP-dependent correlation, target values . . . . . . . . . . . . . . . . . . . . . . 142
7.22. CP-dependent correlation, hourly values with time shift, Oct-Mar . . . . . . . 143
7.23. CP-dependent correlation, hourly values with time shift, Sep-Apr . . . . . . . 144
8.1. Significance test by Monte Carlo simulation of the regional trend signals in
Baden-Württemberg from 1991 to 2003. . . . . . . . . . . . . . . . . . . . . . . 151
X List of Tables
8.2. Significance test by Monte Carlo simulation of the regional trend signals in
Bavaria from 1990 to 2006. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.3. Significance test by Monte Carlo simulation of the regional trend signals in
Bavaria from 1991 to 2003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
List of Abbreviations
APREGE French Global Circulation Model
AR(n) model Auto Regressive Model of order n
CDF Cumulative Distribution Function
CP Circulation Pattern for weather type classification
DDF Depth-Duration-Frequency Curve
DOF Degree of Fulfillment (of Fuzzy Logic rule)
DWD Deutscher Wetterdienst, German meteorological service
ECHAM4 / EHAM5 Global Circulation Model of the Max Planck Institute
Hamburg, Germany
GCM Global Circulation Model
GK X Easting in Gauss Krüger 3 Coordinates
GK Y Northing in Gauss Krüger 3 Coordinates
HadCM3 Global Circulation Model of the Hadley Center for
Climate Prediction and Research, UK
IPCC Intergovernmental Panel on Climate Change of the UN
ITCZ Intertropical Convergence Zone
KOSIM KOntinuierliches Langzeit- SIMulationsmodell,
hydraulic model for sewage system simulations
KOSTRA KOordinierte STarkniederschlags-Regionalisierungs
Auswertungen, official German statistics of extreme
precipitation
Lat Latitude
Lon Longitude
LUBW Landesanstalt für Umwelt, Messungen und Naturschutz
Baden-Württemberg, environmental authority
MLM Maximum Likelihood Method
MSLP Mean Sea Level Pressure
NCAR National Center for Atmospheric Research
of the United States
NCEP National Centers for Environmental Prediction
of the United States
QQ-Transformation Quantile-Quantile Transformation
RCM Regional Circulation Model
SST Sea Surface Temperature
TMPA TRMM Multiple Precipitation Analysis
TRMM Tropical Rainfall Measurement Mission
WMO World Meteorological Organization
Notation
The following table shows the significant symbols used in this work. Local notations are
explained in the text.
Greek Letters:
α significance of hypothesis test
α, β parameters of Beta-distribution
γ variogram
∆ scale, aggregation interval
κ, λ parameters of Weibull Distribution
λ factor of scale change
λi weight of the measurement at location i in interpolation
µ, σ average and standard deviation of Normal distributed random variable
ρxy Spearman’s rank correlation between x and y
Σ covariance matrix
Φ distribution function of the Standard Normal Distribution
Ω support of a distribution function
Latin Letters:
h scaling exponent
C copula
c copula density
D (X) expectation of standard deviation of X
dij distance between station pair i, j
d0 reference distance in exponential regression
E (X) expectation of X
F (X) theoretical distribution function
f(X) probability density
Fn (x) observed non-exceedance frequency of x in a sample of size n
g skewness
H precipitation depth
Hwet precipitation depth of wet time steps
h relative frequency
Kq statistical moment of order q
n number of elements, e. g. in a time series
O objective function
P probability
Notation XIII
P1h rainfall probability on hourly temporal resolution
P24h rain day probability
p observed frequency
R,Z radar reflectivity and calculated rain rate
r correlation matrix
rxy Pearson correlation between x and y
rl observed autocorrelation of lag l
r (x) , q (y) ranks of x, y
s standard deviation
swet observed standard deviation of wet time steps
sxy covariance
sym symmetry of a bivariate copula
T Simulated Annealing Temperature
T time period
U, V uniformly distributed random variables
V CP defining Fuzzy rule set
vmax normalized maximum of x
w cascade generator in fractal scaling models
X,Y random variables
Xˆ best estimate for x, e. g. by regression
x observed data (realization of X)
x¯ average of x
Z standard normal transformed random variable
z realisation of Z

Zusammenfassung
Einleitung
Eine ausreichende Versorgung mit Trinkwasser ist die Grundlage alles menschlichen Le-
bens. Die Hauptquellen für Trinkwasser sind Grundwasserleiter oder Oberflächengewäs-
ser, die durch Niederschläge gespeist werden. Darum ist die natürliche Variabilität des
Niederschlags eine ständige Bedrohung: Ein Mangel an Niederschlag führt zu Dürren und
kann im schlimmsten Fall Hungersnöte auslösen. Ein Zuviel an Niederschlag führt zu Über-
schwemmungen, die nicht nur hohe Infrastruktur Schäden bringen, sondern regelmäßig zu
Todesopfern führen. In den USA beispielsweise sterben jedes Jahr rund 100 Menschen bei
Überschwemmungen (Ashley and Ashley, 2008). Darüber hinaus können intensive Nieder-
schläge Ernten vernichten und führen zu Bodenerosion. Weitere Risiken sind wasserbürtige
Krankheiten oder gefährliche Infektionen durch kontaminiertes Trinkwasser. Die zivilisato-
rische Entwicklung ist stark davon abhängig, diese Risiken beherrschbar zu machen. Grö-
ßere Siedlungen sind ohne wasserbauliche Strukturen für eine stetige Versorgung mit Trink-
wasser, für den Hochwasserschutz und für die Entsorgung von Abwasser nicht möglich.
Solche Strukturen, wie z. B. städtische Kanalnetze sind sehr langfristige und kostenintensive
Investitionen. Die richtige Dimensionierung ist darum entscheidend. Überdimensionierung
führt zu unnötigem Ressourceneinsatz, Unterdimensionierung erhöht das Versagensrisiko
des Systems. Bei der Auslegung hydraulischer Strukturen wird deshalb versucht, den opti-
malen Kompromiss zwischen den Konstruktions- und Unterhaltskosten einerseits und den
Schadenskosten durch nicht beherrschte Extremereignisse andererseits zu finden.
Heutzutage werden viele Dimensionierungsprobleme mit dem Einsatz hydraulischer oder
hydrologischer Modelle, wie z. B. Niederschlagsabflussmodelle für Flusseinzugsgebiete ge-
löst. Die eingesetzten Modelle werden dabei kontinuierlich verbessert, indem immer mehr
Prozesse des natürlichen Abflussregimes numerisch abgebildet werden. Daraus entsteht je-
doch ein neues Problem: Je detaillierter das numerische Modell ist, desto höher ist dessen
Bedarf an beobachteten Messdaten als Eingangsgrößen. Limitierender Faktor ist darum häu-
fig das Datenangebot, so dass die existierenden Modelle gar nicht in voller Komplexität ge-
nutzt werden können.
Besonders deutlich wird diese Einschränkung in der Simulation von Kanalnetzsystemen.
Kanalnetzsysteme reagieren sehr schnell auf Niederschlagereignisse mit Konzentrations-
zeiten von teilweise unter einer Stunde. Eine adäquate Simulation bedarf entsprechend
hochaufgelöster Niederschlagszeitreihen als Eingangsdaten. Da die Reaktion eines Kanal-
netzes von der Vorfüllung der Kanäle abhängt, reicht die Simulation eines einzelnen Nie-
derschlagsereignisses nicht aus. Für eine korrekte Abschätzung von Rückstauwahrschein-
lichkeiten, ist eine Langzweitsimulation von idealerweise mehreren Jahrzehnten notwendig.
ZUSAMMENFASSUNG
Betrachtet man Niederschlag in hoher zeitlicher Auflösung, so ist die räumliche Variabilität
sehr hoch. Eine beobachtete Zeitreihe von 1 Stundenwerten ist nur für eine sehr begrenz-
te Fläche wirklich repräsentativ. Selbst ein engmaschiges Niederschlagsmessnetz wie das
in Baden-Württemberg kann deshalb die benötigten langen und hochaufgelösten Nieder-
schlagszeitreihen nicht flächendeckend liefern.
Als Ersatz können statistische Modelle eingesetzt werden, um synthetische Zeitreihen zu
erzeugen, die tatsächlich beobachtetem Niederschlag möglichst genau entsprechen.
Statistische Modelle zur Erzeugung von synthetischen Niederschlagszeitreihen
Es existiert eine Vielzahl statistischer Konzepte zur Erzeugung synthetischer Niederschlags-
zeitreihen. Die Konzeption der meisten Modelle basiert darauf, dass die als grundlegend
angesehenen statistischen Eigenschaften des Niederschlags im Modell nachgebildet wer-
den. Erfassen diese die (unbekannten) physikalischen Prozesse des Niederschlagsgesche-
hens möglichst genau, so die Annahme, werden auch nicht in der Modellkonzeption ver-
wendete Eigenschaften des Niederschlags richtig abgebildet.
Markov-Ketten Modelle beispielsweise gehen vom Gedächtnis der Niederschlagszeitreihe
als grundlegende Eigenschaft aus. Markov Ketten simulieren den Niederschlag in diskreten
Zeitschritten. Das Auftreten von Regen zu einem Zeitschritt wird als Zufallsprozess verstan-
den, dessen Wahrscheinlichkeit vom Zustand (Regen / kein Regen) vorheriger Zeitschritte
bestimmt wird. Die Niederschlagshöhe für die nassen Zeitschritte wird dabei meist sepa-
rat modelliert (z. B. Katz, 1977). Die Anwendung von Markov Ketten Modellen auf zeitlich
hochaufgelöste Zeitreihen ist sehr komplex, weil hierbei sehr viele vergangene Zeitschritte
miteinbezogen werden müssen (Katz and Parlange, 1995). Dies führt zu einer hohen Anzahl
an Parametern und einer sehr datenintensiven Kalibrierung.
Punktprozess-Modelle simulieren den Niederschlag als eine Abfolge von zufällig eintreffen-
den niederschlagsaktiven Einheiten, welche hierarchisch organisiert sind: Die größte Einheit
bilden “Stürme”, in denen eine zufällige Anzahl von niederschlagsaktiven “Zellen“ einge-
lagert sind (Rodriguez-Iturbe et al., 1987). In der Simulierung hochaufgelöster Zeitreihen
können die Zellen noch in “Pulse” unterteilt werden (Cowpertwait et al., 2007). Da sich die
Grundeinheiten des Modells überlagern können, sind sie in gemessenen Zeitreihen nicht
direkt beobachtbar. Die Kalibrierung von Punktprozessmodellen erfolgt daher meist über
numerische Methoden. Eine korrekte Parametrisierung ist noch immer Gegenstand aktuel-
ler Forschung (Vanhaute et al., 2012).
In Alternating-Renewal Modellen wird die Überlappung der Grundeinheiten nicht zugelas-
sen, was die Kalibrierung vereinfacht. Die Zeitreihe besteht aus sich abwechselnden Nass-
und Trockenzeiten, wobei in den Nasszeiten weiter in Regenimpulse unterteilt werden,
zwischen denen kurze Regenunterbrechungen eingelagert sind (Haberlandt et al., 2008).
Alternating-Renewal Modelle hängen stark von der präzisen Definition der Regenereignis-
se ab, die zur Kalibrierung herangezogen werden. Entscheidend ist vor allem die Frage,
ab welcher Dauer der Unterbrechung zwei Regenereignisse als unabhängig angenommen
werden können.
XVI
ZUSAMMENFASSUNG
Auf hoher zeitlicher Auflösung gemessene Niederschlagszeitreihen haben, statistisch ge-
sehen, eine sehr komplexe Struktur. Sie sind hochvariabel, jedoch nicht komplett zufällig.
Stündliche Niederschlagszeitreihen beispielweise haben eine deutliche Persistenz. Wenn es
in einer Stunde regnet, ist die Wahrscheinlichkeit für Regen auch in der nächsten Stunde er-
höht. Gleichzeitig treten aber abrupte Wechsel auf zwischen längeren nassen und trockenen
Phasen. Dieses Verhalten ist mit den üblichen statistischen Maßzahlen, wie z. B. der Auto-
korrelation, schwer zu beschreiben. Die verschiedenen niederschlagserzeugenden Prozesse
treten alle in typischen räumlichen und zeitlichen Skalen auf. Gewitterzellen beispielswei-
se sind generell kleinräumige Phänomene und in ihrer Lebensdauer auf wenige Stunden
begrenzt, während sich die Fronten eines Tiefdrucksystems über mehrere Tausend Qua-
dratkilometer erstrecken und entsprechend länger anhaltenden und gleichförmigeren Nie-
derschlag erzeugen können. Durch das Zusammenspiel der unterschiedlichen Prozesse zei-
gen Niederschlagszeitreihen charakteristische Skalierungseigenschaften zwischen den zeit-
lichen Aggregierungen, z. B. zwischen den statistischen Momenten von Stunden-, 6h- und
Tageswerten.
Konzeptionelle Modelle können die komplexe statistische Struktur nur begrenzt nachbil-
den. Die korrekte Abbildung der Skalierungseigenschaften ist in den meisten Modellen nur
mit sehr hohem Parametrisierungsaufwand möglich (Koutsoyiannis, 2006). Es besteht die
Gefahr, dass ein Modell auf die zeitliche Skala angepasst ist, die zur Kalibrierung verwen-
det wurde und die Variabilität auf anderen Aggregierungen unterschätzt (Koutsoyiannis
and Foufoula-Georgiou, 1993; Katz and Parlange, 1995).
Ein datenbasierter Ansatz zur Erzeugung von synthetischen Niederschlags-
zeitreihen in stündlicher Auflösung
In dieser Arbeit wurde ein datenbasierter Ansatz gewählt, wie er in Bárdossy (1998) be-
schrieben ist. Dieser Ansatz macht möglichst wenig konzeptionelle Einschränkungen, ledig-
lich die Verteilung der stündlichen Niederschlagswerte wird vorgegeben. Außerdem wer-
den statistische Zielvorgaben an eine Zeitreihe formuliert (z. B. der Autokorrelation oder der
Skalierungseigenschaften auf unterschiedlichen Dauerstufen), die aus beobachteten Zeitrei-
hen abgeleitet werden. Anhand der Verteilung wird eine zufällige Zeitreihe initialisiert, die
anschließend durch Tauschen von Werten so lange optimiert wird, bis sie die statistischen
Zielvorgaben möglichst weitgehend erfüllt.
Der Ansatz ist sehr datenintensiv, weil die statistischen Vorgaben von beobachteten Nie-
derschlagszeitreihen abgeleitet werden müssen. Umgekehrt ist er für die Simulation in Re-
gionen mit einem engmaschigen Messnetz, wie das Projektgebiet von Baden-Württemberg,
besonders geeignet, denn damit wird die Information aus den vorhandenen Messwerten
optimal genutzt. Ein weiterer Vorteil liegt in der einfachen Regionalisierung. Mit mit geo-
statistischen Methoden können die Zielstatistiken auf jeden beliebigen Raumpunkt des Pro-
jektgebiets übertragen werden. Genauso kann eine Konditionierung auf höher aggregierte
Werte erfolgen, wenn diese als Zielwerte der Optimierung vorgegeben werden. Deshalb
eignet sich die Methode auch für Downscaling, z. B. von Klimamodelldaten.
Bezüglich des Ansatzes von Bárdossy (1998) wurde der Algorithmus vor allem in Bezug
auf die Generierung der Initialreihe modifiziert, die an das Optimierungsschema überge-
XVII
ZUSAMMENFASSUNG
ben wird. Die Initialreihe wird durch die Niederschlagswahrscheinlichkeit definiert, welche
die Anzahl an auftretenden Regenstunden steuert, sowie durch die Parameter der theoreti-
schen Verteilungsfunktion, welcher die stündlichen Niederschlagssummen folgen. Als Ver-
teilungsfunktion wurde in dieser Arbeit die Weibullverteilung gewählt. Die Wahrscheinlich-
keitsdichte f und die theoretische Verteilungsfunktion F sind durch folgende Gleichungen
gegeben:
f (X) =
κ
λ
(
X
λ
)κ−1
e−(x/λ)
κ
(1)
F (X) = 1− e−(x/λ)κ (2)
Die richtige Anpassung der Verteilungsfunktion ist entscheidend für die Güte der gene-
rierten Zeitreihen. Aufgrund der hohen Schiefe stündlicher Niederschlagssummen wird die
Anpassung der Verteilungsparameter λ und κ durch die hohe Anzahl sehr niedriger Wer-
te dominiert. Dadurch besteht die Gefahr, dass die Wahrscheinlichkeit extrem hoher Werte
unterschätzt wird. Um die Variabilität in den Extremwerten zu erhöhen und um jahreszeitli-
che Effekte mit einzubeziehen, wurden die 1h-Regenwahrscheinlichkeit und die Parameter
λ und κ monatsweise bestimmt.
Wie in dieser Arbeit gezeigt werden konnte, hängen die für die Erzeugung der Initialreihe
maßgeblichen Statistiken der stündlichen Regenwahrscheinlichkeit P1h, der mittleren Nie-
derschlagssumme aller nassen Stunden E (Hwet) und der Standardabweichung der nassen
Stunden D (Hwet) alle von der Monatssumme Hmon ab. Der Zusammenhang ist universal
für das gesamte Projektgebiet und kann mit einer vierdimensionalen Gaussschen Copula
beschrieben werden.
Die Copula misst den Zusammenhang der Variablen unabhängig von den absoluten Werten.
Es ist die multivariate Verteilungsfunktion im [0, 1]-Raum der gleichverteilt transformierten
Variablen Ui:
C : [0, 1]ndim → [0, 1]
Die Copula beschreibt die multivariate Nicht-Überschreitungswahrscheinlichkeit aller ndim
Variablen:
C
(
U∗1 , U
∗
2 , . . . , U
∗
ndim
)
= F
(
U1 ≤ U∗1 , U2 ≤ U∗2 , . . . , Undim ≤ U∗ndim
)
(3)
Die Gaussche Copula ist identisch mit einer multivariaten Normalverteilung, wenn alle Va-
riablen Ui in standardnormalverteilte Zufallsvariablen transformiert werden:
C (U1, U2, . . . , Undim) = Φr
(
Φ−1 (U1) ,Φ−1 (U2) , . . . ,Φ−1 (Undim)
)
(4)
Hierbei is Φ−1 die Umkehrfunktion der Standardnormalverteilung und Φr die multi-
variate Standardnormalverteilung. Der einzige Parameter einer Gausschen Copula ist
die Matrix der paarweisen Korrelationen r zwischen allen Variablenkombinationen im
standardnormal-transformierten Raum. Abb. 1 zeigt die empirische Copuladichte zwischen
10000 Zufallszahlen, die einer bivariaten Gaussschen Copula mit einer Korrelation r im
standardnormal-transformierten Raum von r = 0.75 folgen.
XVIII
ZUSAMMENFASSUNG
4 3 2 1 0 1 2 3 4
Z(x)
4
3
2
1
0
1
2
3
4
Z
(y
)
(a) Scatterplot
0.0 0.2 0.4 0.6 0.8 1.0
F(x)
0.0
0.2
0.4
0.6
0.8
1.0
F(
y
)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) empirische Copuladichte
Abb. 1.: Beispiel einer bivariaten empirischen Copula von 10000 standardnormalverteilten
Zufallszahlen mit r = 0.75. Links: Skatterplot im standardnormalen Raum, rechts:
Empiriche Copuladichte
0.0 0.2 0.4 0.6 0.8 1.0
Monatssumme
0.0
0.2
0.4
0.6
0.8
1.0
1
h
-R
e
g
e
n
w
a
h
rs
ch
e
in
lic
h
ke
it
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) Januar
0.0 0.2 0.4 0.6 0.8 1.0
Monatssumme
0.0
0.2
0.4
0.6
0.8
1.0
1
h
-R
e
g
e
n
w
a
h
rs
ch
e
in
lic
h
ke
it
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) July
Abb. 2.: Empirische Copuladichte zwischen der Monatssumme und der 1h-
Niederschlagswahrscheinlichkeit
Als Beispiel für den vierdimensionalen Zusammenhang zwischen den Parametern ist in
Abb. 2 die empirische Copuladichte zwischen der Monatssumme des Niederschlags und
der Niederschlagswahrscheinlichkeit dargestellt. Der Zusammenhang zeigt die für die
Gaussche Copula charakteristische elliptische Form der Copuladichte. Generell ist der Zu-
sammenhang im Sommer niedriger, weil kleinräumige konvektive Niederschlagsereignisse
auftreten.
Anhand der vierdimensionalen Copula können zufällige Werte für P0, E (Hwet) und
D (Hwet) gezogen werden, die sowohl untereinander, als auch zur Monatssumme die rich-
XIX
ZUSAMMENFASSUNG
tige Abhängigkeitsstruktur zeigen. Dafür wird der jeweils aktuelle Messwert der Monats-
summe über die Randverteilung in einen standardnormalverteilten Wert Zmon umgerech-
net und die vierdimensionale Korrelationsmatrix r auf diesen Wert konditioniert. Über die
Standardnormalverteilung und die Umkehrung der Randverteilungen für P0, E (Hwet) und
D (Hwet) werden die im standardnormal-transformierten Raum gezogenen Werte Zi wie-
der in absolute Werte Xi zurück gerechnet. Um jahreszeitliche Effekte zu berücksichtigen,
wurden die vierdimensionale Copula und die Randverteilungen jeweils für dreimonati-
ge Zeiträume aufgestellt. Als Randverteilung für die 1h-Niederschlagswahrscheinlichkeit
wurde die Beta-Verteilung gewählt. Für die anderen Größen wurden mehrere rechtsschie-
fe Verteilungen getestet. Die Monatssumme wird am besten mit der Weibullverteilung be-
schrieben, die Momente der nassen Stunden E (Hwet) und D (Hwet) jeweils mit der Log-
Normalverteilung. Alle Verteilungen wurden mit der Maximum-Likelihood-Methode an-
gepasst.
Aus den gezogenen WertenE (Hwet) undD (Hwet) für jeden Monat werden die Parameter λ
und κ der Weibullverteilung nach der Momentenmethode berechnet. Anschließend wird an-
hand von P0 und den Weibullparametern die Initialzeitreihe Monat für Monat generiert. Die
vollständig initialisierte Zeitreihe für ein Jahr wird dann dem Optimierungsschema überge-
ben.
0 5 10 15 20 25 30 35 40 45
[mm/h]
0
10
20
30
40
50
60
70
80
x= 3501820 y= 5384490
Jaehrliche Serie - 10 Werte pro Jahr
beobachtet 1977 - 1992
NaN= 3.5%
simuliert 1977 - 1992
(a) Stundensummen
0 10 20 30 40 50 60 70 80
[mm/day]
0
10
20
30
40
50
60
70
x= 3501820 y= 5384490
Jaehrliche Serie - 10 Werte pro Jahr
beobachtet 1977 - 1992
NaN= 3.5%
simuliert 1977 - 1992
(b) Tagessummen
Abb. 3.: Empirische Verteilung der zehn höchsten Niederschlagssummen in der beobache-
ten und simulierten Zeitreihe von Holzgerlingen
Als Test der Modellgüte wurden die erzeugten Zeitreihen mit beobachteten Messreihen in
Holzgerlingen verglichen, wo lange Datenreihen mit geringer Fehlwertquote vorliegen. Es
zeigte sich, dass die monatsweise Anpassung der Verteilungsfunktion zu guten Ergebnis-
sen führt. Sowohl in den grundlegenden Statistiken (Regenwahrscheinlichkeit, Mittelwert
und Schiefe) als auch in den Extremwerten auf unterschiedlichen Aggregierungen (Abb. 3)
stimmen die generierten Zeitreihen gut mit den beobachteten überein. Gleiches gilt für die
Verteilung der Längen von Nass- und Trockenphasen. Eine Stärke des Generierungsalgo-
rithmus im Hinblick auf die Regionalisierung ist die Konditionierung der Zeitreihen auf
die Monatssumme des Niederschlags. Die Monatssumme ist räumlich weit weniger varia-
XX
ZUSAMMENFASSUNG
bel als Statistiken auf höherer zeitlicher Auflösung. Aus diesem Grund ist die Interpolation
auf Raumpunkte, an denen keine Messstationen vorliegen, mit einer geringen Unsicherheit
verknüpft.
Die Erzeugung von simultanen, synthetischen Niederschlagszeitreihen
Für die hydraulische Simulation größerer Kanalnetze ist die Verwendung einer Nieder-
schlagszeitreihe als Eingangsdaten nicht mehr ausreichend, da die räumliche Heterogenität
über dem Einzugsgebiet zu groß wird. Sollen mehrere synthetische Zeitreihen verwendet
werden, so muss die räumlich-zeitliche Abhängigkeit zwischen den Zeitreihen bei der Ge-
nerierung berücksichtigt werden.
Der für die Generierung von Einzelzeitreihen entwickelte Algorithmus kann für die Erzeu-
gung simultaner Zeitreihen nicht übernommen werden. Der Rechenaufwand des Optimie-
rungsschemas ist abhängig von der Anzahl an möglichen Kombinationen in der zu optimie-
renden Zeitreihe. Bei der Erzeugung simultaner Zeitreihen steigt dieser überproportional
an, da die möglichen Kombinationen der 1h-Niederschlagswerte aller Zeitreihen berück-
sichtigt werden müssen. Dadurch wird das ursprüngliche Generierungsschema schon bei
wenigen Stationen impraktikabel. Es wurde deshalb ein neuer Ansatz entwickelt, bei dem
zuerst simultane, räumlich-zeitlich abhängige Zeitreihen von Tageswerten erzeugt werden.
Die Tageswerte werden dann in einem zweiten Schritt auf 24 Stundenwerte verteilt. Diese
Disaggregierung funktioniert ebenfalls über eine Optimierung, bei der kleine Inkremente
der Niederschlagssummen zwischen den Stunden eines Tages ausgetauscht werden. Wie
bisher auch wird der Zustand der zu optimierenden Zeitreihen mit einer Zielfunktion be-
wertet. Neben den statistischen Zielwerten für die Einzelreihen werden dabei auch Vorga-
ben bezüglich des räumlichen Zusammenhangs gemacht.
Analog zur Generierung von Einzelreihen wird die Initialreihe auf die Monatssumme kon-
ditioniert. Im Unterschied zur Einzelgenerierung hat die Zeitreihe nun jedoch eine zeitliche
Auflösung von 24 h. Die Parameter zur Generierung der Initialreihe sind somit die tägliche
Niederschlagswahrscheinlichkeit P24h, sowie der Mittelwert E (Hwet) und die Standardab-
weichung D (Hwet) des Niederschlags an nassen Tagen. Neben der Abhängigkeit der Ge-
nerierungsparameter an jeder Station muss für eine simultane Generierung ebenso berück-
sichtigt werden, wie die Generierungsparameter verschiedener Stationen miteinander zu-
sammen hängen, z. B. wie die Monatssumme an Station i mit der Regenwahrscheinlichkeit
an Station j zusammen hängt. Unter Annahme eines Gausschen Copula Zusammenhangs
ergibt sich dadurch eine Korrelationsmatrix r im standardnormal-transformierten Raum mit
4nstat×4nstat Einträgen, wobei nstat für die Anzahl an Stationen in der simultanen Generie-
rung steht. Die Korrelationswerte zwischen Parametern von verschiedenen Stationen i 6= j
werden mittels einer exponentiellen Regressionsfunktion über den Abstand zwischen den
Stationen geschätzt.
Wie bei der Generierung von Einzelzeitreihen beschrieben, wird die Korrelationsmatrix r
auf den Vektor der standardnormal-transformierten Monatssummen an allen Stationen kon-
ditioniert und es werden für jede Station i und jeden Monat zufällige Werte für die Nieder-
schlagswahrscheinlichkeit P i24h sowie die Momente E
i (Hwet) und Di (Hwet) gezogen. Über
XXI
ZUSAMMENFASSUNG
die Momente wird für jede Station und jeden Monat eine Weibullverteilung angepasst und
die tägliche Initialreihe gezogen. Die Initialreihen werden wiederum über ein Permutations-
schema optimiert, wobei neben den statistischen Zielwerten für die einzelnen Zeitreihen die
Korrelation der Zeitreihen untereinander in die Zielfunktion eingeht.
Um die räumliche Abhängigkeit zwischen den Zeitreihen auf einer kürzeren zeitlichen Ska-
la zu berücksichtigen, wird eine Wetterlagenklassifikation (“Circulation Pattern”, im fol-
genden CP) herangezogen. Für die Definition der Wetterlagen wird auf Bárdossy (2010)
verwiesen. Der Zielwert der Korrelation zu jeder Wetterlage für die Disaggregierung wird
anhand des Abstandsvektors zwischen den Zielpunkten der Generierung festgelegt. Dabei
wird nicht nur die Korrelation gleichzeitiger Werte betrachtet, sondern auch die Korrelation
bei einstündigem Zeitversatz zwischen den Zeitreihen.
(a) CP2 (b) CP3
Abb. 4.: Durchschnittliche Luftdruckanomalie aller Tage mit CP2 (links) und CP3 (rechts)
als 1hPa Isobaren
Abb. 4 zeigt die mittlere Luftdruckanomalie als Isobarenkarte für zwei der 17 CPs. In Abb. 5
und in Abb. 6 sind für diese CPs die Korrelationen stündlicher Niederschlagswerte in Ab-
hängigkeit des Abstandsvektors zwischen den Messstationen dargestellt. Jede Rasterzel-
le in den Diagrammen repräsentiert die durchschnittliche Korrelation aller Stationspaare,
deren Abstandsvektor in die Rasterzelle fällt. Die absolute Lage der Stationen zueinander
wird also ignoriert, entscheidend ist lediglich die räumliche Konfiguration der Stationen
zueinander. Die Korrelation wurden auf Basis von 137 Niederschlagszeitreihen aus Baden-
Württemberg berechnet.
Es zeigt sich, dass die räumliche Abhängigkeit stark von der CP beeinflusst wird. Wäh-
rend Hochdruckwetterlagen wie CP2, mit schwachem atmosphärischem Austausch, entste-
hen die meisten Niederschlagsereignisse lokal und die Korrelationen sind generell niedrig
(Abb. 5). Während Tiefdrucklagen sind die Korrelationen viel höher und zeigen eine aus-
geprägte Anisotropie. Die Korrelationen sind parallel zur atmosphärischen Strömungsrich-
tung, bei CP3 von West nach Ost, deutlich höher (Abb. 6). Wenn man die Zeitreihe am Ende
des Abstandsvektors für die Berechnung der Korrelation um eine Stunde verschiebt (zweite
XXII
ZUSAMMENFASSUNG
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte - Tage mit CP2
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte - Tage mit CP2
(b) April to September
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte, 1h-Zeitversatz - CP2
(c) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte, 1h-Zeitversatz - CP2
(d) April to September
Abb. 5.: Richtungsabhängige paarweise Korrelation von 137 1h-Niederschlagsstationen in
Baden-Württemberg für alle Tage mit CP2; erste Reihe: Korrelation der gleichzeiti-
gen Werte, zweite Reihe: Korrelation mit 1h Zeitversatz
Zeile von Abb. 5 und Abb. 6), zeigt sich bei den Tiefdrucklagen, dass die Korrelation in Strö-
mungsrichtung höher ist als entgegen der Strömungsrichtung. Gleichzeitig gibt es einen
deutlichen jahreszeitlichen Effekt. Durch die konvektiven Niederschlagsereignisse in den
Sommermonaten sind die Korrelationen zwischen April und September generell niedriger
als im advektiv geprägten Winterhalbjahr.
Die paarweisen Korrelationen der gleichzeitigen und versetzten Zeitreihen dienen neben
den statistischen Vorgaben für die Einzelreihen als Zielwerte im Optimierungsalgorithmus
der Disaggregierung.
Im Vergleich mit beobachteten Zeitreihen zeigt das Generierungsschema für simultane
Zeitreihen vergleichbar gute Ergebnisse wie die Einzelreihengenerierung. Die Möglichkeit
räumliche Zusammenhänge zu berücksichtigen, ist hingegen noch begrenzt. Den Copula-
Zusammenhang zwischen den Generierungsparametern und die Korrelation der 1h-Werte
XXIII
ZUSAMMENFASSUNG
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte - Tage mit CP3
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte - Tage mit CP3
(b) April to September
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte, 1h-Zeitversatz - CP3
(c) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
Abstand O-W [km]
300
200
100
0
100
200
300
A
b
st
a
n
d
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Korr 1h-Werte, 1h-Zeitversatz - CP3
(d) April to September
Abb. 6.: Richtungsabhängige paarweise Korrelation von 137 1h-Niederschlagsstationen in
Baden-Württemberg für alle Tage mit CP3; erste Reihe: Korrelation der gleichzeiti-
gen Werte, zweite Reihe: Korrelation mit 1h Zeitversatz
über den Abstand zwischen den Stationen zu schätzen, ist mit einer hohen Streuung ver-
bunden. Es gibt jedoch keine Anzeichen, dass die Streuung auf Anistropien oder Instationa-
ritäten zurückzuführen ist. Eine weitere Einschränkung liegt darin, dass nicht-lineare Ab-
hängigkeiten, wie z. B. eine erhöhte Korrelation zwischen den extremsten Niederschlagser-
eignissen nicht in der Generierung berücksichtigt sind.
Auswirkungen des Klimawandels auf Niederschlag in hoher zeitlicher Auflö-
sung
Werden die Daten der entwickelten Niederschlagsgeneratoren in der Planung wasserbau-
licher Strukturen eingesetzt, so hängen davon sehr langfristige Investitionen ab. Bei der
Dimensionierung städtischer Kanalnetze beispielsweise geht man davon aus, dass die gene-
rierten Zeitreihen repräsentativ für den Niederschlag während der gesamten Lebensdauer
XXIV
ZUSAMMENFASSUNG
von mehreren Jahrzehnten sind. Implizit steckt darin die Annahme, dass die klimatischen
Bedingungen aus dem Kalibrierungszeitraum auch für die Zukunft gelten. Unter sich wan-
delndem Klima ist diese Annahme mehr und mehr fraglich, je weiter man in die Zukunft
blickt.
Im zwanzigsten Jahrhundert stieg die jährliche Durchschnittstemperatur weltweit um et-
wa 0.8◦C an. Ergebnisse von Klimamodellläufen lassen darauf schließen, dass sich dieser
Temperaturanstieg in diesem Jahrhundert fortsetzen wird (IPCC, 2007). Da sowohl die Ver-
dunstung, als auch die Kapazität der Atmosphäre Wasserdampf aufzunehmen temperatu-
rabhängig sind, kann man davon ausgehen, dass dieser Temperaturanstieg weltweit auch
den Niederschlag verändert (z. B. Wentz et al., 2007). Im regionalen Maßstab lassen sich
solche Veränderungen schon beobachten. Hundecha and Bárdossy (2005) konnten anhand
täglicher Niederschlagsmessungen der letzten Jahrzehnte im Rheineinzugsgebiet einen An-
stieg der Niederschlagssumme im Winterhalbjahr nachweisen, gleichzeitig nahm der Nie-
derschlag in den Sommermonaten an einigen Stationen leicht ab. Außerdem stieg der Anteil
an der Niederschlagssumme, der während Extremereignissen fällt, deutlich an. Es zeigt sich
auch, dass die Wirkung des klimatischen Wandels auf den Niederschlag von der betrach-
teten zeitlichen Skala abhängt. Bei höherer atmosphärischer Temperatur muss mit weniger
häufigen, aber intensiveren Niederschlagsereignissen gerechnet werden (Trenberth, 1999).
So kann es sein, dass es trotz allgemein trockeneren Sommern zu mehr Extremereignissen
kommt (Christensen and Christensen, 2004).
In dieser Arbeit wurden deshalb Methoden entwickelt, mit denen mögliche Auswirkungen
des Klimawandels auf den Niederschlag in den für viele wasserbauliche Fragestellungen
entscheidenden kurzen Dauerstufen abgeschätzt werden können.
Eine Abschätzung der zukünftigen klimatischen Verhältnisse kann prinzipiell auf zwei Ar-
ten erfolgen, entweder durch eine Extrapolation beobachteter Niederschlagstrends oder
durch den Einsatz globaler oder regionaler Klimamodelle. Die Trendfortschreibung basiert
auf der Annahme, dass ein beobachteter Trend sich in der Zukunft auf die gleiche Weise
fortsetzen wird. Es ist eine rein statistische Methode, die keinerlei atmosphärenphysikali-
sche Zusammenhänge berücksichtigt, sondern lediglich davon ausgeht, dass Änderungen
in den physikalischen Prozessen einer gewissen Trägheit unterliegen. Aufgrund dieser An-
nahme ist der Zeithorizont von Trendabschätzungen generell begrenzt. Eine Prognose für
das Jahr 2100, wie sie von den Klimamodellen geliefert wird, ist damit nicht möglich. Da-
für kann die hohe Detailliertheit beobachteter Niederschlagsdaten voll ausgenützt werden,
welche von globalen oder regionalen Klimamodellen so nicht abgebildet werden kann.
Für eine regionale Trendanalyse wurden die jährlichen Extremwerte an jeder verfügbaren
Niederschlagsstation in Baden-Württemberg auf das durchschnittliche jährliche Maximum
normiert. Damit können alle Zeitreihen vergleichbar gemacht und gemeinsam analysiert
werden, soweit die Daten für denselben Zeitraum vollständig vorliegen. Auf diese Weise
wurden 40 stündliche Niederschlagsstationen und 104 Tagesstationen im Zeitraum von 1991
bis 2003 in einer regionalen linearen Regression zusammengefasst. In Baden-Württemberg
konnte damit kein signifikanter Trend nachgewiesen werden. Eventuell sind die verfügba-
ren Zeitreihen dafür zu kurz. In Bayern jedoch, wo die Datenbasis mit 54 stündlichen Sta-
tionen und 293 Tagesstationen größer und der mögliche Untersuchungszeitraum von 1991
XXV
ZUSAMMENFASSUNG
bis 2006 länger ist, zeigte sich eine signifikante Zunahme in den 1h-Extremwerten während
des Sommerhalbjahrs.
Der Anstieg in den Niederschlagsintensitäten kurzer Dauer wurde durch eine Analy-
se der Skalierungseigenschaften zwischen Niederschlagssummen unterschiedlicher Dau-
erstufe bestätigt. Zwischen 1958 und 1980 betrug das durchschnittliche jährliche 1h-
Niederschlagsmaximum in Baden-Württemberg 28% des täglichen Maximums. Im Zeit-
raum 1981 bis 2003 stieg dieser Anteil auf 31% an. Das 4h-Maximum stieg im gleichen Zeit-
raum von 50% des Tagesmaximums auf 55% an. Die Zunahme der kurzzeitigen Intensitäten
betrifft nicht nur die Extremwerte, sondern wurde flächendeckend auch für den Mittelwert
der Niederschlagssumme nachgewiesen. Die Ergebnisse der Untersuchungen entsprechen
den aus der Atmosphärenphysik abgeleiteten Voraussagen von Trenberth (1999), dass bei
einer höheren Atmosphärentemperatur mit weniger, aber intensiveren Niederschlägen zu
rechnen ist.
Mit Klimamodellen, welche die physikalischen Prozesse in der Atmosphäre numerisch ab-
bilden, ist eine Prognose für längere Zeithorizonte möglich als mit rein statistischen Me-
thoden. Klimamodelle können jedoch den globalen Wasserkreislauf nicht fehlerfrei abbil-
den (Habemann et al., 2006) und es gibt Indizien, dass der Fehler in den prognostizierten
Niederschlagssummen von der betrachteten räumlichen und zeitlichen Skala abhängt (Ro-
eckner et al., 2006). Extreme Niederschlagsereignisse werden generell unterschätzt, wobei
die Unterschätzung auf kürzerer zeitlicher Skala stärker wird (Fowler et al., 2007). Doch
selbst wenn die Klimamodelle perfekte Vorhersagen liefern würden, könnten die Daten
nicht direkt für wasserbauliche Aufgabenstellungen verwendet werden. Zwischen den Nie-
derschlagswerten der Modelle und beobachteten Werten von Niederschlagsmessstationen
besteht ein grundsätzlicher Unterschied. Klimamodelle berechnen den Gebietsniederschlag
über einer Rasterzelle, während Messstationen Information von einem Raumpunkt liefern.
Die beiden Größen würden nur dann übereinstimmen, wenn der Niederschlag in der Ra-
sterzelle vollständig räumlich homogen wäre.
Klimamodelldaten müssen darum korrigiert und durch sogenanntes Downscaling auf die
lokale Skala übertragen werden. Eine Möglichkeit ist die Quantil-Quantil-Transformation,
mit der die Verteilung der beobachteten lokalen Niederschlagssummen auf die Klimamo-
delldaten aufgeprägt wird. In dieser Arbeit wurde jedoch gezeigt, dass eine solche direkte
Korrektur der Niederschlagsdaten keine zufriedenstellenden Ergebnisse liefert. Selbst nach
einer Korrektur ist die Bandbreite zwischen verschiedenen Klimamodellen höher als die
Bandbreite unterschiedlicher Szenarios, die ja eigentlich die gesamte Unsicherheit der zu-
künftigen Entwicklung abdecken sollten. Zudem sind die meisten direkten Korrekturen nur
für aggregierte Daten, wie z. B. die Monatssummen, anwendbar, weil zeitliche Abhängigkei-
ten zwischen den Werten nicht berücksichtigt werden können.
In dieser Arbeit wurde deshalb eine indirekte Korrekturmethode entwickelt, bei der die
Nutzung von Niederschlagsdaten aus dem Klimamodell vermieden wird. Es werden le-
diglich Luftdruck- und Temperaturfelder verwendet, die als weitaus verlässlicher angese-
hen werden. Anhand von Luftrduckdaten aus den NCEP/NCAR Reanalysen (Kistler et al.,
2001) wurde eine CP-Klassifikation vorgenommen, die bezüglich der durchschnittlichen At-
mosphärentemperatur weiter unterteilt wurde. Mit den Daten der Messstationen in Baden-
XXVI
Württemberg wurde die beobachtete Verteilung der 1h-Niederschlagssummen zu jeder CP-
Temperatur-Kombination ermittelt. Im Zeitraum von 1958 bis 2003 haben diejenigen CPs,
die für Hochdruckwetterlagen stehen, in den Sommermonaten zugenommen, während Tief-
druckwetterlagen seltener wurden. Gleichzeitig reagieren die CPs auf die atmosphärische
Durchschnittstemperatur. Vor allem in den Sommermonaten führt eine höhere Temperatur
zu einer niedrigeren Niederschlagswahrscheinlickeit. Wenn es aber regnet, steigt die Wahr-
scheinlichkeit von Extremereignissen deutlich an.
Unter der Annahme, dass die Verteilung der 1h-Niederschlagssummen zu jeder CP-
Temperaturklasse in der Zukunft konstant ist, wurde die vom Klimamodell ECHAM5 (Ro-
eckner et al., 2003) prognostizierte Verteilung der 1h-Niederschlagswerte für jedes Jahr bis
2060 ermittelt. Es zeigt sich, dass der beobachtete Trend zu höheren Kurzzeitintensitäten
auch vom Klimamodell bestätigt wird. Bis 2060 ist mit einem deutlichen Anstieg der 1h-
Extremwerte zu rechnen. Da dieses Signal des Klimawandels sowohl aus beobachteten
Messwerten, als auch aus Klimamodelldaten isoliert werden konnte und darüber hinaus
aufgrund von atmosphärenphysikalischen Überlegungen vorausgesagt wurde (Trenberth,
1999), muss es als sehr verlässlicher Trend eingestuft werden.

1. Introduction
1.1. Context and Motivation
Live depends on water. A sufficient supply with drinking water is the basis for all human
activity. Drinkable freshwater is mainly found in ground or surface water bodies that are fed
by precipitation. Hence, the natural variability in precipitation has always been a thread.
A lack of precipitation leads to failed harvests and can provoke starvation. An excess of
precipitation causes floods which are not only responsible for high damage to goods and
infrastructure, but also for a high number of casualties. In the United States, for example,
about 100 people are killed each year in flood events (Ashley and Ashley, 2008). Violent
precipitation events can destroy crops and degrade soils. Furthermore, there is the risk of
water borne diseases and dangerous infections from contaminated drinking water.
The development of human society is highly linked to the ability to manage these water
risks. Bigger human settlements are not possible without hydraulic constructions that en-
sure a steady supply with drinking water, protect from flooding and discharge waste water.
Roman aqueducts, as the “Pont Du Gard” in South of France are spectacular early examples
of these efforts.
Hydraulic structures like dikes or sewage channel are expensive and long lasting invest-
ments. Hence, the correct dimensioning has always been an issue. Overdimensioning con-
sumes a lot of resources, too small structures bear the risk of system failure. The design of
hydraulic structures tries to find the optimal trade-off between the construction costs on the
one hand and the avoided potential damage on the other hand. The German law states that
flood protection measurements at river banks should be designed to retain all flood events
with a higher average frequency than once in one hundred years (WHG, 2009).
Without any statistical or physical information, dimensioning was trial and error: if a town
was flooded, the dikes were reinforced. Learning from experience, qualitative rules were
established, e. g. to avoid construction on areas that are prone to flooding. Since system-
atic measurements of river discharge and precipitation are available, empirical rules have
been developed that are based on statistical properties. Using extreme value theory, dis-
tributions, for example of river discharge measurements, can be extrapolated beyond the
observed range. The probability of extreme values, that have not yet been measured, can
be quantified. These rules, however, do not consider any physical knowledge. They are
governed merely by observed distributions.
Today, it is possible to attack many dimensioning problems by sophisticated numerical mod-
eling. Such computer models represent the natural variability of the flow regime. Hydro-
logical models are able to simulate the discharge of a river based on the precipitation over
2 Introduction
the catchment area. Hydraulic models of urban sewage systems calculate the discharges in a
channel network in dependence of channel diameters and reservoir sizes. Numerical mod-
els are continuously improved. More and more processes of the hydrological environment
are incorporated in the model systems. This however, raises a new issue: The more detailed
a numerical model is, the higher is its demand of real world observations as input data.
More often, the performance of numerical simulations is limited by the offer of available
input data and not by the limited complexity of the applied model.
The restriction due to data availability is most remarkable in hydraulic modeling of urban
sewage channel systems. Sewage channel networks are very fast reacting systems with short
concentration times of typically less than one hour. Therefore, accurate hydraulic modeling
requires precipitation input on high temporal resolution of one hour or shorter. Since a
sewage network reacts differently if the channels and reservoirs are still filled from former
precipitation events, it is not sufficient to model one single event. Best practice for a correct
estimation of flood and backwater risks is a hydraulic simulation based on long rainfall time
series of several decades.
Temporally high resolution precipitation data show pronounced spatial variability. There-
fore the representative area of a time series in hourly resolution is very limited. Germany is
covered by several thousand rain gauges, most of them run by the German weather service
DWD. Additionally, the area is scanned by 17 rain radar devices. Nevertheless, the offer of
available data is not sufficient to provide adequate measurements for every target location.
Besides, the observed time series are rarely long enough for correct risk assessment.
In 2000 the synthetic rainfall generator “NiedSim” was developed by Prof. András Bárdossy.
In a data driven approach NiedSim assembles stochastic time series that mimic the statisti-
cal characteristics of observed high resolution precipitation measurements (Bárdossy, 1998).
At any locations in Baden-Württemberg, where no measurements are available, NiedSim
provides a simulated time series that can be used as substitute.
Like any model, NiedSim represents the characteristics of real precipitation only to a certain
extend. It suffers from a limited ability to represent seasonal variability and inconsistencies
in the extreme value statistics. Besides, NiedSim does not consider spatial temporal depen-
dence. Two rain gauges at nearby locations record very similar rainfall time series, whereas
NiedSim generated time series for the same locations are very different. In hydraulic mod-
eling of larger sewage systems, however, the application of one single time series leads to
errors in the modeled channel discharges, due to high spatial temporal variability of precip-
itation. The application of several simultaneous time series on the other hand is only valid
if the spatial temporal interdependencies are taken into account.
This work presents an approach how NiedSim can be modified to overcome these limita-
tions. The spatial temporal dependence of observed precipitation is measured on different
scales and incorporated in the generation scheme.
The calibration of stochastic precipitation models is based on past time observations. There-
fore, the application of simulated time series in the dimensioning of sewage system net-
works implies the assumption that the measurements during the calibration period are rep-
resentative for the future. Urban sewage channels have a life span of 80 to 100 years. Under
1.2 Outline 3
changing climatic conditions the assumption, that the past is representative for the future,
is less and less true, the further the projection goes into the future. The future climatic con-
ditions already have to be considered during planning and construction. Therefore, this
work concludes by presenting several methods for the estimation of climate trends in high
resolution precipitation time series. These methods are based on extrapolation of observed
regional trends and the correction and downscaling of climate model output.
1.2. Outline
In the next chapter (Chapter 2) this work starts with a description of the climatic conditions
of Baden-Württemberg. The precipitation regime is explained in the context of global atmo-
spheric circulation. Regional differences within the study area, mostly due to orographic
effects of mountain ranges, are pointed out.
Chapter 3 is dedicated to the acquisition of precipitation data. Different measurement meth-
ods are explained, from direct measurements on the ground to remote sensing techniques
like rain radar and satellite measurements. The focus is on the accuracy and the potential
error sources of the different measurement methods. Some examples of identified errors are
presented. Next, the precipitation data is described which is used in this work .
In Chapter 4 some of the important statistical properties of precipitation are explained: the
high variability on different spatial and temporal scales, as well as the intermittency in oc-
currence and the scaling behavior of measurements on different spatial or temporal aggre-
gations.
If no measurement data is available, stochastic models are applied to generate synthetic
precipitation data. Chapter 5 gives an overview of the existing modeling concepts. The
basic ideas of each concept are explained, strong points and limitations are pointed out.
In Chapter 6 the stochastic time series generator NiedSim is presented. The necessary mod-
ifications to are explained. A new version of the generation algorithm is set up and tested
by a comparison of an observed and a simulated rainfall time series of the same location.
The comparison is based on extreme value statistics and the length of wet and dry spells as
a measure of temporal persistence.
In Chapter 7 the spatial dependence structure of precipitation is analyzed and it is explained
how the dependence is represented by statistical measures that can be incorporated in the
rainfall generation algorithm. A multi-site generator is set up that is able to simulate several
spatial correlated precipitation time series in hourly temporal resolution. The performance
of the multi-site generator is evaluated by the same statistics as in Chapter 6.
Chapter 8 deals with climate change. The first part of Chapter 8 gives a literature overview
of the climatic changes that are expected for the study area. Then, climate trends that are
already present in the observed data are identified by statistical analysis. Predictions for the
precipitation regime in the future are developed by an indirect method that exploits progno-
sis for the atmospheric circulation from Global Circulation Models to classify precipitation
4 Introduction
measurements from the past. The predicted trend signals of all the different analysis are
compared, and the most reliable trends are pointed out. The chapter concludes with some
ideas how the identified trends could be incorporated in the statistical precipitation models
developed in this work.
2. Characterization of the Hydrological
Conditions in Baden-Württemberg
In Central Europe, people often talk about the weather. The topic is always up to date be-
cause the weather changes frequently enough to keep it interesting. Sometimes they regard
the atmospheric conditions as comfortable (“nice weather”), other times as uncomfortable
(“bad weather”). People are accustomed to the fact that the weather keeps changing in ir-
regular patterns. They might judge it as normal but globally seen, this is a very special case,
restricted to mid-latitudes and there mostly to the west side of the continents.
In other parts of the world precipitation activity exhibits higher regularity. It is either domi-
nated by daily or by seasonal cycles or completely constant as over the subtropical desserts
where it constantly does not rain, sometimes for years. The inner tropics exhibit daytime
climate. In Singapore for example, most of the precipitation falls in the afternoon, some
in the early morning, but it generally does not rain between 20:00 pm and midnight. In
comparison, the changes over the year are of minor importance. The outer tropics and the
subtropics on the other hand are dominated by a yearly cycle of distinct rainy and dry sea-
sons with totally different conditions. Examples are the monsoon climates of southern Asia
on the one hand where almost the complete yearly rainfall comes at the end of the summer,
or the Mediterranean type climates on the other hand were the long summers are as dry as
in the dessert, but the rest of the year is humid (Weischet and Endlicher, 2008).
Central Europe exhibits a pronounced annual cycle in temperature, but less in precipitation.
In the study region of Baden-Württemberg, the highest monthly sum is observed in June
or July, the lowest in February. In general, however, precipitation can occur at any time of
the year. At measurement stations in Baden-Württemberg the average number of rain days
exceeds ten in any month (Fig. 2.1, Fig. 2.2).
The general atmospheric conditions in Baden Württemberg change in patterns of several
days to few weeks. German language has a special word for this phenomenon called “Wit-
terung”. It does not describe the actual weather, which is called “Wetter” in German, but the
average conditions over several days. The distinction between “Wetter” and “Witterung”
shows that climatic variations in Baden-Württemberg take place in different temporal scales,
from subdaily to several days, which is a result of the global atmospheric circulation.
2.1. Context in the Global Atmospheric Circulation
Due to the different exposition angles to the sun, the solar radiation income is higher at the
equator than at the poles. At the equator, the solar heating provokes a massive convective
6 Characterization of the Hydrological Conditions in Baden-Württemberg
uplift of air that results in a high pressure zone in the upper troposphere at about 10km
of altitude (Weischet and Endlicher, 2008). The cooling over the poles, especially in winter
when the poles are permanently shaded from the sun, leads to setting of air masses and so
low pressure in the upper troposphere.
The pressure gradient between Equator and Poles drives a strong compensation movement,
on the northern hemisphere from south to north. Coriolis force caused by the earth’s rota-
tion, deviates the south-north flux. The result is zonal west wind in the upper atmosphere.
This wind field is strongest in the mid latitude around 45◦ where the Coriolis force is strong
and the energy gradient is the highest.
The west wind field blocks the exchange between tropic and northern latitudes. If the west
wind field was stable, the energy gradient between Equator and Pole could never be at-
tenuated. The tropics would continously heat up, the energy gradient would increase and
the wind field permanently become stronger. However, if the Energy gradient in mid lat-
itudes exceeds 6◦ Kelvin per 1000km, the west wind zone becomes instable and starts to
undulate. Wide waves are formed that, embedded in the west wind field, move eastwards.
Where the waves peak to the north, subtropic warm air can move northwards, where they
peak to the south, cold arctic air is transported southwards. In this state, there is a real en-
ergy exchange and the driving temperature gradient is reduced. When it drops below 3.5◦
Kelvin per 1000km, the zonal west wind can recover and the cycle starts anew (Weischet
and Endlicher, 2008).
At the end of the undulating phase, when the waves break down, drops of cold are can be
cut off from the northern side of the interface between tropical warm and cool arctic air and
move southeastwards. The same can happen with warm air from the south. Both can block
the westerly wind field until the gradient builds up again and becomes strong enough to
reestablish the main flow direction (“blocking action”). This is especially frequent in winter,
when cold drops are supported by a cold ground layer over Siberia.
This permanently ongoing cycle governs the atmospheric conditions of Baden-
Württemberg:
• Averaged over the year, the main flow direction is from west-southwest. The energy
and moisture that determine the weather in Baden-Württemberg mostly come from
the Atlantic Ocean.
• The average atmospheric conditions are condemned to change in irregular intervals of
several days. Blocking situations sometimes can last longer up to few weeks or, if the
west wind zone is especially weak, reestablish after short interruptions.
• The state of the west wind field governs the average conditions. If the west wind
field forms a northern wave over Europe, the flux comes from the southwest and the
arriving air masses are rather warm and humid. During a southern wave the flux
comes from the northwest resulting in cool conditions. In blocking situations, the
energy and moisture fluxes are cut and stable dry conditions can establish.
• Longer dry periods are always related to blocking action. If in the same time a high
pressure zone is present further east over Asia, easterly wind fields can establish that
bring dry continental air that is cold in winter and hot in summer.
2.1 Context in the Global Atmospheric Circulation 7
• The energy and moisture of the weather that is experienced in Central Europe is
“imported”, mostly from the Atlantic Ocean, sometimes from the polar Sea or the
Mediterranean. Autochthonal conditions that reflect the local radiation and energy
balance can only establish for short periods during blocking events. In winter, when
the solar heating is low, it often leads to inversion with cold ground air and low
hanging clouds that bring no precipitation. In summer it causes warm and sunny
conditions.
The described large scale circulation in the upper troposphere triggers characteristic meso-
scale circulation patterns in lower levels. These patterns have a smaller spatial temporal
scale and are often referred to as “synoptic”. Over the west Atlantic, offshore of Maine, the
temperature gradient between cold polar and warm subtropical air masses is the most pro-
nounced. This zone, called the polar front, is the griddle of cyclonic depressions that char-
acterize Central European weather conditions. The undulations in the higher troposphere
weaken the stability of the polar front which reacts by a wave movement. In contrast to the
upper troposphere waves, the spatial scale is much smaller with the wavelengths limited to
1000km (Weischet and Endlicher, 2008). The waves move eastwards at the polar front and
start to turn and form an eddy, a cyclonic depression. The polar front recovers and a new
wave can start off within the next few days. In this way, families of mostly four, sometimes
up to six cyclonic depressions are formed (Weischet and Endlicher, 2008) until changes in
the west wind zone of the upper troposphere moves the polar front and triggers the next
series of depression from a new location.
At the polar front, the subtropical warm air has a higher momentum as the polar cold air,
since it comes from lower latitudes were the absolute speed of the earth’s rotation is higher.
When a wave starts to form at the polar front, the warm air, being lighter, glides upon the
cold air at the front side (the east side) of the wave. The forced uplift leads to cooling and
thus to condensation. A warm front is forming, which is characterized by compact strati-
form cloud bands of several tens of kilometers width and up to a few hundred kilometers
length. On the west side of the wave, the cold air from the north moves southwards. Being
heavier, it pushes underneath the warm air and lifts it up. The forced uplift leads to cooling
and high turbulent mixing of the two air masses. A cold front is formed which is, due to the
turbulent mixing, characterized by a scattered band of high convective clouds. The warm
and cold air masses behind the fronts try to fill up the low pressure center of the eddy but are
deviated to the right by Coriolis forcing. Therefore the eddy starts to rotate anti-clockwise
(on the northern hemisphere) forming a cyclonic depression. The cyclone gains speed as
long it is over the sea and then looses its momentum over the continent due to the increase
in ground friction. Over Eastern Europe, the low pressure center fills up and the cyclone
dies.
If the study region of Baden-Württemberg is within the zone of influence of a cyclonic de-
pression, depends on its birth place and the pathway it has taken over the Atlantic ocean.
Due to asymmetric Coriolis force, cyclonic depressions tend to move northwards, which is
the reason why west-southwest (and not west) is the main flow direction in Central Europe
(Weischet and Endlicher, 2008). The storms arriving at one point in Central Europe are clus-
8 Characterization of the Hydrological Conditions in Baden-Württemberg
tering in time (Vitolo et al., 2009). It can be assumed that the first storm of the family defines
the preferential pathway for its successors. So cyclonic depression frequently pass in series
of four to five with intervals of one to two days.
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dez
0
5
10
15
20
25
30
D
a
y
s 
o
f 
P
re
ci
p
it
a
ti
o
n
 H
 >
 0
.5
 m
m
avg= 12.4 d/month
FREIBURG I.BR.-EBNET  (1958-2003)
(a) Number of rain days
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dez
0
20
40
60
80
100
120
P
re
ci
p
it
a
ti
o
n
 [
m
m
/m
o
n
th
]
sum= 1052 mm/a
FREIBURG I.BR.-EBNET  (1958-2003)
(b) Monthly precipitation sum
Figure 2.1.: Average number of rain days with more than 0.5mm and average monthly rain-
fall sum in Freiburg-Ebnet
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dez
0
5
10
15
20
25
30
D
a
y
s 
o
f 
P
re
ci
p
it
a
ti
o
n
 H
 >
 0
.5
 m
m
avg= 11.5 d/month
FELLBACH (1958-2003)
(a) Number of rain days
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dez
0
20
40
60
80
100
120
P
re
ci
p
it
a
ti
o
n
 [
m
m
/m
o
n
th
]
sum=  724 mm/a
FELLBACH (1958-2003)
(b) Monthly precipitation sum
Figure 2.2.: Average number of rain days with more than 0.5mm and average monthly rain-
fall sum in Fellbach
The weather in the study area is to a high extend characterized by the cyclonic activity over
the northern Atlantic. This has the following consequences:
• Most of the precipitation is bound to warm and cold fronts of cyclonic depressions.
2.1 Context in the Global Atmospheric Circulation 9
Legend
elevation [m]
84 - 196
196 - 308
308 - 416
416 - 523
523 - 632
632 - 750
750 - 907
907 - 1 405
(a)
Legend
Yearly Sum [mm]
622 - 700
701 - 800
801 - 900
901 - 1000
1001 - 1100
1101 - 1300
1301 - 1500
1501 - 1600
1601 - 1800
1801 - 2000
(b)
Figure 2.3.: Elevation in 1km × 1km averaged raster and average yearly precipitation sum
for the study area of Baden-Württemberg
10 Characterization of the Hydrological Conditions in Baden-Württemberg
• There are two main types of precipitation: advective precipitation at warm fronts and
convective precipitation at cold fronts with different characteristics in duration, vari-
ability and intensity.
• The warm front of a cyclonic depression always arrives first. The warm front is char-
acterized by stratiform clouds that at first cover the sky at high altitudes and then gain
in thickness by growing downwards. The resulting rainfall is rarely very intensive but
generally long lasting and spatially homogeneous. The following cold front is char-
acterized by turbulent mixing. It only rains under convective channels where the air
is forced upwards by the mixing. Therefore, the precipitation field is highly variable
and scattered. The intensities are generally higher, but the duration is shorter than at
warm fronts. It sometimes rains in violent showers or even thunder storms. Behind
the cold front, the colder polar air masses are in general instable and bring some more,
but sparse and less intensive showers.
• Cyclonic activity is highest in winter when the temperature gradient between the
Equator (constantly warm) and the North Pole (much colder in winter) is the steep-
est. The evaporation over the Atlantic Ocean depends on the water temperature. In
combination of the two factors, the precipitation due to cyclonic activity reaches its
maximum in late autumn.
• In the study region however, the highest monthly precipitation sums are measured
in June (Fig. 2.1 and Fig. 2.2). The additional precipitation in summer is the result
of enhanced convection. The increased solar heating is an additional driving force
for the turbulent mixing at cold fronts resulting in higher precipitation intensities. The
solar heating can become so strong that it even triggers convective precipitation events
which are not bound to an approaching front. For this reason, the highest precipitation
intensities are measured during summer, although the average number of rain days is
not higher.
• Passing fronts react to orographic effects. Mountain ranges are obstacles in the flow
field that force the approaching air masses to move upwards which induces cool-
ing and thus condensation. The release of moisture results in enhanced precipitation
(Weischet and Endlicher, 2008). With the main flow direction of west-south west, oro-
graphic enhancement is most perceptible at the western slopes of mountain ranges.
The regions east of mountain ranges the other hand are shaded from the main flow
direction and exhibit lower precipitation sums.
2.2. Regional Conditions in Baden-Württemberg
Baden-Württemberg has an area of 35752km2, the elevation is between 84m above sea level
in the Rhine valley in the North West and 1493m at the peak of the Feldberg in the south-
ern Black Forest. The Rhine valley covers the complete Western Boarder to France and
Rhineland-Palatinate. East of it is the Black Forest which is the highest and also the widest
mountain range in the study area. Other Mountain ranges are the Swabian Alb with peaks
2.2 Regional Conditions in Baden-Württemberg 11
of more than 1000m located in the East of Baden-Württemberg, as well as the Odenwald in
the very north and the Swabian-Franconian Forest in the northeast whose highest peaks are
around 600m.
The highly structured landscape of Baden-Württemberg finds its expression in the spatial
variability of precipitation. Fig. 2.3 compares a 1km2 average elevation raster with an in-
terpolated field of the average yearly precipitation sum between 1958 and 2003. The Black
Forest is a pronounced obstacle for the flow in west-east direction. Orographic effects are the
strongest here and therefore it receives the highest yearly precipitation. The areas east of it
are shaded by the Black Forest, thus the Neckar valley is the driest region in the study area.
The range of average yearly precipitation reaches from less than 700mm, e. g. in Stuttgart
to about 2000mm, e. g. at the Hornisgrinde, the highest peak in the northern Black Forest.
To a lesser extent, orographic effects can also be found at all the other mountain ranges. At
the Swabian Alb however, due to shading, the average yearly precipitation sums reach only
about 60% of the values from the Black Forest at the same altitude. The Danube valley which
is, seen from the main flow direction, behind Black Forest and Swabian Alb is again rather
dry. Then South of the Danube, the average yearly precipitation sum is gradually increasing.
This is the increasing orographic influence of the nearby Alps – especially affecting fluxes
from the Northwest. The Odenwald, although much lower than the Swabian Alb, is more
humid anyway because it is not shaded by any high mountain range west of it.
The orographic effect of the Black Forest can also be seen in the distribution of precipitation
over the year. The deficit of precipitation in Fellbach (Fig. 2.2), downstream of the Black
Forest, compared to Freiburg (Fig. 2.1), upstream, is of more than 300mm per year. The
deficit is highest in the months of November and December when the cyclonic activity is
highest. The orographic shading is the most effective during that time of the year. During
summer months, when precipitation is dominated by convective events, the difference be-
tween Fellbach and Freiburg is lower. As a result, the ratio between the rainfall sum in the
driest month of February and the wettest month of June is lower in Fellbach. This can be
extended to a rule of thumb: the further east a station is located in Baden-Württemberg, the
higher is the part of the yearly precipitation sum that falls during the summer months.
3. Acquisition of Precipitation Data
The essential basis on any statistical research on precipitation are measured precipitation
data. In principle measuring precipitation is easy. It is sufficient to collect all the precipita-
tion that falls during a defined time interval on a surface of defined area and measure the
volume or the mass of the collected water. Precipitation is generally measured in volume
per area which is equivalent to a depth. The most common unit is `m2 = mm. Referring the
depth to a time interval, results in rainfall intensity, measured for example in mmh .
In reality however, accurate precipitation measurement is challenging. It is difficult to catch
all drops and in the same time not to disturb the process, for example not to alter the wind
field by the measurement device. Measurements on high temporal resolution are especially
difficult as the measured volume becomes very small on short time intervals. Therefore,
small losses can induce a high relative error.
Direct measurements only give information about the precipitation at one point which is a
limitation if for example the water balance over a river catchment is analyzed. For informa-
tion on areal precipitation remote sensing techniques are applied.
3.1. Point Measurements
3.1.1. Rain Gauge after Hellmann
The most common point measurement device in Germany are rain gauges that follow the
design principle developed by Hellmann in 1886. It features a 200cm2 circular collection
funnel (diameter of 15.95cm) made from steal or other non-corroding metal. The collection
funnel is limited by a sharp edged ring to avoid side effects at the fringe and it is equipped
with a metal cross on top that prevents fallen snow from being blown out.
The funnel conducts the precipitation to a container that is collecting the water. In the sim-
plest case, the container is emptied once in 24h either by hand or automatically and the
volume or mass of the collected water is measured. Daily precipitation sums are generally
measured from 7AM to 7AM.
For higher time resolutions instantaneous rainfall intensities have to be measured. There are
several different types of devices:
• Measurement by Water Level
In such devices the water trickling down the funnel is collected in a small cylindrical
container equipped with a floater. The floater registers the rise of the water level.
3.1 Point Measurements 13
Knowing the base area of the cylinder, the rising speed can be transferred into rainfall
intensity. When the cylinder is full, it is automatically emptied into a bigger container
that collects the total precipitation.
• Tipping-Buckets Measurements
In a Tipping Bucket device the precipitation from the funnel drips into a small cup of
defined volume. When the cup is full, it tips over and releases the collected water.
The number of tips is counted and transformed into a rainfall intensity. In general,
the Tipping Bucket has two chambers. When one chamber is emptying, the other one
is in place so that no water gets lost during the emptying. Due to the discrete nature
of Tipping Bucket Measurements, the temporal resolution depends on the rain rate.
No signal is emitted before the bucket tips over. Hence, if it rains less intensive, the
measurement interval is longer.
• Gravimetric Measurements
In gravimetric devices the container collecting the water is mounted on a balance.
The precipitation intensity is calculated by the change in weight over time. Gravimet-
ric Measurements are principally more accurate than Tipping Buckets Measurements.
Being fully continuous, they show no time delay except the time it takes for the water
to trickle down into the container.
Compared to daily stations, high resolution rain gauges require much effort in maintenance
and calibration. The maintenance and calibration standard of the German Weather Service
(DWD) recommends to equip each station with an network connected analysis and alarm
system for remote technical controls (Bartels et al., 1999). To prevent the Tipping Buckets
or weighted containers from freezing, the housing has to be heated. Bartels et al. (1999)
demand thermostat controlled heating to avoid excessive evaporation losses.
3.1.2. Disdrometer
Disdrometer do not measure rainfall intensity but the full spectrum of rain drop sizes. (“Dis-
dro” stands for “distribution”.) This is especially important for the calibration of rain radar
devices (Section 3.2.1). The first disdrometer has been developed by Joss and Waldvogel in
1967. It consists of a styrofoam body that is kept in position by two magnetic coils. If a rain
drop hits the styrofoam body, it transfers its momentum, provoking a small deviation of the
styrofoam body. By electromagnetic induction an electric current is generated at the mag-
netic devices. As the momentum of a raindrop depends on its mass, the current is related to
the size of the raindrop.
Malvern Particle Sizers measure the distribution optically by a laser beam. The beam is
emitted at one side of the device and received at the other side by light sensitive diodes.
Each raindrop that falls through the scanned volume leads to a short interruption of the
laser beam. The drop size is calculated from the extinction amplitude. The time length of
the extinction is a measure for the falling speed of the drop.
14 Acquisition of Precipitation Data
While the disdrometer of Joss and Waldvogel is suited for the detection of big drops, the
Malvern Particle Sizer is especially useful for the detection of small droplets with a diameter
of less than 0.1mm (Emeis, 2010).
3.1.3. Error Sources
To attain high accuracy in short interval precipitation measurements is an ambitious task.
The calibration outline of the German Weather Service defines a spread of 5% in three
measurements as acceptable for Tipping Bucket devices. In gravimetric measurements the
spread in three measurements must not exceed 0.4mm (Bartels et al., 1999). Groisman and
Legates (1994) estimates that the total error in accumulated annual precipitation of US rain
gauges is between 5% and 25% with larger errors in higher elevation and in winter time.
Main error source in rain gauge measurements is wind. It prevents the rain drops from land-
ing in the funnel. The World Meteorological Organization (WMO) defined a standard wind
protected reference rain gauge. It is either buried into the ground or protected by a dou-
ble wind shield (Sevruk et al., 2009). Comparing measurements from ordinary, unshielded
gauges to the reference, Sevruk et al. (2009) estimate a wind induced error of about 3% to
6%. Wind errors can be corrected by empirical formulas based on numerical simulation.
Another important source of error is snow. According to Sevruk et al. (2009),at a wind speed
of 6m/s an ordinary rain gauge catches only about 20% of the precipitation when it is snow-
ing and only 50% of mixed precipitation of rain and snow. The loss due to evaporation on
the other hand is comparably small (Groisman and Legates, 1994). However, it affects espe-
cially the smallest values which are important in the estimation of precipitation probability.
Most susceptible for evaporation losses are graviemtric devices since the water remains in
the container for a long time.
Other errors concern the mechanics. The rain gauge funnel can get clogged by debris or the
reed switches of Tipping Buckets can be blocked (Ciach and Krajewski, 2006). To detect and
fix such errors it is required that rain gauges are controlled in short maintenance intervals.
Recording and data transmission are further potential error sources. When for example
the baseline signal of a recording station is slightly drifting, the rain gauge gives out small
rainfall increments instead of zero values. This is not important in terms of the overall
precipitation volume but can be a severe problem in the estimation of rainfall probability.
Finally, for spatial correlation analysis, it is important that all precipitation stations are syn-
chronized. If the time stamp at one station is wrong (for example shifted by some minutes)
the observed spatial correlations to all other stations is altered.
3.1.3.1. Examples of Recording and Data Transmission Errors
Fig. 3.1 shows an error that is probably caused by data transmission. It was found at the
station in Wiesloch (Gauss-Krüger 3 of x = 3478250 and y = 5462670). The picture is an edit
of the time series delivered from DWD. The seven digit code at the beginning of each record
3.1 Point Measurements 15
Figure 3.1.: Example of an error in a precipitation time series, probably occurred during data
transmission
is the time stamp. It states the day in the year (first three digits) and the daytime (next four
digits). What is particular about this series are the isolated high values that always occur at
half past midnight. It is assumed that these are errors, due to their regularity, the crisp value
of exactly 10.0mm and the periodic behavior. The reasons are less clear. Perhaps it was
caused by a misinterpretation of an end of line character when daily files were combined to
longer time series or a signal error in data transmission that altered the code for “new day”.
The data presented in Fig. 3.2 and Fig. 3.3 are from the station of Craislheim (X = 3577650;
Y = 5446865). The time series attracted attention due to the highly monthly sum in July
2003, which was a rather dry month at other stations. The values on the first day, the 14th of
July, all sum up to the same amount of 15mm (Fig. 3.2) the values of the second day to about
30mm Fig. 3.3. Most probably a service team was at the station and checked the accuracy.
Later, it was forgotten to cancel out the values or the report got lost. It is very unlikely
that these values are really rainfall because of the isolated, unrealistic high intensities. The
hypothesis of maintenance is more likely. The values on the next morning, now the 16th of
July, are less suspicious. However, in combination with the unrealistic rain depths 24h and
48h before, they were sorted out.
16 Acquisition of Precipitation Data
Figure 3.2.: Example of an error in a precipitation time series, probably due to unreported
calibration and maintenance
3.1.3.2. Some Aspects of Error Identification
Many errors can be detected by simple cross checking. Therefore, Ciach and Krajewski
(2006) recommend to always run two rain gauges at the same location. If the stations in
Crailsheim and Wiesloch featured two independent measurement devices, the error could
have been canceled out immediately: One of the devices would have reported precipitation
and the other would not. To control for measurement errors all new DWD stations have to
be equipped with two measurement devices, either with Tipping Bucket and gravimetric
devices or with water level and gravimetric devices (Bartels et al., 1999).
It is hard to set up a fully automatic algorithm for error detection. Many errors are obvious
for a human being looking at the data but difficult to translate in purely mathematical logic.
A good strategy is a semi-automated method. Many errors can be detected by unrealistic
high yearly, monthly or daily precipitation sums. However, not all extreme sums are due to
measurement errors. The search for suspicious values can be performed by a computer al-
gorithm. Then this sample of potential data errors has to be checked by eye. In this manner,
the errors reported in Section 3.1.3.1 were found.
The person in charge with error correction is in a dilemma. As Emil Julius Gumbel, one
3.1 Point Measurements 17
Figure 3.3.: Example of an error in a precipitation time series, probably due to unreported
calibration and maintenance (continued)
of the founders of extreme value statistics, stated: “It’s impossible that the improbable will
never happen.” (Gumbel, 1958). So it is not a good strategy to sort out all the values that
seem unrealistic. In this way one would systematically eliminate all real extremes events too,
which is problematic because it leads to a sever underestimation of potentially dangerous
precipitation events.
However, not all the errors are as obvious as the two examples stated in Section 3.1.3.1.
Sometimes error detection demands for a little fantasy. It often helps to check on the internet.
If an extreme rainfall event occurred, then maybe there was an article in a local newspaper
or a note in the chronicle of the fire brigade that can be found on the web.
18 Acquisition of Precipitation Data
3.2. Remote Sensing
Remote sensing can be defined as a technique that allows to measure “without direct con-
tact between instrument and object” (Emeis, 2010). For precipitation, this is mainly the mea-
surement of precipitation related radiation. There are two different measurement principles:
passive and active measurement. Passive measurement devices only detect naturally emit-
ted radiation, while active measurement devices send out a signal and measure the reflection
(e. g. the fraction of reflected energy).
The two main types of operational remote sensing devices are terrestrial rain radar and
satellite remote sensing. Rain radar measurements can give detailed images of spatial het-
erogeneous rain fields in temporal resolutions up to few minutes. Terrestrial rain radar mea-
surements are active measurements. Satellite remote sensing combines different actively
and passively measured signals as microwave radiation and infrared radiation in several
wavelengths. Satellite images are limited in spatial-temporal resolution, but are the most
important source of information about precipitation intensities in sparsely gauged regions
and over the ocean.
3.2.1. Radar Measurements
Electromagnetic radiation gets scattered by any particles that are much smaller than the
wavelength. The fraction of scattered energy depends on the 6th power of the droplet di-
ameter (Emeis, 2010) and on the inverse of the 4th power of the wavelength. This phenom-
ena, called Rayleigh scattering, is the measurement principle of rain radar devices. Water
droplets and ice crystals scatter the energy of a radar beam that was emitted into the at-
mosphere. The scattered energy dissipates in every direction. Part of it is reflected back to
where the beam was sent out from. A functional dependence can be set up between the re-
flected fraction of energy Z and the rainfall intensity R in the scanned atmospheric volume.
It is called the Z-R relation.
A rain radar consists of an emitter sending out a distinct beam of known duration and de-
fined amount of energy and an receiver that measures the energy reflected back to the radar
station. Emitter and receiver are synchronized. From the travel-time between emission and
reception the reflection distance is calculated. Applying the Z-R-relationship, transforms
the reflectivity profile into a rainfall intensity profile along the beam axis. The intensity is an
average over the diameter of the radar beam. In general, radar beams of a certain aperture
angle, typically 1◦ are sent out. This means that the beam measures the intensity in a conic
volume whose diameter is increasing with the distance from the radar station. Thus, the
spatial resolution depends on the distance to the radar station.
There are different modes of rain radar scans. The simplest application is a one dimensional
scan along the axis of a radar with fixed orientation. This is mostly used for vertical profiles
of the atmosphere, e. g. to identify height levels on which precipitation is formed. Two
dimensional scans can be performed if the radar emitter sends out horizontal beams accord-
ing to a fixed pattern, e. g. every degree of angle, while rotating on its vertical axis, e. g.
3.2 Remote Sensing 19
once in five minutes. The profile of all beams from one rotation can be composed to two di-
mensional images. For a volumetric scan the rotations are repeated with varying elevation
angle.
Different radar bands are used for rain radar, namely theX ,C and S band with wavelengths
of 3.2cm, 5cm and 10cm. X band radar achieves the highest spatial resolution but, com-
pared to the other two bands, the range is lowest since the energy dissipation is higher for
shorter wavelengths. The S band radar signal becomes the least attenuated when traveling
through the atmosphere but needs longer antennas and more energy to give a reasonable
spatial resolution (Delrieu et al., 2000).
Radar images are the only source of information that gives a high resolution continuous pic-
ture of the spatial patterns in precipitation fields. Detailed investigations of spatial structure
and dependence in precipitation could not start before the first radar images were avail-
able. Quantitative precipitation measurement, however, is difficult since rain radar mea-
surements are affected by many different error sources:
• Errors due to variations in the Z-R relation.
A priori, the relation between reflectivity and rain rate is a physical law Since the
reflectivity depends on the rain drop surface and not directly on the volume and due
to the fact that the falling velocity increases with the drop size, the Z-R relation is
highly non-linear. Generally it is described by a power law function of the form
Z = a ·Rb (3.1)
(e. g. Klazura et al. (1999), Delrieu et al. (2000) or Lee and Zawadski (2005)).
The non-linear function can only be determined if the drop size distribution is fully
identified. Since the drop size distribution is changing in time, the Z-R relation is
non-stationary.
The real drop size distribution within the volume that is scanned by the radar beam
cannot be observed. It can only be measured at the ground and only point-wise. The
Z-R relation is usually set up for the climatological average of the drop size distribu-
tion (e.g. Klazura et al., 1999). This purely statistical function averages out any vari-
ability in the drop size distribution, for example due to differences in rainfall forma-
tion. Rainfall from melted snow for example leads to different drop sizes than warm
precipitation as snowflakes aggregate in the formation and break up in the melting
(Mitra et al., 1990). Precipitation from stratiform clouds consists of smaller drops than
rainfall from convective events (Lee and Zawadski, 2005).
For Montreal, Canada, the average relative error that is induced by using a climato-
logical average Z-R relation was quantified by 41% (Lee and Zawadski, 2005). Daily
averaged Z − R relations induced a relative error of 28%. The drop size distribution
can change even within one rainfall event, e. g. upstream or downstream of a passing
front (Lee and Zawadski, 2005). Therefore, recent publications investigate statistical
tools for a better identification of the drop size distribution, by classifying events ac-
cording to the rainfall formation (Ignaccolo and Michele, 2012a) or by the presence of
orographic effects (Ignaccolo and Michele, 2012b).
20 Acquisition of Precipitation Data
• Bright band
Related to the errors in the Z-R relation, however more specific, is the problem of
bright band occurrence. The water on melting snow flakes or hail leads to stronger
reflection than expected by their water content. The radar “sees” the diameter of the
snow flake and assumes a droplet of this size. The result is a strong overestimation of
the precipitation rate if the radar beam hits the melting layer in a precipitation field
(Germann and Joss, 2002) or in the presence of hail (Austin, 1987). Since hail is corre-
lated with very high precipitation intensities, bright band correction can lead to a loss
of information on most extreme events (Baeck and Smith, 1998).
• Ground clutter and beam blockage.
Raindrops are not the only objects that reflect the radar beams. Beside other flying
objects as birds or airplanes, the reflection at high towers, trees or mountain peaks is a
real issue. If such objects block the beam completely, the rain radar is blind for what is
located behind. If the beam is only blocked partly, the radar assumes a high rain rate
at the clutter point.
Clutter identification can be done by pixel-wise accumulation of rainfall sum (e. g.
during one month), which is highly overestimated at clutter points, or geographically
with the help of a precise digital elevation model. More sophisticated data driven
clutter correction algorithms are based on Fuzzy logic or Bayesian framework (Rico-
Ramirez and Cluckie, 2008) or on aggregated statistics of the radar signal (Hubbert
et al., 2009).
• Errors due to vertical variability
The issue of clutter and beam blockage is especially severe in mountainous regions. To
avoid excessive losses in scanning range, rain radar stations are often built on moun-
tain peaks. However, the higher elevation comes at the price of reinforced errors due
to the vertical variability in the rainfall intensity (Germann and Joss, 2002). If the rain
clouds are (partly) at a lower altitude than the lowest scanning level, the radar beam
does not measure the full rainfall intensity since part of the precipitation is forming
below the scanned volume (“overshooting”). According to Germann and Joss (2002),
this is the most severe error source in alpine regions where the radar beams can reach
altitudes of 2000 or 3000m. Andrieu and Creutin (1995) developed a method to correct
for vertical variability errors, but it requires at least data from two different elevation
angles. Germann and Joss (2002) propose a pixel-wise correction factor relating the
radar measured intensities to gauge measurements at the ground.
• Attenuation
The raindrops falling through the radar beam do not only reflect the energy back to the
radar station but dissipate it in all possible directions. Each detected raindrop dimin-
ishes the amount of energy remaining for the following measurement farther out on
the radar beam. The higher the rainfall intensity, the higher the attenuation. Since the
rain rate is calculated from the fraction of radar energy reflected back to the receiver,
attenuation leads to underestimation of the rainfall intensity in areas that are hidden
behind intensive rainfall events. S-band radar is much less subject to attenuation than
C-band radar. In heavy rainfall events C-band rain rates can be more than 50% lower
3.2 Remote Sensing 21
(Delrieu et al., 2000).
The correction for attenuation errors is very difficult since the error is proportional
to the integral over the reflectivity measured along the beam. There is an analytical
solution to calculate the path integrated attenuation, but, due to the dependence on
the measurement signal, it is very sensitive to errors and can become instable (Delrieu
et al., 2000).
Radial fields from two dimensional radar scans are stored in a polar coordinate system that
is centered on the radar stations. In overlapping regions, the segments of the fields from two
different stations are not identical. For the use of these measurements in hydrological appli-
cations, e.g. flash flood warning, the measurements are interpolated on a regular, Cartesian
grid. Besides, errors are detected and corrected. Many national meteorological services
give out radar data in interpolation products (e. g. RADOLAN for Germany, Bartels et al.,
2004). RADOLAN consists of five minute sums of the areal precipitation on a 1km × 1km
grid. It is calculated from the two dimensional scans of 17 C band radar stations. Clutter is
eliminated and the rainfall amounts are adjusted with data of several hundred precipitation
stations from all over Germany. Such products, however, are still subject to errors. Baeck
and Smith (1998) for example find a general underestimation in the WSR-88D-III product
emitted by the National Weather Service of the United States of America.
Radar and rain gauge measurements, however, are not directly comparable. Even if both
methods produced perfect error free measurements, there would be deviations. Rain gauges
measure rainfall intensities or accumulations at one point at ground level. Radar measures
instantaneous rainfall intensities in a scanned atmospheric volume that are accumulated to
precipitation sums of a certain time interval (e. g. five minute values) over a defined area
(e. g. a square of 1km× 1km). In the absence of measurement errors, both would only give
identical values if the precipitation was spatially homogeneous over the area of the radar
cell. Besides, radar measures the precipitation at a certain altitude and it is unknown where
or when the same precipitation can be measured on the ground because the raindrops need
some time to fall and drift in the wind during their travel.
Rain gauge and radar measurement can only be compared in accumulations over long time
intervals during which spatial variability is assumed to cancel out.
The difference between point and areal precipitation makes it difficult to correct for errors in
measured radar fields. The true areal precipitation, which would be needed as a reference to
develop error correction algorithms, is unknown. Rain gauge measurements are generally
considered as more accurate but only give point values. The radar exhibits more scatter but
gives a detailed picture of the spatial heterogeneity. Merging algorithms (Ehret et al., 2008)
try to benefit from both features by embossing the variability of the radar measurements on
the station values. A test of different merging algorithms can be found in Goudenhoofdt
and Delobbe (2009).
Germann et al. (2009) point out that the correlation structure in the error is very important
for any hydrological application that uses radar data as input. Hydrological models to some
extend integrate over the catchment area, and therefore uncorrelated errors are mostly can-
celing out in the hydrological response whereas highly correlated errors are adding up.
22 Acquisition of Precipitation Data
When radar measurement came up, there was a hope that radar measurements could one
day substitute for the expensive ground station network. Collier (1986) reports: “There
has been considerable debate in recent years as to whether measurement techniques based
upon remote sensing technology, using radar and/or satellite systems, can complement,
or indeed replace, rain gauge measurements.” Despite of the advances in radar technology
and radar correction during the last 25 year, a replacement of station measurements by radar
data did not take place. In reality the opposite is the case; rain radars depend on ground
measurements. Full benefit from radar data can only be achieved if it is combined with
rain gauge data. Firstly, as all indirect measurement methods, rain radar measurements
depend on calibration that is only possible with directly measured ground data. Secondly,
radar correction algorithms depend on the additional information of rain gauge data. The
denser the station network is, the better is already the spatial representation in the point
measurement network and the higher is the accuracy of the spatial-temporal precipitation
fields combining the two sources of information.
3.2.2. Satellite Remote Sensing
Satellites used for precipitation estimation are equipped with different devices that mea-
sure precipitation related signals in radiation. The satellites feature sensors for detection
of infrared radiation of different wavelengths and for microwave radiation. The newest
precipitation satellite, the Tropical Rainfall Measurement Mission (TRMM) satellite, is also
equipped with a rain radar device (JAXA, 2012).
The detection of precipitation by infrared and microwave measurements is very indirect.
Primarily, the radiation scans give information on the shape and thickness of clouds which
in turn is used for the estimation of precipitation rates (Kidd, 2001). As the Earth’s back-
ground radiation is different over sea than over land and as the rainfall formation processes
vary with latitude, the signal proceeding has to be adapted on the geographic location.
Precipitation detection satellites are on Low Earth Orbits (LEO) of several hundred kilome-
ters above ground. They surround the Earth in on or two hours (Kidd, 2001). Seen from the
Earth, they are constantly moving. The satellites scan the surface in swaths of few hundred
kilometers width which is depending on the measurement sensor. As a consequence, each
point on the Earth is visited once in a certain satellite specific interval.
As a result of the radar movement the nature of radar measurements is discontinuous. The
radar images for one location are instantaneous snapshots with long regular gaps in be-
tween. For continuous time series, radar data have to be aggregated and interpolated. The
interpolation is a trade off between temporal resolution and spatial coverage. Products, as
for example the TRMM Multisite Precipitation Analysis (TMPA) combine data from differ-
ent satellites and also from different sensor types. The temporal resolution of TMPA data
is 3h, the spatial resolution of 0.25◦ × 0.25◦, which is equivalent to 27.8km at the equator
(Huffman et al., 2007).
Radar measurements are of great benefit for previously ungauged regions as remote parts
of the continents and the ocean. Radar data help to complete the picture of global precipi-
tation and are therefore essential in research on global atmospheric circulation and climate.
3.3 The Data Set of Precipitation Measurements 23
For many hydrological applications however, for example rainfall runoff simulations for
small scale river catchments or urban sewage systems, radar data is sill too coarse. Experts
state that a spatial-temporal resolution of 4km and 30min “can be achieved in the near fu-
ture” (Sorooshian et al., 2011) but accuracy issues remain. Sorooshian et al. (2011) point out
that there is still demand for research in validation, uncertainty analysis and bias removal.
AghaKouchak et al. (2011) analyzed several satellite precipitation products for the Central
United States. They state that “no single procuct can be considered ideal for detecting ex-
treme events”. They all tend to “miss a significant volume of rainfall”. A limitation that up
to now prevents the design of radar based flood warning systems (Sorooshian et al., 2011).
3.3. The Data Set of Precipitation Measurements
The research of this work is based on precipitation measurements from ground sta-
tions. The precipitation time series were delivered by the environmental authority of
Baden-Württemberg (“Landesanstalt für Umwelt, Messungen und Naturschutz Baden-
Württemberg” – LUBW), except for eight stations that were delivered by the Swiss mete-
orological service, MeteoSwiss. The data was provided in numeric form. The network of
rain gauges in Baden-Württemberg includes stations maintained by the LUBW and the Ger-
man meteorological service (“Deutscher Wetterdienst”, DWD). According to LUBW the data
was verified, but a close manual inspection revealed some errors in the time series. Some
examples can be seen in Section 3.1.3. The data set was already used in former studies and
originally consisted of 295 high resolution rainfall stations (Brommundt, 2008). Three sta-
tions however, had to be sorted out completely due to data problems that were detectable
in aggregated statistics as yearly precipitation sum but could not be located in the time se-
ries (see Section 3.1.3). These were the stations in Oberburken with DWD-Number 61347,
Buchen with DWD-Number 116558 and Oberrotweil-Vogstburg with DWD-Number 70293.
The rainfall stations are rain gauges after Hellmann (Section 3.1.1). Most high resolution
stations are tipping bucket rain gauges. The stations inaugurated after 1997 are equipped
with two measurement devices for cross checking, either a gravimetric and a water level de-
vice or a gravimetric and a tipping bucket measurement device. The average spatial density
is one rain gauge in 70km2 for the daily stations and one station in 122km2 for the high res-
olution rain gauges. The average length in the daily time series is 28.9 years (Brommundt,
2008) in the hourly time series of about 9 years. The longest time series in hourly resolution
is the station in Karlsruhe (DWD-Nb. 70621) with 33 years of available data from 1958 to
1990 but with a high fraction of missing values. Due to the simpler measurement procedure
for daily precipitation values and the lower effort in maintenance, the daily time series are
not only longer and more frequent but also exhibit a better quality with less missing values.
3.3.1. Subsets
The number of available stations varies over the years. Many of the longer high resolution
time series end in the late 1980s or early 1990s. The situation improves again by the year
24 Acquisition of Precipitation Data
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Legend
!R daily stations
elevation [m]
84 - 196
196 - 308
308 - 416
416 - 523
523 - 632
632 - 750
750 - 907
907 - 1 405
(b)
Figure 3.4.: Location of the daily and hourly precipitation stations used in this work
3.3 The Data Set of Precipitation Measurements 25
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Legend
!H 1h station 1997-2003
elevation [m]
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308 - 416
416 - 523
523 - 632
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907 - 1 405
Figure 3.5.: Subset of 137 high resolution rain gauges with continuous measurements from
1997 to 2003
of 1997 when the DWD inaugurated many new rain gauges. For analysis of spatial depen-
dencies it is required that all measurement stations have the same measurement period. For
such tasks, only the sub-period from 1997 to 2003 is considered. During this time, data from
137 of the 292 high resolution rain gauges is available. The location of all stations in the
subset can be found in Fig. 3.5.
Another station subset of special importance is the subset of precipitation stations with
30min measurement interval where the fraction of missing values is particularly low. It
consists of 30 stations. For for the measurement period from 1991 to 2003 the data availabil-
ity is close to 100% (Fig. 3.6).
The red crosses in Fig. 3.4a indicate the two stations in Holzgerlingen. These two stations
were in operation from 1977 to 1992. With only about 4% percent of missing values, the data
quality is high. The two stations are located near the center of the study region in a dis-
tance of 3km to each other in approximately east west direction (Gaus-Krüger 3 coordinates
of: X=3498810, Y=5383440 and X=3501820, Y=5384490). Due to the high data quality, the
vicinity and the long common measurement period, the two stations are regarded as an rep-
resentative example for spatial correlated precipitation time series and used for verification
purposes throughout this work.
26 Acquisition of Precipitation Data
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elevation [m]
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308 - 416
416 - 523
523 - 632
632 - 750
750 - 907
907 - 1 405
Figure 3.6.: Subset of 30 high resolution rain gauges with continuous measurements from
1991 to 2003 and low number of missing values.
4. Statistical Properties of Precipitation
The German weather service DWD operates several thousand rain gauges in Germany.
Their temporal resolution ranges from daily scale to five minute values. Additionally, the
whole area of Germany is covered by a network of 17 precipitation radar devices. Despite
these efforts, there is a lack between the demand for precipitation data as input for hydrolog-
ical and hydraulic simulation and the offer of available measurement data. This lack of data
is due to statistical properties of precipitation: spatial temporal heterogeneity, intermittency
and scaling.
4.1. Spatial and Temporal Heterogeneity
0 20 40 60 80 100 120 140
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Figure 4.1.: Dependence of correlation between rainfall time series on the station distance.
Example for Westernheim (GK3: x = 3546150; y = 5375070) and five surround-
ing rainfall stations. Data from 1998.
a) correlation of the daily time series versus distance.
b) correlation depending on the temporal aggregation from five minutes to 24
hours
One of the statistical characteristics of precipitation is a high spatial and temporal variability
and the fact that variation in space and in time are connected. A typical example is displayed
in Fig. 4.1. The figure shows the correlation of the time series for 1998 at five rainfall stations
(Wiesensteig, Muensingen-Apfelstetten, Baltmannsweiler-Hohengehren, Schwäbisch Hall
28 Statistical Properties of Precipitation
and Durbach-Ebersweier) to the station of Westerheim on the Swabian Alb (Gauss-Krüger
3 coordinates of x = 3546150; y = 5375070). The stations and the target year were chosen
arbitrarily. The only criteria were a big range of distances (between 6km and 120km) and
100% data availability. Fig. 4.1a displays the correlation of the daily time series at these
stations to the time series at Westerheim. The correlation can be seen as a measure of their
similarity. From Fig. 4.1a it becomes clear that the similarity is higher if the distance between
the stations is low. The correlation values are within a typical range for Middle Europe.
Osborn and Hulme (1997) analyzed 185 stations pairs in Middle and Western Europe and
found an average distance of 150km to 200km for a correlation of 0.5.
However, independently of the distance between the rainfall stations, the correlation also
depends on the temporal resolution of the records (Fig. 4.1b). At least at short distances, the
daily values are strongly related. The hourly values show more freedom in their variations
while five minute values have a very low correlation if the distance between the stations is
higher than a few kilometers. If one thinks of a horizontal line through Fig. 4.1b at the 0.6
correlation level one can say (for this particular set of stations - but similarly for all other
stations in Baden-Württemberg) that two five minute time series of a few kilometer distance
differ as much from each other as two time series of daily values in more than one hundred
kilometer distance.
This permits the reversal conclusion that the area for which a rainfall record is representative
is more and more limited, the shorter the considered temporal resolution is. For a complete
spatial representation, the network of five minute rain gauges should therefore be much
denser than the network of daily rain gauges. Unfortunately, the opposite is the case in
Germany due to the much higher maintenance requirement (Section 3.1) for high resolution
measurement stations.
Two stations at a longer distance have a lower overall level of correlation but show the same
decline towards shorter aggregation time intervals. The different curves in Fig. 4.1b are more
or less parallel. Different scales in rainfall statistics are influenced by different meteorologic
factors. Whether one month is wet or dry depends on the large scale atmospheric conditions
during longer periods. If for example one rain gauge records an especially high monthly
rainfall sum, it is very likely that all surrounding stations do the same, since they were
influenced by the same atmospheric conditions during that month.
On daily scale it is less true. Daily rainfall values are more sensitive to local effects at the rain
gauge as monthly sums. One example for such local effects is orographic uplift at hill slopes
enhancing rainfall intensities. Weston and Roy (1994) proofed for the Scottish Highlands
that orographic effects depend on the flow direction. Daily rainfall sums can vary by a
factor of two or more depending on whether the rain gauge is situated at the upwind or
downwind side of the mountain range (Weston and Roy, 1994). Since the flow direction
can vary considerably between one day and another, it can provoke differences in the daily
precipitation sums at nearby station with different exposition.
Another phenomena which induces high heterogeneity, especially in daily and subdaily
temporal scales, is the presence of convective rainfall events. In middle European climate
estival thunder storms can provoke very high intensities over very small areas. The flooding
4.1 Spatial and Temporal Heterogeneity 29
at the Starzel river in South West Germany, 2008, is an extreme example (Ruiz-Villanueva
et al., 2012). It is possible that one rainfall stations is hit by such an event while the the
surrounding stations in few kilometer distance do not even record any precipitation. Over
a longer time range this effect averages out to some extent. Therefore convection induced
heterogeneity is hardly visible in monthly sums but effects records of daily or subdaily time-
scales.
Free convection is caused by heat induced uplift of air and moisture and occurs mostly in
summer months. Due to the presence of convection, the spatial correlation in precipitation
on daily or subdaily scale is always lower during summer.
In hourly rainfall data local effects are more dominant than in daily data. In high temporal
resolution, the location of the stations relative to the current flow direction becomes impor-
tant. For example, if one station is located ten kilometers upwind of another station, it will
be hit earlier by an approaching rainfall event. As a result the hourly records of the two
stations will differ considerably but show similar daily sums since it is the same rainfall that
passed over both stations. However, if the distance vector of the two stations is parallel to
an approaching front, both will be attained by the approaching rainfall approximately at the
same time and the hourly records will be similar (Brommundt, 2008).
Since rainfall radar images have been available, the spatial and temporal heterogeneity
could be examined more closely. Austin and Houze (1972) examined the spatial patterns
in rain fields that were recorded during heavy rain storms in Massachusetts. They distin-
guish different active zones with different characteristic lifespan: “Large Meso Scale” areas
as the largest structure, like for example the whole rainfall zone of an approaching front con-
taining “Small Meso Scale Areas” which are especially active. In each of these one can find
one or several “Rain Cells”, which exhibit the highest rainfall intensities. The distinction is
more conceptual than physical but gives a qualitative picture of the variability, both in time
and space. Since all rain events are moving, the spatial heterogeneity leads to temporal vari-
ations when the rain event passes over a certain point. The structures are not stable and the
lifespan of spatially smaller structures is generally shorter (Austin and Houze, 1972) which
is also contributing to temporal heterogeneity.
The airport of Oklahoma City features a dense and regular rain gauge network located in
a flat and open area in which all stations are doubled for cross checking. The network is
used for experimental analysis of small scale variability. The spatial correlations between
the network stations are neither isotropic nor stationary (Ciach and Krajewski, 2006). Fur-
thermore, the correlations depend on the rain rate. Storms with higher peak intensities show
more spatial coherence. The highest influence on the correlations, however, comes from the
width of the aggregation interval (Ciach and Krajewski, 2006). The shorter the time span is,
over which the rainfall amount is summed up (e. g. hourly values compared to ten minute
values), the lower the correlation is at a certain distance between the stations.
Berne et al. (2004) examine the consequences of the spatial and temporal variability on ur-
ban hydrology taking the city of Marseilles, France, as an example where a dense rain gauge
network is available, as well as vertical and horizontal rain radar. Berne et al. (2004) com-
pare the recorded intensities with discharges measured in the urban sewage system by 100
30 Statistical Properties of Precipitation
Telemeter stations. They conclude that the spatial density of the rain gauge network was not
high enough to capture the spatial variability in the six minute rainfall sums. An empirical
formula is given to approximate the representative range of a rainfall station depending on
its temporal resolution. By comparing the measurement resolution with the concentration
time of a catchment, the spatial and temporal resolution of rainfall data can be calculated
that is necessary for hydraulic modeling of the discharges in the catchment. The smaller
the catchment is, the higher is the required spatial and temporal resolution (Berne et al.,
2004). It is found that most networks of recording rainfall stations in the Southern France
are spatially too coarse (Berne et al., 2004). Rain radar on the other hand, according to Berne
et al. (2004), delivers a sufficient spatial resolution but is limited temporally by the scanning
interval (usually some minutes) to be of full benefit in hydrological modeling of small scale
catchments. The findings, however, are limited to areas with similar Mediterranean type
climate.
4.2. Intermittency and Scaling
If only one temporal scale is regarded, the occurrence and amount of precipitation is tem-
porally persistent. On the scale of very short time intervals, e. g. five minute values, it
very likely that it will be raining in the next time step if it is raining now. The same is true
on daily scale: To assume that the average weather tomorrow will be “about the same as
today” is a forecast that is more likely to be true than false for the temperate climate of
Baden-Württemberg – even if the weather can be very variable during one day with periods
of sunny and rainy conditions.
On the other hand, there are sudden changes. Within a few hours the weather can switch
to a completely new situation that then again is stable for a longer time. The changes are
triggered by processes on different spatial-temporal scales, e. g. shifts in the large scale
westerly flow field of the upper troposphere or the synoptic scale of cyclonic depression
(see Section 2.1). As a result, there is a high variety of different events from estival thunder
storms to the passing of warm and cold fronts that all have a typical range of temporal
scales. A thunder storm for example is physically bound to thermal uplift by solar heating
and therefore will not last more than a few hours. Drizzle from a dying front can last for one
or two days. The same holds true for dry periods. The information of “no rain” can mean
very different things depending on the meteorological situation. An hour with zero rainfall
in a time series from Central Europe can either be just a gap within a longer precipitation
event, the start of a dry period lasting for several hours, e. g. between the passing of warm
and cold front of a cyclonic depression, or the start of a high pressure situation that in some
cases prevents rainfall during several weeks.
It is difficult to attack this behavior by statistical tools. Autocorrelation for example is an
approbate measure to describe the persistence within a precipitation event, but it is not able
to account for the sudden change when the event is ending. The fact that the occurrence
of precipitation is at the same time temporally persistent and highly variable is statistically
paradoxical. It is named by the specific term of intermittency. Koutsoyiannis (2006) shows
4.3 Consequences 31
how difficult it is, to describe this behavior with standard statistical methods. The fit of a
time series model for example, that describes the sequence of wet and dry time steps, is only
possible with high effort in parametrization.
Since spatial and temporal variations are linked, the same is true for the spatial structure
of precipitation. Rainfall fields are highly variable but not arbitrarily scattered. Rainfall
clusters in active zones on different spatial scales. The spatial variability is bound by physi-
cal processes in precipitation formation: Stratiform precipitation arrives in wide bands due
to large scale frontal uplift; convective precipitation occurs in scattered showers because
the convective uplift channel that generates the condensation is fed by the moisture from a
wider surrounding area (Weischet and Endlicher, 2008).
As a direct consequence of intermittency precipitation exhibits a very characteristic scaling
behavior. Scaling means how statistical properties change when the data is aggregated for
longer time intervals or over wider areas. Extreme precipitation on different temporal scales
for example is caused by different type of events. In Baden-Württemberg, extreme hourly
intensities are generally caused by convective precipitation events, whereas extreme daily
precipitation sums are mainly due to cyclonic depressions (Section 2.1). During one hour,
the rainfall intensity can be much higher than during a whole day. As a result, the relation
between maximum precipitation and the aggregation time is a concave function between 1h
and 24h. Each statistical parameter, e. g. the rainfall intensity, the average and the standard
deviation of precipitation sums or the rainfall probability has its own scaling properties.
It depends on the spatial and temporal variability and so on the typical scales of different
rainfall producing elements present in the time series.
Closely related to scaling is the question whether precipitation exhibits long term memory.
The presence of long time memory means that extremely high values attract other extremely
high values, and therefore, it has a strong effect on extreme value probability. (The same for
extremely low values.) Long term memory can be measured by the Hurst exponent which
is the exponent of a power function describing the increase in range when aggregating the
values of a time series. Bunde et al. (2005) find long term memory in reconstructed precipi-
tation records of New Mexico and relate it to the clustering of extreme events. Arnaud et al.
(2007) find clustering in the most extreme precipitation events of many regions of France.
However, strictly speaking, the presence of long term memory could only be proved by an
infinite long time series (Lovejoy and Mandelbrot, 1985). With limited data, the Hurst ex-
ponent can only be approximated. The question whether precipitation exhibits long term
memory is still subject to ongoing discussion.
4.3. Consequences
Due to the spatial-temporal heterogeneity, none of the three principle precipitation mea-
surement tools rain gauges, rain radar and satellite remote sensing can give a complete high
resolution picture of areal precipitation. Rain gauges are principally seen as the most accu-
rate, but they can only give point values. The surrounding area for which the measurements
are representative is very limited, especially for high temporal resolution. Rain radar on the
32 Statistical Properties of Precipitation
other hand can give a spatially detailed picture but are highly affected by measurement
errors. Most problematic in the application of radar data is the risk of underestimating ex-
treme events. Satellite remote sensing too is subject to many error sources and it does not
deliver the spatial and temporal resolution required, e. g. for applications in urban sewage
system simulation.
There is a demand for high resolution precipitation data, e, g. for urban sewage system
modeling, that cannot be met by any observed precipitation data. The only solution is to
bridge the gap by synthetic precipitation data generated by either physical or statistical
precipitation models.
Any statistical model has to account for the high statistical complexity of precipitation. In-
termittency and scaling require an adequate response in the model structure. Simple models
are not able to catch all characteristic statistical features (Koutsoyiannis, 2002). On the other
hand, a more complex model exhibits more parameters that have to be fitted by observed
data, which makes the application to sparsely gauged regions more and more difficult. A
trade-off has to be found between adequate model complexity and data demand for calibra-
tion.
Due to the scaling properties of precipitation, it is not sufficient if an applied model repro-
duces the statistical characteristics of precipitation at one scale. The extreme value proba-
bilities on different temporal or spatial aggregations depend highly on the scaling behavior.
Besides, the scale of the events that are crucial in terms of flood risk assessment and damage
protection depends on the concentration time of the catchment. In small catchments, high
local rainfall intensities during few hours are the most dangerous (Ruiz-Villanueva et al.,
2012). The lower courses of big continental streams are threatened by flooding if it rains
for several days over a big part of the catchment area (see for example the description of
the Elbe flooding of 2002 in Haberlandt (2007)). The flood producing events in both cases
are not the same. The events that bring the highest intensities are different from the events
that bring the highest total volume. A model that is only adapted to one scale cannot rep-
resent all of these events at once. Hence, there is a high risk that it will fail to estimate the
probabilities of the crucial extreme events correctly.
5. Modeling of Precipitation
Modeling of precipitation is the attempt to find a numerical representation of the natural
precipitation process. This representation, as in every model, necessarily implies simplifi-
cation. The physical processes in the formation of precipitation are way too complex, and
partly still unknown, to be completely described by a model. And even if a complete physi-
cal description was possible, it would not be feasible to provide all necessary measurement
data to define the initial state of all variables.
There are generally two types of precipitation models. Physical models, also called dynam-
ical models and conceptual / statistical models.
Physical models try to capture the physical laws that govern precipitation as completely as
possible. The major simplification concerns the spatial resolution of such models, restricted
to a spectral grid. Since the formation of precipitation is linked to many other atmospheric
variables as well, e. g. temperature, humidity or the wind field, a full meteorological model
is required for this task. Due to the high complexity, physical precipitation modeling can
only be provided by public or private weather services that have both, the required physical
knowledge to represent the atmospheric processes in the model and the necessary CPU
power to run it. Regarding the effort in data acquisition by ground stations and satellites
for accurate precipitation forecast and that forecasts are still limited to several hours or few
days (Bliefernicht, 2010), one can estimate the complexity of this task.
The main application of physical models is in (quantitative) precipitation forecast for
weather reports, reservoir management or as a service for farmers. In reanalysis projects,
measurements from different data sources are integrated in a physical model for spatial ex-
trapolation from point sources to areal precipitation on a homogeneous regular grid. Global
and Regional Circulation models (GCM and RCM) used in climate research are also physical
meteorological models but, due to the long forecasting time of up to 100 years, on a reduced
spatial-temporal resolution. It should be mentioned that physical models are not purely
physical. They also depend on conceptual parameters for quantities that cannot directly be
measured. Examples are the leaf area index and fraction of green vegetation in GCM that are
necessary to describe the average albedo due to plants (Roeckner et al., 2003, for the model
ECHAM5).
Conceptual precipitation models do not consider any physical laws. They are based on the
statistical characteristics of precipitation. Most models try to mimic certain statistical prop-
erties of precipitation that are assumed to be the most essential. This can be for example the
memory in precipitation time series (Section 5.3) or the scaling between precipitation statis-
tics on different spatial-temporal resolutions (Section 5.6). The idea is that if the considered
statistical characteristic reflect the (unknown) physical nature of precipitation as close as
34 Modeling of Precipitation
possible also other properties that were not used in the model set up will be reproduced
correctly by the model. Conceptual models require parameter calibration for the actual cli-
matic conditions that can only be done by observed precipitation data.
As physical models are not applied in the generation of long synthetic precipitation time
series, which is the aim of this work, the following review of precipitation models will only
consider conceptual models.
5.1. Purpose of Precipitation Modeling
The modeling of precipitation is always performed in the focus of a hydrological or hy-
draulic application that uses the synthetic precipitation data, e. g. for dimensioning of hy-
draulic structures or flood frequency calculations.
According to the complex scaling of precipitation, the performance of precipitation models
is scale dependent. Since they can only give an incomplete picture of the precipitation pro-
cess, precipitation models usually perform best on the scales that were used in calibration to
observed precipitation and worse on scales of very different spatial or temporal aggregation.
As a consequence, the applied precipitation model has to be chosen carefully in regard of
the hydrological or hydraulic application that it is used for. Many models are even designed
for one specific purpose.
The application also defines how closely the precipitation model can stick to measured pre-
cipitation data. In the order of increasing abstraction from observed data, these tasks are the
following:
1. extension of time series, ensembles of different realizations
In some cases it is only necessary to increase the data volume of a given precipitation
time series. Either the time series is extended, e. g. for a better extreme value estima-
tion, or an ensemble of different possible time series is created, that can be used for
example in probabilistic precipitation discharge models.
From the precipitation modeler’s point of view, this is the easiest task since the param-
eters of the precipitation model can be derived from the time series itself. No indirect
parameter estimation, neither the transfer from other locations nor from higher aggre-
gated measurements is needed.
2. disaggregation and downscaling
Precipitation measurements on coarser scales are by far better available than measure-
ments on finer scales. Reanalysis for example provide long time series of gridded
areal precipitation that cover wide regions or even the whole globe. An example are
the NCEP/NCAR reanalysis which provide a continuous time series of global precip-
itation fields from 1948 till today for 24h or 6h intervals on a 2.5◦ × 2.5◦ grid (Kistler
et al., 2001). The precipitation time series from climate models are available on simi-
lar scales. For most hydrological or hydraulic applications however, the resolution of
these data sets is not sufficient.
5.2 Spatial Interpolation of Precipitation Data 35
It is a common situation that coarse scale data has to be brought to a finer scale for
the use in a hydrological or hydraulic application. A spatial scale change is called
“downscaling”, a temporal scale change “disaggregation”. A precipitation model used
for disaggregation or downscaling depends on parameters that link the coarser scale
statistics to the finer scale. It is often the case that fine scale measurements are only
available for a short time period or for one locations. Thus, the model parameters
are estimated by means of the available data and considered constant in time and/or
transferable to new locations.
Not all precipitation models can directly be used for downscaling. Some models, gen-
erate free simulations that cannot be conditioned on coarse scale observations. A pos-
sible work-around, e. g. in temporal disaggregation, is to run long simulations and cut
out pieces of the time series that by chance fit to the coarse scale values.
3. generation for ungauged areas
This is the most ambitious task in precipitation modeling. It consists not only of the
setup of a suitable precipitation model but also in the transfer of all necessary param-
eters of the model to the new, ungauged location. Unlike the task of downscaling and
disaggregation, no further information on the precipitation regime at the new location
is available that guides the simulation. The modeler might enjoy the freedom of sim-
ulating precipitation time series without any constraints, but he or she should keep in
mind that the risk of generating inadequate simulations is highest in this task.
It becomes even more challenging for spatial-temporal simulation, for example of sev-
eral, spatially correlated time series. In this case not only the model parameters have
to be transfered, but also the spatial dependence between the time series has to be
estimated at the new, ungauged locations.
As a consequence of the spatial-temporal heterogeneity of precipitation and the scaling be-
havior, modeling becomes the more challenging, the higher the required spatial-temporal
resolution is. This is common to all models and all modeling tasks. For most model con-
cepts, the number of parameters increases dramatically with increasing temporal resolution.
As a result, many publications can be found on the generation of daily precipitation time se-
ries. Models for hourly or even five minute time series are much less frequent.
5.2. Spatial Interpolation of Precipitation Data
Being confronted with a lack of precipitation data, the first idea one would probably come
up with is interpolation: If a location is ungauged, it might be possible to approximate the
missing values by taking a weighted average of the values from surrounding stations. Doing
this for each time step, one could set up an interpolated time series.
An appropriate interpolation method is Kriging, which has been developed for the mining
industry. A typical problem in mining is that there are some point measurements from test
drillings, e. g. of the thickness of an metal ore deposit and one has to decide from these if
36 Modeling of Precipitation
the deposit is worth exploiting. With Kriging the point measurements can be interpolated
to locations that have not been probed.
Kriging is a linear estimator. The interpolated value z¯0 at the target location is a linear
combination of the measurement values at surrounding measurement points:
zˆ (x0) =
n∑
i=1
λi · z (xi) (5.1)
with zˆ (x0) estimation for location x0
z (xi) measurement at location xi
λi weight of the measurement at location xi
Eq. (5.1) holds true for many linear interpolation methods. The difference between them, is
how the weights λ are defined.
Kriging can be applied if the underlying field Z from which the samples z are drawn ex-
hibits intrinsic stationarity. It means that the expectation of Z is independent of the location
and that the expectation of the variance between two measurements at locations xi and xj
depends only on their Euclidean separation distance hij :
E [Z (xi)− Z (xj)] = 0 (5.2)
1
2
E
[
(Z (xi)− Z (xj))2
]
= γ (hij) (5.3)
where γ is called the variogram.Under the condition that intrinsic stationarity in Z is ful-
filled, the weights λ are chosen so that the estimation is unbiased and that the expected
variance between the true value Z (x0) and the estimated value Zˆ0 is minimal:
E
[
Zˆ0 − Z0
]
!
= 0 (5.4)
E
[(
Zˆ0 − Z0
)2] !
= min (5.5)
It follows from the unbiasedness condition Eq. (5.4) that the weights λ have to sum up to
one.
n∑
i=1
λi = 1 (5.6)
Inserting Eq. (5.1) and exploiting the definition of the variogram γ (Eq. (5.3)), Eq. (5.6) can
be transformed to
E
[(
Zˆ0 − Z0
)2]
= −
n∑
i=1
n∑
j=1
λiλjγ (hij) + 2
n∑
i=1
λiγ (hi0) (5.7)
where n is the number of measurements that are used in the interpolation.
5.2 Spatial Interpolation of Precipitation Data 37
Minimizing Eq. (5.7) as a function of the weights λi under the unbiasedness condition
Eq. (5.4) leads to the linear equation system of Ordinary Kriging
−
n∑
j=1
λjγ (hij) + ν = −γ (hi0) , i = 1, 2, . . . , n (5.8)
n∑
i=1
λi = 1 (5.9)
The Kriging System consists of n + 1 equations with n + 1 unknowns. The variable ν is
the Lagrange multiplier which is necessary to link the unbiasedness condition. For the vari-
ogram γ empirical values can be used or it can be estimated from a theoretical function that
is fit to the observed data.
Minimizing the sample variance, Kriging is the best unbiased linear estimator (“BLUE”) for
the unknown value at x0 under the assumption that intrinsic stationarity is fulfilled and that
the error variance (Eq. (5.5)) is a meaningful estimator. The latter condition implies that the
distribution of Z should be symmetric and approximately Gaussian.
To apply Kriging on precipitation fields is difficult since precipitation amounts exhibit right
skewed distributions. The possibilities to account for the skewness are limited, e. g. by log-
arithmic transformation of the data. Therefore, Kriging works best for higher aggregated
data like monthly sums where the skewness is lower. On short aggregation intervals, the
occurrence of zero values becomes a problem. Like any interpolation the Kriging estima-
tions zˆ have a lower variance than the underlying field Z. Hence, the Kriging estimations
underrate the fraction of zero values.
A possible solution to the skewness problem is Indicator Kriging. Instead of the value of Z,
the indicator IC (xi) is interpolated that indicates if the value z (xi) is within the class C or
not. If “yes”, IC (xi) = 1; else IC (xi) = 0. The interpolation result at x0 is the probability that
z0 ∈ C. If Indicator Kriging is performed for several rainfall amount classes, the distribution
of Z at x0 can be approximated (Haberlandt, 2007, , for the interpolation of the precipitation
field causing the Elbe Flooding in 2002).
Kriging can also be improved by incorporating the additional information of an external
variable Y that is linked to Z and available at any measurement location xi as well as the
target location x0. In External Drift Kriging it is assumed that Y are linked linearly:
E [Z (x0) |Y (x0)] (5.10)
where E [Z (x0) |Y (x0)] is the expectation of Z (x0) given that the value Y (x0) is known.
38 Modeling of Precipitation
The linear equation system for External Drift Kriging is
−
n∑
j=1
λjγ (hij) + ν1 + ν2Y (xi) = −γ (hi0) , i = 1, 2, . . . , n (5.11)
n∑
i=1
λi = 1 (5.12)
n∑
i=1
λiY (xi) = Y (x0) (5.13)
where ν1 and ν2 are Lagrange multipliers.
Haberlandt (2007) uses the external information of rain radar fields for External Drift Krig-
ing of the precpitation amounts measured during the Elbe flooding in 2002. During this
event, the rain radar fields showed severe underestimation up to a factor of six (Haberlandt,
2007). Nevertheless, even if the absolute values of the radar data are wrong, the information
on spatial variability improves the interpolation.
Indicator Kriging and External Drift Kriging can be combined: Whether it is raining at x0 or
not, is decided by Indicator Kriging. If yes, the amount of precipitation is interpolated by
External Drift Kriging (Verworn and Haberlandt, 2011).
It should be noted that the cited investigations dealt with the interpolation of single precipi-
tation fields. For the generation of rainfall time series interpolation is not appropriate. Krig-
ing is not able to take temporal persistence into account. The generation of a time series by
stepwise interpolation results in a time series that overestimates the autocorrelation. At the
same time, the variance reduction leads to an underestimation of extreme events (Bárdossy
et al., 2000). Due to the scaling behavior, the deviations are highest for short aggregation
intervals (e. g. five minute values).
The variance reduction can be compensated by stochastic simulation that randomly disturbs
the estimated values to restore the original variability. However, the simulation has to take
the skewness of precipitation into account, as well as the temporal persistence.
5.3. Markov Chain Models
Markov Chain models do not represent the continuous nature of precipitation but discrete
time series of equal measurement intervals (e. g. daily values or hourly values). The Markov
Chain produces a binary sequence of rainfall occurrences expressed as the states “rain” and
“no rain” at each time step. For a complete simulated precipitation time series the rainfall
amounts for the wet intervals have to be modeled, e. g. by drawing from a distribution
function.
Let X be the random variable of rainfall occurrence that takes two values 0 and 1:
Xt =
{
1 if it rains at time step t
0 if not
(5.14)
5.3 Markov Chain Models 39
Then a Markov chain is defined by the following equation:
Pr(Xt = i) =
∑
i,j,k...w
P
(
Xt = i
∣∣Xt−1 = j,Xt−2 = k, . . .Xt−n = w) (5.15)
Pr probability for X = i (“rain” or “no rain”) at time step t
i, j, k . . . w states of X at the n previous time-steps
Pr(Xt = i) is a conditional probability which depends on the state of X at n previous time
steps. n is the order of the Markov Chain model. More descriptive n is the number of
previous time steps that have to be known to determine the rainfall probability at time step
t. The conditional probabilities are assumed to be constant over time. The Markov Chain
Model for the occurrence of precipitation is a two-state Markov chain since the random
variable X can only take two values, 0 and 1. In this case, the sum in Eq. (5.15) consists of
2n+1 elements that one has to estimate to set up a Markov Chain model. They are called the
transition probabilities (e. g. Chin, 1977) because they describe the transition from one state
(“rain” or “no rain”) to another state at the next time step. The transition probabilities can be
approximated by counting the transition frequencies in observed precipitation time series.
Thereby, some of the transition probabilities can be deduced by complementary probabilities
which reduces the number of independent parameters so 2n−1 where n is the order of the
Markov Chain (Koutsoyiannis, 2006). For a first order Markov Chain Model, the transition
probabilities can also be deduced from the unconditional probabilities for the states of “rain”
and “no rain”.
The Markov Chain Model determines many statistical properties of the generated bivariate
time series of the rainfall occurrence. For a first oder Markov Chain the expected number
of consecutive wet (or dry) time steps follows an geometric distribution (Gabriel and Neu-
mann, 1962). The autocorrelation on different time lags follows an exponentially declining
function. These characteristics can be used to verify a Markov Chain Model. For more de-
tails on the properties of first order Markov Chain Models one might refer to Katz (1974) or
Lloyd (1974).
The computation of first order Markov Chain time series is straight forward and fast. These
qualities made the models very popular when numerical calculations were not yet as avail-
able as today. Gabriel and Neumann (1962) applied a first order Markov Chain model to
the daily rainfall occurrence in Tel Aviv. They used a threshold of 0.1mm for the limit be-
tween “rain” and “no rain”. Since the model assumes constant transition probabilities, it is
restricted to the rainy season in winter months.
Todorovic and Woolhiser (1975) present a full model for daily precipitations including a
simulation of the rainfall amounts which are drawn from an exponential distribution. The
amounts of consecutive days are assumed as independent. The model is verified by the
distribution of the maximum rainfall amount in n consecutive days, as well as the mean
and variance of the total precipitation sum during n days for which they deduced analytical
solutions. At the test location in Austin, Texas, the model shows deviations in the statistics
for longer aggregation intervals. The average 10 day precipitation sum is overestimated, the
40 Modeling of Precipitation
distribution of the 20 day and 30 day sums have lack of variance compared to the obser-
vations. Katz (1977) modified the approach by assuming precipitation amounts as Gamma
distributed with two different sets of parameters: one for the first day in a row of consecu-
tive wet days and one for the following days.
Katz and Parlange (1993) refined the Markov Chain Model by using information on atmo-
spheric circulation to improve the adjustment of the transition probabilities. The approach
was tested with the rainfall data from several daily stations in California. Two amtmospheric
Circulation Patterns (CPs) are defined based on the mean sea level pressure map. The tran-
sition probabilities pi of the occurrence model as well as the parameters of distribution de-
scribing the rainfall amounts were set up for each CP separately. The model was verified by
the simulated and observed variance of monthly rainfall sums. Using the CP classification,
reduces the lack of variance in higher aggregation sums remarkably compared to a Markov
Chain model which is not conditioned by the CP.
The model proposed by Haan et al. (1976) avoids the division into an occurrence and an in-
tensity model part. The advantage is that the memory of the time series is directly expressed
in the rainfall amounts and not only in the occurrences. It is realized by defining six rainfall
amount classes. Adding the dry state, a first order Markov Chain with seven different states
is set up leading to 7× 7 transition probabilities. The disadvantage of this model is the high
sensitivity to sampling errors. To avoid seasonal effects the model is set up for each month
separately which leads to 12× 7× 7 = 588 transition probabilities that have to be estimated
from the observed time series. Even with time series of several decades as data input the
estimation of the rarest transition probabilities is not stable (Haan et al., 1976).
If the statistical properties of observed rainfall differ significantly from first order Markovian
properties, a Markov Chain of higher order has to be applied. Higher order Markov Chains
are able to represent more complex temporal persistence. The autocorrelation over different
time lags is more flexible than in the first order chain were it is bound to exponential decline.
The higher flexibility helps to reduce deviation in aggregated properties over n consecutive
time steps as in Todorovic and Woolhiser (1975).
In an extensive study with more than one hundred 25 year long time series from rainfall
stations all across the United States of America Chin (1977) proofed that a first order Markov
Chain is not alway sufficient to describe daily rain fall occurrence. The required Markov
Chain order depends on the region and most of all on the seasons. In summer months
a first order chain is sufficient in most of the United States. In winter months however,
the use of a second or third order model is required. In summer months rainfall comes
primarily from local thunder storms with low temporal persistence. During winter season,
the rainfall in the US is mostly due to polar fronts and cyclonic depressions which have a
characteristic life cycle of about four days which leads to higher order autocorrelation in the
rainfall occurrence (Chin, 1977).
Precipitation on the sub-daily temporal scale generally shows higher autocorrelation than
daily values and a more complex autocorrelation structure. Therefore, modeling requires
sophisticated higher order Markov Chain Models. Katz and Parlange (1995) construct a
third order Markov Chain Model to simulate time series of hourly precipitation. Daily cy-
cles in the precipitation activity, e. g. during summer months, can be taking into account
5.3 Markov Chain Models 41
by applying a sinus-cosinus-function with a period of 24h to the transition probabilities.
The amounts in consecutive hours are assumed as autocorrelated. Due to the higher or-
der Markov Chain and the additional parameters for daily cycle and persistence, the model
is heavy in the number of parameters. The estimation is burdensome since the parameters
cannot directly be derived from the observations. Katz and Parlange (1995) verify the model
by the rainfall probability and rainfall sum on daily aggregation. While the daily averages
are modeled correctly, the simulated time series exhibit a lack of variance in daily values.
Katz and Parlange (1995) conclude that “virtually all conceptual models tend to underes-
timate the variance of daily total precipitation”, a phenomena that is often referred to as
“overdispersion” (e. g. in Katz and Parlange (1998) or Koutsoyiannis (2006)).
Katz and Parlange (1998) test different methods to reduce overdispersion in daily Markov
Chain Models. One possibility is to draw autocorrelated rainfall values, another to apply
different distributions at the first and the following rain days (as in Katz, 1977) or by ap-
plying a higher order Markov Chain occurrences model. The overdispersion is tested by
different statistics on longer (e. g. monthly) aggregation as the distribution of the number
of wet days as well as the mean and variance of monthly rainfall sums. The lack of vari-
ance in the number of wet days can be reduced by applying a higher Order Markov Chain.
However, the lack of variance in aggregated sums, does persist (Katz and Parlange, 1998).
All models presented so far simulate single time series for one location. There are only
few approaches published that extent the Markov Chain concept to simulate simultaneous,
spatially and temporally correlated time series (space-time models). Wilks (1998) sets up
a Markov-Chain Model for spatially correlated daily precipitation values. The model is
restricted to precipitation stations whose time series can be described by a first order Markov
Chain, which is the basis of his approach. Spatially correlated random numbers are drawn
which decide over the state of rainfall occurrence at all sites simultaneously. The rainfall
amounts are simulated by a mixed exponential distribution. The mixing weight is linked to
the rainfall occurrence. Thus, for stations at the edge of a rainfall field the expected amount
is lower than for a station in the middle of a rainfall field, which is consistent with real
world observations. However, the model has the issue that some of the parameters cannot
be calibrated directly from observations which necessitates empirical estimation functions.
Furthermore, the model assumes that the rainfall amounts are temporal uncorrelated, which
leads to an underestimation of the spatial correlations with time lag between the stations.
Markov Chain Models are conceptually simple. In the same time, the model structure de-
termines many statistical properties that can be used for verification. It is a strong point that
the model can be extended by additional information as diurnal cycles (Katz and Parlange,
1995) or weather type information (Katz and Parlange, 1993). The fact that the model pro-
duces discrete time series is an advantage in parameter fitting. The model can directly be
linked to discretely measured precipitation time series by the observed transition frequen-
cies. The parameters are not “hidden” as it is the case for Point Process Models, which are
discussed in the next section (Section 5.4).
Markov Chain Models are most suitable for the generation of point rainfall. The extension to
spatial-temporal modeling is difficult. In areal precipitation the amount of rainfall is depen-
dent on the size and shape of the rainfall field. The separate simulation of rainfall occurrence
42 Modeling of Precipitation
and rainfall amounts that is applied in most Markov Chain Models causes problems in this
case.
Most applications of Markov Chain Models deal with daily rainfall. In most climates the
sequence of wet and dry days can well be described by a Markov Chain of first or second
order. On the sub-daily scale however, the temporal pattern of rainy and dry time steps
becomes more complex. The occurrence of intermittency and long term persistence (Section
4.2) requires higher order Markov Chain Models. However, the fit of a Markov Chain Model
of order n involves the estimation of 2n−1 parameters (Koutsoyiannis, 2006). The demand
for observed data to fit all these parameters increases over-linearly with the model order.
Since the model assumes stable conditions, it has to be fit to each month separately if the
climate exhibits an annual cycle. The high demand of observation data is a weak point of
Markov Chain Models. Geng et al. (1986) state “it usually is believed that at least twenty
years of data should be used to provide reliable estimates.” Chin (1977) states that the model
order is underrated if the data series is too short for an adequate fit. In long time series, on
the other hand, it becomes doubtful if the assumption of constant transition frequencies can
still be accepted.
Additionally, the high number of parameters makes it difficult to use Markov Chain Mod-
els in ungauged areas. None of the cited publications describes a method to transfer the
transition probabilities found at one rain gauge to a new location.
5.4. Point Process Models
In contrast to Markov Chain Models, Point Process Models describe rainfall (as a temporal
process at one site or as a spatial-temporal process over an area) in its continuous nature.
Rainfall is seen as an overlay of several elementary rain “cells”. The arrival of the cells, as
well as the duration and intensity of each cell are seen as random variables and stochas-
tically modeled. To account for the intermittency of precipitation, Point Process Models
exhibit a hierarchical structure of “storms” and “cells”. Storms arrive with a certain Poisson
distributed arrival rate. Within each storm a random number of rainfall “cells” is active. (A
zero-rainfall interval in this model can either be a break within a storm with a short expected
duration or a possibly longer break between two storms.) The instantaneous rainfall rate at
any time point t0 is the sum of the intensities of all cells that are active (Rodriguez-Iturbe
et al., 1987). If the cells are assumed to have a constant intensity over their whole life span,
the model is called a “Rectangular Pulse Model” (Rodriguez-Iturbe et al., 1987).
Two very similar models of this kind are the Bartlett-Lewis Rectangular Pulse Model and
the Neyman-Scott Rectangular Pulse Model (Rodriguez-Iturbe et al., 1987). The difference
between the Bartlett-Lewis Rectangular Pulse Model and the Neyman-Scott Rectangular
Pulse Model is only in the definition of the rain cell arrival. In the Bartlett-Lewis model cells
arrive according to a Poisson process as long as the current storm is active. In the Neyman-
Scott Model at first the number of cells belonging to each storm is drawn from a random
distribution. Then the location, symmetric around the storm center, is modeled (Rodriguez-
Iturbe et al., 1987).
5.4 Point Process Models 43
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Figure 5.1.: Schematic for Point Process Rectangular Pulse Model.
On top: storm and cell arrival in the Bartlett-Lewis and the Neyman-Scott Pro-
cess. Below: intensity superposition in Point Process Models.
Scheme by Onof et al. (2000)
Since the rainfall generating cells are overlaying, they cannot be identified in observed pre-
cipitation time series. Therefore, the parameters of Point Process Models (e. g. of the dis-
tributions for the rain cell intensity and the duration) cannot be derived from observations
directly. Generally, Point Process Models are fitted by linking the parameters to observ-
able statistical properties. Rodriguez-Iturbe et al. (1987) for example set up a Neyman-Scott
Model assuming the number of cells per storm as geometrically distributed, cell intensity
and duration as exponentially distributed. In this form of the model the mean, the variance
and the autocorrelation of the generated time series can be expressed as a function of the
generation parameters and fitted to observed statistics. Another possibility is to vary the
model parameters in a numerical optimization scheme until the properties of the simulated
time series are in agreement with the observations (Onof and Wheater, 1993).
However, the model of Rodriguez-Iturbe et al. (1987) differs to real world observations in the
scaling properties. If the model is fitted repeatedly with the same data but aggregated on dif-
ferent scales (from 1h to 1day), it results in different model parameters (Foufoula-Georgiou
and Guttorp, 1986). They conclude that “there is not a unique Neyman-Scott Model” which,
when fitted to one time scale, reproduces the observed statistical moments on other tempo-
ral scales.
To improve the consistency between modeled statistics at different temporal aggregations,
44 Modeling of Precipitation
Calenda and Napolitano (1999) propose an alternative parameter fitting procedure. The
model parameters are expressed as a function of the scaling in the variance of the rainfall
intensity over different aggregations. Parameter fit is done by non-linear regression over the
scales.
Onof and Wheater (1993) modify the generation algorithm of the Neyman-Scott Model by
replacing the exponential parameter in the distribution of the rain cell duration by another
random function. As a result, the cells of each storm follow a different cell duration distri-
bution. The increased variability improves the representation on scales that are not used in
the fitting. Cowpertwait (1994) enhanced the variability of the model by introducing differ-
ent cell types. He distinguishes between “heavy short-duration convective cells” and “light
long-duration stratiform cells” with different distribution parameters for duration and in-
tensity.
Evin and Favre (2008) link the intensity and duration of the rain cells, which are assumed
to be independent in the original set-up, by cubic copulas. The deduction of method of
moment estimators for the model parameters is extremely tedious and only possible if it is
assumed that both, duration and intensity follow an exponential distribution. However, the
benefit of the model concept extension, e. g. on the simulation of extreme values on different
scales, is small.
For a better representation of the intemittency of five minute values Cowpertwait et al.
(2007) extended the model concept to a three level structure: The rainfall time series is mod-
eled as a succession of “storms” that contain a random number of “cells” which consist of
several precipitation “pulses”. The model in this form has eleven parameters that are fit by
the coefficient of variation, the autocorrelation, the skewness on four different aggregation
levels.
Arnaud and Lavabre (1999) enlarged the concept of a Point Process Model in a less concep-
tual but more heuristic sense. Instead of modeling complete time series they concentrated
on extreme rainfall events. The presented model, named “SHYPRE”´, is designed for the use
in hydraulic dimensioning and flood protection measures. The aim is to simulate very long
time series (e. g. 10000 years) and to calculate extreme value statistics based on the empirical
frequencies in these simulations. All rainfall events (defined as a period of consecutive rain
days) with less than 20mm are ignored. Compared to former Neyman-Scott approaches, the
model has several new parameters. Most important, it does not work with rectangular but
triangular pulses. Therefore the ratio of peak intensity to average intensity and the relative
peak position in each rain cell have to be modeled as random variables. Furthermore, it
distinguishes between “major storms” and “minor storms” which have different parameter
distributions. This is necessary to adjust the model to observed temporal persistence.
In 2002 SHYPRE was combined with a discharge model to investigate extremes in river dis-
charges and to analyze errors in the whole modeling chain from rainfall to runoff (Arnaud,
2002). Arnaud et al. (2007) proved the performance of the generator in all available climate
zones of France and its oversea territories. Arnaud et al. (2007) point out that the model
performance is highly dependent on the right parametrization of storm persistence. They
show that major storms in one event are often connected, the presence of the first major
5.4 Point Process Models 45
storm enhances the probability for another one. Ignoring this connection, leads to a (po-
tentially dangerous) underestimation of extremes. Muller et al. (2009) made an extensive
study of SHYPRE’s sensitivity to parametrization and fitting data. Uncertainties in the pa-
rameters were modeled in a Bayesian framework by applying a prior distribution to each
parameter and checking the range of the extremes in the resulting time series. Lack of data
was simulated by leaving out some of the calibration data and checking for differences in
the simulation results. It could be shown that SHYPRE give reasonable results in higher
annuities up to 100 years with 20 years of data.
However, SHYPRE suffers from some minor drawbacks. Firstly, it does not model a com-
plete time series but only the most extreme events in the year. This limits the possible ap-
plications, e. g. it is not suitable for sewage system design where more frequent events are
more important than extremes of high annuity. Secondly, its heuristic character can be seen
as a conceptual weakness. Especially the definition of “major” and “minor” storms and
their persistence parameters are rather arbitrarily and not justified by any physically based
or phenomenological concept of rainfall.
The extension of the Point Process Model into spatial domain is conceptually easy. For
spatial modeling it is sufficient to consider storms and cells as two dimensional features
(Rodriguez-Iturbe et al., 1986). Storms arrival is modeled by a spatial two dimensional Pois-
son Process. The cells cluster around the storm centers according to the Neyman-Scott Pro-
cess with a random number of cells per storm. The cells are seen as circular discs with
a maximum intensity at the cell center and declining intensity towards the edge. The de-
cline can be modeled as a random function as well (Rodriguez-Iturbe et al., 1986). For a
fully temporal-spatial model the movement and the evolution of storms and cells has to be
taken into account. Rodriguez-Iturbe and Eagleson (1987) extend the model in this sense by
adding another exponential decline function describing the evolution of rain cell intensity
in time.
Cowpertwait (2002) follow the same idea to set up a space-time rainfall model for the UK.
They assume the rainfall intensity in each cell to be constant and to follow a Weibull distri-
bution. The fit of the model is done by observed statistics like the variance, the coefficient
of variation, the skewness, the autocorrelation and the cross correlation in observed rainfall
records. It is not straight forward and involves an iteration scheme that passes by the fit of
single site models which are, in a second step, linked by means of spatial statistics.
Cowpertwait (2006) applied the model in a multipurpose scheme to the Thames Catchment
in the UK. It is designed for disaggregation, gap-filling in measured data series and the gen-
eration of time series at ungauged locations. However, the model cannot directly be used
for gap-filling and disaggregation because it is not possible to condition the model on ob-
served values (e. g. for the daily rainfall sum). Therefore an indirect method is applied in
which 200 years of data are generated in a free simulation and then slices are cut out ac-
cording to similarity measures (Cowpertwait, 2006). This technique might become difficult
for extreme events where only few similar situations are available in the generated time se-
ries. The model is available under the name of “RAINSIM”, commercially distributed in a
software package called “STORMPAC”. It is widely used by the UK water industry but also
applied in alpine climate in Italy or Switzerland, in the Dutch Lowlands and the arid climate
46 Modeling of Precipitation
of Spain (Burton et al., 2008).
Willems (2001) proposes a spatial-temporal generation scheme that takes the movement of
cells and storms explicitly into account. The rain field is represented as Small Meso Scale
Areas (the cells) clustered in Large Meso Scale areas (the storms) within a Large Synoptic
Scale Areas (e. g. the front of a cyclonic depression). The movement of the cells is processed
similarly to mass-conservative diffusion: As a cell moves, it becomes broader and its max-
imum intensity decreases. For the model fit the rain cells have to be identified in observed
precipitation time series. The propagation velocity is determined by tracking the rain cell
movement from gauge to gauge (Willems, 2001). Therefore, the model is very data demand-
ing. Correct tracking requires a very dense rain gauge network that is at the same time large
enough to represent the Large Synoptic Scale Areas completely.
Point Process Models are frequently applied and well tested in a wide range of different cli-
mates (see Onof et al., 2000, for an overview). Their strong point is the continuous nature of
the simulated rainfall process. The model can be fit with aggregated data (e.g. in daily reso-
lution) and still deliver time series on high temporal resolution (e. g. hourly data). Another
quality is that the extension to spatial-temporal modeling is conceptually easy. The same
holds true for the regionalization of the model. It is sufficient to transfer the statistics of the
aggregated data that were used in the parameter fitting to the ungauged locations. Cow-
pertwait et al. (1996) for example regionalized a Neyman-Scott Model over Great Britain by
linking the parameters to geographic related variables like altitude, distance to the distance
to the coast or a location index for the west or the east side of Britain.
A weak point of Point Process Models is the parameter estimation, which is burdensome
since the model parameter values are not directly observable. At the beginning, much ef-
fort was made to find analytical equations (or equation systems) that link the model pa-
rameters to observable statistical properties (e. g. Rodríguez-Iturbe et al. (1987). With the
improvement in computer power in recent year the focus shifted towards parameter fitting
by numerical optimization (Burton et al., 2008). Vanhaute et al. (2012) compared different
optimization methods and their performance in the parameter fit of a Bartlett-Lewis model.
It shows that 25 years after the first use of Point Process Models for rainfall time series,
the right calibration is still an issue. Katz and Parlange (1995) critisize that Point Process
Models assume climatic stationarity and cannot account for additional information as djur-
nal cycles. They state that it would “make already difficult parameter estimation problems
more complex, if not infeasible”.
Another issue is, that the concept of “storms” and “cells” gives a rather rough picture of the
real physical rainfall process and that the modeling results, especially in extreme rainfall, are
sensitive to the assumed persistence in storm and cell arrival (Arnaud et al., 2007). Over the
Eastern Atlantic Ocean and most of Europe, the storms are clustering (Mailier et al., 2006).
The assumption of Poisson distributed storm arrival rates is doubtful. Mailier et al. (2006)
tracked the paths of all cyclonic storms over the North Atlantic and over Europe during
the winter seasons of 53 years. From the emerging storms over the Western Atlantic, not
all storms are able to reach the European continent. However, if the first storm manages to
pass through, it enhances the probability for its successors. Vitolo et al. (2009) prove that the
clustering is even more pronounced in the most intensive storm events.
5.5 Alternating Renewal Models 47
There are approaches that take the persistence or the movement of events into account.
However, they rely on methods and parameters that are more heuristic than physically jus-
tified. An example is the division into “major” and “minor storm” in the SHYPRE model
family (Arnaud and Lavabre, 1999) or the rain cell identification algorithm proposed by
Willems (2001).
5.5. Alternating Renewal Models
In contrast to Point Process Models, Alternating Renewal Models do not simulate the arrival
rate of overlaying storms and cells but explicitly the duration of “wet” and “dry spells”.
Each wet spell is followed by a dry spell. The length of both are drawn randomly from a
theoretic distribution function. Similar to the hierarchic concept of storms and cells in Point
Process Models, Alternating Renewal Model exhibit an external and an internal structure.
The external structure is the sequence of wet and dry spells. While the dry spells are com-
pletely dry, the wet spells consist of different rain pulses that can be separated by short dry
periods in between. The internal structure is characterized by the rainfall frequency within
the wet spell, the shape of the pulses, as well as the distribution of intensity and duration of
the pulses.
The model of Acreman (1990) for an 1h precipitation time series in Farnborough, UK is an
early example of an Alternating Renewal Model. In his concept dry spell lengths are mod-
eled by a Generalized Pareto Distribution, wet spells lengths by an Exponential Distribution.
The internal structure is kept simple as dry intervals within the wet spells are not consid-
ered. The wet spells are assumed to be bell shaped, the total precipitation sum to follow a
Gamma distribution conditioned on the duration. The model was validated in terms of the
extreme values on different aggregation levels ranging from 1h to 48h. On any aggrega-
tion level, the model captures the shape of the extreme value distribution but systematically
underestimates the extremes of high annuity.
Bernardara et al. (2007) set up Alternating Renewal Models for different 5min and 1h pre-
cipitation stations in Italy. They model the external structure by two Generalized Pareto
Distributions for the length of wet and dry spells, and the additional parameter of a thresh-
old length that determines if a zero rain interval is regarded as a gap in a wet spell or as
its own dry spell. The internal structure is consists of a mixed distribution of Generalized
Pareto type representing the intensities and the rainfall probability within the wet spell. The
distribution of the rainfall pulse intensities is modeled by its spectrum. The spectrum loads
are assumed to be power-law dependent on the spectrum frequency. In validation by the
observed time series the model exhibits a slight overestimation of the annual precipitation
volume and the expected average total volume per event. The standard deviation of the
total event volume is highly overestimated. On the other hand, the distribution of wet frac-
tions is well represented as well as the extreme value behavior in terms of depth duration
frequency curves over a wide range of scales from 1h to 36h.
The model of Haberlandt et al. (2008) uses the Alternating Renewal Approach to simulate 1h
precipitation time series for the Bode catchment in the Harz mountains in Middle Germany.
48 Modeling of Precipitation
The wet spell length is modeled by a General Extreme Value Distribution, the length of dry
spells by a Weibull distribution. The intensity of the rainfall pulses is modeled by a Kappa
Distribution. A particular feature of this model is that the rain pulse duration and its average
intensity are assumed to be dependent, modeled by a two dimensional Frank Copula.
Compared to Point Process Models, Alternating Renewal Model have the clear advantage
that there is no overlap between the model elements. The length of wet and dry spells as well
as the properties of the rainfall pulses within the wet spells can be observed in recorded pre-
cipitation time series. Therefore, the model calibration is straight forward allowing a higher
flexibility in the model concept, for example in the shape of the rainfall pulses. Regional-
ization of the generation parameters works principally in the same way as for Point Process
Models by spatial interpolation of the model parameters (Haberlandt et al., 2008).
The model structure considers no memory in the sequence of alternating wet and dry spells.
Per definition two following wet spells are independent events. However, whether this con-
dition is fulfilled, depends on the length of the dry spell in between. Therefore, Alternating
Renewal Model react very sensitively on the threshold of the minimum dry spell length. In
reality, the time period after which two rain events can be assumed as independent may
vary with the weather situation (Acreman, 1990), wich is not considered in the model. The
empirical distributions of wet and dry spell length in observed time series are very vul-
nerable to measurement errors. A slight shift in the zero-rainfall baseline for example, can
strongly affect the derived model parameters.
One reason why Alternating Renewal Models are not used as frequently as Point Process
Models is that they are restricted to the simulation of point rainfall. An extension of the
model concept to spatial-temporal simulation is difficult. One among the few existing ap-
proaches of spatial-temporal modeling is the “string of beads” model by Pegram and Cloth-
ier (2001). Here, the alternating renewal process simulates the sequence of two dimensional
radar fields. The radar pictures are classified into three categories: dry, scattered rain and
rain according to the fraction of wet pixels in the image. The external structure is the se-
quence of “rain”, “scattered rain” or “dry” radar blocks. The Alternating Renewal Process
merely refers to the temporal sequence. The spatial structure within the rain fields is sim-
ulated separately as the internal structure of the uninterrupted “rain” and “scattered rain”
events.
Haberlandt et al. (2008) simulates several spatial correlated time series by permutation of
the order of wet ad dry spells in an optimization scheme. The objectives of the optimization
are the joint rainfall probability between the stations, the Pearson correlation of the intensity
and the expectation of the rainfall amount conditioned on whether it rains on surrounding
stations or not. The optimization is performed sequentially, every new station is optimized
in relation to the stations that already have been optimized before.
5.6. Scaling Models
As it was pointed out in Section 4.1 and in Section 4.2, the statistical characteristics of
precipitation (e. g. mean, variance or autocorrelation) depend highly on the spatial and tem-
5.6 Scaling Models 49
poral aggregation that is analyzed. One of the main challenges for Markov-Chain as well as
Point-Process Models is to reproduce these characteristics equally well on different spatial
or temporal scales (see e. g. Rodriguez-Iturbe et al. (1986), Foufoula-Georgiou and Guttorp
(1986), Katz and Parlange (1995)). This fact let to the idea to create models directly on the
spatial and/or temporal scaling behavior of rainfall. The assumption is that scaling is one of
the main features of precipitation and that an adequate representation of the scaling behav-
ior in a precipitation model also leads to an adequate representation of the main physical
processes.
Scaling models follow the basic idea of “fractal scaling” which was extensively researched
by Mandelbrot (for more details see Mandelbrot, 1983). Fractal scaling objects or quantities
are “self-similar”. It means that they exhibit the same characteristics on any regarded scale.
(In most publications the word “fractal” is omitted and the therm “scaling” already stands
for fractal scaling. This section will adapt to this conduct.) Mathematically, scaling can be
infinite. Regarding real world objects and variables, scaling is always limited to a certain
range of scales with a lower and upper scale limit ∆min and ∆max.
Concerning precipitation, scaling is assumed in the distribution of precipitation sums or
intensities on different spacial or temporal aggregations. For the intensity this can be ex-
pressed as:
I (λ∆)
d
= λhI (∆) (5.16)
where I (∆) rainfall intensity averaged over scale ∆
I (λ∆) rainfall intensity averaged over scale λ∆
λ factor of scale change
h scaling exponent
d
= equality in distribution
The factor λ takes the value of any integer or the reciprocal of any integer as long as the two
dimensions ∆ and λ∆ do not exceed the limit scales ∆min and ∆max. ∆ can be a temporal or
spacial scale. In the latter case the intensity becomes the average areal precipitation over a
certain time interval scaling with the area.
If precipitation is scaling, it can only be described by self-similar distribution functions that
fulfill Eq. (5.16). Since precipitation measurements generally exhibit positively skewed dis-
tributions, possible choices are for example the Pareto Distribution (Lovejoy and Mandel-
brot, 1985) or Log-Normal Distribution (Burlando and Rosso, 1996).
Lovejoy and Mandelbrot (1985) present several empirical arguments to support that rainfall
fields exhibit fractal scaling (over a certain range in space and time): The energy fluxes
and velocities in a fully turbulent flow field are described by the Navier-Stokes equation
which is a fractal scaling function. Since the atmosphere is a turbulent flow field, it is fractal
scaling and rainfall should be scaling too since it is mainly governed by the atmospheric
flow. Another indication is that the boundaries of cloud fields derived from satellite images
are fractal scaling (Lovejoy and Mandelbrot, 1985).
50 Modeling of Precipitation
If the distribution of the rainfall intensity is scaling, then the statistical moments of the dis-
tribution are scaling too. There is a distinction between simple and multi fractal scaling. In
a simple scaling system the statistical moments of order q scale all according to the scaling
exponent h of the underlying distribution. In the calculation of the qth-moment every value
of I is multiplied q-times with itself. Changing the scale from ∆ to λ∆ the factor of scale
change λ is raised q-times by the scaling exponent h. This leads to the following equation:
E (Kq (λ∆)) = λ
hq · E (Kq (∆)) (5.17)
In multi scaling systems the scaling exponents of moments q > 1 are lower:
E (Kq (λ∆)) = λ
φ(q)h · E (Kq (∆)) (5.18)
where Kq statistical moment of order q on scale ∆
h scaling exponent of the underlying distribution
φ(q) non-linear, concave function
According to Harris et al. (1997) “multifractality is associated with the presence of spatial
correlation of order higher than two”. This means that not all peaks show the same spatial
correlation, but “more intense peaks cluster more” (Harris et al., 1997). Therefore, multi-
fractality has severe consequences in hydrology: it means that one extreme event can attract
the next, which increases flood risks. Gupta and Waymire (1990) find multi scaling in the
statistical moments of river discharges and rain radar images.
If the scaling assumption is fulfilled, the statistical properties of all scales between ∆min and
∆max are linked by the scaling exponent h. The relation can be exploited for statistical mod-
eling. Koutsoyiannis and Foufoula-Georgiou (1993) for example use the scaling properties
of duration and intensity of rain storms in observed rainfall time series. A stochastic model
is setup for the instantaneous rainfall intensity given the storm duration. In their definition
a storm is a rainfall event that includes all periods of zero intensity shorter than seven hours.
The model can be used to disaggregate rainfall data from point measurements if the total
storm duration in known. It can be easily extended to a full rainfall model for point mea-
surements by combining it with a distribution for the storm duration and with a stochastic
model of the storm arrival rate (e.g. as a Poisson process).
Burlando and Rosso (1996) use the concept of scaling to derive depth-duration-frequency
(short: DDF) curves, which describe the dependence between rainfall duration and rainfall
depth for different exceedance frequencies (e. g. once a year, once in ten years). If the ana-
lyzed rainfall records exhibits scaling, the exceedance frequencies can be estimated for one
duration and then be transfered to all the other duration by the scaling relation.
By a similar approach Menabde et al. (1999) showed that under the assumption of simple
scaling the rainfall depth to any extreme event is simply the product of a power-law function
of duration and a scale invariant function of exceedance frequency. This relation holds true
for any distribution function that can be normalized by its parameters in the form F (I) =
F ∗
(
I−µ
σ
)
, e.g. the Gumbel distribution. This means that extremes of any duration within
the scaling limits ∆min and ∆max can be calculated if the extreme value distribution at one
5.6 Scaling Models 51
scale and the scaling exponent are known. Menabde et al. (1999) tested the approach on time
series from South Africa and Australia. The observed scaling limits were ∆min = 30min
and ∆max = 24h which means precipitation extremes on high temporal resolution can be
calculated from daily data with generally much better availability.
The Sum of Fractal Pulses Model (Lovejoy and Mandelbrot, 1985) is a deviation of the Point
Process Model discussed in Section 5.4. The pulses correspond to what is called rain cells
in the Point Process Model. In the Fractal Pulses Model the intensity of a pulse is fully
dependent on its duration and calculated by the scaling relation. In one dimension this
model can be used to simulate a time series of point rainfall at one measurement location.
For spatial modeling the pulses are seen as two dimensional discs with a certain range. The
intensity is modeled as function of the disc size. For spatial heterogeneity the discs can be
modeled as ellipses. Although the model is strikingly versatile in producing different spatial
patterns, it cannot account for all features in real world rainfall fields like fronts and bands
(Lovejoy and Mandelbrot, 1985). More details on the mathematic properties of the Sum of
Fractal Pulses Model can be found in Mandelbrot (1995).
Multi fractal scaling can ge generated by a Multiplicative Cascade Model (Gupta and
Waymire, 1990). This means that the modeled quantity X - this can be the the total pre-
cipitation volume - is entered into the system on the upper limit scale ∆max and then con-
secutively passed down to finer scales. In each step one coarser parent cell is linked to a
number of b subcells and the amount of X received from the parent cell is divided among
these subcells. This procedure is repeated until the lower limiting scale ∆min is reached. The
function that controls which fraction ofX is distributed to which subcell is called the cascade
generator w. The number b of subcells at each scale jump is called the branch number.
In contrast to the additive Point Process Model the cascade is a multiplicative process. To
calculate the rainfall depth of any cell i on the finest scale, one has to track the path from
∆max down to ∆min and at each scale jump multiplyXsum with the cascade generator for the
respective subcell. Two separated cells are correlated to each other as long as they share one
of the coarser cells on the higher resolution (see for example Marsan et al., 1996)). In this way,
correlations of higher order are introduced between the values on the finest scale. Higher
order correlation means that the value at a time step t cannot be explained sufficiently only
by the preceding time step t− 1.
The performance of a Mutliplicative Cascade Model depends strongly on the distribution
function of the cascade generator w (Molnar and Burlando, 2005). For some distributions of
w an analytical formulation can be given for the moment term φ (q) in the multiple scaling
equation Eq. (5.18) (Gupta and Waymire, 1993), e. g. for the frequently used log-normal
distributed cascade generator (Molnar and Burlando, 2005). This relation can be exploited
in the parameter fit of w, as for example in Harris et al. (1997).
Multiplicative Cascade Models are probably the most frequently used among scaling mod-
els. They are especially popular for spatial or temporal downscaling purposes if the rain-
fall sum on the coarsest scale is known. Lovejoy and Schertzer (1986) propose a Cascade
Model for spatial rainfall fields. The rainfall sum is introduced on the coarsest resolution in
a rectangular grid and then cascaded down to a sub-grid of n times m new rectangles (in
52 Modeling of Precipitation
North-South and East-West direction) in each step. Choosing n 6= m one can account for
anisotropy. Intermittency is considered by allowing the cascade generator w to become zero
with a certain probability P0.
Marsan et al. (1996) extend this approach to spatial-temporal modeling. The temporal scale
is just added as a new dimension. To account for differences in the spatial-temporal scaling
properties, the branch number b in space and time is different, e. g. b = 3 in space and
b = 2 in time. One consequence of this approach is that spatially extended structures are
also temporally long lasting (Marsan et al., 1996). This corresponds well to the observations
of Austin and Houze (1972) discussed in Section 4.1.
Onof et al. (2005) developed a cascade model for temporal disaggregation of 1h-values into
5min minute values as a possible extension to the RAINSIM-model (Cowpertwait, 2006, see
Section 5.4). The model can be calibrated without any 5min-data just by extrapolating the
scaling behavior of 1h to 3h data, which is important since RAINSIM is applied to ungauged
locations.
Scaling models are versatile and flexible, e. g. in their ability to model higher order au-
tocorrelations, which are difficult to represent in Point Process or Markov Chain Models.
Nevertheless, Scaling Models are hardly used in operational rainfall generation systems.
One reason may be that the estimation of the scaling exponent h (or φ (q)h in multifractal
scaling) is very sensitive to bias for several reasons.
Scaling models are based on statistical moments of different orders of magnitudes. In an
observed time series of given length the sampling error in the estimation depends on the
aggregation. To adjust a multiplicative cascade generatorw based on seven scale levels, data
from ten minute scale to 10 · 27 = 1280min scale (almost daily scale) is needed. A one year
long record consists of more than 50000 ten minute values but only of about 400 values on
the 1280min scale. Therefore, the calculated statistical moments on the finest scale are much
less subject of sampling errors than in the coarsest scales. At a certain aggregation level
∆max the sampling errors become predominant due to the limited number of observations
and the scaling “breaks”. A correct estimation of the scaling range ∆min to ∆max is essential
for an unbiased estimation of the scaling exponent. Hence, the scaling exponent h is severely
affected by the systematic sampling error asymmetry (Harris et al., 1997).
The sampling error regarding different statistical moments q show a similar systematic. The
higher the moment, the more weight is given to the most extreme values in the time series.
If extreme events are randomly “missing” due to the limited number of records in the time
series, the effect on the moment increases non-linearly with the moment order q (Harris
et al., 1997). As a result the scaling in higher order moment q of a limited sample from
a perfectly scaling distribution will flatten out and become increasingly linear instead of
power law shaped. In very low moment orders q < 1 on the other hand, measurement errors
gain more and more weight because they especially affect very low values (high relative
error). Therefore, scaling is not only limited to the aggregation between ∆min and ∆max,
but there is also range in the moment order between qmin and qmax for which the scaling
behavior is observable (Harris et al., 1997). qmin and qmax depend on the length of the time
series.
5.6 Scaling Models 53
The revealed systematic in sampling errors lead to the question if scaling behavior can be
detected at all by the usually applied methods as spectrum analysis (Marsan et al., 1996) or
box counting (Mandelbrot, 1983) in the presence of sampling errors. In a theoretical exper-
iment Harris et al. (1997) additively mix data from processes with different scaling expo-
nents. The additive (instead of multiplicative) mixing should destroy the scaling all together.
Instead, a scaling relationship is detected that is closer to the more intermittent of the two
data sets – which is a problem regarding precipitation fields. Passing cold fronts for exam-
ple exhibit higher intermittency in the postfrontal than in the prefrontal rainfall. Therefore,
the parameter estimation has a dilemma. On the one hand, the scaling estimation should
be restricted to physically homogeneous precipitation processes, on the other hand the time
series should be as long as possible to avoid sampling errors, especially in the calculation of
higher moments and in the higher aggregated data.
Ferraris et al. (2002) have doubts if observed fractal scaling is a proof for a multiplicative
process in precipitation formation and higher order dependencies in rainfall field. They
build random fields from from Fourier transformed linearly correlated data that show sim-
ilar scaling behavior as rain radar fields. This means that the scaling can be mimicked by
non-linear filtered random fields that do not exhibit higher order correlation structures.
5.6.1. Autoregressive Models
Autoregressive models assume a memory between the elements in a rainfall time series
of discrete time steps. In this respect, they are related to Markov-Chain models (Section
5.3). The memory, however, is described differently. It is not expressed as the transition
probability between defined discrete states (e. g. “rain” / “no rain”) but as a fraction of
information in each new value that is herited from the last n preceding values. The number
n of preceding values that have to be considered is called the order of the autoregressive
model. In short term, it is called an AR(n) model.
In a temporal AR(1) model the autocorrelation rl declines exponentially with the lag l be-
tween two considered time steps, since it is always the same part of information that is
passed. A deviation of the observed autocorrelation from exponential decline indicates that
a higher order autoregressive model is required. The parameters for the set-up of higher
order models cannot be accessed directly from an observed time series because the autocor-
relations parameters of different lags l all influence the time series at the same time. They
have to be derived from the observed autocorrelations by a linear equation system.
AR(1) models can be extended to spatial-temporal modeling if the autocorrelation param-
eter is seen as a n × n matrix that describes how the values at each station depend on the
previous values of all n stations. A second matrix is needed for the description of the cross
correlations at simultaneous time steps.
Bárdossy and Plate (1992) present a spatial-temporal model for daily precipitation sums
based on an autoregressive model. The daily rainfall sums are described by a power trans-
form of the truncated normal distribution whereas the truncation is based on the rainfall
probability. The daily rainfall sums are assumed to follow a spatial-temporal AR(1) model
54 Modeling of Precipitation
in the normal transformed space. The interdependencies between all modeled locations are
considered in a crosscorrelation matrix and a lag-one crosscorrelation matrix, linking the
values at the different locations to the subsequent time step. The rainfall probability and
spatial-temporal correlations are estimated separately for 11 different CP groups to account
for persistence and long term memory in the time series. The model is able to represent the
statistical moments, the autocorrelations and the distribution of wet and dry spell lengths
(Bárdossy and Plate, 1992). The spatial representation of intermittency is checked by calcu-
lating the statistical moments conditioned on whether the precipitation sums at surrounding
stations exceed a certain threshold or not. Bárdossy and Plate (1992) also describe how the
model can be transformed to previously ungauged locations.
Koutsoyiannis et al. (2003) proposes a model for spatially correlated disaggregation e. g.
of several spatially correlated daily precipitation series to hourly time series. The model
needs at least one hourly time series as a starting point, which can for example be generated
by a purely temporal (and though simpler) disaggregation model. The spatial-temporal
correlation between the hourly time series are described by a non-Gaussian transform of an
AR(1) model. The correlations on the hourly scale are estimated by a matrix multiplication
that is coupling the correlation between the existing hourly time series to the daily time
series at the same location and the correlation between the daily time series of the different
locations. The model has the shortcoming that it can produce negative values that have to
be set to zero which disturbs the aggregated statistics (the moments at the daily scales).
6. Generation of Point Rainfall
6.1. The Synthetic Rainfall Time Series Generator NiedSim
The conceptual rainfall generation models presented in Chapter 5 rebuild the statistical
characteristics of precipitation only to a limited extend. Markov Chain Models work very
well on daily scale, but the application on time series of shorter aggregation is difficult due
to the persistence and intermittency of precipitation. A regionalization to ungauged areas
is hardly possible (Section 5.3). Point Process Models have problems in simulating the
statistical characteristic on all temporal aggregations equally well. A correct assessment of
the scaling is very challenging and requires a high number of model parameters, as well as
burdensome parameter fitting routines (Section 5.4). Scaling models, have a high risk of
bias if the amount of available data for the model fit is limited (Section 5.6).
Data driven time series generators try to avoid these problems by making as few concep-
tual specifications as possible. Instead, precipitation is merely described by its statistical
appearance.
Since the year 2000 the data driven model NiedSim (Bárdossy, 1998) is operational. It is used
by the authorities of several federal states in South Germany for the generation of rainfall
time series in hourly or five minute temporal resolution. The only conceptual specifica-
tion in NiedSim concerns the theoretical distribution of precipitation values. An initial time
series is set up that has the right intensity distribution but differs from real precipitation
in statistics related to the temporal sequence of values. In a second step, the temporal se-
quence is optimized by subsequently swapping arbitrarily chosen pairs of values (Fig. 6.1).
The performed swaps are evaluated by an objective function O that measures the similar-
ity between the simulated time series and statistical target values derived from real world
observations, for example the autocorrelation on different lags, the extreme value behav-
ior or the daily rainfall probability. The optimization stops when the different between the
simulated statistics and the target values is minimized.
This generation principle is very data demanding because the statistical target values have to
be derived from data of observed measurement station. On the other hand, the method takes
maximum benefit from the high density of the German rain gauge network. In principle,
the detailedness of the generated precipitation time series is only limited by the accuracy
of the applied theoretical distribution and the number of statistical target values that are
considered in the objective function O.
As another strong point NiedSim is well suited for the generation at ungauged locations. For
a regionalization it is sufficient to transfer the statistical target values from the observed rain
56 Generation of Point Rainfall
t
hN
t
hN
S
im
ul
at
ed
 
A
nn
ea
lin
g
S
im
ul
at
ed
 
A
nn
ea
lin
g
t
hN
O(Z)  < O(Z)  ?new old
initial time-series
optimization by swapping values
Figure 6.1.: Principle of the permutation scheme in NiedSim
gauges by geostatistical methods. The same is true for downscaling. The generation can be
easily conditioned on higher aggregated values, e. g. monthly sums by implementing them
as target values in the objective function.
The temporal resolution of NiedSim is one hour. In higher temporal resolution, the num-
ber of possible combination of the values from one year soon becomes unfeasible in the
optimization. For a further refinement to 5min values, NiedSim is combined with a disag-
gregation scheme in which the hourly values are distributed among the twelve five minute
intervals of the hour. The disaggregation is again performed by an optimization that ex-
changes small increments of precipitation between the five minute intervals.
The generation scheme of NiedSim is the basis for this work. The experiences in operation
has shown the need for several adaptations.
6.1.1. Necessary Adaptations to NiedSim
1. Mixing of different distribution functions
Hourly precipitation values in Baden-Württemberg exhibit very skewed distributions.
Most hourly precipitation values are smaller than 1mm while on the other hand ex-
6.1 The Synthetic Rainfall Time Series Generator NiedSim 57
tremes can go up to 50mm. The fit of a distribution function that represents the low
values and the extremes equally well is difficult. In Niedsim, three different distribu-
tion functions are used. The maximum is drawn from a Gumbel distribution the three
next highest values of the year from an Exponential distribution. Both distributions
are fitted to the theoretical one yearly and one hundred yearly extreme values accord-
ing to the official extreme value statistics KOSTRA of the German Weather Service
DWD (Bartels et al., 2005). The remaining values are drawn from a Gamma distribu-
tion whose parameters are fit to observed precipitation time series of 292 rain gauges
(Section 3.3) and then interpolated on the target location by Kriging.
0 5 10 15 20 25 30 35 40 45
[mm/h]
0
20
40
60
80
100
x= 3517340 y= 5450720
annual series - 10 members per year
observed 1974 - 1989
NaN= 1.6%
simulated 1974 - 1989
(a) empirical absolute frequencies
0 10 20 30 40
[mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3517340 y= 5450720
annual series - 10 members per year
observed 1974 - 1989
NaN= 1.6%
simulated 1974 - 1989
(b) empirical cumulative distribution
Figure 6.2.: Comparison of the observed and NiedSim-simulated empirical distribution of
the ten highest 1h precipitation values per year at the rain gauge in Neckarsulm
Since the distributions are derived from different data sets, they do not correspond
very well at every location. A generation for the rain gauge in Neckarsulm is pre-
sented as an example (Fig. 6.2). The histogram of the simulated time series (red line in
Fig. 6.2a) shows a remarkable gap in the frequency of values between six and twelve
millimeters per hour that cannot be observed in the measurements. This leads to a
particularly flat part in the cumulative distribution between six and twelve millime-
ters of hourly rainfall (Fig. 6.2b). The bimodal distribution is not realistic but a sta-
tistical artifact of the mixing. It could be avoided if the three distributions functions
were replaced by one single distribution. However, applying only one distribution
would reduce the variability. Either the highest values would be underestimated or
the expectation of the mean 1h precipitation amount would be overestimated.
2. Variability between wet and dry periods
In Niedsim every month of the year is generated with the same distribution param-
eters. Whether the month is wet or dry, the distribution of 1h precipitation values
with non-zero precipitation F (H | H > 0mm) is constant. In real world observations
however, the difference between wet and dry periods can have a direct effect on the
distribution of the 1h precipitation values. Besides, the distribution of hourly rainfall
58 Generation of Point Rainfall
sums depends on the season. Such dependencies of the distribution parameters are
not yet considered in the operational version of NiedSim.
3. Uncertainties in the regionalization of the simulation parameters
Most parameters of NiedSim are derived for observed rain gauges and regionalized by
External Drift Kriging, (Section 5.2) with the square root of the elevation used as drift
variable. The fact that parameter interdependencies are not considered can lead to
inconsistencies. For example, if there is a hidden, non-linear interdependence between
the shape and the scale parameter of the Gamma distribution, it will be altered by the
interpolation and the distribution at the target location will be biased.
To overcome these short comings, the generation scheme will be modified in this work.
The generation of the initial time series will be restricted to only one distribution function,
namely the Weibull distribution. To enhance the variability, the distribution parameters
will be conditioned on the monthly precipitation sum. In this way, wet months will exhibit
a different distribution than dry months. At the same time, the interdependencies of the
distribution parameters are taken into account. The interdependencies are described by a
Gaussian copula. To account for annual cycles, the description of the interdependencies is
done for each season separately.
6.2. Statistical Parameters of Hourly Precipitation Time Series
The two parametric Weibull distribution can be fit by the first two statistical moments, av-
erage and standard deviation (method of moments). Assuming that the 1h precipitation
values follow the Weibull distribution, it can be defined by the following statistics:
• rainfall probability, that controls the fraction of wet time steps
P1h = 1− F (H = 0mm) (6.1)
where H is the 1h rainfall sum
• the expected average precipitation of wet hours E (Hwet)
E (Hwet) = E (H | H > 0mm) (6.2)
• the expectation of the standard deviation
D (Hwet) =
[
E (Hwet − E (Hwet))2
] 1
2 (6.3)
of the rainfall amounts of wet hours
The rainfall probability P1h, as well as the expectationsE (Hwet) andD (Hwet) are theoretical
values that are strictly speaking only observable in a sample of infinite size. With data from
real world measurements, where the number of observations is limited, one can only give
estimates of these theoretical quantities.
6.3 Methods for the Statistical Description of Distributions 59
In the following the variables X,Y, . . . (U, V, . . . for uniform distributions) in capital letters
are dedicated to random variables in the theoretical world of infinite samples. The use of
lower case letter x, y . . ., (u, v, . . .) indicates observed real world data sets of limited length
n.
The values of P1h, E (Hwet) and D (Hwet) can be approximated by the rainfall frequency
p1h, as well as the average H¯wet and the standard deviation swet of wet hour precipitation
amounts calculated from an observed precipitation time series. For the estimation of the
statistical parameters of one month they are calculated in the following manner:
p1h =
1
nmon
nmon∑
i=1
1{H(i)>0mm} (6.4)
=
nwet
nmon
(6.5)
H¯wet =
1
nwet
nmon∑
i=1
H (i) (6.6)
swet =
1
nwet − 1
nmon∑
i=1
1{H(i)>0}
(
H (i)− H¯wet
)2 (6.7)
(6.8)
where nmon number of hours in current month
1{H(i)>0mm}
{
1 if H (i) > 0
0 if H (i) = 0
nwet number of wet hours in current month
The observed values of p1h, H¯wet and swet vary from month to month. Even if the underly-
ing theoretical expectations were constant, there would be spread due to sampling effects.
In addition, there is variability resulting from the different rainfall characteristic of each
month and due to seasonal effects. The resulting variability can be described by theoretical
distribution functions F (X).
6.3. Methods for the Statistical Description of Distributions
For a quantity X defined on the domain ΩX the distribution function F (X) assigns the non-
exceedance probability P (X ≤ X0) to each value X0 within the domain. It can be seen as
the limit of the observed non-exceedance frequency F (x) if it was possible to draw a sample
of infinite size. ΩX is also called the “event space” of X .
F (X0) = P (X ≤ X0) , X ∈ ΩX (6.9)
= lim
n→∞F (x0) (6.10)
60 Generation of Point Rainfall
There are different families of theoretical distribution functions that can be fit to an observed
distribution x. The choice of the distribution depends on the domain ΩX and on the shape
of the observed distribution. For some configurations of x the choice of the distribution
function can be justified by probability theory.
6.3.1. Weibull Distribution
The Weibull Distribution was first described in the context of life spans in material control.
The probability density function (PDF) is given by:
f (X) =
κ
λ
(
X
λ
)κ−1
e−(X/λ)
κ
(6.11)
Its cumulative distribution function (CDF) is given by:
F (X) = 1− e−(X/λ)κ (6.12)
The support of the Weibull Distributions are the positive rational numbers X ∈ [0,∞].
The parameter κ defines the shape of the distribution, λ defines the scale. For κ = 1 the
Weibull Distribution is equivalent to the Exponential Distribution. For κ ≤ 1 the density is
highest at X = 0, for κ > 1 the PDF shows a maximum at X > 0.
The first two moments of the Weibull Distribution are given by the following equations:
E (X) = λΓ
(
1 +
1
κ
)
(6.13)
D2 (X) =
1
λ2
[
Γ
(
1 +
2
κ
)
− Γ2
(
1 +
1
κ
)]
(6.14)
where the Γ-function is:
Γ (x) =
∫ ∞
0
tx−1e−tdt (6.15)
The Weibull Distribution can be linearized by a double logarithmic transformation:
ln (− ln (1− F (x))) = −κ lnλ+ κ lnX (6.16)
W = a∗ + b∗V (6.17)
This relation is used to check if the data of the precipitation stations follow a Weibull Dis-
tribution or not. If the observed data pairs (x|Fn (x)) from an observed precipitation time
series spread symmetrically around a straight line in the v, w-plot where
v = lnx (6.18)
w = ln (− ln (1− Fn (x))) (6.19)
the assumption of Weibull distributed data is valid.
Fn (x) is the observed non-exceedance frequency that is calculated by the ranks r (x) =
[1, 2, . . . n] of the observed data:
Fn (x) =
r (x)
n
(6.20)
6.3 Methods for the Statistical Description of Distributions 61
To account for sample affects in the transformed plot, Fn (x) will not be calculated according
to Eq. (6.20) if the data volume n is lower than 50 but with the following formula:
Fn (x) =
r − 0.3
n+ 0.4
if n < 50 (6.21)
where r rank of x according to Eq. (6.46)
n number of observations
6.3.2. Log-Normal Distribution
A quantity X exhibits a Log-Normal Distribution if the transformed quantity V = lnX is
normally distributed.
The central limit theorem states that any mean or sum of a large number of identically dis-
tributed random variables approximately follow a Normal Distribution (see for example
Hartung, 2009). In the logarithmic transformed space any product is transformed into a
sum. The sum of logarithms is equal to the logarithm of the product. Applying the cen-
tral limit theorem, it follows that any product of a large number of identically distributed
random variables is approximately Log-Normal distributed.
On that account the Log-Normal Distribution is a reasonable choice for random variables
that arise from multiplicative processes. According to the theory of multifractal scaling (see
Section 5.6), this is an argument that the Log-Normal distribution could be an appropriate
choice for precipitation related statistics.
The Log-Normal Distribution is characterized by the following equations:
f (X) =
1
X
√
2piσ2
e−
(lnX−µ)2
2σ2 (6.22)
F (X) =
∫
f (X) dX (6.23)
The parameters µ and σ can be interpreted as the average and standard deviation in the
logarithmic transformed space. The integral F (X) can only be estimated numerically.
The fit of the Log-Normal Distribution can be checked graphically in the linearized space of
v, w:
v = ln (x) (6.24)
w = φ−1 (Fn (x)) · σ + µ (6.25)
(6.26)
where φ−1 inverse of the Standard Normal distribution
µ, σ parameters of the fitted Log-Normal distribution
62 Generation of Point Rainfall
6.3.3. Pareto Distribution
The highest extremes in hydrological variables (e. g. precipitation amounts or discharges)
are responsible for the most dangerous situations. Therefore, it is important that the applied
theoretical distribution exhibits a sufficiently high skewness to avoid potentially dangerous
underestimations of the extremes. Distributions that attribute a high probability to extreme
values are often called “thick tailed”.
An especially “thick tailed” theoretical distribution function is the Pareto Distribution.
Koutsoyiannis (2005) argues that it was the principal distribution to describe extreme pre-
cipitation frequencies. Like the Log-Normal Distribution it is self-similar and in accordance
with the theory of fractal scaling.
The PDF and the CDF are given by:
f (X) =
αXαm
Xα+1
, X ≥ Xm (6.27)
F (X) = 1−
(
Xm
X
)α
, X ≥ Xm (6.28)
The support of the Pareto distribution is X ∈ [Xm;∞]. Xm is the basic scale of the distri-
bution α is the shape parameter. All probabilities are expressed in relation to Xm. This is
a problem, when X can become zero. For Xm = 0 the distribution is not defined. In that
case it is necessary to shift the distribution so that the support starts at zero. The resulting
distribution is called the Lomax Distribution. PDF and CDF are given by:
f (X) =
α
λ
(
1 +
X
λ
)−(α+1)
, X ≥ 0 (6.29)
Its CDF is given by:
F (X) = 1−
(
1 +
X
λ
)−α
, X ≥ 0 (6.30)
The equations 6.29 and 6.30 can be derived from the CDF and PDF of the Pareto Distribution
by setting λ = Xm and substituting X by X +Xm.
The Lomax distribution can be linearized if the scale parameter λ is known:
ln (1− F (X)) = − ln (λ)− α ln (X + λ) (6.31)
W = a∗ + b∗V (6.32)
where the transformed observed values are
v = ln (x+ λ) (6.33)
w = ln (1− Fn (x)) (6.34)
Fn (x) is calculated by Eq. (6.20).
6.3 Methods for the Statistical Description of Distributions 63
6.3.4. Beta Distribution
The Beta Distribution is given by the following equations for the PDF and CDF:
f (X,α, β) =
Xα−1 (1−X)β−1
B (α, β)
, X ∈ (0, 1) (6.35)
F (X,α, β) =
BX (α, β)
B (α, β)
, X ∈ (0, 1) (6.36)
where B (α, β) is the Beta Function and BX (α, β) the incomplete Beta Function:
B (α, β) =
∫ 1
0
tα−1 (1− t)β−1 dt (6.37)
BX (α, β) =
∫ X
0
tα−1 (1− t)β−1 dt (6.38)
Since there are no analytical solutions to the integrals, both equations can only be solved
numerically.
The support for the beta distribution is Ω = (0, 1) while the limits of the interval are ex-
cluded. For α = 1 and β = 1 the density is constant over Ω leading to a uniform distribution.
If α < 1 and β < 1 the PDF is highest for X → 0 and X → 1. If α > 1 and β > 1 the density
is unimodal. For α = β the distribution is symmetric to X = 0.5.
6.3.5. Comparison of Theoretical Distributions
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0
log(x) observed
4
3
2
1
0
1
2
lo
g
(x
) 
th
e
o
re
ti
ca
l
weibull distribution 
 mistaken for lognormal distribution
(a) Weibulll Distribution
0 2 4 6 8 10 12 14
log(x) observed
10
5
0
5
10
15
lo
g
(x
) 
th
e
o
re
ti
ca
l
lomax distribution 
 mistaken for lognormal distribution
(b) Lomax Distribution
Figure 6.3.: Effect of fitting a Log-Normal Distribution by Maximum Likelihood Method to
a distribution of 1000 random numbers drawn from a different theoretical dis-
tribution
The three distributions Weibull Distribution, Log-Normal Distribution and Lomax Distribu-
tion differ the most in the probabilities attributed to extreme precipitation. Fig. 6.3 illustrates
64 Generation of Point Rainfall
the differences. 1000 random values are drawn from a Weibull Distribution (Fig. 6.3a) or a
Lomax Distribution (Fig. 6.3b) and are plotted in the linearized Log-Normal Plot. For Log-
Normal distributed data, the red crosses of the observation would spread symmetrically
around the blue line.
Data from the Weibull distribution, leads to a convexly bending point cloud. The Log-
Normal distribution overestimated very low and very high values. Therefore the Log-
Normal distribution exhibits higher extreme value probabilities while the Weibull distribu-
tion allocates more density in the vicinity of the median. The concave curbing of the Lomax
distributed observations indicates the opposite effect (Fig. 6.3b). The Lomax distribution has
a higher skewness. In this case, the application of the Log-Normal Distribution leads to an
underestimation in the extremes. The values are generally very low, except for the extremes.
The density around the median is low compared to the Normal Distribution.
6.3.6. Maximum Likelihood Method for the Fit of Distribution Functions
Theoretical distribution functions exhibit parameter that have to be adjusted to the observed
distribution. On possible estimation is the Maximum Likelihood Method (MLM). It consists
of finding the configurations of the parameters α, β . . . of the tested distribution functions
for which the observed values of x are associated with the highest PDF-values. In other
words: One searches the configuration of the distribution function for which the observed
distribution x is most likely to occur. (More on MLM in Wilks, 2011)
L (x|α, β . . .) =
n∏
i=1
f (xi|α, β, . . .)→ max! (6.39)
Transforming Eq. (6.39) into logarithmic space does not change the location of the maximum.
In logarithmic form, however, the product can be transferred to a sum:
lnL (x|α, β, . . .) =
n∑
i=1
ln f (xi|α, β, . . .)→ max! (6.40)
This sum is maximized by numerical optimization.
6.4. Methods for the Description of Multivariate Dependencies
6.4.1. Covariance and Correlation
A standard statistical tool to measure the linear dependence between two observed quanti-
ties x and y is the correlation rxy
rxy =
sxy
sxsy
(6.41)
calculated by covariance between x and y
sxy =
1
n− 1
n∑
i=1
(xi − x¯) (yi − y¯) (6.42)
6.4 Methods for the Description of Multivariate Dependencies 65
and the standard deviations in x and y
sx =
(
1
n− 1
n∑
i=1
(xi − x¯)2
) 1
2
(6.43)
where x¯ the average of the observations in x
n number of data pairs
The covariance and the correlation can easily be extended to the multivariate case:
Σ = [sij ] where i, j ∈ [1, nvar] (6.44)
r = [rij ] where i, j ∈ [1, nvar] (6.45)
where nvar the number of correlated variables. The bold typing of r and Σ indicates that
these are matrices. The diagonal elements rii i ∈ [1, nvar] are all equal to one. The diago-
nal elements Σii are equal to the respective variances s2i .(for more details, see for example
Hartung, 2009).
Despite the wide spread use of multivariate covariance and correlation, it has considerable
draw backs. The data pairs that are far away from the point of the average (x¯ | y¯) have a
higher weight in the calculation of the covariance. The correlation therefore is especially
governed by the highest values which can be a problem in skewed distributions like many
precipitation related statistics. Besides, the restriction to linear dependence is somehow
contradictory. Any non-linear transformation of x or y changes the value of the correlation
rxy. However, if x or y are multiplied by a deterministic non-linear function, why should
this affect the “real” dependence between the two quantities? If x and y follow a functional
relation y = f (x) the correlation can take any value −1 ≤ rxy ≤ 1 although y is completely
determined by x. As a consequence, the correlation rxy is highly sensitive to non-linear
measurement errors in the correlated quantities.
6.4.2. Spearman’s Rank Correlation
A more robust measure of dependence is the Spearman’s rank correlation ρxy. The rank of
an observed distribution is calculated by sorting the data in ascenting order according to
their value and then replacing each value by its position number
x = [x1, x2, x3 . . . xn] where x1 ≤ x2 ≤ x3 . . . ≤ xn
Rank (x) = [1, 2, 3 . . . n] (6.46)
ρxy is then calculated as the correlation from the position numbers r = Rank (x) and q =
Rank (y).
ρxy =
srq
srsq
(6.47)
where the covariance srq and the standard deviations sr and sq are calculated by Eq. (6.42)
and Eq. (6.43) just be replacing x and y by their respective ranks r and q.
66 Generation of Point Rainfall
Spearman’s ρ only depends on the configuration of the data and not on the distribution.
As long as the order of values is kept unchanged, the rank correlation is constant. It is
not affected by any monotonic transformation of the data. For this reason Spearman’s ρ is
insensitive to any measurement errors, as long as the order of the data is not affected. |ρ|
becomes 1 for any functional dependence y = f (x) between x and y.
6.4.3. Copulas as a Measure of Dependence
Spearman’s ρ cannot measure all aspects of multivariate dependence. Environmental vari-
ables may exhibit asymmetry in the correlation structure, so that a subsample of higher-
ranked data pairs show a different rank correlation than a subsample of lower ranked pairs.
While the rank correlation reduces the whole data range to one specific number and cannot
take such effects into account, copulas and copula density are tools to measure the depen-
dence structure over the whole range of the distributions.
Due to its ability to measure complex dependence structures, copulas have been used more
and more frequently in precipitation analysis for the last ten years. Theoretical copulas of
the Archimedean family for example are applied to relate volume, duration and intensity
of rain storm events and to calculate joint probabilities for simultaneous extremes in these
three quantities (Zhang and Singh, 2007). It is shown that the estimated probabilities are
more realistic than joint probabilities calculated from multivariate correlation (Zhang and
Singh, 2007).
AghaKouchak et al. (2010) use a V-transformed copula to describe the dependence between
1h precipitation amounts from rain gauge measurements and areal precipitation amounts
derived from rain radar. The relationship is used to generate ensembles of rainfall fields that
are conditioned on the rain gauge measurements and exhibit the spatial variability and the
error structure of the radar fields.
van den Berg et al. (2011) exploit empirical, non parametric copulas for the downscaling
of areal 1h precipitation sums from a coarse scale of 19.1km × 19.1km to a fine scale of
600m× 600m. It is found that the distribution on the finer scale is not constant but depends
on the coarse scale rainfall sum. The effect can be taken into account be the copulas.
Salvadori and De Michele (2010) give an overview of how copulas can be applied in the
estimation of joint extreme value return periods. As an example they model joint return
periods of extreme discharge events at four river gauging stations in the Spey catchment,
UK.
A copula represents the cumulative distribution function of ndim joint variables if the vari-
ables all follow a uniform distribution in the interval [0, 1]
C : [0, 1]ndim → [0, 1] (6.48)
where the hypercube [0, 1]ndim is the support for the copula. The copula describes the joint
non-exceedance probability in the ndim variables.
C
(
U∗1 , U
∗
2 , . . . , U
∗
ndim
)
= F
(
U1 ≤ U∗1 , U2 ≤ U∗2 , . . . , Undim ≤ U∗ndim
)
(6.49)
6.4 Methods for the Description of Multivariate Dependencies 67
The copula becomes zero if one of the ndim variables is zero. It is equal to U if one of the
variables has a value of U and the others are one.
C (U1, . . . , Ui−1, 0, Ui+1, . . . , Undim) = 0 (6.50)
C (1, . . . , 1, Ui, 1, . . . , 1) = Ui (6.51)
For any interval B of dimension ndim in the hypercube [0, 1]ndim the probability that U =
[U1, U2, . . . , Un] is within B is non-negative.
P (U ∈ B) =
∫
B
dC (U) ≥ 0 (6.52)
The derivative c = dC is called the copula density. It is a multivariate extension of the
probability density f (X) for joint uniformly distributed variables U .
According to the Sklar’s Theorem (see for example Nelsen, 2006) the joined distribution of
ndim random variables X1, X2, . . . Xndim can be calculated by a copula if the distribution
functions F1 (X1) , F2 (X2) , . . . Fndim (Xndim) are known:
F (X1, X2, . . . Xndim) = C (F1 (X1) , F2 (X2) , . . . Fndim (Xndim)) (6.53)
In this way, the analysis of the dependence between the variables can be separated from the
analysis of their distributions, which are from now on called the marginal distributions.
6.4.3.1. Empirical Copulas
Since copulas are describing probabilities they can, strictly speaking, only be observed for
an infinite number of data tuples. Nevertheless, a copula can be easily approximated by real
world data since the joined non-exceedance frequency Fn (x) of a data set x with n values is
a uniform distribution in the interval [0, 1] for any possible distribution.
The estimation of the empirical copula density follows a five step proceeding. For the bi-
variate case of x and y data with n data pairs:
1. Replacing the data by its rank according to Eq. (6.46): x→ r (x), y → q (x)
2. Transforming the ranks into the uniform distribution of non-exceedance frequencies u
and v:
u = Fn (x) =
r (x)− 0.5
n
(6.54)
(v respectively by transforming y)
3. Mapping the pairs of of ui and vi in the unit square [0, 1]2.
4. Dividing the square into nbins × nbins sub-squares. By counting the number of pairs
(ui | vi) in each square and by dividing the number by the size of the sub-square area,
the empirical copula density is calculated.
cij = n {u ∈ δi; v ∈ δj} · n2bins (6.55)
68 Generation of Point Rainfall
where cij empirical average copula density in cell ij
i, j cell number in u and v-direction, i, j ∈ [1, nbins]
δi = [
i−1
n ,
i
n ] borders of the i
th sub-square in u-direction
δj = [
j−1
n ,
j
n ] borders of the j
th sub-square in v-direction
1/n2bins area of one sub-square
5. Finally, the copula can be calculated by summation over the sub-squares in u and v
direction:
C(u, v) =
nbins∑
i=1
nbins∑
j=1
1{i/nbins≤u}1{j/nbins≤v}cij (6.56)
where 1{i/nbins≤u}
{
1 if i/nbins ≤ u
0 if i/nbins > u
6.4.3.2. Theoretical Copulas
In probability calculations observed non-exceedance frequencies F (x) of finite samples are
used to fit theoretical distribution functions F (X). It is assumed that F (X) is the limiting
case for the observed distribution F (x) if the sample size could be extended to infinity:
F (X) = 1 lim
n→∞F (x) (6.57)
Assuming a certain shape of the theoretical distribution, e. g. Exponential, Normal or
Gamma distribution, it can be used to estimate non-exceedance probabilities for values that
have not yet been observed in the fitting sample or that even exceed the range of the ob-
served distribution.
Applying the same concept, a theoretical copula is fit to a given data set. Instead of the
observation x, ndim dimensional tuples of uniformly transformed observations u are used.
C (U1, U2, . . . , Undims) = limn→∞C (u1, u2, . . . , undims) (6.58)
There are different choices of theoretical copulas that can be fit to the data. If the marginal
distributions are known, it is common to perform a maximum likelihood fit based on the
copula densities (Schoelzel and Friederichs, 2008). For some copula families predefined
maximum likelihood estimators exist.
One of the main difference between the copula families is the tail dependence structure
(AghaKouchak et al., 2010). Tail dependence means that the highest or lowest extreme val-
ues show a different dependence structure than the other, non-extreme values. In this work
the Gaussian copula is exploited. The Gaussian copula assumes absence of tail dependence.
The dependence structure is constant over the whole range of all uniformly transformed
random variables U1, U2, . . . Undim .
6.4 Methods for the Description of Multivariate Dependencies 69
6.4.3.3. The Gaussian Copula
The Gaussian copula can be described by the following equation:
C (U1, U2, . . . , Undim) = Φr
(
Φ−1 (U1) ,Φ−1 (U2) , . . . ,Φ−1 (Undim)
)
(6.59)
where Φr Multivariate normal distribution of dimension ndim
with multivariate correlation r
Φ−1 inverse of the Standard Normal distribution
The copula density is given by:
c (U1, U2, . . . , Undim) =
1√|r| ·
(
−1
2
ZT
(
r−1 − I)Z) (6.60)
where r−1 the inverse of the correlation matrix
I the identity matrix
Z = φ−1 (U) Standard Normal transform of U
The Gaussian copula is constructed from any tuples of random variables X1, X2, . . . Xndim
by first transforming the random variables in the unit space using their inverse distribution
Ui = F
−1 (Xi) and further transforming them to Standard Normally distributed values Zi =
Φ−1 (Ui) and by finally applying a multivariate normal distribution.
If the variables X and Y are transformed into Standard Normal space, the fit of a Gaus-
sian copula is reduced to the calculation of the multivariate correlation r according to
Eq. (6.41) (Renard and Lang, 2007). It can be estimated by the Spearman’s rank correlation
ρij (Eq. (6.47)) between any combination of variables i and j ∈ ndim:
ri,j = 2 sin
(pi
6
ρi,j
)
(6.61)
The departures of ri,j from ρi,j are in the range of a few percent.
Fig. 6.4 shows an example of a bivariate Gaussian dependence structure. The scatter plot
Fig. 6.4a exhibits the Standard Normal transformed values of 10000 random pairs drawn
from Gaussian copula with a rank correlation of ρ = 0.5. Fig. 6.4b displays the empirical
copula density for a grid with 16×16 sub-squares. The distribution of Y does not depend on
X except for the average. The probability density on any vertical (or horizontal) line through
Fig. 6.4a follows the Standard Normal distribution. Transformed to the copula space, it
results in the typical elliptic shaped density of the Gauss copula.
The Gauss copula is symmetric. The symmetry of a bivariate Copula can be checked by:
Sym =
1
n
n∑
i=1
(
(Fn (xi)− 0.5)2 (Fn (yi)− 0.5) + (Fn (xi)− 0.5) (Fn (yi)− 0.5)2
)
(6.62)
70 Generation of Point Rainfall
(a) Standard Normal transformed
0.0 0.2 0.4 0.6 0.8 1.0
F(random x)
0.0
0.2
0.4
0.6
0.8
1.0
F(
ra
n
d
o
m
 y
)
sym= 0.00002
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) copula density
Figure 6.4.: Scatter plot in the Standard Normal transformed space and empirical copula
density of 10000 random pairs with rank correlation ρ = 0.5 and Gaussian de-
pendence structure
where Fn (xi) observed non-exceedance frequency of x according to Eq. (6.54)
Fn (yi) observed non-exceedance frequency of x according to Eq. (6.54)
n number of data pairs
(See Haslauer (2011))
The name symmetry is somehow misleading as a perfectly symmetric copula would exhibit
Sym = 0. Generally the values of sym are small. A value exceeding 0.01 would indicate a
significantly skewed copula.
6.5. Analysis of Observed Precipitation Data
6.5.1. Multivariate Dependence in Observed Data
This section will explore the interdependencies of monthly rainfall sum Hmon, rainfall fre-
quency p1h, average H¯wet and standard deviation swet of wet hour precipitation amounts
in the available observed data from 292 rain gauges (Section 3.3). It is assumed that the
dependence structure between the statistical characteristics is independent of the location
throughout the whole study region and that it is an expression of the overall climatic condi-
tions of the region.
The dependence structure is described by a copula (Section 6.4.3) in the four dimensional
space of the [0, 1]-transformed ranks from Hmon, P1h, E (Hwet) and D (Hwet). Since copulas
exploit the ranks, they do not depend on absolute values. The absolute values, seen as an
expression of the local effects, are modeled by the marginal distribution, which is explored
in the next section (Section 6.5.2).
6.5 Analysis of Observed Precipitation Data 71
A four dimensional Copula cannot be displayed on two dimensional paper. Instead, the
dependence between Hmon, P1h, E (Hwet) and D (Hwet) is analyzed pairwise. The main
question is whether the bivariate Copulas are Gaussian or not. Only if all bivariate Copulas
are of Gaussian type, it can be assumed that the four dimensional Copula might also be
Gaussian. This, however, is only a necessary condition but not a sufficient condition.
All 292 stations in the study area are examined at once. The empirical statistics Hmon, p1h,
H¯wet and swet are calculated for each month in the whole measurement period from 1958
to 2003. Each month at each station contributes with one data pair to the bivariate copula
densities. The overall data volume is of n = 27788 station-months which is an average
of about 95 months per station. The empirical copulas are calculated according to the five
point scheme in Section 6.4.3.1. To account for seasonal effects, the copulas are calculated
for each month separately. Table 6.1 gives an overview of the rank correlations values ρ
(Eq. (6.46)) and the values of the symmetry measure sym (Eq. (6.62)). In the following the
empirical bivariate copula densities for all parameter combinations are displayed (Fig. 6.5
to Fig. 6.10). The two plots of each figure feature the months that differ the most in terms of
the empirical copula density.
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
1
h
 r
a
in
fa
ll 
fr
e
q
u
e
n
cy
sym= 0.00113
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) March
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
1
h
 r
a
in
fa
ll 
fr
e
q
u
e
n
cy
sym= 0.00255
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) July
Figure 6.5.: Empirical bivariate copula between monthly rainfall sum and 1h rainfall fre-
quency calculated from month-wise censored data
There is a significant dependence between the monthly rainfall sumHmon and the 1h rainfall
frequency p1h (Fig. 6.5). The rank correlation is highest in March and October. The cyclonic
activity over the Atlantic Ocean is also highest in spring and autumn and lowest during
summer (Weischet and Endlicher, 2008). It can be assumed that advective precipitation
from cyclonic fronts enhances the dependence (Fig. 6.5a), whereas free convective precipi-
tation with high intensity and short duration in Summer months decreases the dependence
(Fig. 6.5b). In summer, the empirical copula density is slightly asymmetric. The dependence
between pairs with high monthly sums and high average precipitation is weaker.
The dependence of the average rainfall amount of wet hours H¯wet on the monthly sumHmon
in comparison is lower. A higher monthly sum is to a greater part generated by more rainy
72 Generation of Point Rainfall
Table 6.1.: Rank correlation and symmetry of the empirical copula from monthly data for
bivariate combinations of all parameters
Month Hmon − p1h Hmon − H¯wet Hmon − swet
ρ Sym ρ Sym ρ Sym
January 0.85 -0.0017 0.56 -0.0017 0.55 -0.0028
February 0.81 -0.0023 0.60 0.0016 0.59 0.0008
March 0.86 0.0011 0.57 -0.0019 0.48 -0.0043
April 0.80 0.0023 0.49 -0.0004 0.41 0.0001
May 0.83 -0.0010 0.33 -0.0063 0.20 -0.0075
June 0.73 -0.0027 0.21 -0.0041 0.10 -0.0049
July 0.55 0.0026 0.40 -0.0046 0.37 -0.0068
August 0.75 -0.0020 0.46 -0.0021 0.45 -0.0034
September 0.80 0.0004 0.16 -0.0042 0.10 -0.0054
October 0.86 0.0001 0.57 -0.0054 0.48 -0.0062
November 0.84 -0.0002 0.63 -0.0016 0.59 -0.0017
December 0.77 0.0021 0.64 -0.0002 0.57 -0.0003
Month p1h − H¯wet p1h − swet H¯wet − swet
ρ Sym ρ Sym ρ Sym
January 0.40 -0.0015 0.44 -0.0019 0.86 -0.0019
February 0.37 -0.0009 0.40 -0.0014 0.90 0.0004
March 0.35 -0.0033 0.30 -0.0063 0.84 -0.0011
April 0.18 0.0008 0.18 0.0010 0.85 -0.0008
May 0.04 -0.0073 -0.02 -0.0081 0.85 -0.0003
June -0.21 -0.0054 -0.22 -0.0055 0.82 -0.0002
July -0.18 -0.0030 -0.11 -0.0040 0.88 -0.0012
August 0.10 -0.0037 0.19 -0.0042 0.84 -0.0009
September -0.22 -0.0028 -0.20 -0.0036 0.85 -0.0001
October 0.31 -0.0068 0.25 -0.0068 0.86 -0.0020
November 0.41 -0.0014 0.40 -0.0005 0.85 -0.0013
December 0.32 0.0033 0.29 0.0028 0.90 -0.0004
hours than by higher 1h precipitation amounts. The correlations are lowest in the beginning
of the convective summer season from Mai to June and in September. They are highest in
November, December and February. Despite the high difference in rank correlation (ρ = 0.16
in September, ρ = 0.59 in November) the shape of the copula density does not look very
different (Fig. 6.6a and Fig. 6.6b). Both exhibit the typical lens shape of the Gaussian copula.
The rank correlation between Hmon and H¯wet on the one hand and between Hmon and swet
on the other hand shows the same yearly cycle (Table 6.1). It is the same months that exhibit
the highest and lowest rank correlations. The absolute values, however, are slightly lower
for the combination of Hmon and swet. In both cases, the shape of the empirical copulas
between Hmon and swet is close to Gaussian shape (Fig. 6.7).
6.5 Analysis of Observed Precipitation Data 73
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00416
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) September
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00159
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) November
Figure 6.6.: Empirical bivariate copula between monthly rainfall sum and average 1h rain-
fall amount of wet hours calculated from month-wise censored data
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00543
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) September
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00170
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) November
Figure 6.7.: Empirical bivariate copula between monthly rainfall sum and standard devia-
tion of 1h rainfall amounts of wet hours calculated from month-wise censored
data
The average H¯wet and the standard deviation swet also react very similar in their dependence
to the 1h rainfall probability. In the convective summer season the correlations are negative
(except for August). It is not during the rainiest summers (with the most rainy hours) that
the highest intensities are expected. Generally, long lasting advective events have lower
rainfall intensities than convective events of short duration. In absence of convective events,
between September and April, there is a clear positive correlation between P1h and H¯wet
(and between P1h and swet). In the empirical copula density Fig. 6.8b and Fig. 6.9b one can
see the clear concentration of points in the upper left corner. These points represent months
74 Generation of Point Rainfall
0.0 0.2 0.4 0.6 0.8 1.0
1h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00154
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) January
0.0 0.2 0.4 0.6 0.8 1.0
1h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00544
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) June
Figure 6.8.: Empirical bivariate copula between 1h rainfall probability and average 1h rain-
fall amount of wet hours calculated from month-wise censored data
0.0 0.2 0.4 0.6 0.8 1.0
1h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00186
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) January
0.0 0.2 0.4 0.6 0.8 1.0
1h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00550
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) June
Figure 6.9.: Empirical bivariate copula between 1h rainfall probability and standard devia-
tion of 1h rainfall amounts of wet hours calculated from month-wise censored
data
with very few but intensive rainfall events. A combination that is much less frequent in the
plots of the advective seasons (Fig. 6.8a and Fig. 6.9a).
The rank correlation between the average H¯wet and the standard deviation swet of wet hour
precipitation shows only minor variations between the months (Fig. 6.1). The rank corre-
lation is lowest during May, June and July and highest in December. The empirical copula
plots look all very similar (Fig. 6.10). They exhibit a high symmetry (a low value of sym)
and the elliptical shape of the Gaussian copula.
6.5 Analysis of Observed Precipitation Data 75
0.0 0.2 0.4 0.6 0.8 1.0
monthly average of 1h precipitation amount
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00030
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) May
0.0 0.2 0.4 0.6 0.8 1.0
monthly average of 1h precipitation amount
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
1
h
 p
re
ci
p
it
a
ti
o
n
 a
m
o
u
n
t
sym= -0.00036
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) December
Figure 6.10.: Empirical bivariate copula between average and standard deviation of 1h rain-
fall amounts of wet hours calculated from month-wise censored data
With the exception of the dependence between the average and the standard deviation
of precipitation amounts of wet hours, distinct differences between the seasons can be
found. The cyclonic activity which is highest in spring and autumn enhance the depen-
dence of P1h, H¯wet, swet on the monthly sum, the occurrence of convective events from May
to September lead to a decrease in rank correlation. The strongest yearly cycle is observed
in the correlation of the monthly sum Hmon to the average H¯wet and standard deviation swet
of precipitation amounts of wet hours.
The statistics of p1h, H¯wet, swet all exhibit a positive rank correlation to the monthly rainfall
sum. These findings are in favor of the idea that the rainfall generation can be conditioned
on the monthly sum. Assuming a Gaussian copula leads to some minor distortions of the
real dependence structure, most of all in the dependence of the average amount and the
standard deviation on the rainfall frequency. However, the dependence is anyway weak in
these combinations and so the departures from a Gaussian copula are of minor importance.
6.5.1.1. Gaussian Copula Dependence used for Rainfall Generation
The described dependencies will be exploited in the rainfall simulations to draw values of
P1h, E (Hwet) and D (Hwet) that are conditioned on the observed monthly precipitation sum
Hmon. The dependence structure will be calculated for three monthly seasons (December to
February, March to Mai, June to August and September to November). Table 6.2 displays
the respective correlations r in the Standard Normal transformed space.
6.5.2. Observed Marginal Distributions
According to Sklar’s theorem, the description of the dependence structure and of the dis-
tributions can be separated. In the last section the copula dependence between the rainfall
76 Generation of Point Rainfall
Table 6.2.: Correlation in the Standard Normal transformed space of monthly precipitation
sum, hourly rainfall probability and monthly average and standard deviation of
wet-hour precipitation amounts
December to February
Hmon P1h E (Hwet) D (Hwet)
Hmon 1.000 0.873 0.623 0.589
P1h 0.873 1.000 0.387 0.405
E (Hwet) 0.623 0.387 1.000 0.895
D (Hwet) 0.589 0.405 0.895 1.000
March to May
Hmon P1h E (Hwet) D (Hwet)
Hmon 1.000 0.838 0.503 0.423
P1h 0.838 1.000 0.113 0.092
E (Hwet) 0.503 0.113 1.000 0.898
D (Hwet) 0.423 0.092 0.898 1.000
June to August
Hmon P1h E (Hwet) D (Hwet)
Hmon 1.000 0.786 0.329 0.296
P1h 0.786 1.000 -0.118 -0.054
E (Hwet) 0.329 -0.118 1.000 0.861
D (Hwet) 0.296 -0.054 0.861 1.000
September to November
Hmon P1h E (Hwet) D (Hwet)
Hmon 1.000 0.854 0.418 0.352
P1h 0.854 1.000 0.029 0.014
E (Hwet) 0.418 0.029 1.000 0.898
D (Hwet) 0.352 0.014 0.898 1.000
governing parameters Hmon, P1h, E (Hwet) and D (Hwet) was explored. Besides, it is neces-
sary to estimate the marginal distribution of the four statistics for the transformation of real
world values into the copula space (and back).
Since the dependence structure was estimated for all stations at once, it is required that one
unique marginal distribution can be found to each of the statistics.
The analyzed observed statisticsHmon, H¯wet and swet exhibit right skewed distributions. The
values by their nature cannot become smaller than zero. Therefore, the Weibull, Log-Normal
and the Pareto Distribution will be tested. To the monthly 1h precipitation frequencies the
6.5 Analysis of Observed Precipitation Data 77
Beta Distribution will be fit since it is a distribution with support X ∈ (0, 1).
For each statistical parameter it is tested if all 292 station agree on one distribution function.
If so, the distribution is fit to the whole data set of all 27788 station months between 1958
to 2003. In the following the distributions at the station in Holzgerlingen (Gaus-Krüger 3
coordinates of X=3501820, Y=5384490, see Section 3.3) are shown as examples of the single
site fit (Fig. 6.12 to 6.15).
• Monthly Precipitation Sum
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
x*
7
6
5
4
3
2
1
0
1
2
y
*
station  23    n= 552
 x=3501820  y=5384490  z= 513
empirical CDF
weibull-fit
(a) Weibull Distribution
50 100 150 200 250 300 350 400
msum
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
re
l.
 f
re
q
. 
/ 
p
d
f
station  23    n= 552
 x=3501820  y=5384490  z= 513
(b) Histogram - Weibull PDF
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
log(x) observed
1
2
3
4
5
6
lo
g
(x
) 
th
e
o
re
ti
ca
l
station  23    n= 552
 x=3501820  y=5384490  z= 513
empirical distribution
log-normal
(c) Log-Normal Distribution
0.00000 0.00005 0.00010 0.00015 0.00020 0.00025
x*
8
7
6
5
4
3
2
1
0
1
y
*
station  23    n= 552
 x=3501820  y=5384490  z= 513
empirical CDF
lomax-fit
(d) Lomax Distribution
Figure 6.11.: Observed and theoretical distribution in the linearized space of the monthly
precipitation sum in Holzgerlingen. Fit of three different theoretical distribu-
tions. Histogram and PDF of the best fitting Weibull Distribution.
In Fig. 6.11 the MLH-fit of the three tested distribution functions Weibull Distribution,
Log-Normal Distribution and Lomax Distribution as well as the observed histogram
and the PDF of the best fitting Weibull Distribution is shown. The vertical line in
Fig. 6.11b indicates the average value of the distribution. The fit is performed to the
78 Generation of Point Rainfall
monthly sums kriged from the daily stations. In this way, a complete time series is
available between 1958 and 2003. Using kriged values is a reasonable choice because
the developed simulation scheme will also be based on kriged values of the monthly
sum.
In Holzgerlingen, the Weibull Distribution fits best. The other two systematically over-
estimate the probabilities of very high monthly sums. The distributions from all 291
other stations agree on this result. At only a handful of stations the Weibull Distri-
bution slightly underestimates the probabilities of the wettest months but remains the
best choice among the three distributions.
• Average Precipitation Amount of Wet Hours
1.5 1.0 0.5 0.0 0.5 1.0
x*
6
5
4
3
2
1
0
1
2
3
y
*
station  23    n= 185
 x=3501820  y=5384490  z= 513
empirical CDF
weibull-fit
(a) Weibull Distribution
0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 0.00014 0.00016
x*
6
5
4
3
2
1
0
1
y
*
station  23    n= 185
 x=3501820  y=5384490  z= 513
empirical CDF
lomax-fit
(b) Lomax Distribution
1.5 1.0 0.5 0.0 0.5 1.0
log(x) observed
1.5
1.0
0.5
0.0
0.5
1.0
lo
g
(x
) 
th
e
o
re
ti
ca
l
station  23    n= 185
 x=3501820  y=5384490  z= 513
empirical distribution
log-normal
(c) Log-Normal Distribution
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
pavg
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
re
l.
 f
re
q
. 
/ 
p
d
f
station  23    n= 185
 x=3501820  y=5384490  z= 513
(d) Histogram - Log-Normal PDF
Figure 6.12.: Observed and theoretical distribution in the linearized space of the average
precipitation amount of wet hours in Holzgerlingen. Fit of three different theo-
retical distributions. Histogram and PDF of the best fitting Log-Normal Distri-
bution.
The distribution of the average precipitation amount of wet hours is explained best
6.5 Analysis of Observed Precipitation Data 79
by the Log-Normal distribution (Fig. 6.12). The Weibull distribution underestimates
the frequencies in the highest events. The Lomax distribution slightly overshoots in
the highest events and also overestimates the probabilities of very low precipitation
amounts.
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
pavg
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
re
l.
 f
re
q
. 
/ 
p
d
f
station  30    n= 148
 x=3482980  y=5347040  z= 661
(a) Histogram - Log-Normal PDF
1.0 0.5 0.0 0.5 1.0 1.5
log(x) observed
1.5
1.0
0.5
0.0
0.5
1.0
1.5
lo
g
(x
) 
th
e
o
re
ti
ca
l
station  30    n= 148
 x=3482980  y=5347040  z= 661
empirical distribution
log-normal
(b) Log-Normal Distribution
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
pavg
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
re
l.
 f
re
q
. 
/ 
p
d
f
station  30    n= 148
 x=3482980  y=5347040  z= 661
(c) Histogram - Lomax PDF
0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008
x*
7
6
5
4
3
2
1
0
1
y
*
station  30    n= 148
 x=3482980  y=5347040  z= 661
empirical CDF
lomax-fit
(d) Lomax Distribution
Figure 6.13.: Comparison of Weibull Distribution and Log-Normal Distribution fitted by
Maximum Likelihood Method to the observed average precipitation of wet
hours from Geislingen Waldhof
Holzgerlingen is again exemplary for the whole of the study region. Only few stations
show minor deviations to the Log-Normal distribution, most of them an underestima-
tion in the highest amounts. Fig. 6.13 displays a comparison of the histograms and the
linearized distributions of Log-Normal and Lomax Distribution at station of Geislin-
gen Waldhof where the fit of the Log-Normal Distribution performs worst. However,
the Log-normal Distribution remains the best choice. The Lomax distribution puts
more weight on the highest precipitation amounts but turns the underestimation into
an overestimation. At the same time, there is a clear deviation in the observed and
theoretical frequencies of the lower values.
80 Generation of Point Rainfall
• Standard Deviation of Precipitation Amounts of Wet Hours
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5
x*
6
5
4
3
2
1
0
1
2
3
y
*
station  23    n= 185
 x=3501820  y=5384490  z= 513
empirical CDF
weibull-fit
(a) Weibull Distribution
0.00000 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012
x*
6
5
4
3
2
1
0
1
y
*
station  23    n= 185
 x=3501820  y=5384490  z= 513
empirical CDF
lomax-fit
(b) Lomax Distribution
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5
log(x) observed
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
lo
g
(x
) 
th
e
o
re
ti
ca
l
station  23    n= 185
 x=3501820  y=5384490  z= 513
empirical distribution
log-normal
(c) Log-Normal Distribution
0 1 2 3 4 5 6 7 8
pstd
0.0
0.2
0.4
0.6
0.8
1.0
re
l.
 f
re
q
. 
/ 
p
d
f
station  23    n= 185
 x=3501820  y=5384490  z= 513
(d) Histogram - Log-Normal PDF
Figure 6.14.: Observed and theoretical distribution in the linearized space of the standard
deviation of precipitation amounts of wet hours. Fit of three different theoreti-
cal distributions. Histogram and PDF of the best fitting Log-Normal Distribu-
tion.
One outcome of the multivariate dependence analysis in Section 6.5.1 is the close rela-
tion between the average and standard deviation of the hourly precipitation amounts
of wet hours. They are very similar in terms of the marginal distribution, too: The best
suited theoretical distribution function for the standard deviation is the Log-Normal
Distribution as well (Fig. 6.14).
• Hourly Precipitation Probability
The monthly frequency of 1h precipitation can be very well represented by a Beta-
Distribution. The fit for the station in Holzgerlingen shown in Fig. 6.15 is represen-
tative for the whole study area. None of the 292 stations exhibits visible deviation
between the CDF F (X) and the observed frequency Fn (x).
6.5 Analysis of Observed Precipitation Data 81
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0
2
4
6
8
10
re
l.
 f
re
q
. 
/ 
p
d
f
station  23    n= 367
 x=3501820  y=5384490  z= 513
(a) Histogram + PDF
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0.0
0.2
0.4
0.6
0.8
1.0
y
station  23    n= 367
 x=3501820  y=5384490  z= 513
empirical CDF
beta-fit
(b) CDF
Figure 6.15.: Observed distribution and fitted Beta Distribution of the monthly 1h precipita-
tion frequency of wet hours at the station in Holzgerlingen
6.5.2.1. Marginal Distributions used for Rainfall Generation
In all of the four parameters the distributions at all 292 rain gauges can be described by
the same theoretical distribution functions. Minor deviations, mostly in the distribution of
the highest values, only occur at a few stations. It is, therefore, reasonable to apply the
respective distribution to all station months at once. Fig. A.1 to Fig. A.8 in the appendix to
this chapter show the histogram and the (linearized) theoretical and observed distributions.
For consistency with other generation parameters the fit is performed season wise.
Table 6.3.: Distribution functions used in the parameter estimation for the rainfall generation
Parameter Distribution Parameter December March-May
Function -February
Hmon Weibull Distribution λ, κ 81.366 1.456 88.616 1.767
P1h Beta-Distribution α, β 2.938 20.919 2.826 22.507
E (Hwet) Log-Normal Distribution µ, σ -0.427 0.340 -0.187 0.346
D (Hwet) Log-Normal Distribution µ, σ -0.436 0.465 -0.093 0.502
Parameter Distribution Parameter June-August September
Function -November
Hmon Weibull Distribution λ, κ 106.643 2.263 83.733 1.665
P1h Beta-Distribution α, β 3.451 33.722 3.026 23.958
E (Hwet) Log-Normal Distribution µ, σ 0.214 0.329 -0.145 0.338
D (Hwet) Log-Normal Distribution µ, σ 0.526 0.474 -0.025 0.487
Fit to the data of all 292 stations, the distributions exhibit about the same behavior as in
the fit to single stations. Fig. A.1 and Fig. A.2 show the histogram, the empirical and the-
oretical distributions of the monthly sum for all four seasons, which are used in the gen-
82 Generation of Point Rainfall
eration scheme. The fit of the Weibull distribution is best during the winter months from
December to February. In spring season the scatter cloud bends concavely, indicating an
underestimation of the highest observed monthly sums. The overall marginal distribution
of the standard deviation and the monthly average (Fig. A.3 to Fig. A.6) show a slight un-
derestimation in the highest values, especially during the winter months from December to
February. However, the linearized plots overemphasis these deviations since the density of
points in the middle of the point cloud is several times higher than near the edges.
The fit of the Beta Distribution to the 1h rainfall frequencies is good in all four seasons.
No systematic deviations can be found neither in the histogram (Fig. A.7) nor in the CDF
(Fig. A.8).
6.6 The Generation Algorithm for 1h Precipitation Time Series 83
6.6. The Generation Algorithm for 1h Precipitation Time Series
In Section 6.5.1 it was shown that the dependence structure between the observed monthly
sum, the 1h precipitation frequency, the monthly average and standard deviation of wet
hours can be approximated by a Gaussian Copula. In the last section a universal marginal
distribution for the whole study area was deduced for each of the parameters. Based on
these findings, the parameter estimation of the NiedSim algorithm will be set up anew.
The generation parameters governing the distribution of 1h precipitation values will be
conditioned on the time series of observed monthly sums.
6.6.1. Parameter Estimation for the Initial Time Series
It is assumed that the 1h rainfall amounts follow a Weibull Distributions. Its parameters
λ (Hwet) and κ (Hwet) are calculated for each month from stochastic values of E (Hwet) and
D (Hwet) according to Eq. (6.13) and Eq. (6.14). Additionally, the value of the rainfall proba-
bility P1h has to be estimated which governs the number of wet hours per month.
1. Starting point is the time series of monthly rainfall sums Hmon at the target location
of the simulation. If the monthly sum is not available, Hmon is estimated from sur-
rounding stations by External Drift Kriging (Eq. (5.11)) with the smoothed altitude as
external information.
2. This target valueHmon is transformed into the copula spaceU of uniformly distributed
values between 0 and 1 by its marginal distribution F :
umon = F (Hmon) (6.63)
umon can be seen as the non-exceedance probability of Hmon in reference to the whole
study region and simulation time period. The parameters of the marginal distribution
for the monthly sum can be found in Table 6.3.
3. The non-exceedance probabilities of 1h rainfall probability UP , expectation of the
mean UH and expectation of the standard deviation Us are drawn conditioned on umon
from the four dimensional Gaussian copula describing the parameter interdependen-
cies. Drawing from a Gaussian copula is identical to drawing values from a correlation
matrix Σ in the Standard Normal transformed space. In the following, the index con-
vention for the parameters is 1=ˆHmon, 2=ˆP1h, 3=ˆE (Hwet) and 4=ˆD (Hwet).
z|a ∼ N
(
µ|a,Σ|a
)
(6.64)
a = Φ−1 (umon) (6.65)
where z|a =
[
zi|a, i = 2, 3, 4
]T correlated conditioned random numbers
in the Standard Normal transformed space
N (µ,Σ) multivariate normal distribution
with mean vector µ and covariance matrix Σ
µ|a =
[
µi|a , i = 2, 3, 4
]
mean vector conditioned on x1 = a
Σ|a covariance matrix conditioned on x1 = a
84 Generation of Point Rainfall
The conditioned mean vector µ|a is calculated by multiplying the values r1j , j = 2, 3, 4
of the first column of the multivariate correlation matrix (Table 6.2) with the condition-
ing value a. The conditioned correlation matrix Σ|a can be calculated by inverting the
overall correlation matrix Σ according to Table 6.2, dropping the row i = 1 and col-
umn j = 1 belonging to Hmon and inverting back. (For more mathematical details, see
Section A.2 in the appendix).
The conditioned, Standard Normal transformed random numbers z|a are drawn by
means of a vector z = of uncorrelated random numbers and lower triangle matrix L
of the Cholesky decomposition of Σ|a
z|a = µ|a + L · z (6.66)
Σ|a = LLT (6.67)
4. The estimated Standard Normal transformed values x∗i representing the rainfall prob-
ability, as well as the average and the standard deviation of precipitation amounts of
wet hours are back-transformed to absolute values in two steps.
ui = Φ
(
zi|a
)
(6.68)
x∗i = F
−1
i (ui) (6.69)
where x∗i estimated target value of parameter i = 2, 3, 4
Φ Standard Normal Distribution Function
zi|a conditioned random number in the Standard Normal
transformed space drawn for parameter i
ui uniform transformed of zi|a
F−1i inverse of the theoretical distribution function of parameter i
according to Table 6.3
5. The parameters λ (Hwet) and κ (Hwet) of the Weibull Distribution for the 1h rainfall
amounts of wet hours are calculated from the drawn statisticsE∗ (Hwet) andD∗ (Hwet)
by the method of moment (Eq. (6.13) and Eq. (6.14)).
To account for seasonal effects, the parameters of the marginal distribution (Table 6.3) and
the Gaussian copula (Table 6.2) are depending on the season (December to February, March
to May, June to August, September to November)
6.6.2. Rainfall Generation
The generation is performed year by year.
1. Given the monthly series of rainfall probabilities P1h and the Weibull Distribution pa-
rameters λ (Hwet) and κ (Hwet) according to the proceeding described in the last sec-
tion, an initial time series is generated. For each time-step in the year
6.6 The Generation Algorithm for 1h Precipitation Time Series 85
a) a random number u is drawn from a uniform distribution in the interval [0, 1]. If
u is not greater than the rainfall probability of the current month (u ≤ P1h), the
current time step is assumed to be a rainy hour.
b) If the current hour is wet, another random number u is drawn. By the inverse of
the Weibull Distribution with the parameters λ (Hwet) and κ (Hwet) of the current
month, u is transformed into a 1h precipitation amount H (t).
H (t) = F−1 (u, λ, κ) (6.70)
c) 1a to 1b are repeated for the next time step.
The initial time series exhibits the right distribution of the 1h precipitation amounts
and follows the expectation of the monthly precipitation sums. It is passed to the
permutation scheme for the optimization of the temporal sequence of values.
2. The optimization is performed by a Simulated Annealing algorithm (Aarts and van
Laarhoven, 1989). It is based on an objective function O that measures the deviations
of the current state from the statistical target state for a perfect simulation. The better
the simulated time series reproduces the statistical features expected for a measured
time series at the same location, the smaller the value of O.
O = w1O1 + w2O2 + w3O3 + . . .+ wnOn → min! (6.71)
Oi = (ρi,sim − ρi,tar)2 (6.72)
where ρi,sim value of the simulated series for the ith statistical feature
ρi,tar target value for the ith statistical feature
The weights w (i) of the objective function parts are chosen depending on possible
range in Oi. By the weights w (i) it is assured that the different statistics i have more
or less the same influence on the overall objective function O.
The Simulated Annealing algorithm consists of the following steps:
a) Randomly choose one value of the time series i and a second value j within the
same month so that
max (xi, xj) > 0
(If both values are zero, the exchange has no effect.)
Exchange the two values and calculate the state of the objective function for the
altered series O∗.
b) If the exchange improves the state of the objective function
O∗ < Oold
keep the changes.
Oold = O
∗
Go to 2a and perform the next exchange.
86 Generation of Point Rainfall
c) If the swap worsens the time series (O∗ > Oold), it is kept anyhow with a certain
probability. This feature is implemented to avoid that the optimization gets stuck
in a local optimum. The probability of keeping a “bad” swap depends on the
difference in the objective function and an optimization parameter T called the
annealing temperature.
Pkeep = exp
(
O∗ −Oold
T
)
(6.73)
For each bad swap, a uniform random number U ∈ [0, 1] is drawn. If U ≤ Pkeep,
the swap is kept. The algorithm continues at 2a by choosing the next two time
steps for a swap.
d) The steps 2a to 2c are repeated nT times. Then the temperature is decreased
Tnew = dT · Told (6.74)
where dT < 1.
e) Each temperature decrease reduces the probability of accepting a “bad” swap.
At the beginning of the simulation the system has much freedom in performing
different swaps. Later mostly “good” swaps are accepted. The generation stops
when the number of performed swaps drops below a pre-defined value (e. g. one
swap in 10000 trials).
6.6.2.1. Statistical Parameters in the Objective Function
The different parts Oi of the objective function evaluating the simulated time series are:
• 24 h extreme value according to KOSTRA statistics
The official German extreme value statistics KOSTRA (Bartels et al., 2005) assumes that
the extreme precipitations of the annual series (the highest precipitations of each year)
follows a Gumbel Distribution. The parameters of this Gumbel distribution can be re-
established by the yearly and 100 yearly extreme value from KOSTRA. The restored
distribution is used to draw the target value for the highest 24h precipitation sum in
the simulation year.
• scaling of the statistical moments between different aggregation levels
The kth statistical moment mk of an empirical distribution is defined as:
mk =
1
n
n∑
i=1
x (i) (6.75)
where n number of values in the distribution (here: number of time steps
in the simulated time series for the current year)
6.6 The Generation Algorithm for 1h Precipitation Time Series 87
The characteristics in the scaling of the kth statistical moments over different aggrega-
tion times ∆t are regarded. ∆t ranges from 1h to 24h. The scaling of the kth moments
is described by a non-linear regression function:
mk (∆t) = a ·∆bkt (6.76)
If the simulated time series exhibit the right scaling behavior, the exponent bk of the
simulated data will correspond to the target value. The focus is not on the absolute
values but on the relation of the values from different aggregations. Therefore, the pa-
rameter a is ignored and only b is considered as a target value in the objective function.
• autocorrelation on different aggregation levels
r (l) =
∑n−l
i=1 w (i) · (xi − x¯) (xi+l − x¯)∑n
i=1 · (xi − x¯)
(6.77)
where w (i) a seasonal weighting factor
n number of values in the time series (here the number of time steps
in the simulated time series of the current year).
k lag of the autocorrelation.
The autocorrelation is a measure for the memory of the time series. In the generation
the autocorrelation to the preceding time step is considered, therefore k = 1. It is
calculated for six different aggregations of the yearly time series x (∆t) with ∆t =
1h, 2h, 3h, 6h, 12h and 24h. The weighting factor is set to w = 1.
Additionally, a seasonally weighted autocorrelation is calculated for the time series of
the 1h rainfall amounts, the weighting factor w (i) depends on the relative position of
the time step in the year:
w (i) = 1 + sin
(
i
nyear
· 2pi + 1
6
)
(6.78)
The target values of the autocorrelation depend on the generation year.
• average daily precipitation amount and frequency for three different weather type
classes
The weather classes are found by a Fuzzy Logic Based classification algorithm that
divides the days by their mean sea level pressure patterns (see Section 7.5.2). The
originally 17 different weather types are regrouped into three classes: dry, medium
and humid. The average of daily precipitation amount and frequency for the three
groups can be found in Table 7.6 to Table 7.8. The statistics are calculated from 575
daily rain gauges (Section 3.3).
Since the monthly precipitation sums do not correspond to the observed values but
are drawn randomly, the average precipitation is not considered as absolute values. It
is normed by the simulated monthly rainfall sum.
88 Generation of Point Rainfall
6.7. Comparison of Simulated and Observed Rainfall Time
Series
The performance of the generator is tested by a comparison of the statistical characteristics
of observed and simulated time series at the station in Holzgerlingen (Gaus-Krüger 3 co-
ordinates of X=3501820, Y=5384490, see Section 3.3). The time period of the comparison
ranges from 1977 to 1992.
0 50 100 150 200 250 300 350
[mm/month]
0
10
20
30
40
50
x= 3501820 y= 5384490
monthly rainfall sum, 1977 - 1992
observed(NaN= 4.2%)
simulated
(a) Histogram
0 50 100 150 200 250 300
[mm/month]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
monthly rainfall sum, 1977 - 1992
observed
simulated
(b) CDF
Figure 6.16.: Empirical Distribution of the simulated and observed monthly precipitation
sum.
Since the generation is conditioned on the monthly precipitation sums, the generation re-
stores the distribution of the monthly precipitation sums (Fig. 6.16) as well as the average
yearly sum (727mm in the simulations, 735mm observed). The small fluctuations are due
to sampling variability.
The comparison of hourly statistics presented Table 6.4 reflects the accuracy in the distri-
bution of the 1h time series. The daily statistic reflect not only the distribution but also
the temporal sequence in the generated time series. The calculation of hourly mean, stan-
dard deviation and skewness is based on all values exceeding a threshold of 0.1mm. The
threshold for daily values is 0.5mm.
Over all, the statistics of the hourly time series are modeled correctly Table 6.4. Signifi-
cant deviations concern the rainfall frequency p>0.1mm and the average precipitation of wet
hours. Since the generation respects the yearly sum, the underestimation of the rainfall fre-
quency results in an overestimation of the average hourly rainfall. The deviations, however,
only concern the smallest values, the frequencies with higher threshold are all modeled cor-
rectly.
In daily aggregation too the main difference between the observed and simulated time series
is found in the rainfall frequency. The fewer number of rain days indicates that the simu-
lated rainfall events are more concentrated than in reality. Another deviation concerns the
6.7 Comparison of Simulated and Observed Rainfall Time Series 89
Table 6.4.: Basic statistics of the simulated and observed time series in Holzgerlingen, 1977
to 1992
hourly values daily precipitation sum
simulated observed simulated observed
mean [mm] 1.108 0.879 8.655 5.432
stdev. [mm] 1.498 1.382 7.485 6.403
skewness [-] 7.643 6.605 0.870 3.199
p>0.1mm [-] 7.34% 9.04% 32.3% 43.4%
p>0.5mm [-] 4.34% 4.32% 22.6% 36.8%
p>2mm [-] 1.07% 0.85% 16.3% 24.4%
p>5mm [-] 0.164% 0.153% 12.6% 12.6%
p>10mm [-] 0.017% 0.033% 9.22% 5.39%
p>20mm [-] 0.0004% 0.0005% 0.84% 1.37%
Hmax [mm] 42.9 27.2 42.9 69.2
skewness. The frequency of small daily sums is underestimated, the frequency of p>10mm is
overestimated.
As a representation of the extreme value behavior the empirical distribution of the ten high-
est precipitation amounts is calculated (Fig. 6.17). On hourly aggregation the distribution of
the simulated time series is reasonable, although moderate extremes (between 10mm and
20mm) are slightly underestimated. The same is true for the six hourly extremes. On daily
scale, the new generation scheme exhibits a slight lack of variability. The frequency of val-
ues around 15mm is overestimated. The most extreme events, on the other hand, are lower
than the observed. The lack of very high values is possibly caused by the KOSTRA statis-
tics, which restricts the depth of the 24h precipitation sums (see Section 6.6.2.1). This is
supported by the fact that the simulated maximum hourly and daily value are the same.
Calculating the annual series of 1h values for each month separately reveals the seasonal
variability in the simulated time series. Fig. 6.18 displays the distributions for January and
July. The generation is able to distinguish between the seasons. In both months, the distri-
bution is close to the observed. The representation of seasonal variability is a new feature.
In NiedSim the same distribution is applied during the whole year.
Fig. 6.19 presents the distribution of the wet spell lengths. Gaps of one hour are not consid-
ered as dry spells and therefore do not interrupt a wet spell. This threshold is introduced
to avoid that longer, dependent precipitation events are interrupted due to a short break
in rainfall. To avoid a high influence of measurement errors, only hours with no less than
0.1mm rainfall are considered as wet.
The generated time series counts 2515 wet spells which is in the same range as the observed
time series with 2380 wet spells. The frequencies are generally captured well,except of very
short wet spells that are too frequent. The simulated time series slightly underestimates
the duration of the longest wet spells (Fig. 6.20a). However, the distribution is very sensi-
tive to the definition of a wet spell. If gaps up to three hours are accepted, the duration is
90 Generation of Point Rainfall
0 10 20 30 40 50
[mm/h]
0
10
20
30
40
50
60
70
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(a) Histogram - 1 h
0 10 20 30 40 50
[mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) CDF - 1 h
0 10 20 30 40 50
[mm/6h]
0
10
20
30
40
50
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(c) Histogram - 6 h
0 10 20 30 40 50
[mm/6h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(d) CDF - 6 h
0 10 20 30 40 50 60 70 80
[mm/day]
0
20
40
60
80
100
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(e) Histogram - 24 h
0 10 20 30 40 50 60 70
[mm/day]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(f) CDF - 24 h
Figure 6.17.: Empirical distribution of the ten highest rainfall amounts in the observed and
the simulated time series
6.7 Comparison of Simulated and Observed Rainfall Time Series 91
0 5 10 15 20 25 30 35
[mm/h]
0
10
20
30
40
50
60
70
80
x= 3501820 y= 5384490
annual series  - 10 members per year
Jan observed 1977 - 1992
NaN= 6.3%
Jan simulated 1977 - 1992
Jul observed 1977 - 1992
NaN= 0.0%
Jul simulated 1977 - 1992
(a) Histogram
0 5 10 15 20 25 30
[mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series  - 10 members per year
Jan observed 1977 - 1992
NaN= 6.3%
Jan simulated 1977 - 1992
Jul observed 1977 - 1992
NaN= 0.0%
Jul simulated 1977 - 1992
(b) CDF
Figure 6.18.: Empirical distribution of the ten highest 1h rainfall amounts in the observed
and simulated time series during January and July for the station in Holzger-
lingen
0 10 20 30 40 50
length [h]
0
200
400
600
800
1000
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(a) Histogram
0 10 20 30 40 50
length [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) CDF
Figure 6.19.: Empirical distribution of the wet spell length in hourly resolution. Gaps up to
a length of 1h do not interrupt a wet spell.
overestimated (Fig. 6.20b).
A dry spell is defined as a consecutive sequence of dry hours that is not interrupted by a
precipitation event with more than 0.5mm. The simulated distribution of dry spell lengths
is displayed in Fig. 6.21 as histogram, logarithmic histogram and CDF. The simulated dis-
tribution of dry spell lengths is more skewed than the observed. The shortest rain breaks of
only one to two hours are overestimated, as well as long dry spells of more than seven days.
The distribution of the longest 100 dry spells is shifted by about four to five days (Fig. 6.22).
Whereas the length of the longest reported dry spell is comparable in the observations and
the simulations.
92 Generation of Point Rainfall
0 10 20 30 40 50
length [h]
0
5
10
15
20
25
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(a) Histogram - 1h gabs
0 10 20 30 40 50 60 70 80
length [h]
0
5
10
15
20
25
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) Histogram - 3h gabs
Figure 6.20.: Empirical distribution of the 100 longest wet spells in hourly resolution. Gaps
up to a length of 1h (left) / 3h (right) do not interrupt a wet spell.
0 200 400 600 800 1000
length [h]
0
500
1000
1500
2000
2500
3000
3500
4000
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(a) Histogram
0 200 400 600 800 1000
length [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) CDF
Figure 6.21.: Empirical distribution of the dry spell length in hourly resolution. Rainfall
events with less ore equal 0.5mm cumulated precipitation sum do not inter-
rupt a dry spell.
It has to be noticed that the definitions of wet and dry spells are not complementary. If it
was so, the number of wet and dry spells would differ at most by one. In this analysis an
overlap is accepted. There are indeed wet hours that do not interrupt a dry spell but are
counted within a wet spell.
6.8 Conclusions 93
0 200 400 600 800 1000
length [h]
0
5
10
15
20
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(a) Histogram
0 200 400 600 800
length [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) CDF
Figure 6.22.: Empirical distribution of the 100 longest dry spells in hourly resolution. Rain-
fall events with less ore equal 0.5mm cumulated precipitation sum do not in-
terrupt a dry spell.
6.8. Conclusions
The new generation algorithm is based on the assumption that the dependence between
monthly precipitation sum, rainfall probability and the statistical moments of wet hours is
constant over the study regions. It is modeled by a Gaussian Copula. The local effects are
expressed by the marginal distributions of the parameters. Since all statistics are related to
the monthly precipitation sum, the generation is conditioned on observed monthly sums.
While the dependence structure shows some minor deviation to a Gaussian copula, the fit
of the marginal distributions is very close.
The developed generation algorithm is able to represent the statistical characteristics of pre-
cipitation reasonably well. The restriction to one single distribution function, compared to
NiedSim, does not lead to reduced variability. On all tested aggregations the extreme values
are represented reasonably well. The strongest deviation to observed precipitation is found
in the rainfall frequency. It shows that a correct estimation of the rainfall probability is cru-
cial for the performance of the model. Unfortunately, the assessment of observed rainfall
frequencies is strongly effected by measurement errors. It depends on a correct assessment
of the smallest values, which exhibit the highest relative measurement errors.
The analysis of wet and dry spell lengths proves that the generation is able to reproduce
the temporal persistence of observed rainfall time series. Deviations concern an overestima-
tion in the frequency of the shortest rainfall events and at the same time of very short dry
spells. However, longer wet and dry spells are modeled correctly. Hence, it is not subject to
“overdispersion” unlike most conceptual models (see Section 4.2 in Chapter 4 and Section
5.3). The opposite is the case. The rainfall events cluster more than in reality, which leads to
an underestimation of the number of rain day.
7. Spatial Generation
Hydraulic sewage system models require a correct estimation of the total storm water in-
flow, which is the sum of precipitation volume that falls on sealed areas. An exact estima-
tion would only be possible if the areal precipitation was known. However, time series of
rain gauge measurements, as well as synthetic time series generated by NiedSim are point
information. In most sewage system models, like for example in KOSIM (ITWH, 2009), the
assigned time series is transformed into aerial precipitation of the same intensity. This ap-
proximation is only valid for small catchments. Spatial effects, e. g. that only a part of the
network is affected by a storm event or a time shift in the peak of rainfall intensity over
different subareas, are not considered. Modeling larger areas, it will lead to errors, typically
an overestimation of the storm water discharges.
If more than one precipitation time series is required and no rain gauge data is available, the
only alternative is the application of several simulated time series. The operational Nied-
Sim generator, however, is not able to consider spatial correlation. Therefore, NiedSim time
series of two nearby locations exhibit very similar statistical characteristics (for example the
extreme value statistics) but independent temporal sequences. For example, they do not
feature simultaneous extreme events. For realistic hydraulic simulations of larger sewage
channel networks simultaneous, spatially correlated synthetic rainfall time series are re-
quired.
7.1. A High Dimensional Problem
The NiedSim approach cannot be applied without modifications to the generation of several,
spatially correlated time series. A permutation based generation reaches its limits if it is
used for the generation of simultaneous time series. The reason will be explained in a small
example.
If it is supposed that there is a time series with three entries that has to be optimized by
ressampling, the number of possible sequences is factorial three:
ncomb = 3! (7.1)
= 3 · 2 · 1 = 6 (7.2)
One can choose among three values for the first position in the time series, then among the
two remaining for the second position. The optimization algorithm has to choose the best
of these six different sequences.
7.2 Generation Principle 95
If the task is to generate simultaneous time series, the intercorrelations have to be taken
into account. Therefore, it is not sufficient to search for the best permutations individually.
Instead, the optimum hast to be found in the combination of them all. This increases the
number of possible combinations to
ncomb = (3!)
nstat (7.3)
= 6nstat (7.4)
where nstat is the number of stations. Thus, for the simulation of two stations, this leads
to 36 possible combinations, for three stations 216, for four stations as much as 1296. If it is
assumed that optimization algorithm has to try a constant (very small) portion of all possible
combinations to find the optimum sequence, the required CPU time increases by the power
of the number of stations.
The factorial is an extremely fast growing function. For the simulation of an hourly time
series of one day, assuming no identical values (e.g. zero values), one gets 24! = 6.2 · 1022
combinations, for the simulation of a one week long time series in hourly data 2.5 · 10302
different combinations. The simulation algorithm that was set up for the generation of one
single time series (Chapter 6) is based on month-wise resampling. The number of possible
combinations for one month in hourly values is 720! (assuming no identical values). This
number is so huge that it cannot be represented in double precision floating point. For
the simulation of simultaneous time series, it is raised to the power of nstat. As a result,
the number of possible combination becomes unfeasible when the number of simultaneous
stations is increasing.
7.2. Generation Principle
It is obvious that the number of possible combinations has to be drastically restricted. This
is realized by a two step approach. At first, daily precipitation sums are simulated and
optimized by Simulated Annealing. Then the daily time series are disaggregated into 24
one hour values.
1. Generation of Spatial Correlated Daily Time Series
The generation of daily time series follows the same principle as the single site simula-
tion (see Section 6.6 in Chapter 6). Initial time series for the nstat locations are defined
by the rainfall probability as well as the average and the standard deviation of precip-
itation amounts during wet time steps – but now on a 24h temporal resolution. The
statistical moments define the parameters of a Weibull Distribution where the daily
precipitation values are drawn from.
The parameter interdependencies at the nstat target locations are represented by a
Gaussian copula. Compared to the simulation at a single location, the correlation ma-
trix is extended by the interdependencies of the generation parameters at the different
locations. The question is for example “How does the average daily precipitation at
station A depend on the monthly sum at station B?”.
96 Spatial Generation
The optimization of the time series by simulated Annealing is performed for all sta-
tions simultaneously. The objective function consists of the statistical target values at
each location and information on the spatial dependence among the stations.
2. Disaggregation into Spatial Correlated Hourly Values
The optimized daily time series is disaggregated by randomly distributing the daily
precipitation sum of each day among the 24 hourly values of that day. In the next
step, an optimization is performed. Small precipitation increments are exchanged by
subtracting them from the value of one time step and adding them to another. To
avoid that the optimized daily sums are altered, the chosen time steps are restricted to
be from the same day in the year. The objective function of the optimization evaluates
the statistical characteristics at each location, as well as the spatial dependence on the
hourly time scale.
The statistics that are used to represent the spatial dependencies in the generation are de-
scribed in Section 7.5. Next, the developed generation algorithm is explained (Section 7.6).
Finally, the statistical characteristics of simulated and observed time series are compared in
terms of the statistical characteristics (Section 7.7).
7.3. Observed Multivariate Dependencies in Daily Precipitation
Data
Like in the single site generation, the simulation will be conditioned on the monthly pre-
cipitation sums at the chosen generation locations. Therefore, an estimation of the bivariate
Gaussian dependencies of the monthly sum Hmon, the rainfall probability P24h, the average
rainfall E (Hwet) and the standard deviation D (Hwet) on daily time scale is required. The
statistics are estimated by the observed frequency of rain days p24h, as well as the average
precipitation sum H¯wet and the standard deviation swet of wet day precipitation. In the
coarser daily resolution one can benefit from the broader data base of the 575 daily stations
(Section 3.3).
The interdependencies are analyzed for monthly censored data. The values are ranked
(Eq. (6.46)) and transformed into the [0, 1]-space of uniform distributions by the non-
exceedance frequencies (Eq. (6.54)). Then the bivariate copula densities are regarded and
compared to the Gaussian copula.
Table 7.1 gives an overview of the rank correlation ρ according to Eq. (6.46) and the sym-
metry measure sym according to Eq. (6.62). For all parameter combinations the monthly
differences in rank correlations and symmetry are smaller than in the hourly data. It means
that daily precipitation values are less influenced by seasonal effects than hourly values.
Nevertheless, the rank correlations ρ are generally lower in the summer months with con-
vective precipitation, but not all parameter combinations follow a clear seasonal cycle.
Compared to hourly resolution, the correlation between monthly sum Hmon and average
precipitation H¯wet increases. it is an effect of the temporal resolution. The monthly sum
7.3 Observed Multivariate Dependencies in Daily Precipitation Data 97
Table 7.1.: Rank correlation and symmetry of the empirical copula from monthly data for
bivariate combinations of all parameters; data in daily resolution
Month Hmon − p24h Hmon − H¯wet Hmon − swet
ρ Sym ρ Sym ρ Sym
January 0.71 -0.00562 0.87 0.00101 0.83 0.00027
February 0.70 -0.00498 0.84 0.00189 0.77 0.00173
March 0.69 -0.00039 0.88 0.00072 0.82 -0.00007
April 0.73 0.00410 0.80 0.00022 0.72 -0.00083
May 0.71 -0.00331 0.85 -0.00012 0.76 -0.00064
June 0.61 -0.00408 0.78 -0.00020 0.67 -0.00115
July 0.59 -0.00431 0.78 -0.00081 0.69 -0.00186
August 0.69 -0.00215 0.84 -0.00011 0.74 -0.00162
September 0.71 -0.00058 0.72 -0.00118 0.68 -0.00180
October 0.67 -0.00033 0.81 -0.00179 0.79 -0.00198
November 0.60 -0.00268 0.84 0.00006 0.76 -0.00080
December 0.65 -0.00128 0.87 0.00105 0.82 0.00062
Month p24h − H¯wet p24h − swet H¯wet − swet
ρ Sym ρ Sym ρ Sym
January 0.33 -0.00386 0.37 -0.00501 0.90 0.00030
February 0.26 -0.00182 0.25 -0.00216 0.90 0.00062
March 0.31 0.00133 0.32 -0.00017 0.90 0.00025
April 0.22 0.00217 0.21 0.00055 0.88 0.00011
May 0.26 -0.00328 0.27 -0.00248 0.86 -0.00103
June 0.04 -0.00368 0.06 -0.00341 0.84 -0.00102
July 0.02 -0.00543 0.04 -0.00621 0.85 -0.00064
August 0.23 -0.00241 0.26 -0.00418 0.82 -0.00043
September 0.07 -0.00132 0.16 -0.00291 0.84 0.00039
October 0.17 -0.00265 0.27 -0.00280 0.88 -0.00041
November 0.13 -0.00184 0.14 -0.00247 0.88 0.00044
December 0.24 0.00041 0.30 -0.00026 0.90 -0.00008
consists of only 28 to 31 daily values and thus the relation exhibits a lower freedom than
in hourly resolution with 672 to 744 values per month. On the hourly scale precipitation
frequency and the statistical moments were almost uncorrelated. On daily scale there is a
significant positive correlation, except for May and June.
The respective copula densities are presented in Fig. B.1 to Fig. B.6 in the appendix to this
chapter. Each figure features the plots of the two most different months in terms of the
respective parameter combination. Generally, the deviations to bivariate Gaussian copulas
are low. The combinations p24h and H¯wet, as well as p24h and swet , however, are asymmetric
with a high density in the lower left corner (Fig. B.4 and Fig. B.5): In a month with only one
or two wet days, the variations in the daily rainfall amount are very limited.
98 Spatial Generation
Table 7.2.: Correlation in the Standard Normal transformed space of monthly precipitation
sum, daily rainfall probability as well as monthly average and standard deviation
of wet day precipitation amounts applied in the simulation of daily precipitation
amounts.
December to February
Hmon P24h E (Hwet) D (Hwet)
Hmon 1.000 0.710 0.875 0.831
P24h 0.710 1.000 0.310 0.343
E (Hwet) 0.875 0.310 1.000 0.911
D (Hwet) 0.831 0.343 0.911 1.000
March to May
Hmon P24h E (Hwet) D (Hwet)
Hmon 1.000 0.696 0.867 0.797
P24h 0.696 1.000 0.255 0.262
E (Hwet) 0.867 0.255 1.000 0.900
D (Hwet) 0.797 0.262 0.900 1.000
June to August
Hmon P24h E (Hwet) D (Hwet)
Hmon 1.000 0.669 0.813 0.716
P24h 0.669 1.000 0.132 0.154
E (Hwet) 0.813 0.132 1.000 0.852
D (Hwet) 0.716 0.154 0.852 1.000
September to November
Hmon P24h E (Hwet) D (Hwet)
Hmon 1.000 0.672 0.796 0.761
P24h 0.672 1.000 0.103 0.183
E (Hwet) 0.796 0.103 1.000 0.884
D (Hwet) 0.761 0.183 0.884 1.000
As in the hourly data, the ranks in all parameters are positively correlated to the ranks in the
monthly precipitation sum. Therefore, it is possible to condition the generation parameters
on the monthly sum. Table 7.2 displays the bivariate correlation coefficients in the Standard
Normal transformed space that are used to model the Gaussian copula in the rainfall gener-
ation. The deviations to Gaussian copula density in the parameter combination p24h versus
H¯wet and p24h versus swet are of minor importance. If the rainfall probability is anyhow al-
most zero, the expectation of the average and standard deviation used for the simulation
have a very low influence on the result.
7.4 Marginal Distributions of Daily Precipitation Data 99
7.4. Marginal Distributions of Daily Precipitation Data
5 10 15 20 25 30
pavg
0.00
0.05
0.10
0.15
0.20
0.25
0.30
re
l.
 f
re
q
. 
/ 
p
d
f
station 220    n= 525
 x=3504382  y=5387828  z= 431
(a) Histogram - Log-Normal PDF
0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(x) observed
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
lo
g
(x
) 
th
e
o
re
ti
c
a
l
station 220    n= 525
 x=3504382  y=5387828  z= 431
(b) Log-Normal Distribution
Figure 7.1.: Observed and theoretical Log-Normal Distribution in the linearized space of the
average wet day precipitation amount in Schönaich.
0 5 10 15 20 25 30
pstd
0.00
0.05
0.10
0.15
0.20
0.25
re
l.
 f
re
q
. 
/ 
p
d
f
station 220    n= 525
 x=3504382  y=5387828  z= 431
(a) Histogram - Log-Normal PDF
2 1 0 1 2 3 4
log(x) observed
2
1
0
1
2
3
4
lo
g
(x
) 
th
e
o
re
ti
c
a
l
station 220    n= 525
 x=3504382  y=5387828  z= 431
(b) Log-Normal Distribution
Figure 7.2.: Observed and theoretical Log-Normal Distribution in the linearized space of the
standard deviation of the wet day precipitation amount in Schönaich.
The marginal distributions of the new statistical parameters P24h, E (Hwet) and D (Hwet)
have to be estimated. They are used for the transformation into the copuloa space and for
the back transformation into real world values.
Like in hourly resolution the average and standard deviation of wet day precipitation show
right skewed empirical distributions at all 575 stations. Therefore, the same theoretical dis-
tributions functions as in Section 6.5.2 are tested: Weibull Distribution, Log-Normal Distri-
bution and Lomax Distribution (Section 6.3). It is found that the Log-Normal distribution
100 Spatial Generation
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
re
l.
 f
re
q
. 
/ 
p
d
f
station 220    n= 525
 x=3504382  y=5387828  z= 431
(a) Histogram
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.2
0.4
0.6
0.8
1.0
y
station 220    n= 525
 x=3504382  y=5387828  z= 431
empirical CDF
beta-fit
(b) Log-Normal Distribution
Figure 7.3.: Observed and theoretical Beta Distribution of the rainfall frequency in daily res-
olution in Schönaich.
is suited best for the description of the average wet day precipitation as well as for the
standard deviation of wet day precipitation. The rainfall probability is described by a Beta-
distribution (Section 6.3.4).
Fig. 7.1, Fig. 7.2 and Fig. 7.3 show the empirical and theoretical distribution in Schönaich as
an example. This station is situated four kilometers Northeast of Holzgerlingen, so that the
results of the hourly and daily fit can be compared.
The histograms and CDFs of the seasonal fit to all daily precipitation stations can be found
in the appendix to this chapter (Section B.2). The fit for the average and standard deviation
of wet day precipitation is closer to the observed than the fit for the hourly data, especially
in the highest values (compare Fig. A.4 with Fig. B.8 and Fig. A.6 with Fig. B.10). Deviations
only concern the lowest values.
The fit of the Beta distribution, on the other hand, is better for the hourly data. This, how-
ever, is a sampling effect due to the limited number of possible values in the observed fre-
quencies. E. g. for a month with 31 days and a rainfall probability of P24h = 0.95 there is a
probability of 20% that all days are wet and that the observed rainfall frequency p24h is one.
Table 7.3 resumes the parameters of the theoretical distribution function used to describe the
marginal distributions in the generation of simultaneous daily time series. The distribution
of the monthly sum is unchanged as it does not depend on the temporal resolution of the
observed data.
7.5 Representation of Spatial Dependencies 101
Table 7.3.: Distribution functions used in the parameter estimation for the generation of si-
multaneous daily rainfall time series
Parameter Distribution Parameter December March-May
Function -February
Hmon Weibull Distribution λ, κ 81.366 1.456 88.616 1.767
P24h Beta-Distribution α, β 3.700 3.418 4.807 4.759
E (Hwet) Log-Normal Distribution µ, σ 1.373 0.573 1.486 0.483
D (Hwet) Log-Normal Distribution µ, σ 1.448 0.628 1.541 0.563
Parameter Distribution Parameter June-August September
Function -November
Hmon Weibull Distribution λ, κ 106.643 2.263 83.733 1.665
P24h Beta-Distribution α, β 5.525 6.440 3.585 4.186
E (Hwet) Log-Normal Distribution µ, σ 1.795 0.413 1.577 0.513
D (Hwet) Log-Normal Distribution µ, σ 1.866 0.496 1.657 0.560
0 10 20 30 40 50
distance [km]
0
200
400
600
800
1000
1200
1400
1600
1800
a
b
so
lu
te
 f
re
q
u
e
n
cy
Figure 7.4.: Absolute frequency of the distances between 575 daily precipitation stations
7.5. Representation of Spatial Dependencies
7.5.1. Observed Interdependencies of Statistical Parameters at Different
Measurement Locations
Exploiting the results of the last section, it is possible to draw interdependent values of daily
rainfall probability P24h, of expected average wet day precipitationE (Hwet) and of expected
standard deviation of wet day precipitation D (Hwet) that are conditioned on the monthly
sum Hmon. With these information, it is possible to set up a generation scheme for daily
precipitation time series analogous to the scheme for hourly precipitation described in the
102 Spatial Generation
last chapter. For the generation of simultaneous time series, however, the information is not
sufficient yet. The missing link is, how the statistical parameters at different stations are
influencing each other.
The four statistical parameters Hmon, P24h, E (Hwet) and D (Hwet) at different locations are
interdependent. If for example the rainfall frequency is high at one station, it will be high at
any neighboring station too. Besides, the information of a high rain day frequency does not
only condition the rainfall frequency at other station. Due to the interdependencies of the
statistics, the distribution of the average or standard deviation at the surrounding stations
will react too.
The strength of the rank correlation between two stations depends on the parameter combi-
nation considered, the distance between the two stations and the season. Fig. 7.5 and Fig. 7.6
display the rank correlation of two of parameter combinations as a function of the distance.
The graphs of the remaining parameter combinations can be found in the appendix to this
chapter (Fig. B.13 to Fig. B.20). Each station combination in the data set of daily stations at-
tributes one correlation value to each of the diagrams. In total there are 575× 575 = 330625
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
4
0
8
0
1
2
0
1
6
0
2
0
0
2
4
0
2
8
0
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
3
0
6
0
9
0
1
2
0
1
5
0
1
8
0
2
1
0
2
4
0
2
7
0
nb of values per cell
(b) June to August
Figure 7.5.: Observed rank correlation between the rain day frequency at two stations de-
pending on the separation distance; for all station pairs closer than 50km. Black
line: best exponential regression fit; blue dashed lines: empirical 90% confidence
interval.
7.5 Representation of Spatial Dependencies 103
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
1
2
0
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
nb of values per cell
(b) June to August
Figure 7.6.: Observed rank correlation between the average wet day precipitation at one sta-
tion and the standard deviation of wet day precipitation at another station de-
pending on the separation distance for all station pairs closer than 50km. Black
line: best exponential regression fit; blue dashed lines: empirical 90% confidence
interval.
combinations. All station pairs with less than 12 overlapping values in the respective season
are excluded, which leads to about 290000 remaining pairs. In respect to the generation of
simulation time series the focus is on small distances. For this reason, the following analysis
are restricted to all station pairs of less than 50km distance.
For a better readability, the plots are not presented as scatter plots but as rasters of point
densities. The black line in Fig. 7.5 and Fig. 7.6 indicates an exponential regression function
that gives the best estimation of the correlation depending on the distance:
ρˆ (d) = ρ0 · exp
(
−
(
d
d0
)s0)
(7.5)
where ρˆ (d) best estimation for the rank correlation at distance d
ρ0 rank correlation between parameters at zero distance
d0 reference distance
s0 scale parameter
104 Spatial Generation
The parameters ρ0, d0 and s0 of the regression are estimated based on numerical minimiza-
tion of the Mean Square Error between the observed and estimated rank correlations:
E =
n∑
i=1
n∑
j=1
(ρij − ρˆ (dij))2 → min! (7.6)
where ρij observed rank correlation between the station pair i, j
ρˆ (dij) estimated rank correlation for the station pair i, j
dij distance between the stations i and j
As a robust measure of the spread, the 90% confidence interval is indicated by two dashed
blue lines. It is calculated empirically for classes of 5km distance. The total range, which is
most visible in the scatter plots, is highly affected by the number of available data pairs.
Fig. 7.5 and Fig. 7.6 are chosen as examples because the graphs illustrate the range of dif-
ferent spatial behavior in the parameter interdependence. The correlation of the rainfall
frequency p24h at two stations can be seen as a regional characteristic. It does not depend
on the distance between the station pair. In winter months the best estimation for the rank
correlation at 5km distance is ρˆ = 0.89, at 50km distance it is still ρˆ = 0.86. In the sum-
mer months, the correlation is only slightly lower. ρˆ = 0.86 at 5km and ρˆ = 0.83 at 50km
distance. Seasonal effects can hardly be seen.
The monthly sum (Fig. B.13), as well as the combination of monthly sum and daily rainfall
frequency (Fig. B.16) show similar characteristics. The rank correlation of p24 versus H¯wet
and of p24 versus s¯wet do not depend on the distance either. The correlations, however, are
much lower and the spread is higher (Fig. B.19 and Fig. B.20). In any distance uncorrelated
stations can be found, as well as station pairs with ρ > 0.7
The rank correlation of H¯wet versus swet, on the other hand, shows a strong dependence of
the correlation on the distance as well as a strong seasonal difference in correlation (Fig. 7.6).
The average and the standard deviations are both statistics that are related to local effects
and therefore the dependence weakens rapidly with growing station distance. In winter the
best estimation for the correlation is ρˆ = 0.79 at 5km distance, at 50km distance it is ρˆ = 0.64.
In summer it drops from ρˆ = 0.66 to ρˆ = 0.35. Similar characteristics are found in the rank
correlations of the average and the standard deviation of wet day precipitation (Fig. B.14
and Fig. B.15), as well as in the combination of monthly sum versus average (Fig. B.17) and
monthly sum versus standard deviation (Fig. B.18).
In all parameter combinations, due to the presence of small scale convective events, the rank
correlation are lower in summer.
7.5.1.1. Analysis of Anisotropy
For the configuration of several target locations of the time series generation the rank cor-
relations ρ between the statistical parameters are unknown. The true correlation between
7.5 Representation of Spatial Dependencies 105
two locations is substituted by the regression estimation ˆrho (dij). The spread of the corre-
lation around the regression functions is discarded. For some of the relations, this is not a
problem since the spread is low. These are principally the relation of the monthly sum and
the rain day probability to themselves as well as the combination of these two parameters.
The rank correlations involving the average or standard deviation of wet day precipitation
on the other hand show high spread, the deviation to the regression function can go up to
∆ρ = 0.2 or more.
1 NNE
4 SSE
2 WSW
3 WNW
4 NNW
1 SSW
2 ENE
3 ESE
Figure 7.7.: Sketch of the sector division for rank correlation analysis
It is checked if the spread is due to anisotropy in the relation between distance and cor-
relation. For this analysis the station pairs are classified into 4 sectors according to their
geographic orientation (Fig. 7.7). The results are presented for the parameter combination
of Hmon and swet as an example. It is chosen because this combination exhibits the widest
spread in the correlation over distance relation. The results of other parameter combinations
lead to the same conclusions.
The main atmospheric flow direction in the study area is from west-southwest to east-
northeast (Section 2.1). It was found in a former study that two stations a more related,
regarding absolute precipitation values, if the vector joining the two stations is parallel to
the main atmospheric flow direction than if it is perpendicular to the flow direction (Brom-
mundt, 2008). In the presence of anisotropy a classification of the station pairs according
their to geographic orientation should alter the shape of the point clouds and the regression
functions. According to the work of Brommundt (2008) the correlations should be highest if
the vector linking the two stations falls in sector 2 of Fig. 7.7 and lowest for sector 4. How-
ever, the results of the classification (Fig. 7.8) do not support the idea. The graphs for the
four sectors are hardly distinguishable. The regression line for the best estimation in rank
correlation ρˆ takes the same course as in Fig. B.18 for any of the sectors. Hence, the spread
in the correlations does not come from an anisotropic dependence. The confidence inter-
vals for the sector classified data are by no means smaller than for the whole data set with
all station combinations. Hence, the anisotropy found in the correlation of absolute daily
106 Spatial Generation
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
4 8 1
2
1
6
2
0
2
4
2
8
3
2
3
6
4
0
nb of values per cell
(a) December to February – sector 2
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
4 8 1
2
1
6
2
0
2
4
2
8
3
2
nb of values per cell
(b) December to February – sector 4
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
3 6 9 1
2
1
5
1
8
2
1
2
4
2
7
3
0
nb of values per cell
(c) June to August sector 2
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
4 8 1
2
1
6
2
0
2
4
2
8
nb of values per cell
(d) June to August sector 4
Figure 7.8.: Observed rank correlation during winter months between the monthly precipi-
tation sum at one station and the standard deviation of wet day precipitation at
another station depending on the separation distance and the sector
7.5 Representation of Spatial Dependencies 107
values (Brommundt, 2008) cannot be identified in the rank correlation of deduced statistical
parameters.
7.5.1.2. Spatial Rank Correlations Used for the Generation of Simultaneous Daily
Time Series
The rank correlation between the parameters at the locations chosen for simultaneous time
series generation will be estimated by the respective regression functions according to
Eq. (7.5). The parameters of this function for each combination of Hmon, P24h, E (Hwet)
and D (Hwet) can be found in Table 7.4 and Table 7.5.
Table 7.4.: Parameters of the regression functions according to Eq. (7.5) for the estimation of
parameter rank correlation over distance - same parameter at both stations
Parameter at December to February March to May
Station 1 Station 2 r0 d0 s0 r0 d0 s0
Hmon Hmon 0.956 1503.4 0.751 0.982 753.3 0.614
P24h P24h 0.917 5724.3 0.584 0.912 1613.0 0.770
E (Hwet) E (Hwet) 0.937 433.3 0.524 1.000 277.2 0.468
D (Hwet) D (Hwet) 1.000 262.1 0.490 1.000 185.2 0.489
Parameter at June to August September to November
Station 1 Station 2 r0 d0 s0 r0 d0 s0
Hmon Hmon 1.000 337.4 0.483 0.982 918.4 0.644
P24h P24h 0.873 990.2 1.003 0.908 2527.9 0.672
E (Hwet) E (Hwet) 1.000 64.9 0.454 0.972 261.9 0.488
D (Hwet) D (Hwet) 1.000 50.1 0.421 1.000 175.4 0.505
108 Spatial Generation
Table 7.5.: Parameters of the regression functions according to Eq. (7.5) for the estimation of
parameter rank correlation over distance - combination of different parameters
Parameter at December to February March to May
Station 1 Station 2 r0 d0 s0 r0 d0 s0
Hmon P24h 0.766 1042.7 1.521 0.695 1330.1 0.830
Hmon E (Hwet) 0.776 695.2 0.772 0.876 421.1 0.487
Hmon D (Hwet) 0.735 729.7 0.637 0.784 395.2 0.484
P24h E (Hwet) 0.420 528.9 2.372 0.339 18366.0 0.264
P24h D (Hwet) 0.367 642.6 1.986 0.406 176400.0 0.064
E (Hwet) D (Hwet) 0.879 384.7 0.519 0.913 264.2 0.483
Parameter at June to August September to November
Station 1 Station 2 r0 d0 s0 r0 d0 s0
Hmon P24h 0.661 1451.2 0.713 0.718 2307.6 0.767
Hmon E (Hwet) 1.000 60.1 0.355 0.781 430.9 0.496
Hmon D (Hwet) 0.907 47.1 0.324 0.676 374.8 0.675
P24h E (Hwet) 0.350 11842.3 0.074 0.211 5743.0 10.536
P24h D (Hwet) 0.166 1091.2 0.907 0.222 6334.9 0.599
E (Hwet) D (Hwet) 1.000 46.0 0.376 0.877 249.3 0.513
7.5 Representation of Spatial Dependencies 109
7.5.2. Spatial Correlation According to Atmospheric Circulation Patterns
The effect of spatial dependencies in precipitation depends on the regarded temporal scale
(Section 4.1). On monthly scale, the dependence is described by the parameter interdepen-
dencies derived in the last section. The investigation of the spatial dependence on subdaily
temporal scale is based on weather type classification.
The dependence of two rain gauges is affected by the current atmospheric flow direction.
Brommundt (2008) uses radar data to define principal flow directions depending on the
Circulation Pattern (CP) as a measurement of the overall atmospheric condition. The corre-
lation of two 1h precipitation time series is different if the two stations are located parallel
to the flow direction or orthogonal to the flow direction (Brommundt, 2008). In this section
it is analyzed how the CP effects the spatial correlations in all possible orientations.
7.5.2.1. Definition of the Circulation Pattern Classification System
The Circulation Pattern used in this examination were taken from Bárdossy (2010). They
result from an automated fuzzy rule based weather type classification system. The setup of
the classification is described in Bárdossy and Filiz (2005). Data basis of the classification
are mean sea level pressure fields (MSLP) from NCEP/NCAR reanalysis that provide an
archive of homogenized and gridded atmospheric variables (Kistler et al., 2001). The tem-
poral resolution is 2.5◦ by 2.5◦. The classification is based on normalized anomalies. At each
grid point, first, the mean value for this point is subtracted. Then, the difference is divided
by the standard deviation of the grid point resulting in a dimensionless number. Positive
numbers indicate high pressure, negative numbers low pressure zones.
Based on the pressure anomalies, five fuzzy states are defined:
1. large positive anomalies
2. medium positive anomalies
3. medium negative anomalies
4. large negative anomalies
5. arbitrary
The fuzzy rule set defining the k CPs specifies the required fuzzy state for each of the grid
points. It can be described by a matrix:
V = v (i, j) ; i = 1, . . . npts; j = 1, . . . k (7.7)
The definition of one CP class j consists of the state index v = [1, 5] to each grid point i
of the npts grid point in the NCEP/NCAR anomalie field of the Europe and the northern
Atlantic Ocean. The state “arbitrary” is assigned to all points that are indifferent to the
current weather type definition. The principle of the weather classification is explained
110 Spatial Generation
+ + +  - - -
+ +  - - - -
+ +
- - -  + +
- -
 
 
- - - + + +
- -  + +
+ 
+ + +
+ + +
 - - -
- -  + +
+ + + +
- - -  + +
- -
+ + - -
+ + - -
+ + +
- - -  + 
-
CP1 CP2 CP3
normalized SLP 
anomalies grid 
of day t 
(2.5° x 2.5°)
highest DOF!
CP1 is CP-class of the day t
- -  
+ + 
negative /positive 
pressure anomalies
compare with the fuzzy rule of each CP
Figure 7.9.: Principle of the fuzzy rule based CP classification
graphically by the sketch in Fig. 7.9. For each day, the anomaly field is compared with the
CP-defining fuzzy rule set. The CP with the highest resemblance, resulting in the highest
degree of fulfillment of the fuzzy rule, is chosen as the CP of the day.
The CP definition, which is the choice of the states v to each grid point i, is done by an auto-
mated system based on Simulated Annealing (for a description of the Annealing algorithm,
see Section 6.6.2). The Annealing algorithm arbitrarily chooses grid points and assigns
fuzzy anomaly states. If the resulting classification is of any sense, the resulting CPs should
be different in their precipitation characteristics. This is measured by the objective function
O of the Annealing algorithm. The optimization is performed by maximizing the objective
function O.
The classification used in this study was set up by Bárdossy (2010). O is defined by the
fraction of stations that exceed a certain precipitation threshold:
O (η) =
√√√√ 1
T
T∑
t=1
(pc (CP (t))i − p¯ci)2 (7.8)
η spatial coverage taken as a threshold, expressed as the number
of observing stations with non-zero rainfall; η ∈ [0, 1]
pc (CP (t))i relative frequency of the coverage exceeding the threshold η
in the neighborhood of station i at days with a given CP
7.5 Representation of Spatial Dependencies 111
p¯ci relative frequency of coverage threshold exceedance for all days
without classification
T time period of the calibration.
See Bárdossy (2010) for further explanations.
To identify wet and dry situations equally well, the objective functionO for the classification
considers two thresholds:
O = O(η = 0.0) +O(η = 0.999) → max! (7.9)
(Bárdossy, 2010)
The classification system optimizes the CP definitions but not the number of different CPs. It
has to be chosen by the user, e. g. by fitting classification systems with different numbers
of CPs and choosing the best by a performance measure. For the cited objective function
Bárdossy (2010) estimated 17 + 1 as the optimal CP number. “+1” stands for ”CP99” which
is the CP class for all days that cannot be assigned to any other class.
7.5.2.2. CP Dependent Precipitation Statistics
The objective function O in the CP class definition evaluates the hydrological response and
not the weather. However, similar rainfall characteristics are the result of similar atmo-
spheric conditions. If the mean sea level pressure anomalies of all days belonging to the
same CP are averaged, the result represents common weather situations. Fig. 7.10 shows
three examples. CP2 is the driest among the 18 CPs. The rain day frequency averaged over
575 daily stations is about 12% and the average daily precipitation 0.3mm. The average
pressure anomalies of all days belonging to CP2 explain why. CP2 exhibits anticyclonic con-
ditions over Baden-Württemberg. In average, the center of the high pressure zone is over
Denmark. CP3 is exactly the inverse situation and therefore, the wettest CP in the classifi-
cation system. Southwest Germany is under the influence of a cyclonic depression whose
center is close to Baden-Württemberg, over the North sea. The average rain day frequency
is 85% and the average amount about 7mm per day. CP15 is somehow inbetween. It is a
typical situation that sometimes is refered to as “Shottland Low”. The distance of the center
of the depression to the study region is higher than for CP3, and the cyclonic depression is
pushed on a track in northern direction by the high pressure zone over eastern Europe. Over
Baden-Württemberg the approaching warm and cold fronts are generally less intensive due
to the higher distance to the center of the low. Therefore, the rain day frequency of 55% and
the average daily precipitation of 5.2mm are lower than for CP3. Table 7.6 to Table 7.8 list
the basic daily statistics of all CPs in the classification.
7.5.2.3. Analysis of Spatial Correlation
Considering the gradients resulting from the average pressure anomalies and the effect of
Coriolis force, the main flow directions can be identified. During CP3 precipitation arrives
112 Spatial Generation
(a) CP2 (b) CP3
(c) CP15
Figure 7.10.: Average pressure anomalies of all days belonging to the same CP, lines are 1hPa
isobars; red – positive anomalies; blue – negative anomalies
by western flux and during CP15 by southwestern flux. The anticyclonic conditions of CP2
lead to a calm situation with low atmospheric exchange. An analysis of spatial correlation
reveals anisotropies caused by the flow direction.
The analysis is based on 575 daily stations and 137 stations sharing the same measurement
period from 1997 to 2003 (Section 3.3). The stations are paired and the correlations Eq. (6.41)
between the time series of the station pairs are calculated. A pair of daily stations is consid-
ered if it shares at least 30 daily values in the respective CP class. Hourly station pairs are
considered if they share at least 100 hourly values.
The resulting correlations are examined in perspective of the spatial configuration of the
station pairs. Fig. 7.11 to Fig. 7.13 display the correlation in daily precipitation time series
according to the distance vector (x kilometers to the east and y kilometers to the north)
separating the two stations. Each raster cell in these plots represents the average correlation
of all station pairs that have approximately the same distance vector. The absolute location
7.5 Representation of Spatial Dependencies 113
Table 7.6.: Relative frequency, rainfall frequency, average and standard deviation of wet day
precipitation for all anticyclonic dry CPs
CP October to March April to September
rel. Freq. p24h H¯24h s24h rel. Freq. p24h H¯24h s24h
1 0.0454 0.2559 0.8736 2.7229 0.0581 0.3132 1.6683 4.5967
2 0.0617 0.1141 0.173 0.9379 0.0608 0.1328 0.5507 2.7702
7 0.052 0.2704 0.8775 2.9705 0.0688 0.2953 1.4761 4.4296
8 0.0511 0.1944 0.5137 1.9051 0.0445 0.1932 0.9964 3.8954
10 0.0315 0.2618 0.6121 2.0257 0.0396 0.2149 0.8654 3.1074
11 0.0822 0.2022 0.4929 1.8411 0.0639 0.2042 1.2645 4.7771
12 0.0753 0.2348 0.7793 2.8574 0.0811 0.2565 1.3752 4.5016
13 0.0657 0.2796 0.8924 2.6953 0.0638 0.2324 0.9634 3.2042
gr1 0.4649 0.2228 0.6439 2.2387 0.4806 0.2338 1.1743 3.9772
Table 7.7.: Relative frequency, rainfall frequency, average and standard deviation of wet day
precipitation for all cyclonic moderate CPs
CP October to March April to September
rel. Freq. p24h H¯24h s24h rel. Freq. p24h H¯24h s24h
5 0.0914 0.6671 2.6319 4.3994 0.1038 0.5819 2.8309 5.2148
6 0.045 0.6626 3.5101 5.7622 0.0575 0.5976 3.7232 6.3764
14 0.0401 0.3938 1.1529 2.7727 0.0448 0.3951 2.203 5.2943
15 0.0192 0.5376 2.2298 4.5033 0.0349 0.5885 3.9771 6.9979
99 0.0274 0.4573 2.5435 5.4805 0.022 0.4952 3.459 7.3874
gr2 0.2231 0.5801 2.4977 4.5238 0.263 0.5471 3.1237 5.9006
Table 7.8.: Relative frequency, rainfall frequency, average and standard deviation of wet day
precipitation for all cyclonic humid CPs
CP October to March April to September
rel. Freq. p24h H¯24h s24h rel. Freq. p24h H¯24h s24h
3 0.0866 0.8954 7.8709 9.623 0.0633 0.804 6.1722 7.9476
4 0.071 0.8186 5.5872 7.8459 0.0356 0.7341 4.7911 7.3703
9 0.0719 0.6321 3.2564 5.7177 0.0665 0.7233 5.1617 8.1486
16 0.0276 0.7558 3.1647 4.6589 0.0344 0.7158 5.8871 9.0076
17 0.055 0.8169 4.85 6.3503 0.0564 0.7932 5.9318 7.8464
gr3 0.3121 0.7911 5.3399 7.3036 0.2564 0.7591 5.6268 8.0397
of the station pairs is ignored.
The results of the correlation analysis for all days belonging to CP2 are presented in Fig. 7.11.
114 Spatial Generation
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Correlation of daily values - days with CP2
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Correlation of daily values - days with CP2
(b) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Correlation of 1h-values - days with CP2
(c) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Correlation of 1h-values - days with CP2
(d) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Corr of 1h-values - 1h offset - CP2
(e) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Corr of 1h-values - 1h offset - CP2
(f) April to September
Figure 7.11.: Direction dependent correlation of daily and hourly precipitation stations for
two seasons and all days belonging to CP2
7.5 Representation of Spatial Dependencies 115
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Correlation of daily values - days with CP3
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Correlation of daily values - days with CP3
(b) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Correlation of 1h-values - days with CP3
(c) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Correlation of 1h-values - days with CP3
(d) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Corr of 1h-values - 1h offset - CP3
(e) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Corr of 1h-values - 1h offset - CP3
(f) April to September
Figure 7.12.: Direction dependent correlation of daily and hourly precipitation stations for
two seasons and all days belonging to CP3
116 Spatial Generation
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Correlation of daily values - days with CP15
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Correlation of daily values - days with CP15
(b) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Correlation of 1h-values - days with CP15
(c) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Correlation of 1h-values - days with CP15
(d) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Corr of 1h-values - 1h offset - CP15
(e) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.08
0.16
0.24
0.32
0.40
0.48
Corr of 1h-values - 1h offset - CP15
(f) April to September
Figure 7.13.: Direction dependent correlation of daily and hourly precipitation stations for
two seasons and all days belonging to CP15
7.5 Representation of Spatial Dependencies 117
In the left column only days during the six months period from October to March are con-
sidered, in the right column from April to September. The graphs in the first row display
the correlations of daily values. The plots in the second row show the correlations of the 1h
rainfall time series. The plots are point-symmetrical to the origin. (The correlation between
the stations i and j is the same as between the stations j and i.) The last row shows the hourly
correlation if the time-series at the second station is shifted by ∆t = 1h. Due to the shift, the
fields in the last row are not symmetric. For the shifted correlation it makes a difference if
station i in the pair i, j is chosen as starting or end point of the linking vector. The correlation
is higher in flow direction than against it.
CP2 does not contribute much precipitation and rather inhibits atmospheric exchange.
Therefore, the spatial correlations are low, as well as the anisotropy. A preferential flow
direction can only be seen in daily values, from west in winter and from west-southwest in
summer months. The correlation for stations in high distance is slightly higher in winter
(Fig. 7.11a and Fig. 7.11b).
The days belonging to CP3 exhibit a very different spatial structure. The correlations are
overall much higher, especially in the winter half-year. A clear preferential flow direction
can be seen, between October and March from west-southwest and between September and
April from southwest. The spatial correlation is stronger in winter months, at the same
distance about twice as high. The shifted correlation (Fig. 7.12e and Fig. 7.12f) exhibit strong
asymmetry. In winter months the correlation against the flow direction drops by 50% within
the first 20 kilometers.
The highest correlation in daily precipitation values occur at days belonging to CP15
(Fig. 7.13a and Fig. 7.13b). Besides, CP15 exhibits the particularity of having two prefer-
ential flow directions during the winter half-year (Fig. 7.13c and Fig. 7.13e). What might
seem paradox in the first moment, can be explained by the typical wind field of a cyclonic
depression. The passing of a warm or cold front is generally accompanied by a turn in the
wind direction. An approaching cyclonic depression is often preceded by winds from south-
west. During the passage of the warm front, the wind direction turns to west, during the
passage of the cold front to northwest (Weischet and Endlicher, 2008).
The generally higher range in daily correlations is related to the average atmospheric flow
velocity. During 24h, passing fronts of the cyclonic depression have much time to travel
and can generally reach both stations of a pair, even if the separation distance is 100km
or more. In one hour however, an approaching front can only propagate by several tens
of kilometers, depending on the wind speed. Therefore, station pairs at higher distances
remain uncorrelated in the hourly time series.
The number of available data pairs is very different depending on the configuration. Gen-
erally, the number is higher for closer distances. However, there are hardly any stations at
distances of less than 10km. The low number of values at high distances explains some of
the variations at fringes of the correlation plots.
Fig. 7.14 to Fig. 7.16 display the standard deviation of the correlation values in each grid
cell. It measures the spread in the correlation distance relation. For anti-cyclonic CPs, as
CP2 (Fig. 7.14), the standard deviation depends on the correlation value. At short distances,
118 Spatial Generation
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP2
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP2
(b) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP2
(c) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP2
(d) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - 1h offset - CP2
(e) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - 1h offset - CP2
(f) April to September
Figure 7.14.: Standard deviation of all correlation values in the same grid cell for two seasons
and all days belonging to CP2
7.5 Representation of Spatial Dependencies 119
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP3
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP3
(b) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP3
(c) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP3
(d) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - 1h offset - CP3
(e) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - 1h offset - CP3
(f) April to September
Figure 7.15.: Standard deviation of all correlation values in the same grid cell for two seasons
and all days belonging to CP3
120 Spatial Generation
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP15
(a) October to March
30
0
20
0
10
0 0
10
0
20
0
30
0
distance E-W [km]
300
200
100
0
100
200
300
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP15
(b) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP15
(c) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - days with CP15
(d) April to September
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - 1h offset - CP15
(e) October to March
20
0
10
0 0
10
0
20
0
distance E-W [km]
200
100
0
100
200
d
is
ta
n
ce
 N
-S
 [
km
]
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
Stdev of Correlation values - 1h offset - CP15
(f) April to September
Figure 7.16.: Standard deviation of all correlation values in the same grid cell for two seasons
and all days belonging to CP15
7.5 Representation of Spatial Dependencies 121
where the correlation is high, the spread is high too. The pictures of correlation and standard
deviation look very similar. The relation between the distance vector and the correlation is
heteroscedastic.
For the cyclonic CPs (Fig. 7.15 and Fig. 7.16), the standard deviation does not depend on
the value. It shows only marginal spatial structure and the values are generally low. An
exception is the secondary direction in CP15 that causes some spread. At very low sepa-
ration distances the cyclonic CPs show a slight increase in standard deviation during the
summer months (the second columns of (Fig. 7.15 and Fig. 7.16). At a lower level, it is a
similar structure as for anticyclonic CPs. It is caused by convective, small scale precipitation
events that are the main source of precipitation during anticyclonic CPs but can also occur
during cyclonic CPs.
7.5.2.4. CP Dependent Correlations Used For the Generation of Simultaneous Time
Series
X
X
X
Observed Station
Location for Simulation
Configuration Vector
Distance Vector for Kriging 
Weights
Figure 7.17.: Principle of the Kriging interpolation of CP dependent correlations
The generation of simultaneous precipitation time series requires the estimation of the CP
dependent spatial correlations between ungauged locations. It is performed by ordinary
kriging (Eq. (5.8)) according the the scheme in Fig. 7.17. The correlation values are interpo-
lated from observed station pairs according to the configuration of the simulation locations.
This means the x and y component of the vector linking each station pair (gray dashed ar-
rows in Fig. 7.17). Like in the correlation analysis the starting point of the vector linking the
generation locations is projected onto the origin (0|0). Thus, the absolute location within
the study area is ignored. The configuration of the simulation locations (black crosses) is
compared with the configurations of all station pairs of the observed data set. The twenty
pairs with the most similar configuration are chosen. Then the values for each CP are inter-
polated by ordinary Kriging. The Kriging weights are assigned by an empirical variogram
γ (h) (Eq. (5.3)) according to the distances h between the configuration vector end points
(black arrows).
122 Spatial Generation
7.6. Generation Algorithm for Simultaneous 1h Precipitation
Time Series with Spatial Temporal Dependence
The generation of nstat simultaneous, spatial interdependent time series is realized by a
three step simulation passing by the intermediate of simultaneous daily time series. In the
following, the superscript i ∈ [1, nstat] is used to identify the generation location. It is not an
exponent, but an identifier.)
1. First an initial time series in daily temporal resolution is set up for each of the nstat
generation locations. The required parameters are shape κi (Hwet) and scale λi (Hwet)
of the Weibull Distribution for daily precipitation values and daily rainfall probabil-
ity P i24h. κ
i (Hwet) and λi (Hwet) are calculated according to the equations Eq. (6.13)
and Eq. (6.14) from the expected values for average E
(
H iwet
)
and standard deviation
D
(
H iwet
)
of wet day precipitation.
In the same way as for the single site simulation, the rain day probability P i24h, as well
as the expectations E
(
H iwet
)
and D
(
H iwet
)
are conditioned on the monthly precipita-
tion sumH imon at each location i. Additionally, the parameter interdependencies at the
different stations are considered.
2. The temporal order of the daily time series is optimized by Simulated Annealing for
all nstat stations simultaneously.
3. The daily values are disaggregated into hourly values. The daily precipitation sum
of each day is randomly distributed among the 24 associated hours. Next, the hourly
series is optimized by exchanging small precipitation increments among the hours of
the same day. The optimization is performed by Simulated Annealing.
7.6.1. Parameter Estimation for the Initial Daily Time Series
The estimation of the parameters defining the initial daily time series is performed month
by month.
1. Starting point are the time series of monthly rainfall sums H imon at the nstat target lo-
cations of the simulation. If the monthly sum at a station i is not available, H imon is es-
timated from surrounding stations by External Drift Kriging Eq. (5.11) with smoothed
altitude as external information.
2. The target values H imon are transformed into the copula space U of uniformly dis-
tributed values between 0 and 1 by the marginal distribution F . F is depending on the
season (December to February, March to May, June to August, September to Novem-
ber):
uimon = F
(
H imon
)
(7.10)
3. To condition the generation parameters on the monthly sum, the matrix Σ of param-
eter interdependencies in the standard transformed space has to be set up. The cor-
relations that have to be estimated are ρijkl with i, j ∈ [1, nstat] defining the station
7.6 Generation Algorithm for Simultaneous 1h Precipitation Time Series 123
pair and k, l ∈ [1, 4] defining the parameter combination of parameter k at station i
and parameter l at station j. Therefore, the matrix Σ of parameter interdependencies
has 4nstat × 4nstat entries. The convention for the order of the parameters k and l is:
1=ˆHmon; 2=ˆP24h; 3=ˆE (Hwet) ; 4=ˆD (Hwet)
Station 1 Station 2 Station n︷ ︸︸ ︷ ︷ ︸︸ ︷ ︷ ︸︸ ︷
Σ =

r1111 r1112 r1113 r1114
r1121 r1122 r1123 r1124
r1131 r1132 r1133 r1134
r1141 r1142 r1143 r1144
rˆ1211 rˆ1212 rˆ1213 rˆ1214
rˆ1221 rˆ1222 rˆ1223 rˆ1224
rˆ1231 rˆ1232 rˆ1233 rˆ1234
rˆ1241 rˆ1242 rˆ1243 rˆ1244
· · ·
rˆ1n11 rˆ1n12 rˆ1n13 rˆ1n14
rˆ1n21 rˆ1n22 rˆ1n23 rˆ1n24
rˆ1n31 rˆ1n32 rˆ1n33 rˆ1n34
rˆ1n41 rˆ1n42 rˆ1n43 rˆ1n44
rˆ2111 rˆ2112 rˆ2113 rˆ2114
rˆ2121 rˆ2122 rˆ2123 rˆ2124
rˆ2131 rˆ2132 rˆ2133 rˆ2134
rˆ2141 rˆ2142 rˆ2143 rˆ2144
r2211 r2212 r2213 r2214
r2221 r2222 r2223 r2224
r2231 r2232 r2233 r2234
r2241 r2242 r2243 r2244
· · ·
rˆ2n11 rˆ2n12 rˆ2n13 rˆ2n14
rˆ2n21 rˆ2n22 rˆ2n23 rˆ2n24
rˆ2n31 rˆ2n32 rˆ2n33 rˆ2n34
rˆ2n41 rˆ2n42 rˆ2n43 rˆ2n44
· · · · · · · · · · · ·
rˆn111 rˆn112 rˆn113 rˆn114
rˆn121 rˆn122 rˆn123 rˆn124
rˆn131 rˆn132 rˆn133 rˆn134
rˆn141 rˆn142 rˆn143 rˆn144
rˆn211 rˆn212 rˆn213 rˆn214
rˆn221 rˆn222 rˆn223 rˆn224
rˆn231 rˆn232 rˆn233 rˆn234
rˆn241 rˆn242 rˆn243 rˆn244
· · ·
rnn11 rnn12 rnn13 rnn14
rnn21 rnn22 rnn23 rnn24
rnn31 rnn32 rnn33 rnn34
rnn41 rnn42 rnn43 rnn44

 Station 1
 Station 2
 Station n
1
Figure 7.18.: Matrix of the parameter correlations in the standard normal transformed space
for the generation of simultaneous time series
Fig. 7.18 shows a sketch of this matrix for the generation of n stations. The parameter
interdependencies at the same site (i = j), which are the four times four blocks around
the diagonal of Σ, are assumed to be constant. In the same way as for the single site
generation, the correlations riikl, i ∈ [1, nstat] are taken from Table 6.2 (according to
the current season). The parameter dependencies between the different locations rˆijkl
(where i 6= j) are calculated by the distance dij using the regression function (see
Table 7.4 and Table 7.5) for the respective parameter combination k, l and Eq. (6.61)
for the transformation of the rank correlations ρijkl into Standard Normal correlations
rijkl.
4. The matrix Σ is being conditioned on the vector of Standard Normal transformed
monthly sums.
a =
[
Φ
(
uimon
)
, i = 1 . . . nstat
]T (7.11)
where Φ is the Standard Normal Distribution Function.
This leads to a normal distribution X ∼ N (µ|a,Σ|a) with conditioned mean vector
µ|a and conditioned covariance matrix Σ|a. µ|a and Σ|a are calculated according to
Eq. (A.5) and Eq. (A.6) in Section A.2 of the appendix. The conditioned correlation
matrix Σ|a can be calculated by inverting the overall correlation matrix Σ (Fig. 7.18),
dropping all the rows and columns belonging to H imon and inverting back.
The conditioned, Standard Normal transformed random number vector z|a represent-
ing the parameters P24h,E (Hwet) andD (Hwet) at all stations nstat are drawn by means
of a vector z of uncorrelated random numbers and lower triangle matrix L of the
124 Spatial Generation
Cholesky decomposition of Σa
z|a = µ|a + L · z (7.12)
Σ|a = LLT (7.13)
z|a and z have 3 · nstat elements.
It is a necessary condition for the applied Cholesky decomposition that the matrix
Σ is positive definite. In theory, any correlation matrix is positive definite. If not,
the pairwise correlations cannot be valid all at once without contradiction. However,
due to estimation errors in the regression, it can happen that the estimated correlation
matrix Σ is not positive definite. In this case, it is corrected to the nearest positive
definite matrix according to the algorithm described in Higham (2002).
It has been tested how much the correction according to Higham (2002) deforms the
estimated correlation matrix. In none of the tested configurations, the correlations
were changed by more than 5%.
5. The drawn values, x∗ representing the rainfall probability, the average and the stan-
dard deviation of wet day precipitation are now Standard Normally distributed. They
are back-transformed to absolute values in two steps.
u
(
P i24h
)
= Φ
(
zi|a
(
P i24h
))
(7.14)
P ∗i24h = F
−1 (u (P i24h)) (7.15)
where z|a (P24h) conditioned random number in the Standard Normal trans-
formed space for the 24h rainfall probability in the current month
u
(
P i24h
)
uniform transform of the 24h rainfall probability
Φ standard normal distribution function
F−1 (u (P24h)) inverse of the theoretical distribution function of the 24h
rainfall probability according to Table 7.3
P ∗i24h target value for the 24h rainfall probability of the current
month at station i
The proceeding for the target values of the average rain day precipitation sum
E∗
(
H iwet
)
and the standard deviation D∗
(
H iwet
)
it the same. The parameters of the
distribution functions applied for back-transformation can be found in Table 7.3.
6. The parameters λ (Hwet) and κ (Hwet) of the Weibull Distribution for the 1h rainfall
amounts in wet hours are calculated from the drawn statistics E∗ (X) and D ∗ (X) by
the method of moment (Eq. (6.13) and Eq. (6.14)).
7.6.2. Generation of Simultaneous Daily Precipitation Time Series
The generation of daily time series follows the same steps as the generation of hourly time
series for single locations (see Section 6.6.2):
7.6 Generation Algorithm for Simultaneous 1h Precipitation Time Series 125
Given the monthly series of rainfall probabilities P i24h and the Weibull Distribution param-
eters λ
(
H iwet
)
and κ
(
H iwet
)
, an initial daily time series is generated at each location i sepa-
rately.
For each day, a uniform random number u between 0 and 1 is drawn. It is compared with
the rainfall probability of the current month P i24h. If the day is chosen to be a rain day, the
daily precipitation amount is drawn according to Eq. (6.70) from the inverse of the Weibull
Distribution.
The nstat initial time series exhibit the right distribution of the daily precipitation amounts
and, at each location i, they follow the expectation of the monthly precipitation sums. They
are passed to the resampling scheme for optimization of the temporal sequence of values.
The optimization is performed by a Simulated Annealing algorithm similar to the single site
simulation scheme. In each step, one of the stations is randomly chosen. Then two time steps
of the same month are selected for a swap. All nstat time series are evaluated simultaneously
by one global objective function O according to Eq. (6.71). The consequences of the swap for
all nstat stations is evaluated by O.
The statistical parameter that are incorporated in the objective function O are the following:
• Average daily precipitation amount and precipitation frequency of the three CP
classes defined in Section 7.5.2
The average values of the three groups can be found in Table 7.6 to Table 7.8. The
statistics are calculated from 575 daily stations.
Since the monthly precipitation sums do not correspond to the observed values but
are drawn randomly, the average precipitation is not considered in absolute values. It
is normed at each station by the simulated monthly rainfall sum.
• Cross correlation between the daily time series of all station pairs
The target value of the cross correlations between the station pair ij, where i, j ∈
[1, nstat] is estimated in the four dimensional space (xs, xe, ys, ye) of the start and end-
point of the vectors linking the station pair ij.
The applied method is proposed by Brommundt (2008). The estimated correlation rˆij
is the sum of a regression estimation based on the four dimensional distance d4D of the
stations and a residual estimated by ordinary kriging of the observed residuals in the
four dimensional space. If ~ij and ~kl are two vectors linking the stations pairs ij and
kl, four dimensional distance d4D is defined as:
d4D =
√
(xi − xk)2 + (xj − xl)2 + (yi − yk)2 + (yj − yl)2 (7.16)
The x component is the easting in Gauss-Krüger 3 coordinates of the station location.
y is the northing. More details on the method can be found in Brommundt (2008).
• The autocorrelation of 24h precipitaton values according to Eq. (6.77)
126 Spatial Generation
7.6.3. Disaggregation to hourly values
The initial hourly time series is set up by distributing the amount of each day randomly
among the 24 associated hours. To achieve a time series with realistic scaling behavior as a
starting point, the values are distributed in the following manner: One hour of the day is
randomly chosen. It receives 40% of the daily rainfall but at least 0.1mm. The next hour is
chosen and receives again 40% of the remaining daily sum but at least 0.1mm. One hour can
be picked several times. As soon as the daily sum is completely distributed, the procedure
continues on the next day.
The Simulated Annealing algorithm in the disaggregation scheme exchanges small precipi-
tation increments between the time steps of each day. For each swap at first the day is chosen
followed by the two hours of the day. At the beginning, the precipitation increments are in a
range between 0.75mm and 0.05mm. An alternation between series of big and small incre-
ments is applied to avoid that the structure is dominated by big rainfall amounts. Towards
the end of the optimization small increments of 0.01mm are used.
The objective function O consists of two parts. One part is evaluating the performance at
each side individually and the other part measures the spatial dependence. At the begin-
ning, the Simulated Annealing algorithm alternates between evaluating each station sep-
arately and evaluating all station simultaneously. The number of single site evaluation at
each Annealing Temperature T drops proportional to the temperature T . The alternation is
required to give the individual time series enough freedom in resampling. Without single
site optimizations, one runs the risk that a positive change at a single site is prevented due
to the relation with the other stations whose sequence is not yet perfect either.
The objective function parts Oi evaluating the single site performance are mostly the same
as in Section 6.6.2.1:
• 1 h extreme value according to KOSTRA statistics
In the new setup the daily precipitation values are completely defined by the Weibull
Distribution. Therefore, the objective function according to the 24h precipitation ex-
treme cannot be applied any more. It is replaced by the 1h precipitation extreme ac-
cording to the official German extreme value statistics KOSTRA (Bartels et al., 2005).
The Gumbel Distribution of KOSTRA is re-established at each simulation location by
the yearly and 100 yearly extreme value from KOSTRA. The restored distribution is
used to draw the target value for the highest 1h precipitation amount in the simula-
tion year. To consider the high spatial correlation in the potential extremes, the 1h
extreme values at all stations are drawn with one single random number. (This means
that the maximum value at nearby simulated stations is similar. However, it does not
mean that maximum is reached at all stations in the same time step.)
• scaling of the statistical moments between different aggregation levels
The scaling exponent b (Eq. (6.76)) is evaluated for the first three statistical moments of
the time series (Eq. (6.75)). The aggregation intervals ∆t are ranging from 1h to 24h.
• autocorrelation on different aggregation levels
The autocorrelation according to Eq. (6.77) is evaluated for the same aggregation levels
7.7 Comparison of Simulated and Observed Simultaneous Time Series 127
as in the single site simulation. The aggregations are ∆t = 1h, 2h, 3h, 6h and 12h. The
24h autocorrelation has already been exploited in the generation of the intermediate
daily time series (Section 7.6.2).
Additionally, the seasonally weighted autocorrelation of hourly values is evaluated.
The target values of the autocorrelation depend on the generation year.
• average hourly precipitation frequency for three different weather type classes
The CP classes are the same as in the optimization of the intermediate daily time series
(Section 7.6.2). In the daily generation not only the precipitation frequency, but also
the average precipitation is used to evaluate the time series. For the disaggregation,
however, the average is of no information any more. Since no rainfall volume is ex-
changed between the days and thus between the CPs, the average on any regarded
temporal aggregation is constant and completely defined by the daily generation.
The objective functions evaluating the spatial dependence are the CP-dependent correla-
tions estimated in Section 7.5.2. The correlations are measured pairwise and for two seasons
from March to October and April to September, accounting for differences due to convective
events in the summer months.
As a representation of the flow direction, two types of correlations estimated for each station
pair i, j: the correlation of simultaneously measured time steps xi (t) and xj (t) as well as
the correlation between xi (t) and xj (t+ 1h) with one hour time shift.
The result is a series of 72 matrices with nstat×nstat elements; one matrix per season, CP and
for simultaneous or shifted correlation. The target values of the correlations are determined
by interpolation (see Fig. 7.17).
7.7. Comparison of Simulated and Observed Simultaneous Time
Series
The performance of the algorithm will be tested by a comparison of simulated and observed
time series. First, the same statistics as for the single site generation (Chapter 6) are in-
vestitaged. Then, the representation of spatial dependence is tested.
Table 7.9.: Coordinates of target locations for the generation of simultaneous time series
Station x-GK3 [m] y-GK3 [m]
1 3498810 5383440
2 3501820 5384490
3 3499810 5386440
4 3500810 5381940
Since the generator is primarily designed for urban sewage systems, the stations should
lie in a spatial configuration that is of realistic scale for sewage system simulations. Even
128 Spatial Generation
34
98
00
0
34
99
00
0
35
00
00
0
35
01
00
0
35
02
00
0
35
03
00
0
x-GK3 [m]
53
81
00
0
53
82
00
0
53
83
00
0
53
84
00
0
53
85
00
0
53
86
00
0
53
87
00
0
y
-G
K
3
 [
m
]
1000 1000 1010
4
3
1
0
5
0
1
5
0
0
1
9
5
0
1
2
measured +
generated
generated
Figure 7.19.: Overview of the target location for the generation of simultaneous time series
in big cities of Baden-Württemberg, the sewage system rarely exceeds a dimension of one
hundred square kilometers. Therefore, the maximum distance between the stations should
not exceed several kilometers.
The verification is done for two stations in Holzgerlingen that are three kilometers apart.
The two stations share a long common data period from 1977 to 1992 and exhibit both a very
low number of missing values (less than 10%). To make the simulation more realistic for a
real sewage system setup and to add more constraints, the generation is not only performed
for the two measurement stations but for a total of four stations. The coordinates of all four
stations can be found in Table 7.9. Observed data is available at stations 1 and 2. Station
2 is the one that was used for the verification of the single site generation (Section 6.7 in
Chapter 6). The spatial configurations of the stations is displayed in Fig. 7.19.
7.7.1. Comparison of Observed and Simulated Distributions
In this section the same statistics as in Section 6.7 for the single site generator are tested. The
statistical moment, the exceedance frequencies to different threshold in hourly and in daily
resolution, the empirical annual series as a representation of the extreme value behavior on
different aggregations and the length of wet and dry spells as a measure of the temporal
persistence in the generated time series.
7.7 Comparison of Simulated and Observed Simultaneous Time Series 129
Table 7.10.: basic hourly statistics of the simultaneous simulated and observed time series in
Holzgerlingen, 1977 to 1992
Station 1 Station 2 Station 3 Station 4
simulated observed simulated observed simulated simulated
mean [mm] 0.532 0.857 0.526 0.879 0.572 0.537
stdev. [mm] 0.864 1.288 0.907 1.382 0.950 0.902
skewness [-] 12.119 6.133 12.465 6.605 10.006 10.613
p>0.0mm 27.74% 25.29% 28.78% 27.16% 27.98% 29.91%
p>0.1mm 15.78% 8.20% 16.31% 9.04% 16.19% 16.47%
p>0.5mm 4.638% 3.812% 4.662% 4.313% 4.850% 4.777%
p>2mm 0.499% 0.796% 0.544% 0.859% 0.588% 0.591%
p>5mm 0.073% 0.130% 0.086% 0.153% 0.102% 0.074%
p>10mm 0.015% 0.021% 0.016% 0.033% 0.021% 0.012%
p>20mm 0.0028% 0.0030% 0.0050% 0.0052% 0.0028% 0.0028%
Hmax [mm] 30.3 27.8 29.0 27.2 29.650 29.035
Table 7.11.: basic daily statistics of the simultaneous simulated and observed time series in
Holzgerlingen, 1977 to 1992
Station 1 Station 2 Station 3 Station 4
simulated observed simulated observed simulated simulated
mean [mm] 5.654 5.125 5.641 5.432 5.899 5.668
stdev. [mm] 6.283 6.006 6.804 6.403 6.910 6.655
skewness [-] 2.822 3.219 4.905 3.199 3.835 3.668
p>0.1mm 41.6% 41.4% 42.5% 43.4% 42.2% 43.3%
p>0.5mm 36.7% 34.6% 38.0% 36.8% 37.8% 37.8%
p>2mm 25.9% 22.3% 25.9% 24.5% 26.6% 26.2%
p>5mm 13.6% 11.7% 14.0% 12.6% 14.8% 14.3%
p>10mm 5.3% 4.6% 5.5% 5.4% 6.2% 6.0%
p>20mm 1.5% 1.0% 1.3% 1.4% 1.5% 1.4%
In the comparison between Table 6.4 on one hand and Table 7.10 and Table 7.11 on the other
hand, some effects of the different generation algorithms become apparent. In single site
generation the hourly statistics are closer to the statistics of observed time series whereas in
multi site generation the daily values show less differences to observed time series.
The representation of the observed time series by the generation is generally good. On
hourly aggregation (Table 7.10), the rainfall probability is modeled well if 0.0mm is taken
as threshold, however, overestimated if 0.1mm is taken. The observed time series exhibits
a high fraction of very small rainfall amounts that cannot be reproduced by the multi site
generation. Since the deviation concerns only the smallest values, it does not have a serious
impact. The probabilities of more intensive precipitation are all very well modeled. The
skewness of hourly values is overestimated.
130 Spatial Generation
On daily scale all statistics are modeled very well. The only exception is the fraction of
very small values with less than 0.5mm per day that are underestimated. The very different
generated daily precipitation maxima reveal the high possible variability in the generated
statistics. While the observed maxima are very close to each other, the simulated differ by
a factor of more than two. From this data, it seems that the simulation overestimates the
variability in the highest 24h extremes. However, referring to only one single values, the
sampling variance in the maximum is high. With the available data it cannot be judged if
the deviation is random or systematic. From the practical point of view, an overestimation
in the extreme value variability is at least better than an underestimation because it prevents
from “bad surprises” of catastrophically high events.
Fig. 7.20 and Fig. 7.21 compare the observed and simulated empirical distributions at the
two observing stations in Holzgerlingen. The annual series are calculated from the ten high-
est hourly precipitation amounts of each year, so 120 values in total. In all aggregation the
range is realistic. On hourly scale the simulated frequency of the lowest values between two
and four millimeters is too high. Medium values between eight and twelve millimeters are
less frequent than in the observations. For longer aggregation intervals, coming closer to the
daily values that are explicitly modeled by the distribution function, the variability in the
annual series increases. Therefore, the distributions for 6h and 12h extremes are closer to
the observed.
Although the two stations are only three kilometers apart, the observed annual series differ
considerably. At station 2 the average extreme is significantly higher. Only on the 24h
aggregation level (last line of Fig. 7.20 and Fig. 7.21) the differences between the simulated
annual series reach the same extend.
The analyzes of the annual series is repeated monthwise (Fig. 7.22). Compared to the single
site generator, the generation algorithm for multiple time series has a lower ability to dis-
tinguish between the seasons The actual differences in the distributions for January and Juli
are not completely represented in the simulated time series.
The overestimation in the rainfall probability on hourly scale (Table 7.10) leads to an over-
estimation in the length and the absolute number of wet spells (Fig. 7.23). At Station 1, the
overestimation is of 3146 simulated wet spells instead of 2213 spells observed in the sixteen
years from 1977 to 1992, at Station 2 of 3246 instead of 2380. The shape of the distribution
on the other hand is realistic. Nevertheless, the frequency of very long wet spells over two
days is overestimated at both stations.
The distribution of wet spell lengths is very sensitive to the definition of the end of a spell.
Changing the maximum length of gaps that are ignored in the wet spell length estimation
from 1h to 3h, the overestimation of the total number of wet spells can be reduced to real-
istic values. This means that the the simulated occurrence of precipitation is too scattered.
Compared to the observations, short gaps of one or two hours are more frequent.
Fig. 7.24 shows the histogram and the empirical CDF of the 100 longest observed and sim-
ulated wet spells at both stations. Gaps up to 1h are ignored. In contrast to the single site
generator that slightly underestimates the lengths, the simulated distribution is shifted to
7.7 Comparison of Simulated and Observed Simultaneous Time Series 131
0 5 10 15 20 25 30 35
[mm/h]
0
10
20
30
40
50
60
70
x= 3498810 y= 5383440
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(a) Histogram – 1 h
0 5 10 15 20 25 30
[mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(b) CDF – 1 h
0 10 20 30 40 50
[mm/6h]
0
10
20
30
40
50
x= 3498810 y= 5383440
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(c) Histogram – 6 h
0 10 20 30 40 50
[mm/6h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(d) CDF - 6 h
0 10 20 30 40 50 60 70 80
[mm/day]
0
10
20
30
40
x= 3498810 y= 5383440
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(e) Histogram – 24 h
0 10 20 30 40 50 60 70
[mm/day]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(f) CDF – 24 h
Figure 7.20.: Empirical Distribution of the ten highest 1h rainfall amounts in the observed
and simulated time series in Holzgerlingen, Station 1, as part of a spatial tem-
poral simulation with four simultaneous time series
132 Spatial Generation
0 5 10 15 20 25 30
[mm/h]
0
10
20
30
40
50
60
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(a) Histogram – 1 h
0 5 10 15 20 25 30
[mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) CDF – 1 h
0 10 20 30 40 50 60 70 80
[mm/6h]
0
10
20
30
40
50
60
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(c) Histogram – 6 h
0 10 20 30 40 50 60 70 80
[mm/6h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(d) CDF – 6 h
0 20 40 60 80 100 120 140
[mm/day]
0
10
20
30
40
50
60
70
80
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(e) Histogram – 24 h
0 20 40 60 80 100 120 140
[mm/day]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series - 10 members per year
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(f) CDF – 24 h
Figure 7.21.: Empirical Distribution of the ten highest 1h rainfall amounts in the observed
and simulated time series in Holzgerlingen, Station 2, as part of a spatial tem-
poral simulation with four simultaneous time series
7.7 Comparison of Simulated and Observed Simultaneous Time Series 133
0 5 10 15 20 25
[mm/h]
0
10
20
30
40
50
60
70
x= 3498810 y= 5383440
annual series  - 10 members per year
Jan observed 1977 - 1992
NaN= 6.3%
Jan simulated 1977 - 1992
Jul observed 1977 - 1992
NaN= 0.0%
Jul simulated 1977 - 1992
(a) Histogram - Station 1
0 5 10 15 20 25 30
[mm/h]
0
10
20
30
40
50
60
70
80
x= 3501820 y= 5384490
annual series  - 10 members per year
Jan observed 1977 - 1992
NaN= 6.3%
Jan simulated 1977 - 1992
Jul observed 1977 - 1992
NaN= 0.0%
Jul simulated 1977 - 1992
(b) Histogram - Station 2
0 5 10 15 20
[mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
annual series  - 10 members per year
Jan observed 1977 - 1992
NaN= 6.3%
Jan simulated 1977 - 1992
Jul observed 1977 - 1992
NaN= 0.0%
Jul simulated 1977 - 1992
(c) CDF - Station 1
0 5 10 15 20 25 30
[mm/h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
annual series  - 10 members per year
Jan observed 1977 - 1992
NaN= 6.3%
Jan simulated 1977 - 1992
Jul observed 1977 - 1992
NaN= 0.0%
Jul simulated 1977 - 1992
(d) CDF - Station 2
Figure 7.22.: Empirical distribution of the ten highest 1h rainfall amounts in the observed
and simulated time series during January and July in Holzgerlingen as part of
a spatial temporal simulation with four simultaneous time series
higher lengths. In average, the 100 longest wet spells are about ten hours longer in the sim-
ulation. Extremes go up to more than four days of rainfall without a gap that is longer than
one hour – which is not unrealistic but almost twice as high as the observed extremes.
Regarding the dry spell lengths (Fig. 7.25), the overestimation of very short dry spells be-
comes obvious. The simulated number of short gaps up to a few hours is more than three
times as high as the observed number. This is another indication that the occurrence of pre-
cipitation in the simulated time series is too scattered. The shape of the distribution, on the
other hand, is realistic. The distribution of the longest dry spells (Fig. 7.26) is shifted to the
right compared to the observations. The 100th longest dry spell is about eight days instead of
nine at Station 1 and seven days instead of eight at Station 2. Overall, the dry spell length are
realisticly modeled. The highest difference in modeled and observed exceedance frequency
is around 300h (Fig. 7.26c and Fig. 7.26d).
134 Spatial Generation
0 10 20 30 40 50 60 70
lenght [h]
0
100
200
300
400
500
600
700
800
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(a) Histogram - Station 1
0 10 20 30 40 50 60 70
lenght [h]
0
100
200
300
400
500
600
700
800
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) Histogram - Station 2
0 10 20 30 40 50 60
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(c) CDF - Station 1
0 10 20 30 40 50 60
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(d) CDF - Station 2
Figure 7.23.: Empirical Distribution of the wet spell length in hourly resolution; time series
of Holzgerlingen as part of a spatial temporal simulation with four simultane-
ous time series. Gaps up to a length of 1h do not interrupt a wet spell.
(The length of wet and dry spells is not the same as the definitions of wet and dry spell are
not complementary.)
7.7 Comparison of Simulated and Observed Simultaneous Time Series 135
0 20 40 60 80 100 120 140
lenght [h]
0
5
10
15
20
25
30
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(a) Histogram - Station 1
0 20 40 60 80 100
lenght [h]
0
5
10
15
20
25
30
35
40
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) Histogram - Station 2
0 20 40 60 80 100 120
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(c) CDF - Station 1
0 20 40 60 80 100 120
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(d) CDF - Station 2
Figure 7.24.: Empirical Distribution of the 100 longest wet spells in hourly resolution; time
series of Holzgerlingen as part of a spatial temporal simulation with four si-
multaneous time series. Gaps up to a length of 1h do not interrupt a wet spell.
136 Spatial Generation
0 100 200 300 400 500 600
lenght [h]
0
1000
2000
3000
4000
5000
6000
7000
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(a) Histogram - Station 1
0 100 200 300 400 500 600
lenght [h]
0
1000
2000
3000
4000
5000
6000
7000
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) Histogram - Station 2
0 100 200 300 400 500
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(c) CDF - Station 1
0 100 200 300 400 500
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(d) CDF - Station 2
Figure 7.25.: Empirical distribution of the dry spell length in hourly resolution; time series of
Holzgerlingen as part of a spatial temporal simulation with four simultaneous
time series. Rainfall events with less or equal 0.5mm cumulated precipitation
sum do not interrupt a dry spell.
7.7 Comparison of Simulated and Observed Simultaneous Time Series 137
0 200 400 600 800 1000 1200
lenght [h]
0
5
10
15
20
25
30
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(a) Histogram - Station 1
0 200 400 600 800 1000
lenght [h]
0
5
10
15
20
25
30
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(b) Histogram - Station 2
0 200 400 600 800 1000
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3498810 y= 5383440
observed 1977 - 1992
NaN= 3.7%
simulated 1977 - 1992
(c) CDF - Station 1
0 200 400 600 800
lenght [h]
0.0
0.2
0.4
0.6
0.8
1.0
x= 3501820 y= 5384490
observed 1977 - 1992
NaN= 3.5%
simulated 1977 - 1992
(d) CDF - Station 2
Figure 7.26.: Empirical Distribution of the 100 longest dry spells in hourly resolution; time
series of Holzgerlingen as part of a spatial temporal simulation with four si-
multaneous time series. Rainfall events with less or equal 0.5mm cumulated
precipitation sum do not interrupt a dry spell.
138 Spatial Generation
7.7.2. Representation of the Spatial Dependence
The two stations in Holzgerlingen that are analyzed in this section are not used in the estima-
tion of the generation parameters describing the spatial dependence structure. The param-
eter interdependencies (Section 7.5.1) are estimated from the set of daily stations (Fig. 3.4b
in Section 3.3). The CP dependent correlation were calculated for the subset of 137 stations
with homogeneous time periods (Fig. 3.5 in Section 3.3), in which the stations in Holzger-
lingen are not included. The two stations are, therefore, suitable to test if the generation is
able to model the spatial dependence in a realistic manner.
Table 7.12.: Correlation of daily values between the four simulated time series
Station 1 2 3 4
1 1 0.7887 0.793 0.8046
2 0.7887 1 0.7798 0.7714
3 0.793 0.7798 1 0.7891
4 0.8046 0.7714 0.7891 1
Table 7.13.: Target values for the simulation of daily correlation
Station 1 2 3 4
1 1 0.903 0.904 0.911
2 0.903 1 0.911 0.913
3 0.904 0.911 1 0.886
4 0.911 0.913 0.886 1
The correlation (Eq. (6.41)) on daily temporal resolution (Table 7.12) is a measure of the
spatial dependence between the daily time series at the four simulation locations displayed
in Fig. 7.19. As a reverence the target values of daily correlation are given in Fig. 7.13.
Measurements are only available at location 1 and 2, thus only the correlation r12 can be
compared with an observed value. In the measurements r12 = 0.93 which is very close to
the target values of the simulation (Table 7.13). The correlations that are actually modeled,
however, are lower than the target values. The optimization is not able to reproduce such
high correlations by resampling. The individual daily time series are already too different.
The target values are missed by about 12 to 18%.
This is a special problem of very highly related time series where the random drawing of
monthly generation parameters (Section 7.6.1) induces too much freedom in the distribu-
tion of the daily precipitation values. Table 7.14 and Table 7.15 show the daily correlations
for a simulation with ten times higher distances between the four locations (but the same
configuration). In this setup the target values for daily correlation are fulfilled for most sta-
tion combinations. Only the combinations involving station 4 are underestimated by 9 to
18%. However, the correlations remain biased. Deviations between the simulated corre-
lations and the target values always concern underestimations. The correlations are never
7.7 Comparison of Simulated and Observed Simultaneous Time Series 139
overestimated. (Unfortunately, there are no observed stations in in this configuration that
could be used to verify the simulated correlations for the higher distances.)
Table 7.14.: Simulated spatial correlation of daily values for a simulation with 10 times
higher distances between the station locations.
Station 1 2 3 4
1 1 0.721 0.725 0.639
2 0.721 1 0.716 0.635
3 0.725 0.716 1 0.602
4 0.639 0.635 0.602 1
Table 7.15.: Target values for the simulation of daily correlation for a simulation with 10 time
higher distances between the station locations.
Station 1 2 3 4
1 1 0.735 0.724 0.732
2 0.735 1 0.731 0.775
3 0.724 0.731 1 0.664
4 0.732 0.775 0.664 1
Table 7.16.: Correlation of hourly values between the four simulated time series
Station 1 2 3 4
1 1 0.6185 0.6119 0.6209
2 0.6185 1 0.6155 0.6227
3 0.6119 0.6155 1 0.5941
4 0.6209 0.6227 0.5941 1
The spatial correlations on hourly scale are not defined by any generation parameter. They
depend on the CP sequence and the CP dependent correlations. If the correlation of the
simulated time series is close to the observed correlation, the average over all CP dependent
correlations is realistic. The observed correlation between Station 1 and 2 is r12 = 0.6938,
which is about 10% higher than the modelled correlation of r12 = 0.6185 (Table 7.16). Com-
pared to daily scale, the underestimation is reduced.
Table 7.17 and Table 7.18 present the hourly correlation values with one hour time shift be-
tween the time series r(xi(t);xj(t + 1h)). The line of the table indicates the first station i,
the column the second station j where the time series is shifted. Due to the shift, the matrix
is not symmetric. Which of the correlations ij and ji is higher, depends on the preferential
flow direction. The diagonal elements represent the autocorrelation (Eq. (6.77)) of hourly
values with lag k = 1h. The autocorrelation at each station is a target value in the Simulated
Annealing optimization, therefore, the simulated values are close to the observed ones. The
cross correlation between Station 1 and Station 2 on the other hand are too low in the simu-
140 Spatial Generation
Table 7.17.: Correlation of hourly values with one hour time shift between the four simulated
time series
Station 1 2 3 4
1 0.449 0.393 0.444 0.336
2 0.348 0.461 0.390 0.356
3 0.369 0.342 0.461 0.351
4 0.389 0.434 0.408 0.456
Table 7.18.: Correlation of hourly values with one hour time shift between the observed time
series
Station 1 2 3 4
1 0.444 0.456 – –
2 0.422 0.425 – –
3 – – – –
4 – – – –
lated time series. The underestimation of about 10% of the order of magnitude is the same
as of the correlation without time lag. The anisotropy, however, is well represented. Both, in
the observations and the simulations, the correlation is slightly higher, if the time series of
Station 2 is shifted by one hour. This means that the simulation assumes a preferential flow
direction from west to east, which is realistic for the climatic context of Baden-Württemberg
(see Chapter 2.1).
The anisotropy is not very pronounced, neither in the observed nor in the simulated time
series because of the short distances between the stations. Assuming purely advective flow,
the highest anisotropy in the correlation of the hourly time series of Station 1 and Station 2
would occur if the rainfield traveled at 3km per hour, which is very slow. During most of the
time the advection velocities are higher and, therefore, the time shift between the stations is
shorter. Hence, it cannot be represented to full extend in the time series of hourly resolution.
Table 7.19.: Correlation of hourly values with one hour time shift for a simulation with ten
times higher distances between the station locations.
Station 1 2 3 4
1 0.425 0.336 0.259 0.345
2 0.263 0.452 0.251 0.283
3 0.309 0.341 0.440 0.253
4 0.218 0.273 0.207 0.463
In a simulation with the same station configuration as in Fig. 7.19 but with ten times higher
distances a time shift of 1h between Station 1 and Station 2 corresponds to an advection ve-
locity of 30km/h, which is closer to typical rain field travel times. Therefore, the anisotropy
7.7 Comparison of Simulated and Observed Simultaneous Time Series 141
in the simulated correlations with one hour time shift is more pronounced in this case (Ta-
ble 7.19).
Table 7.20.: Correlation between the hourly time series from 1977 to 1992 depending on the
CP type
CP October to March April to September
frequency [d] simulated observed frequency [d] simulated observed
1 150 0.586 0.655 168 0.777 0.590
2 174 0.609 0.801 191 0.388 0.559
3 270 0.817 0.751 194 0.701 0.748
4 217 0.719 0.780 85 0.750 0.794
5 260 0.516 0.702 325 0.505 0.643
6 118 0.693 0.724 182 0.702 0.711
7 141 0.631 0.811 193 0.706 0.578
8 164 0.678 0.851 152 0.363 0.298
9 203 0.800 0.824 203 0.641 0.640
10 91 0.786 0.577 134 0.625 0.771
11 217 0.655 0.805 191 0.468 0.754
12 196 0.641 0.734 212 0.572 0.773
13 221 0.877 0.833 180 0.547 0.494
14 109 0.755 0.731 117 0.745 0.693
15 61 0.764 0.793 89 0.698 0.550
16 82 0.749 0.812 85 0.605 0.602
17 157 0.689 0.774 167 0.726 0.741
99 85 0.849 0.824 60 0.629 0.686
avg 0.712 0.766 0.619 0.646
stdev 0.096 0.070 0.124 0.125
For the examination of the CP dependent correlations only the combination of Station 1 and
Station 2 is considered where the simulation results can be compared to the observed data
(Table 7.20). In average over all CPs the spatial dependence is modeled correctly. There is
only a slight bias between the observed and simulated correlations. Between October and
March the simulated correlations are higher, between April and September the observed
correlations. This means that the difference between the seasons is slightly higher in the
simulations. The variability among the CPs, measured by the standard deviation of the
correlation values, is modeled in a realistic manner. The standard deviations are close to
the observed in both seasons. The correlation between the pairs of observed and simulated
CP dependent correlations is r = 0.48 in the summer half-year and r = 0.24 in the winter
half-year. Hence, there is at least a part of the CP dependent signal in the spatial correlation
that is correctly modeled. During summer months, the representation is better.
The differences between the simulated CP dependent correlations and the target values for
the Simulated Annealing algorithm are low (Table 7.21). Deviations only occur if a CP is
142 Spatial Generation
Table 7.21.: Comparison of the simulated CP dependent correlation between the hourly time
series from 1977 to 1992 and the target values
CP October to March April to September
simulated target value simulated target value
1 0.586 0.687 0.777 0.807
2 0.609 0.682 0.388 0.612
3 0.817 0.850 0.701 0.793
4 0.719 0.900 0.750 0.800
5 0.516 0.666 0.505 0.701
6 0.693 0.773 0.702 0.762
7 0.631 0.708 0.706 0.724
8 0.678 0.742 0.363 0.499
9 0.800 0.874 0.641 0.724
10 0.786 0.862 0.625 0.671
11 0.655 0.746 0.468 0.582
12 0.641 0.697 0.572 0.621
13 0.877 0.863 0.547 0.695
14 0.755 0.795 0.745 0.786
15 0.764 0.894 0.698 0.797
16 0.749 0.79 0.605 0.787
17 0.689 0.832 0.726 0.765
99 0.849 0.883 0.629 0.755
rare during one year so that the correlation is calculated from very few precipitation val-
ues, especially if it is a very dry CP. Therefore, the correlation of anticyclonic CPs like CP5
and CP17 is the most underestimated. Since the target values are well represented in the
simulation, the deviations between observed and simulated time series comes mainly from
the estimation of the target values. Station pairs with very short distances are rare. There-
fore, the interpolation of the CP dependent correlations (see Section 7.5.2) is linked to high
uncertainty.
In Table 7.22 and Table 7.23 the CP dependent correlation with one hour time shift are listed.
The time series at the station that is named last is the one that is shifted by one hour. In
average the correlations r12 from west to east are higher than the correlations r21 from east
to west. Overall, the representation of the observed correlations in the simulations is better
than for the CP dependent correlations without time shifts. In the winter half-year the sim-
ulated correlations are 10% to 20% too low, in the summer half-year they are a little higher
than the observed correlation. This means that the simulated seasonal effect on the correla-
tion is lower than the observed. The simulated variability in the CP dependent correlation
values, measured by the standard deviation, is in the same range as the observed. In av-
erage it is about 5% lower. The correlation between the pairs of simulated and observed
correlation of all CPs is r12 = 0.29 and r21 = 0.29 in the winter half-year and r12 = 0.60 and
r21 = 0.62 in the summer half-year. The shifted CP dependent correlation between April
7.8 Conclusion 143
Table 7.22.: Correlation between the hourly time series from 1977 to 1992 with one hour time
shift, data from October to March
CP frequency Station 1 to Station 2 Station 2 to Station 1
[d] simulated observed simulated observed
1 150 0.328 0.353 0.317 0.366
2 174 0.264 0.524 0.277 0.635
3 270 0.599 0.556 0.524 0.489
4 217 0.522 0.517 0.487 0.468
5 260 0.327 0.635 0.335 0.545
6 118 0.493 0.489 0.377 0.503
7 141 0.466 0.580 0.386 0.610
8 164 0.502 0.694 0.478 0.653
9 203 0.657 0.746 0.598 0.580
10 91 0.572 0.883 0.473 0.334
11 217 0.535 0.561 0.531 0.751
12 196 0.496 0.559 0.478 0.676
13 221 0.693 0.453 0.647 0.688
14 109 0.444 0.685 0.465 0.440
15 61 0.512 0.731 0.472 0.538
16 82 0.573 0.684 0.482 0.621
17 157 0.459 0.628 0.412 0.557
99 85 0.624 0.492 0.602 0.599
avg 0.497 0.605 0.455 0.556
stdev 0.113 0.127 0.097 0.114
and September is the spatial information that is represented best by the simulation.
7.8. Conclusion
For the generation of simultaneous time series the resampling approach of NiedSim can-
not be applied without modifications. The high number of possible combinations in the
temporal sequence of precipitation values at all stations has to be restricted. Therefore, the
generation algorithm is changed, passing by an intermediate time series of daily precipita-
tion values. The modifications, however, affect the capacity of the generator to reproduce
the statistical characteristics of observed precipitation time series.
In the single site generation (Section 6.6) the distribution of hourly precipitation amounts
is directly defined by the parameters of the Weibull Distribution that are estimated for each
month. Higher aggregated values have more freedom because they vary with the sequence
of the hourly values. In the multi site generation the opposite is the case. The distribution of
daily values is directly defined by the theoretical Weibull Distribution while the values on
144 Spatial Generation
Table 7.23.: Correlation between the hourly time series from 1977 to 1992 with one hour time
shift, data from September to April
CP frequency Station 1 to Station 2 Station 2 to Station 1
[d] simulated observed simulated observed
1 168 0.330 0.228 0.377 0.371
2 191 0.407 0.363 0.245 0.171
3 194 0.336 0.401 0.437 0.441
4 85 0.551 0.336 0.497 0.436
5 325 0.393 0.321 0.291 0.347
6 182 0.293 0.412 0.276 0.311
7 193 0.439 0.270 0.393 0.272
8 152 0.294 0.410 0.222 0.281
9 203 0.499 0.437 0.395 0.357
10 134 0.576 0.766 0.453 0.295
11 191 0.695 0.503 0.299 0.342
12 212 0.399 0.338 0.331 0.251
13 180 0.196 0.197 0.287 0.272
14 117 0.394 0.257 0.401 0.310
15 89 0.375 0.344 0.376 0.378
16 85 0.495 0.475 0.418 0.484
17 167 0.462 0.394 0.399 0.323
99 60 0.465 0.441 0.461 0.434
avg 0.420 0.379 0.359 0.332
stdev 0.121 0.130 0.078 0.077
shorter aggregation have more freedom in their variations. Therefore, the statistics on daily
scale are represented best by the multi site generator.
The distributions of extreme values on shorter than daily temporal aggregations are mainly
governed by the scaling exponents b of the statistical moments (Section 7.6.3). In the ana-
lyzed example, the shorter time scale is correctly represented by the simulated time series.
The modeled distributions of annual series on hourly, three hourly and six hourly aggrega-
tion correspond well to the observed distributions.
On the other hand, the necessary modifications in the generation principle reduce the ca-
pacity of the generator to model seasonal differences between summer and winter months.
The highest intensities in summer months are generally caused by convective events (see
Chapter 2). These event are hardly ever longer than a few hours. Therefore, they are less
visible in the average and the standard deviation of daily precipitation amounts than in
the statistics of hourly precipitation. In the generation the main seasonal signal is induced
by the monthwise estimates of the parameters of the Weibull Distribution for the precipita-
tion amounts. For the multi-site generator this information is only available on daily scale.
Therefore, the multi-site generator underestimates the seasonal variability in short aggre-
7.8 Conclusion 145
gation intervals. The new generation scheme with intermediate daily time series comes at
the price of reduced seasonal variability because the seasonal information on hourly scale
cannot be used.
The distribution of all hourly values depends not only on the scaling exponents b but is
also highly influenced by the estimates of the hourly rainfall probability for the different CP
groups (Section 7.6.3). This can be problematic. The relative error in observed precipitation
measurements is highest for the smallest values. Therefore, the estimation of observed rain-
fall frequencies is subject to high measurement uncertainty. In combination with the mod-
eled daily sums this can induce errors in the distribution of hourly precipitation amounts.
In the tested examples the modeled hourly rainfall probability is too high (Table 7.10). Since
the daily statistics are correctly modeled, the overestimation in rainfall probability causes
an underestimation in the average wet hour precipitation amount.
In Section 7.7 it is found that the simulated precipitation is too scattered. There are too
many very short rainfall events and at the same time too many short interruptions in longer
precipitation events. Probably, the scatter is caused by a contradiction between the scaling
exponent and the erroneous estimate of the hourly rainfall probability.
The spatial dependence in the multi site generation is considered at three temporal scales.
On monthly scale, the generation parameters of daily rainfall probability, as well as the aver-
age and the standard deviation of wet day precipitation are assumed as spatially correlated
in the standard normal transformed space. They are linked to the monthly rainfall sums at
the simulation locations. Daily scale correlation is used in the optimization of the tempo-
ral sequence of daily precipitation values. On the hourly scale, the spatial dependence is
represented by the CP dependent correlation of simultaneous and temporally shifted time
series.
If the time series are used in sewage system modeling, the distances between the simulation
locations are generally short, in the range of a few kilometers. In the data set that is used for
the estimation of the CP dependent correlations the distances between stations are generally
higher. For short distances the sampling uncertainty in the CP dependent correlation is high.
In the analyzed example the average correlation of hourly precipitation values is modeled
in a realistic manner. The characteristics of the different CPs, however, are not represented
very well in the simulated time series. It can be assumed that this is an effect of the high
estimation uncertainty at short distances.
At very short distances the correlations on daily scale are underestimated by simulated time
series. If the target values of daily correlation are higher than about r = 0.9, they cannot be
reached in the optimization because the differences in the initial daily time series cannot be
fully compensated by the optimization of the temporal sequences in the time series. It seems
that the generation parameter interdependencies do not capture all the spatial information.
The individual values of the different time series exhibit too much freedom. Probably, the
problem could be attacked by drawing the daily values of all time series simultaneously
with correlated random numbers.
8. Some Aspects of Climate Change
Statistical precipitation models all rely on observed precipitation data. The data is re-
quired to estimate conceptual model parameters, for example the transition frequencies in a
Markov Chain model. The resampling-based generators developed in this work, are linked
to observations by the monthly precipitation sums, the precipitation frequencies, the pa-
rameters of the distributions for the 1h precipitation values and the target statistics of the
Simulated Annealing algorithm like the autocorrelation and the scaling exponent.
Data from synthetic precipitation generators is frequently used in planning and design is-
sues. NiedSim, for example, is dedicated to the dimensioning of urban sewage systems. If
the design of a hydraulic structure is based on NiedSim data, the precipitation during the
aspired life span of the planned structures is represented by the measurement period of the
calibration data. It implies the assumption that the climatic conditions in the future are the
same as during the calibration period. Under changing climatic conditions this assumption
is less and less true, the further the projection looks into the future.
The fourth assessment report of the International Pannel on Climate Change (IPCC) states
that the climate is about to change and that this change is most likely due to anthropogenic
effects (IPCC, 2007). From 1900 to 2005 the global mean temperature increased by about
0.8◦C (IPCC, 2007). The increase is spatially heterogeneous. The alps for example warmed
up more rapidly than the average. The temperature rise during the 20th century in the alpine
region is of 1.3◦C (Auer et al., 2007). The faster temperature rise is related to an increase in
average atmospheric pressure, probably due to a shift of the subtropic high pressure zone
to the north (Auer et al., 2007). It is an indication that climate change does not only concern
atmospheric temperature but also affects atmospheric circulation.
The global water cycle is sensitive to the average global temperature. Warmer surface tem-
peratures enhance evaporation, especially over the Oceans. Besides, the capacity of air
masses to carry moisture is increasing as a function of temperature according to the Clausius
Clapeyron Relation. If more water is evaporated and carried, it has to fall down somewhere
so that the mass balance is fulfilled. Wentz et al. (2007) state that an increase in atmospheric
temperature by 1◦C leads to an increase in total global precipitation volume of about 7.4%
(with a possible error of ±2.6%). Held and Soden (2006) analyze the Global Circulation
Models that were used in IPCC’s fourth assessment report and estimate that the global pre-
cipitation increases by 1 - 3% per degree Celcius of global warming.
However, an increase in total precipitation volume does not say anything about the spatial-
temporal distribution. If the average yearly precipitation amount increases, but it falls in
fewer and more intensive events, with higher seasonality or higher interannual variability,
147
flood risks and the probability of water stress can rise considerably – even both at the same
time.
On the regional scale Hundecha and Bárdossy (2005) found season specific trends in daily
precipitation from the German part of the Rhine catchment. The study is based on records
of 632 daily precipitation stations. Hundecha and Bárdossy (2005) investigate several pre-
cipitation related statistics as the 10% and 90% quantile of daily precipitation sum or the
maximum five day total precipitation. During the recording period from 1958 to 2001, most
of the precipitation related indices have been increasing. The increase is strongest in winter
months from December to February. The summer months, from June to August are the only
season where the indices of daily precipitation show a slightly decreasing trend. The trend
signal is most pronounced for extreme events exceeding the 90% quantile (Hundecha and
Bárdossy, 2005). Except for the summer season, the percentage of total precipitation volume
that falls during extreme events is increasing. That means that the highest daily precipitaton
sums rise faster than the average and that there is a shift in the distribution.
Haberlandt et al. (2010) conduct similar investigations for 263 daily rain gauges in Lower
Saxony. The trend signals in this regions confirm the findings from the Rhine catchment.
Daily precipitation is increasing in winter and decreasing in summer. Daily extreme precip-
itation shows a positive trend. Additionally, Haberlandt et al. (2010) analyze the length of
dry spells and find a significant increase in summer months.
Haylock and Goodess (2004) link the trend signal in daily extreme precipitation to a trend in
the North Atlantic Oscillation (NAO) which is a measure of the meridional pressure gradient
over the Atlantic Ocean and hence of the activity of the mid-latitude west wind zone (see
Chapter 2). Thus, the increase in daily precipitation during the second half of the last
century is, to a large extend, caused by stronger atmospheric fluxes from the Atlantic Ocean.
Predictions from Global Circulation Models for Central Europe lead to the conclusion that
the ongoing changes in precipitation will continue. Based on model runs of ECHAM4, a
global circulation model run by the Max Planck Institute in Hamburg, Germany, Arpe and
Roeckner (1999) predict a precipitation increase in Middle Europe that is at the expense of
precipitation over the Mediterranean countries. The simulations of the AM3P model, run
by the Met Office Hadley Centre in the UK, make similar predictions (Kendon et al., 2010).
Déqué et al. (2007) summarize the climate projections from 19 different combinations of
Global and Regional Circulation models for the time period from 2071 to 2100. For Central
Europe the ensemble of all simulations agrees on a precipitation increase in winter and a
decrease in summer months.
Christensen and Christensen (2004) focus on the extremes in climate model projections.
They show that time series from Regional Circulation Models can have trend signals of
different signs in the average and extreme precipitation. In regions where the estival daily
precipitation decreases in average, there can be an increasing signal in the extreme events.
The cited trend analyzes are all on daily or monthly scale. For the target scale of the pre-
sented stochastic rainfall generator the information on monthly or daily precipitation is of
little use. Since extreme events on different scales are caused by different type of events
148 Some Aspects of Climate Change
(Section 4.2) they exhibit different climate change signals. From regional, daily information
it cannot be deduced how local extremes in hourly time series react to global warming.
In Central Europe extreme precipitation sums over longer periods of one day or of several
days are caused by cyclonic depressions that carry moisture from the Atlantic Ocean or the
Mediterranean. Extreme short time intensities, however, mainly occur during convective
precipitation events in summer months. There are meso-scale circulation patterns that are
in favor of heavy convective precipitation events, but the events also reflect the local heat
flux and the availability of moisture.
The ocean and the land masses of the continents react very differently to global warming.
Over the ocean a higher sea surface temperature (SST), in any case, leads to higher evapo-
ration. Over the continents the evaporation may be limited by the available soil moisture,
especially during summer. Thus, the portion of incoming solar radiation that is transformed
into latent heat is smaller over land. The atmosphere heats up faster than over the ocean.
In summer months, the strong increase in storage capacity of the heated air masses over the
continent can overcompensate the increase of evaporation over the ocean. As a result the
relative humidity may be lower, even if the absolute volume of atmospheric water is higher
(Kendon et al., 2010).
If the relative humidity of the atmosphere is lower, fewer occurrence of precipitation can
be expected. If at the same time the absolute water content is higher, the intensity of pre-
cipitation is expected to increase (Trenberth, 1999). As a result, more water can rain down
in shorter time during fewer events. The effect, however, is only detectable on sub daily
temporal scale. For this reason, Trenberth (1999) states that “more datasets of hourly precip-
itation rates and frequency need to be developed and analyzed” for the evaluation of global
climate change.
Global Circulation Models are spatially too coarse to represent the regional processes lead-
ing to extreme precipitation (Christensen and Christensen, 2004) and the temporal resolu-
tion is not able to image the increasing intensity of events. The question is, how the available
GCM information can anyhow be used to estimate trends on sub-daily scale and which other
sources of information are available.
8.1. Trend Signals in Observed Precipitation Time Series
One possibility to predict future climate conditions is the extrapolation of observed trend
signals. Trend estimation is based on the assumption that an observed trend will continue
in the same way in the future. It is a purely statistical method that does not use any informa-
tion of atmospheric physics, but it assumes some sort of inertness in the physical processes
governing the trend.
For this assumption, statistical trend estimation does not allow to look far into the future.
A time horizon of the year 2100, as in Global Circulation Models, is not possible. On the
other hand, it allows to benefit from the full detailedness of observed precipitation data that
future predictions from GCMs or RCMs cannot deliver.
8.1 Trend Signals in Observed Precipitation Time Series 149
As there is indication that the climate trend signal in precipitation is scale dependent, the
focus in the following investigations is on hourly precipitation data.
8.1.1. Regional Trend in Extreme Precipitation
If only one representative station was picked out for trend analysis, the results would be
highly influenced by local effects and sampling uncertainty. Since precipitation is highly
variable in space (Section 4.1), it is possible that the trend at one station is opposite to the
regional trend signal. The use of just one station might lead to the wrong conclusion.
To enable the use of data from all available rain gauges in the same analysis, the measure-
ments have to be made comparable. One possibility is to reference all values on the station
average of the respective statistics.
H¯max =
1
nyr
nyr∑
i=1
Hmax (i) (8.1)
vmax (i) =
Hmax (i)
H¯max
(8.2)
(8.3)
where Hmax (i) maximum precipitation observed in year i
H¯max average yearly maximum precipitation
vmax (i) normalized maximum of year i
nyr number of years in reference period
The normalized maxima vmax of all stations can be joined together in the same statistics.
The average over all years is one. If there is for example a positive trend in the maxima, the
average will be below one in the beginning of the analyzed period and above one towards
the end. The trend signal can be estimated by linear regression.
vˆmax (i = x) = a0 + b0x (8.4)
where vˆmax (i = x) best guess for the normalized annual maximum in year x
a0 intersect of the linear regression function
b0 slope of the linear regression function
The parameters of the regression are estimated by minimizing the least square error between
the observed values vmax and the best regression estimate vˆmax.
The normalization is only valid if the used average maximum Hmax at all stations refers to
the same time period. Therefore, not all stations and not the full time period of the data set
(Section 3.3) can be used for the regional trend analysis. It is restricted to a homogeneous set
in which all stations are available in every year. As a trade off between the longest possible
time period and the highest possible number of available stations the time period from 1991
150 Some Aspects of Climate Change
to 2003 was used. In this period 40 stations with hourly time resolution and 104 stations
with daily resolution are available.
For a broader data basis the analysis is repeated twice, at first with the annual maximum
and then again with the five highest values per year. To account for seasonal effects, it was
performed with censored data from two seasons, October to March and April to September.
1990 1992 1994 1996 1998 2000 2002 2004
year
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
n
o
rm
e
d
 m
a
x
observed
y =  11.952 + -0.0055x
(a) hourly data – October to March
1990 1992 1994 1996 1998 2000 2002 2004
year
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
n
o
rm
e
d
 m
a
x
observed
y =   0.288 +  0.0004x
(b) hourly data – April to September
1990 1992 1994 1996 1998 2000 2002 2004
year
0.0
0.5
1.0
1.5
2.0
2.5
n
o
rm
e
d
 m
a
x
observed
y = -15.214 +  0.0081x
(c) daily data – October to March
1990 1992 1994 1996 1998 2000 2002 2004
year
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
n
o
rm
e
d
 m
a
x
observed
y =  15.564 + -0.0073x
(d) daily data – April to September
Figure 8.1.: Regional trend in the five highest hourly and daily precipitation sums at 40
hourly and 104 daily rain gauges in Baden-Württemberg from 1991 to 2003
Due to the broader data basis, the trend signal in the data of the five highest values are seen
as more reliable. They are presented in Fig. 8.1. (The results for the annual maximum are
comparable.) In both seasons the extreme values do not change much over the years. There
is a slightly decreasing trend in the hourly extremes from October to March (Fig. 8.1a) and in
the daily extreme values from April to September. The daily extremes in the winter half-year
are increasing.
The significance of the trend signal is tested by a non-parametric Monte Carlo method. If
the data exhibits a significant temporal trend, it should be destroyed, if the data from the
8.1 Trend Signals in Observed Precipitation Time Series 151
different years is randomly mixed. If the trend signal is just a sampling effect due to inter-
annual variability, it is possible that the random trend signal is more pronounced after the
mixing.
The portion of linear regression slopes calculated from mixed data that have a steeper slope
(of the same sign) than the regression calculated from the observed data can be seen as the
level of significance in the trend signal. The only restriction of this significance testing is
that the number of trials in the mixing should be much lower than the number of possible
combinations. With 13 years of available data and hence 13! different combinations, which
is more than six billion, the Monte Carlo method is performed with 10000 trials.
The significance test is performed as a single sided test. The tested hypothesis H0 and H1
depend on the slope b0 of regression line in the observed data:
• H0: β0 ≤ 0
H1: β0 > 0 if b0 > 0
• H0: β0 ≥ 0
H1: β0 < 0 if b0 < 0
where β0 is the “true” slope wich would be calculated if the regression was based on an
infinite number of data pairs. The significance α is the probability of making a mistake
when H0 is rejected in favor of H1. It can be approximated by the fraction of random slopes
b that are steeper than b0.
Table 8.1.: Significance test by Monte Carlo simulation of the regional trend signals in Baden-
Württemberg from 1991 to 2003.
Hourly Data Daily Data
October April to October April to
to March September to March September
b0 -0.0132 0.0004 0.0081 -0.0073
brand > b0 8283 4855 2147 7434
α 17.17% 48.55% 21.47% 25.66%
The results of the significance testing in Table 8.1 show that none of the trend signals is
highly significant. For the most pronounced trend in the hourly data of the winter half-year,
one out of six regression lines calculated from randomly mixed data have a steeper slope.
The data period reaching over thirteen years is probably too short for a significant trend
detection. As a second trial data from Bavaria is used, which is the direct eastern neighbor of
the federal state of Baden-Württemberg. In Bavaria 54 hourly stations and 293 daily stations
are available in the time period from 1990 to 2006. If the longer time series in Bavaria exhibit
significant trend signals, the same could be expected for Baden-Württemberg if longer time
series were available.
In Bavaria the daily extreme values between 1990 and 2006 are constant (Fig. 8.2). The hourly
extreme values. on the other hand, are increasing in both seasons. The results presented in
152 Some Aspects of Climate Change
1990 1995 2000 2005
year
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
n
o
rm
e
d
 m
a
x
observed
y = -20.756 +  0.0109x
(a) hourly data – October to March
1990 1995 2000 2005
year
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
n
o
rm
e
d
 m
a
x
observed
y = -28.433 +  0.0147x
(b) hourly data – April to September
(c) daily data – October to March (d) daily data – April to September
Figure 8.2.: Regional trend in the five highest hourly and daily precipitation sum at 40
hourly and 104 daily rain gauges in Bayern from 1990 to 2006
Table 8.2.: Significance test by Monte Carlo simulation of the regional trend signals in
Bavaria from 1990 to 2006.
Hourly Data Daily Data
October April to October April to
to March September to March September
b0 0.0109 0.0147 0.0037 -0.0013
brand > b0 940 52 2868 5842
α 9.40% 0.52% 28.68% 41.58%
Fig. 8.2 refer to the data set of five extremes per year. (For the annual series of only one
extreme, the trends go in the same direction but are less pronounced.) The trend towards
higher hourly values in the summer season has a high significance (Table 8.2). Only 52 out
8.1 Trend Signals in Observed Precipitation Time Series 153
of the 10000 randomly mixed data sets exhibit a higher slope. In winter the significance is
still over 10%
Table 8.3.: Significance test by Monte Carlo simulation of the regional trend signals in
Bavaria from 1991 to 2003.
Hourly Data Daily Data
October April to October April to
to March September to March September
b0 0.0062 0.0129 0.0057 -0.0096
brand > b0 2855 769 3476 8202
α 28.55% 7.69% 34.76% 17.98%
To judge the effect of limited data, the analysis for Bavaria is repeated with a data set that
is restricted to the time interval from 1991 to 2003 (Table 8.3). Cutting down the data period
from 17 to 13 years reduces the significance considerably. The slope of the regression lines
is lower too. In winter months it reduces to one half. The overall tendency however is the
same in the shorter data set. The hourly extreme values are still increasing with a stronger
increase between April and September.
For the same time period the trend signal in Bavaria and Baden-Württemberg show some
deviations. Between October and March hourly extreme precipitation is decreasing in
Baden-Württemberg but increasing further east in Bavaria. The significance of the trends
is however low. It is possible that the difference is only due to sampling effects and not to
regional differences in the climatic conditions.
For a final judgment whether there is a trend signal in hourly precipitation extremes, the
time period of 17 years is too short. In such a short time period climatic cycles with several
year long periods, as for example the “El-Nino” – “La-Ninja” cycle (Weischet and Endlicher,
2008), do not average out and can lead to misinterpretations. The increasing branch of such
a periodic signal might be mistaken for an increasing trend. On the other hand, the findings
from Bavaria are supported by the results of Trenberth (1999) who states that global atmo-
spheric warming will lead to less but more intensive precipitation events due to the higher
moisture capacity of the atmosphere. Thus, there is a physically meaningful interpretation
in the observed increase of hourly precipitation extremes.
8.1.2. Change in Scaling
Normalization by the scaling is another possibility to make the precipitation data from dif-
ferent stations comparable. If the hourly extreme values are increasing while the daily
extremes are constant, as it is found for the Bavarian data, the scaling of precipitation
must change. If it is presumed that an increasing trend in extremes is present in Baden-
Württemberg but could not be measured due to lack of data, the related change in scaling
might be detectable. Scaling, in this case, means the statistical properties of precipitation
154 Some Aspects of Climate Change
data aggregated over different time intervals (for example, hourly data, daily data, see Sec-
tion 4.2).
H¯max (∆t) =
1
nyr
nyr∑
i=1
Hmax (i,∆t) (8.5)
v¯max (∆t) =
H¯max (∆t)
H¯ (24h)
(8.6)
where Hmax (∆t) average yearly maximum at aggregation level ∆t
H¯max (24h) average daily maximum
As in the last subsection, the analysis is based on the five highest extremes of each year.
The normed average maximum v¯max at each station is between 0 and 1. It is the portion of
the daily extreme that can fall in a shorter aggregation interval ∆t. If the scaling behavior
is changing over time, vmax of all aggregations ∆t < 24h is changing too. Trends in daily
precipitation, on the other hand, are eliminated by the normalization. Since it requires data
from several years for a reliable calculation of the average maximum on every aggregation
level, it is not possible to make a trend estimation on yearly basis. Instead, the available data
was divided into two periods and the scaling characteristics were compared. Fig. 8.3 shows
the average of the normed extremes vmax over all rain 292 hourly rain gauges in the data set
(Section 3.3) from 1958 to 1980 and from 1981 to 2003.
0 4 8 12 16 20 24
Duration [h]
0.0
0.2
0.4
0.6
0.8
1.0
re
l.
 I
n
te
n
si
ty
 [
-]
1958-1980
1981-2003
(a) average scaling
0 4 8 12 16 20 24
Duration [h]
0
2
4
6
8
10
12
re
l.
 D
if
fe
re
n
ce
 [
%
]
(b) rel. difference 1981-2003 to 1958-1980
Figure 8.3.: Average of the five highest precipitation events per year depending on the tem-
poral aggregation normed on the daily extreme. Average over all stations in the
data set for two time periods.
In average over all stations, about 50% of the daily extreme precipitation can fall within 4
hours, 20% wihin one hour. Comparing the data from the two sub periods, it is noticeable
that the dashed line from the later data period is always higher. The right plot in Fig. 8.3
8.1 Trend Signals in Observed Precipitation Time Series 155
indicates the relative changes at each aggregation interval between the two periods. It is
always positive and the relative increase is highest in the shortest aggregation intervals.
The results, again, support the thesis of Trenberth (1999) that the precipitation over short
time periods is intensifying with global warming. Compared to the daily extremes, the
short time intervals up to six hours increased by about 10% between the two time periods.
An increase of 10% does not seem dramatic during a measurement period of 46 years, but
one should keep in mind that the trend was revealed by coarsely dividing the data set in
two halfs of 23 years. On this coarse scale much of the actual trend will be averaged out
within the subperiod. A further division into more intervals, however, is not possible due to
the short average time series length of the precipitation stations. If the data is divided any
further, the calculation of the average extremes will become uncertain.
The changes in scaling, however, cannot only be detected in extreme events. One of the
target statistics of the precipitation generators developed in this study is the exponent of
a non-linear regression function through the statistical moments on different aggregation
intervals (see Eq. (6.76) in Section 6.6.2.1). Therefore, it is analyzed how the scaling exponent
b in Eq. (6.75) is changing between the two time periods from 1958 to 1980 and from 1981 to
2003.
0
1
2
3
4
5
6
0 120 240 360 480 600 720 840 960 1080 1200 1320 1440
aggregation intervall [min]
1s
t  m
o m
e n
t  [
m
m
]
 1. moment -changed
 1. moment - observed
 theoretical fit
y =  0.25 * x 0.34
(a) original scaling exponent
0
1
2
3
4
5
6
0 120 240 360 480 600 720 840 960 1080 1200 1320 1440
aggregation intervall [min]
1s
t  m
o m
e n
t  [
m
m
]
 1. moment - changed
 theoretical fit y =  0.98 * x0.22
(b) changed scaling exponent
Figure 8.4.: Principle of the change in scaling exponent.
By increasing the averages over shorter aggregation intervals, the scaling expo-
nent b is getting smaller.
Fig. 8.4 illustrates how the scaling exponent b for the first moment reacts on a scale change.
The light blue bars indicate the original measured average precipitation sum over different
aggregation intervals. For the blue bars, the averages of all aggregations shorter than 24h
were artificially increased. The relative change is higher in the shorter aggregations. The
equation in Fig. 8.4a represents the regression function trough the original data, the equation
in Fig. 8.4b the regression function through the changed data set. The scale exponent of
the regression drops from b = 0.34 to b = 0.22. The scale exponent is always a positive
number between 0 and 1. If b = 0, the regression function would be horizontal which would
156 Some Aspects of Climate Change
mean that all daily precipitation would fall within one hour of the day. If b = 1, it would
permanently rain with constant intensity.
Abs. Change
-0,82 - -0,70
-0,69 - -0,60
-0,59 - -0,50
-0,49 - -0,40
-0,39 - -0,30
-0,29 - -0,20
-0,19 - -0,10
-0,09 - 0,00
0,01 - 0,10
0,11 - 0,20
0,21 - 0,30
0,31 - 0,40
(a) October to March
Abs. Change
-0,82 - -0,70
-0,69 - -0,60
-0,59 - -0,50
-0,49 - -0,40
-0,39 - -0,30
-0,29 - -0,20
-0,19 - -0,10
-0,09 - 0,00
0,01 - 0,10
0,11 - 0,20
0,21 - 0,30
0,31 - 0,40
(b) April to September
Figure 8.5.: Absolute change in scaling exponent b between the two time periods from 1958
to 1980 and from 1981 to 2003
Fig. 8.5 illustrates the changes in scale exponent between the two time periods. Colors from
dark blue to green-yellow indicate a decrease of the scale exponent b and thus an increase of
hourly precipitation values relative to daily values. The colors from orange to red indicate
an increase of b. The spatial detailed maps are calculated by the difference of two 1km×1km
raster fields of interpolated scaling exponent values. The intermediate step of interpolation
is necessary since not all stations are available in both time periods. Therefore, some of the
spatial characteristics can be artifacts due to interpolation uncertainty.
Nevertheless, hardly all of Baden-Württemberg shows a decrease in scale exponent and thus
a relative increase in short term intensities. The only exceptions are in the vicinity of some
precipitation station in the upper Neckar valley and, during the summer months, the south-
ernmost part of Baden-Württemberg in the vicinity of the Alps (Fig. 8.5b). Generally, the
decresase is more pronounced in summer months, which is in accordance with the physical
explication by Trenberth (1999).
Both regions with positive scaling exponent exhibit a climatic particularity. The Upper
Neckar valley is the driest region in the study area that is the most shaded by the Black For-
est. Perhaps heavy convective precipitation events with high intensities are especially rare
in this part of Baden-Württemberg. The regions close to the Alps receive the highest pre-
cipitation sums when the flux comes from northwest to north due to orographic effects on
the high mountains further south. In the rest of Baden-Württemberg, the highest sums are
8.2 Global and Regional Circulation Models 157
received from western to south western direction. It is possible that generally cool northern
air masses react the most on the increase in evaporation caused by global warming. Most
likely, the positive scaling exponent in the pre-alpine region is not caused by decreasing
short term intensities but because the intensification is overcompensated by an increase in
daily precipitation sum.
8.2. Global and Regional Circulation Models
Global Circulation Models are physically based models that solve equations describing at-
mospheric processes as for example energy and mass balance. Most models work with
discretized degree grid cells in several vertical layers. The physical equations are solved
for the interfaces of the grid boxes. There are several GCM by different climatic research
groups. Examples are the ECHAM model family developed by the Max Planck Institute for
Meteorology in Hamburg, Germany (Roeckner et al., 2003) or the models of the Met Office
Hadley Center in the UK (Johns et al., 2003).
In principle, GCM are weather forecast models for the whole globe with a very long tempo-
ral horizon. Due to the heavy computational effort, the resolution is limited, spatial to about
one to two degrees of latitude and longitude (which is equivalent a few hundred kilometers
at the equator), temporally to six hour intervals.
Although the models are physically based, there are quantities that have to be represented
by conceptual parameters because they vary on smaller scales than the GCM resolution. Ex-
amples are the leaf area index, representing the seasonally changing albedo due to crowing
plants or the reflection of solar radiation by clouds that is modeled by a statistical cloud
cover scheme in the ECHAM5 model (Roeckner et al., 2003).
Global circulation models can be divided into two groups: atmospheric models and coupled
atmospheric oceanic models. In atmospheric models the oceans are represented as huge
water reservoirs that interact with the atmosphere on the surface. Lateral energy and mass
fluxes within the oceans are ignored. Coupled models explicitly take the ocean circulation
into account. This increases considerable the number of grid points. HadCM3, the coupled
model of the Hadley Center (Johns et al., 2003), has six ocean grid points per atmospheric
grid point in the horizontal. To compensate for the higher computational demand that is
necessary for the better process representation, the spatial resolution in coupled models is
lower.
There are two possibilities to refine the resolution on certain target areas, either by using
models with variable grid sizes or nesting of Regional Climate Models (RCM). Regional
models have grid lengths of several tens of kilometers and more complex representation of
the physical processes than GCMs. The higher spatial resolution requires a limitation of the
modeled area. Typically, RCMs are set up for one continent or one part of a continent. The
RCMs are nested in the GCM by the boundary conditions. The RCMs receive values for
all state variables, for example pressure, energy or moisture flux at the boundary grid cells
from the governing GCM. An example for an RCM is the REMO model by the Max Planck
Institute (Elizalde et al., 2011) that has been set up for Europe and West Africa among others.
158 Some Aspects of Climate Change
8.2.1. Emission Scenarios
Figure 8.6.: Scenarios for Green House Gas (GHG) emissions from 2000 to 2100 (Nakicenovic
and Swart, 2000)
Climatic predictions from Global and Regional Circulation Models depend on state vari-
ables that are external to atmospheric simulation. An example is the albedo of land surface
that depends mostly on human agricultural activity or, the most important, the concentra-
tion of CO2 in the atmosphere. Before a global circulation model can be run, the future
values of the external variables have to be predicted.
The prediction of CO2 is based on an ensemble of scenarios. Each scenario is defined by a
story line that describes a way in which the Earth will probably evolve. The story line in-
cludes the evolution of the Earth’s population, as well as prognosis about technical progress
and how the energy demand of the world population will be met in the Future.
In 2000, the IPCC published a Special Report on Emission Scenarios (SRES) (Nakicenovic
and Swart, 2000). The SRES scenarios are composed of four different scenario families (A1,
A2, B1, B2) with corresponding story lines. These are subdivided into scenario groups with
six marker scenarios (A1FI, A1T, A1B, A2, B1, B2). The first letter indicates if the overall
orientation on economic growth (A) or environmental issues (B), the following one digit
number indicates if the economic development on the planet is homogeneous and globally
linked (1) or regionally diverse (2). The last letters in the “A” scenarios indicates how the
energy demand is satisfied, e.g. by fossil energy sources (FI) or a mix of fossil and renewable
sources (B).
8.2 Global and Regional Circulation Models 159
The ensenmble of scenarios claims to cover the whole range of possible developments in
the CO2 concentration. Among the marker scenarios, B1 is the most “optimistic”, with the
lowest CO2 concentration, A2FI the most “pessimistic”.
8.2.2. Model Errors in Global and Regional Circulation Models
To cover the full range of possible future climate conditions, GCM models are run for differ-
ent CO2 emission scenarios. Additionally, control runs are performed for past time climate,
e. g. the 20th century runs of the ECHAM and Hadley Center models. In the control runs
the models are forced with historic initial conditions and the observed CO2 concentration
time series. The resulting simulation of 20th century climate can be compared to observed
measurements of climatic variables.
GCM and RCM can only give a simplified representation of the atmospheric processes. Be-
sides, the limited resolution leads to model errors. To estimate the model errors, climate
model runs are generally compared to reanalysis data, which are gridded precipitation fields
in similar spatial-temporal resolution deduced from a worldwide set of observed precipita-
tion records (see also Section 7.5.2).
Habemann et al. (2006) examines the global water cycle simulated by ECHAM5. The com-
parison of monthly precipitation sums to GPCP reanalysis data from the Max Planck Insti-
tute for the control period from 1979 to 1999 reveals some deviations. Seasonal differences
are generally overemphasized by the GCM. Over the Oceans an during Monsoon season in
Asia the precipitation values are generally too high.
Roeckner et al. (2006) shows that the performance depends highly on the spatial resolution
of the model. The simulated precipitation react especially sensitive to the vertical resolution,
which means the number of atmospheric levels present in the model. The fact that the spatial
resolution has a high influence on simulated precipitation values leads to the conclusion that
GCMs induce model errors by the scale difference of GCM and real world precipitation.
Kendon et al. (2010) check for the reliability of precipitation predictions from the Hadley
Center Model HadAM3P in Europe by separating different mechanism influencing future
climate. For example, the model is run with boundary conditions of the control period ex-
cept for the soil moisture that is taken from a future scenario run. In this way, she can judge if
the different mechanism induce trends in the same direction, which makes estimated trends
reliable, or if they work against each other.
Analyzing several precipitation related indices as the 95% quantile, she jugded the trend
signal to warmer and more humid winters in Central Europe as reliable. The trend in the
extremes is more pronounced than in the average. During summer central Europe is in a
transition zone between a increasing trend in Scandinavia and an decreasing trend in the
Mediterranian and, therefore, the trend is less reliable (Kendon et al., 2010). The analysis,
however, is only qualitative and concentrates merely on the sign of the trend signal. She
states that the trend in winter precipitation can be altered by changes in the circulation pat-
tern that are not considered in the model.
160 Some Aspects of Climate Change
In an extensive study, including six different combinations of four RCM nested in three
different GCM, Fowler et al. (2007) estimate the possible error range in climate model sim-
ulations for the European continent. All GCM-RCM combinations of the ensemble exhibit
similar error characteristics. On the one hand, they typically overestimate the number of
wet days and, on the other hand, underestimate the average daily precipitation as well as
the standard deviation. The reduced standard deviation indicates that the models are not
able to represent the full climatic variability in precipitation. Thus, the underestimation is
the most pronounced in extreme values, as for example the daily precipitation extreme with
five year return period (Fowler et al., 2007). Besides, the error in extremes depends on the
scale of the temporal aggregation. The underestimation in five day precipitation sums is less
sever than in daily values.
The results of a similar study of the alpine region confirm the findings of Fowler et al..
Frei et al. (2003) compare the simulation results from five RCMs with gridded observed
precipitation from more than 6000 rain gauges in the alpine region. All models considerably
underestimate the standard devaition of precipitation intensities (by 16% to 42%) as well as
the frequency of heavy events. In this study the RCMs are forced by reanalysis data, which
means by observations and not by a Global Circulation Model. Therefore, the quantified
error signals only concern the RCMs. In a climate simulation the error would add up with
the model error from the GCM simulation govering the RCM.
8.2.3. Correction and Downscaling of GCM Precipitatioin Data
Due to the various error sources listed in the last subsection, GCM and RCM data must not
be applied as it is in any precipitation modeling or hydraulic or hydrological simulation.
Especially the reduced variability is dangerous in terms of the underestimation of extreme
value frequencies. However, even if GCM and RCM produced perfect forecasts, there would
be the scale difference between the circulation models and rain gauge measurements. GCM
and RCM calculate the areal average precipitation for each grid cell, which is not the same
as point values simulated by a rainfall generator or observed by a rain gauge.
Willems and Vrac (2011) presents different methods that can be used to correct and down-
scale precipitation data from GCMs or RCMs. Correction methods are generally based on
a comparison of the distributions from the climate model and observations from reanalysis
or rain gauges during a control period in the past. Willems and Vrac (2011) distinguishes
between two different types of downscaling approaches that either try to directly correct the
simulated precipitation time series or use other GCM variables, as the simulated sea surface
pressure, for a classification of observed precipitation.
8.2.3.1. Direct Correction of GCM Precipitation Data
A direct correction method is the Delta-Change approach. Knowing that the absolute values
of the GCM precipitation is wrong, only the trend in different quantiles is considered. To
calculate a future precipitation scenario, the relative trend in each quantile is multiplied
8.2 Global and Regional Circulation Models 161
to the observed precipitation value of the same quantile (Willems and Vrac, 2011). This
approach is not unique when there are several time steps with identical precipitation values.
Values of the same rank have to be ordered according to additional criteria, for example the
amount of precipitation in preceding and following time steps (Willems and Vrac, 2011).
Another issue is that the delta factor can take a value of infinity for some quantiles if the
rainfall probability is increasing in the future. The problem can be avoided if the ranks are
calculated only on the basis of wet time steps, but then the future rainfall probability has to
be corrected separately.
The QQ-transformation is a direct correction method that is based on a comparison between
the simulated distributions of a GCM and the observed distribution of a ground reference.
It is able to correct for non-linear and non-proportional errors. It means that an individual
correction factor is applied at every quantile of the distribution. Sennikovs and Bethers
(2009) use this kind of transformation in a climate change study for the Baltic sea region.
Boé et al. (2007) apply a QQ-Transformation to data from the French GCM ARPEGE during
a study on the hydrology of the Seine river in Central France.
Mathematically, the QQ-Transformation is described by:
Hcorr = F
−1
true(Ferr(Herr)) (8.7)
dcorr =
Hcorr
Herr
(8.8)
where Hcorr corrected monthly precipiation sum
Herr original value from the erroneous distribution
Ferr non-exceedance probability of Herr according to the erroneous distri-
bution
F−1true Inverse of the cumulative distribution function of the ’true’ distribution
from the ground reference
dcorr correction factor for the respective quantile
QQ-transformation is calibrated by the distributions of GCM and ground reference data
during a control period, typically 30 years. The transformation according to Eq. (8.7) is
applied to the whole range of the GCM distribution. The GCM distribution is transferred
into the distribution of the ground reference and thus the errors in the control period data
are eliminated. The transformation factors to each quantile are kept. Assuming that the
error characteristic of the GCM is constant over time, the correction factors are applied to
data from future runs of the climate model. The result is a corrected distribution that is
comparable with past time observations from the ground reference.
To account for potential seasonal cycles in the error structure, QQ-Transformation can be set
up for each month of the year separately.
QQ-Transformation is not only a correction but also a downscaling tool. The data from
one GCM grid cell is transformed differently at each point. Within one GCM grid cell, the
distribution Ferr of the GCM data is constant, whereas the distribution on the finer gridded
ground reference Ftrue is spatially varying due to local effects. Since the correction factors
162 Some Aspects of Climate Change
dcorr depend on both distributions, the spatial resolution of the ground truth is impressed
on the coarser GCM data.
Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
0
50
100
150
200
250
300
350
a
v
g
. 
m
o
n
th
ly
ra
in
fa
ll 
[m
m
]
Rulindo - Lat:   1.70 ◦ S Lon: 29.90 ◦ E
observed (1961-1990)
MPEH5 CTR (1961-1990)
HADCM3 CTR (1961-1990)
(a) Raw data
Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
0
50
100
150
200
250
a
v
g
. 
m
o
n
th
ly
 r
a
in
fa
ll 
[m
m
]
Rulindo - Lat:   1.70 ◦ S Lon: 29.90 ◦ E 
 2021-2050
(b) QQ- corrected predictions
Figure 8.7.: Example of the effect of QQ-Transformation. a) average annual cycle in raw
data; b) minimum to maximum range of future predictions according to three
scenarios A1B, A2 and B1 in HadCM3 (blue) and ECHAM5 (green) data, black
line: ground references used for downscaling.
In a case study for Central Africa, monthly precipitation sums from two GCM, the atmo-
spheric model ECHAM5 and the coupled model HadCM3, are downscaled on the 0.5◦×0.5◦
latitude-longitude grid of three different reanalysis products that are used as reference for
the true observed precipitation.
Fig. 8.7 demonstrate the effect of the QQ-transformation on the average annual cycle for a
grid cell near Kigali, Rwanda. Fig. 8.7a compares the average annual cycle of the raw data
for the control period from 1961 to 1990. Untransformed, the GCM are not able to model the
data correctly. There is a strong bias in both models, HadCM3 shows an overestimation and
8.2 Global and Regional Circulation Models 163
ECHAM5 sever underestimation. The solid black lines in Fig. 8.7b represent the average
annual cycle of the reanalysis that are used as ground reference in the QQ-Transformation.
Fig. 8.7b displays the minimum to maximum range in the ensemble of future predictions
for the average annual cycle from 2021 to 2050. The range spans over an ensemble of three
scenarios combined with the three reanalysis products. The color indicates the GCM model.
The chosen scenarios (A1B, A2 and B1) cover a wide range of possible CO2 emissions. A2
is one of the scenarios with the highest CO2 concentration in the regarded time period from
2021 to 2050, B1 is one of the scenarios with the lowest concentration. Nevertheless, the ex-
pected change in the annual cycle does not principally depend on the scenario. The choice of
the GCM model has about the same effect on the future prediction, in some months (Febru-
ary, March) the influence of the GCM is higher. As both GCM predict the same future if the
same scenario is applied, any deviation can be interpreted as model uncertainty. Due to the
huge deviations in the original data (see 8.7a), the QQ-transformation cannot eliminate all
errors. The model error in the uncorrelated data is several times higher than the predicted
changes. The different characteristics of the GCM are still visible in the QQ-Transformed
data.
For precipitation on daily or subdaily scale the presented direct correction methods are not
suitable since it cannot consider any information on temporal persistence. Strictly speaking,
the application is limited to temporally independent values. In European climate, the values
on daily or sub-daily temporal resolution are significantly autocorrelated. As each time step
is corrected differently, depending on its quantile, the autocorrelation structure is ignored.
The persistence structure of the GCM is altered, errors in the persistence are not corrected.
Besides, the methods are not able to correct the frequency of zero values, which is important
on daily or subdaily scale. If the GCM generally underestimates the rainfall frequency, the
QQ-Transformation is not unique. It cannot be judged if a certain zero value should be kept
or corrected to a positive precipitation value. If a zero value is corrected to non-zero rainfall,
a value has to be chosen, e.g. by stochastic simulation – which induces further uncertainty
in the predicted GCM time series.
Finally, it should be noted that the assumption of direct methods, that the error in GCM pre-
cipitation is constant over time, cannot be checked. During QQ-Transformation, a potential
drift in the relation between GCM and ground reference is multiplied by the transformation
factors dcorr. The induced error, if the relation between GCM and reanalysis is changing
over time, is amplified by the strong correction that has to be applied to the original GCM
output.
8.2.3.2. Indirect Correction by Additional Information
As an indirect method Willems and Vrac (2011) present the Analogues Method based on
mean sea level pressure fields that are judged as a more reliable than GCM precipitation. To
each day in the GCM output an analogue day is searched among all days of the same month
and the same weather type in an archive of mean sea level pressure fields from reanaly-
sis data. The observations of the analogue day is seen as the future prediction. Additional
164 Some Aspects of Climate Change
criteria, as for example a sub-classification according to temperature can be applied. A short-
coming of this approach is that a refinement in more classes reduces the number of values
in each class, hence induces sampling uncertainty and might lead to an underestimation of
very rare events (Willems and Vrac, 2011). Another limitation is that the future predictions
cannot be extrapolated beyond what was measured in the past. The latter issue can be at-
tacked in a very rough manner by multiplying each value by a correction factor depending
on temperature, e. g. 7% per degree of temperature rise(Willems and Vrac, 2011).
Willems and Vrac (2011) test the method with precipitation data from the rain gauge in Uc-
cle, Belgium, which is one of the longest high resolution precipitation time series in Europe.
They use an ensemble of 17 ECHAM5 runs as GCM data. The model approach is calibrated
on a 30 year period and tested on the next 30 years. During the validation period extreme
precipitation is systematically underestimated after the correction. It is found that the error
is scale dependent and decreases for higher aggregation intervals. Compared to the Delta-
Change approach (Section 8.2.3.1) the Analogues Methods predicts lower changes. Willems
and Vrac (2011) suspect that not all of the trend signal can be explained by changes in the
weather type sequence (Willems and Vrac, 2011).
Another indirect method that exploits weather type information is to use the distribution of
hourly precipitation amounts conditioned on different weather type classes as a prediction
for future rainfall. Such a method will be described in the next section.
8.3. Prediction of Future Precipitation by GCM Defined
Circulation Patterns
In physical atmospheric modeling the representation of precipitation is the most difficult. It
depends on a high number of other variables that have to be calculate before, as for example
the available atmospheric moisture from evaporation, the air temperature and the wind
field. In weather forecast, the exact predictions of precipitation is the most difficult task (see
for example Bliefernicht, 2010). It has been illustrated in the last section that precipitation
predictions of GCM are related with high modeling uncertainties.
Many publications indicate that the change in Central European climate is at least partly
due to changes in meso-scale atmospheric conditions (Haylock and Goodess (2004), Kendon
et al. (2010) or Auer et al. (2007)). By a weather type classification, the changes in atmo-
spheric circulation can be related to changes in precipitation.
To estimate the influence of changes in the atmospheric circulation on short term precip-
itation intensities, a fuzzy rule based CP-classification system is set up. The response in
precipitation of each CP is calculated from observed rain gauge data. The application of
uncertain precipitation data from GCM is avoided in this way. From the GCM only predic-
tions of the Mean Sea Level Pressure (MSLP) are considered to calculate a CP sequence for
the Future. The changes in precipitation are seen as the result of changes in the CP sequence
as well as the reaction of the CPs to changes in air temperature.
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 165
The applied CP classification is based on NCEP/NCAR reanalysis (Kistler et al., 2001). The
definition of CP classes follows the approach by simulated annealing described in Bárdossy
(2010) but applies a different objective function. (See also Section 7.5.2 for a detailed de-
scription of the method.) To examine the effects of climate change in hourly precipitation in-
tensities, the objective function is designed to detect differences in the distribution of hourly
precipitation amounts.
According to the sketch in Fig. 8.8, the optimal CP classification is found when distribution
of 1h precipitation amounts of all the days belonging to one CP differ the most from the
climatic average. The difference is measured by the variable χ2 which is equal to the squared
sum of the residuals in relative frequency between the histogram for all days belonging to
one CP and the histogram for the climatic average without classification. (In Fig. 8.8 χ2 is
equal to the sum of the squared lengths of the arrows.)
χ2 =
ncp∑
i=1
k∑
j=1
(hi (j)− hclim (j))2 → max! (8.9)
where hi (j) observed relative frequency of intensity class j during all days
belonging to CP i
hclim (j) relative frequency of class j in the calibration period from 1991 to 2003
without CP classification
ncp number of different Circulation Patterns in the classification
k number of histogram classes
Data basis for the histograms calculation is a homogeneous and close to complete subset of
30 hourly precipitation stations of the NiedSim data set (see Fig. 3.6 in Section 3.3). The
calibration period is from 1991 to 2003. For comparability reasons the histogram classes
are not based on absolute values but on the quantiles at each rain gauge location. This
CPi
histogram for the entire period
1991-2003
1h prec. sum 
[mm]
relative 
frequency
relative 
frequency
1h prec. sum 
[mm]
0 030 30
conditional histogram for CPi
Figure 8.8.: Principle of the objective function for the CP definition in the simulated anneal-
ing scheme.
166 Some Aspects of Climate Change
means that the values within one class have different absolute amounts, depending on the
location, but lie within the same range of the CDF at their respective location. To shift the
focus towards extreme events, the classes are not equidistant but smaller in the higher CDFs.
The upper limits of the histogram classes are F = 0.6, F = 0.8, F = 0.9, F = 0.95, F = 0.98,
F = 0.99 and F = 1. The CDFs are calculated from all wet hours with no less than 0.1mm
of precipitation.
The histograms are calculated for two seasons, from September to April and from May to
August to account for convective precipitation enhancement during summer months. As a
trade-off between a good representation of the possible variability in atmospheric circula-
tion and a sufficient number of precipitation events in each CP class, the number of CPs is
restricted to ncp = 12 + 1 – the last CP reserved for all days that cannot be assigned to any
of the other twelve classes.
Fig. 8.15 displays the average pressure anomalies in the NCEP-NCAR data set of all days be-
longing to three CPs with high potential for extreme precipitation exceeding the 95% quan-
tile. It is surprising that CP2 (Fig. 8.9a) is among the extreme CPs. It is a high pressure
situation that generally leads to dry condition. The point is that is rarely rains on days with
CP2, but if it rains, the precitation can reach high intensities during summer months.
Fig. 8.15 displays the relative exceedance frequencies of the 95% and 99% quantile of all wet
hour precipitation amounts. In average over all CPs and both seasons the frequency is 5%
respectively 1%. Extreme intensities are generally more likely in summer. The exceedance
freqencies from May to August are higher in all CPs. However, the seasonal response of the
CPs is varying. CP2 only has an elevated extreme value potential in summer months, while
the frequencies of CP11 are among the highest in both seasons. CPs with westerly fluxes as
CP7 show the lowest increase in extreme intensity frequency during summer months. It is
less influenced by convective intensity enhancement.
(a) CP2 (b) CP7 (c) CP11
Figure 8.9.: Mean sea level pressure anomalies of CPs related to high precipitation intensities
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 167
0.0%
4.0%
8.0%
12.0%
16.0%
20.0%
CP1 CP2 CP3 CP4 CP5 CP6 CP7 CP8 CP9 CP10 CP11 CP12 CP99
Sep-Apr
May-Aug
(a) 95% quantile
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
CP1 CP2 CP3 CP4 CP5 CP6 CP7 CP8 CP9 CP10 CP11 CP12 CP99
Sep-Apr
May-Aug
(b) 99% quantile
Figure 8.10.: Exceedance frequency of the 95% and 99% quantile for two seasons.
Quantiles and frequencies are calculated for all hours with H ≥ 0.1mm
8.3.1. Changes in CP-Frequencies According to GCM and Reanalysis Data
For the analysis of the occurrence frequencies the CPs are grouped according to their meteo-
rological characteristics. Fig. 8.11 shows the evolution of the frequency of anticyclonic CPs.
During the calibration period from 1991 to 2003, one of these four CPs occurred on 35.6%
of all days, but only 17.5% of all rainy hours at the 30 rain gauges used for calibration were
counted during that time. In Fig. 8.12 the absolute frequency of cyclonic CPs is displayed.
168 Some Aspects of Climate Change
1960 1970 1980 1990 2000
year
0
20
40
60
80
100
120
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
(a) September to April
1960 1970 1980 1990 2000
year
0
10
20
30
40
50
60
70
80
90
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
(b) May to August
Figure 8.11.: Absolute frequency of anticyclonic CPs (CP2, CP3, CP5 and CP8) according to
NCEP/NCAR reanalysis during two seasons
1960 1970 1980 1990 2000
year
0
20
40
60
80
100
120
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
(a) September to April
1960 1970 1980 1990 2000
year
0
10
20
30
40
50
60
70
80
90
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
(b) May to August
Figure 8.12.: Absolute frequency of cyclonic CPs (CP7, CP10, CP11 and CP12) according to
NCEP/NCAR reanalysis during two seasons
From 1991 to 2003, one of these CPs occurred on 30.0% of all days. They are responsible for
55.4% of all hours with non-zero precipitation.
Fig. 8.11 and Fig. 8.12 suggest that the atmospheric circulation has shifted between 1958 and
2004. Especially in summer months, the number of days with antyclonic CPs has increased,
the number of days with cyclonic CPs decreased. In Fig. 8.13 and Fig. 8.14 the trend signal
is quantified by linear regression. The value of “alpha” below the equation of the regression
function is the p-value of a two sided hypothesis testing that the slope of the true regression
line is zero. If alpha is low, it is safe to reject the hypothesis, thus the increasing or decreasing
trend is significant.
Taking 5% as a limit, there is a significant inccreasing trend in high pressure situations in
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 169
1960 1970 1980 1990 2000
year
0
20
40
60
80
100
120
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=-1093.180 +  0.596x 
 alpha= 0.024 %
(a) NCEP/NCAR - September to April
1960 1970 1980 1990 2000
year
0
10
20
30
40
50
60
70
80
90
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=-707.102 +  0.380x 
 alpha= 0.033 %
(b) NCEP/NCAR - May to August
1960 1970 1980 1990 2000
year
0
20
40
60
80
100
120
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=-320.072 +  0.205x 
 alpha=21.302 %
(c) ECHAM5 - September to April
1960 1970 1980 1990 2000
year
0
10
20
30
40
50
60
70
80
90
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=-3.809 +  0.025x 
 alpha=82.660 %
(d) ECHAM5 - May to August
Figure 8.13.: Trends in the frequency of anticyclonic CPs (CP2, CP3, CP5 and CP8) accord-
ing to NCEP/NCAR reanalysis and the 20th century control run of ECHAM5
during two seasons
both seasons and a decreasing trend in low pressure situations for the non-convective season
from September to April. The trends are derived from NCEP/NCAR MSLP data that is seen
as the observation reference. (Other reanalysis products were tested too. The results stay the
same, for example using ERA40 data.)
The second line of Fig. 8.13 and Fig. 8.14 presents CP-frequencies predicted by the ECHAM5
model. The data is from the 20th century run. This is a free climate simulation but initiated
with observed starting values and forced by the real measured land use and CO2 emissions
during the last century. The CPs are defined on the basis of the available MSLP data from
1961 to 2000. Since the GCM is run in a free simulation and not forced by any measure-
ments (except for the external quantities of land use and CO2) it cannot be expected that
the absolute CP-frequencies are the same as in the reanalysis data. In average over several
years, however, the frequencies should be comparable if the GCM is able to simulate the
atmospheric circulation correctly.
170 Some Aspects of Climate Change
1960 1970 1980 1990 2000
year
0
20
40
60
80
100
120
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=167.183 + -0.043x 
 alpha=76.797 %
(a) NCEP/NCAR - September to April
1960 1970 1980 1990 2000
year
0
10
20
30
40
50
60
70
80
90
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=680.465 + -0.320x 
 alpha= 0.065 %
(b) NCEP/NCAR - May to August
1960 1970 1980 1990 2000
year
0
20
40
60
80
100
120
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=676.824 + -0.299x 
 alpha= 3.792 %
(c) ECHAM5 - September to April
1960 1970 1980 1990 2000
year
0
10
20
30
40
50
60
70
80
90
a
b
s.
 f
re
q
u
e
n
cy
 [
d
]
y=292.579 + -0.127x 
 alpha=26.576 %
(d) ECHAM5 - May to August
Figure 8.14.: Trends in the frequency of cyclonic CPs (CP7, CP10, CP11 and CP12) accord-
ing to NCEP/NCAR reanalysis and the 20th century control run of ECHAM5
during two seasons
It can be seen that ECHAM5 has limited capacities in reproducing the atmospheric condi-
tions. Average frequencies of cyclonic CPs are correctly modeled (Fig. 8.13c and Fig. 8.13d),
but ECHAM5 misses most of the trend signal. In the non-convective season the trend
is underestimated by two thirds (Fig. 8.13c), in summer months it is completely missed
(Fig. 8.13d) although anticyclonic CPs increased by almost two days in every five years be-
tween 1958 to 2004. For the anticyclonic CPs ECHAM5 models a decreasing trend from
September to April (Fig. 8.14c) that cannot be seen in the reanalysis data. The observed
decreasing trend in summer, on the other hand is underestimated (Fig. 8.14d).
The CPs are defined on the basis of normalized pressure anomalies and not on absolute
pressure values, hence the problems of ECHAM5 in reproducing the CP frequencies do not
come from different mean conditions. They express differences in the temporal sequence
of tbe MSLP-fields. The deviation between NCEP/NCAR and ECHAM5 data in CP condi-
tioned pressure anomalies are low. The average pressure anomaly maps in Fig. 8.15 derived
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 171
(a) CP2 (b) CP7 (c) CP11
Figure 8.15.: Mean sea level pressure anomalies according to ECHAM5 data of CPs related
to high precipitation intensities
from the GCM data look very similar to the pressure maps from the reanalysis (Fig. 8.9).
The centers of the high and low pressure zones are at the same locations and the shapes are
comparable.
8.3.2. Temperature Sensitivity of CP-related Precipitation
The capacity of the atmosphere to carry moisture is increasing with temperature, the evap-
oration over the ocean and over land too. It can be expected that the precipitation response
of the CPs is temperature dependent. By the average observed daily air temperature at 156
measurement stations in Baden-Württemberg the CP classes were further divided into five
temperature classes. Since temperature is a far less varying quantity than precipitation, the
set of these point measurements can be seen as representative for the average temperature
in the low level atmosphere.
The temperature subdivision is made by the observed quantile. The 20% with coldest aver-
age temperature in each CP class are declared “cold”, the next 20% cool, followed by “avg”,
“warm” and “hot” conditions. To avoid seasonal effects, the quantiles are calculated on
basis of anomalies from the average yearly cycle.
The temperature reaction is tested by the hourly precipitation frequency with a threshold
of 0.1mm and the exceedance frequency of the 95% and 99% quantile. The quantiles are
calculated based on all values exceeding the treshold. The exceedance frequenccies refer to
all wet hours (H > 0.1mm) of the respective CP.
All three tested CPs react on temperature (Fig. 8.16 to Fig. 8.18) and the reaction is in general
different for the non-convective and convective season. The results for the summer months
support the findings of Trenberth (1999) that the rainfall frequency was decreasing and the
peak intensities increasing with global atmospheric temperature. In the summer months, all
172 Some Aspects of Climate Change
hourly precipitation frequency
CP2 - Sep-Apr
0
0.02
0.04
0.06
0.08
0.1
0.12
cold cool avg warm hot
temperature class
(a) Sep to April
hourly precipitation frequency
CP2 - May-Aug
0
0.02
0.04
0.06
0.08
0.1
0.12
cold cool avg warm hot
temperature class
(b) May to August
exceedance frequency of 95%-quantile
CP2 - Sep-Apr
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
cold cool avg warm hot
temperature class
(c) 95% Quantile – Sep to April
exceedance frequency of 95%-quantile
CP2 - May-Aug
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
cold cool avg warm hot
temperature class
(d) 95% Quantile – May to August
exceedance frequency of 99%-quantile
CP2 - Sep-Apr
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
cold cool avg warm hot
temperature class
(e) 99% QuantileSep to April
exceedance frequency of 99%-quantile
CP2 - May-Aug
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
cold cool avg warm hot
temperature class
(f) 99% Quantile – May to August
Figure 8.16.: Precipitation frequency and frequency of extreme 1 h precipitation amounts
during all days belonging to CP2 as a function of temperature.
The extreme value frequency is referred to the number of wet hours
(H > 0.1mm) during CP2
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 173
hourly precipitation frequency
CP7 - Sep-Apr
0
0.05
0.1
0.15
0.2
0.25
0.3
cold cool avg warm hot
temperature class
(a) Sep to April
hourly precipitation frequency
CP7 - May-Aug
0
0.05
0.1
0.15
0.2
0.25
0.3
cold cool avg warm hot
temperature class
(b) May to August
exceedance frequency of 95%-quantile
CP7 - Sep-Apr
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
cold cool avg warm hot
temperature class
(c) 95% Quantile – Sep to April
exceedance frequency of 95%-quantile
CP7 - May-Aug
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
cold cool avg warm hot
temperature class
(d) 95% Quantile May to August
exceedance frequency of 99%-quantile
CP7 - Sep-Apr
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
cold cool avg warm hot
temperature class
(e) 99% Quantile – Sep to April
exceedance frequency of 99%-quantile
CP7 - May-Aug
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
cold cool avg warm hot
temperature class
(f) 99% Quantile – May to August
Figure 8.17.: Precipitation frequency and frequency of extreme 1 h precipitation amounts
during all days belonging to CP7 as a function of temperature.
The extreme value frequency is referred to the number of wet hours
(H > 0.1mm) during CP7
174 Some Aspects of Climate Change
hourly precipitation frequency
CP11 - Sep-Apr
0
0.05
0.1
0.15
0.2
0.25
0.3
cold cool avg warm hot
temperature class
(a) Sep to April
hourly precipitation frequency
CP11 - May-Aug
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
cold cool avg warm hot
temperature class
(b) May to August
exceedance frequency of 95%-quantile
CP11 - Sep-Apr
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
5.0%
cold cool avg warm hot
temperature class
(c) 95% Quantile – Sep to April
exceedance frequency of 95%-quantile
CP11 - May-Aug
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
cold cool avg warm hot
temperature class
(d) 95% Quantile – May to August
exceedance frequency of 99%-quantile
CP11 - Sep-Apr
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
cold cool avg warm hot
temperature class
(e) 99% Quantile – Sep to April
exceedance frequency of 99%-quantile
CP11 - May-Aug
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
cold cool avg warm hot
temperature class
(f) 99% Quantile – May to August
Figure 8.18.: Precipitation frequency and frequency of extreme 1 h precipitation amounts
during all days belonging to CP11 as a function of temperature.
The extreme value frequency is referred to the number of wet hours
(H > 0.1mm) during CP11
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 175
CPs show a pronounced drop in precipitation frequency with increasing temperature while
during the rest of the year the rainfall frequency is constant (CP7 and CP11) or slightly
increasing (CP2).
The extreme intensity reacts more sensitively on temperature than the rainfall frequency,
especially in summer months. If it is raining, the exceedance frequency of the 99% quantile
on CP7 “hot” days is more than three times higher than the average over all days belong-
ing to CP7 (Fig. 8.17f). And it seems that the highest extremes are affected the most. The
differences in the 99% quantile are far more pronounced than in the 95% quantile. During
CP7 and CP11 the increase in wet hour extreme frequency overcompensates the decrease in
precipitation frequency. Refered to all hours, the frequency of extrem hourly precipitation is
higher than average on “hot” days – even if it rains rarely.
CP2, however, reacts differently. The highest extreme value frequencies, referred to all
hours, are measured during days of average temperature. There are several probable ex-
plications. Either the increasing moisture storage capacity of hot air inhibits precipitation
(as it is described in Kendon et al. (2010)). However, it is not clear why this should espe-
cially effect the most extreme events. Or it depends on the available soil moisture. The
evaporation consumes energy and thus reduces the air temperature. Hence, the days with
the highest energy level might not be the hottest days because part of the energy goes into
latent heat. If the water availability to evaporation is limited, the temperature can rise more
and precipitation is reduced. The effect is not present in the other CPs since they receive
moisture by advective transport from the Atlantic Ocean (CP7) or the North Sea and Polar
Sea (CP11).
Besides, intense precipitation is always linked to cooling. Much of the raindrop water in
estival thunder storms is directly evaporated before the drop reaches the ground (Weischet
and Endlicher, 2008). The energy for evaporation is delivered by the surrounding air leading
to a drop in temperature. Since most thunderstorms occur in the afternoon, the cooling
affects the average daily air temperature. A more representative measure in this respect
would be the average temperature during the time that the thunderstorm builds up, or the
maximum just before it begins to rain, which both is not available.
During CP11, the coldest days have particularly high extreme value frequencies (Fig. 8.18d
and Fig. 8.18d). This is another indication that the average daily temperature cannot mea-
sure all effects of temperature sensitivity. During CP11 the atmospheric flow comes from
the north. If the atmosphere was warm before, the drop in temperature when the northern
flux sets in releases much water, due to the decreasing storage capacity of the atmosphere.
The amount of released atmospheric water is a function of the temperature difference and
depends on the amount of available water that is present from the days before. Therefore, it
could be beneficial to consider the temperature history before a precipitation event.
8.3.3. Expected Trend in 1h Precipitation Amounts due to Changes in
Atmospheric Circulation
During the last forty years, the CP sequence was significantly changing. The CPs in turn
have a high influence on the distribution of 1h precipitation amounts. In the same time, CP-
176 Some Aspects of Climate Change
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Figure 8.19.: Expected precipitation sum and expected extreme value probability estimated
by the CP sequence based on ECHAM5 data for 1961 to 2060, CO2 emission
scenario A1B
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 177
related precipitation is reacting on temperature. The question is, how the different mecha-
nism are combining in the future.
For a test it is assumed that the precipitation response of each CP in each temperature class
is constant over time. The CP sequence of the future can be estimated by applying the fuzzy
rule set developed on basis of the NCEP/NCAR reanalysis on MLSP data from GCM runs
for the future. The atmospheric temperature level can directly be derived from the GCM.
Temperature is seen as the most reliable quantity in GCM output, (see for example Kendon
and Clark, 2008).
If the time series of the combined CP-temperature classes is known, the expectation of differ-
ent precipitation related statistics can be calculated for the future. The method is illustrated
by the example of the 95% precipitation quantile: It is assumed that a day i between May
and August in the year j is assigned to CP2 and the temperature level to “warm”. Then
the 95% extreme value probability is assumed to be 13.6% and the rainfall probability 18.6%
(see Fig. 8.16d and Fig. 8.16b) according to the average observed values in the calibration
period. In reference to all hour (wether it is raining or not), the 95% extreme precipitation
probability is P95% = 0.136 ·0.186 = 2.53% in each hour of day i. Doing the same calculation
for all days i ∈ [1, 365] of one year (with the statistics of the respective season) and averaging
over the year, one can calculate the expectation of the 95% extreme value probability for the
respective year. (Since the calculation of the 95% quantile is based on wet hour only, the
expected exceedance frequencies in respect to all hours in the year is lower than 5%.)
The described method is applied to ECHAM5 data of the time period from 1961 to 2060.
Until 2000, data from the 20th century run is used. The future climate predictions start in
the year 2001. The considered scenario is A1B. Yearly expectations of the 95% and 99%
extreme probability are calculated and similarly of the yearly preciptiation sum (in this case
not by averaging but by adding up the contributions of each day). An additional analysis is
performed for the convective summer months from May to August.
Compared to the reanalysis, the GCM tends to dryer CP-temperature combinations
(Fig. 8.19a and Fig. 8.19b). The average precipitation in the control period from 1961 to
2000 is underestimated . It spreads around 600mm per year. The observed average over
the 30 precipitation station used for calibration is about 150mm higher. In all the graphs
of Fig. 8.19, the step from year 2000 to 2001, when the data basis is changing from control
period to scenario run, is marked by a large jump.
Dividing the GCM data into control and scenario run, trend signals can be found that are
hidden when the data is combined (Fig. 8.20 and Fig. 8.21). The exceedance frequencies
of the 95% and 99% quantile are increasing. The increase is strongest in the exceedance
of the 99% quantile in summer months (Fig. 8.21e and Fig. 8.21f), indicating a shift in the
distribution towards the highest events. In the same time the precipitation sum during
summer months is decreasing (Fig. 8.21a and 8.21b) while the yearly precipitation sum stays
about the same (Fig. 8.20a and Fig. 8.20b), indicating a shift in the precipitation activity
towards winter.
The trend signals in the control period and the scenario run correspond well. Except for
the summer precipitation sum, the slopes of the linear regression function in both periods
178 Some Aspects of Climate Change
1960 1970 1980 1990 2000
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Figure 8.20.: Linear trend in the theoretic expectation of yearly precipitation sum and
yearly extreme value probability according to the CP sequence derived from
ECHAM5 MSLP field
8.3 Prediction of Future Precipitation by GCM Defined Circulation Patterns 179
1960 1970 1980 1990 2000
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(f) 99% quantile – May to August
Figure 8.21.: Linear trend in the theoretic expectation of precipitation sum and extreme value
probability for the summer months according to the CP sequence derived from
ECHAM5 MSLP field
180 Some Aspects of Climate Change
do not differ by more than some percent. On the other hand, the absolute values are dis-
continuous. At the break point between control and scenario run, from 2000 to 2001, the
extreme value probabilities drop again to the level of 1960 to start increasing a new. Only in
the precipitation sum, the end of the regression function of the control period is close to the
starting point for the future period.
8.4. Conclusions
8.4.1. Reliability of Estimated Trend Signals
• The capacity of GCMs to model realistic precipitation time series is very limited. It is
reported from literature that the variability and extreme value probability are under-
estimated. As the model uncertainties are scale dependent, it should be avoided to use
GCM precipitation data in high temporal resolution. The use of aggregated data, for
example monthly precipitation sums, is preferable.
• By QQ-Transformation, monthly GCM or RCM precipitation sums can be corrected
according to observed precipitation values from a control period. However, QQ-
Transformation, as most direct correction methods, is only valid on higher temporal
aggregations. The correction of daily or subdaily data is not recommended.
• GCM predictions of Mean Sea Level Pressure (MSLP) and temperature are considered
as more reliable than precipitation data. An alternative to the direct use of GCM or
RCM precipitation is to relate predictions of MSLP and temperature to an expected
precipitation distribution by an atmospheric Circulation Pattern (CP) classification
system.
Applying the classification, information on hourly temporal scale is derived from the
GCM merely by using daily MLSPE and temperature values. Hence, it can be seen as
a temporal downscaling technique whereas the precipitation distribution is estimated
indirectly by a classification of past time observations. The underlying assumption is
that the future precipitation response of each CP-temperature class is constant. Trends
that are not related to CP or temperature or will be missed. The same can happen
if the future brings new atmospheric conditions that where not observed during the
calibration period and do not fit in the CP classification.
• If a trend in precipitation is found by different methods in different data sets, for ex-
ample in observed rainfall records as well as in precipitation from GCM simulations
or the expected precipitation according to the CP sequence in the GCM, it can be con-
sidered as a reliable signal and used for future prognosis.
The increase in winter precipitation and decrease in summer precipitation, that is pre-
dicted by the CP-temperature sequence (Section 8.3), is confirmed by observed trends
(Hundecha and Bárdossy, 2005) as well as by direct analysis of RCM precipitation
(Déqué et al., 2007). The increase in extreme precipitation and the related shift in scal-
ing toward more intensive short term events that has been deduced from observed
8.4 Conclusions 181
precipitation measurements (see Section 8.1) is also found in the expected 1h precipi-
tation statistics according to the CP-temperatures classification for the future (Section
8.3). In a physically sound interpretation, these purely statistical results can be linked
to higher atmospheric temperature (Trenberth, 1999).
Therefore, these trend signals are considered as reliable predictions of the future. If so,
the potential for water stress and catastrophic extreme events in summer are both in-
creasing in the same time. Especially agriculture will suffer from the expected changes
in summer. The effort for irrigation will raise, but more crops will be destroyed in ex-
treme events (e. g. hail storms) too.
• Although the general tendencies of the trend signals are confirmed by different stud-
ies, high uncertainty remains about the absolute values. The CP sequence modeled by
the GCM for example misses some of the trend signal in the observed CP time series.
Especially, the decreasing trend in cyclonic situations during summer is ignored. If the
trend to less cyclonic CPs continues, it will have two effects. On the one hand, the pre-
cipitation sum in summer will decrease faster than predicted by the CP sequence of the
GCM. On the other hand, the extreme value probability will be lower than predicted
because extreme precipitation is more frequent during cyclonic CPs.
8.4.2. Potential for Climate Change Adaptations in Rainfall Generation
In perspective of the estimated trends, there ares several possibilities for modifications of
the resampling based rainfall generators presented in Chapter 6 and Chapter 7.
• The time series of monthly precipitation sum could be replaced by a QQ-Transformed
time series of monthly precipitation from a GCM model.
• The daily rainfall frequency and daily average precipitation of the CP-groups that are
used as a target function in the resampling scheme (see Section 6.6.2.1 and Section
7.6.2) could be replaced by a CP and temperature dependent classification according
to Section 8.3. The CP and temperature time series could be derived from GCM data.
• The target values of the scaling exponent (Eq. (6.76)) in the first three statistical mo-
ments of precipitation on different aggregations from 1h to 24h, that is part of the ob-
jective function in the resampling scheme of the single site generation (Section 6.6.2.1)
or the disaggregation from daily to hourly values in generation of simultaneous time
series (Section 7.6.3), could be replaced by a time dependent function that extrapolates
the trend observed between the two time periods from 1958 to 1980 and from 1981 to
2003 (Section 8.1.2)
Future predictions are not available for all statistics used in the presented resampling based
rainfall generator. For example, it has to be assumed that the autocorrelation structure is not
altered in the future. Replacing only some parameters by future projections induces hetero-
geneity in the parameter set which can alter the interactions in the generation algorithm.
The basic idea of the generators presented in Chapter 6 and Chapter 7 is that the interde-
pendencies between the precipitation probability, as well as the average and the standard
182 Some Aspects of Climate Change
deviation of the precipitation amounts in wet time steps can be expressed as a function of
the the monthly sum. If the time series of the observed monthly sum was replaced by GCM
data, it would imply two assumptions; firstly that the relation between these four statistics
is the same in the future as it was in the observed past and secondly that the relation is not
altered by errors in the GCM time series. Especially the second assumption is problematic,
considering the high errors in uncorrected GCM precipitation. Correction methods, like
QQ-Transformation, can only eliminate a part of the errors.
In the observations there is obviously a close relation between the monthly precipitation
sum and the CP related precipitation statistics as each month consists of a 28 to 31 days
long CP sequence. In the future predictions, the monthly precipitation sum and the CP
related precipitation origin from different sources. While the monthly precipitation is a
direct output of the GCM, the CP related precipitation arise from observed distributions that
are only conditioned on the GCM. This can lead to conflicts in the permutation scheme. If
the monthly precipitation sum derived from the CGM is far off the expected sum according
to the CP sequence, the objective function part evaluating the CP dependent statistics will
demand for unrealistic values and the optimisation will fail.
Finally, and most problematic for the use in rainfall generation, it is doubtful weather the
ECHAM5 data set is homogeneous. The jump in most statistics between the control pe-
riod and the future period indicates that it might not be the case. If there is a break point
between the control run and the future scenario run, the error structure in the scenario is dif-
ferent than in the control run. This affects not only the hydrological response of the CP tem-
perature sequence in the future but also breaks the assumption of the QQ-Transformation,
Delta-Chance Method or other correction methods that the relation between model and ob-
servation of the control period is constant in the future.
8.4.3. Future Predictions as an Ensembles
A remaining problem is the choice of the emission scenario and most of all of the Circulation
Model from which the data is taken. For the scenarios, one could argue by the aim of the
simulation, for example to always take the A2 emission scenario for a “worst case” study
with the most pronounced changes. However, such reasoning is not possible for choice of
the Circulation Model as there is no general order from the model with the least changes to
the model with the most severe consequences. A priori all models have the same probability
of giving a realistic picture of the future. All claim to represent the major physical processes
involved in the movement of air and moisture in the atmosphere. Which of the models
performs best cannot be judges from today’s point of view. If only one model scenario
combination is chosen, it is very likely that it is not the combination giving the best prediction
of future climate conditions.
A good way to deal with the uncertainty is to consider all the different data sets as an en-
semble. It means that the rainfall generation is run several times with all available data
sets from the different model-scenario combinations. The resulting time series are used as a
range of possible future precipitation. All following applications, as for example hydraulic
sewage system modeling, have to be run with the whole ensemble too. The result is not one
8.4 Conclusions 183
crisp prognosis of the future conditions but a range of possible outcomes considering the
uncertainty in the applied models and future scenarios. For the hydraulic application, for
example the sewage system dimensioning, the user can judge the uncertainties and account
for it by engineering solutions.
Of course, this is an ambitious task as it multiplies the effort for the computationally de-
manding rainfall generation and all following hydrological or hydraulic applications. If the
same GCM and reanalysis data as in Africa case study was available, it would imply the
set up 18 different data bases for the rainfall generator (with 18 different time series of the
monthly precipitation sums and of the sequence of CP and temperature classes) and to run
it 18 times for each location.
However, even with the ensemble data set, there is no guarantee that the future lies within
the predicted range. If all the models share a common error, for example in the representa-
tion of the physical processes or the initial conditions, they can all be off target.
9. Conclusions and Outlook
The monthly precipitation sum is identified as one of the main statistical characteristics of
precipitation. The rainfall probability, as well as the mean and standard deviation are to
some extend determined by the monthly rainfall sum. Therefore, the distribution of hourly
rainfall depths can be deduced if the monthly sum is known. The relation is exploited
for the generation of hourly rainfall time series that are conditioned on the time series of
monthly sums. Since the monthly sum exhibits low spatial variability, the regionalization
to ungauged locations is easily conducted by geostatistical methods.Therefore, the devel-
oped time series generator can be applied at every location in the study region of Baden-
Württemberg.
The interdependencies of the four statistical parameters monthly sum, rainfall probability,
mean and standard deviation are described by a Gaussian copula in the uniform [0, 1] space.
The interdependencies are a characteristic of the regional climatic conditions and thus re-
garded as constant over the study region. Local effects find their expression in the absolute
values. They are described by the marginal distributions.
The developed time series generator performs a two step simulation. First, an initial time
series of hourly values is set up. Then, it is optimized in a permutation scheme according to
target values of different statistics concerning the temporal persistence, the extreme value
probabilities and the temporal scaling over different aggregations.
A comparison between simulated and observed time series shows that the generator deliv-
ers realistic simulations. It was tested by extreme value statistics on different temporal ag-
gregations and by the distribution of wet and dry spell lengths as a measure of the temporal
persistence. Since a data driven approach takes maximum benefit from the high density of
rain gauges in Baden-Württemberg, it produces realistic results on a wide range of tempo-
ral scales from 1h to 24h. This is a clear advantage compared to most conceptual rainfall
models that perform best on the scale of the calibration data, but lack of variance in higher
temporal aggregations. As another feature, the developed generator is able to model the
difference in distribution between summer and winter months. It represents the seasonal
variability in a realistic manner.
The generation scheme was extended for the generation of several, spatially interdependent,
simultaneous time series. The spatial interdependencies are considered on three different
scales: on the monthly scale by a Gaussian copula that describes the interdependencies of the
generation parameters at all simulation locations, by the correlation of daily precipitation
sums and by the correlation of hourly values conditioned on the atmospheric Circulation
Pattern (CP).
185
The spatial temporal interdependencies of observed precipitation measurements are statis-
tically very complex and cannot be considered without simplifications. The generations for
locations of very short distance (few kilometers) are especially difficult. For some configura-
tions of the station locations there are inconsistencies between the parameter interdependen-
cies and the target values of daily correlation. Secondly, the estimation of the CP dependent
correlations is subject to high uncertainty due to the low number of observed rain gauges
in short distances. Generally, there is high spread in the parameter interdependencies of
different locations. Systematic effects, however, that are responsible for the high observed
variability, e. g. anisotropy or non-stationarity in the relations, could not be identified.
In recent years, researchers have explored new methods for the description of dependence.
Bárdossy and Pegram (2009) analyze the joint exceedance probabilities of different thresh-
old in triplets of rain gauge records by means of the Shannon entropy. Observed triplets ex-
hibit lower Shannon entropy than triplets modeled by correlation matrices or copula. Since
the Shannon entropy is a measure of disorder, it means that there is a part of the interde-
pendence information that is ignored by pairwise correlations, e. g. a higher probability
of simultaneous extreme events. Incorporating such information, especially of the tail de-
pendence in the highest events, could further improve the generation of simultaneous time
series.
For small sewage systems hourly temporal resolution is not yet sufficient and the generated
time series have to be further disaggregated to five minute values. Hydraulic sewage sys-
tem models react very sensitively to persistence on short temporal scale. The way in which
the disaggregation is performed has a high influence on the modeling result. Therefore,
the development of an accurate disaggregation scheme for simultaneous time series should
be the next research task in this context. The representation of spatial dependence in five
minute values, however, will be difficult. Probably, the interdependencies are limited to a
close range of several kilometers that is very poorly sampled in the existing data set. Fur-
thermore, the simultaneous five minute time series are highly affected by synchronization
errors between the rain gauges.
Another important research task is the estimation of rainfall probability, which is one of the
main parameters in the developed rainfall generation algorithms. If the estimated rainfall
probability is wrong, it affects many of the other statistical parameters too, e. g. the average
and the standard deviation of precipitation of wet time steps or the scaling behavior. Un-
fortunately, the relative error in rain gauge measurements is highest in the smallest values.
Therefore, observed rainfall frequencies are subject to high uncertainty. For accurate cali-
bration of the rainfall generators a more robust estimation of rainfall probability would be
a clear improvement. A possible approach could be to extrapolate the rainfall probability
from the exceedance frequencies of higher thresholds that are less susceptible to measure-
ment errors.
A strong point of the developed rainfall models is the applicability in climate change stud-
ies. In the last decade climate model output was evaluated in various studies. It has been
shown that GCMs model real world climatic conditions insufficiently, especially in terms of
precipitation. Although GCMs are permanently improved, the global climate is probably
way too complex to be ever modeled correctly. Therefore, studies are promising that change
186 Conclusions and Outlook
the focus from “best guess” for a certain target year in the future to “what if” like the work
of Kendon et al. (2010) that was presented in Chapter 8. Such studies contribute to a better
understanding of the climate system, independently of the precise outcome of the climate
model.
In this context, data driven approaches can be of great benefit for the understanding of cli-
mate change effects on precipitation of high temporal resolution. The structure of the devel-
oped rainfall generation models allows to modify every generation parameter separately.
Since all parameters have clear statistical meanings, they can be adapted to the results of
climate change studies. In Section 8.1.2 for example, it was shown that climate change ef-
fects the scaling of precipitation on different aggregations. This could be easily modeled by
the modification of the respective parameter in the objective functions. In the same way, a
higher variability in monthly sums, a change in distribution or a reduced rainfall probabil-
ity in summer months could be modeled. A comparison of the model results can be used
to quantify the effect of each possible change signal as well as potential reinforcement or
compensation effects in their combination.
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R. Vitolo, D. B. Stephenson, I. M. Cook, and K. Mitchell-Wallace. Serial clustering of in-
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W. Weischet and W. Endlicher. Einführung in die allgemeine Klimatologie. Gebrüder Born-
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F. J. Wentz, L. Ricciardulli, K. Hilburn, and C. Mears. How much more rain will global
warming bring? Science, 317:233–235, 2007.
K. J. Weston and M. G. Roy. The directional-dependence of the enhancement of rainfall over
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1994.
WHG. Wasserhaushaltsgesetz der bundesrepublik deutschland, 2009.
D. Wilks. Multisite generalization of a daily stochastic precipitation generation model. Jour-
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D. S. Wilks. Statistical Methods in Atmospheric Sciences. Academic Press, third edition, 2011.
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P. Willems and M. Vrac. Statistical precipitation downscaling for small-scale hydrological
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L. Zhang and V. P. Singh. Bivariate rainfall frequency distributions using archimedean cop-
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A. Generation of Point Rainfall
A.1. Marginal Distributions of All 292 Stations
A.1.1. Monthly Precipitation Sum
50 100 150 200 250 300 350 400
msum
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
re
l.
 f
re
q
. 
/ 
p
d
f
n=  40294
(a) December - February
50 100 150 200 250 300 350 400
msum
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
re
l.
 f
re
q
. 
/ 
p
d
f
n=  40294
(b) March - May
50 100 150 200 250 300 350 400
msum
0.000
0.002
0.004
0.006
0.008
0.010
0.012
re
l.
 f
re
q
. 
/ 
p
d
f
n=  40295
(c) June - August
50 100 150 200 250 300 350 400
msum
0.000
0.002
0.004
0.006
0.008
0.010
0.012
re
l.
 f
re
q
. 
/ 
p
d
f
n=  40291
(d) September - November
Figure A.1.: Histogram and Weibull PDF of the monthly precipitation sum at 292 stations
198 Generation of Point Rainfall
4 2 0 2 4 6 8
x*
14
12
10
8
6
4
2
0
2
4
y
*
n=  40294
empirical CDF
weibull-fit
(a) December - February
1 2 3 4 5 6 7
x*
12
10
8
6
4
2
0
2
4
6
y
*
n=  40294
empirical CDF
weibull-fit
(b) March - May
1 2 3 4 5 6 7
x*
12
10
8
6
4
2
0
2
4
y
*
n=  40295
empirical CDF
weibull-fit
(c) June - August
0 1 2 3 4 5 6 7
x*
12
10
8
6
4
2
0
2
4
y
*
n=  40291
empirical CDF
weibull-fit
(d) September - November
Figure A.2.: Observed distribution and Weibull fit of the monthly precipitation sum at 292
stations in the linearized space
A.1 Marginal Distributions of All 292 Stations 199
A.1.2. Average Precipitation in Wet Hours
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
pavg
0.0
0.5
1.0
1.5
2.0
2.5
re
l.
 f
re
q
. 
/ 
p
d
f
n=   6697
(a) December - February
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
pavg
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
re
l.
 f
re
q
. 
/ 
p
d
f
n=   6823
(b) March - May
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
pavg
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7046
(c) June - August
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
pavg
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7033
(d) September - November
Figure A.3.: Histogram and Log-Normal PDF of the average precipitation in wet hours at
292 stations
200 Generation of Point Rainfall
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5
log(x) observed
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   6697
empirical distribution
log-normal
(a) December - February
3 2 1 0 1 2 3
log(x) observed
3
2
1
0
1
2
3
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   6823
empirical distribution
log-normal
(b) March - May
1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5
log(x) observed
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   7046
empirical distribution
log-normal
(c) June - August
2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0
log(x) observed
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   7033
empirical distribution
log-normal
(d) September - November
Figure A.4.: Observed distribution and Log-Normal fit of the average precipitation in wet
hours at 292 stations in the linearized space
A.1 Marginal Distributions of All 292 Stations 201
A.1.3. Standard Deviation of Precipitation Amounts in Wet Hours
0 1 2 3 4 5 6 7 8
pstd
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
re
l.
 f
re
q
. 
/ 
p
d
f
n=   6696
(a) December - February
0 1 2 3 4 5 6 7 8
pstd
0.0
0.2
0.4
0.6
0.8
1.0
1.2
re
l.
 f
re
q
. 
/ 
p
d
f
n=   6823
(b) March - May
0 1 2 3 4 5 6 7 8
pstd
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7046
(c) June - August
0 1 2 3 4 5 6 7 8
pstd
0.0
0.2
0.4
0.6
0.8
1.0
1.2
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7033
(d) September - November
Figure A.5.: Histogram and Log-Normal PDF of the standard deviation of the precipitation
amounts in wet hours at 292 stations
202 Generation of Point Rainfall
3 2 1 0 1 2 3
log(x) observed
3
2
1
0
1
2
3
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   6696
empirical distribution
log-normal
(a) December - February
4 3 2 1 0 1 2 3
log(x) observed
4
3
2
1
0
1
2
3
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   6823
empirical distribution
log-normal
(b) March - May
2 1 0 1 2 3
log(x) observed
2
1
0
1
2
3
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   7046
empirical distribution
log-normal
(c) June - August
4 3 2 1 0 1 2 3
log(x) observed
4
3
2
1
0
1
2
3
lo
g
(x
) 
th
e
o
re
ti
ca
l
n=   7033
empirical distribution
log-normal
(d) September - November
Figure A.6.: Observed distribution and Log-Normal fit of the standard deviation of precipi-
tation amounts in wet hours at 292 stations in the linearized space
A.1 Marginal Distributions of All 292 Stations 203
A.1.4. Hourly Rainfall Probability
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0
1
2
3
4
5
6
7
re
l.
 f
re
q
. 
/ 
p
d
f
n=  17408
(a) December - February
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0
1
2
3
4
5
6
7
8
re
l.
 f
re
q
. 
/ 
p
d
f
n=  17483
(b) March - May
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0
2
4
6
8
10
12
re
l.
 f
re
q
. 
/ 
p
d
f
n=  17751
(c) June - August
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0
1
2
3
4
5
6
7
8
re
l.
 f
re
q
. 
/ 
p
d
f
n=  17540
(d) September - November
Figure A.7.: Histogram and Beta PDF of the monthly 1h rainfall probability at 292 stations
204 Generation of Point Rainfall
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=  17408
empirical CDF
beta-fit
(a) December - February
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=  17483
empirical CDF
beta-fit
(b) March - May
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=  17751
empirical CDF
beta-fit
(c) June - August
0.0 0.2 0.4 0.6 0.8 1.0
P(1h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=  17540
empirical CDF
beta-fit
(d) September - November
Figure A.8.: Observed distribution and Beta fit of the monthly 1h rainfall probability at 292
stations
A.2 Drawing Conditioned Values from a Multivariate Normal Distribution 205
A.2. Drawing Conditioned Values from a Multivariate Normal
Distribution
A multivariate Gaussian random vectorX can be generated from a vector of multivariate in-
dependent standard normal random numbers if the covariance matrix and the mean vector
are known:
X ∼ N (µ+ Σ) (A.1)
with X multivariate Gaussian random variable
N Normal distribution function
µ vector of averages in X
Σ matrix of covariances in X
Dividing X in two vectors X = [X1, X2]T , Σ and µ can be written as:
µ =
[
µ1
µ2
]
with sizes
[
q × 1
(N − q)× 1
]
(A.2)
Σ =
[
Σ11 Σ12
Σ21 Σ22
]
with sizes
[
q × q q × (N − q)
(N − q)× q q × q
]
(A.3)
The distribution of X conditioned on a value of X2 = a is given by:
(X1|X2 = a) = µ1|a + Σ1|aZ1 (A.4)
with Z1 multivariate standard normal random variable of size q × 1
µ1|a average vector of X1 conditioned on X2 = a of size q × 1
Σ1|a covariance matrix of X1 conditioned on X2 = a of size q × q
Applying the convention of Eq. (A.3) the conditional mean and covariance can be written
as:
µ1|a = µ1 + Σ12Σ−122 (a− µ2) (A.5)
Σ1|a = Σ11 −Σ12Σ−122 Σ21 (A.6)
The conditioned covariance Σ1|a is independent of the conditioning value a. The shape and
spread of the distribution is independent of a, only its location is changing according to
Eq. (A.5).
Correlated random numbers from the conditioned normal distribution is are drawn by
means of a vector Z of uncorrelated standard normal random variables and the lower trian-
gle matrix L of the Cholesky decomposition of Σa
x∗ = µ1|a + L · z (A.7)
Σ1|a = LLT (A.8)
B. Spatial Generation
B.1. Observed Multivariate Dependencies in Daily Precipitation
Data
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
2
4
h
 r
a
in
fa
ll 
fr
e
q
u
e
n
cy
sym= 0.00041
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) April
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
2
4
h
 r
a
in
fa
ll 
fr
e
q
u
e
n
cy
sym= -0.00431
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) July
Figure B.1.: Empirical bivariate copula between monthly precipitation sum and rainfall fre-
quency in daily resolution calculated from month-wise censored data
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
2
4
h
 p
re
ci
p
it
a
ti
o
n
 s
u
m
sym= 0.00072
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) March
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
2
4
h
 p
re
ci
p
it
a
ti
o
n
 s
u
m
sym= -0.00020
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) June
Figure B.2.: Empirical bivariate copula between monthly precipitation sum and average wet
day precipitation sum calculated from month-wise censored data
B.1 Observed Multivariate Dependencies in Daily Precipitation Data 207
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
2
4
h
 p
re
ci
p
ip
it
a
ti
o
n
 s
u
m
sym= -0.00007
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) March
0.0 0.2 0.4 0.6 0.8 1.0
monthly rainfall sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
2
4
h
 p
re
ci
p
ip
it
a
ti
o
n
 s
u
m
sym= -0.00115
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) June
Figure B.3.: Empirical bivariate copula between monthly precipitation sum and standard
deviation of wet day precipitation sum calculated from month-wise censored
data
0.0 0.2 0.4 0.6 0.8 1.0
24h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
2
4
h
 p
re
ci
p
it
a
ti
o
n
 s
u
m
sym= -0.00386
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) January
0.0 0.2 0.4 0.6 0.8 1.0
24h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 a
v
e
ra
g
e
 o
f 
2
4
h
 p
re
ci
p
it
a
ti
o
n
 s
u
m
sym= -0.00543
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) July
Figure B.4.: Empirical bivariate copula between monthly rainfall frequency in daily resolu-
tion average and wet day precipitation sum calculated from month-wise cen-
sored data
208 Spatial Generation
0.0 0.2 0.4 0.6 0.8 1.0
24h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
2
4
h
 p
re
ci
p
ip
it
a
ti
o
n
 s
u
m
sym= -0.00501
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) January
0.0 0.2 0.4 0.6 0.8 1.0
24h rainfall frequency
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
2
4
h
 p
re
ci
p
ip
it
a
ti
o
n
 s
u
m
sym= -0.00621
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) July
Figure B.5.: Empirical bivariate copula between monthly rainfall frequency in daily reso-
lution and standard deviation of wet day precipitation sum calculated from
month-wise censored data
0.0 0.2 0.4 0.6 0.8 1.0
monthly average of 24h precipitation sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
2
4
h
 p
re
ci
p
ip
it
a
ti
o
n
 s
u
m
sym= 0.00030
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(a) January
0.0 0.2 0.4 0.6 0.8 1.0
monthly average of 24h precipitation sum
0.0
0.2
0.4
0.6
0.8
1.0
m
o
n
th
ly
 s
td
e
v
 o
f 
2
4
h
 p
re
ci
p
ip
it
a
ti
o
n
 s
u
m
sym= -0.00043
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b) August
Figure B.6.: Empirical bivariate copula between average and standard deviation of wet day
precipitation sum calculated from month-wise censored data
B.2 Marginal Distribution in Daily Resolution 209
B.2. Marginal Distribution in Daily Resolution
B.2.1. Average Precipitation in Wet Hours
5 10 15 20 25 30
pavg
0.00
0.05
0.10
0.15
0.20
0.25
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49936
(a) December - February
5 10 15 20 25 30
pavg
0.00
0.05
0.10
0.15
0.20
0.25
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49954
(b) March - May
5 10 15 20 25 30
pavg
0.00
0.05
0.10
0.15
0.20
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49910
(c) June - August
5 10 15 20 25 30
pavg
0.00
0.05
0.10
0.15
0.20
0.25
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49857
(d) September - November
Figure B.7.: Histogram and Log-Normal PDF of the average wet day precipitation at 575
stations
210 Spatial Generation
(a) December - February (b) March - May
(c) June - August (d) September - November
Figure B.8.: Observed distribution and Log-Normal fit of the average wet day precipitation
575 stations in the linearized space
B.2 Marginal Distribution in Daily Resolution 211
B.2.2. Standard Deviation of Precipitation Amounts in Wet Hours
0 5 10 15 20 25 30
pstd
0.00
0.05
0.10
0.15
0.20
0.25
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49875
(a) December - February
0 5 10 15 20 25 30
pstd
0.00
0.05
0.10
0.15
0.20
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49954
(b) March - May
0 5 10 15 20 25 30
pstd
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49903
(c) June - August
0 5 10 15 20 25 30
pstd
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
re
l.
 f
re
q
. 
/ 
p
d
f
n=  49814
(d) September - November
Figure B.9.: Histogram and Log-Normal PDF of the standard deviation of wet day precipi-
tation amounts at 575 stations
212 Spatial Generation
(a) December - February (b) March - May
(c) June - August (d) September - November
Figure B.10.: Observed distribution and Log-Normal fit of the standard deviation of wet day
precipitation amounts at 575 stations in the linearized space
B.2 Marginal Distribution in Daily Resolution 213
B.2.3. Daily Rainfall Probability
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.5
1.0
1.5
2.0
2.5
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7237
(a) December - February
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.5
1.0
1.5
2.0
2.5
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7561
(b) March - May
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7969
(c) June - August
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
re
l.
 f
re
q
. 
/ 
p
d
f
n=   7565
(d) September - November
Figure B.11.: Histogram and Beta PDF of the monthly 24h rainfall frequency at 575 stations
214 Spatial Generation
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=   7237
empirical CDF
beta-fit
(a) December - February
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=   7561
empirical CDF
beta-fit
(b) March - May
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=   7969
empirical CDF
beta-fit
(c) June - August
0.0 0.2 0.4 0.6 0.8 1.0
P(24h)
0.0
0.2
0.4
0.6
0.8
1.0
y
n=   7565
empirical CDF
beta-fit
(d) September - November
Figure B.12.: Observed distribution and Beta fit of the monthly 24h rainfall frequency at 575
stations
B.3. Representation of Spatial Dependencies
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
4
0
8
0
1
2
0
1
6
0
2
0
0
2
4
0
2
8
0
3
2
0
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
2
0
4
0
6
0
8
0
1
0
0
1
2
0
1
4
0
1
6
0
1
8
0
nb of values per cell
(b) June to August
Figure B.13.: Observed rank correlation between the monthly precipitation sum at two sta-
tions depending on the separation distance
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
1
2
0
1
3
5
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
nb of values per cell
(b) June to August
Figure B.14.: Observed rank correlation between the average wet day precipitation at two
stations depending on the separation distance
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
1
2
0
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
nb of values per cell
(b) June to August
Figure B.15.: Observed rank correlation between the standard deviation of wet day precipi-
tation at two stations depending on the separation distance
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
3
0
6
0
9
0
1
2
0
1
5
0
1
8
0
2
1
0
2
4
0
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
1
2
0
1
3
5
nb of values per cell
(b) June to August
Figure B.16.: Observed rank correlation between the monthly precipitation sum at one sta-
tion and the rain day frequency at another station depending on the separation
distance
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
2
0
4
0
6
0
8
0
1
0
0
1
2
0
1
4
0
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
nb of values per cell
(b) June to August
Figure B.17.: Observed rank correlation between the monthly precipitation sum at one sta-
tion and the average wet day precipitation at another station depending on the
separation distance
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
1
2
0
1
3
5
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
nb of values per cell
(b) June to August
Figure B.18.: Observed rank correlation between the monthly precipitation sum at one sta-
tion and the standard deviation of wet day precipitation at another station de-
pending on the separation distance
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
5
3
0
4
5
6
0
7
5
9
0
1
0
5
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
nb of values per cell
(b) June to August
Figure B.19.: Observed rank correlation between the rain day frequency at one station and
the average wet day precipitation at another station depending on the separa-
tion distance
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
1
0
0
nb of values per cell
(a) December to February
0
1
0
2
0
3
0
4
0
5
0
d
ista
n
ce
 [km
]
0
.0
0
.2
0
.4
0
.6
0
.8
1
.0
correlation [-]
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
nb of values per cell
(b) June to August
Figure B.20.: Observed rank correlation between the rain day frequency at one station and
the average wet day precipitation at another station depending on the separa-
tion distance
Curriculum Vitae
born on May 10, 1979 in Backnang, Germany
Experiences
2006 - 2012 University of Stuttgart,
Institute for Modelling Hydraulic and Environmental Systems
Research assistant at the department of Hydrology and Geohy-
drology
Lecturer for statistics in the degree programs of Civil
Engineering and Environmental Engineering
Assisted the setup of a large scale experiment at the Research
Facility for Subsurface Remediation – VEGAS
2004 - 2005 Institut polytechnique de Grenoble,
Laboratoire des Transfert en Hydrologie et Environnement
Internship in the domain of probabilistic weather forecast
Research Topics
2010-2012 Generation of simultaneous synthetic rainfall time series
of high temporal resolution
2010-2011 Downscaling of Precipitation Data from Global Circulation
Models for Equatorial Africa
2006-2009 Generation of synthetic rainfall time series
under changing climatic conditions
Education
1998 - 2005 University of Stuttgart
Master degree of Water Resources Engineering and Management
(WAREM)
Diplom engineer of Umweltschutztechnik (Environmental
Engineering)
2001 - 2002 Ecole Nationale des Travaux Public de l’Etat
Exchange studies in Vaulx-en-Velin, France
 
 
 
 
 
 
 
Institut für Wasser- und 
Umweltsystemmodellierung 
Universität Stuttgart 
 
 
Pfaffenwaldring 61 
70569 Stuttgart (Vaihingen) 
Telefon (0711) 685 - 64717/64749/64752/64679 
Telefax (0711) 685 - 67020 o. 64746 o. 64681 
E-Mail: iws@iws.uni-stuttgart.de 
http://www.iws.uni-stuttgart.de 
 
 
 
 
 
Direktoren 
Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy 
Prof. Dr.-Ing. Rainer Helmig 
Prof. Dr.-Ing. Silke Wieprecht 
 
 
Vorstand (Stand 01.04.2009) 
Prof. Dr. rer. nat. Dr.-Ing. A. Bárdossy 
Prof. Dr.-Ing. R. Helmig 
Prof. Dr.-Ing. S. Wieprecht 
Jürgen Braun, PhD 
Dr.-Ing. H. Class 
Dr.-Ing. S. Hartmann 
Dr.-Ing. H.-P. Koschitzky 
PD Dr.-Ing. W. Marx  
Dr. rer. nat. J. Seidel 
 
Emeriti 
Prof. Dr.-Ing. habil. Dr.-Ing. E.h. Jürgen Giesecke 
Prof. Dr.h.c. Dr.-Ing. E.h. Helmut Kobus, PhD 
 
Lehrstuhl für Wasserbau und  
Wassermengenwirtschaft 
Leiter: Prof. Dr.-Ing. Silke Wieprecht 
Stellv.:  PD Dr.-Ing. Walter Marx, AOR 
 
Versuchsanstalt für Wasserbau 
Leiter:   Dr.-Ing. Sven Hartmann, AOR 
 
Lehrstuhl für Hydromechanik  
und Hydrosystemmodellierung 
Leiter:   Prof. Dr.-Ing. Rainer Helmig 
Stellv.:  Dr.-Ing. Holger Class, AOR 
 
Lehrstuhl für Hydrologie und Geohydrologie 
Leiter: Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy 
Stellv.:  Dr. rer. nat. Jochen Seidel 
 
VEGAS, Versuchseinrichtung zur  
Grundwasser- und Altlastensanierung 
Leitung:  Jürgen Braun, PhD 
               Dr.-Ing. Hans-Peter Koschitzky, AD 
 
 
 
Verzeichnis der Mitteilungshefte 
 
 
1 Röhnisch, Arthur: Die Bemühungen um eine Wasserbauliche Versuchsanstalt an 
der Technischen Hochschule Stuttgart, und 
Fattah Abouleid, Abdel: Beitrag zur Berechnung einer in lockeren Sand geramm-
ten, zweifach verankerten Spundwand, 1963 
 
2 Marotz, Günter: Beitrag zur Frage der Standfestigkeit von dichten Asphaltbelägen 
im Großwasserbau, 1964 
 
3 Gurr, Siegfried: Beitrag zur Berechnung zusammengesetzter ebener Flächen-
tragwerke unter besonderer Berücksichtigung ebener Stauwände, mit Hilfe von 
Randwert- und Lastwertmatrizen, 1965 
 
4 Plica, Peter: Ein Beitrag zur Anwendung von Schalenkonstruktionen im Stahlwas-
serbau, und Petrikat, Kurt: Möglichkeiten und Grenzen des wasserbaulichen Ver-
suchswesens, 1966 
2 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
5 Plate, Erich: Beitrag zur Bestimmung der Windgeschwindigkeitsverteilung in der 
durch eine Wand gestörten bodennahen Luftschicht,  
und 
Röhnisch, Arthur; Marotz, Günter: Neue Baustoffe und Bauausführungen für den 
Schutz der Böschungen und der Sohle von Kanälen, Flüssen und Häfen; Geste-
hungskosten und jeweilige Vorteile, sowie Unny, T.E.: Schwingungs-
untersuchungen am Kegelstrahlschieber, 1967 
 
6 Seiler, Erich: Die Ermittlung des Anlagenwertes der bundeseigenen Bin-
nenschiffahrtsstraßen und Talsperren und des Anteils der Binnenschiffahrt an die-
sem Wert, 1967 
 
7 Sonderheft anläßlich des 65. Geburtstages von Prof. Arthur Röhnisch mit Beiträ-
gen von Benk, Dieter; Breitling, J.; Gurr, Siegfried; Haberhauer, Robert; Hone-
kamp, Hermann; Kuz, Klaus Dieter; Marotz, Günter; Mayer-Vorfelder, Hans-Jörg; 
Miller, Rudolf; Plate, Erich J.; Radomski, Helge; Schwarz, Helmut; Vollmer, Ernst; 
Wildenhahn, Eberhard; 1967 
 
8 Jumikis, Alfred: Beitrag zur experimentellen Untersuchung des Wassernachschubs 
in einem gefrierenden Boden und die Beurteilung der Ergebnisse, 1968 
 
9 Marotz, Günter: Technische Grundlagen einer Wasserspeicherung im natürlichen 
Untergrund, 1968 
 
10 Radomski, Helge: Untersuchungen über den Einfluß der Querschnittsform wellen-
förmiger Spundwände auf die statischen und rammtechnischen Eigenschaften, 
1968 
 
11 Schwarz, Helmut: Die Grenztragfähigkeit des Baugrundes bei Einwirkung vertikal 
gezogener Ankerplatten als zweidimensionales Bruchproblem, 1969 
 
12 Erbel, Klaus: Ein Beitrag zur Untersuchung der Metamorphose von Mittelgebirgs-
schneedecken unter besonderer Berücksichtigung eines Verfahrens zur Bestim-
mung der thermischen Schneequalität, 1969 
 
13 Westhaus, Karl-Heinz: Der Strukturwandel in der Binnenschiffahrt und sein Einfluß 
auf den Ausbau der Binnenschiffskanäle, 1969 
 
14 Mayer-Vorfelder, Hans-Jörg: Ein Beitrag zur Berechnung des Erdwiderstandes un-
ter Ansatz der logarithmischen Spirale als Gleitflächenfunktion, 1970 
 
15 Schulz, Manfred: Berechnung des räumlichen Erddruckes auf die Wandung kreis-
zylindrischer Körper, 1970 
 
16 Mobasseri, Manoutschehr: Die Rippenstützmauer. Konstruktion und Grenzen ihrer 
Standsicherheit, 1970 
 
17 Benk, Dieter: Ein Beitrag zum Betrieb und zur Bemessung von Hochwasser-
rückhaltebecken, 1970 
Verzeichnis der Mitteilungshefte 3  
 
 
18 Gàl, Attila: Bestimmung der mitschwingenden Wassermasse bei überströmten 
Fischbauchklappen mit kreiszylindrischem Staublech, 1971, 
 
19 Kuz, Klaus Dieter: Ein Beitrag zur Frage des Einsetzens von Kavitationserschei-
nungen in einer Düsenströmung bei Berücksichtigung der im Wasser gelösten Ga-
se, 1971,  
 
20 Schaak, Hartmut: Verteilleitungen von Wasserkraftanlagen, 1971 
 
21 Sonderheft zur Eröffnung der neuen Versuchsanstalt des Instituts für Wasserbau 
der Universität Stuttgart mit Beiträgen von Brombach, Hansjörg; Dirksen, Wolfram; 
Gàl, Attila; Gerlach, Reinhard; Giesecke, Jürgen; Holthoff, Franz-Josef; Kuz, Klaus 
Dieter; Marotz, Günter; Minor, Hans-Erwin; Petrikat, Kurt; Röhnisch, Arthur; Rueff, 
Helge; Schwarz, Helmut; Vollmer, Ernst; Wildenhahn, Eberhard; 1972 
 
22 Wang, Chung-su: Ein Beitrag zur Berechnung der Schwingungen an Kegelstrahl-
schiebern, 1972 
 
23 Mayer-Vorfelder, Hans-Jörg: Erdwiderstandsbeiwerte nach dem Ohde-
Variationsverfahren, 1972 
 
24 Minor, Hans-Erwin: Beitrag zur Bestimmung der Schwingungsanfachungs-
funktionen überströmter Stauklappen, 1972,  
 
25 Brombach, Hansjörg: Untersuchung strömungsmechanischer Elemente (Fluidik) 
und die Möglichkeit der Anwendung von Wirbelkammerelementen im Wasserbau, 
1972,  
 
26 Wildenhahn, Eberhard: Beitrag zur Berechnung von Horizontalfilterbrunnen, 1972 
 
27 Steinlein, Helmut: Die Eliminierung der Schwebstoffe aus Flußwasser zum Zweck 
der unterirdischen Wasserspeicherung, gezeigt am Beispiel der Iller, 1972 
 
28 Holthoff, Franz Josef: Die Überwindung großer Hubhöhen in der Binnenschiffahrt 
durch Schwimmerhebewerke, 1973 
 
29 Röder, Karl: Einwirkungen aus Baugrundbewegungen auf trog- und kastenförmige 
Konstruktionen des Wasser- und Tunnelbaues, 1973 
 
30 Kretschmer, Heinz: Die Bemessung von Bogenstaumauern in Abhängigkeit von 
der Talform, 1973 
 
31 Honekamp, Hermann: Beitrag zur Berechnung der Montage von Unterwasserpipe-
lines, 1973 
 
32 Giesecke, Jürgen: Die Wirbelkammertriode als neuartiges Steuerorgan im Was-
serbau, und Brombach, Hansjörg: Entwicklung, Bauformen, Wirkungsweise und 
Steuereigenschaften von Wirbelkammerverstärkern, 1974 
 
4 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
33 Rueff, Helge: Untersuchung der schwingungserregenden Kräfte an zwei hinterein-
ander angeordneten Tiefschützen unter besonderer Berücksichtigung von Kavita-
tion, 1974 
 
34 Röhnisch, Arthur: Einpreßversuche mit Zementmörtel für Spannbeton - Vergleich 
der Ergebnisse von Modellversuchen mit Ausführungen in Hüllwellrohren, 1975 
 
35 Sonderheft anläßlich des 65. Geburtstages von Prof. Dr.-Ing. Kurt Petrikat mit Bei-
trägen von:  Brombach, Hansjörg; Erbel, Klaus; Flinspach, Dieter; Fischer jr., Ri-
chard; Gàl, Attila; Gerlach, Reinhard; Giesecke, Jürgen; Haberhauer, Robert; Haf-
ner Edzard; Hausenblas, Bernhard; Horlacher, Hans-Burkhard; Hutarew, Andreas; 
Knoll, Manfred; Krummet, Ralph; Marotz, Günter; Merkle, Theodor; Miller, Chris-
toph; Minor, Hans-Erwin; Neumayer, Hans; Rao, Syamala; Rath, Paul; Rueff, Hel-
ge; Ruppert, Jürgen; Schwarz, Wolfgang; Topal-Gökceli, Mehmet; Vollmer, Ernst; 
Wang, Chung-su; Weber, Hans-Georg; 1975 
 
36 Berger, Jochum: Beitrag zur Berechnung des Spannungszustandes in rotations-
symmetrisch belasteten Kugelschalen veränderlicher Wandstärke unter Gas- und 
Flüssigkeitsdruck durch Integration schwach singulärer Differentialgleichungen, 
1975 
 
37 Dirksen, Wolfram: Berechnung instationärer Abflußvorgänge in gestauten Gerin-
nen mittels Differenzenverfahren und die Anwendung auf Hochwasserrückhalte-
becken, 1976 
 
38 Horlacher, Hans-Burkhard: Berechnung instationärer Temperatur- und Wärme-
spannungsfelder in langen mehrschichtigen Hohlzylindern, 1976 
 
39 Hafner, Edzard: Untersuchung der hydrodynamischen Kräfte auf Baukörper im 
Tiefwasserbereich des Meeres, 1977, ISBN 3-921694-39-6 
 
40 Ruppert, Jürgen: Über den Axialwirbelkammerverstärker für den Einsatz im Was-
serbau, 1977, ISBN 3-921694-40-X 
 
41 Hutarew, Andreas: Beitrag zur Beeinflußbarkeit des Sauerstoffgehalts in Fließge-
wässern an Abstürzen und Wehren, 1977, ISBN 3-921694-41-8,  
 
42 Miller, Christoph: Ein Beitrag zur Bestimmung der schwingungserregenden Kräfte 
an unterströmten Wehren, 1977, ISBN 3-921694-42-6 
 
43 Schwarz, Wolfgang: Druckstoßberechnung unter Berücksichtigung der Radial- und 
Längsverschiebungen der Rohrwandung, 1978, ISBN 3-921694-43-4 
 
44 Kinzelbach, Wolfgang: Numerische Untersuchungen über den optimalen Einsatz 
variabler Kühlsysteme einer Kraftwerkskette am Beispiel Oberrhein, 1978,  
ISBN 3-921694-44-2 
 
45 Barczewski, Baldur: Neue Meßmethoden für Wasser-Luftgemische und deren An-
wendung auf zweiphasige Auftriebsstrahlen, 1979, ISBN 3-921694-45-0 
Verzeichnis der Mitteilungshefte 5  
 
 
46 Neumayer, Hans: Untersuchung der Strömungsvorgänge in radialen Wirbelkam-
merverstärkern, 1979, ISBN 3-921694-46-9 
 
47 Elalfy, Youssef-Elhassan: Untersuchung der Strömungsvorgänge in Wirbelkam-
merdioden und -drosseln, 1979, ISBN 3-921694-47-7 
 
48 Brombach, Hansjörg: Automatisierung der Bewirtschaftung von Wasserspeichern, 
1981, ISBN 3-921694-48-5 
 
49 Geldner, Peter: Deterministische und stochastische Methoden zur Bestimmung der 
Selbstdichtung von Gewässern, 1981, ISBN 3-921694-49-3,  
 
50 Mehlhorn, Hans: Temperaturveränderungen im Grundwasser durch Brauchwas-
sereinleitungen, 1982, ISBN 3-921694-50-7,  
 
51 Hafner, Edzard: Rohrleitungen und Behälter im Meer, 1983, ISBN 3-921694-51-5 
 
52 Rinnert, Bernd: Hydrodynamische Dispersion in porösen Medien: Einfluß von Dich-
teunterschieden auf die Vertikalvermischung in horizontaler Strömung, 1983, ISBN 
3-921694-52-3,  
 
53 Lindner, Wulf: Steuerung von Grundwasserentnahmen unter Einhaltung ökologi-
scher Kriterien, 1983, ISBN 3-921694-53-1,  
 
54 Herr, Michael; Herzer, Jörg; Kinzelbach, Wolfgang; Kobus, Helmut; Rinnert, Bernd: 
Methoden zur rechnerischen Erfassung und hydraulischen Sanierung von Grund-
wasserkontaminationen, 1983, ISBN 3-921694-54-X 
 
55 Schmitt, Paul: Wege zur Automatisierung der Niederschlagsermittlung, 1984, ISBN 
3-921694-55-8,  
 
56 Müller, Peter: Transport und selektive Sedimentation von Schwebstoffen bei ge-
stautem Abfluß, 1985, ISBN 3-921694-56-6 
 
57 El-Qawasmeh, Fuad: Möglichkeiten und Grenzen der Tropfbewässerung unter be-
sonderer Berücksichtigung der Verstopfungsanfälligkeit der Tropfelemente, 1985, 
ISBN 3-921694-57-4,  
 
58 Kirchenbaur, Klaus: Mikroprozessorgesteuerte Erfassung instationärer Druckfelder 
am Beispiel seegangsbelasteter Baukörper, 1985, ISBN 3-921694-58-2 
 
59 Kobus, Helmut (Hrsg.): Modellierung des großräumigen Wärme- und Schadstoff-
transports im Grundwasser, Tätigkeitsbericht 1984/85 (DFG-Forschergruppe an 
den Universitäten Hohenheim, Karlsruhe und Stuttgart), 1985,  
ISBN 3-921694-59-0,  
 
60 Spitz, Karlheinz: Dispersion in porösen Medien: Einfluß von Inhomogenitäten und 
Dichteunterschieden, 1985, ISBN 3-921694-60-4, 
 
61 Kobus, Helmut: An Introduction to Air-Water Flows in Hydraulics, 1985,  
ISBN 3-921694-61-2 
6 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
62 Kaleris, Vassilios: Erfassung des Austausches von Oberflächen- und Grundwasser 
in horizontalebenen Grundwassermodellen, 1986, ISBN 3-921694-62-0 
 
63 Herr, Michael: Grundlagen der hydraulischen Sanierung verunreinigter Poren-
grundwasserleiter, 1987, ISBN 3-921694-63-9 
 
64 Marx, Walter: Berechnung von Temperatur und Spannung in Massenbeton infolge 
Hydratation, 1987, ISBN 3-921694-64-7 
 
65 Koschitzky, Hans-Peter: Dimensionierungskonzept für Sohlbelüfter in Schußrinnen 
zur Vermeidung von Kavitationsschäden, 1987, ISBN 3-921694-65-5 
 
66 Kobus, Helmut (Hrsg.): Modellierung des großräumigen Wärme- und Schadstoff-
transports im Grundwasser, Tätigkeitsbericht 1986/87 (DFG-Forschergruppe an 
den Universitäten Hohenheim, Karlsruhe und Stuttgart) 1987, ISBN 3-921694-66-3 
 
67 Söll, Thomas: Berechnungsverfahren zur Abschätzung anthropogener Tempera-
turanomalien im Grundwasser, 1988, ISBN 3-921694-67-1 
 
68 Dittrich, Andreas; Westrich, Bernd: Bodenseeufererosion, Bestandsaufnahme und 
Bewertung, 1988, ISBN 3-921694-68-X,  
 
69 Huwe, Bernd; van der Ploeg, Rienk R.: Modelle zur Simulation des Stickstoffhaus-
haltes von Standorten mit unterschiedlicher landwirtschaftlicher Nutzung, 1988, 
ISBN 3-921694-69-8,  
 
70 Stephan, Karl: Integration elliptischer Funktionen, 1988, ISBN 3-921694-70-1 
 
71 Kobus, Helmut; Zilliox, Lothaire (Hrsg.): Nitratbelastung des Grundwassers, Aus-
wirkungen der Landwirtschaft auf die Grundwasser- und Rohwasserbeschaffenheit 
und Maßnahmen zum Schutz des Grundwassers. Vorträge des deutsch-franzö-
sischen Kolloquiums am 6. Oktober 1988, Universitäten Stuttgart und Louis Pas-
teur Strasbourg (Vorträge in deutsch oder französisch, Kurzfassungen zwei-
sprachig), 1988, ISBN 3-921694-71-X 
 
72 Soyeaux, Renald: Unterströmung von Stauanlagen auf klüftigem Untergrund unter 
Berücksichtigung laminarer und turbulenter Fließzustände,1991, 
ISBN 3-921694-72-8 
 
73 Kohane, Roberto: Berechnungsmethoden für Hochwasserabfluß in Fließgewäs-
sern mit überströmten Vorländern, 1991, ISBN 3-921694-73-6 
 
74 Hassinger, Reinhard: Beitrag zur Hydraulik und Bemessung von Blocksteinrampen 
in flexibler Bauweise, 1991, ISBN 3-921694-74-4, 
 
75 Schäfer, Gerhard: Einfluß von Schichtenstrukturen und lokalen Einlagerungen auf 
die Längsdispersion in Porengrundwasserleitern, 1991, ISBN 3-921694-75-2 
 
76 Giesecke, Jürgen: Vorträge, Wasserwirtschaft in stark besiedelten Regionen; Um-
weltforschung mit Schwerpunkt Wasserwirtschaft, 1991, ISBN 3-921694-76-0 
Verzeichnis der Mitteilungshefte 7  
 
 
77 Huwe, Bernd: Deterministische und stochastische Ansätze zur Modellierung des 
Stickstoffhaushalts landwirtschaftlich genutzter Flächen auf unterschiedlichem 
Skalenniveau, 1992, ISBN 3-921694-77-9, 
 
78 Rommel, Michael: Verwendung von Kluftdaten zur realitätsnahen Generierung von 
Kluftnetzen mit anschließender laminar-turbulenter Strömungsberechnung, 1993, 
ISBN 3-92 1694-78-7 
 
79 Marschall, Paul: Die Ermittlung lokaler Stofffrachten im Grundwasser mit Hilfe von 
Einbohrloch-Meßverfahren, 1993, ISBN 3-921694-79-5, 
 
80 Ptak, Thomas: Stofftransport in heterogenen Porenaquiferen: Felduntersuchungen 
und stochastische Modellierung, 1993, ISBN 3-921694-80-9, 
 
81 Haakh, Frieder: Transientes Strömungsverhalten in Wirbelkammern, 1993,  
ISBN 3-921694-81-7 
 
82 Kobus, Helmut; Cirpka, Olaf; Barczewski, Baldur; Koschitzky, Hans-Peter: Ver-
sucheinrichtung zur Grundwasser und Altlastensanierung VEGAS, Konzeption und 
Programmrahmen, 1993, ISBN 3-921694-82-5 
 
83 Zang, Weidong: Optimaler Echtzeit-Betrieb eines Speichers mit aktueller Abflußre-
generierung, 1994, ISBN 3-921694-83-3, 
 
84 Franke, Hans-Jörg: Stochastische Modellierung eines flächenhaften Stoffeintrages 
und Transports in Grundwasser am Beispiel der Pflanzenschutzmittelproblematik, 
1995, ISBN 3-921694-84-1 
 
85 Lang, Ulrich: Simulation regionaler Strömungs- und Transportvorgänge in Karst-
aquiferen mit Hilfe des Doppelkontinuum-Ansatzes: Methodenentwicklung und Pa-
rameteridentifikation, 1995, ISBN 3-921694-85-X, 
 
86 Helmig, Rainer: Einführung in die Numerischen Methoden der Hydromechanik, 
1996, ISBN 3-921694-86-8, 
 
87 Cirpka, Olaf: CONTRACT: A Numerical Tool for Contaminant Transport and 
Chemical Transformations - Theory and Program Documentation -, 1996,  
ISBN 3-921694-87-6 
 
88 Haberlandt, Uwe: Stochastische Synthese und Regionalisierung des Niederschla-
ges für Schmutzfrachtberechnungen, 1996, ISBN 3-921694-88-4 
 
89 Croisé, Jean: Extraktion von flüchtigen Chemikalien aus natürlichen Lockergestei-
nen mittels erzwungener Luftströmung, 1996, ISBN 3-921694-89-2,  
 
90 Jorde, Klaus: Ökologisch begründete, dynamische Mindestwasserregelungen bei 
Ausleitungskraftwerken, 1997, ISBN 3-921694-90-6,   
 
91 Helmig, Rainer: Gekoppelte Strömungs- und Transportprozesse im Untergrund - 
Ein Beitrag zur Hydrosystemmodellierung-, 1998, ISBN 3-921694-91-4,   
8 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
92 Emmert, Martin:  Numerische Modellierung nichtisothermer Gas-Wasser Systeme 
in porösen Medien, 1997, ISBN 3-921694-92-2 
 
93 Kern, Ulrich: Transport von Schweb- und Schadstoffen in staugeregelten Fließge-
wässern am Beispiel des Neckars, 1997, ISBN 3-921694-93-0,   
 
94 Förster, Georg:  Druckstoßdämpfung durch große Luftblasen in Hochpunkten von 
Rohrleitungen 1997, ISBN 3-921694-94-9 
 
95 Cirpka, Olaf: Numerische Methoden zur Simulation des reaktiven Mehrkomponen-
tentransports im Grundwasser, 1997, ISBN 3-921694-95-7,   
 
96 Färber, Arne: Wärmetransport in der ungesättigten Bodenzone: Entwicklung einer 
thermischen In-situ-Sanierungstechnologie, 1997, ISBN 3-921694-96-5  
 
97 Betz, Christoph: Wasserdampfdestillation von Schadstoffen im porösen Medium: 
Entwicklung einer thermischen In-situ-Sanierungstechnologie, 1998,  
ISBN 3-921694-97-3 
 
98 Xu, Yichun: Numerical Modeling of Suspended Sediment Transport in Rivers, 
1998, ISBN 3-921694-98-1,  
 
99 Wüst, Wolfgang: Geochemische Untersuchungen zur Sanierung CKW-
kontaminierter Aquifere mit Fe(0)-Reaktionswänden, 2000, ISBN 3-933761-02-2 
 
100 Sheta, Hussam: Simulation von Mehrphasenvorgängen in porösen Medien unter 
Einbeziehung von Hysterese-Effekten, 2000, ISBN 3-933761-03-4 
 
101 Ayros, Edwin: Regionalisierung extremer Abflüsse auf der Grundlage statistischer 
Verfahren, 2000, ISBN 3-933761-04-2,  
 
102 Huber, Ralf: Compositional Multiphase Flow and Transport in Heterogeneous 
Porous Media, 2000, ISBN 3-933761-05-0 
 
103 Braun, Christopherus: Ein Upscaling-Verfahren für Mehrphasenströmungen in po-
rösen Medien, 2000, ISBN 3-933761-06-9 
 
104 Hofmann, Bernd: Entwicklung eines rechnergestützten Managementsystems zur 
Beurteilung von Grundwasserschadensfällen, 2000, ISBN 3-933761-07-7 
 
105 Class, Holger: Theorie und numerische Modellierung nichtisothermer Mehrphasen-
prozesse in NAPL-kontaminierten porösen Medien, 2001,  
ISBN 3-933761-08-5 
 
106 Schmidt, Reinhard: Wasserdampf- und Heißluftinjektion zur thermischen Sanie-
rung kontaminierter Standorte, 2001, ISBN 3-933761-09-3 
 
107 Josef, Reinhold:, Schadstoffextraktion mit hydraulischen Sanierungsverfahren un-
ter Anwendung von grenzflächenaktiven Stoffen, 2001, ISBN 3-933761-10-7 
 
Verzeichnis der Mitteilungshefte 9  
 
 
108 Schneider, Matthias: Habitat- und Abflussmodellierung für Fließgewässer mit un-
scharfen Berechnungsansätzen, 2001, ISBN 3-933761-11-5 
 
109 Rathgeb, Andreas: Hydrodynamische Bemessungsgrundlagen für Lockerdeckwer-
ke an überströmbaren Erddämmen, 2001, ISBN 3-933761-12-3 
 
110 Lang, Stefan: Parallele numerische Simulation instätionärer Probleme mit adapti-
ven Methoden auf unstrukturierten Gittern, 2001, ISBN 3-933761-13-1 
 
111 Appt, Jochen; Stumpp Simone: Die Bodensee-Messkampagne 2001, IWS/CWR 
Lake Constance Measurement Program 2001, 2002, ISBN 3-933761-14-X 
 
112 Heimerl, Stephan: Systematische Beurteilung von Wasserkraftprojekten, 2002, 
ISBN 3-933761-15-8,  
 
113 Iqbal, Amin: On the Management and Salinity Control of Drip Irrigation, 2002, ISBN 
3-933761-16-6 
 
114 Silberhorn-Hemminger, Annette: Modellierung von Kluftaquifersystemen:  Geosta-
tistische Analyse und deterministisch-stochastische Kluftgenerierung, 2002, ISBN 
3-933761-17-4  
 
115 Winkler, Angela: Prozesse des Wärme- und Stofftransports bei der In-situ-
Sanierung mit festen Wärmequellen, 2003, ISBN 3-933761-18-2 
 
116 Marx, Walter: Wasserkraft, Bewässerung, Umwelt -  Planungs- und Bewertungs-
schwerpunkte der Wasserbewirtschaftung, 2003, ISBN 3-933761-19-0 
 
117 Hinkelmann, Reinhard: Efficient Numerical Methods and Information-Processing 
Techniques in Environment Water, 2003, ISBN 3-933761-20-4 
 
118 Samaniego-Eguiguren, Luis Eduardo: Hydrological Consequences of Land Use / 
Land Cover and Climatic Changes in Mesoscale Catchments, 2003,  
ISBN 3-933761-21-2 
 
119 Neunhäuserer, Lina: Diskretisierungsansätze zur Modellierung von Strömungs- 
und Transportprozessen in geklüftet-porösen Medien, 2003, ISBN 3-933761-22-0 
 
120 Paul, Maren: Simulation of Two-Phase Flow in Heterogeneous Poros Media with 
Adaptive Methods, 2003, ISBN 3-933761-23-9 
 
121 Ehret, Uwe: Rainfall and Flood Nowcasting in Small Catchments using Weather 
Radar, 2003, ISBN 3-933761-24-7 
 
122 Haag, Ingo: Der Sauerstoffhaushalt staugeregelter Flüsse am Beispiel des Ne-
ckars - Analysen, Experimente, Simulationen -, 2003, ISBN 3-933761-25-5 
 
123 Appt, Jochen: Analysis of Basin-Scale Internal Waves in Upper Lake Constance, 
2003, ISBN 3-933761-26-3 
 
10 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
124 Hrsg.: Schrenk, Volker; Batereau, Katrin; Barczewski, Baldur; Weber, Karolin und 
Koschitzky, Hans-Peter: Symposium  Ressource Fläche und  VEGAS - Statuskol-
loquium 2003, 30. September und  1. Oktober 2003, 2003, ISBN 3-933761-27-1 
 
125 Omar Khalil Ouda: Optimisation of Agricultural Water Use: A Decision Support 
System for the Gaza Strip, 2003, ISBN 3-933761-28-0 
 
126 Batereau, Katrin: Sensorbasierte Bodenluftmessung zur Vor-Ort-Erkundung von 
Schadensherden im Untergrund, 2004, ISBN 3-933761-29-8 
 
127 Witt, Oliver: Erosionsstabilität von Gewässersedimenten mit Auswirkung auf den 
Stofftransport bei Hochwasser am Beispiel ausgewählter Stauhaltungen des Ober-
rheins, 2004, ISBN 3-933761-30-1 
 
128 Jakobs, Hartmut: Simulation nicht-isothermer Gas-Wasser-Prozesse in komplexen 
Kluft-Matrix-Systemen, 2004, ISBN 3-933761-31-X 
 
129 Li, Chen-Chien: Deterministisch-stochastisches Berechnungskonzept zur Beurtei-
lung der Auswirkungen erosiver Hochwasserereignisse in Flussstauhaltungen, 
2004, ISBN 3-933761-32-8 
 
130 Reichenberger, Volker; Helmig, Rainer; Jakobs, Hartmut; Bastian, Peter; Niessner, 
Jennifer: Complex Gas-Water Processes in Discrete Fracture-Matrix Systems: Up-
scaling, Mass-Conservative Discretization and Efficient Multilevel Solution, 2004, 
ISBN 3-933761-33-6  
 
131 Hrsg.: Barczewski, Baldur; Koschitzky, Hans-Peter; Weber, Karolin; Wege, Ralf:  
VEGAS - Statuskolloquium 2004, Tagungsband zur Veranstaltung am 05. Oktober 
2004 an der Universität Stuttgart, Campus Stuttgart-Vaihingen, 2004, ISBN 3-
933761-34-4 
 
132 Asie, Kemal Jabir: Finite Volume Models for Multiphase Multicomponent Flow 
through Porous Media. 2005, ISBN 3-933761-35-2 
 
133 Jacoub, George:  Development of a 2-D Numerical Module for Particulate 
Contaminant Transport in Flood Retention Reservoirs and Impounded Rivers, 
2004, ISBN 3-933761-36-0  
 
134 Nowak, Wolfgang: Geostatistical Methods for the Identification of Flow and 
Transport Parameters in the Subsurface, 2005, ISBN 3-933761-37-9 
 
135 Süß, Mia: Analysis of the influence of structures and boundaries on flow and 
transport processes in fractured porous media, 2005, ISBN 3-933761-38-7 
 
136 Jose, Surabhin Chackiath: Experimental Investigations on Longitudinal Dispersive 
Mixing in Heterogeneous Aquifers, 2005, ISBN: 3-933761-39-5 
 
137 Filiz, Fulya: Linking Large-Scale Meteorological Conditions to Floods in Mesoscale 
Catchments, 2005, ISBN 3-933761-40-9 
 
Verzeichnis der Mitteilungshefte 11  
 
 
138 Qin, Minghao: Wirklichkeitsnahe und recheneffiziente Ermittlung von Temperatur 
und Spannungen bei großen RCC-Staumauern, 2005, ISBN 3-933761-41-7 
 
139 Kobayashi, Kenichiro: Optimization Methods for Multiphase Systems in the 
Subsurface - Application to Methane Migration in Coal Mining Areas, 2005,  
ISBN 3-933761-42-5 
 
140 Rahman, Md. Arifur: Experimental Investigations on Transverse Dispersive Mixing 
in Heterogeneous Porous Media, 2005, ISBN 3-933761-43-3 
 
141 Schrenk, Volker: Ökobilanzen zur Bewertung von Altlastensanierungsmaßnahmen, 
2005, ISBN 3-933761-44-1 
 
142 Hundecha, Hirpa Yeshewatesfa: Regionalization of Parameters of a Conceptual 
Rainfall-Runoff Model, 2005, ISBN: 3-933761-45-X 
 
143 Wege, Ralf: Untersuchungs- und Überwachungsmethoden für die Beurteilung na-
türlicher Selbstreinigungsprozesse im Grundwasser, 2005, ISBN 3-933761-46-8 
 
144 Breiting, Thomas: Techniken und Methoden der Hydroinformatik - Modellierung 
von komplexen Hydrosystemen im Untergrund, 2006, 3-933761-47-6 
 
145 Hrsg.: Braun, Jürgen; Koschitzky, Hans-Peter; Müller, Martin: Ressource Unter-
grund: 10 Jahre VEGAS: Forschung und Technologieentwicklung zum Schutz von 
Grundwasser und Boden, Tagungsband zur Veranstaltung am 28. und 29. Sep-
tember 2005 an der Universität Stuttgart, Campus Stuttgart-Vaihingen, 2005, ISBN 
3-933761-48-4 
 
146 Rojanschi, Vlad:  Abflusskonzentration in  mesoskaligen  Einzugsgebieten unter 
Berücksichtigung des Sickerraumes,  2006, ISBN 3-933761-49-2  
 
147 Winkler, Nina Simone:  Optimierung der Steuerung von Hochwasserrückhaltebe-
cken-systemen, 2006, ISBN 3-933761-50-6 
 
148 Wolf, Jens:  Räumlich differenzierte Modellierung der Grundwasserströmung allu-
vialer Aquifere für mesoskalige Einzugsgebiete, 2006, ISBN: 3-933761-51-4 
 
149 Kohler, Beate: Externe Effekte der Laufwasserkraftnutzung,  2006,  
ISBN 3-933761-52-2 
 
150 Hrsg.: Braun, Jürgen; Koschitzky, Hans-Peter; Stuhrmann, Matthias: VEGAS-
Statuskolloquium 2006, Tagungsband zur Veranstaltung am 28. September 2006 
an der Universität Stuttgart, Campus Stuttgart-Vaihingen, 2006,  
ISBN 3-933761-53-0 
 
151 Niessner, Jennifer: Multi-Scale Modeling of Multi-Phase - Multi-Component Pro-
cesses in Heterogeneous Porous Media, 2006, ISBN 3-933761-54-9 
 
152 Fischer, Markus: Beanspruchung eingeerdeter Rohrleitungen infolge Austrocknung 
bindiger Böden, 2006, ISBN 3-933761-55-7 
12 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
153 Schneck, Alexander: Optimierung der Grundwasserbewirtschaftung unter Berück-
sichtigung der Belange der Wasserversorgung, der Landwirtschaft und des Natur-
schutzes, 2006, ISBN 3-933761-56-5 
 
154 Das, Tapash: The Impact of Spatial Variability of Precipitation on the Predictive 
Uncertainty of Hydrological Models, 2006, ISBN 3-933761-57-3 
 
155 Bielinski, Andreas: Numerical Simulation of CO2 sequestration in geological 
formations, 2007, ISBN 3-933761-58-1 
 
156 Mödinger, Jens: Entwicklung eines Bewertungs- und Entscheidungsunterstüt-
zungssystems für eine nachhaltige regionale Grundwasserbewirtschaftung, 2006, 
ISBN 3-933761-60-3 
 
157 Manthey, Sabine: Two-phase flow processes with dynamic effects in porous 
media - parameter estimation and simulation, 2007, ISBN 3-933761-61-1 
 
158 Pozos Estrada, Oscar: Investigation on the Effects of Entrained Air in Pipelines, 
2007, ISBN 3-933761-62-X 
 
159 Ochs, Steffen Oliver: Steam injection into saturated porous media – process 
analysis including experimental and numerical investigations, 2007,  
ISBN 3-933761-63-8 
 
160 Marx, Andreas: Einsatz gekoppelter Modelle und Wetterradar zur Abschätzung 
von Niederschlagsintensitäten und zur Abflussvorhersage, 2007, 
ISBN 3-933761-64-6 
 
161 Hartmann, Gabriele Maria: Investigation of Evapotranspiration Concepts in Hydro-
logical Modelling for Climate Change Impact Assessment, 2007,  
ISBN 3-933761-65-4 
 
162 Kebede Gurmessa, Tesfaye: Numerical Investigation on Flow and Transport 
Characteristics to Improve Long-Term Simulation of Reservoir Sedimentation, 
2007, ISBN 3-933761-66-2 
 
163 Trifković, Aleksandar: Multi-objective and Risk-based Modelling Methodology for 
Planning, Design and Operation of Water Supply Systems, 2007,  
ISBN 3-933761-67-0 
 
164 Götzinger, Jens: Distributed Conceptual Hydrological Modelling - Simulation of 
Climate, Land Use Change Impact and Uncertainty Analysis, 2007,  
ISBN 3-933761-68-9 
 
165 Hrsg.: Braun, Jürgen; Koschitzky, Hans-Peter; Stuhrmann, Matthias: VEGAS – 
Kolloquium 2007,  Tagungsband zur Veranstaltung am 26. September 2007 an der 
Universität Stuttgart, Campus Stuttgart-Vaihingen, 2007, ISBN 3-933761-69-7 
 
166 Freeman, Beau: Modernization Criteria Assessment for Water Resources 
Planning; Klamath Irrigation Project, U.S., 2008, ISBN 3-933761-70-0 
Verzeichnis der Mitteilungshefte 13  
 
 
167 Dreher, Thomas: Selektive Sedimentation von Feinstschwebstoffen in Wechselwir-
kung mit wandnahen turbulenten Strömungsbedingungen, 2008,  
ISBN 3-933761-71-9 
 
168 Yang, Wei: Discrete-Continuous Downscaling Model for Generating Daily 
Precipitation Time Series, 2008, ISBN 3-933761-72-7 
 
169 Kopecki, Ianina: Calculational Approach to FST-Hemispheres for Multiparametrical 
Benthos Habitat Modelling, 2008, ISBN 3-933761-73-5 
 
170 Brommundt, Jürgen: Stochastische Generierung räumlich zusammenhängender 
Niederschlagszeitreihen, 2008, ISBN 3-933761-74-3 
 
171 Papafotiou, Alexandros: Numerical Investigations of the Role of Hysteresis in 
Heterogeneous Two-Phase Flow Systems, 2008, ISBN 3-933761-75-1 
 
172 He, Yi: Application of a Non-Parametric Classification Scheme to Catchment 
Hydrology, 2008, ISBN 978-3-933761-76-7 
 
173 Wagner, Sven: Water Balance in a Poorly Gauged Basin in West Africa Using 
Atmospheric Modelling and Remote Sensing Information, 2008, 
ISBN 978-3-933761-77-4 
 
174 Hrsg.: Braun, Jürgen; Koschitzky, Hans-Peter; Stuhrmann, Matthias; Schrenk, Vol-
ker: VEGAS-Kolloquium 2008  Ressource Fläche III, Tagungsband zur Veranstal-
tung am 01. Oktober 2008 an der Universität Stuttgart, Campus Stuttgart-
Vaihingen, 2008, ISBN 978-3-933761-78-1 
 
175 Patil, Sachin: Regionalization of an Event Based Nash Cascade Model for Flood 
Predictions in Ungauged Basins, 2008, ISBN 978-3-933761-79-8 
 
176 Assteerawatt, Anongnart: Flow and Transport Modelling of Fractured Aquifers 
based on a Geostatistical Approach, 2008, ISBN 978-3-933761-80-4 
 
177 Karnahl, Joachim Alexander: 2D numerische Modellierung von multifraktionalem 
Schwebstoff- und Schadstofftransport in Flüssen, 2008, ISBN 978-3-933761-81-1 
 
178 Hiester, Uwe: Technologieentwicklung zur In-situ-Sanierung der ungesättigten Bo-
denzone mit festen Wärmequellen, 2009, ISBN 978-3-933761-82-8 
 
179 Laux, Patrick: Statistical Modeling of Precipitation for Agricultural Planning in the 
Volta Basin of West Africa, 2009, ISBN 978-3-933761-83-5 
 
180 Ehsan, Saqib: Evaluation of Life Safety Risks Related to Severe Flooding, 2009,  
ISBN 978-3-933761-84-2 
 
181 Prohaska, Sandra: Development and Application of a 1D Multi-Strip Fine 
Sediment Transport Model for Regulated Rivers, 2009, ISBN 978-3-933761-85-
9 
14 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
182  Kopp, Andreas: Evaluation of CO2 Injection Processes in Geological Formations 
for Site Screening, 2009, ISBN 978-3-933761-86-6 
 
183 Ebigbo, Anozie: Modelling of biofilm growth and its influence on CO2 and water 
(two-phase) flow in porous media, 2009, ISBN 978-3-933761-87-3 
 
184 Freiboth, Sandra: A phenomenological model for the numerical simulation of 
multiphase multicomponent processes considering structural alterations of 
porous media, 2009, ISBN 978-3-933761-88-0 
 
185 Zöllner, Frank: Implementierung und Anwendung netzfreier Methoden im Kon-
struktiven Wasserbau und in der Hydromechanik, 2009, 
ISBN 978-3-933761-89-7 
 
186 Vasin, Milos: Influence of the soil structure and property contrast on flow and 
transport in the unsaturated zone, 2010, ISBN 978-3-933761-90-3 
 
187 Li, Jing: Application of Copulas as a New Geostatistical Tool, 2010, ISBN 978-3-
933761-91-0 
 
188 AghaKouchak, Amir: Simulation of Remotely Sensed Rainfall Fields Using 
Copulas, 2010, ISBN 978-3-933761-92-7 
 
189 Thapa, Pawan Kumar: Physically-based spatially distributed rainfall runoff 
modelling for soil erosion estimation, 2010, ISBN 978-3-933761-93-4 
 
190 Wurms, Sven: Numerische Modellierung der Sedimentationsprozesse in Retenti-
onsanlagen zur Steuerung von Stoffströmen bei extremen Hochwasserabflusser-
eignissen, 2011, ISBN 978-3-933761-94-1 
 
191 Merkel, Uwe: Unsicherheitsanalyse hydraulischer Einwirkungen auf Hochwasser-
schutzdeiche und Steigerung der Leistungsfähigkeit durch adaptive Strömungs-
modellierung, 2011, ISBN 978-3-933761-95-8 
 
192 Fritz, Jochen: A Decoupled Model for Compositional Non-Isothermal Multiphase 
Flow in Porous Media and Multiphysics Approaches for Two-Phase Flow, 2010, 
ISBN 978-3-933761-96-5 
 
193 Weber, Karolin (Hrsg.): 12. Treffen junger WissenschaftlerInnen an Wasserbauin-
stituten, 2010, ISBN 978-3-933761-97-2 
 
194 Bliefernicht, Jan-Geert: Probability Forecasts of Daily Areal Precipitation for Small 
River Basins, 2011, ISBN 978-3-933761-98-9 
 
195 Hrsg.: Koschitzky, Hans-Peter; Braun, Jürgen: VEGAS-Kolloquium 2010 In-situ-
Sanierung - Stand und Entwicklung Nano und ISCO -, Tagungsband zur Veran-
staltung am 07. Oktober 2010 an der Universität Stuttgart, Campus Stuttgart-
Vaihingen, 2010, ISBN 978-3-933761-99-6 
 
Verzeichnis der Mitteilungshefte 15  
 
 
196 Gafurov, Abror: Water Balance Modeling Using Remote Sensing Information - 
Focus on Central Asia, 2010, ISBN 978-3-942036-00-9 
 
197 Mackenberg, Sylvia: Die Quellstärke in der Sickerwasserprognose: 
Möglichkeiten und Grenzen von Labor- und Freilanduntersuchungen, 
2010, ISBN 978-3-942036-01-6 
 
198 Singh, Shailesh Kumar: Robust Parameter Estimation in Gauged and Ungauged 
Basins, 2010, ISBN 978-3-942036-02-3 
 
199 Doğan, Mehmet Onur: Coupling of porous media flow with pipe flow, 2011, 
ISBN 978-3-942036-03-0 
 
200 Liu, Min: Study of Topographic Effects on Hydrological Patterns and the 
Implication on Hydrological Modeling and Data Interpolation, 2011, 
ISBN 978-3-942036-04-7 
 
201 Geleta, Habtamu Itefa: Watershed Sediment Yield Modeling for Data Scarce 
Areas, 2011, ISBN 978-3-942036-05-4 
 
202 Franke, Jörg: Einfluss der Überwachung auf die Versagenswahrscheinlichkeit von 
Staustufen, 2011, ISBN 978-3-942036-06-1 
 
203 Bakimchandra, Oinam: Integrated Fuzzy-GIS approach for assessing regional soil 
erosion risks, 2011, ISBN 978-3-942036-07-8 
 
204 Alam, Muhammad Mahboob: Statistical Downscaling of Extremes of 
Precipitation in Mesoscale Catchments from Different RCMs and Their Effects 
on Local Hydrology, 2011, ISBN 978-3-942036-08-5 
 
205 Hrsg.: Koschitzky, Hans-Peter; Braun, Jürgen: VEGAS-Kolloquium 2011 Flache Ge-
othermie - Perspektiven und Risiken, Tagungsband zur Veranstaltung am 06. Ok-
tober 2011 an der Universität Stuttgart, Campus Stuttgart-Vaihingen, 2011, ISBN 
978-3-933761-09-2 
 
206 Haslauer, Claus: Analysis of Real-World Spatial Dependence of Subsurface 
Hydraulic Properties Using Copulas with a Focus on Solute Transport 
Behaviour, 2011, ISBN 978-3-942036-10-8 
 
207 Dung, Nguyen Viet: Multi-objective automatic calibration of hydrodynamic 
models – development of the concept and an application in the Mekong Delta, 
2011, ISBN 978-3-942036-11-5 
 
208 Hung, Nguyen Nghia: Sediment dynamics in the floodplain of the 
Mekong Delta, Vietnam, 2011, ISBN 978-3-942036-12-2 
 
209 Kuhlmann, Anna: Influence of soil structure and root water uptake on flow in the 
unsaturated zone, 2012, ISBN 978-3-942036-13-9 
16 Institut für Wasser- und Umweltsystemmodellierung * Universität Stuttgart * IWS   
 
 
210 Tuhtan, Jeffrey Andrew: Including the Second Law Inequality in Aquatic 
Ecodynamics: A Modeling Approach for Alpine Rivers Impacted by Hydropeaking, 
2012, 
ISBN 978-3-942036-14-6 
 
211 Tolossa, Habtamu: Sediment Transport Computation Using a Data-Driven 
Adaptive Neuro-Fuzzy Modelling Approach, 2012, ISBN 978-3-942036-15-3 
 
212 Tatomir, Alexandru-Bodgan: From Discrete to Continuum Concepts of Flow in 
Fractured Porous Media, 2012, ISBN 978-3-942036-16-0 
 
213 Erbertseder, Karin: A Multi-Scale Model for Describing Cancer-Therapeutic 
Transport in the Human Lung, 2012, ISBN 978-3-942036-17-7 
 
214 Noack, Markus: Modelling Approach for Interstitial Sediment Dynamics and 
Reproduction of Gravel Spawning Fish, 2012, ISBN 978-3-942036-18-4 
 
215 De Boer, Cjestmir Volkert: Transport of Nano Sized Zero Valent Iron Colloids 
during Injection into the Subsurface, 2012, ISBN 978-3-942036-19-1 
 
216 Pfaff, Thomas: Processing and Analysis of Weather Radar Data for Use in Hydro-
logy, 2013, ISBN 978-3-942036-20-7 
 
217 Lebrenz, Hans-Henning: Addressing the Input Uncertainty for Hydrological 
Modeling by a New Geostatistical Method, 2013, ISBN 978-3-942036-21-4 
 
218 Darcis, Melanie Yvonne: Coupling Models of Different Complexity for the Simula-
tion of CO2 Storage in Deep Saline Aquifers, 2013, ISBN 978-3-942036-22-1 
 
219 Beck, Ferdinand: Generation of Spatially Correlated Synthetic Rainfall Time Series 
in High Temporal Resolution - A Data Driven Approach, 2013, 
ISBN 978-3-942036-23-8 
 
 
Die Mitteilungshefte ab der Nr. 134 (Jg. 2005) stehen als pdf-Datei über die Homepage 
des Instituts: www.iws.uni-stuttgart.de zur Verfügung.