Browsing by Author "Li, Jing"

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Open Access
Application of copulas as a new geostatistical tool
(2010) Li, Jing; Bárdossy, András (Prof. Dr. rer. nat. Dr.-Ing.)
In most geostatistical analysis, the spatial variability is solely described with a variogram or a covariance function. These measures have three major problems. First of all, they imply a Gaussianity assumption for the spatial dependence structure. This is, however, often violated in reality as manifested by the dataset investigations in this study. Second, they are sensitive to outliers and thus can be easily polluted by measurement anomalies. Third, they describe the spatial dependence as an integral over the whole marginal distribution of the parameter values. The change of the dependence strength with different quantiles of the parameter is not reflected by these measures, hence not considered in the subsequent interpolation or simulation procedures. But this aspect can be of vital importance for some prediction and design purposes. For example, to predict the flow or transport behavior in a heterogeneous subsurface, where cracks or connected paths, whose spatial continuity deviates largely from the mean trend, exist. Or to extend a monitoring network for noncompliance with environmental standards, where analysis of estimation uncertainty plays a central role and the difference in the uncertainties for different parameter values cannot be overlooked. To overcome the above addressed problems, the concept of copulas is borrowed in this study as an alternative to the traditional geostatistical tools for spatial description and modeling. As a counterpart of the shortcomings of variogram/covariance function, the main advantages of using copula are also threefold. First of all, it captures the {\it pure} dependence among the random variables separately from their univariate distributions and thus the influence of measurement outliers or very skewed distributions vanishes. Second, since it describes the dependence as a full distribution instead of the mean behavior, the variation of dependence strength for different quantiles is revealed, which considerably improves the estimation and prediction quality. Last but not the least, non-Gaussian theoretical copulas which are suitable for spatial modeling can be developed so that the Gaussianity assumption is no more a must and the real dependence structure can be mimicked better. In this study, methodology of using copulas for spatial modeling is established, including making the basic hypothesis, defining empirical copulas as a substitute for variogram/covariance, adopting and devising scale invariant measures for quantification of spatial dependence. Theoretical non-Gaussian copulas which are suitable for spatial modeling are derived and the model inference approach which is a combination of maximum likelihood and multiple-point statistics is proposed as well. The application of the methodology breaks down into three parts. The first part focuses on spatial interoplation, where the procedure of interpolation based on conditional copula is developed. An example of spatial interpolation of groundwater parameters in Baden-Württemberg (Germany) shows that the copula based approach gives better cross validation results than the ordinary and indicator kriging methods. Validation of the confidence intervals estimated from conditional copulas indicates that they are more realistic than the estimation variances obtained from ordinary and indicator kriging. The second part deals with the topic of spatial simulations. In this part, simulation algorithms of generating realizations of multivariate non-Gaussian dependence are developed for both unconditional and conditional cases. Spatial analysis of the hydraulic conductivity measurements of Las Cruces Trench Site shows that the spatial dependence of this dataset exhibit clear anisotropic and non-Gaussian behavior. Statistical tests of the realizations from three copula models parameterized on this dataset, i.e., the Gaussian copula, the v-transformed normal copula and the maximum normal copula, also indicate that the Gaussian copula is most likely to be rejected, while the maximum normal copula is proved to be the most suitable one. The significance of multivariate dependence structure described by copulas to the flow and transport behavior is studied indrectly by investigating the topological/connectivity characteristics of the realizations from different copula models. In the third part, the approach of spatial modeling based on copulas is applied to facilitate observation network design where the estimated conditional copulas describing the probabilistic structure of the values at the unsampled locations are embedded into the utility function which acts as the objective function to be maximized in order to select the optimal location for new measurement. The application to expand the observation network of groundwater quality parameters in a sub-region of Baden-Wüttemberg demonstrates the potentialities of the methodology.