Asymmetric dependence based spatial copula models : empirical investigations and consequences on precipitation fields

Abstract

Hydrologic system analysis plays an important role in various projects in water resources related issues, such as floods and drought control, as well as in planning, design, and operation of such projects. The role of this system is to produce output from given inputs. Precipitation is one of the major inputs of hydrologic systems. In reality, precipitation data are unfortunately frequently inadequate observation datasets, regarding both record length and completeness due to issues such as a small number of spatial observations and instrument error. Insufficient precipitation data regarding time series length and spatial coverage can lead to serious problems in hydrological analysis, resulting in either underestimation or overestimation of hydrological design values. To overcome the lack of high quality of precipitation observation data, spatial and temporal models of precipitation are required to fill in missing values, extend the length of data, and interpolate and simulate spatial data at unobserved locations. Many precipitation models have been developed over the past half-century by hydrologists in order to bridge the gap between the need for high-quality precipitation data and the lack of available data in reality. Most precipitation models are developed with either an explicit or implicit underlying Gaussian dependence assumption, which can bias the estimation of reality. One of the main characteristics of Gaussian models is that the observation data are assumed to exhibit symmetric dependence structures, for instance, between low and high values. The first goal of this study is to empirically investigate the behaviour of the spatial dependence of precipitation fields. This would determine whether the Gaussian assumption is fulfilled in regard to the symmetric spatial dependence structure between low and high precipitation values. The second target is to then quantify the consequences of an asymmetric spatial dependence on the spatial extremes of areal precipitation amounts, where are frequently required for hydrological design. In order to complete the first goal of this study, an asymmetry function which can incorporate zero precipitation amounts is introduced on the basis of empirical bivariate copulas. Copulas are new tools for multivariate modelling which have been broadly implemented into precipitation applications over the last decade. Copulas are multivariate distributions with uniform marginal distributions used to describe the dependence structure between random variables without information on the univariate marginal distributions. The asymmetric function is then used for the investigations. Investigations of asymmetric spatial dependence are carried out in the regions of Bavaria, Baden-Württemberg and Singapore. In order to achieve the second target, the symmetric Gaussian dependence based models are evaluated in the context of spatial extremes of areal precipitation amounts over a regular grid and compared to the asymmetric spatial dependence based models. Both models are implemented in the regions of Singapore and Bavaria using daily precipitation. Gaussian copulas are chosen to represent the symmetric spatial dependence based models because the model is very popular and simple where the dependence structure is completely determined by the correlation coefficient matrix. The V-copulas are selected to represent the asymmetric spatial dependence based models which are constructed from Gaussian copulas through a non-monotonic transformation with the parameters m and k. Both Gaussian and V-copulas are fitted to the empirical copulas using standard maximum likelihood methods, where zero precipitation amounts are treated as latent variables of a continuous distribution. Zero-inflated precipitation data frequently occur, especially at higher time resolutions (e.g. hourly or daily scales). Precipitation is modelled using a continuous distribution. Dry locations correspond to values not exceeding a threshold in the continuous distribution. Investigation results prove that precipitation events tend to follow the positive asymmetric spatial dependence structure, in particular at short separating distances. This implies that precipitation with higher intensities tends to be more spatially correlated than lower intensities. This is very interesting since spatial interpolation is commonly carried out using nearby points. Consequently, spatial precipitation models based on symmetric Gaussian dependence could result in underestimation of actual precipitation extremes. The V-transformed normal copulas provide a possible solution to model the natural processes of precipitation which follow asymmetric spatial dependence structures reasonably well within high multidimensional problems. Empirical investigations focusing on the spatial extremes of gridded areal precipitation amounts reveal that the Gaussian copulas frequently exhibit lower spatial extremes of mean areal gridded precipitation values than the V-copulas. This is an indication that extreme precipitation occurrences, which typically behave in a clustering manner, cannot be modelled reasonably by Gaussian copulas. As a result, Gaussian copulas would yield an underestimation of flood risks and should therefore be implemented with care in the wider practice of flood designs.