Geostatistical interpolation using copulas

Abstract

In many applications of geostatistical methods, the dependence structure of the investigated parameter is described solely with the variogram or covariance functions, which are susceptible to measurement anomalies and implies the assumption of Gaussian dependence. Moreover the kriging variance respects only observation density, data geometry and the variogram model. To address these problems, we borrow the idea from copulas, to depict the dependence structure without the influence of the marginal distribution. The methodology and basic hypotheses for application of copulas as geostatistical methods are discussed and the Gaussian copula as well as a non-Gaussian copula are used in this paper. Copula parameters are estimated using a division of the observations into multipoint subsets and a subsequent maximization of the corresponding likelihood function. The interpolation is carried out with two different copulas, where the expected and median values are calculated from the copulas conditioned with the nearby observations. The full conditional copulas provide the estimation distributions for the unobserved locations and can be used to define confidence intervals which depend on both the observation geometry and values. Observations of a large scale groundwater quality measurement network in Baden-Württemberg are used to demonstrate the methodology. Five groundwater quality parameters: chloride, nitrate, pH, sulfate and dissolved oxygen are investigated. All five parameters show non-Gaussian dependence. The copula-based interpolation results of the five parameters are compared to the results of conventional ordinary and indicator kriging. Different statistical measures including mean squared error, relative differences and probability scores are used to compare cross validation and split sampling results of the interpolation methods. The non-Gaussian copulas give better results than the geostatistical interpolations. Validation of the confidence intervals shows that they are more realistic than the estimation variances obtained by ordinary kriging.