Application of Copulas in Geostatistics

Abstract

This paper demonstrates how empirical copulas can be used to describe and model spatial dependence structures of real-world environmental datasets in the purest form and how such a copula model can be employed as the underlying structure for interpolation and associated uncertainty estimates.

Using copulas, the dependence of multivariate distributions is modelled by the joint cumulative distribution of the variables using uniform marginal distribution functions. The uniform marginal distributions are the effect of transforming the marginal distributions monotonically by using the ranks of the variables. Due to the uniform marginal distributions, copulas express the dependence structure of the variables independent of the variables’ marginal distributions which means that copulas display interdependence between variables in its purest form. This property also means that marginal distributions of the original data have no influence on the spatial dependence structure and can not “cover up” parts of the spatial dependence structure. Additionally, differences in the degree of dependence between different quantiles of the variables are readily identified by the shape of the contours of an empirical copula density.

Regarding the quantification of uncertainties, copulas offer a significant advantage: the full distribution function of the interpolated parameter at every interpolation point is available. The magnitude of uncertainty does not depend on the density of the observation network only, but also on the magnitude of the measurements as well as on the gradient of the magnitude of the measurements. That means for the same configuration of the observation network, interpolating two events with very similar marginal distribution, the confidence intervals look significantly different for both events.