Robustness and Bayesian analysis of spatial interpolation.

Abstract

Kriging is a well known spatial interpolation technique widely used in earth science and environment sciences, the variogram plays a central role in the kriging predictor. In this dissertation, we will mainly study two problems which are closely related to the kriging predictor. The first one is how the variogram affects the kriging predictor and how this effect is qualified. The second one is how to approach kriging with an uncertain variogram, which includes both the functional form and the parameters in the variogram. For the first problem, some investigation of robustness of kriging predictor have been done by some authors. And for the second one, two frameworks have been used to approach the kriging with uncertain variogram in recent years. For the formal approach, the Bayesian framework is used to achieve the goal, and for the latter one, the fuzzy set theory is used, which mainly means that the kriging with an uncertain variogram is represented by the calculated membership function for each kriged value. The object of this dissertation is to extend the robustness results of kriging, to generalize the robustness concept to the cross-validation method, and to study the robustness of the cross-validation. We define the influence function of kriging and cross-validation technique and derive their influence functions in terms of perturbation of variogram and sample configuration. We will derive some different Bayesian kriging models under different assumptions and study their properties. We will also modify the fuzzy kriging model. Moreover we discuss the relationship between Bayesian kriging and fuzzy kriging and relate the fuzzy kriging to the robustness of kriging. Finally, in this work, we will show the power of Bayesian kriging and display its advantage as an interpolation technique for the analysis of spatial data. This is done through the presentation of a case study.