The first chapter of this dissertation discusses besides my own contributions to Bayesian geostatistics the most important papers in this area since 1989.
I wrote this chapter with special emphasis on how to take account of the uncertainty of the covariance function by means of the Bayesian approach. Another point that seemed important to me was to formulate models that go away from and weaken the usual Gaussian assumption in geostatistics.
Chapter 2 is devoted to a frequentist approach to taking account of the uncertainty of the covariance function developed by myself and called covariance robust minimax kriging. In contrast to the Bayesian approach of chapter 1, where the covariance functions are weighted according to their prior distributions, here we look for a predictor minimizing the worst possible mean square error of prediction among a class of equally possible covariance functions.
Chapter 3 is about spatial sampling design. Besides my own contribution to this theory in the form of approximating a stochastic process by a linear regression model with stochastic coefficients and then using classical Bayesian experimental design theory to calculate spatial sampling designs, I give here also a survey of the most recent results in this nice field. Especially also here we discuss some approaches of how to take the uncertainty of the covariance function into account still during spatial sampling design.