The emergence of dense spatial data sets allows us to examine spatial processes on a local level. This thesis analyzes local prediction and local estimation of the covariance model for a Gaussian process observed on a single dense regular grid. We assume a smooth mean and make the assumption that locally, the covariance function is stationary and approximately an even polynomial plus a principal irregular term. This covariance model satisfies a large class of processes including some which are locally stationary, but nonstationary globally, and processes with locally stationary increments. Some examples include multifractional Brownian motion, the Matern autocovariance and the deformation model.
We justify the use of a Kriging estimator which relies on the covariance only through the principal term. Then we consider local estimation of the principal term through a local linear smoother and prove infill asymptotic convergence results. We prove a central limit theorem with a rate matching the optimal nonparametric rate assuming two derivatives and prove an almost sure uniform convergence result with a rate slightly slower than optimal. Simulation results are provided that validate our theory and we explore additional problems such as estimation at the boundary and missing data.