| Gaussian random fields (GRFs) can be used to model many physical processes in space. In this work we study two topics about GRFs: spatial sampling design and covariance parameter estimation.; For the estimation problem, we study the use of increments to estimate the fractal dimension of a two-dimensional fractal Brownian surface observed on a regular grid. Linear filters are used to describe differencing of two-dimensional surfaces and generalized variograms are defined based on them. We examine the practical performance of least square estimators based on different filters and a comparison to the restricted maximum likelihood estimator is also provided.; For the design problem, we first study sampling designs for estimating the fractal dimension of Gaussian processes, in which observations are on one of two regular grids of different spacing. We show it is possible to estimate the fractal dimension with mean squared error tending to 0 arbitrarily quickly as the sample size increases when the process can be observed pointwise without error. We next study both spatial sampling designs for covariance parameter estimation and designs for prediction of stationary isotropic Gaussian models with estimated parameters of the covariance functions. Several possible design criteria are discussed and an annealing algorithm is used to search for optimal designs of small sample size. When the sample size is moderately large, the computational cost for the annealing algorithm is prohibitive and a two-step algorithm is proposed to solve this problem. |
