Spartan Gibbs Random Field Models for Geostatistical Applications

Abstract

The inverse problem of determining the spatial dependence of random fields (inference of the spatial model) from experimental samples is a central issue in geostatistics. We propose a computationally efficient approach based on Spartan Gibbs random fields. Their probability density function depends on a small set of parameters that can be determined by matching sample constraints with corresponding model constraints based on the stochastic moments. We investigate a specific Spartan probability density with spatial dependence derived from generalized gradient and Laplacian operators, and we derive permissibility conditions for the model parameters. The optimal values of the model parameters are determined by minimizing a normalized metric measuring the "distance" between stochastic moments and the respective sample constraints. The computational complexity of the minimization depends on the number of parameters but not on the sample size. The computational complexity of the constraint calculation increases linearly with the sample size. In contrast, the classical variogram calculation is considerably less efficient and its complexity increases as the square of the sample size. We propose a numerical approach for inferring the Spartan model parameters and illustrate it using simulated (synthetic) samples for regular (lattice) and irregular sample distributions. Based on our analysis, Spartan Gibbs random fields provide computationally efficient spatial models, which are especially useful if the sample size is large or reliable estimation of the variogram is not possible. Estimation of the field values at unsampled positions, conditional simulations, anisotropic spatial dependence, and non-Gaussian probability densities are briefly discussed.