Spartan Gibbs Random Field Models for Geostatistical Applications

GEOSTATISTICAL APPLICATIONS Abstract: The inverse problem of determining the spatial dependence of random fields from an experimental sample is a central issue in Geostatistics. We propose a computationally efficient approach based on Spartan Gibbs random fields. Their probability density function is determined by a small set of parameters, which can be estimated by enforcing sample-based constraints on the stochastic moments. The computational complexity of calculating the constraints increases linearly with the sample size. We investigate a specific Gibbs probability density with spatial dependence derived from generalized gradient and Laplacian operators, and we derive permissibility conditions for the model parameters. The optimal values for the parameters are determined by minimizing a normalized metric measuring the ÒdistanceÓ between stochastic moments and the sample constraints. The computational complexity of the minimization depends on the number of the model parameters, but not on the sample size. We illustrate the method using simulated control samples with different types of spatial dependence. Spartan Gibbs random fields are useful for modeling large samples, and when reliable estimation of the variogram is not possible. Estimation of the field values at non-sampled positions, conditional simulations, as well as extensions to anisotropic spatial dependence, non-Gaussian probability densities, and off-lattice distributions are briefly discussed.