Conditional Simulation and Estimation of Gauss-Markov Random Fields Using the Bayesian Nearest Neighbor Method

Abstract

In this paper a specialized method for generating Markovian random fields, with or without conditioning, is presented. Here, the prior fields are assumed to be stationary second-order Gauss-Markov random fields in N-dimensional (N-D) euclidian space. The unconditional Markov fields are generated as numerical solutions of white-noise driven PDEs, for the solution of which finite difference or finite volume discretization on a regular grid yields an algorithm commonly known as the Nearest Neighbor Method (NNM). In addition, starting with the continuous space PDE-driven field, we develop a generalization of the NNM algorithm for the generation and optimal interpolation (estimation) of conditional random fields in the Bayesian sense. This generalized NNM algorithm is called Bayesian-NNM, or B-NNM, and is essentially similar to the classical NNM-type algorithms. The B-NNM method can be used for conditional simulation of random fields, as well as the optimal estimation of spatial fields assuming Markovian prior statistics, based on a collection of measurements or data. These data may be defined either at points or, more importantly, on continuous subdomains of various sizes and shapes, such as lineaments, faults, layers, boreholes, etc. In this paper, we also present exact closed form relations for linear estimation of I-D Markovian fields, and this without recourse to space discretization. The complete analytical results are used for testing the multi-dimensional BNNM code in the 1-D case.