Multilevel Discretized Random Field Models with "Spin" Correlations for the Simulation of Environmental Spatial Data

A problem of practical significance is the analysis of large, spatially distributed data sets. The problem is more challenging for variables that follow non-Gaussian distributions. We show that the spatial correlations between variables can be captured by interactions between "spins". The spins represent multilevel discretizations of the initial field with respect to a number of pre-defined thresholds. The spatial dependence between the "spins" is imposed by means of short-range interactions. We present two approaches, inspired by the Ising and Potts models, that generate conditional simulations from samples with missing data. The simulations of the "spin system" are forced to respect locally the sample values and the system statistics globally. We compare the two approaches in terms of their ability to reproduce the sample statistical properties, to predict data at unsampled locations, as well as in terms of their computational complexity. We discuss the impact of relevant simulation parameters, such as the domain size, the number of discretization levels, and the initial conditions.