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

A current problem of practical significance is the analysis of large, spatially distributed, environmental data sets. The problem is more challenging for variables that follow non-Gaussian distributions. We show by means of numerical simulations that the spatial correlations between variables can be captured by interactions between “spins”. The spins represent multilevel discretizations of environmental variables 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 of spatially distributed variables from samples with missing data. Currently, the sampling and simulation points are assumed to be at the nodes of a regular grid. The conditional simulations of the “spin system” are forced to respect locally the sample values and the system statistics globally. The second constraint is enforced by minimizing a cost function representing the deviation between normalized correlation energies of the simulated and the sample distributions. In the approach based on the Nc−state Potts model, each point is assigned to one of Nc classes. The interactions involve all the points simultaneously. In the Ising model approach, a sequential simulation scheme is used: the discretization at each simulation level is binomial (i.e., ±1.) Information propagates from lower to higher levels as the simulation proceeds. We compare the two approaches in terms of their ability to reproduce the target statistics (e.g., the histogram and the variogram of the sample distribution), to predict data at unsampled locations, as well as in terms of their computational complexity. The comparison is based on a non-Gaussian data set (derived from a digital elevation model of the Walker lake area, Nevada, USA). We discuss the impact of relevant simulation parameters, such as the domain size, the number of discretization levels, and the initial conditions.