Multilevel discretized random field models with ‘spin’ correlations for the simulation ofenvironmental spatial data

Abstract

A current problem of practical significance is how to analyze large, spatially distributed,environmental data sets. The problem is more challenging for variables that follownon-Gaussian distributions. We show by means of numerical simulations that the spatialcorrelations between variables can be captured by interactions between ‘spins’. The spinsrepresent multilevel discretizations of environmental variables with respect to anumber 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 ofspatially distributed variables from samples with missing data. Currently, thesampling and simulation points are assumed to be at the nodes of a regular grid. Theconditional simulations of the ‘spin system’ are forced to respect locally the samplevalues and the system statistics globally. The second constraint is enforced byminimizing a cost function representing the deviation between normalized correlationenergies of the simulated and the sample distributions. In the approach based on theNc-state Potts model, each point is assigned to one ofNc classes. The interactions involve all the points simultaneously. In the Ising model approach, asequential 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. Wecompare 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 unsampledlocations, as well as in terms of their computational complexity. The comparison isbased on a non-Gaussian data set (derived from a digital elevation model of theWalker Lake area, Nevada, USA). We discuss the impact of relevant simulationparameters, such as the domain size, the number of discretization levels, and the initialconditions.