Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set

This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions, constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate directing function and a coding process with a separable covariance, or by including drift components in the directing function. Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the median and for a spatial clustering of high pollutant concentrations.