Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set
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Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set Xavier Emery
Received: 20 April 2005 / Accepted: 23 August 2007 / Published online: 15 December 2007 © International Association for Mathematical Geology 2007
Abstract 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. Keywords Conditional simulation · Isofactorial bivariate distribution · Bivariate Gaussian distribution · Bivariate gamma distribution · Gibbs sampler 1 Introduction An important aspect in the analysis of regionalized variables is the modeling of local uncertainty and its incorporation in decision-making processes. In polluted site management, the planner is interested in mapping the probability that the concentration of a pollutant exceeds a regulatory threshold, given the information available at data locations. This problem can be solved by using nonlinear kriging techniques X. Emery () Department of Mining Engineering, University of Chile, Avenida Tupper 2069, Santiago, Chile e-mail: [email protected]
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Math Geosci (2008) 40: 83–99
like indicator, disjunctive, or multi-Gaussian kriging (Emery 2006a; Journel 1984; Matheron 1976a; Oliver et al. 1996). In general, the decision-making process involves a more complex transfer function of the pollutant concentration. In particular, this concentration must be upscaled from the data support to that of remediation units. One possibility is to define a changeof-support model and to combine it with one of the previous nonlinear kriging techniques (Emery and Soto-Torres 2005; Matheron 1976b, 1984). Another possibility is to use conditional simulation, which provides alternative realizations of the pollutant concentration that can be averaged to the support of the remediation units. This approach is flexible as it can be used when the upscaling differs from an arithmetic averaging. For instance, the decision-mak
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