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Random Field, Gaussian

Stochastic Modeling and Environmental Change

  1. Keith J. Worsley

Published Online: 15 SEP 2006

DOI: 10.1002/9780470057339.var008

Encyclopedia of Environmetrics

Encyclopedia of Environmetrics

How to Cite

Worsley, K. J. 2006. Random Field, Gaussian. Encyclopedia of Environmetrics. 5.

Author Information

  1. McGill University, Québec, Canada

Publication History

  1. Published Online: 15 SEP 2006


The Gaussian random field Y(t), t ∈ T, is one of the most common models used to describe spatial stochastic processes. In many applications, the domain T is a subset of D-dimensional Euclidean space (usually D = 2 or D = 3), and the function Y(t) is almost surely continuous or smooth in t. The definition is simple: the Gaussian random field must be multivariate Gaussian at all finite sets of points, that is, [Y(t1), …, Y(tn)] must be multivariate Gaussian for all n > 0 and all tj ∈ T. Since the multivariate Gaussian is specified uniquely by its mean vector and variance matrix, then the Gaussian random field is defined uniquely by its mean function μ(t) = E[Y(t)] and its covariance function C(s, t) = cov[Y(s), Y(t)].