Random Field, Gaussian
Stochastic Modeling and Environmental Change
Published Online: 15 SEP 2006
Copyright © 2002 John Wiley & Sons, Ltd
Encyclopedia of Environmetrics
How to Cite
Worsley, K. J. 2006. Random Field, Gaussian. Encyclopedia of Environmetrics. 5.
- 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)].