The semivariogram analysis of the thick data set in Theoretical Semivariogram Model Fitting of the VARIOGRAM procedure considered the spatial random field (SRF) of the thickness values to be free of surface trends. The expected value is then a constant , which suggests that you can work with the original thickness data rather than residuals from a trend surface fit. In fact, a reasonable approximation of the spatial process generating the coal seam data is given by

where is a Gaussian SRF with Gaussian covariance structure

Note that the term "Gaussian" is used in two ways in this description. For a set of locations , the random vector

has a multivariate Gaussian or normal distribution . The (,)th element of is computed by , which happens to be a Gaussian functional form.

Any functional form for that yields a valid covariance matrix can be used. Both the functional form of and the parameter values

are estimated by using PROC VARIOGRAM and PROC NLIN in Theoretical Semivariogram Model Fitting in the VARIOGRAM procedure. Specifically, the expected value is reported in the VARIOGRAM procedure OUTV output data set, and the parameters and are estimates derived from a weighted least squares fit.

The choice of a Gaussian functional form for is simply based on the data, and it is not at all crucial to the simulation. However, it is crucial to the simulation method used in PROC SIM2D that be a Gaussian SRF. For details, see the section Computational and Theoretical Details of Spatial Simulation.