A semivariance analysis of the coal seam thickness `thick`

data set is performed in Getting Started: VARIOGRAM Procedure in Chapter 106: The VARIOGRAM Procedure. The analysis considers the spatial random field (SRF) of the `Thick`

variable 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

Of note, 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 (i,j) 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 in section Theoretical Semivariogram Model Fitting in Chapter 106: 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.