The SIM2D Procedure
A semivariance analysis of the coal seam thickness thick data set is performed in Getting Started: VARIOGRAM Procedure of the VARIOGRAM procedure. The analysis considers the spatial random field (SRF) 
of the Thick variable to be free of surface trends. The expected value ![$E[Z(\bm {s})]$](images/statug_sim2d0016.png){kind=small}
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
![\[ Z(\bm {s}) = \mu + \varepsilon (\bm {s}) \]](images/statug_sim2d0018.png){kind=small}
where 
is a Gaussian SRF with Gaussian covariance structure
![\[ C_ z(\bm {h}) = c_0\exp \left(-\frac{h^2}{a_0^2}\right) \]](images/statug_sim2d0020.png){kind=small}
Of note, the term “Gaussian” is used in two ways in this description. For a set of locations 
, the random vector
![\[ \bm {Z}(\bm {s}) = \left[ \begin{array}{c} Z(\bm {s}_1) \\ Z(\bm {s}_2) \\ \vdots \\ Z(\bm {s}_ n) \\ \end{array} \right] \]](images/statug_sim2d0022.png)
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 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.