The VARIOGRAM Procedure

Theoretical Semivariogram Model Fitting

PROC VARIOGRAM features automated semivariogram fitting. In particular, the procedure selects a theoretical semivariogram model to fit the empirical semivariance and produces estimates of the model parameters in addition to a fit plot. You have the option to save these estimates in an item store, which is a binary file format that is defined by the SAS System and that you cannot modify. Then, you can retrieve this information at a later point from the item store for future analysis with PROC KRIGE2D or PROC SIM2D.

The coal seam thickness empirical semivariogram in Figure 102.7 shows first a slow, then rapid, rise from the origin. This behavior suggests that you can approximate the empirical semivariance with a Gaussian-type form

\[  \gamma _ z(h)=c_0\left[1-\exp \left(-\frac{h^2}{a_0^2}\right)\right]  \]

as shown in the section Theoretical Semivariogram Models. Based on this remark, you choose to fit a Gaussian model to your classical semivariogram. Run PROC VARIOGRAM again and specify the MODEL statement with the FORM=GAU option. By default, PROC VARIOGRAM uses the weighted least squares (WLS) method to fit the specified model, although you can explicitly specify the METHOD= option to request the fitting method. You want additional information about the estimated parameters, so you specify the CL option in the MODEL statement to compute their 95% confidence limits and the COVB option of the MODEL statement to produce a table with their approximate covariances. You also specify the STORE statement to save the fitting outcome into an item store file with the name SemivStoreGau and a desired label. You run the following statements:

proc variogram data=thick outv=outv;
   store out=SemivStoreGau / label='Thickness Gaussian WLS Fit';
   compute lagd=7 maxlag=10;
   coordinates xc=East yc=North;
   model form=gau cl / covb;
   var Thick;

ods graphics off;

After you run the procedure you get a series of output objects from the fitting analysis. In particular, Figure 102.10 shows first a model fitting table with the name and a short label of the model that you requested to use for the fit. The table also displays the name and label of the specified item store.

Figure 102.10: Semivariogram Model Fitting General Information

Spatial Correlation Analysis with PROC VARIOGRAM

The VARIOGRAM Procedure
Dependent Variable: Thick
Angle: Omnidirectional
Current Model: Gaussian

Semivariogram Model Fitting
Name Gaussian
Label Gau
Item Store Label Thickness Gaussian WLS Fit

If you specify no parameters, as in the current example, then PROC VARIOGRAM initializes the model parameters for you with default values based on the empirical semivariance; for more details, see the section Theoretical Semivariogram Model Fitting. The initial values provided by the VARIOGRAM procedure for the Gaussian model are displayed in the table in Figure 102.11.

Figure 102.11: Semivariogram Fitting Model Information

Model Information
Parameter Initial
Nugget 0
Scale 6.7992
Range 34.9635

Otherwise, in PROC VARIOGRAM you can specify initial values for parameters with the PARMS statement. Alternatively, you can specify fixed values for the model scale and range with the SCALE= and RANGE= options, respectively, in the MODEL statement. A nugget effect is always used in model fitting. Unless you explicitly specify a fixed nugget effect with the NUGGET= option in the MODEL statement or initialize the nugget parameter in the PARMS statement, the nugget effect is automatically initialized to zero. See the section Syntax: VARIOGRAM Procedure for more details about how the MODEL statement and the PARMS statement handle model parameters.

The output in Figure 102.12 comes from the optimization process that takes place during the model parameter estimation. The optimizer produces an optimization information table, information about the optimization technique that is used, optimization-related results, and notification about the optimization convergence.

Figure 102.12: Fitting Optimization Information

Optimization Information
Optimization Technique Dual Quasi-Newton
Parameters in Optimization 3
Lower Boundaries 3
Upper Boundaries 0
Starting Values From PROC

Spatial Correlation Analysis with PROC VARIOGRAM

The VARIOGRAM Procedure
Dependent Variable: Thick
Angle: Omnidirectional
Current Model: Gaussian

Dual Quasi-Newton Optimization

Dual Broyden - Fletcher - Goldfarb - Shanno Update (DBFGS)

Hessian Computed by Finite Differences (Using Analytic Gradient)

Optimization Results
Iterations 12 Function Calls 45
Gradient Calls 0 Active Constraints 1
Objective Function 11.433894152 Max Abs Gradient Element 3.0128744E-8
Slope of Search Direction -3.986332E-8    

Convergence criterion (GCONV=1E-8) satisfied.

The fitting process is successful, and the parameters converge to the estimated values shown in Figure 102.13. For each parameter, the same table also displays the approximate standard error, the degrees of freedom, the t value, the approximate p-value, and the requested 95% confidence limits.

Figure 102.13: Semivariogram Fitting Parameter Estimates

Parameter Estimates
Parameter Estimate Approx
Std Error
Approximate 95% Confidence
DF t Value Approx
Pr > |t|
Lower Upper
Nugget 0 0 0 0 8 . .
Scale 7.4599 0.2621 6.8555 8.0643 8 28.46 <.0001
Range 30.1111 1.1443 27.4724 32.7498 8 26.31 <.0001

The approximate covariance matrix of the estimated parameters is displayed in Figure 102.14.

Figure 102.14: Approximate Covariance Matrix of Parameter Estimates

Approximate Covariance Matrix
Parameter Nugget Scale Range
Nugget 0.0000 0.0000 0.0000
Scale 0.0000 0.0687 0.2326
Range 0.0000 0.2326 1.3094

The fitting summary table in Figure 102.15 displays statistics about the quality of the fitting process. In particular, the table shows the weighted error sum of squares in the Weighted SSE column and the Akaike information criterion in the AIC column. See more information about the fitting criteria in section Quality of Fit.

Figure 102.15: Semivariogram Model Fitting Summary

Fit Summary
Model Weighted
Gau 11.43389 6.42556

Figure 102.16 demonstrates the fitted theoretical semivariogram against the empirical semivariance estimates with the weighted least squares method. The fit seems to be more accurate closer to the origin $\bm {h}=0$, and this is explained as follows: A smaller $\bm {h}$ corresponds to smaller semivariance; in turn, this corresponds to smaller semivariance variance, as shown in the section Theoretical and Computational Details of the Semivariogram. By definition, the WLS optimization weights increase with decreasing variance, which leads to a more accurate fit for smaller distances $\bm {h}$ in the WLS fitting results.

Figure 102.16: Fitted Theoretical and Empirical Semivariogram for Coal Seam Thickness

 Fitted Theoretical and Empirical Semivariogram for Coal Seam Thickness