The VARIOGRAM Procedure

Fitting with Matérn Forms

When you use a Matérn form in the fitting process, it is possible that the fitting optimizer might encounter numerical difficulties if it tries to push the smoothing parameter $\nu $ towards increasingly high values. The VARIOGRAM procedure addresses this issue by imposing an amply elevated upper bound of 1,000,000 on the smoothness values it processes. The section Characteristics of Semivariogram Models mentions that $\nu \rightarrow \infty $ gives the Gaussian model. In the scenario of progressively increasing smoothness values, PROC VARIOGRAM acknowledges that the Matérn form behavior tends asymptotically to become Gaussian and replaces automatically the Matérn with a Gaussian form in the model. Subsequently, fitting resumes with the resulting model.

If you explore fitting of multiple models, then any duplicate models that might occur due to Matérn-to-Gaussian form conversions are fitted only once. Also, if a nested model has more than one Matérn form, then the fitting process checks one of them at a time about whether they need to be replaced by a Gaussian form. Consequently, following the switch of one Matérn form, the fitting process starts anew with the resulting model before any decisions for additional form conversions are made.

Replacement of the Matérn form with the Gaussian form occurs by default when $\nu > 10,000$. However, you can control this threshold value with the MTOGTOL= parameter of the MODEL statement. Practically, the Matérn form starts to resemble the Gaussian behavior for $\nu $ values that are about $\nu > 10$. If you encounter such conversions of the Matérn form into Gaussian and you prefer to set a lower $\nu $ threshold for the conversion than the default, you might experience improved code performance because computation of the Matérn semivariance can be numerically demanding.