Assuming a dataset has been examined, transformed where necessary and semivariances computed, modelling of the variogram may proceed. A large variety of alternative models have been proposed over recent years, and many of the more popular models are incorporated into GIS packages and geostatistical software. Table 6‑3 shows a selection of the principal models provided within such packages and Figure 6‑41 provides sample graphs for each of those listed. The majority of these models are included within ArcGIS Geostatistical Analyst (which also includes a number of other models), Vertical Mapper for MapInfo and Surfer, and in packages that are based on or utilise GStat, such as Idrisi, PCRaster and GRASS. Other products, such as GS+, GSLIB and Variowin (a free, interactive variogram analysis tool) may provide a smaller range of models but offer considerable flexibility in the modelling and display process (see also, the R spatial projects page on the SAL website). In Table 6‑3 the distance variable, h, is pre-scaled by the effective range, a. In much of the literature and product documentation sets this scaling is not shown, so where we show h they may show h/a, or a similar ratio. Linear combinations of models are widely supported, but selecting and combining the component functions may be difficult.

Selecting and fitting variograms is something of an art rather than a science. Many packages provide a default model and attempt to find the best set of parameters to fit the dataset, whilst others apply this process for all models they support and select the one with the highest correlation coefficient or lowest residual sum of squares. Selecting only models that are asymptotic to the sill (i.e. to 1 in the diagrams shown) provides a useful first level of discrimination between functions. Both the selection of active lag distance to be considered, and the lag interval to be used, will affect the model fitted — the use of values that are too large in either case may result in over-large estimates of nugget variance. Examining the profiles of these functions it is clear that the main differences between them apply in the first quarter of the range, hence close examination of the data within this scale (a/4) is advisable. Anisotropy may be automatically modelled or require selection, with attributes such as size and type of sectors being specified. Optimised selection of model and/or parameters under anisotropy may well alter the major axis of anisotropy identified to some extent, depending on the model selected. Most packages do not allow interactive alteration of models and parameters (one exception being Variowin), but all will allow exploration by trial-and-error, re-calculating the model to user-defined specifications. One difficulty with these approaches is that it is very difficult to replicate results across packages, even when selected parameters are “forced” by the user.