Choosing functions for semi-variograms of soil properties and fitting them to sampling estimates

The semi-variogram is central to geostatistics and the single most important tool in geo-statistical applications to soil. Mathematical functions for semi-variograms must be conditional negative semi-definite, and there are only a few families of simple function that meet this demand. These include the transitive models with finite a priori variance deriving from moving average processes. The spherical and exponential schemes are the most often encountered members. The other major group is that of unbounded models in which the variance appears to increase without limit. The linear model is the most common in this group. If more complex models are needed they can be formed by combining two or more simple models. The usual estimator of the semi-variance is often considered inefficient and to be sensitive to departures from normality in the data. It is compared with a robust estimator and shown to be generally preferable in being unbiased and having confidence intervals that are no wider. For routine analysis, fitting models to sample semi-variograms by weighted least squares approximation, with weights proportional to the expected semi-variance, is preferred to the more elaborate and computationally demanding statistical procedures of generalized least squares and maximum likelihood. The Akaike information criterion is recommended for selecting the best model from several plausible ones to describe the observed variation in soil, though for kriging it may be desirable to validate the chosen model. Examples of models fitted to soil semi-variograms are shown and compared.