A tutorial guide to geostatistics: Computing and modelling variograms and kriging

TL;DR: This work guides readers through computing the sample variogram and modelling it by weighted least-squares fitting and explains how to choose the most suitable functions by a combination of graphics and statistical diagnostics.

Abstract: Many environmental scientists are analysing spatial data by geostatistical methods and interpolating from sparse sample data by kriging to make maps. They recognize its merits in providing unbiased estimates with minimum variance. Several statistical packages now have the facilities they require, as do some geographic information systems. In the latter kriging is an option for interpolation that can be done at the press of a few buttons. Unfortunately, the ease conferred by this allows one to krige without understanding and to produce unreliable and even misleading results. Crucial for sound kriging is a plausible function for the spatial covariances or, more widely, of the variogram. The variogram must be estimated reliably and then modelled with valid mathematical functions. This requires an understanding of the assumptions in the underlying theory of random processes on which geostatistics is based. Here we guide readers through computing the sample variogram and modelling it by weighted least-squares fitting. We explain how to choose the most suitable functions by a combination of graphics and statistical diagnostics. Ordinary kriging follows straightforwardly from the model, but small changes in the model function and its parameters can affect the kriging error variances. When kriging is automated these effects remain unknown. We explain the choices to be made when kriging, i.e. whether the support is at points or over blocks, and whether the predictions are global or within moving windows.