Two contrasting spatial processes with a common variogram: inference about spatial models from higher-order statistics

Geostatistical analysis of soil properties is undertaken to allow prediction of values of these properties over regions or at unsampled locations. A key step in geostatistical analysis is the estimation of a variogram function that describes the spatial covariance structure of the variable in question. If it can be assumed plausibly that the data are a realization of a second-order stationary multivariate normal random function then this function is entirely characterized by its mean (expectation) and spatially dependent covariance. Because of this, the variogram is sometimes computed as a general 'descriptor' of spatial variation, and used, for example, to compare the spatial structure at within-aggregate scales of soils under different management, or to compare soils from different land uses with respect to the spatial structure of their microbial populations. The objective of this paper is to draw attention to the limited value of the variogram for characterizing spatial variation (as opposed to deriving best linear unbiased predictions). Specifically, it is shown how two contrasting processes, one of which gives rise to a multivariate normal random function (a convolution filter applied to independent identically distributed random values) and one which does not (a partition of space into random sets), may have the same variogram function. A diagnostic is proposed that indicates which of these two processes is most plausible as a model for a data set. This will allow the spatial analysis of soil data to give greater insight into the factors underlying the variation of a soil property, and may permit more realistic simulation of soil properties.