Cokriging particle size fractions of the soil

It is often necessary to predict the distribution of mineral particles in soil between size fractions, given observations at sample sites. Because the contents in each fraction necessarily sum to 100%, these values constitute a composition, which we may assume is drawn from a random compositional variate. Elements of a D-component composition are subject to non-stochastic constraints; they are constrained to lie on a D - 1 dimensional simplex. This means we cannot treat them as realizations of unbounded random variables such as the multivariate Gaussian. For this reason, there are theoretical reasons not to use ordinary cokriging (or ordinary kriging) to map particle size distributions. Despite this, the compositional constraints on data on particle size fractions are not always accounted for by soil scientists. The additive log-ratio (alr) transform can be used to transform data from a compositional variate into a form that can be treated as a realization of an unbounded random variable. Until now, while soil scientists have made use of the alr transform for the spatial prediction of particle size, there has been concern that the simple back-transform of the optimal estimate of the alr-transformed variables does not yield the optimal estimate of the composition. A numerical approximation to the conditional expectation of the composition has been proposed, but we are not aware of examples of its application and it has not been used in soil science. In this paper, we report two case studies in which we predicted clay, silt and sand contents of the soil at test sites by ordinary cokriging of the alr-transformed data followed by both the direct (biased) back-transform of the estimates and the unbiased back-transform. We also computed estimates by ordinary cokriging of the untransformed data (which ignores the compositional constraints on the variables) for comparison. In one of our case studies, the benefit of using the alr transform was apparent, although there was no consistent advantage in using the unbiased back-transform. In the other case study, there was no consistent advantage in using the alr transform, although the bias of the simple back-transform was apparent. The differences between these case studies could be explained with respect to the distribution on the simplex of the particle size fractions at the two sites.