Ordinary multigaussian kriging for mapping conditional probabilities of soil properties

This paper addresses the problem of assessing the risk of deficiency or excess of a soil property at unsampled locations, and more generally of estimating a function of such a property given the information monitored at sampled sites. It focuses on a particular model that has been widely used in geostatistical applications: the multigaussian model, for which the available data can be transformed into a set of Gaussian values compatible with a multivariate Gaussian distribution. First, the conditional expectation estimator is reviewed and its main properties and limitations are pointed out; in particular, it relies on the mean value of the normal scores data since it uses a simple kriging of these data. Then we propose a generalization of this estimator, called "ordinary multigaussian kriging" and based on ordinary kriging instead of simple kriging. Such estimator is unbiased and robust to local variations of the mean value of the Gaussian field over die domain of interest. Unlike indicator and disjunctive kriging, it does not suffer from order-relation deviations and provides consistent estimations. An application to soil data is presented, which consists of pH measurements on a set of 165 soil samples. First, a test is proposed to check the suitability of the multigaussian distribution to the available data, accounting for the fact that the mean value is considered unknown. Then four geostatistical methods (conditional expectation, ordinary multigaussian, disjunctive and indicator kriging) are used to estimate the risk that the pH at unsampled locations is less than a critical threshold and to delineate areas where liming is needed. The case study shows that ordinary multigaussian kriging is close to the ideal conditional expectation estimator when the neighboring information is abundant, and departs from it only in under-sampled areas. In contrast, even within the sampled area, disjunctive and indicator kriging substantially differ from the conditional expectation; the discrepancy is greater in the case of indicator kriging and can be explained by the loss of information due to the binary coding of the pH data