Two robust estimators of the cross-variogram for multivariate geostatistical analysis of soil properties

If we wish to describe the coregionalization of two or more soil properties for estimation by cokriging then we must estimate and model their auto- and cross-variogram(s). The conventional estimates of these variograms, obtained by the method-of-moments, are unduly affected by outlying data which inflate the variograms and so also the estimates of the error variance of cokriging predictions. Robust estimators are less affected. Robust estimators of the auto-variogram and the pseudo cross-variogram have previously been proposed and used successfully, but the multivariate problem of estimating the cross-variogram robustly has not yet been tackled. Two robust estimators of the cross-variogram are proposed. These use covariance estimators with good robustness properties. The robust estimators of the cross-variogram proved more resistant to outliers than did the method-of-moments estimator when applied to simulated fields which were then contaminated. Organic carbon and water content of the soil was measured at 256 sites on a transect and the method-of-moments estimator, and the two robust estimators, were used to estimate the auto-variograms and cross-variogram from a prediction subset of 156 sites. The data on organic carbon included a few outliers. The method-of-moments estimator returned larger values of the auto- and cross-variograms than did either robust estimator. The organic carbon content at the 100 validation sites on the transect was estimated by cokriging from the prediction data plus a set of variograms fitted to the method-of-moments estimates and two sets of variograms fitted to the robust estimates. The ratio of the actual squared prediction error to the cokriging estimate of the error variance was computed at each validation site. These results showed that cokriging using variograms obtained by the method-of-moments estimator overestimated the error variance of the predictions. By contrast, cokriging with the robustly estimated variograms gave reliable estimates of the error variance of the predictions.