A comparison of some robust estimators of the variogram for use in soil survey

The standard estimator of the variogram is sensitive to outlying data, a few of which can cause overestimation of the variogram. This will result in incorrect variances when estimating the value of a soil property by kriging or when designing a sampling grid to map the property to a required precision. Several robust estimators of the variogram, based on location and scale estimation, have been proposed as improvements. They seem to be suitable for analysis of soil data in circumstances where the standard estimator is likely to be affected by outliers. Robust estimators are based on assumptions about the distribution of the data which will not always hold and which need not be made in kriging or in estimating the variogram by the standard estimator. The estimators are reviewed. Simulation studies show that the robust estimators vary in their susceptibility to moderate skew in the underlying distribution, but that the effects of outliers are generally greater. The estimators are applied to some soil data, and the resulting variograms used for ordinary kriging at sites in a separate validation data set. In most cases the variograms derived from the standard estimator gave kriging variances which appeared to overestimate the mean squared error of prediction (MSEP). Kriging with variograms based on robust estimators sometimes gave kriging variances which underestimated the MSEP or did not differ significantly from it. Estimates of kriging variance and the MSEP derived from the validation data were generally close to estimates from cross-validation on the prediction set used to derive the variograms. This indicates that variogram models derived from different estimators could be compared by cross-validation.