Estimation of the Variogram Using Kendall's Tau for a Robust Geostatistical Interpolation

The estimation of an appropriate variogram is a crucial step toward the description of spatial dependence, the geostatistical interpolation of environmental variables, and the subsequent hydrological engineering. The classical variogram in the literature ideally necessitates a normal distribution of the variable and is not robust against outliers within the data. These presumptions are hardly given under empirical conditions and, therefore, a new estimation method is proposed for the variogram. The new method is based on the description of spatial dependence by the robust rank coefficient tau and generalizes the method from the Gaussian to the general case of empirical distributions. The conversion of the robust estimate using a Monte-Carlo simulation and subsequent quantile-quantile transformation with the empirical marginal distribution performs the generalization. Monthly precipitation data from South Africa serve as the variable and were artificially contaminated with outliers. The effects on the variogram and subsequent geostatistical interpolation were investigated for the proposed, classical, and four existing robust variogram models in this comparative study. The investigation revealed that the proposed variogram describes a distinct spatial dependence structure under empirical conditions, which is robust against outliers. The cross validation of the linear estimator demonstrates that the proposed variogram tends to improve the bias and spread of the resulting error distribution, and hence the quality of the geostatistical interpolation.