Determining the effect of asymmetric data on the variogram. I. Underlying asymmetry

Matheron's usual variogram estimator can result in unreliable variograms when data are strongly asymmetric or skewed. Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. This paper examines the effects of underlying asymmetry on the variogram and on the accuracy of prediction, and the second one examines the effects arising from outliers. Standard geostatistical texts suggest ways of dealing with underlying asymmetry; however, this is based on informed intuition rather than detailed investigation. To determine whether the methods generally used to deal with underlying asymmetry are appropriate, the effects of different coefficients of skewness on the shape of the experimental variogram and on the model parameters were investigated. Simulated annealing was used to create normally distributed random fields of different size from variograms with different nugget:sill ratios. These data were then modified to give different degrees of asymmetry and the experimental variogram was computed in each case. The effects of standard data transformations on the form of the variogram were also investigated. Cross-validation was used to assess quantitatively the performance of the different variogram models for kriging. The results showed that the shape of the variogram was affected by the degree of asymmetry, and that the effect increased as the size of data set decreased. Transformations of the data were more effective in reducing the skewness coefficient in the larger sets of data. Cross-validation confirmed that variogram models from transformed data were more suitable for kriging than were those from the raw asymmetric data. The results of this study have implications for the 'standard best practice' in dealing with asymmetry in data for geostatistical analyses. (C) 2007 Elsevier Ltd. All rights reserved.