Estimating variograms of soil properties by the method-of-moments and maximum likelihood

Variograms of soil properties are usually obtained by estimating the variog ram for distinct lag classes by the method-of-moments and fitting an approp riate model to the estimates. An alternative is to fit a model by maximum l ikelihood to data on the assumption that they are a realization of a multiv ariate Gaussian process. This paper compares the two using both simulation and real data. The method-of-moments and maximum likelihood were used to estimate the vari ograms of data simulated from stationary Gaussian processes. In one example , where the simulated field was sampled at different intensities, maximum l ikelihood estimation was consistently more efficient than the method-of-mom ents, but this result was not general and the relative performance of the m ethods depends on the form of the variogram. Where the nugget variance was relatively small and the correlation range of the data was large the method -of-moments was at an advantage and likewise in the presence of data from a contaminating distribution. When fields were simulated with positive skew this affected the results of both the method-of-moments and maximum likelih ood. The two methods were used to estimate variograms from actual metal concentr ations in topsoil in the Swiss Jura, and the variograms were used for krigi ng. Both estimators were susceptible to sampling problems which resulted in over- or underestimation of the variance of three of the metals by kriging . For four other metals the results for kriging using the variogram obtaine d by maximum likelihood were consistently closer to the theoretical expecta tion than the results for kriging with the variogram obtained by the method -of-moments, although the differences between the results using the two app roaches were not significantly different from each other or from expectatio n. Soil scientists should use both procedures in their analysis and compare the results.