Fluctuations in method-of-moments variograms caused by clustered sampling and their elimination by declustering and residual maximum likelihood estimation

Soil data accumulated in national and regional archives derive from many sources and tend to be concentrated in zones of particular interest. Experimental variograms computed from such data by the usual method of moments can appear highly erratic, and therefore models fitted to them are likely to be unreliable. We have explored two methods of avoiding the effects, one by computing declustering weights and incorporating them into the method of moments, the other using residual maximum likelihood. The methods are illustrated with data on bulk density, exchangeable magnesium, cation exchange capacity and organic carbon of 4182 samples of soil from numerous soil surveys in the whole of Australia and stored in the CSIRO's national archive. The experimental variograms of all four variables are erratic. Cell declustering produced much smoother sequences of estimates to which plausible models could be fitted with confidence. The residual maximum likelihood models closely matched those models over several hundred km. Finally values were simulated at the same sampling points from the residual maximum likelihood models, reproducing spiky' experimental variograms such as those computed from the data. The simulation showed that clustered design of sampling causes spiky artefacts. We conclude that where data are clustered experimental variograms should be computed with declustered weighting or variogram models be fitted by residual maximum likelihood.