The basic maximum likelihood method for estimating spatial covariance parameters has some limitations: it is based on the assumption that the experimental data follow a multi-dimensional Gaussian distribution, it yields biased estimates, it is impractical for large data sets and it usually assumes a polynomial trend. On the other hand, it has the important advantages of parametric estimation such as easy evaluation of parameter uncertainty, no information lost in binning (typical in the method of moments estimation) and there is the possibility of including additional information using a Bayesian framework. In this paper we provide extensions to overcome the disadvantages of the maximum likelihood method whilst maintaining the advantages. Most Earth Sciences experimental data follow a non-Gaussian distribution and thus overcoming this limitation will significantly increase the general applicability of the maximum likelihood method. The problem solved in the paper is that of obtaining covariance parameters of the raw experimental data from parameters estimated from the set of normal scores of the experimental data. This is the first extension of the method. Although restricted maximum likelihood yields unbiased estimates of spatial covariance parameters it does so at the cost of increasing the estimation variance. In a second extension of the method we present an alternative that gives unbiased estimates whilst keeping the estimation variance constant. The cost of doing so is increased computing time. The third extension overcomes the practical restriction of the maximum likelihood to small data sets of no more than around 1,000 data. This is done with an approximation to the complete maximum likelihood approach and which has been tried successfully with sets of up to 90,000 experimental data. The method has the additional advantage of automatically accommodating troublesome sampling configurations such as samples from clusters of sparse drill holes. Finally, the fourth extension expands the polynomial drift to other functional forms which expands the possibilities of dealing with the two most usual scales of interest: a regional variation with a superimposed local variation. The extensions are illustrated with several synthetic and real data sets. |
