Nested sampling for estimating spatial semivariograms compared to other designs

In spatial studies, use is commonly made of nested sampling plans. By applying such plans, one takes observations according to a hierarchical scheme, with decreasing distances between observations. As observed by Miesch (1975), the cumulative sum of variance components provided by the nested sampling plan may be used in some situations to obtain semivariogram values. In this article, proofs are given for both balanced and unbalanced designs. Different estimation procedures for obtaining semivariogram values are compared with each other. The paper is illustrated with two numerical examples, one on actual soil pH data and one on simulated random fields. Mean squared pair differences are shown to be inferior to expected mean squares and restricted maximum likelihood for variance component estimation and several other spatial sampling plans may be superior to the nested sampling plan for estimating the spatial semivariogram.