Robust Resampling Confidence Intervals for Empirical Variograms

The variogram function is an important measure of the spatial dependenciesof a geostatistical or other spatial dataset. It plays a central role in kriging, designingspatial studies, and in understanding the spatial properties of geological andenvironmental phenomena. It is therefore important to understand the variability attachedto estimates of the variogram. Existing methods for constructing confidenceintervals around the empirical variogram either rely on strong assumptions, such asnormality or known variogram function, or are based on resampling blocks and subjectto edge effect biases. This paper proposes two new procedures for addressingthese concerns: a quasi-block-bootstrap and a quasi-block-jackknife. The new methodsare based on transforming the data to decorrelate it based on a fitted variogrammodel, resampling blocks from the decorrelated data, and then recorrelating. Thecoverage properties of the new confidence intervals are compared by simulation to anumber of existing resampling-based intervals. The proposed quasi-block-jackknifeconfidence interval is found to have the best properties of all of the methods consideredacross a range of scenarios, including normally and lognormally distributed dataand misspecification of the variogram function used to decorrelate the data.