Approximation of confidence limits on sample semivariograms from single realizations of spatially correlated random fields

Abstract

A fundamental requirement for geostatistical analyses of spatially correlated environmental data is the estimation of the sample semivariogram to characterize spatial correlation. Selecting an underlying theoretical semivariogram based on the sample semivariogram is an extremely important and difficult task that is subject to a great deal of uncertainty. Current standard practice does not involve consideration of the confidence associated with semivariogram estimates, largely because classical statistical theory does not provide the capability to construct confidence limits from single realizations of correlated data, and multiple realizations of environmental fields are not found in nature. The jackknife method is a nonparametric statistical technique for parameter estimation that may be used to estimate the semivariogram. When used in connection with standard confidence procedures, it allows for the calculation of closely approximate confidence limits on the semivariogram from single realizations of spatially correlated data. The accuracy and validity of this technique was verified using a Monte Carlo simulation approach which enabled confidence limits about the semivariogram estimate to be calculated from many synthetically generated realizations of a random field with a known correlation structure. The synthetically derived confidence limits were then compared to jackknife estimates from single realizations with favorable results. Finally, the methodology for applying the jackknife method to a real‐world problem and an example of the utility of semivariogram confidence limits were demonstrated by constructing confidence limits on seasonal sample variograms of nitrate‐nitrogen concentrations in shallow groundwater in an approximately 12‐mi2 (∼30 km2) region in northern Illinois. In this application, the confidence limits on sample semivariograms from different time periods were used to evaluate the significance of temporal change in spatial correlation. This capability is quite important as it can indicate when a spatially optimized monitoring network would need to be reevaluated and thus lead to more robust monitoring strategies.