Bayesian variogram modeling

Date of Completion

January 1997






The variogram is a basic tool in geostatistics. It expresses the variability between pairs of observations as a function of their separation vector. When a spatial process is isotropic, customary approaches to variogram modeling create an empirical variogram to which one customarily fits a standard variogram model. We model standard variograms and using a flexible class of discrete mixtures of Bessel functions employing a Bayesian perspective introduce a utility based variogram model choice criterion that encourages parsimony. Scallop catches in the Atlantic Ocean during 1993 serve as an illustrative dataset throughout this work, while an earlier 1990 dataset from the same region supplies useful prior information.^ Exploratory data analysis (EDA) techniques such as directional empirical semivariograms and the rose diagram are widely used to detect departures from isotropy. We develop the empirical semivariogram contour (ESC) plot in R$\sp2$ and model geometric anisotropy which can be viewed as a linear transformed isotropic process. The Bayesian methodology allows simultaneously estimation of the linear transformation and the other semivariogram parameters. We demonstrate how useful prior information such as a priori knowledge of geometric anisotropy can be incorporated into the model.^ In R$\sp2,$ isotropy is depicted by circular variogram contours while geometric anisotropy has elliptical contours; however, both are special cases of range anisotropy. We introduce a simple yet flexible range anisotropic correlation structure that allows modeling of general variogram contours. Modeling implications for a nonconstant mean, a trend surface in the absence of non-geographical covariates, are discussed. We model the trend surface and covariance parameters simultaneously and introduce the detrended variogram. In analyzing the 1993 scallop data, we withhold ten sites to compare the accuracy and precision of a noiseless version of the predictive distribution for two range anisotropic models. Lastly, we predict locally and globally on the original scale of the data for a particular subregion of interest. ^