THE VARIANCE-BASED CROSS-VARIOGRAM - YOU CAN ADD APPLES AND ORANGES

The variancc-based cross-variogram between two spatial processes, Z(1) (.) and Z(2)(.), is var (Z(1)(u) - Z(2)(v)), expressed generally as a bivariate function of spatial locations u and v. It characterizes the cross-spatial dependence between Z(1)(.) and Z(2)(.) and can be used t o obtain optimal multivariable predictors (cokriging). It has also bee n called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z(1) (u) - Z(1)(v), Z(2)(u) - Z(2)(v)). One concern with the variance-based cross-variogram has been that Z(1)(.) and Z(2)(.) might be measured i n different units (''apples'' and ''oranges''). In this note, we show that the cokriging predictor based on variance-based cross-variograms can handle any units used for Z(1)(.) and Z(2)(.); recommendations are given for an appropriate choice of units. We review the differences b etween the variance-based cross-variogram and the covariance-based cro ss-variogram and conclude that the former is more appropriate for cokr iging. In practice, one often assumes that variograms and cross-variog rams are functions of u and v only through the difference u - v. This restricts the types of models that might be fitted to measures of cros s-spatial dependence.