The Variance-Based Cross-Variogram: You Can Add Apples and Oranges

The variance-based cross-variogram between two spatial processes, Z₁ (·) and Z₂ (·), is var (Z₁ ( u ) – Z₂ ( v )), expressed generally as a bivariate function of spatial locations uandv. It characterizes the cross-spatial dependence between Z₁ (·) and Z₂ (·) and can be used to obtain optimal multivariable predictors (cokriging). It has also been called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z₁ ( u ) – Z₁ ( v ), Z₂ ( u ) – Z₂ ( v )). One concern with the variance-based cross-variogram has been that Z₁ (·) and Z₂ (·) might be measured in 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 Z₁ (·) and Z₂ (·); recommendations are given for an appropriate choice of units. We review the differences between the variance-based cross-variogram and the covariance-based cross-variogram and conclude that the former is more appropriate for cokriging. In practice, one often assumes that variograms and cross-variograms are functions of uandv only through the difference u – v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.