The Variance-Based Cross-Variogram: You Can Add Apples and Oranges1

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Abstract

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 u and v.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 u and v only through the difference u – v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.

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