The Variance-Based Cross-Variogram: You Can Add Apples and Oranges
- Cite this article as:
- Cressie, N. & Wikle, C.K. Mathematical Geology (1998) 30: 789. doi:10.1023/A:1021770324434
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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 locationsuandv. 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 ofuandvonly through the differenceu − v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.