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

DOI: 10.1023/A:1021770324434

- Cite this article as:
- Cressie, N. & Wikle, C.K. Mathematical Geology (1998) 30: 789. doi:10.1023/A:1021770324434

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## Abstract

*The variance-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 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 (*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 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* Z_{1}*(·) and* Z_{2}*(·); 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.*