Mathematical Geology

, Volume 30, Issue 7, pp 789-799

First online:

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

  • Noel CressieAffiliated withDepartment of Statistics, Iowa State University
  • , Christopher K. WikleAffiliated withNational Center for Atmospheric Research

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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 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 uv. This restricts the types of models that might be fitted to measures of cross-spatial dependence.

cokriging equivariance pseudo cross-variogram