crossvariogram

A crossvariogram is a geostatistical tool used to describe the spatial cross-dependence between two regionalized variables. It measures how the joint variability of two variables changes with separation distance, capturing the relationship between their spatial structures. Crossvariograms are especially important in cokriging, where information from one variable is used to improve the estimation of another.

Mathematically, for two variables Z1 and Z2 defined over a region and a lag vector h, the crossvariogram is defined (for a second-order stationary field with centered data) as

gamma12(h) = (1/2) E[(Z1(x) − Z1(x + h)) (Z2(x) − Z2(x + h))].

In practice, means are removed from the data, and the expectation is replaced by averages over pairs of observations whose locations differ by h. The crossvariogram can be estimated for a set of lag distances by summing the products of paired differences and normalizing by the number of pairs.

Relation to cross-covariance is given by gamma12(h) = C12(0) − (1/2)[C12(h) + C12(−h)], where C12(h) is the cross-covariance function. If the cross-covariance is symmetric (C12(h) = C12(−h)), this reduces to gamma12(h) = C12(0) − C12(h). Crossvariograms are modeled together with single-variable variograms, often within the linear model of coregionalization, to form a coherent, multi-variable spatial model. They must satisfy certain mathematical constraints to yield valid cokriging systems.

Applications of crossvariograms span environmental science, mining, and any field requiring joint spatial estimation of related variables. Limitations include data sparsity and the need for appropriate modeling to ensure positive definite predictions.