Multivariate Correlation in the Framework of Support and Spatial Scales of Variability

This paper extends the concept of dispersion variance to the multivariate case where the change of support affects dispersion covariances and the matrix of correlation between attributes. This leads to a concept of correlation between attributes as a function of sample supports and size of the physical domain. Decomposition of dispersion covariances into the spatial scales of variability provides a tool for computing the contribution to variability from different spatial components. Coregionalized dispersion covariances and elementary dispersion variances are defined for each multivariate spatial scale of variability. This allows the computation of dispersion covariances and correlation between attributes without integrating the cross-variograms. A correlation matrix, for a second-order stationary field with point support and infinite domain, converges toward constant correlation coefficients. The regionalized correlation coefficients for each spatial scale of variability, and the cases where the intrinsic correlation hypothesis holds are found independent of support and size of domain. This approach opens possibilities for multivariate geostatistics with data taken at different support. Two numerical examples from soil textural data demonstrate the change of correlation matrix with the size of the domain. In general, correlation between attributes is extended from the classic Pearson correlation coefficient based on independent samples to a most general approach for dependent samples taken with different support in a limited domain.