Model-Based Sampling Design for Multivariate Geostatistics

The quality of inferences made from geostatistical data is affected significantly by the spatial locations, or design, of the sites that are sampled. A large body of published work exists on sampling design for univariate geostatistics, but not for multivariate geostatistics. This article considers multivariate spatial sampling design based on criteria targeted at classical co-kriging (prediction with known covariance parameters), estimation of covariance (including cross-covariance) parameters, and empirical co-kriging (prediction with estimated covariance parameters). Through a combination of analytical results and examples, we investigate the characteristics of optimal designs with respect to each criterion, addressing in particular the design's degree of collocation. We also consider the robustness of the optimal design to the strength of spatial correlation and cross-correlation; the effects of smoothness and/or separability of the sampled process on the optimal design; the relationship between optimal designs for the multivariate problems considered here and univariate problems considered previously; and the efficiency of optimal collocated designs. One key finding is that optimal collocated designs are highly efficient in many cases. Supplementary materials are available online.