On the Validity and Identi ability of SpatialDeformation Models for HeterogeneousSpatial Correlation

Many environmental processes are heterogeneous in space (spatially non-stationary), due to factors such as topography, local pollutant emissions, and meteorology. Much of the commonly used spatial statistical methodology depends on simplifying assumptions such as spatial isotropy. Violations of these assumptions can cause problems, including incorrect error assessment of spatial estimates. This paper demonstrates important properties of the spatial deformation model of Sampson and Guttorp (1992) and Guttorp and Sampson (1994) for heterogeneous anisotropic spatial correlation structure. The modeling approach utilizes a deformation of the geographic coordinate space into a new coordinate system (known as the D-space, or D-plane in two dimensions) where isotropic spatial correlation structure is modeled. We provide proofs of two fundamental properties of the model: validity and invariance of the modeled correlations to translating, scaling and rotating operations on a D-space representation. We also prove two identi ability results. We prove rst that two non-trivial variograms, and two corresponding a ne transformations of the geographic coordinate system yield the same modeled dispersions between all pairs of locations in a region if and only if the variogram models and a ne mappings are identical. Second we prove that two strictly increasing D-space variograms and corresponding bijective transformations of the geographic coordinate system yield the same modeled dispersions between all pairs of locations in a region if and only if the variogram models and deformation mappings are identical.