Spatial Covariance Estimation by Scaled-metric Scaling and Biorthogonal Grids

A new, nonparametric approach is proposed for global estimation of the spatial covariance structure of a random function X(z) observed repeatedly at a finite number of sampling stations zi' i=l, 2, ... ,n, in the plane. The true covariance structure is assumed to be neither isotropic nor stationary. Using a new variant of multidimensional scaling, termed scaled-metric scaling, one may compute a two-dimensional representation for the sampling stations in which interpoint distances approximate sample estimates of the spatial dispersions Var(X(zi)-X(Zj))' (These variances are.usually.fItteclhyparametric. models fer-the •. variogram.) ..The .Il1ethods ofuninimumLaplacian interpolation andbiorthogonal grids, introduced by Bookstein in the field of morphometries, are applied to compute and depict a smooth mapping of the geographic representation of the sampling stations onto the scaled-metric scaling representation. This yields a graphical representation of the global spatial covariance structure and a method for estimating Var(X(a)--X(b)) for-any two unsampledlocatioIls a artdb irithe geographic pla.ne. Preliminary applications to solar radiation data and rainfall acidity data are discussed.