Mathematical Geology, Vol. 25, No. 3, 1993 Spatial Orthogonality of the Principal Components Computed from Coregionalized Variables 1 P. Goovaerts 2 Within the frame of the linear model of coregionalization, this paper sets up equations relating the variogram matrix of the principal components extracted from the variance-covariance matrix to the diagonal variogram matrices of the regionalized factors. The spatial orthogonality of the principal components is investigated in three situations: the intrinsic correlation, two basic structures with independent nugget components, three basic structures with independent nugget components and uncorrelated subsets of variables. Two examples point out that the correlation between the principal components may be nonnegligible at short distances, especially if the correlation structure changes according to the spatial scale considered. For one of the two case studies, an orthogonal varimax rotation of the first principal components is found to greatly reduce the spatial correlation between some of them. KEY WORDS: variogram matrix, linear model of coregionalization, principal component analy- sis, intrinsic correlation, orthogonal rotation. INTRODUCTION Besides its use in mining or petroleum field, geostatistics is now widely applied in areas such as geochemical exploration or soil science where the information is mainly multivariate. When dealing with multivariate datasets, difficulties es- sentially arise from the fitting of a valid model of coregionalization and the solving of large cokriging systems. Similar problems are encountered in non- parametric geostatistics where a vector of indicator variables is available at each sampled point. To reduce the modeling and computational effort, several authors (Davis and Greenes, 1983; Suro-Prrez and Journel, 1991) have proposed rather similar methods that involve: computing the principal components of the co- variance function matrix for h = 0 or small lags h, kriging them and estimating the initial variables as linear combinations of the kriged principal components. ~Received 18 February 1992; accepted 13 October 1992. 2Unit6 de Biomrtrie, Universit6 Catholique de Louvain, Place Croix du Sud 2 Bte i6, B-1348 Louvain-la-Neuve, Belgique. Present address: Department of Applied Earth Science, Stanford University, Stanford, California 94305. 281 0882-8121/93/0400"0281,¢[ff}7.00/1 © 1993 International Association for Mathematical Geology