Simulation models for calculating moisture deficits for areas of land require interpolation procedures to arrive from point observations to area-covering statements. We use both (1) calculate first, interpolate later (CI) procedures, which interpolate calculated model results for test locations, and (2) procedures which interpolate basic soil data toward test locations, followed by model calculations, interpolate first, calculate later (IC) procedures. In this study several CI and IC procedures which simulate moisture deficits are compared by means of a test set of 100 observations, yielding the mean standard error (MSE). CI procedures consistently produced lower MSE values than IC procedures. Parameters of the pseudocovariance function (PCF), which models the spatial structure of bivariate increments in universal cokriging, were estimated by means of the restricted maximum likelihood procedure. Compared to universal kriging, universal cokriging yielded comparable MSE values, but a lower mean variance of the prediction error. Best results in this study were obtained by pointwise simulation model calculations, followed by statistical interpolations.

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A large part of the ideas developed here is due to long discussions and exchange of views of the first author with Dr. K. R. GABRIEL of the Hebrew University at Jerusalem who kindly gave permission to include that material in this paper and to whose present and future publicaticns [2, 3, 41 reference has been made in gratitude. In particular the idea that Wishart matrices are involved here is his.The content of section 1 is essentially present in HOUSEHOLDER and YOUNG [6]. The modification in section 2 is incorrectly or incompletely discussed upon by GOLLOB [5] and MANDEL [8]. About the statistical problem of testing for multiplicative effects in a two-way layout in the sense of this paper non-trivial contributions are presented in WILLIAMS [9] and MANDEL [7] in addition to the Gollob and Mandel papers mentioned before. Approximation of a matrix by a matrix productThe first problem considered is the least squares approximation of the m by n matrix X of rank at least k by AB' where A is m by k , and B is n by k, while k < m and k < n. In other words, minimize with respect to A and B where IICI12 = tr(C'C).The solution proceeds in two stages: i) Minimize!, given A, with respect to B. Let thej-th column ofX and B' be denoted xi and b,, respectively ( j = I , . . .,n). Then f = z;= l ( x j -Ab,)2 is minimized by choosing each b, such that Ab, equals the orthogonal projection, P < * > x j , of xi on the space < A > spanned by the columns of A. Hence P,,,X is the unique optimal value for AB'. From the fact that ( x , -A~, )~ will now be equal to xi' -(Ab,)' it follows that f will be equal to llXllz -IIAB'II as well.

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