ON THE CONDITION NUMBER OF COVARIANCE MATRICES IN KRIGING, ESTIMATION, AND SIMULATION OF RANDOM-FIELDS

The numerical stability of linear systems arising in kriging, estimati on, and simulation of random fields, is studied analytically and numer ically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditi onal random field generation by the superposition method, which is bas ed on kriging, and the multivariate Gaussian method, which requires fa ctoring a covariance matrix. A large condition number corresponds to a n ill-conditioned, numerically unstable system. In the case of station ary covariance matrices and uniform grids, as occurs in kriging of uni formly sampled data, the degree of ill-conditioning generally increase s indefinitely with sampling density and, to a limit, with domain size . The precise behavior is, however, highly sensitive to the underlying covariance model Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussi an. This list reflects an approximate ranking of the models, from ''be st'' to ''worst'' conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such represent ative analyses, conducted in this work, include the spherical and peri odic hole-effect (hole-sinusoidal) covariance models. The effect of sm all-scale variability (nugget) is addressed and extensions to irregula r sampling schemes and higher dimensional spaces are discussed.