Book,
Interpolation of spatial data
Springer Series in Statistics Springer-Verlag, New York, (1999)Some theory for Kriging.
DOI: 10.1007/978-1-4612-1494-6
Abstract
MR: A usual practice in geostatistics "is to estimate the covariance structure (of the random field observed) using the same data that will be used for interpolation. The properties of interpolants based on an estimated covariance structure are not well understood.'' The author attacks this complicated and deep problem in a systematic manner and comes at least to partial solutions, which may be a valuable basis for future research.
Chapter 1 contains basic facts on linear prediction. Section 1.7 presents some of the author's messages: Üse the Matérn model. Calculate and plot likelihood functions for unknown parameters of models for covariance structures. Do not put too much faith in empirical semivariograms.'' Chapter 2 presents fundamentals of the theory of random fields, simulation methods and facts on autocovariance functions, including many formulae for spectra of covariance models. The author gives convincing arguments against the popular Gaussian autocovariance function and the spherical variogram. In contrast, he prefers the Matérn model, a three-parameter family of autocovariance functions.
Chapter 3 considers asymptotic properties of linear estimators, in particular the behaviour of linear prediction under some incorrect second-order structure. The term äsympotic'' is used in the sense of "fixed-domain asymptotic''. Interpolation and extrapolation are influenced in different ways by the covariance structure. This chapter also discusses measurement errors in detail.
Chapter 4 discusses relationships between Gaussian measures, such as equivalence and orthogonality, using ideas of I. A.Ibragimov and Y. A. Rozanov Gaussian random processes, Translated from the Russian by A. B. Aries, Springer, New York, 1978; MR0543837 (80f:60038). They are needed for the study of different covariance structures and their effect in the asymptotic behaviour of linear prediction. Chapter 5 gives results for the problem of prediction of integrals of random fields, in particular asymptotics for the simple average and improvements of this estimation method.
The final Chapter 6 "provides theorems, heuristic derivations, numerical calculations and a simulated example concerning the estimation of autocovariance functions and prediction of random fields based on these estimates''. The reviewer sees these results as valuable partial solutions but not yet as the final solution. The simulated example may disappoint some readers. It is based on one linear sample of 23 values only, a simulation that is the "first one'' the author ran.
It is unusual to have exercises for the reader in such a research monograph. Most of them are well chosen, but some of them (see page 198) may frustrate the reader.
The book contains many original and interesting ideas and should be of great value for all (theoretical) geostatisticians. En passant, some traditional concepts of geostatistics are called into question from a mathematical point of view, which may change the way of thinking of many geostatisticians. For the reviewer, the book is a landmark in the development of geostatistics.
