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**Interpolation of spatial data. Some theory for kriging.**
*(English)*
Zbl 0924.62100

Springer Series in Statistics. New York, NY: Springer. xvii, 247 p. (1999).

Classical linear predictors for stationary random fields usually presume that the covariance structure of the fields is known. In difference to this, kriging often relies on an estimate for the covariance structure using the same data that will be used for interpolation. A number of applications of kriging is readily provided by geostatistics and general statistical analysis of spatial processes. The book develops a mathematically rigorous theory of kriging when the covariance structure is partially known, usually in the form of a parametric family of covariance functions.

The author covers a number of mathematical issues that concern the kriging approach to Gaussian stationary random fields, including asymptotic properties of linear predictors, equivalence/orthogonality of Gaussian measures and prediction and estimation of integrals of random fields. Some of the recurrent themes include contrast between interpolation and extrapolation; importance of the local behaviour of the random field for kriging; fixed-domain asymptotic results that, instead of working with growing windows, presume that more and more observations are taken within a fixed window; aspects of the model important for spatial interpolation in relation to other aspects estimable from the data.

Throughout the book, the author advocates use of the MatĂ©rn model of the covariance function that is flexible enough proving random fields with any required degree of differentiability. A particular emphasis is put on best linear predictors at the cost of less emphasis on modelling the mean and using best linear unbiased predictors. The author develops inference techniques for differentiable random fields that are not based on the empirical semivariogram. A number of presented results are new.

Although the presentation is succinct and quite technical overall, the book is well-readable. Primarily designed as a research monograph, the book can be used for a (rather advanced) course for graduate students. The author has done a nice work equipping the main text with numerous exercises and some simulation results. The book can be viewed as a mathematical extension of the broad overview of kriging provided by N.A.C. Cressie [Statistics for spatial data. (1991; Zbl 0799.62002)]. Many future readers of this book will come this way looking for a mathematical theory for kriging. The book will also be of interest for probabilists and mathematical statisticians accustomed to the classical interpolation theory for random fields. They will find the text much easier to read with a number of open ends that call for future work in order to attack some of the unresolved problems that would lead to a complete mathematical theory of kriging with estimated covariance structure.

The author covers a number of mathematical issues that concern the kriging approach to Gaussian stationary random fields, including asymptotic properties of linear predictors, equivalence/orthogonality of Gaussian measures and prediction and estimation of integrals of random fields. Some of the recurrent themes include contrast between interpolation and extrapolation; importance of the local behaviour of the random field for kriging; fixed-domain asymptotic results that, instead of working with growing windows, presume that more and more observations are taken within a fixed window; aspects of the model important for spatial interpolation in relation to other aspects estimable from the data.

Throughout the book, the author advocates use of the MatĂ©rn model of the covariance function that is flexible enough proving random fields with any required degree of differentiability. A particular emphasis is put on best linear predictors at the cost of less emphasis on modelling the mean and using best linear unbiased predictors. The author develops inference techniques for differentiable random fields that are not based on the empirical semivariogram. A number of presented results are new.

Although the presentation is succinct and quite technical overall, the book is well-readable. Primarily designed as a research monograph, the book can be used for a (rather advanced) course for graduate students. The author has done a nice work equipping the main text with numerous exercises and some simulation results. The book can be viewed as a mathematical extension of the broad overview of kriging provided by N.A.C. Cressie [Statistics for spatial data. (1991; Zbl 0799.62002)]. Many future readers of this book will come this way looking for a mathematical theory for kriging. The book will also be of interest for probabilists and mathematical statisticians accustomed to the classical interpolation theory for random fields. They will find the text much easier to read with a number of open ends that call for future work in order to attack some of the unresolved problems that would lead to a complete mathematical theory of kriging with estimated covariance structure.

Reviewer: I.S.Molchanov (Glasgow)

### MSC:

62M40 | Random fields; image analysis |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

60G60 | Random fields |

86A32 | Geostatistics |