##
**Geostatistics. Modeling spatial uncertainty.
2nd ed.**
*(English)*
Zbl 1256.86007

Wiley Series in Probability and Statistics. Hoboken, NJ: John Wiley & Sons (ISBN 978-0-470-18315-1/hbk; 978-1-118-13617-1/ebook). xv, 699 p. (2012).

Geostatistics is a well established subject, that has shown its potential over many years. Although originally developed by G. Matheron as a methodology for ore reserve estimation, it is nowadays applied in many fields, including environmental studies and medicine. It requires a good background in statistics, mathematics, and a deep understanding of estimation and uncertainty. The methods of geostatistics are numerous in the many variants published in journals and reports, and have seen many advances in the last decade. The thoroughly revised and updated second edition of the book under review brings the state of the art in geostatistics, integrating new material and removing chapters which were, as described in the preface, too detailed for the casual reader and too incomplete for the specialist, like e.g. Chapter 8 of the first edition. The preface presents a pretty detailed overview of the main changes.

The second edition comprises 7 chapters, in addition to the Introduction and a very useful list of abbreviations. Chapter 1, devoted to preliminaries, addresses random functions, discusses the objectivity of probabilistic statements, and introduces transitive theory.

Chapter 2 deals with structural analysis, in particular with the concepts of variogram cloud, sample variogram, mathematical properties of the variogram, regularisation, nugget effect, and variogram models, the problem of fitting models, and how to deal with the presence of a drift. Compared to the first edition, complements on practical questions have been included in this chapter.

Kriging is dealt with in Chapter 3. It includes kriging with known and unknown mean, estimation of spatial average, selection of a neighbourhood, and dealing with errors and outliers. The external drift model as a variant of the universal kriging model with polinomial drift has been included in this chapter. The last section of this chapter, 3.9 Kriging under inequality constraints, will surely see new developments in the near future. Some hints can be found in Section 5.6, in particular in Subsection 5.7.4 Compositional data, based on a specific algebraic-geometric structure of the sample space. This structure allows working in coordinates and that way get rid of some constraints.

The intrinsic model of order \(k\) is analysed, with only few changes, in Chapter 4, while Chapter 5 on multivariate methods, has been extended to include collocated cokriging and space-time models.

The chapter on nonlinear methods (Chapter 6) has been updated and expanded, in particular with respect to transformations to normality, and the presentation of the change of support.

Chapter 7, the last one, is a thorough update of conditional simulation methods, incorporating the numerous advances over the last decade. It includes simulation of the fractional Brownian motion, a Gibbs propagation algorithm, pluri-Gaussian simulations, stochastic process based simulation, multi-point simulation, and gradual deformation.

As stated by the authors in the preface to the first edition, their ambition is to provide the reader with a unified view of geostatistics, with an emphasis on Methodology. The authors have succeeded in their goals. The assertion, stated in a review of the first edition by [J.-P. Chilès and P. Delfiner, Geostatistics: Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics. New York, NY: Wiley. (1999; Zbl 0922.62098)], [Book review: M. E. Hohn, Math. Geol. 35, No. 3, 353–355 (2003)], that it is a readable, comprehensive volume that belongs on the desk, close at hand, of any serious researcher or practitioner, remains true. It is adequate as a guide for teaching basic courses, for PhD students and post-docs who need frequently to consult issues in the subject, for advanced and senior researchers looking for fundamental methodological issues and references to previous work, for practitioners needing advice on practical issues. In summary, a worthwhile investment.

The second edition comprises 7 chapters, in addition to the Introduction and a very useful list of abbreviations. Chapter 1, devoted to preliminaries, addresses random functions, discusses the objectivity of probabilistic statements, and introduces transitive theory.

Chapter 2 deals with structural analysis, in particular with the concepts of variogram cloud, sample variogram, mathematical properties of the variogram, regularisation, nugget effect, and variogram models, the problem of fitting models, and how to deal with the presence of a drift. Compared to the first edition, complements on practical questions have been included in this chapter.

Kriging is dealt with in Chapter 3. It includes kriging with known and unknown mean, estimation of spatial average, selection of a neighbourhood, and dealing with errors and outliers. The external drift model as a variant of the universal kriging model with polinomial drift has been included in this chapter. The last section of this chapter, 3.9 Kriging under inequality constraints, will surely see new developments in the near future. Some hints can be found in Section 5.6, in particular in Subsection 5.7.4 Compositional data, based on a specific algebraic-geometric structure of the sample space. This structure allows working in coordinates and that way get rid of some constraints.

The intrinsic model of order \(k\) is analysed, with only few changes, in Chapter 4, while Chapter 5 on multivariate methods, has been extended to include collocated cokriging and space-time models.

The chapter on nonlinear methods (Chapter 6) has been updated and expanded, in particular with respect to transformations to normality, and the presentation of the change of support.

Chapter 7, the last one, is a thorough update of conditional simulation methods, incorporating the numerous advances over the last decade. It includes simulation of the fractional Brownian motion, a Gibbs propagation algorithm, pluri-Gaussian simulations, stochastic process based simulation, multi-point simulation, and gradual deformation.

As stated by the authors in the preface to the first edition, their ambition is to provide the reader with a unified view of geostatistics, with an emphasis on Methodology. The authors have succeeded in their goals. The assertion, stated in a review of the first edition by [J.-P. Chilès and P. Delfiner, Geostatistics: Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics. New York, NY: Wiley. (1999; Zbl 0922.62098)], [Book review: M. E. Hohn, Math. Geol. 35, No. 3, 353–355 (2003)], that it is a readable, comprehensive volume that belongs on the desk, close at hand, of any serious researcher or practitioner, remains true. It is adequate as a guide for teaching basic courses, for PhD students and post-docs who need frequently to consult issues in the subject, for advanced and senior researchers looking for fundamental methodological issues and references to previous work, for practitioners needing advice on practical issues. In summary, a worthwhile investment.

Reviewer: Vera Pawlowsky-Glahn (Girona)

### MSC:

86A32 | Geostatistics |

86-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geophysics |

62P12 | Applications of statistics to environmental and related topics |

62Hxx | Multivariate analysis |

62M30 | Inference from spatial processes |