Epilogue

The book we have edited was written many years ago, but it still remains a topical book. Many borrowings of the book can be found in the current literature in geostatistics. There is another reason why this book remains pertinent today. We can indeed consider some branches which originate from the trunk of linear geostatistics and which are today full of life. We will demonstrate it by means of four examples....

Transitive Methods

Chapter 2, devoted to deterministic transitive methods, introduces the notion of a transition phenomenon by using the example of the indicator function of a set, which is described by means of its geometrical covariogram. This notion is then extended to any numerical function f of the space. The next step consists in studying in detail the operation of ‘grading’ in the isotropic case, where the initial pointwise regionalised variable is replaced by its regularised average along a line, as would occur during drilling. Simple rules for passing from the pointwise covariogram to the graded one are given. When the regionalised variable is recognised by a regular grid whose origin has a random setting, the integral of function f becomes a random variable. Its estimation variance is expressed in terms of the covariogram, and the rules about regularisation allow the numerical computation of this variance. The case of set indicators is treated in detail.

Universal kriging

This chapter discusses universal kriging, the kriging of a random function Z(x) which is not intrinsic and exhibits an expectation E [Z(x)] = m(x) variable over the space. For the function m(x), called the drift of Z(x), a model has to be chosen, usually polynomial of degree 1, 2, or 3. Then, the best polynomial coefficients are provided by a Lagrange approach similar to that used for simple kriging. However, drift estimation is not required for the true kriging, that is, for the best estimate of Z itself from a given sampling. The associated system of equations is still linear and can be formally calculated in dimension 1 for the usual drifts and variograms. Universal kriging is compared with least squares methods, and a few variants are studied, including universal kriging in the case of an implicit drift, and universal kriging of a vector random function, or cokriging.

Introduction

This brief introductory chapter begins by providing some notation. This is followed by a reminder on the convolution (moving average) operation, which is presented here as a regularisation procedure. The notion of a regionalised variable is then introduced, with both its structured and its random aspects. The definition of a regionalised variable requires the data of its support and of its field. The chapter ends by comparing the deterministic approach, which uses transitive methods, and the stochastic approach, which uses intrinsic theory. Note that both approaches lead to equivalent results. Indeed, they mainly differ by their domains of application, the first one being more adapted to global questions, and the second to local ones. However, both techniques involve some statistical inference, and here the minimal requirement turns out to be the quasi-intrinsic assumption.

Kriging

Chapter 4 discusses kriging; one calls kriging, or simple kriging, of the random function Y in a panel P the best linear estimator Y K of Y by N samples Y α‎. This optimum is given by starting from the extension variance of Y − Y K, where Y K = λ‎ α‎ Y α‎, and by calculating its minimum with respect to the λ‎ α‎ upon the condition that the sum of these coefficients equals 1 (the non-bias condition). This results in the kriging equations, that is, a Lagrange linear system of equations in λ‎ α‎. When the panel P reduces to one point x moving into the space, then the kriging yields the best estimation map of the random function Y. The approach extends to the case when the samples form a continuous set. In dimension 1, the kriging system can be formally calculated, and the corresponding expressions are given for linear variograms and de Wijsian variograms.

Theory of Intrinsic Random Functions

Chapter 3 discusses intrinsic random functions Y(x) of the space variable x, i.e. functions whose mean and variance of the increments Y(x + h) − Y(x) depend on h only. Half of this variance defines the variogram γ‎(h). The behaviour of the variogram near the origin, such as continuity, the nugget effect, etc., expresses the regularity of these functions. Regularisations of them, by grading and convolutions, produce new variograms, via the same rules as for transitive methods. When Z(v) and Z(v′) are the averages of these functions in v and v′, respectively, then the variance of Z(v) − Z(v′) by attributing to v′ the grade in v is called extension variance. Its formal expression is given and calculated for various patterns of sampling v, in dimensions 1, 2, or 3, via the de Wijsian scheme and the spherical scheme, and for various models of variograms, such as the semi-variogram.