Multivariate Geostatistics

Geostatistics is a rapidly evolving branch of applied mathematics which originated in the mining industry in the early fifties to help improve ore reserve calculation. The first steps were taken in South Africa, with the work of the mining engineer DG Krige and the statistician HS Sichel (see reference number [136] in the bibliography).

In this chapter the elementary concepts of mean, variance and covariance are presented. The expectation operator is introduced, which serves to compute these quantities in the framework of probabilistic models.

Linear regression is presented in the case of two variables and then extended to the multivariate case. Simple kriging is a transposition of multiple regression in a spatial context. The algebraic problems generated by missing values in the multivariate case serve as a motivation for introducing a covariance function in the spatial case.

The mean value of samples from a geographical space can be estimated, either using the arithmetic mean or by constructing a weighted average integrating the knowledge of the spatial correlation of the samples. The two approaches are developped and it turns out that the solution of kriging the mean reduces to the arithmetic mean when there is no spatial correlation between data locations.

The data provide information about regionalized variables, which are simply functions z(x) whose behavior we would like to characterize in a given region of a spatial or time continuum. In applications the regionalized variables are usually not identical with simple deterministic functions and it is therefore of advantage to place them into a probabilistic framework.

Pairs of sample values are evaluated by computing the squared difference between the values. The resulting dissimilarities are plotted against the separation of sample pairs in geographical space and form the variogram cloud. The cloud is sliced into classes according to separation in space and the average dissimilarities in each class form the sequence of values of the experimental variogram.

The experimental variogram is a convenient tool for the analysis of spatial data as it is based on a simple measure of dissimilarity. Its theoretical counterpart reveals that a broad class of phenomena are adequately described by it, including phenomena of unbounded variation. When the variation is bounded, the variogram is equivalent to a covariance function.

We present a few models of covariance functions. They are defined for isotropic (i.e. rotation invariant) random functions. On the graphical representations the covariance functions are plotted as variograms using the relation γ(h) = C(0) − C(h).

Experimental calculations can reveal a very different behavior of the experimental variogram in different directions. This is called an anisotropic behavior. As variogram models are defined for the isotropic case, we need to examine transformations of the coordinates which allow to obtain anisotropic random functions from the isotropic models. In practice anisotropies are detected by inspecting experimental variograms in different directions and are included into the model by tuning predefined anisotropy parameters.

Measurements can represent averages over volumes, surfaces or intervals, called their support. The computation of variances depends intimately on the supports that are involved as well as on a theoretical variogram associated to a point-wise support. This is illustrated with an application from industrial hygienics. Furthermore, three simple sampling designs are examined from a geostatistical perspective.

We include an account about the geometrical meaning of two measures of dispersion: the variance and the selectivity. This is based on Matheron [176][177] and Lantuéjoul [143][144]. The context is mining economics, but a parallel to analog concerns in the evaluation of time series from environmental monitoring is drawn at the end of the chapter.

Ordinary kriging is the most widely used kriging method. It serves to estimate a value at a point of a region for which a variogram is known, using data in the neighborhood of the estimation location. Ordinary kriging can also be used to estimate a block value. With local second-order stationarity, ordinary kriging implicitly evaluates the mean in a moving neighborhood. To see this, first a kriging estimate of the local mean is set up, then a simple kriging estimator using this kriged mean is examined.

The behavior of kriging weights in 2D space is discussed with respect to the geometry of the sample/estimation locations, with isotropy or anisotropy and in view of the choice of the type of covariance function. The examples are borrowed from the thesis of Rivoirard [213] which contains many more.

Kriging can be used as an interpolation method to estimate values on a regular grid using irregularly spaced data. Spatial data may be treated locally by defining a neighborhood of samples around each location of interest in the domain.

A regionalized phenomenon can be thought as being the sum of several independent subphenomena acting at different characteristic scales. A linear model is set up which splits the random function representing the phenomenon into several uncorrelated random functions, each with a different variogram or covariance function. In subsequent chapters we shall see that it is possible to estimate these spatial components by kriging, or to filter them. The linear model of regionalization brings additional insight about kriging when the purpose is to create a map using data irregularly scattered in space.

The components of regionalization models can be extracted by kriging. The extraction of a component of the spatial variation is a complementary operation to the filtering out (rejection) of the other components. This is illustrated by an application on geochemical data.

How smooth are estimated values from kriging with irregularly spaced data in a moving neighborhood? By looking at a few typical configurations of data around nodes of the estimation grid and by building on our knowledge of how spatial components are kriged, we can understand the way the estimated values are designed in ordinary kriging.

Principal component analysis is the most widely used method of multivariate data analysis owing to the simplicity of its algebra and to its straightforward interpretation.

In the previous chapter we have seen how principal component analysis was used to determine linear orthogonal factors underlying a set of multivariate measurements. Now we turn to the more ambitious problem to compare two groups of variables by looking for pairs of orthogonal factors, which successively assess the strongest possible links between the two groups.

Canonical analysis investigates, so to speak, the correspondence between the factors of two groups of quantitative variables. The same approach applied to two qualitative variables, each of which represents a group of mutually exclusive categories, is known under the evocative name of “correspondence analysis”.

The cross covariance between two random functions can be computed not only at locations x but also for pairs of locations separated by a vector h. On the basis of an assumption of joint second-order stationarity a cross covariance function between two random functions is defined which only depends on the separation vector h.

It is actually difficult to characterize directly a covariance function matrix. This becomes easy in the spectral domain on the basis of Cramer’s generalization of the Bochner theorem, which is presented in this chapter. We consider complex covariance functions.

Is the multivariate correlation structure of a set of variables independent of the spatial correlation? When the answer is positive, the multivariate correlation is said to be intrinsic. It is an interesting question in applications to know whether or not a set of variables can be considered intrinsically correlated, because the answer may imply considerable simplifications for subsequent handling of this data.

The cokriging procedure is a natural extension of kriging when a multivariate variogram or covariance model and multivariate data are available. A variable of interest is cokriged at a specific location from data about itself and about auxiliary variables in the neighborhood. The data set may not cover all variables at all sample locations. Depending on how the measurements of the different variables are scattered in space we distinguish between isotopic and heterotopic data sets. After defining these situations we examine cokriging in the heterotopic case.

When a set of variables is intrinsically correlated, cokriging is equivalent to kriging for each variable, if all variables have been measured at all sample locations. For a particular variable in a given data set in the isotopic case, cokriging can be equivalent to kriging, even if the variable set is not intrinsically correlated. This is examined in detail in this chapter.

The multivariate regionalization of a set of random functions can be represented with a spatial multivariate linear model. The associated multivariate nested variogram model is easily fitted to the multivariate data. Several coregionalization matrices describing the multivariate correlation structure at different scales of a phenomenon result from the variogram fit. The relation between the coregionalization matrices and the classical variance-covariance matrix is examined.

The geostatistical analysis of multivariate spatial data can be subdivided into two steps the analysis of the coregionalization of a set of variables leading to the definition of a linear model of coregionalization;the cokriging of specific factors at characteristic scales.

The question of how to model and krige a complex variable has been analyzed by Lajaunie & Béjaoui [140] and Grzebyk [111]. The covariance structure can be approached in two ways: either by modeling the real and imaginary parts of the complex covariance or by modeling the coregionalization of the real and complex parts of the random function. This opens several possibilities for (co-)kriging the complex random function.

The linear model of coregionalization of real variables implies even cross covariance functions and it was thus formulated with variograms in the framework of intrinsic stationarity. The use of cross variograms however excludes deferred correlations (due to delay effects or phase shifts). A more general model was set up by Grzebyk [111] [112] which allows for non even cross covariance functions.

We start the study of non-stationary methods with a multivariate method that is applicable to auxiliary variables that are densily sampled over the whole domain and linearly related to the principal variable. Such auxiliary variables can be incorporated into a kriging system as external drift functions.

The external drift approach has left questions about the inference of the variogram unanswered. The theory of universal kriging, i.e. kriging with several universality conditions, will help deepen our understanding. The universal kriging model splits the random function into a linear combination of deterministic functions, known at any point of the region, and a random component, the residual random function. It turns out that this model is in general difficult to implement, because the underlying variogram of the random component can only be inferred in exceptional situations.

The characterization of the drift by a linear combination of translation-invariant deterministic functions opens the gate to the powerful theory of the intrinsic random functions of order k. A more general tool of structural analysis than the variogram can be defined in this framework: the generalized covariance function. The constraints on the weights in the corresponding intrinsic kriging appear as conditions for the positive-definiteness of the generalized covariance. Parallels between this formulation and the spline approach to interpolation can be drawn.