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This chapter discusses, from a theoretical point of view, how the geostatistical approach relates to other commonly-used models for inversion or data assimilation in the petroleum industry. The formal relationship between point Kriging and splines or radial basis functions is first presented. The generalizations of Kriging to the estimation of average values or values affected by measurement errors are also addressed. Two algorithms are often used for conditional simulation: the “rough plus smooth” approach consists of adding a smooth correction to a non-conditional simulation, whilst sequential Gaussian simulation allows the point-by-point construction of the realizations. As with Kriging, conditional simulation can be applied to average values or to data affected by measurement errors. Geostatistical inversion generates high-resolution realizations of vertical impedance traces constrained by seismic amplitudes. If the relationship between impedance and amplitude data is linearized, geostatistical inversion is a particular case of Bayesian inversion. Because of the non-linearity of production data vis-à-vis the variables of the earth model, their assimilation is harder than that of seismic data. Ensemble Kalman filtering, if considered from a geostatistical viewpoint, consists of using a large number—or ensemble—of realizations to calculate empirical covariances between the dynamic data and the parameters of the geostatistical model. These covariances are then used in the equations for interpolating the mismatch between simulated and new production data using a coKriging-like formalism. Interestingly, most of these techniques can be expressed using the same generic equation by which an initial model not honouring some newly arrived data is made conditional to these data by adding a (co-)Kriged interpolation of the data mismatches to the initial model. In spite of their similar equations, Bayesian inversion, geostatistics and ensemble Kalman filtering have a different approach to the inference of the covariance models used by these equations.

We compare and contrast several statistical methods for predicting the occurrence of mineral deposits on a regional scale. Methods include logistic regression, Poisson point process modelling, maximum entropy, monotone regression, nonparametric curve estimation, recursive partitioning, and ROC (Receiver Operating Characteristic) curves. We discuss the use and interpretation of these methods, the relationships between them, their strengths and weaknesses from a statistical standpoint, and fallacies about them. Potential improvements and extensions include models with a flexible functional form; techniques which take account of sampling effort, deposit endowment and spatial association between deposits; conditional simulation and prediction; and diagnostics for validating the analysis.

While tests for pairwise conditional independence of random variables have been devised, testing joint conditional independence of several random variables seems to be a challenge in general. Restriction to categorical random variables implies in particular that their common distribution may initially be thought of as contingency table, and then in terms of a log-linear model. Thus, Hammersley–Clifford theorem applies, and provides insight in the factorization of the log-linear model corresponding to assumptions of independence or conditional independence. Such assumptions simplify the full joint log-linear model, and in turn any conditional distribution. If the joint log-linear model corresponding to the assumption of joint conditional independence given the conditioning variable is not sufficiently large to explain some data according to a standard log-likelihood test, its null–hypothesis of joint conditional independence may be rejected with respect to some significance level. Enlarging the log-linear model by some product terms of variables and running the log-likelihood test on different models may provide insight which variables are lacking conditional independence. Since the joint distribution determines any conditional distribution, the series of tests eventually provides insight which variables and product terms a proper logistic regression model should comprise.

Compositions describe parts of a whole and carry relative information. Compositional data appear in all fields of science, and their analysis requires paying attention to the appropriate sample space. The log-ratio approach proposes the simplex, endowed with the Aitchison geometry, as an appropriate representation of the sample space. The main characteristics of the Aitchison geometry are presented, which open the door to statistical analysis addressed to extract the relative, not absolute, information. As a consequence, compositions can be represented in Cartesian coordinates by using an isometric log-ratio transformation. Standard statistical techniques can be used with these coordinates.

In the absence of empirical data that allows resolution of the vexing problem of how to address probabilistic dependencies among and between elements of large sets of geologic random variables data we need methods that refocus and streamline expert geological judgment inputs along with analytical methods for modeling dependencies that go beyond pairwise correlation and its cousins. Some possibilities are reviewed.

Remote sensing has become an essential component of the geosciences (the study of Earth and its system components). Remote sensing measurements are almost always energies measured in selected parts of the electro-magnetic spectrum. That is, the geophysical variable of interest is only observed indirectly; a forward model relates the energies to the variable(s) of interest and other elements of the state. The first derivative of that forward model with respect to the state is known as the Jacobian. In this chapter, we review the importance of the Jacobian to inferring the state, and we use it to diagnose which state elements may be difficult to estimate. We develop the Statistical Significance Filter and flag those state elements that consistently fail to get through the filter.

Geostatistical simulation of mineral deposits is becoming commonplace. The methodology and software are well established and professionals have access to the training and checking steps required for reliable application. Managing multiple realizations, however, remains daunting and unclear for many: (1) the non-uniqueness of multiple realizations is disturbing; (2) many calculations including mine planning algorithms are aimed at a single block model; and (3) there are concerns of excessive computational requirements. The correct approach to managing multiple realizations is reviewed: consider all realizations all the time and base decisions on the appropriate expected value. The principles of simulation and decision making are reviewed for resource management.

Paleoecologists and paleogeographers still make use of binary coefficients in multivariate analysis decades after being introduced to the geosciences. Among the main groups, similarity, matching and association, selecting a particular coefficient remains a confusing and sometimes empirical process. Coefficients within groups tend to correlate highly when applied to datasets. With increasing interest in a probabilistic approach to grouping taxa or faunal lists, the Raup-Crick measure of association is closely related in purpose and empirically to coefficients of association and works well in cluster analysis and ordination. A reasonable strategy is to compare dendrograms and ordinations calculated with several coefficients, care being taken to select coefficients with different performance characteristics. Above all, the practitioner should understand the purpose of each coefficient.

The mathematical method called Plurigaussian Simulations was invented in France in the 1990s for simulating the internal architecture of oil reservoirs. It rapidly proved useful in other domains in the earth sciences: mining, hydrology and history matching. In this chapter we use complex dynamic networks first developed in statistical mechanics to track the diffusion of the method within academia, using citation data from Google Scholar. Since governments and funding agencies want to know whether ideas developed in research projects have a positive effect on the economy, we also studied how plurigaussian simulations diffused from academia to industry. The literature on innovation usually focusses on patents but as there were few on plurigaussian simulations, we needed criteria for deciding whether an innovation had been adopted by industry. Three criteria were identified: The second criterion revealed how important master’s level courses are in training geoscientists in the latest techniques. Their role in transferring knowledge to industry is undervalued in current procedures for evaluating university departments.

In the first part of the chapter, the status of the discipline of mathematical geosciences (MG) is reviewed and a new definition of MG as an interdisciplinary field of science is suggested. Similar to other disciplines such as geochemistry and geophysics, mathematical geosciences or geomathematics is the science of studying mathematical properties and processes of the Earth (and other planets) with prediction of its resources and changing environments. In the second part of the chapter, original research results are presented. The new concepts of fractal density and local singularity are introduced. In the context of fractal density and singularity a new power-law model is proposed to associate differential stress with depth increments at the phase transition zone in the Earth’s lithosphere. A case study is utilized to demonstrate the application of local singularity analysis for modeling the clustering frequency—depth distribution of earthquakes from the Pacific subduction zones. Datasets of earthquakes with magnitudes of at least 3 were selected from the Ring of Fire, subduction zones of Pacific plates. The results show that datasets from the Pacific subduction zones except from northeastern zones depict a profound frequency—depth cluster around the Moho. Further it is demonstrated that the clusters of frequency—depth distributions of earthquakes in the colder and older southwestern boundaries of the Pacific plates generally depict stronger singularity than those obtained from the earthquakes in their hotter and younger eastern boundaries.