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[EN] In the mid-1980s, still in his young 40s, Andre Journel was already recognized as one of the giants of geostatistics. Many of the contributions from his new research program at Stanford University had centered around the indicator methods that he developed: indicator kriging and multiple indicator kriging. But when his second crop of graduate students arrived at Stanford, indicator methods still lacked an approach to conditional simulation that was not tainted by what Andre called the 'Gaussian disease'; early indicator simulations went through the tortuous path of converting all indicators to Gaussian variables, running a turning bands simulation, and truncating the resulting multi-Gaussian realizations. When he conceived of sequential indicator simulation (SIS), even Andre likely did not recognize the generality of an approach to simulation that tackled the simulation task one step at a time. The early enthusiasm for SIS was its ability, in its multiple-indicator form, to cure the Gaussian disease and to build realizations in which spatial continuity did not deteriorate in the extreme values. Much of Stanford's work in the 1980s focused on petroleum geostatistics, where extreme values (the high-permeability fracture zones and the low-permeability shale barriers) have much stronger anisotropy, and much longer ranges of correlation in the maximum continuity direction, than mid-range values. With multi-Gaussian simulations necessarily imparting weaker continuity to the extremes, SIS was an important breakthrough. The generality of the sequential approach was soon recognized, first through its analogy with multi-variate unconditional simulation achieved using the lower triangular matrix of an LU decomposition of the covariance matrix as the multiplier of random normal deviates. Modifying LU simulation so that it became conditional gave rise to sequential Gaussian simulation (SGS), an algorithm that shared much in common with SIS. With nagging implementation details like the sequential path and the search neighborhood being common to both methods, improvements in either SIS or SGS often became improvements to the other. Almost half of the contributors to this Special Issue became students of Andre in the classes of 1984-1988, and several are the pioneers of SIS and SGS. Others who studied later with Andre explored and developed the first multipoint statistics simulation procedures, which are based on the same concept that underlies sequential simulation. Among his many significant intellectual accomplishments, one of the cornerstones of Andre Journel's legacy was sequential simulation, built one step at a time., The first author wishes to acknowledge the financial contribution of the Spanish Ministry of Science and Innovation through Project Number PID2019-109131RB-I00.

First proposed in the early 1990s, the geostatistical algorithm known as multiple-point statistics (MPS) now enjoys widespread use, particularly in petroleum studies. It has become part of the toolkit that new practitioners are trained to use in several oil companies; it has been incorporated into commercial software; and research programs in many universities continue to tap into the central MPS idea of extracting statistical information directly from a training image. The inspiration for the development of a proof-of-concept MPS prototype code owes much to several different researchers and research programs in the late 1980s and early 1990s: the sequential algorithms pioneered at Stanford University, the work of Chris Farmer, then at UK Atomic Energy, and the growing use of outcrop studies by several oil companies. This largely accidental confluence of divergent theoretical perspectives, and of distinct practical workflows, serves as an example of how science often advances through the intersection of ideas that are not only disparate but even contradictory.

Petroleum reservoir geological models are usually built in two steps. First, a 3-D model of geological bodies is computed, within which rock properties are expected to be stationary and to have low variability. Such geological domains are referred to as “facies” and are often “electrofacies” obtained by clustering petrophysical log curves and calibrating the results with core data. It can happen that log responses of different types of rock are too similar to enable satisfactory estimation of the facies. In such situations, taking into account the spatial aspect of the data might help the discriminative process. Since the clustering algorithms that are used in this context usually fail to do so, we propose a method to overcome such limitations. It consists in post-calibrating the estimated probabilities of the presence of each facies in the samples, using geological trends determined by experts. The final facies probability is estimated by a simple kriging of the initial ones. Measurement errors reflecting the confidence in the clustering algorithms are added to the model, and the target mean is taken as the aforementioned geological trend. Assets and liabilities of this approach are reviewed; in particular, theoretical and practical issues about stationarity, neighborhood choice, and possible generalizations are discussed. The estimation of the variance to be assigned to each data point is also analyzed. As the class probabilities sum up to one, the classes are not independent; solutions are proposed in each context. This approach can be applied for extending class probabilities in 3-D.

Castelo de Sonhos, a gold deposit in Para State, Brazil, has seen several phases of exploration since the mid-1990s. These programs have provided drill hole data, surface mapping of outcrops, geophysical surveys, geochemical surveys of soil samples, and preliminary metallurgical test work. All available data from these exploration programs have been integrated with recent advances in paleo-plate reconstructions, in modeling sedimentary depositional systems, in geostatistical simulation, and in data mining. This integration of ideas and methods from petroleum geostatistics, from classical statistics, and from plate tectonics makes it possible to predict the range of the project’s potential tonnage and grade and to assess the project’s upside and downside risk. This leads to an exploration target range that is probabilistically quantified, that is well grounded in data, in field observations and science, and that is testable through drilling. Not only does this quantitative risk assessment improve analysis of the project’s technical and economic viability but also, importantly, it builds confidence among investors whose support is critical for advancing the project.

An overview of the geostatistical toolkit is presented, from data analysis through estimation and simulation, with a focus on problems that typically arise in the assessment and development of coal deposits. Geostatistical procedures for the data analysis are described, leading to a discussion of the importance of spatial variation and the variogram. The most common geostatistical estimation procedure, ordinary kriging, is presented as an improvement to inverse-distance methods; two ways are presented of understanding kriging without recourse to the underlying mathematics. Estimation and simulation are compared and contrasted, and the benefits of a family of equally likely scenarios are covered. The paper concludes with brief summaries of the 16 additional papers in the International Journal of Coal Geology's Special Issue on Geostatistics, and provides two indexes to guide the reader to papers according to the problems they address and according to the tools they use.

Authenticity of a document alleged to be a transcript of a 26 November 1941 telephone call between Churchill and Roosevelt is assessed using a ‘diplomatics’ approach. If genuine, this document would have transforming historical significance since it is dated well before the attack on Pearl Harbor and, taken at face value, serves as evidence that Roosevelt had specific prior warning of the impending attack. A detailed analysis of form, content and provenance establishes that the document is not authentic.

In many precious metals mining operations, resource estimation and mine planning require a model of the geometry of the high-grade lenses. These models are often developed by a manual interpretation of the drill hole data, leading to a “wireframe” solid that provides a single deterministic description of the major geological control on mineralization. An alternative approach is proposed, one that treats the problem probabilistically. Like the conventional approach, this alternative honors drill hole data; it also allows the user to represent their geological model as a wireframe solid. Two demonstrations of the method are provided, one that treats the drill hole data as “hard” information that must be honored exactly, and the other that treats some of the drill hole data as “soft” information that is honored on average over many realizations. These examples make use of a variogram model that offers the user flexible control of the degree of spatial continuity of the lenses. The results demonstrate that the technique offers all that the conventional approach offers—a wireframe that honors drill hole data—and has the advantage of offering multiple renditions of the ore lens geometry that can be used to study the impact of geologic uncertainty on resource and reserve estimates.

The indicator conditional simulation technique provides stochastic simulations of a variable that (i) honor the initial data and (ii) can feature a richer family of spatial structures not limited by Gaussianity. The data are encoded into a series of indicators which then are used to estimate the conditional probability distribution (cpdf) of the variable under study at any unsampled location. Once the cpdf has been estimated, any particular simulated value is obtained by straightforward Monte-Carlo drawing. Each new simulated value is included in the conditioning data set so that the next simulated values at other locations be conditioned to it. This technique has the advantage over other more traditional techniques such as the turning bands method in that it is not multiGaussian related. The user has full control of the bivariate (2-point) statistics imposed on the simulated field instead of controlling a mere covariance model. The source code is provided in C according to the ANSI standard.

The practical advantages and theoretical disadvantages of P-field simulation are reviewed in the light of more than a decade of application and research since it was first introduced. A case study example highlights the enduring attractions of the algorithm: its flexibility and speed. When first introduced, probability field simulation was well-suited to certain types of problems that were not well handled by other simulation algorithms available. In particular, it adapted well to the situation where a priori local distributions were available. As it rapidly gained practical acceptance, largely because of its speed, “P-field” simulation was also dismissed by some as a procedure lacking a proper theoretical foundation — more of a clever algorithmic trick than a properly conceived approach to stochastic spatial simulation. In the past decade, the advantages and shortcomings of the procedure have been illuminated through continued widespread application and theoretical research. This paper begins with an overview of the theoretical background and the usual practical implementation of P-field simulation. It then discusses theoretical concerns and assesses their practical implications. A mining case study example illustrates two enduring strengths of P-field simulation: flexibility and speed. 2 Overview and implementation Let F [u; z] denote the cumulative distribution function (cdf) at location u of an attribute Z. Any simulated value, zsim, represents a specific quantile of this local cdf: the z-value at which F [u; z] reaches a probability p(u): zsim = F −1 [u; p(u)] (1) The p values are not spatially independent; this would preclude reproduction of almost any desired spatial autocorrelation in Z .I nstead, thep values must be regarded as a realization of a random function P (u), and simulated with an appropriate pattern of spatial continuity.

It strikes me that there is an inconsistency between the first and third guidelines given by Journel for modelling uncertainty.

Since Lemmer first introduced the notion of the mononodal cutoff in the mid-1980’ I have always had a basic problem believing that it really worked. With this mos recent contribution, however, I finally have some sense for why it works (and why would not work in some situations).

Simulated annealing offers tremendous flexibility in the kind of information that can be honoured in a conditional simulation. Several researchers have shown that the speed of annealing (and, therefore, its practical utility) is related to whether or not the energy function used by the annealing procedure can be readily updated. If it is readily updateable, and does not have to be recalculated from scratch, annealing can be as fast as other commonly used geostatistical techniques for conditional simulation.

Solow and Ratick present an example of the use of conditional simulation for determining whether additional sample information is valuable. When used for this type of risk analysis, geostatistical conditional simulation is simply an adaptation of classical Monte Carlo methods to a spatial setting. There is an implicit assumption with such methods that the outcomes used in the calculations are equiprobable. For example, Solow and Ratick’s equation for approximating the expected net benefit is an equally-weighted average of the net benefit calculated on Kindependent realizations: $$E\left( {E*\left( Z \right)|x} \right) = \frac{1}{K}\sum\limits_{{j = 1}}^{K} {E*\left( {z_{{|x}}^{j}} \right)}$$ . Such a calculation is perfectly reasonable if any one of the Krealizations is as likely as any other one. If the realizations are not equiprobable—if the computer code that generates them is not fairly sampling the full space of uncertainty—then the whole approach is compromised. A non-representative set of outcomes will lead to a biased calculation and, possibly, to erroneous decisions.

Traditionally, geostatistical models are conditioned only on univariate and bivariate statistics such as the sample histogram and covariance or indicator covariances. Higher order sample statistics such as three, four, multi-point covariances, as obtained, for example, from a training image, would improve considerably stochastic images if they could be reproduced.

Geostatistics has traditionally used a probabilistic framework, one in which expected values or ensemble averages are of primary importance. The less familiar deterministic framework views geostatistical problems in terms of spatial integrals. This paper outlines the two frameworks and examines the issue of which spatial continuity measure, the covarianceC (h) or the variogram γ(h), is appropriate for each framework. AlthoughC (h) and γ(h) were defined originally in terms of spatial integrals, the convenience of probabilistic notation made the expected value definitions more common. These now classical expected value definitions entail a linear relationship betweenC (h) and γ(h); the spatial integral definitions do not. In a probabilistic framework, where available sample information is extrapolated to domains other than the one which was sampled, the expected value definitions are appropriate; furthermore, within a probabilistic framework, reasons exist for preferring the variogram to the covariance function. In a deterministic framework, where available sample information is interpolated within the same domain, the spatial integral definitions are appropriate and no reasons are known for preferring the variogram. A case study on a Wiener-Levy process demonstrates differences between the two frameworks and shows that, for most estimation problems, the deterministic viewpoint is more appropriate. Several case studies on real data sets reveal that the sample covariance function reflects the character of spatial continuity better than the sample variogram. From both theoretical and practical considerations, clearly for most geostatistical problems, direct estimation of the covariance is better than the traditional variogram approach.

The variogram often suffers in practice from the effects of heteroscedasticity and clustering. “Relative” variograms have enjoyed practical success despite their uncertain theoretical pedigree. These relative variograms achieve their robustness by scaling the traditional variogram by a function of the mean. The mean which determines the scaling factor can be chosen in several different ways, giving rise to several different relative variograms. Other practically successful alternatives to the variogram include the covariance, which achieves its robustness by accounting for the lag means, and the correlogram, which incorporates the lag variances. A simulated data set is used to explore the performance of the traditional variogram and four alternatives. The results of these studies lead to the conclusion that the traditional variogram should not be used to describe spatial continuity in the presence of skewed distributions which have been preferentially sampled.