Substitution Random Fields with Gaussian and Gamma Distributions: Theory and Application to a Pollution Data Set

This paper presents random field models with Gaussian or gamma univariate distributions and isofactorial bivariate distributions, constructed by composing two independent random fields: a directing function with stationary Gaussian increments and a stationary coding process with bivariate Gaussian or gamma distributions. Two variations are proposed, by considering a multivariate directing function and a coding process with a separable covariance, or by including drift components in the directing function. Iterative algorithms based on the Gibbs sampler allow one to condition the realizations of the substitution random fields to a set of data, while the inference of the model parameters relies on simple tools such as indicator variograms and variograms of different orders. A case study in polluted soil management is presented, for which a gamma model is used to quantify the risk that pollutant concentrations over remediation units exceed a given toxicity level. Unlike the multivariate Gaussian model, the proposed gamma model accounts for an asymmetry in the spatial correlation of the indicator functions around the median and for a spatial clustering of high pollutant concentrations.     

Approximate local confidence intervals under change of support

Approximate local confidence intervals are constructed from uncertainty models in the form of the conditional distribution of the random variable Z given values of variables [Z_i, i=1,...,n]. When the support of the variable Z is any support other than that of the data, the conditional distributions require a change of support correction. This paper investigates the effect of change of support on the approximate local confidence intervals constructed by cumulative indicator kriging, class indicator kriging, and probability kriging under a variety of conditions. The conditions are generated by three simulated deposits with grade distributions of successively higher degree of skewness; a point support and two different block supports are considered. The paper also compares the confidence intervals obtained from these methods using the most used measures of confidence interval effectiveness.     

T -distributed Random Fields: A Parametric Model for Heavy-tailedWell-log Data1

Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they are layer specific and include uncertainty caused by using variance estimates.     

Isofactorial models for granulodensimetric data

Existing isofactorial models developed for disjunctive kriging using a cutoff grade on one variable are extended to the bivariate case which arises when dealing with granulo-densimetric data, such as are obtained from coal washing or mineral processing.     

Two Ordinary Kriging Approaches to Predicting Block Grade Distributions

Multigaussian kriging aims at estimating the local distributions of regionalized variables and functions of these variables (transfer or recovery functions) at unsampled locations. In this paper, we focus on the evaluation of the recoverable reserves in an ore deposit accounting for a change of support and information effect caused by ore/waste misclassifications. Two approaches are proposed: the multigaussian model with Monte Carlo integration and the discrete Gaussian model. The latter is simpler to use but requires stronger hypotheses than the former. In each model, ordinary multigaussian kriging gives unbiased estimates of the recoverable reserves that do not utilize the mean value of the normal score data. The concepts are illustrated through a case study on a copper deposit which shows that local estimates of the metal content based on ordinary multigaussian kriging are close to the optimal conditional expectation when the data are abundant and are not dominated by the global mean when the data are scarce. The two proposed approaches (Monte Carlo integration and discrete Gaussian model) lead to similar results when compared to two other geostatistical methods: service variables and ordinary indicator kriging, which show strong deviations from conditional expectation.     

Stepwise Conditional Transformation for Simulation of Multiple Variables

Most geostatistical studies consider multiple-related variables. These relationships often show complex features such as nonlinearity, heteroscedasticity, and mineralogical or other constraints. These features are not handled by the well-established Gaussian simulation techniques. Earth science variables are rarely Gaussian. Transformation or anamorphosis techniques make each variable univariate Gaussian, but do not enforce bivariate or higher order Gaussianity. The stepwise conditional transformation technique is proposed to transform multiple variables to be univariate Gaussian and multivariate Gaussian with no cross correlation. This makes it remarkably easy to simulate multiple variables with arbitrarily complex relationships: (1) transform the multiple variables, (2) perform independent Gaussian simulation on the transformed variables, and (3) back transform to the original variables. The back transformation enforces reproduction of the original complex features. The methodology and underlying assumptions are explained. Several petroleum and mining examples are used to show features of the transformation and implementation details.     

Resource Model Updating For Compositional Geometallurgical Variables

In the field of mineral resources extraction, one main challenge is to meet production targets in terms of geometallurgical properties. These properties influence the processing of the ore and are often represented in resource modeling by coregionalized variables with a complex relationship between them. Valuable data are available about geometalurgical properties and their interaction with the beneficiation process given sensor technologies during production monitoring. The aim of this research is to update resource models as new observations become available. A popular method for updating is the ensemble Kalman filter. This method relies on Gaussian assumptions and uses a set of realizations of the simulated models to derive sample covariances that can propagate the uncertainty between real observations and simulated ones. Hence, the relationship among variables has a compositional nature, such that updating these models while keeping the compositional constraints is a practical requirement in order to improve the accuracy of the updated models. This paper presents an updating framework for compositional data based on ensemble Kalman filter which allows us to work with compositions that are transformed into a multivariate Gaussian space by log-ratio transformation and flow anamorphosis. This flow anamorphosis, transforms the distribution of the variables to joint normality while reasonably keeping the dependencies between components. Furthermore, the positiveness of those variables, after updating the simulated models, is satisfied. The method is implemented in a bauxite deposit, demonstrating the performance of the proposed approach.

     

Geometallurgical Modeling at Olympic Dam Mine, South Australia

Modeling of geometallurgical variables is becoming increasingly important for improved management of mineral resources. Mineral processing circuits are complex and depend on the interaction of a large number of properties of the ore feed. At the Olympic Dam mine in South Australia, plant performance variables of interest include the recovery of Cu and U₃O₈, acid consumption, net recovery, drop weight index, and bond mill work index. There are an insufficient number of pilot plant trials (841) to consider direct three-dimensional spatial modeling for the entire deposit. The more extensively sampled head grades, mineral associations, grain sizes, and mineralogy variables are modeled and used to predict plant performance. A two-stage linear regression model of the available data is developed and provides a predictive model with correlations to the plant performance variables ranging from 0.65–0.90. There are a total of 204 variables that have sufficient sampling to be considered in this regression model. After developing the relationships between the 204 input variables and the six performance variables, the input variables are simulated with sequential Gaussian simulation and used to generate models of recovery of Cu and U₃O₈, acid consumption, net recovery, drop weight index, and bond mill work index. These final models are suitable for mine and plant optimization.     

Variograms of Order ω: A Tool to Validate a Bivariate Distribution Model

The multigaussian model is used in mining geostatistics to simulate the spatial distribution of grades or to estimate the recoverable reserves of an ore deposit. Checking the suitability of such model to the available data often constitutes a critical step of the geostatistical study. In general, the marginal distribution is not a problem because the data can be transformed to normal scores, so the check is usually restricted to the bivariate distributions. In this work, several tests for diagnosing the two-point normality of a set of Gaussian data are reviewed and commented. An additional criterion is proposed, based on the comparison between the usual variogram and the variograms of lower order: the latter are defined as half the mean absolute increments of the attribute raised to a power between 0 and 2. This criterion is then extended to other bivariate models, namely the bigamma, Hermitian and Laguerrian models. The concepts are illustrated on two real data-sets. Finally, some conditions to ensure the internal consistency of the variogram under a given model are given.     

Disjunctive Kriging with Hard and Imprecise Data

This paper presents a methodology for assessing local probability distributions by disjunctive kriging when the available data set contains some imprecise measurements, like noisy or soft information or interval constraints. The basic idea consists in replacing the set of imprecise data by a set of pseudohard data simulated from their posterior distribution; an iterative algorithm based on the Gibbs sampler is proposed to achieve such a simulation step. The whole procedure is repeated many times and the final result is the average of the disjunctive kriging estimates computed from each simulated data set. Being data-independent, the kriging weights need to be calculated only once, which enables fast computing. The simulation procedure requires encoding each datum as a pre-posterior distribution and assuming a Markov property to allow the updating of pre-posterior distributions into posterior ones. Although it suffers some imperfections, disjunctive kriging turns out to be a much more flexible approach than conditional expectation, because of the vast class of models that allows its computation, namely isofactorial models.     

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

Geostatistics for Power Models of Gaussian Fields

This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions.     

A generalized probabilistic approach for slope analysis

Summary A new probabilistic approach is introduced for slope stability analysis, which is general in types of variable distributions and correlations or dependency between variables, and flexible enough to include any adverse impact analysis for blasting vibrations and groundwater conditions.The material strength within a slope area, given in terms of the internal friction angle (ø) and cohesion (c), is randomized in the bivariate joint probability analysis. To be a completely general engineering method, the new probabilistic approach employs the random variable transformation technique: the Hermite model of the Gaussian transformation function, which transforms the experimental histogram of shear strength parameters to the standard Gaussian distribution (=0, ²=1.0).Because a binormal joint probability is analysed on the true probability region projected on the plane of the Gaussian transformed variables, it is an exact solution of slope stability based on the available sample data. No assumption on the shape of the experimental histogram or independency between two random variables is made as in the current probability methods of slope analysis.     

Geostatistical case studies of the advantages of lognormal-de Wijsian kriging with mean for a base metal mine and a gold mine

Parallel variogram analyses, block kriging, and follow-up studies were effected for the lead content of part of the Prieska copper-zinc ore body and for the gold content of the highly variable Breef in a section of the Loraine gold mine, based first on untransformed values and second on logarithmically transformed values using the lognormal-de Wijsian model. For both models the effect was also analyzed of using the population mean or ignoring it. Practical follow-up comparisons confirm theoretical considerations and show that on these mines conditional biases can be eliminated conveniently by kriging with mean; also that the lognormal-de Wijsian model with mean gives the best results.     

Models for Support and Information Effects: A Comparative Study

The recoverable reserves in an ore deposit depend on several factors, in particular the size of the selective mining units (support effect) and the misclassifications when sending these units to mill or dump according to their estimated grade (information effect). Both effects imply a loss of selectivity and have to be correctly forecasted. In this work, several models are reviewed and applied to a synthetic ore deposit characterized by a highly skewed grade histogram and a spatial connectivity of high grades. The affine correction, mosaic correction, and discrete Gaussian model are compared when assessing the global recoverable reserves, whereas local estimations are performed by indicator kriging with affine correction, bigaussian disjunctive kriging, and multigaussian conditional expectation. Despite their convenience and simplicity, distribution-free methods like affine correction or indicator kriging have a poorer accuracy than the other methods. In the global framework, the discrete Gaussian model is a better alternative and is based on mild assumptions. Local estimations are not accurate and may be improved by resorting to a more suitable parametric model or to conditional simulations.     

Direct forecasting of reservoir performance using production data without history matching

The conventional paradigm for predicting future reservoir performance from existing production data involves the construction of reservoir models that match the historical data through iterative history matching. This is generally an expensive and difficult task and often results in models that do not accurately assess the uncertainty of the forecast. We propose an alternative re-formulation of the problem, in which the role of the reservoir model is reconsidered. Instead of using the model to match the historical production, and then forecasting, the model is used in combination with Monte Carlo sampling to establish a statistical relationship between the historical and forecast variables. The estimated relationship is then used in conjunction with the actual production data to produce a statistical forecast. This allows quantifying posterior uncertainty on the forecast variable without explicit inversion or history matching. The main rationale behind this is that the reservoir model is highly complex and even so, still remains a simplified representation of the actual subsurface. As statistical relationships can generally only be constructed in low dimensions, compression and dimension reduction of the reservoir models themselves would result in further oversimplification. Conversely, production data and forecast variables are time series data, which are simpler and much more applicable for dimension reduction techniques. We present a dimension reduction approach based on functional data analysis (FDA), and mixed principal component analysis (mixed PCA), followed by canonical correlation analysis (CCA) to maximize the linear correlation between the forecast and production variables. Using these transformed variables, it is then possible to apply linear Gaussian regression and estimate the statistical relationship between the forecast and historical variables. This relationship is used in combination with the actual observed historical data to estimate the posterior distribution of the forecast variable. Sampling from this posterior and reconstructing the corresponding forecast time series, allows assessing uncertainty on the forecast. This workflow will be demonstrated on a case based on a Libyan reservoir and compared with traditional history matching.     

The Bi-Gaussian approach: A simple method for recovery estimation

A new and simple method is proposed to obtain estimates of recovery functions: the Bi-Gaussian approach. Existing methods estimate recovery functions with conditional distributions where the conditioning set is all the data available. Here instead the simple kriging estimate of the Gaussian transform is proposed to be used. Results in the point recovery case are identical to the multi-Gaussian approach of Verly (1983, 1984), whereas in the non-point-support situation, an approximation is derived which saves computer time as compared to employing the strict multi-Gaussian hypothesis. Two examples compare favorably with the well-established disjunctive kriging method (discrete Gaussian model).     

Semivariogram Models Based on Geometric Offsets

Kriging-based geostatistical models require a semivariogram model. Next to the initial decision of stationarity, the choice of an appropriate semivariogram model is the most important decision in a geostatistical study. Common practice consists of fitting experimental semivariograms with a nested combination of proven models such as the spherical, exponential, and Gaussian models. These models work well in most cases; however, there are some shapes found in practice that are difficult to fit. We introduce a family of semivariogram models that are based on geometric shapes, analogous to the spherical semivariogram, that are known to be conditional negative definite and provide additional flexibility to fit semivariograms encountered in practice. A methodology to calculate the associated geometric shapes to match semivariograms defined in any number of directions is presented. Greater flexibility is available through the application of these geometric semivariogram models.     

Transition probability-based indicator geostatistics

Traditionally, spatial continuity models for indicator variables are developed by empirical curvefitting to the sample indicator (cross-) variogram. However, geologic data may be too sparse to permit a purely empirical approach, particularly in application to the subsurface. Techniques for model synthesis that integrate hard data and conceptual models therefore are needed. Interpretability is crucial. Compared with the indicator (cross-) variogram or indicator (cross-) covariance, the transition probability is more interpretable. Information on proportion, mean length, and juxtapositioning directly relates to the transition probability: asymmetry can be considered. Furthermore, the transition probability elucidates order relation conditions and readily formulates the indicator (co)kriging equations.