Geostatistical Simulation with a Trend Using Gaussian Mixture Models

Geostatistics applies statistics to quantitatively describe geological sites and assess the uncertainty due to incomplete sampling. Strong assumptions are required regarding the location independence of statistical parameters to construct numerical models with geostatistical tools. Most geological data exhibit large-scale deterministic trends together with short-scale variations. Such location dependence violates the common geostatistical assumption of stationarity. The trend-like deterministic features should be modeled prior to conventional geostatistical prediction and accounted for in subsequent geostatistical calculations. The challenge of using a trend in geostatistical simulation algorithms for the continuous variable is the subject of this paper. A stepwise conditional transformation with a Gaussian mixture model is considered to provide a stable and artifact-free numerical model. The complex features of the regionalized variable in the presence of a trend are removed in the forward transformation and restored in the back transformation. The Gaussian mixture model provides a seamless bin-free approach to transformation. Data from a copper deposit were used as an example. These data show an apparent trend unsuitable for conventional geostatistical algorithms. The result shows that the proposed algorithm leads to improved geostatistical models.     

Categorization of Mineral Resources Based on Different Geostatistical Simulation Algorithms: A Case Study from an Iron Ore Deposit

Mineral resource classification plays an important role in the downstream activities of a mining project. Spatial modeling of the grade variability in a deposit directly impacts the evaluation of recovery functions, such as the tonnage, metal quantity and mean grade above cutoffs. The use of geostatistical simulations for this purpose is becoming popular among practitioners because they produce statistical parameters of the sample dataset in cases of global distribution (e.g., histograms) and local distribution (e.g., variograms). Conditional simulations can also be assessed to quantify the uncertainty within the blocks. In this sense, mineral resource classification based on obtained realizations leads to the likely computation of reliable recovery functions, showing the worst and best scenarios. However, applying the proper geostatistical (co)-simulation algorithms is critical in the case of modeling variables with strong cross-correlation structures. In this context, enhanced approaches such as projection pursuit multivariate transforms (PPMTs) are highly desirable. In this paper, the mineral resources in an iron ore deposit are computed and categorized employing the PPMT method, and then, the outputs are compared with conventional (co)-simulation methods for the reproduction of statistical parameters and for the calculation of tonnage at different levels of cutoff grades. The results show that the PPMT outperforms conventional (co)-simulation approaches not only in terms of local and global cross-correlation reproductions between two underlying grades (Fe and Al₂O₃) in this iron deposit but also in terms of mineral resource categories according to the Joint Ore Reserves Committee standard.

     

Geological Modeling of Gold Deposit Based on Grade Domaining Using Plurigaussian Simulation Technique

Mineral resource evaluation requires defining grade domains of an ore deposit. Common practice in mineral resource estimation consists of partitioning the ore body into several grade domains before the geostatistical modeling and estimation at unsampled locations. Many ore deposits are made up of different mineralogical ensembles such as oxide and sulfide zone: being able to model the spatial layout of the different grades is vital to good mine planning and management. This study addresses the application of the plurigaussian simulation to Sivas (Turkey) gold deposits for constructing grade domain models that reproduce the contacts between different grade domains in accordance with geologist’s interpretation. The method is based on the relationship between indicator variables from grade distributions on the Gaussian random functions chosen to represent them. Geological knowledge is incorporated into the model by the definition of the indicator variables, their truncation strategy, and the grade domain proportions. The advantages of the plurigaussian simulation are exhibited through the case study. The results indicated that the processes are seen to respect reproducing complex geometrical grades of an ore deposit by means of simulating several grade domains with different spatial structure and taking into account their global proportions. The proposed proportion model proves as simple to use in resource estimation, to account for spatial variations of the grade characteristics and their distribution across the studied area, and for the uncertainty in the grade domain proportions. The simulated models can also be incorporated into mine planning and scheduling.     

Public attitudes and policies toward mineral resources on the brink of the 21st century

In this issue, we feature an article by W. David Menzie, a research geologist with the U.S. Geological Survey, Reston, Virginia. Dr. Menzie is a leading expert on quantitative mineral-resource assessment. He has made significant contributions to quantitative assessment methodologies through the development of spatial mineral deposit density models, grade and tonnage models, and the design of metrics for describing mineral deposit occurrences. He has also studied the geology and mineral resources of the Circle quadrangle, Alaska. Dr. Menzie earned a B.S. degree in geology from Dickinson College, an M.S. in geology, an M.A. in statistics, and a Ph.D. in Geology from the Pennsylvania State University.     

Sequential Simulation Approach to Modeling of Multi-seam Coal Deposits with an Application to the Assessment of a Louisiana Lignite

There are multiple ways to characterize uncertainty in the assessment of coal resources, but not all of them are equally satisfactory. Increasingly, the tendency is toward borrowing from the statistical tools developed in the last 50 years for the quantitative assessment of other mineral commodities. Here, we briefly review the most recent of such methods and formulate a procedure for the systematic assessment of multi-seam coal deposits taking into account several geological factors, such as fluctuations in thickness, erosion, oxidation, and bed boundaries. A lignite deposit explored in three stages is used for validating models based on comparing a first set of drill holes against data from infill and development drilling. Results were fully consistent with reality, providing a variety of maps, histograms, and scatterplots characterizing the deposit and associated uncertainty in the assessments. The geostatistical approach was particularly informative in providing a probability distribution modeling deposit wide uncertainty about total resources and a cumulative distribution of coal tonnage as a function of local uncertainty.     

Geology-based conditional simulation in the Athabasca oil sands deposit,Alberta, Canada

The Athabasca oil sands deposit, Alberta, Canada, is one of the largest known hydrocarbon accumulations. The efficient exploitation of this deposit, as well as other oil sand accumulations throughout the world, is based onin situ recovery and surface mining methods. Quantitative modeling of deposit heterogeneity provides a valuable engineering tool. In the present study, conditional simulation was used to model oil-saturated zones in part of the Athabasca deposit. This technique generates equiprobable models of thein situ variability of essential deposit attributes that honor the available data and their spatial statistics. The application of the technique is based on the delineation of geologically homogeneous zones within the host McMurray Formation, their statistical validity, and the integration of geological interpretations. The geological framework is developed, and subsequently, high resolution conditionally simulated models of three identified hydrocarbon-bearing zones are generated, in terms of the zone boundaries and the percent weight of oil saturation. These models serve as “what-if” tools for risk assessment and future planning.     

Monte Carlo Simulations of Product Distributions and Contained Metal Estimates

Estimation of product distributions of two factors was simulated by conventional Monte Carlo techniques using factor distributions that were independent (uncorrelated). Several simulations using uniform distributions of factors show that the product distribution has a central peak approximately centered at the product of the medians of the factor distributions. Factor distributions that are peaked, such as Gaussian (normal) produce an even more peaked product distribution. Piecewise analytic solutions can be obtained for independent factor distributions and yield insight into the properties of the product distribution. As an example, porphyry copper grades and tonnages are now available in at least one public database and their distributions were analyzed. Although both grade and tonnage can be approximated with lognormal distributions, they are not exactly fit by them. The grade shows some nonlinear correlation with tonnage for the published database. Sampling by deposit from available databases of grade, tonnage, and geological details of each deposit specifies both grade and tonnage for that deposit. Any correlation between grade and tonnage is then preserved and the observed distribution of grades and tonnages can be used with no assumption of distribution form.     

Modelling uncertainty and spatial dependence: Stochastic imaging

Abstract

The most vibrant area of research in geostatistics is stochastic imaging, that is, the modelling of spatial uncertainty through alternative, equiprobable, numerical representations (maps) of spatially distributed phenomena. These stochastic images are conditioned to a variety of data accounting for their specific measurement scale and reliability.

Any geostatistical prediction is built on a prior model of spatial correlation that ties data to unsampled values and, equally importantly, unsampled values at different locations together. Since a major goal in the exercise of mapping is to display organization in space, spatial correlation is a necessity. As for uncertainty it is so pervasive that it is imperative to account for it.     

Probabilistic Estimates of Number of Undiscovered Deposits and Their Total Tonnages in Permissive Tracts Using Deposit Densities

Empirical evidence indicates that processes affecting number and quantity of resources in geologic settings are very general across deposit types. Sizes of permissive tracts that geologically could contain the deposits are excellent predictors of numbers of deposits. In addition, total ore tonnage of mineral deposits of a particular type in a tract is proportional to the type’s median tonnage in a tract. Regressions using size of permissive tracts and median tonnage allow estimation of number of deposits and of total tonnage of mineralization. These powerful estimators, based on 10 different deposit types from 109 permissive worldwide control tracts, generalize across deposit types. Estimates of number of deposits and of total tonnage of mineral deposits are made by regressing permissive area, and mean (in logs) tons in deposits of the type, against number of deposits and total tonnage of deposits in the tract for the 50th percentile estimates. The regression equations (R ² = 0.91 and 0.95) can be used for all deposit types just by inserting logarithmic values of permissive area in square kilometers, and mean tons in deposits in millions of metric tons. The regression equations provide estimates at the 50th percentile, and other equations are provided for 90% confidence limits for lower estimates and 10% confidence limits for upper estimates of number of deposits and total tonnage. Equations for these percentile estimates along with expected value estimates are presented here along with comparisons with independent expert estimates. Also provided are the equations for correcting for the known well-explored deposits in a tract. These deposit-density models require internally consistent grade and tonnage models and delineations for arriving at unbiased estimates.     

Typing Mineral Deposits Using Their Grades and Tonnages in an Artificial Neural Network

A test of the ability of a probabilistic neural network to classify deposits into types on the basis of deposit tonnage and average Cu, Mo, Ag, Au, Zn, and Pb grades is conducted. The purpose is to examine whether this type of system might serve as a basis for integrating geoscience information available in large mineral databases to classify sites by deposit type. Benefits of proper classification of many sites in large regions are relatively rapid identification of terranes permissive for deposit types and recognition of specific sites perhaps worthy of exploring further.Total tonnages and average grades of 1,137 well-explored deposits identified in published grade and tonnage models representing 13 deposit types were used to train and test the network. Tonnages were transformed by logarithms and grades by square roots to reduce effects of skewness. All values were scaled by subtracting the variable's mean and dividing by its standard deviation. Half of the deposits were selected randomly to be used in training the probabilistic neural network and the other half were used for independent testing. Tests were performed with a probabilistic neural network employing a Gaussian kernel and separate sigma weights for each class (type) and each variable (grade or tonnage).Deposit types were selected to challenge the neural network. For many types, tonnages or average grades are significantly different from other types, but individual deposits may plot in the grade and tonnage space of more than one type. Porphyry Cu, porphyry Cu-Au, and porphyry Cu-Mo types have similar tonnages and relatively small differences in grades. Redbed Cu deposits typically have tonnages that could be confused with porphyry Cu deposits, also contain Cu and, in some situations, Ag. Cyprus and kuroko massive sulfide types have about the same tonnages. Cu, Zn, Ag, and Au grades. Polymetallic vein, sedimentary exhalative Zn-Pb, and Zn-Pb skarn types contain many of the same metals. Sediment-hosted Au, Comstock Au-Ag, and low-sulfide Au-quartz vein types are principally Au deposits with differing amounts of Ag.Given the intent to test the neural network under the most difficult conditions, an overall 75% agreement between the experts and the neural network is considered excellent. Among the largestclassification errors are skarn Zn-Pb and Cyprus massive sulfide deposits classed by the neuralnetwork as kuroko massive sulfides—24 and 63% error respectively. Other large errors are the classification of 92% of porphyry Cu-Mo as porphyry Cu deposits. Most of the larger classification errors involve 25 or fewer training deposits, suggesting that some errors might be the result of small sample size. About 91% of the gold deposit types were classed properly and 98% of porphyry Cu deposits were classes as some type of porphyry Cu deposit. An experienced economic geologist would not make many of the classification errors that were made by the neural network because the geologic settings of deposits would be used to reduce errors. In a separate test, the probabilistic neural network correctly classed 93% of 336 deposits in eight deposit types when trained with presence or absence of 58 minerals and six generalized rock types. The overall success rate of the probabilistic neural network when trained on tonnage and average grades would probably be more than 90% with additional information on the presence of a few rock types.     

Selection of Optimal Thresholds for Estimation and Simulation Based on Indicator Values of Highly Skewed Distributions of Ore Data

This study strives to outline a geostatistics model for estimation and simulation of the Qolqoleh gold ore deposit located in Saqqez, NW of Iran. Considering that this gold deposit contains high-grade values, accurate evaluation of such values is of high importance, and therefore different methods based on indicator values, such as full indicator kriging (FIK) and sequential indicator simulation (SIS), have been employed to improve the accuracy of estimation and simulation of high-grade values. FIK and SIS cover the full range of grades based on several thresholds on the indicator data. The cumulative distribution function (CDF) is typically used for selection of threshold values. Given the highly skewed distribution of gold grade and its intense fluctuations, the number of thresholds is increased using CDF, which in turn results in a whole lot of calculations. To reduce the volume of calculations, the number–size (N–S) fractal model has been used to select thresholds. From such a model, all optimal thresholds are chosen with respect to geology and the unnecessary thresholds are excluded from selection. Thus, a study of the selection of optimal thresholds for estimation and simulation of a gold ore resource by means of FIK and SIS, respectively, based on thresholds selected using the N–S fractal model is presented. Finally, it is proved that results of these geostatistical methods based on thresholds selection from the N–S model appear to be better-positioned to explain ore grade variability compared to thresholds selected from the CDF and threshold selection from the N–S model is more effective for reducing the volume of required calculations.     

Additional Samples: Where They Should Be Located

Information for mine planning requires to be close spaced, if compared to the grid used for exploration and resource assessment. The additional samples collected during quasimining usually are located in the same pattern of the original diamond drillholes net but closer spaced. This procedure is not the best in mathematical sense for selecting a location. The impact of an additional information to reduce the uncertainty about the parameter been modeled is not the same everywhere within the deposit. Some locations are more sensitive in reducing the local and global uncertainty than others. This study introduces a methodology to select additional sample locations based on stochastic simulation. The procedure takes into account data variability and their spatial location. Multiple equally probable models representing a geological attribute are generated via geostatistical simulation. These models share basically the same histogram and the same variogram obtained from the original data set. At each block belonging to the model a value is obtained from the n simulations and their combination allows one to access local variability. Variability is measured using an uncertainty index proposed. This index was used to map zones of high variability. A value extracted from a given simulation is added to the original data set from a zone identified as erratic in the previous maps. The process of adding samples and simulation is repeated and the benefit of the additional sample is evaluated. The benefit in terms of uncertainty reduction is measure locally and globally. The procedure showed to be robust and theoretically sound, mapping zones where the additional information is most beneficial. A case study in a coal mine using coal seam thickness illustrates the method.     

A case study of model selection and parameter inference by maximum likelihood with application to uncertainty analysis

One of the uses of geostatistical conditional simulation is as a tool in assessing the spatial uncertainty of inputs to the Monte Carlo method of system uncertainty analysis. Because the number of experimental data in practical applications is limited, the geostatistical parameters used in the simulation are themselves uncertain. The inference of these parameters by maximum likelihood allows for an easy assessment of this estimation uncertainty which, in turn, may be included in the conditional simulation procedure. A case study based on transmissivity data is presented to show the methodology whereby both model selection and parameter inference are solved by maximum likelihood.     

Space Deformation Non-stationary Geostatistical Approach for Prediction of Geological Objects: Case Study at El Teniente Mine (Chile)

This article addresses the problem of the prediction of the breccia pipe elevation named Braden at the El Teniente mine in Chile. This mine is one of the world’s largest known porphyry-copper ore bodies. Knowing the exact location of the pipe surface is important, as it constitutes the internal limit of the deposit. The problem is tackled by applying a non-stationary geostatistical method based on space deformation, which involves transforming the study domain into a new domain where a standard stationary geostatistical approach is more appropriate. Data from the study domain is mapped into the deformed domain, and classical stationary geostatistical techniques for prediction can then be applied. The predicted results are then mapped back into the original domain. According to the results, this non-stationary geostatistical method outperforms the conventional stationary one in terms of prediction accuracy and conveys a more informative uncertainty model of the predictions.     

Nonparametric Geostatistical Simulation of Subsurface Facies: Tools for Validating the Reproduction of,and Uncertainty in,Facies Geometry

Delineation of facies in the subsurface and quantification of uncertainty in their boundaries are significant steps in mineral resource evaluation and reservoir modeling, which impact downstream analyses of a mining or petroleum project. This paper investigates the ability of nonparametric geostatistical simulation algorithms (sequential indicator, single normal equation and filter-based simulation) to construct realizations that reproduce some expected statistical and spatial features, namely facies proportions, boundary regularity, contact relationships and spatial correlation structure, as well as the expected fluctuations of these features across the realizations. The investigation is held through a synthetic case study and a real case study, in which a pluri-Gaussian model is considered as the reference for comparing the simulation results. Sequential indicator simulation and single normal equation simulation based on over-restricted neighborhood implementations yield the poorest results, followed by filter-based simulation, whereas single normal equation simulation with a large neighborhood implementation provides results that are closest to the reference pluri-Gaussian model. However, some biases and inaccurate fluctuations in the realization statistics (facies proportions, indicator direct and cross-variograms) still arise, which can be explained by the use of a single finite-size training image to construct the realizations.

     

Practical Incorporation of Multivariate Parameter Uncertainty in Geostatistical Resource Modeling

There is a need to bridge theory and practice for incorporating parameter uncertainty in geostatistical simulation modeling workflows. Simulation workflows are a standard practice in natural resource and recovery modeling, but the incorporation of multivariate parameter uncertainty into those workflows is challenging. However, the objectives can be met without considerable extra effort and programming. The sampling distributions of statistics comprise the core theoretical notion with the addition of the spatial degrees of freedom to account for the redundancy in the spatially correlated data. Prior parameter uncertainty is estimated from multivariate spatial resampling. Simulation-based transfer of prior parameter uncertainty results in posterior distributions which are updated by data conditioning and the model domain extents and configuration. The results are theoretically tractable and practical to achieve, providing realistic assessments of uncertainty by accounting for large-scale parameter uncertainty, which is often the most important component impacting a project. A simulation-based multivariate workflow demonstrates joint modeling of intrinsic shale properties and uncertainty in estimated ultimate recovery in a shale gas project. The multivariate workflow accounts for joint prior parameter uncertainty given the current well locations and results in posterior estimates on global distributions of all modeled properties. This is achieved by transferring the joint prior parameter uncertainty through conditional simulations.     

Grade-tonnage and other models for diamond kimberlite pipes

Grade-tonnage and other quantitative models help give reasonable answers to questions about diamond kimberlite pipes. Diamond kimberlite pipes are those diamondiferous kimberlite pipes that either have been worked or are expected to be worked for diamonds. These models are not applicable to kimberlite dikes and sills or to lamproite pipes. Diamond kimberlite pipes contain a median 26 million metric tons (mt); the median diamond grade is 0.25 carat/metric ton (ct/mt). Deposit-specific models suggest that the median of the average diamond size is 0.07 ct and the median percentage of diamonds that are industrial quality is 67 percent. The percentage of diamonds that are industrial quality can be predicted from deposit grade using a regression model (log[industrial diamonds (percent)]=1.9+0.2 log[grade (ct/mt)]). The largest diamond in a diamond kimberlite pipe can be predicted from deposit tonnage using a regression model (log[largest diamond (ct)]=–1.5+0.54 log[size (mt]). The median outcrop area of diamond pipes is 12 hectares (ha). Because the pipes have similar forms, the tonnage of the deposits can be predicted by the outcrop area (log[size (mt)]=6.5+1.0 log[outcrop area (ha)]). Once a kimberlite pipe is identified, the probability is approximately .005 that it can be worked for diamonds. If a newly discovered pipe is a member of a cluster that contains a known diamond kimberlite pipe, the probability that the new discovery can be mined for diamonds is 56 times that for a newly discovered kimberlite pipe in a cluster without a diamond kimberlite pipe. About 30 percent of pipes with worked residual caps at the surface will be worked at depth. Based on the number of discovered deposits and the area of stable craton rocks thought to be well explored in South Africa, about 10^–5 diamond kimberlite pipes are present per square kilometer. If this density is applicable to the South American Precambrian Shield, more than 70 undiscovered kimberlite pipes are predicted to be present.     

A Chart for Judging Optimal Sample Spacing for Ore Grade Estimation

Mineral deposit grades are usually estimated using data from samples of rock cores extracted from drill holes. Commonly, mineral deposit grade estimates are required for each block to be mined. Every estimated grade has always a corresponding error when compared against real grades of blocks. The error depends on various factors, among which the most important is the number of correlated samples used for estimation. Samples may be collected on a regular sampling grid and, as the spacing between samples decreases, the error of grade estimated from the data generally decreases. Sampling can be expensive. The maximum distance between samples that provides an acceptable error of grade estimate is useful for deciding how many samples are adequate. The error also depends on the geometry of a block, as lower errors would be expected when estimating the grade of large-volume blocks, and on the variability of the data within the region of the blocks. Local variability is measured in this study using the coefficient of variation (CV). We show charts analyzing error in block grade estimates as a function of sampling grid (obtained by geostatistical simulation), for various block dimensions (volumes) and for a given CV interval. These charts show results for two different attributes (Au and Ni) of two different deposits. The results show that similar errors were found for the two deposits, although they share similar features: sampling grid, block volume, CV, and continuity model. Consequently, the error for other attributes with similar features could be obtained from a single chart.     

Forthcoming events

Spatial data uncertainty models (SDUM) are necessary tools that quantify the reliability of results from geographical information system (GIS) applications. One technique used by SDUM is Monte Carlo simulation, a technique that quantifies spatial data and application uncertainty by determining the possible range of application results. A complete Monte Carlo SDUM for generalized continuous surfaces typically has three components: an error magnitude model, a spatial statistical model defining error shapes, and a heuristic that creates multiple realizations of error fields added to the generalized elevation map. This paper introduces a spatial statistical model that represents multiple statistics simultaneously and weighted against each other. This paper's case study builds a SDUM for a digital elevation model (DEM). The case study accounts for relevant shape patterns in elevation errors by reintroducing specific topological shapes, such as ridges and valleys, in appropriate localized positions. The spatial statistical model also minimizes topological artefacts, such as cells without outward drainage and inappropriate gradient distributions, which are frequent problems with random field-based SDUM. Multiple weighted spatial statistics enable two conflicting SDUM philosophies to co-exist. The two philosophies are ‘errors are only measured from higher quality data’ and ‘SDUM need to model reality’. This article uses an automatic parameter fitting random field model to initialize Monte Carlo input realizations followed by an inter-map cell-swapping heuristic to adjust the realizations to fit multiple spatial statistics. The inter-map cell-swapping heuristic allows spatial data uncertainty modelers to choose the appropriate probability model and weighted multiple spatial statistics which best represent errors caused by map generalization. This article also presents a lag-based measure to better represent gradient within a SDUM. This article covers the inter-map cell-swapping heuristic as well as both probability and spatial statistical models in detail.     

Multiple Point Metrics to Assess Categorical Variable Models

Geostatistical models should be checked to ensure consistency with conditioning data and statistical inputs. These are minimum acceptance criteria. Often the first and second-order statistics such as the histogram and variogram of simulated geological realizations are compared to the input parameters to check the reasonableness of the simulation implementation. Assessing the reproduction of statistics beyond second-order is often not considered because the “correct” higher order statistics are rarely known. With multiple point simulation (MPS) geostatistical methods, practitioners are now explicitly modeling higher-order statistics taken from a training image (TI). This article explores methods for extending minimum acceptance criteria to multiple point statistical comparisons between geostatistical realizations made with MPS algorithms and the associated TI. The intent is to assess how well the geostatistical models have reproduced the input statistics of the TI; akin to assessing the histogram and variogram reproduction in traditional semivariogram-based geostatistics. A number of metrics are presented to compare the input multiple point statistics of the TI with the statistics of the geostatistical realizations. These metrics are (1) first and second-order statistics, (2) trends, (3) the multiscale histogram, (4) the multiple point density function, and (5) the missing bins in the multiple point density function. A case study using MPS realizations is presented to demonstrate the proposed metrics; however, the metrics are not limited to specific MPS realizations. Comparisons could be made between any reference numerical analogue model and any simulated categorical variable model.