Fuzzy Modeling for Reserve Estimation Based on Spatial Variability

This article addresses a new reserve estimation method which uses fuzzy modeling algorithms and estimates the reserve parameters based on spatial variability. The proposed fuzzy modeling approach has three stages: (1) Structure identification and preliminary clustering, (2) Variogram analysis, and (3) Clustering based rule system. A new clustering index approach and a new spatial measure function (point semimadogram) are proposed in the paper. The developed methodology uses spatial variability in each step and takes the fuzzy rules from input-output data. The model has been tested using both simulated and real data sets. The performance evaluation indicates that the new methodology can be applied in reserve estimation and similar modeling problems     

Comparing sequential Gaussian and turning bands algorithms for cosimulating grades in multi-element deposits

Stochastic simulation is increasingly used to map the spatial variability in the grades of elements of interest and to assess the uncertainty in the mineral resources and ore reserves. The practical implementation requires specifying a stochastic model, which describes the spatial distribution of the grades, and an algorithm to construct realizations of these grades, viewed as different possible outcomes or scenarios. In the case of the Gaussian random field model, a variety of algorithms have been proposed in the past decades, but their ability to reproduce the model statistics is often unequal. In this paper, we compare two such algorithms, namely the turning bands and the sequential algorithms. The comparison is hold through a synthetic case study and a real case study in a porphyry copper deposit located in southeastern Iran, in which it is of interest to jointly simulate the copper, molybdenum, silver, lead and zinc grades. Statistical testing and graphical validations are realized to check whether or not the realizations reproduce the features of the true grades, in particular their direct and cross variograms. Sequential simulation based on collocated cokriging turns out to poorly reproduce the cross variograms, while turning bands proves to be accurate in all the analyzed cases.     

Digital simulation of multivariate two- and three-dimensional stochastic processes with a spectral turning bands method

The space domain version of the turning bands method can simulate multidimensional stochastic processes (random fields) having particular forms of covariance functions. To alleviate this limitation a spectral representation of the turning bands method in the two-dimensional case has shown that the spectral approach allows simulation of isotropic two-dimensional processes having any covariance or spectral density function. The present paper extends the spectral turning bands method (STBM) even further for simulation of much more general classes of multidimensional stochastic processes. Particular extensions include: (i) simulation of three-dimensional processes using STBM, (ii) simulation of anisotropic two- or three-dimensional stochastic processes, (iii) simulation of multivariate stochastic processes, and (iv) simulation of spatial averaged (integrated) processes. The turning bands method transforms the multidimensional simulation problem into a sum of a series of one-dimensional simulations. Explicit and simple expressions relating the cross-spectral density functions of the one-dimensional processes to the cross-spectral density function of the multidimensional process are derived. Using such expressions the one-dimensional processes can be simulated using a simple one-dimensional spectral method. Examples illustrating that the spectral turning bands method preserves the theoretical statistics are presented. The spectral turning bands method is inexpensive in terms of computer time compared to other multidimensional simulation methods. In fact, the cost of the turning bands method grows as the square root or the cubic root of the number of points simulated in the discretized random field, in the two- or three-dimensional case, respectively, whereas the cost of other multidimensional methods grows linearly with the number of simulated points. The spectral turning bands method currently is being used in hydrologic applications. This method is also applicable to other fields where multidimensional simulations are needed, e.g., mining, oil reservoir modeling, geophysics, remote sensing, etc.     

Simulation of mineral grades with hard and soft conditioning data: application to a porphyry copper deposit

This work deals with the geostatistical simulation of mineral grades whose distribution exhibits spatial trends within the ore deposit. It is suggested that these trends can be reproduced by using a stationary random field model and by conditioning the realizations to data that incorporate the available information on the local grade distribution. These can be hard data (e.g., assays on samples) or soft data (e.g., rock-type information) that account for expert geological knowledge and supply the lack of hard data in scarcely sampled areas. Two algorithms are proposed, depending on the kind of soft data under consideration: interval constraints or local moment constraints. An application to a porphyry copper deposit is presented, in which it is shown that the incorporation of soft conditioning data associated with the prevailing rock type improves the modeling of the uncertainty in the actual copper grades.     

基于DOA估计的阵列式探地雷达逆向投影目标成像方法

本文针对阵列式天线探地雷达系统的目标成像问题,理论模拟分析了最小二乘(LS)估计、Capon估计、幅度相位估计(APES)三种波束形成算法在目标回波到达角(direction of arrival,DOA)估计上的效果和精度,提出了基于DOA估计的阵列式探地雷达逆向投影目标成像方法。该方法综合利用多输入多输出(MIMO)阵列信号估计得到的目标回波信号DOA和阵列空间观测信息对目标体进行成像,通过成像点空间扫描对各测点估计DOA幅度值在成像点的加权积分进行目标定位及反射强度估计。该方法实现简单高效,可以广泛应用于地下简单目标体的快速成像。     

Estimation of base settlement from the surface subsidence profile: Plane‐field of displacements

According to Litwiniszyn's theory, subsidence over a yielding underground geo‐structure is seen as a stochastic (Markov) process. This theory leads to a single, linear parabolic differential equation of diffusion–convection type (D–C equation) in the plane‐field of displacements. If the boundary conditions for the governing D–C equation are prescribed along the shear bands, i.e. at ‘moving’ boundaries—it has been observed from small‐scale model experiments that the subsiding process is always confined between a set of inclined shear bands—then the resulting equation is nonlinear. The inverse problem for this nonlinear equation, i.e. the problem of determining the base displacement using the surface subsidence as ‘initial’ condition, is ill‐posed and estimation of the base displacement from a given surface subsidence profile is not possible. In the present paper the domain of integration of the governing D–C equation is fixed (and bounded)—the boundaries are not evolving. Hence, the governing equation remains linear parabolic. The advantage is that this linear differential equation admits an analytical solution, under the trap‐door mechanism assumption, that enables a direct solution to the inverse problem. Copyright © 2008 John Wiley & Sons, Ltd.     

Forecasting Recoverable Ore Reserves and Their Uncertainty at Morila Gold Deposit, Mali: An Efficient Simulation Approach and Future Grade Control Drilling

Forecasting of recoverable reserves aims to predict the tonnages and grades that will be recovered at the time of mining. The main concern in this forecasting is the imprecision in the selection of ore/waste resulting from both the so-called information effect or information that becomes available during grade control, and the support effect or mining selectivity during mining. Existing approaches to recoverable reserve estimation account for mining selectivity; however, they largely ignore the information effects from future data becoming available through grade control practices. An application at the Morila gold deposit, Mali, is utilized in this paper to document a new simulation-based approach for recoverable reserve forecasting that incorporates the potential effects of future grade control data. This accounts for the information effect as well as changes in data quantity and quality over time. In addition, the case study at the Morila mine elucidates the use of a newer, very efficient, and practical alternative to traditional simulation techniques. This direct block simulation method forecasts recoverable reserves directly into the selective mining unit (support) size under consideration. The case study demonstrates the practical uncertainty assessment of the recoverable reserves within the deposit, so that expected inaccuracies in the selection of ore /waste can be accounted for. This allows for fully informed mining decisions to be made that incorporate the effects of information and selectivity while quantifying the potential impact of uncertainty on the mine operation and its final economic outcome.     

Preliminary Characterisation of New Reference Materials for Microanalysis: Chinese Geological Standard Glasses CGSG‐1, CGSG‐2, CGSG‐4 and CGSG‐5

Four silicate glasses were prepared by the fusion of about 1 kg powder each of a basalt, syenite, soil and andesite to provide reference materials of natural composition for microanalytical work. These glasses are referred to as ‘Chinese Geological Standard Glasses’ (CGSG) ‐1, ‐2, ‐4 and ‐5. Micro and bulk analyses indicated that the glasses are well homogenised with respect to major and trace elements. Some siderophile/chalcophile elements (e.g., Sn, Pt, Pb) may be heterogeneously distributed in CGSG‐5. This paper provides the first analytical data for the CGSG reference glasses using a variety of analytical techniques (wet chemistry, XRF, EPMA, ICP‐AES, ICP‐MS, LA‐ICP‐MS) performed in nine laboratories. Most data agree within uncertainty limits of the analytical techniques used. Discrepancies in the data for some siderophile/chalcophile elements exist, mainly because of possible heterogeneities of these elements in the glasses and/or analytical problems. From the analytical data, preliminary reference and information values for fifty‐five elements were calculated. The analytical uncertainties [2 relative standard error (RSE)] were estimated to be between about 1% and 20%.     

Risk quantification with combined use of lithological and grade simulations: Application to a porphyry copper deposit

The uncertainty in the recoverable tonnages and grades in a mineral deposit is a key factor in the decision-making process of a mining project. Currently, the most prevalent approach to model the uncertainty in the spatial distribution of mineral grades is to divide the deposit into domains based on geological interpretation and to predict the grades within each domain separately. This approach defines just one interpretation of the geological domain layout and does not offer any measure of the uncertainty in the position of the domain boundaries and in the mineral grades. This uncertainty can be evaluated by use of geostatistical simulation methods. The aim of this study is to evaluate how the simulation of rock type domains and grades affects the resources model of Sungun porphyry copper deposit, northwestern Iran. Specifically, three main rock type domains (porphyry, skarn and late-injected dykes) that control the copper grade distribution are simulated over the region of interest using the plurigaussian model. The copper grades are then simulated in cascade, generating one grade realization for each rock type realization. The simulated grades are finally compared to those obtained using traditional approaches against production data.     

An Exploration of the Interplay between the Measurement Uncertainty and the Number of Samples in Contaminated Land Investigations

In the assessment of potentially contaminated land, the number of samples and the uncertainty of the measurements (including that from sampling) are both important factors in the planning and implementation of an investigation. Both parameters also effect the interpretation of the measurements produced, and the process of making decisions based upon those measurements. However, despite their importance, previously there has been no method for assessing if an investigation is fit‐for‐purpose with respect to both of these parameters. The Whole Site Optimised Contaminated Land Investigation (WSOCLI) method has been developed to address this issue, and to allow the optimisation of an investigation with respect to both the number of samples and the measurement uncertainty, using an economic loss function. This function was developed to calculate an ‘expectation of (financial) loss’, incorporating costs of the investigation itself, subsequent land remediation, and potential consequential costs. To allow the evaluation of the WSOCLI method a computer program ‘OCLISIM’ has been developed to produce sample data from simulated contaminated land investigations. One advantage of such an approach is that as the ‘true’ contaminant concentrations are created by the program, these values are known, which is not the case in a real contaminated land investigation. This enables direct comparisons between functions of the ‘true’ concentrations and functions of the simulated measurements. A second advantage of simulation for this purpose is that the WSOCLI method can be tested on many different patterns and intensities of contamination. The WSOCLI method performed particularly well at high sampling densities producing expectations of financial loss that approximated to the true costs, which were also calculated by the program. WSOCLI was shown to produce notable trends in the relationship between the overall cost (i.e., expectation of loss) and both the number of samples and the measurement uncertainty, which are: (a) low measurement uncertainty was optimal when the decision threshold was between the mean background and the mean hot spot concentrations. (b) When the hot spot mean concentration is equal to or near the decision threshold, then mid‐range measurement uncertainties were optimal. (c) When the decision threshold exceeds the mean of the hot spot, mid‐range measurement uncertainties were optimal. The trends indicate that the uncertainty may continue to rise if the difference between hot spot mean and the decision threshold increases further. (d) In any of the above scenarios, the optimal measurement uncertainty was lower if there is a large geochemical variance (i.e., heterogeneity) within the hot spot. (e) The optimal number of samples for each scenario was indicated by the WSOCLI method, and was between 50 and 100 for the scenarios considered generally; although there was significant noise in the predictions, which needs to be addressed in future work to allow such conclusions to be clearer.     

Entropy and spatial disorder

The majority of geostatistical estimation and simulation algorithms rely on a covariance model as the sole characteristic of the spatial distribution of the attribute under study. The limitation to a single covariance implicitly calls for a multivariate Gaussian model for either the attribute itself or for its normal scores transform. The Gaussian model could be justified on the basis that it is both analytically simple and it is a maximum entropy model, i.e., a model that minimizes unwarranted structural properties. As a consequence, the Gaussian model also maximizes spatial disorder (beyond the imposed covariance) which can cause flow simulation results performed on multiple stochastic images to be very similar; thus, the space of response uncertainty could be too narrow entailing a misleading sense of safety. The ability of the sole covariance to adequately describe spatial distributions for flow studies, and the assumption that maximum spatial disorder amounts to either no additional information or a safe prior hypothesis are questioned. This paper attempts to clarify the link between entropy and spatial disorder and to provide, through a detailed case study, an appreciation for the impact of entropy of prior random function models on the resulting response distributions.     

Entropy and spatial disorder

The majority of geostatistical estimation and simulation algorithms rely on a covariance model as the sole characteristic of the spatial distribution of the attribute under study. The limitation to a single covariance implicitly calls for a multivariate Gaussian model for either the attribute itself or for its normal scores transform. The Gaussian model could be justified on the basis that it is both analytically simple and it is a maximum entropy model, i.e., a model that minimizes unwarranted structural properties. As a consequence, the Gaussian model also maximizes spatial disorder (beyond the imposed covariance) which can cause flow simulation results performed on multiple stochastic images to be very similar; thus, the space of response uncertainty could be too narrow entailing a misleading sense of safety. The ability of the sole covariance to adequately describe spatial distributions for flow studies, and the assumption that maximum spatial disorder amounts to either no additional information or a safe prior hypothesis are questioned. This paper attempts to clarify the link between entropy and spatial disorder and to provide, through a detailed case study, an appreciation for the impact of entropy of prior random function models on the resulting response distributions.     

Selection of coherent deposit-type locations and their application in data-driven mineral prospectivity mapping

Data-driven prospectivity mapping can be undermined by dissimilarity in multivariate spatial data signatures of deposit-type locations. Most cases of data-driven prospectivity mapping, however, make use of training sets of randomly selected deposit-type locations with the implicit assumption that they are coherent (i.e., with similar multivariate spatial data signatures). This study shows that the quality of data-driven prospectivity mapping can be improved by using a training set of coherent deposit-type locations. Analysis and selection of coherent deposit-type locations was performed via logistic regression, by using multiple sets of deposit occurrence favourability scores of univariate geoscience spatial data as independent variables and binary deposit occurrence scores as dependent variable. The set of coherent deposit-type locations and three sets of randomly selected deposit-type locations were each used in data-driven prospectivity mapping via application of evidential belief functions. The prospectivity map based on the training set of coherent deposit-type locations resulted in lower uncertainty, better goodness-of-fit to the training set, and better predictive capacity against a cross-validation set of economic deposits of the type sought. This study shows that explicit selection of training set of coherent deposit-type locations should be applied in data-driven prospectivity mapping.     

Imprecise (fuzzy) information in geostatistics

A methodology based on fuzzy set theory for the utilization of imprecise data in geostatistics is presented. A common problem preventing a broader use of geostatistics has been the insufficient amount of accurate measurement data. In certain cases, additional but uncertain (soft) information is available and can be encoded as subjective probabilities, and then the soft kriging method can be applied (Journel, 1986). In other cases, a fuzzy encoding of soft information may be more realistic and simplify the numerical calculations. Imprecise (fuzzy) spatial information on the possible variogram is integrated into a single variogram which is used in a fuzzy kriging procedure. The overall uncertainty of prediction is represented by the estimation variance and the calculated membership function for each kriged point. The methodology is applied to the permeability prediction of a soil liner for hazardous waste containment. The available number of hard measurement data (20) was not enough for a classical geostatistical analysis. An additional 20 soft data made it possible to prepare kriged contour maps using the fuzzy geostatistical procedure.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Kalman filter finite element method applied to dynamic ground motion

The purpose of this paper is to investigate the estimation of dynamic elastic behavior of the ground using the Kalman filter finite element method. In the present paper, as the state equation, the balance of stress equation, the strain–displacement equation and the stress–strain equation are used. For temporal discretization, the Newmark ¼ method is employed, and for the spatial discretization the Galerkin method is applied. The Kalman filter finite element method is a combination of the Kalman filter and the finite element method. The present method is adaptable to estimations not only in time but also in space, as we have confirmed by its application to the Futatsuishi quarry site. The input data are the measured velocity, acceleration, etc., which may include mechanical noise. It has been shown in numerical studies that the estimated velocity, acceleration, etc., at any other spatial and temporal point can be obtained by removing the noise included in the observation. Copyright © 2008 John Wiley & Sons, Ltd.     

Grade estimation using fuzzy- set algorithms

This paper presents a new approach for estimating unknown ore grades within a mining deposit in a fuzzy environment using fuzzy c- means clustering and a fuzzy inference system. Based on a collection of cluster centers obtained from fuzzy c- means, a fuzzy rule base and fuzzy search domains are established to compute grades at these cluster centers. These cluter center- grade pairs act as control information in the fuzzy space- grade system in order to infer unknown grades on the basis of fuzzy interpolation, fuzzy extrapolation, and a defuzzification process of fuzzy control.     

Variogram Model Selection via Nonparametric Derivative Estimation

Before optimal linear prediction can be performed on spatial data sets, the variogram is usually estimated at various lags and a parametric model is fitted to those estimates. Apart from possible a priori knowledge about the process and the user's subjectivity, there is no standard methodology for choosing among valid variogram models like the spherical or the exponential ones. This paper discusses the nonparametric estimation of the variogram and its derivative, based on the spectral representation of positive definite functions. The use of the estimated derivative to help choose among valid parametric variogram models is presented. Once a model is selected, its parameters can be estimated—for example, by generalized least squares. A small simulation study is performed that demonstrates the usefulness of estimating the derivative to help model selection and illustrates the issue of aliasing. MATLAB software for nonparametric variogram derivative estimation is available at http://www-math.mit.edu/~gorsich/derivative.html. An application to the Walker Lake data set is also presented.     

BAM‐S005 Type A and B: New Silicate Reference Glasses for Microanalysis

To test whether the silicate reference glasses BAM‐S005‐A and BAM‐S005‐B from BAM (The Federal Institute for Materials Research and Testing, Germany) are suitable materials for microanalysis, we investigated the homogeneity of these reference glasses using the microanalytical techniques EPMA, LA‐ICP‐MS and SIMS. Our study indicated that all major and most trace elements are homogeneously distributed at micrometre sampling scale in both types of glass. However, some trace elements (e.g., Cs, Cl, Cr, Mo and Ni) seem to be inhomogeneously distributed. We also determined the composition of BAM‐S005‐A and BAM‐S005‐B. The EPMA data of major elements confirmed the information values specified by the certificate. With the exception of Sr, Ba, Ce and Pb, our trace element data by LA‐ICP‐MS were also in agreement with the certified values within the stated uncertainty limits. The reasons for the discrepancy in these four elements are still unclear. In addition, we report new data for twenty‐two further trace elements, for which the concentrations were not certified. Based on our investigation, we suggest that both of these materials are suitable for many microanalytical applications.     

The Impact of Biased Sampling on the Estimation of the Semivariogram Within Fractured Media Containing Multiple Fracture Sets

Monte Carlo simulation was used to examine the error (statistical bias) introduced in estimating a sample semivariogram through application of oriented sampling patterns to variables which are correlated with fracture orientation. Sample semivariograms of the directional components of the water velocity were used to illustrate that oriented sampling schemes can provide biased data sets which result in error in the estimation of the semivariogram, particularly in the estimation of the sill (or variance). Three sampling patterns were used to analyze directional semivariograms of the components of the fluid velocity: sampling along lines parallel to the mean regional hydraulic gradient, sampling among lines perpendicular to the mean regional hydraulic gradient, and sampling along fracture segments. The first two sampling patterns were shown to introduce substantial error in the sills of the velocity variograms. It is argued that this error is due to the combination of unequal sampling of fractures with different orientations (i.e., sampling bias) and the systematic variation in the magnitude of the velocity components with orientation of the fracture. As a consequence, it is suggested that correction factors developed to correct fracture frequency statistics need to be extended to improve estimation of spatial moments of variables which are correlated with fracture orientation.     

Mapping soil erodibility from composed data set in Sele River Basin, Italy

Evaluation of soil erodibility is an important task for Mediterranean lands, in which fertility and crop yield are significantly affected by soil erosion. The soil physicochemical parameters affecting soil erodibility are highly variable in space and, as for many other environmental variables, sample measurements are generally not enough for assessing its spatial variability with an acceptable level of uncertainty at the scales of practical interest. This study illustrates the procedure applied for estimating the pattern of soil erodibility across the Sele Basin (Southern Italy), where soil properties have been measured on a limited number of sparse samples. Sampled data were integrated with other sparse data estimated by local regression functions, which relate soil erodibility to auxiliary variables, such as terrain attributes and land system class memberships. Sampled and estimated data were merged in a composed data set to assess the spatial pattern of soil erodibility by ordinary kriging. The proposed approach offers effective spatial predictions, and it is exportable to regions where financial costs for soil sampling are not feasible.