A correction model for conditional bias in selective mining operations

A nonlinear correction functionK(Z*) is proposed to transform any initial linear grade estimatorZ* into a conditional unbiased estimatorZ**=K(Z*) with reduced conditional estimation variance. Such a corrected estimator allows more accurate prediction of ore reserves at any level of selection performed during the mine lifetime. The correction is based upon an analytical or isofactorial representation of a bivariate distribution model of true gradeZ and its estimatorZ*. This correction model allows derivation of conditional estimation variances for both estimatorsZ* andZ** and provides a solution to the problem of change of support. A case study is presented and performance of the proposed correction model is evaluated in terms of actual conditional bias and mean squared errors. Results obtained stress the practical importance of the correction model in selective mining operations.     

Bootstrapping: A Fast Way to Simulate QTz Curves

The aim of this paper is to present a fast method based on bootstrapping, for simulating recoverable reserves for input to financial Monte Carlo simulations. In mining, the three parameters defining recoverable reserves are the cutoff grade, z, the ore tonnage above cutoff, T, and the metal quantity above cutoff, Q. After introducing the concept of 3-dimensional QTz curves, the statistical technique called bootstrapping is reviewed and applied to a set of South African gold grades. As selective mining is carried out on blocks not points, these curves have to be calculated for blocks. The QTz curves obtained by bootstrapping are compared to those obtained by conditionally simulating the same deposit. The procedure has been extended to incorporate geologists' ideas of the likely size of the ore volume. Lastly, the recoverable reserves obtained by bootstrapping are compared with those obtained by traditional risk analysis (base case ± 10% or 20%).     

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.     

Change of Support for Estimating Local Block Grade Distributions

An important aspect in mineral resource evaluation is the reduction of variance when post-processing the grade distributions defined on the support (volume) of the available data into distributions defined on the support of the proposed selective mining units. Although the volume-variance relationship is well understood for the estimation of global grade distributions, it is still an unsolved issue for local estimation studies based on non-parametric geostatistical methods, such as indicator kriging, for which the support correction is not inherent to the method. To clarify this relationship, the local change of support problem is examined in the scope of two parametric models (multi-Gaussian and discrete Gaussian models). It is shown that the variance reduction factor between point and block-support local distributions depends on the block being considered and is less than the global variance reduction factor. As a consequence, post-processing the local point-support grade distributions on the basis of the latter systematically understates the importance of the change of support at the local scale and makes selective mining appear more economically attractive than it really is. In the light of these results, a methodology is proposed to post-process the local point-support distributions obtained via non-parametric (indicator) methods into block-support distributions. An application to simulated data indicates that this methodology provides an accurate estimation at the block support when dealing with diffusion-type random fields.     

地质统计学储量估算中块尺寸的合理选择

在矿产勘查、矿山设计、生产及管理等各阶段都需要进行资源储量估算的工作。地质统计学方法相对于传统储量估算方法具有明显优势,在资源储量估算中获得了广泛的应用。估值过程中单元块尺寸的合理选择对储量估算结果至关重要,然而,在以往实践中块尺寸通常根据经验或主观判断进行选择,缺乏科学依据。本文在总结前人工作及地质统计学资源储量估算经验的基础上,将块尺寸的选取问题明确地划分为勘探阶段和生产阶段。针对不同阶段的特点,重点讨论了块尺寸与品位、储量、估计精度、选别开采单元的关系,为块尺寸的合理选择提供参考。并以某特大型钼矿为工程背景,进行了深度解析及数据实验,得出了块尺寸不仅影响资源品位分布、矿石量而且影响开采技术经济指标的结论,综合考虑估计精度和选别开采单元,推荐该矿床选择15m×15m×15m的单元块尺寸进行储量估算。     

地质统计学在多金属矿床储量计算中的应用研究

借助国际矿业软件Surpac建立了矿床地质数据库,筛选出矿体内钻孔样品数据并对其进行组合.根据组合样的统计分布特征,应用地质统计学理论建立了品位参数变异规律的数学模型,采用加权多项式回归法自动拟合求得实验变异函数,经交叉验证,优选出理想的实验变异函数参数值.根据理想的实验变异函数参数值,采用普通克立格法对矿体金属元素品位进行了估值,运用估值结果分别按各金属元素和中段进行了储量计算,并与矿山实际勘探获得的储量进行了对比.结果表明,所建模型可靠,计算结果准确,可用于辅助地质及采矿工程师进行资源评估、采矿设计及计划编制等工作.     

An Alternative Measure of the Reliability of Ordinary Kriging Estimates

This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.     

克里格法在离子吸附型稀土矿勘查储量估算中的应用

我国在离子吸附型稀土矿勘查工作中,一般采用地质块段法估算储量,块段法是将矿体划分为不同厚度的块段投影到水平或垂直方向上,块段的划分、各块段的面积和厚度、品位都会影响储量估算结果。本文以赣南某离子吸附型稀土矿床作为研究对象,基于先期勘探钻孔数据资料,运用三维建模软件创建了该矿床钻孔数据库,建立了矿区内矿体的三维DTM模型;采用克里格法对矿体进行稀土氧化物品位分析,将克里格法的储量计算结果与块段法的储量计算结果作对比分析。结果显示,克里格法计算的矿体体积比块段法增加了11.8%,稀土氧化物储量增加了15%,与实际勘探数据相比较,克里格法的计算结果基本合理,且具有快速、准确、方便的特点。本文利用自主开发的以克里格法为基础的三维数字矿山经济评价系统中价格-边界品位敏感性分析模块,动态设置边界品位,灵活圈定不同价格下经济可采的矿体边界,如当精矿的市场价格从10万元/吨变化为12万元/吨时,通过计算获得了此矿山经济可采矿体的空间扩展范围。基于克里格法的三维估算系统能够帮助矿山选择合理的采矿工程布置,有利于满足矿山动态管理的需要以及保证矿产资源的合理利用。     

Determining the best search neighbourhood in reserve estimation, using geostatistical method: A case study anomaly No 12A iron deposit in central Iran

Ordinary kriging and non-linear geostatistical estimators are now well accepted methods in mining grade control and mine reserve estimation. In kriging, the search volume or ‘kriging neighbourhood’ is defined by the user. The definition of the search space can have a significant impact on the outcome of the kriging estimate. In particular, too restrictive neighbourhood, can result in serious conditional bias. Kriging is commonly described as a ‘minimum variance estimator’ but this is only true when the neighbourhood is properly selected. Arbitrary decisions about search space are highly risky. The criteria to consider when evaluating a particular kriging neighbourhood are the slope of the regression of the ‘true’ and ‘estimated’ block grades, the number of kriging negative weights and the kriging variance. Search radius is one of the most important parameters of search volume which often is determined on the basis of influence of the variogram. In this paper the above-mentioned parameters are used to determine optimal search radius.     

Determination of optimum cutoff grade with considering dilution

In open pit mining, cutoff grade is one of the most important factors in production planning, which is simply defined as a grade that discriminates between ore and waste. It is also a sensitive parameter can have a major impact on the net present value and cash flow of the projects. On the other hand, dilution is one of the most important and sensitive parameters in the mining projects, which is closely related to the cutoff grade. Choosing the optimum cutoff grade is of considerable importance, since it has a significant impact on the mining operations. One of the most popular algorithms for determination of the optimum cutoff grade is the Lane’s algorithm. But in the Lane’s algorithm, mining dilution and its cost is not considered during the cutoff grade optimization. In this paper, effects of dilution on the cutoff grade are studied using Lane’s theory. Dilution and its cost is inserted directly into cutoff grade optimization process. The cutoff grades obtained using suggested method will be more realistic rather than ones by using the original form of the Lane’s formulation. Results of the study showed that with an increase of dilution, average grade decreases and consequently the cutoff grade increases. As a result of dilution, the quantity Q _m increases and the quantities Q _c and Q _r decrease. Therefore, the annual profit and NPV of project is very significantly reduced.     

Grade Control Classification of Ore and Waste: A Critical Review of Estimation and Simulation Based Procedures

For a mining operation, minimizing misclassification of ore and waste during grade control is of critical importance. This paper reviews grade control procedures for classifying ore and waste that are based on estimation and conditional simulation approaches. The similarities and differences of the procedures are first discussed in the light of an example. Subsequently, several case studies, including actual operating mines, are presented. The theoretical review shows that conditional simulation is generally better than estimation for ore and waste classification, especially when the profit value of a block is a nonlinear function of the grade of the block. The case studies, however, indicate that, in the context of daily grade control, the value of conditional simulation over good estimation is relatively low compared to the value of improving general practices at mine sites.     

Restricted kriging for mixture of grade models

A modified type of kriging, referred to as restricted kriging (RK), is proposed in this study. The method incorporates constraints on different grade classes to restrict the influence of the samples having different likelihoods in estimation. RK is motivated by the estimation of mineral reserves when grades have highly skewed distributions. Ordinary kriging tends to produce an overly smoothed interpolated surface by underestimating high grades and overestimating low grades. The fact that ordinary kriging gives a uniform prior treatment to all samples independent of their values is a major factor associated with this smoothing effect. The new approach differentiates each grade portion by preselected cutoffs. RK is developed for a single cutoff and then extended into a general form for any finite number of cutoffs. Restricted cokriging (RCK) is also formulated to simultaneously estimate a set of random functions with restriction conditions. Methods are suggested for determination of the probabilities of occurrence of different grade portions. Finally, the new approach is demonstrated on a case study of an epithermal gold deposit.     

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.     

CALCULATION AND PRACTICAL SIGNIFICANCE OF CATEGORY C2 GEOLOGIC RESERVES OF ORE DEPOSITS

Early ideas of rapid decrease in ore in grade and reserves with depth has led to considerable pessimism on calculation of C₂ reserves. Increased efficiency of mining methods, technology, and equipment has made it possible to mine many of the larger low grade deposits, lower grade deposits, and lower grade fringe deposits thereby greatly increasing the life of mines and mining districts. Most old mining districts have had productive lives far beyond known reserves at any one time. Ore zoning, except for unusual type of deposits, only rarely is of much practical importance in considering the depths of deposits. Geologic ore reserves of the C₂ category have been greatly underestimated in Russia. This category is not used for planning the immediate development of deposits and as a result there are many errors of planning and development. C₂ reserves should be calculated much more realistically than the present conservative practice and the results should be used in the proper economic development of ore deposits. - - J. A. Redden.     

Estimation of commercial diamond grades based on microdiamonds: a case study of the Koidu diamond mine,Sierra Leone

This paper documents the application of a microdiamond-based approach to the estimation of diamond grade in the Pipe 1 kimberlite at the Koidu mine in Sierra Leone. A geological model of Pipe 1 was constructed to represent the distribution and volume of the dominant kimberlite units within the pipe. Bulk samples, along with representative microdiamond samples, were collected from these units at surface and were used to define the ratio between microdiamond stone frequency (+212 μm stones per kilogram) and recoverable macrodiamond grade (+1.2 mm carats per tonne; 1 carat = 0.2 g). These ratios were applied to a comprehensive, spatially representative microdiamond sample dataset and were combined with a spatial model of country-rock xenolith dilution within the pipe to estimate +1.2 mm recoverable grades. The resource estimate was reconciled with subsequent production results in the elevation range 160 to 100 m above sea level. Production results for each of the six 10 m benches covering this elevation range were compared to the estimated average grades for these zones in the pipe. For the five cases where most of the kimberlite mass on a given bench is represented in the production data, the results show a maximum discrepancy of 6% between predicted and reported production grade with no indication of any consistent bias. This indicates that, when supported by a sound geological model and suitable microdiamond and macrodiamond data, the microdiamond-based estimation approach can provide reliable constraints on macrodiamond grade, even in the case of geologically complex bodies such as Koidu Pipe 1.

     

Comparison of ordinary and disjunctive kriging in an Indian iron ore deposit

This paper presents the results of disjunctive kriging applied to a supergene iron ore deposit of Bailadila Range of India. Disjunctive kriging is applied firstly to compare estimates of the blocks by ordinary kriging and secondly to estimate benchwise local recoverable reserves of the orebody. Good agreement exists between block estimates by ordinary kriging and disjunctive kriging except for peripheral blocks with less borehole information. Estimation of benchwise reserves shows that the behavior of the distribution of grades is different in various benches. The study shows that disjunctive kriging can be applied successfully for estimation of local recoverable reserves in the case of a good grade hematite iron ore deposit.     

基于三维矿化域模型的泥河铁矿床动态储量估算

基于泥河铁矿床矿体地质特征的详细研究,本文结合边界品位指标以及样条曲线方法对矿体边界进行简化平滑处理,并通过对各剖面的矿体解译边界进行圆滑渐变处理建立控制矿化边界的矿化域模型.基于矿化域模型,用于储量估算的样品分析数据具有更好的连续性和全面性,避免了双指标圈矿带来的矿体形态过度复杂和在三维空间内不可避免的空间占位现象.基于矿化域模型进行的储量估算,可以更快速、合理地获取矿体品位的空间分布特征,从而提高金属矿床储量估算结果的准确性和合理性.     

On the estimation of scale of fluctuation in geostatistics

Describing how soil properties vary spatially is of particular importance in stochastic analyses of geotechnical problems, because spatial variability has a significant influence on local material and global geotechnical response. In particular, the scale of fluctuation θ is a key parameter in the correlation model used to represent the spatial variability of a site through a random field. It is, therefore, of fundamental importance to accurately estimate θ in order to best model the actual soil heterogeneity. In this paper, two methodologies are investigated to assess their abilities to estimate the vertical and horizontal scales of fluctuation of a particular site using in situ cone penetration test (CPT) data. The first method belongs to the family of more traditional approaches, which are based on best fitting a theoretical correlation model to available CPT data. The second method involves a new strategy which combines information from conditional random fields with the traditional approach. Both methods are applied to a case study involving the estimation of θ at three two-dimensional sections across a site and the results obtained show general agreement between the two methods, suggesting a similar level of accuracy between the new and traditional approaches. However, in order to further assess the relative accuracy of estimates provided by each method, a second numerical analysis is proposed. The results confirm the general consistency observed in the case study calculations, particularly in the vertical direction where a large amount of data are available. Interestingly, for the horizontal direction, where data are typically scarce, some additional improvement in terms of relative error is obtained with the new approach.     

多金属矿床工业指标的制定-以中国北方某特大型铜钼矿床为例

多金属矿床工业指标特征的研究表明, 不宜采用单一金属矿床工业指标的制定方法, 必须考虑各种有用金属综合回收的总价值, 制定组合工业指标, 并以此进行各种品位指标的计算。以中国北方某特大型铜钼矿床为例, 介绍了其多金属矿床边界品位、块段最低工业品位、矿区最低平均品位、最小可采厚度和夹石剔除厚度等工业指标的制定过程, 并指出随着企业内部条件和外部环境的变化, 工业指标应是一个动态的标准, 可以修定和优化原有的工业指标。应重视和推广应用地质统计学方法计算储量, 依据市场的变化实行储量和工业指标的动态管理。     

矿石体重回归分析在新疆乌宗布拉克矿区的应用研究

在资源储量估算的过程中,矿石小体重的准确与否将直接影响到资源储量估算的客观程度。对矿石小体重的准确预测将需要通过大量的统计分析工作,传统的方法是利用矿石小体重的算术平均值进行估算矿床资源储量,并没有考虑到多矿种对矿石小体重的影响程度。本文利用Excel软件"数据分析"的"回归"功能模块,对实验室所测定的矿石小体重值与其对应品位进行二元线性回归分析,快速、准确地构建矿石体重与其品位的数学模型,从而为资源储量估算提供了更客观、更科学的矿石体重模型。