Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances

A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features that could not be generated with traditional kriging.     

Evaluating Thickness of Bauxite Deposit Using Indicator Geostatistics and Fuzzy Estimation

Evaluating the geological properties of a mineral deposit is a fundamental task for mine planning and it requires an assessment of reserve parameters such as thickness and grade. This paper presents a linguistic model for estimating bauxite thickness within a deposit in a fuzzy environment using indicator geostatistics and fuzzy modeling. The proposed model consists of two main stages: determining the orebody boundary and estimating the thickness. In order to estimate the thickness, a rule‐based fuzzy inference mechanism has been developed based on data statistics. Results and performance of the model have been compared with that of a well‐known conventional technique, geostatistics (kriging), and it is shown that the proposed model has bigger estimation power. In addition, the fuzzy approach is more flexible than the kriging approach. The fuzzy methodology used in the present paper is convenient for modeling reserve parameters.     

资源量估算的边界分析与矿化体圈定

资源量估算总是在特定的估算域中进行,其边界条件对估值结果有着显著的影响。传统方法采取用工业指标来圈定矿体并进行资源量估算的硬边界条件,过于强调了矿床的经济性,而对矿床的地质和地质统计学规律重视不够。本文介绍了国际上流行的资源量估算域的相关概念及边界分析方法,并通过实际案例来探讨其在资源量估算中的应用,在此基础上提出了利用三维软件进行矿(化)体圈定的基本原则。实践表明,利用软边界约束或圈定矿化体进行品位估计,可能更符合勘查阶段资源量估算的要求,进而降低估算域边界的不确定性带来的估值风险。     

Simple and Ordinary Multigaussian Kriging for Estimating Recoverable Reserves

Multigaussian kriging is used in geostatistical applications to assess the recoverable reserves in ore deposits, or the probability for a contaminant to exceed a critical threshold. However, in general, the estimates have to be calculated by a numerical integration (Monte Carlo approach). In this paper, we propose analytical expressions to compute the multigaussian kriging estimator and its estimation variance, thanks to polynomial expansions. Three extensions are then considered, which are essential for mining and environmental applications: accounting for an unknown and locally varying mean (local stationarity), accounting for a block-support correction, and estimating spatial averages. All these extensions can be combined; they generalize several known techniques like ordinary lognormal kriging and uniform conditioning by a Gaussian value. An application of the concepts to a porphyry copper deposit shows that the proposed “ordinary multigaussian kriging” approach leads to more realistic estimates of the recoverable reserves than the conventional methods (disjunctive and simple multigaussian krigings), in particular in the nonmineralized undersampled areas.     

Reducing the Impact of Outliers in Ore Reserves Estimation

Mining applications commonly faces surprising high values designated as outliers. These values impact dramatically statistical analysis and interpretation. A comprehensive analysis on the causes for the presence of unexpected high values was recommended. However, if an erroneous value was accepted as a part of the solution, some form of correction is recommended. A methodology based on the robust kriging (RoK) algorithm is proposed to be used in exploratory data analysis and also to deal with problems associated with the presence of outliers in the sample data set. The efficiency of RoK method as an interpolator is tested in different types of mineralizations. Importantly, the parent population from which the data was sampled is available, thus allowing direct quantitative assessment of the effectiveness of the estimation technique. The performance of the method is tested in the context of ore reserves estimation. RoK model is compared to models generated by ordinary kriging, median indicator kriging, and lognormal kriging. RoK proved to be more accurate and more precise than those methods reducing substantially the number of misclassified blocks.     

Matheronian geostatistics—Quo vadis?

Components of geostatistical estimation, developed as a method for ore deposit assessment, are discussed in detail. The assumption that spatial observations can be treated as a stochastic process is judged to be an inappropriate model for natural data. Problems of semivariogram formulation are reviewed, and this method is considered to be inadequate for estimating the function being sought. Characteristics of bivariate interpolation are summarized, highlighting kriging limitations as an interpolation method. Limitations are similar to those of inverse distance weighted observations interpolation. Attention is drawn to the local bias of kriging and misplaced claims that it is an “optimal” interpolation method. The so-called “estimation variance,” interpreted as providing confidence limits for estimation of mining blocks, is shown to be meaningless as an index of local variation. The claim that geostatistics constitutes a “new science” is examined in detail. Such novelties as exist in the method are shown to transgress accepted principles of scientific inference. Stochastic modeling in general is discussed, and purposes of the approach emphasized. For the purpose of detailed quantitative assessment it can provide only prediction qualified by hypothesis at best. Such an approach should play no part in ore deposit assessment where the need is for local detailed inventories; these can only be achieved properly through local deterministic methods, where prediction is purely deductive.     

Mapping Curvilinear Structures with Local Anisotropy Kriging

Interpolating geo-data with curvilinear structures using geostatistics is often disappointing. Channels, for example, become disconnected sets of lakes when interpolated from point data. In order to improve the interpolation of geological structures (e.g., curvilinear structures), we present a new form of kriging, local anisotropy kriging (LAK). Local anisotropy kriging combines a gradient algorithm from image analysis with kriging in an iterative way. After an initial standard kriging interpolation, the gradient algorithm determines the local anisotropy for each cell in the grid using a search area around the cell. Subsequently, kriging is carried out with the spatially varying anisotropy. The anisotropy calculation and subsequent kriging steps will then succeed until the result is satisfactory in the way of reproducing the curvilinear structures. Depending on the size of the search area more or less detail in the geological structures can be reproduced with LAK. Using test examples we show that LAK interpolates data with curvilinear structures more realistically than standard kriging. In a real world case, using bathymetric data of the Oosterschelde estuary, LAK also proves to be quantitatively superior to standard kriging. Absolute interpolation errors are decreased by 23%. Local anisotropy kriging only uses information from point data, which makes the method very objective, it only presents “what the data can tell.”     

Area Influence Kriging

This paper presents a modified ordinary kriging technique referred to as the “Area Influence Kriging” (AIK). The method is a simple and practical tool to use for more accurate prediction of global recoverable ore resources in any type of deposit. AIK performs well even in deposits with skewed grade distributions when the ordinary kriging (OK) results are unreasonably smooth. It is robust and globally unbiased like OK. The AIK method is not intended to replace OK, which is a better estimator of the average grade of the blocks. Rather it aims to complement OK with its excellent performance in predicting recoverable resources that have been the major pitfalls of OK in many resource estimation cases. The paper details the methodology of AIK with a couple of examples. It also reports the results from its application to a gold deposit.     

Conditional LAS stochastic simulation of regionalized variables in random fields

Ore reserves forecasts are required to aid in investment decisions, mine design and valuation, short and long term production plans and proper and efficient mill design. In random multivariable fields with limited data and high levels of uncertainty, the kriged block estimates produce a smoothing effect resulting in underestimating high values and overestimating low values. The modified conditional simulation (MCS) methodology solves these problems by simulating the random field to preserve its mean and the variance structure. The simulation model is conditioned to reproduce the data at known sample points to minimize the variability between the simulated data and the true field data. In this study, the authors develop the MCS methodology to simulate ore reserve grades using the best linear unbiased estimation (BLUE) and the local average subdivision (LAS) techniques. The MCS methodology is applied to simulate block grades in a section of the Sabi Gold Project in Zimbabwe. The results are compared with the kriged estimates for this section. Analysis of the results shows that the MCS methodology reproduces the known sample grades with minimum estimation error of 0.001 while the estimation error associated with the kriged estimates is 1.104, a 100% efficiency of the MCS method over the kriging technique.     

基于GMS软件的海砂矿体三维地质建模和资源量估算

本文以南海北部某海域海砂矿区为例,提出了基于GMS软件的海砂矿三维地质建模和资源量估算方法.以钻孔和综合物探数据为源数据,划定了矿区内V1和V2矿体的边界,基于GMS软件系统构建了数据库,综合利用钻孔、综合物探、剖面图和海底地形控制面等资料,建立了三维地质模型,有效实现了海砂矿体三维可视化和体积资源量的估算.三维地质模...     

Using geology to control geostatistics in the Hellyer deposit

The Hellyer orebody, a polymetallic massive sulfide deposit, was discovered in western Tasmania by Aberfoyle in 1983. Delineation diamond drilling was carried out in 1984 on a nominal 50-m square grid pattern to outline the resource. Resource estimation methods were influenced by the requirement to develop a regular block model for conceptual mine planning studies. Detailed geological interpretation indicates that the interpolation technique must take into account several important features to retain geological credibility. The deposit has sharp limits defined by visual geological contacts with virtually barren enclosing rocks. Lateral terminations are rapid with no interfingering internal waste. The dip and strike are variable and a major fault with a measurable displacement cuts acutely through the center of the deposit. Ore grades are reasonably correlateable within specific layers from hole to hole indicating a significant across-dip anisotropy. A hanging wall enriched zone is well-defined throughout the deposit. To overcome the variable geometry problems, a stratigraphic coordinate system was defined arbitrarily to replace the normal z coordinate. This allowed variography in stratigraphic layers. Blocks to be estimated were constrained by hand-drawn and subsequently digitized hanging wall and footwall contours. Each block was ascribed a stratigraphic coordinate by calculating its spatial position in relation to nearby stratigraphic unit boundaries within the massive sulfide body. Estimates were generated for each element by ordinary linear kriging. Despite the relatively sparse data in a large massive deposit, the customized technique developed for Hellyer has provided a reliable model of spatial grade distribution by combining conventional geostatistical methods with careful geological observation and interpretation. Some geometry problems remain which are the subject of ongoing studies.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Comparison of machine learning methods for copper ore grade estimation

In this study, machine learning methods such as neural networks, random forests, and Gaussian processes are applied to the estimation of copper grade in a mineral deposit. The performance of these methods is compared to geostatistical techniques, such as ordinary kriging and indicator kriging. To ensure that these comparisons are realistic and relevant, the predictive accuracy is estimated on test instances located in drill holes that are different from the training data. The results of an extensive empirical study in the Sarcheshmeh porphyry copper deposit in Southeastern Iran illustrate that specially designed Gaussian processes with a symmetric standardization of the spatial location inputs and an anisotropic kernel yield the most accurate predictions. Furthermore, significant improvements are obtained when, besides location, information on the rock type is included in the set of predictor variables. This observation highlights the importance of carrying out detailed studies of the geological composition of the deposit to obtain more accurate ore grade predictions.     

A Functional Data Analysis Approach to Surrogate Modeling in Reservoir and Geomechanics Uncertainty Quantification

Uncertainty quantification for geomechanical and reservoir predictions is in general a computationally intensive problem, especially if a direct Monte Carlo approach with large numbers of full-physics simulations is used. A common solution to this problem, well-known for the fluid flow simulations, is the adoption of surrogate modeling approximating the physical behavior with respect to variations in uncertain parameters. The objective of this work is the quantification of such uncertainty both within geomechanical predictions and fluid-flow predictions using a specific surrogate modeling technique, which is based on a functional approach. The methodology realizes an approximation of full-physics simulated outputs that are varying in time and space when uncertainty parameters are changed, particularly important for the prediction of uncertainty in vertical displacement resulting from geomechanical modeling. The developed methodology has been applied both to a subsidence uncertainty quantification example and to a real reservoir forecast risk assessment. The surrogate quality obtained with these applications confirms that the proposed method makes it possible to perform reliable time–space varying dependent risk assessment with a low computational cost, provided the uncertainty space is low-dimensional.     

矿化域模型在某铷矿床工业指标制定中的运用

以某铷矿为研究对象,依据收集的矿床地质勘查资料,研究了矿化域建模技术方案,应用MICROMINE软件建立了矿化域三维实体模型,并对矿化域进行估值,快速获得了不同边界指标的资源量。与传统手工计算资源量进行比较,结果表明,建立的矿化域模型合理,计算结果准确,其估算的不同边界指标方案的资源量可以用于本矿床工业指标制定。矿化域模型的运用,不仅提高了矿床资源量估算的准确性和合理性,而且大幅度提高了工业指标制定中矿体圈定和资源量估算工作效率。以某铷矿为研究对象,依据收集的矿床地质勘查资料,研究了矿化域建模技术方案,应用MICROMINE软件建立了矿化域三维实体模型,并对矿化域进行估值,快速获得了不同边界指标的资源量。与传统手工计算资源量进行比较,结果表明,建立的矿化域模型合理,计算结果准确,其估算的不同边界指标方案的资源量可以用于本矿床工业指标制定。矿化域模型的运用,不仅提高了矿床资源量估算的准确性和合理性,而且大幅度提高了工业指标制定中矿体圈定和资源量估算工作效率。     

Area Influence Kriging

This paper presents a modified ordinary kriging technique referred to as the Area Influence Kriging (AIK). The method is a simple and practical tool to use for more accurate prediction of global recoverable ore resources in any type of deposit. AIK performs well even in deposits with skewed grade distributions when the ordinary kriging (OK) results are unreasonably smooth. It is robust and globally unbiased like OK. The AIK method is not intended to replace OK, which is a better estimator of the average grade of the blocks. Rather it aims to complement OK with its excellent performance in predicting recoverable resources that have been the major pitfalls of OK in many resource estimation cases. The paper details the methodology of AIK with a couple of examples. It also reports the results from its application to a gold deposit.     

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.     

Conditioning of Multiple-Point Statistics Facies Simulations to Tomographic Images

Geophysical tomography captures the spatial distribution of the underlying geophysical property at a relatively high resolution, but the tomographic images tend to be blurred representations of reality and generally fail to reproduce sharp interfaces. Such models may cause significant bias when taken as a basis for predictive flow and transport modeling and are unsuitable for uncertainty assessment. We present a methodology in which tomograms are used to condition multiple-point statistics (MPS) simulations. A large set of geologically reasonable facies realizations and their corresponding synthetically calculated cross-hole radar tomograms are used as a training image. The training image is scanned with a direct sampling algorithm for patterns in the conditioning tomogram, while accounting for the spatially varying resolution of the tomograms. In a post-processing step, only those conditional simulations that predicted the radar traveltimes within the expected data error levels are accepted. The methodology is demonstrated on a two-facies example featuring channels and an aquifer analog of alluvial sedimentary structures with five facies. For both cases, MPS simulations exhibit the sharp interfaces and the geological patterns found in the training image. Compared to unconditioned MPS simulations, the uncertainty in transport predictions is markedly decreased for simulations conditioned to tomograms. As an improvement to other approaches relying on classical smoothness-constrained geophysical tomography, the proposed method allows for: (1) reproduction of sharp interfaces, (2) incorporation of realistic geological constraints and (3) generation of multiple realizations that enables uncertainty assessment.     

资源储量估算过程中矿业权边界的确定

在垂直纵投影图上如何确定矿业权边界投影位置,对准确圈定资源储量估算范围具有重要的现实意义。在充分研究资源储量估算过程的基础上,结合MapGIS、AutoCAD等常用地质绘图软件,通过构建矿体三维数学模型,进行三角函数公式推导,得出资源储量垂直纵投影图上确定矿业权边界投影位置的公式,可快速、准确地在资源储量垂直纵投影图上确定矿业权边界的投影位置。这一方法既能为合理开采、利用矿业权范围内资源储量提供数据支持,也能为矿政管理部门核准、监管矿业权范围内的资源储量提供技术支持。     

Boundary assessment under uncertainty: A case study

Estimating certain attributes within a geological body whose exact boundary is not known presents problems because of the lack of information. Estimation may result in values that are inadmissible from a geological point of view, especially with attributes which necessarily must be zero outside the boundary, such as the thickness of the oil column outside a reservoir. A simple but effective way to define the boundary is to use indicator kriging in two steps, the first for the purpose of extrapolating control points outside the body, the second to obtain a weighting function which expresses the uncertainty attached to estimations obtained in the boundary region.     

基于三维地质模型的河南韩梁铝土矿成矿分析及资源量估算

依据河南省鲁山—宝丰韩梁地区铝土矿的地质特征,利用Surpac 6.3矿业软件建立了该矿区的地质数据库及地层、矿体、断层和地表模型,形象表达了矿区的地形地貌和矿体的产状及空间关系。推断出矿床形成的环境和成因,应用块体模型和距离幂次反比法对矿体的品位分布和资源量进行了动态估算,表明块体估算资源量的方法可靠准确,提高了储量估算的效率,为勘查工作提供了可靠的参考和科学依据。