土壤水分空间插值的克里金平滑效应修正方法

地统计学的普通克里金法是研究土壤水分空间变异特性和描绘其空间分布的有效方法。但与其它建立在最小二乘标准上的插值方法一样,普通克里金法也存在着平滑效应问题,即估计值的变异程度比实际要小,从而导致估计值往往不能反映出土壤水分真实的空间变化特征。结合实际的土壤水分监测数据,采用Yamamoto提出的一套针对普通克里金估计值进行后处理的方法,较好地解决了普通克里金法平滑效应的问题,在保证局部估计值精度的同时,重现了土壤含水率在空间的分布与变化特征。     

Influence of the ore block model estimation on the determination of the mining cutoff grade policy for sustainable mine production

Strategic mine planning includes different cut-off grade policy depending on economic parameters of mining projects and grade–tonnage distribution of the deposit. Minimizing incorrect classification of ore and waste during grade–tonnage distribution is of critical importance for a mining operation. This article reviews the influence of the ore grade–tonnage distribution over the cut-off grade policy in a given mining operation. In this study, firstly, the interpolation parameters used to characterize the grade–tonnage distribution in the orebody are given. The resulting distribution of ore and waste is used to analyze uncertainty, risk impact, and to justify mine-planning decisions, according to the interpolation technique used and the number of geological settings and sampling scenarios being considered. Then, the working scheme of the cut-off grade policy and economic parameters are compared according to the resulting estimation from the inverse distance and the nearest neighbor methods.     

Designing infill directional drilling in mineral exploration by using particle swarm optimization algorithm

Geostatistical optimization in designing infill boreholes is an important cost-effective approach in increasing the accuracy of the tonnage and grade of an ore deposit. In this research, a new approach is proposed to design the optimum infill directional boreholes. In the proposed approach, the Kriging estimation variance is considered as the objective function and the number and properties of the optimum boreholes are estimated to minimize the objective function. The optimization procedure is implemented by Particle Swarm Optimization (PSO) algorithm. Range of the spatial and directional properties of new boreholes is determined by considering the primary information of the mineralization and administrative constraint of drilling. Then, the PSO algorithm is iteratively applied, and in each iteration, the variation of the estimated Kriging variance after drilling the new boreholes is determined and properties of the new boreholes are updated. The iterative procedure of the algorithm is continued until minimum Kriging variance is satisfied. The approach was applied to the Dalli Cu-Au porphyry deposit in Iran and three new infill directional boreholes were designed by considering six earlier boreholes from the preliminary exploration stage. New optimum boreholes were located where less information from the preliminary exploration stage exists and the highest variance is considered. Two new boreholes are near to vertical (78°) and the third is an inclined with 55° dip. By drilling these three new boreholes, the estimated grade model could be upgraded by 20%. For simplicity, quickness and the ability to search for the required numbers and specifications of a group of directional boreholes in a 3D environment are the most advantages aspects of the proposed approach.     

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.     

黄土高原降水量空间插值方法研究

为了研究黄土高原降水量的空间分布特征,利用ARCGIS软件分别采用多种空间内插方法对黄土高原108个气象站点1980-2009年30年间的平均年降水量进行空降插值。经过正交检验以及内插检验得到最优的插值方法为普通克里金法,最终对利用该方法获得的黄土高原降水量内插图进行分析,得出近30年来黄土高原区降水量分布呈现出明显的由东南向西北递减的态势。     

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%).     

Modelling Short‐Scale Variability and Uncertainty During Mineral Resource Estimation Using a Novel Fuzzy Estimation Technique

For mineral resource assessment, techniques based on fuzzy logic are attractive because they are capable of incorporating uncertainty associated with measured variables and can also quantify the uncertainty of the estimated grade, tonnage etc. The fuzzy grade estimation model is independent of the distribution of data, avoiding assumptions and constraints made during advanced geostatistical simulation, e.g., the turning bands method. Initially, fuzzy modelling classifies the data using all the component variables in the data set. We adopt a novel approach by taking into account the spatial irregularity of mineralisation patterns using the Gustafson–Kessel classification algorithm. The uncertainty at the point of estimation was derived through antecedent memberships in the input space (i.e., spatial coordinates) and transformed onto the output space (i.e., grades) through consequent membership at the point of estimation. Rather than probabilistic confidence intervals, this uncertainty was expressed in terms of fuzzy memberships, which indicated the occurrence of mixtures of different mineralogical phases at the point of estimation. Data from different sources (other than grades) could also be utilised during estimation. Application of the proposed technique on a real data set gave results that were comparable to those obtained from a turning bands simulation.     

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.     

Improvements in Grade Tonnage Curve Prediction via Sequential Gaussian Fractal Simulation

An application of sequential Gaussian fractal conditional simulation is presented, using actual sparse, irregularly spaced open cut gold data, to predict the grade tonnage curve associated with a single mining bench. It is shown that this constitutes an improvement over the prediction of the grade tonnage curves obtained via standard sequential Gaussian simulation and median indicator simulation. The fractal conditional simulation method uses the model of the covariance of increments of fractional Brownian motion together with the fractal co-dimension derived from a truncated power semivariogram model to create conditional simulations from irregularly spaced data. The grades and tonnages above several discrete cut offs are calculated for each of one hundred simulations, and the mean and variance of the grade and tonnage values for each cut off are computed to give an average grade tonnage curve with associated confidence limits.     

A spatiotemporal Bayesian maximum entropy-based methodology for dealing with sparse data in revising groundwater quality monitoring networks: the Tehran region experience

Data inadequacy is a common problem in designing or updating groundwater monitoring systems. The developed methodologies for the optimal design of groundwater monitoring systems usually assume that there is a complete set of data obtained from existing monitoring wells and provide a revised configuration for the system by analyzing the current data. These methodologies are not usually applicable when the current groundwater quantity and quality data are highly sparse. In this paper, a new simulation–optimization approach based on Bayesian maximum entropy theory (BME) is proposed for revising spatial and temporal monitoring frequencies in a sparsely monitored aquifer. The BME is used to simulate the spatial and spatiotemporal variations of groundwater indicators, incorporating the space/time uncertainties due to insufficient data. Comparing the obtained estimations with observations, the best BME model was selected to be linked with an optimization model. The main goal of optimization was to find out the spatial and temporal sampling characteristics of the monitoring stations using the concepts of Entropy theory and a groundwater vulnerability index. The results show the BME estimations are less biased and more accurate than Ordinary Kriging in both spatial and spatiotemporal analysis. The improvements in the BME estimates are mostly related to incorporating hard (accurate) and soft (uncertain) data in the estimation process. The applicability and efficiency of the proposed methodology have been evaluated by applying it to the Tehran aquifer in Iran which is suffering from high groundwater table fluctuations and nitrate pollution. Based on the results, in addition to the existing monitoring wells, seven new monitoring stations have been proposed. Few stations which potentially can be removed or combined with other stations have been identified and a monthly sampling frequency has been suggested.     

Spatial variability estimation and risk assessment of the aquifer level at sparsely gauged basins using geostatistical methodologies

The spatial variability evaluation of the water table level of an aquifer provides useful information in water resources management plans. Three different approaches are applied to estimate the spatial variability of the water table in the study basin. All of them are based on the Kriging methodology. The first is the classical Ordinary Kriging approach, while the second involves information from a secondary variable (surface elevation) and the application of Residual Kriging. The third calculates the probability to lie below a certain groundwater level limit that could cause significant problems in groundwater resources availability. The latter is achieved by means of Indicator Kriging. A recently developed non-linear normalization method is used to transform both data and residuals closer to normal distribution for improved prediction results. In addition, the recently developed Spartan variogram model is applied to determine the spatial dependence of the measurements. The latter proves to be the optimal model, compared to a series of models tested, which provides in combination with the Kriging methodologies the most accurate cross validation estimations. The variogram form is explained with respect to the radius of influence of the pumping wells representing the spatial impact of the pumping activity. Groundwater level and probability maps are developed providing the ability to assess the spatial variability of the groundwater level in the basin and the risk that certain locations have in terms of a safe groundwater level limit that has been set for the sustainability of the groundwater resources of the basin.     

序贯高斯模拟在矿石品位估计中的应用研究

随机模拟是地质统计方法的重要内容。在矿石品位估计方法中克里格方法作为一种无偏估计方法,常被用于矿石品位的估计。但克里格法估值存在平滑效应。作者在分析序贯高斯模拟和普通克里格法基本原理的基础上,运用序贯高斯模拟方法和普通克里格方法对某铁矿体内全铁(TFe)品位进行估计,给出了品位估计结果模型。研究从勘探线方向、垂直勘探线方向和竖直方向分别计算变差函数,对球状模型、指数模型、高斯模型的变差函数拟合效果进行了优选,结果表明球型模型拟合效果最好。针对序贯模拟和克里格品位估值效果进行了分析,结果显示:序贯高斯模拟结果在品位分布形态上更接近样品品位分布形态,其平滑效应更小;克里格方法估计与序贯高斯模拟方法相比仅在品位均值方面更接近样本品位均值。因此,认为序贯高斯模拟方法可以更好地刻画矿体内品位分布状态。     

Methods of Three-part Quantitative Assessments of Undiscovered Mineral Resources: Examples from Victoria,Australia

Quantitative mineral resource assessments following the 3-part form rely on grade and tonnage models and probabilistic estimates of the number of undiscovered deposits. Assessments completed in Victoria, Australia, indicate that undiscovered mineral resources can be effectively estimated using grade and tonnage sub-models constructed using only medium- and large-tonnage deposits. Numbers of undiscovered deposits can be estimated on the basis of expert judgement or entirely by statistical means. Appropriate mathematical aggregation of individual expert views, expressed at interactive expert workshops, provides robust estimates of the number of undiscovered deposits. Underestimation of uncertainty, which is common in expert judgement, can be compensated by the statistical modification of individual interval estimates. In this study, the linear opinion pool was used as a simple and robust method of mathematical aggregation of multiple expert estimates of the number of undiscovered deposits. A general regression model, which estimates numbers of undiscovered deposits based on the size of the geologically permissive area and the median deposit tonnage, provided results generally compatible with those based on expert judgement or local deposit density models.     

Prediction of local ore recoveries and ore-waste ratios at the silver bell uranium mine

Geostatistical estimation techniques were customized to allow forecasting of production figures at the Silver Bell uranium mine (Uravan District).Surface drill hole data were used to provide a block model of kriged estimators of average uranium grades. Figures for recoverable ore grade and the ore-waste ratio are then deduced from regressive curves previously obtained from underground information and production data. Cross-validations of the entire model were performed and were found positive.     

基于地质统计学的NDVI图像估值技术

将疏采样后的NDVI图像作为未受云层影响的已知数据,分别用普通克里格、泛克里格、指示克里格和序贯指示模拟对NDVI图像进行恢复并比较其效果。研究发现,各种克里格法对NDVI图像的估值效果由高到低依次为泛克里格、普通克里格、指示克里格,通常计算方便的普通克里格法就能够满足图像恢复所要求的精度;普通克里格方差和泛克里格方差只能反映数据的构型,不能很好地衡量估值图像的不确定性,指示克里格的条件方差的分布和实际误差的分布基本一致,能够较好地衡量估值图像的不确定性,并且其大小与NDVI影像数据的不确定性大小的分布一致。序贯指示模拟得到的多个等概率实现表现出很大的空间变异性,多个实现的均值图像光滑效应明显,估值精度不高,但是多个实现的方差分布可以很好地表征空间数据的不确定性分布。     

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。     

Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units

This paper presents a methodology to conduct geostatistical variography and interpolation on areal data measured over geographical units (or blocks) with different sizes and shapes, while accounting for heterogeneous weight or kernel functions within those units. The deconvolution method is iterative and seeks the point-support model that minimizes the difference between the theoretically regularized semivariogram model and the model fitted to areal data. This model is then used in area-to-point (ATP) kriging to map the spatial distribution of the attribute of interest within each geographical unit. The coherence constraint ensures that the weighted average of kriged estimates equals the areal datum.This approach is illustrated using health data (cancer rates aggregated at the county level) and population density surface as a kernel function. Simulations are conducted over two regions with contrasting county geographies: the state of Indiana and four states in the Western United States. In both regions, the deconvolution approach yields a point support semivariogram model that is reasonably close to the semivariogram of simulated point values. The use of this model in ATP kriging yields a more accurate prediction than a naïve point kriging of areal data that simply collapses each county into its geographic centroid. ATP kriging reduces the smoothing effect and is robust with respect to small differences in the point support semivariogram model. Important features of the point-support semivariogram, such as the nugget effect, can never be fully validated from areal data. The user may want to narrow down the set of solutions based on his knowledge of the phenomenon (e.g., set the nugget effect to zero). The approach presented avoids the visual bias associated with the interpretation of choropleth maps and should facilitate the analysis of relationships between variables measured over different spatial supports.     

Kriging and Semivariogram Deconvolution in the Presence of Irregular Geographical Units

This paper presents a methodology to conduct geostatistical variography and interpolation on areal data measured over geographical units (or blocks) with different sizes and shapes, while accounting for heterogeneous weight or kernel functions within those units. The deconvolution method is iterative and seeks the pointsupport model that minimizes the difference between the theoretically regularized semivariogram model and the model fitted to areal data. This model is then used in area-to-point (ATP) kriging to map the spatial distribution of the attribute of interest within each geographical unit. The coherence constraint ensures that the weighted average of kriged estimates equals the areal datum.This approach is illustrated using health data (cancer rates aggregated at the county level) and population density surface as a kernel function. Simulations are conducted over two regions with contrasting county geographies: the state of Indiana and four states in the Western United States. In both regions, the deconvolution approach yields a point support semivariogram model that is reasonably close to the semivariogram of simulated point values. The use of this model in ATP kriging yields a more accurate prediction than a na?ve point kriging of areal data that simply collapses each county into its geographic centroid. ATP kriging reduces the smoothing effect and is robust with respect to small differences in the point support semivariogram model. Important features of the point-support semivariogram, such as the nugget effect, can never be fully validated from areal data. The user may want to narrow down the set of solutions based on his knowledge of the phenomenon (e.g., set the nugget effect to zero). The approach presented avoids the visual bias associated with the interpretation of choropleth maps and should facilitate the analysis of relationships between variables measured over different spatial supports.     

地质统计学在羊拉铜矿储量计算中的应用

本文以云南德钦羊拉铜矿床为例, 借助Surpac矿业软件建立了矿床地质数据库, 并运用地质统计学的方法, 通过分析羊拉5号矿体铜品位分布规律, 计算出了厚度、倾向、走向3个方向的实验变异函数, 并进行了理论变异函数的曲线拟合, 确立了矿床的数学模型。在此基础上运用克立格法和距离平方反比法分别进行了储量计算, 再结合传统的块段法计算出的储量, 将三者进行对比分析, 探讨了三种方法的影响因素及其产生误差的根本原因, 最后认为运用克立格法计算储量具有先进性和使用性。文章还分析了铜矿体的品位－吨位曲线图, 为矿山合理确定经济品位和吨位提供了可靠的理论依据。     

Spatial Variability of Rock Depth in Bangalore Using Geostatistical, Neural Network and Support Vector Machine Models

Geospatial technology is increasing in demand for many applications in geosciences. Spatial variability of the bed/hard rock is vital for many applications in geotechnical and earthquake engineering problems such as design of deep foundations, site amplification, ground response studies, liquefaction, microzonation etc. In this paper, reduced level of rock at Bangalore, India is arrived from the 652 boreholes data in the area covering 220 km². In the context of prediction of reduced level of rock in the subsurface of Bangalore and to study the spatial variability of the rock depth, Geostatistical model based on Ordinary Kriging technique, Artificial Neural Network (ANN) and Support Vector Machine (SVM) models have been developed. In Ordinary Kriging, the knowledge of the semi-variogram of the reduced level of rock from 652 points in Bangalore is used to predict the reduced level of rock at any point in the subsurface of the Bangalore, where field measurements are not available. A new type of cross-validation analysis developed proves the robustness of the Ordinary Kriging model. ANN model based on multi layer perceptrons (MLPs) that are trained with Levenberg–Marquardt backpropagation algorithm has been adopted to train the model with 90% of the data available. The SVM is a novel type of learning machine based on statistical learning theory, uses regression technique by introducing loss function has been used to predict the reduced level of rock from a large set of data. In this study, a comparative study of three numerical models to predict reduced level of rock has been presented and discussed.