Block size selection with the objective of minimizing the discrepancy in real and estimated block grade

The valuation of a mining project depends upon the accuracy of geological block model. Sampling density, estimation method, and proper block size mainly affect the accuracy of estimated block. This paper aims to answer three questions: (1) which estimation method is more accurate, (2) what is the relation between sampling density and block size, and (3) what the optimum block size is. Conditional Gaussian simulation (CGS) was used to generate a hypothetical deposit, considered as a real block model. A range of different block dimensions were estimated by ordinary kriging, inverse squared distance, and nearest neighbor methods based on tow-simulated drilling grids database. The comparison of estimated and real block grades reveals that increasing the sampling density results the similar outcomes of geostatistics and deterministic interpolation methods. Furthermore, it was deduced that sampling density could not be a viable alternative in choosing appropriate block dimension and the variogram rang a was suggested as an affective parameter in block size selection. Then a geometrical formula was developed to obtain the block size based on the variogram range. The increment in project value that a mine planner can expected from the additional information of the dense drilling grid was also calculated and it was concluded that the block size obtained based on the suggested formula results acceptable information value. Finally, the database of Chador Malu iron ore mine which is located in 180 km northeast of Yazd city in the central part of Iran were used to validate the suggested formula.     

Characterization and Quantification of Uncertainty in?Coregionalization Analysis

Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.     

An application of the deterministic variogram to design-based variance estimation

When estimating the mean value of a variable, or the total amount of a resource, within a specified region it is desirable to report an estimated standard error for the resulting estimate. If the sample sites are selected according to a probability sampling design, it usually is possible to construct an appropriate design-based standard error estimate. One exception is systematic sampling for which no such standard error estimator exists. However, a slight modification of systematic sampling, termed 2-step tessellation stratified (2TS) sampling, does permit the estimation of design-based standard errors. This paper develops a design-based standard error estimator for 2TS sampling. It is shown that the Taylor series approximation to the variance of the sample mean under 2TS sampling may be expressed in terms of either a deterministic variogram or a deterministic covariance function. Variance estimation then can be approached through the estimation of a variogram or a covariance function. The resulting standard error estimators are compared to some more traditional variance estimators through a simulation study. The simulation results show that estimators based on the new approach may perform better than traditional variance estimators.     

Characterization and Quantification of Uncertainty in Coregionalization Analysis

Coregionalization analysis has been presented as a method of multi-scale analysis for multivariate spatial data. Despite an increasing use of this method in environmental and earth sciences, the uncertainty associated with the estimation of parameters in coregionalization analysis (e.g., sills and functions of sills) is potentially high and has not yet been characterized. This article aims to discuss the theory underlying coregionalization analysis and assess the robustness and limits of the method. A theoretical framework is developed to calculate the ergodic and fluctuation variance-covariance matrices of least-squares estimators of sills in the linear model of coregionalization. To adjust for the positive semidefiniteness constraint on estimated coregionalization matrices, a confidence interval estimation procedure for sills and functions of sills is presented. Thereafter, the relative importance of uncertainty measures (bias and variance) for sills and structural coefficients of correlation and determination is assessed under different scenarios to identify factors controlling their uncertainty. Our results show that the sampling grid density, the choice of the least-squares estimator of sills, the positive semidefiniteness constraint, the presence of scale dependence in the correlations, and the number and range of variogram models, all affect the level of uncertainty, sometimes through multiple interactions. The asymptotic properties of variogram model parameter estimators in a bounded sampling domain impose a theoretical limit to their accuracy and precision. Because of this limit, the uncertainty was found to be high for several scenarios, especially with three variogram models, and was often more dependent on the ratio of variogram range to domain extent than on the sampling grid density. In practice, in the coregionalization analysis of a real dataset, the circular requirement for sill estimates in the calculation of uncertainty measures makes the quantification of uncertainty very problematic, if not impossible. The use of coregionalization analysis must be made with due knowledge of the uncertainty levels and limits of the method.     

Variance–Covariance Matrix of the Experimental Variogram: Assessing Variogram Uncertainty

Assessment of the sampling variance of the experimental variogram is an important topic in geostatistics as it gives the uncertainty of the variogram estimates. This assessment, however, is repeatedly overlooked in most applications mainly, perhaps, because a general approach has not been implemented in the most commonly used software packages for variogram analysis. In this paper the authors propose a solution that can be implemented easily in a computer program, and which, subject to certain assumptions, is exact. These assumptions are not very restrictive: second-order stationarity (the process has a finite variance and the variogram has a sill) and, solely for the purpose of evaluating fourth-order moments, a Gaussian distribution for the random function. The approach described here gives the variance–covariance matrix of the experimental variogram, which takes into account not only the correlation among the experiemental values but also the multiple use of data in the variogram computation. Among other applications, standard errors may be attached to the variogram estimates and the variance–covariance matrix may be used for fitting a theoretical model by weighted, or by generalized, least squares. Confidence regions that hold a given confidence level for all the variogram lag estimates simultaneously have been calculated using the Bonferroni method for rectangular intervals, and using the multivariate Gaussian assumption for K-dimensional elliptical intervals (where K is the number of experimental variogram estimates). A general approach for incorporating the uncertainty of the experimental variogram into the uncertainty of the variogram model parameters is also shown. A case study with rainfall data is used to illustrate the proposed approach.     

Some exact sampling distributions for variogram estimators

For equally spaced observations from a one-dimensional, stationary, Gaussian random function, the characteristic function of the usual variogram estimator for a fixed lag k is derived. Because the characteristic function and the probability density function form a Fourier integral pair, it is possible to tabulate the sampling distribution of a function of a using either analytic or numerical methods. An example of one such tabulation is given for an underlying model that is simple transitive.     

Estimating Variogram Uncertainty

The variogram is central to any geostatistical survey, but the precision of a variogram estimated from sample data by the method of moments is unknown. It is important to be able to quantify variogram uncertainty to ensure that the variogram estimate is sufficiently accurate for kriging. In previous studies theoretical expressions have been derived to approximate uncertainty in both estimates of the experimental variogram and fitted variogram models. These expressions rely upon various statistical assumptions about the data and are largely untested. They express variogram uncertainty as functions of the sampling positions and the underlying variogram. Thus the expressions can be used to design efficient sampling schemes for estimating a particular variogram. Extensive simulation tests show that for a Gaussian variable with a known variogram, the expression for the uncertainty of the experimental variogram estimate is accurate. In practice however, the variogram of the variable is unknown and the fitted variogram model must be used instead. For sampling schemes of 100 points or more this has only a small effect on the accuracy of the uncertainty estimate. The theoretical expressions for the uncertainty of fitted variogram models generally overestimate the precision of fitted parameters. The uncertainty of the fitted parameters can be determined more accurately by simulating multiple experimental variograms and fitting variogram models to these. The tests emphasize the importance of distinguishing between the variogram of the field being surveyed and the variogram of the random process which generated the field. These variograms are not necessarily identical. Most studies of variogram uncertainty describe the uncertainty associated with the variogram of the random process. Generally however, it is the variogram of the field being surveyed which is of interest. For intensive sampling schemes, estimates of the field variogram are significantly more precise than estimates of the random process variogram. It is important, when designing efficient sampling schemes or fitting variogram models, that the appropriate expression for variogram uncertainty is applied.     

Optimized Sample Schemes for Geostatistical Surveys

Sample schemes used in geostatistical surveys must be suitable for both variogram estimation and kriging. Previously schemes have been optimized for one of these steps in isolation. Ordinary kriging generally requires the sampling locations to be evenly dispersed over the region. Variogram estimation requires a more irregular pattern of sampling locations since comparisons must be made between measurements separated by all lags up to and beyond the range of spatial correlation. Previous studies have not considered how to combine these optimized schemes into a single survey and how to decide what proportion of sampling effort should be devoted to variogram estimation and what proportion devoted to kriging An expression for the total error in a geostatistical survey accounting for uncertainty due to both ordinary kriging and variogram uncertainty is derived. In the same manner as the kriging variance, this expression is a function of the variogram but not of the sampled response data. If a particular variogram is assumed the total error in a geostatistical survey may be estimated prior to sampling. We can therefore design an optimal sample scheme for the combined processes of variogram estimation and ordinary kriging by minimizing this expression. The minimization is achieved by spatial simulated annealing. The resulting sample schemes ensure that the region is fairly evenly covered but include some close pairs to analyse the spatial correlation over short distances. The form of these optimal sample schemes is sensitive to the assumed variogram. Therefore a Bayesian approach is adopted where, rather than assuming a single variogram, we minimize the expected total error over a distribution of plausible variograms. This is computationally expensive so a strategy is suggested to reduce the number of computations required     

On the Likelihood and Fluctuations of Gaussian Realizations

The likelihood of Gaussian realizations, as generated by the Cholesky simulation method, is analyzed in terms of Mahalanobis distances and fluctuations in the variogram reproduction. For random sampling, the probability to observe a Gaussian realization vector can be expressed as a function of its Mahalanobis distance, and the maximum likelihood depends only on the vector size. The Mahalanobis distances are themselves distributed as a Chi-square distribution and they can be used to describe the likelihood of Gaussian realizations. Their expected value and variance are only determined by the size of the vector of independent random normal scores used to generate the realizations. When the vector size is small, the distribution of Mahalanobis distances is highly skewed and most realizations are close to the vector mean in agreement with the multi-Gaussian density model. As the vector size increases, the realizations sample a region increasingly far out on the tail of the multi-Gaussian distribution, due to the large increase in the size of the uncertainty space largely compensating for the low probability density. For a large vector size, realizations close to the vector mean are not observed anymore. Instead, Gaussian vectors with Mahalanobis distance in the neighborhood of the expected Mahalanobis distance have the maximum probability to be observed. The distribution of Mahalanobis distances becomes Gaussian shaped and the bulk of realizations appear more equiprobable. However, the ratio of their probabilities indicates that they still remain far from being equiprobable. On the other hand, it is observed that equiprobable realizations still display important fluctuations in their variogram reproduction. The variance level that is expected in the variogram reproduction, as well as the variance of the variogram fluctuations, is dependent on the Mahalanobis distance. Realizations with smaller Mahalanobis distances are, on average, smoother than realizations with larger Mahalanobis distances. Poor ergodic conditions tend to generate higher proportions of flatter variograms relative to the variogram model. Only equiprobable realizations with a Mahalanobis distance equal to the expected Mahalanobis distance have an expected variogram matching the variogram model. For large vector sizes, Cholesky simulated Gaussian vectors cannot be used to explore uncertainty in the neighborhood of the vector mean. Instead uncertainty is explored around the n-dimensional elliptical envelop corresponding to the expected Mahalanobis distance.     

空间分辨率与取样方式对DEM流域特征提取的影响

随着数字水文的兴起和分布式水文模型研究的发展, 利用DEM提取水文特征, 进而进行水文模拟的方法越来越广泛地为水文学者所采用. 空间分辨率的改变与DEM重新取样方式对水文模拟都会产生重要影响. 采取不同取样方法获得多种尺度的DEM, 对不同分辨率下的流域特征值进行了统计分析与比较, 引入熵的概念度量不同分辨率的DEM包含的信息量, 以及不同取样方式对信息量的影响. 并计算了以50 m DEM所包含的信息量为基准, 在不同的信息损失下所要求的最低分辨率.     

Reproduction of the Mean, Variance, and Variogram Model in Spectral Simulation

The application of spectral simulation is gaining acceptance because it honors the spatial distribution of petrophysical properties, such as reservoir porosity and shale volume. While it has been widely assumed that spectral simulation will reproduce the mean and variance of the important properties such as the observed net/gross ratio or global average of porosity, this paper shows the traditional way of implementing spectral simulation yields a mean and variance that deviates from the observed mean and variance. Some corrections (shift and rescale) could be applied to generate geologic models yielding the observed mean and variance; however, this correction implicitly rescales the input variogram model, so the variogram resulting from the generated cases has a higher sill than the input variogram model. Therefore, the spectral simulation algorithm cannot build geologic models honoring the desired mean, variance, and variogram model simultaneously, which is contrary to the widely accepted assumption that spectral simulation can reproduce all the target statistics. However, by using Fourier transform just once to generate values at all the cells instead of visiting each cell sequentially, spectral simulation does reproduce the observed variogram better than sequential Gaussian simulation. That is, the variograms calculated from the generated geologic models show smaller fluctuations around the target variogram. The larger the generated model size relative to the variogram range, the smaller the observed fluctuations.     

多物源条件下的储层地质建模方法

濮城沙三中油藏具有两个主物源,分别为NE向与SE向。油藏数值模拟需要在一套地质网格中对其进行模拟。经典的地质统计学利用变差函数描述区域化变量的空间几何结构特性。变差函数的计算是基于两点进行统计的,对其描述主要涉及方位角、变程、块金值和基台值。为了在一套模拟网格中模拟出多个物源条件下储层的分布特征,必须在不同的位置设置不同的变差函数参数。文中给出了两种方法实现这一目的:一是采用人为分区,把不同物源影响的区域分成不同的区块,分别对不同的区块设置不同的变差函数参数;二是采用变方位角,即根据不同的位置设置不同的变差函数方位角。这两种方法都实现了在一套网格中模拟具有多个物源方向的储层分布,更真实地再现了储层的空间展布特征。     

Optimization of grid-drilling using computer simulation

A computer simulation method has been developed to find efficient drilling grids for mineral deposits. A well-known ore deposit is used as a model to develop an efficient pattern for undiscovered ore bodies in the same area or in other prospects where similar geometry is suspected. The model for this study is the Austinville, Virginia deposit, a Mississippi Valley-type deposit composed of 17 ore bodies totaling 34 million short tons (30 million metric tons). The method employs a computer program that simulates drilling the model deposit with different patterns, including various levels of follow-up drilling. Follow-up holes are drilled in fences at one half the original spacing around holes in the grid that show ore-grade mineralization. Each pattern is drilled 100 times from random starting locations to provide a range of outcomes of drilling, including the best, worst, and most likely. For this study, patterns of 100 drill holes were composed of 10 fences spaced 1000–5000 feet (305–1524 m) apart, each with 10 holes spaced 200–1000 feet (61–305 m) apart. In all, 25 grids were used with zero to three levels of follow-up drilling. The 600/2000 grid, with drill holes spaced 600 feet (183 m) apart in fences spaced 2000 feet (610 m) apart, was compared with the 200/5000 grid because they represented contrasting outcomes. The 600/2000 grid penetrated many ore bodies consistently but with few multiple hits to individual ore bodies; whereas the 200/5000 grid inconsistently penetrated few ore bodies with many multiple hits. The 600/2000 grid was more efficient than the 200/5000 grid at hitting large ore bodies of 1,000,000 short tons or greater (900,000 metric tons or greater) and was made more effective by adding one cycle of follow-up drilling. The 600/2000 grid had a 97% chance of hitting one or more large ore bodies with at least one drill hole per ore body, and the 200/5000 grid had a 64% chance. Once hit, there was an 82% chance that the largest ore body would be penetrated by three or more holes when using the 600/2000 grid and an 88% chance using the 200/5000 grid.     

Use of Variography in Permeability Characterization of Visually Homogeneous Sandstone Reservoirs with Examples from Outcrop Studies

Sandstones of different ages provide economically significant oil, gas, and water reservoirs. In sandstones where heterogeneities are not visually obvious, it is particularly difficult to predict the location of permeability barriers and the scale at which high and low permeability zones occur, yet this is critical in providing information on hydrocarbon reservoir performance. This study uses variogram analysis to investigate spatial variation in permeability in visually homogeneous reservoir sandstone successions. Air permeability measurements were taken using unsteady state probe permeametry following regular grid schemes with centimeter spacing. Spatial variation in permeability was characterized using omnidirectional and directional variograms. This study combines variography with geological interpretation to assess the degree of heterogeneity of permeability in visually homogeneous sandstone successions. Variography indicates spatial dependence and short-range variation at 1 cm grid spacings that is not apparent at a larger 5 cm grid spacing in the visually homogeneous sandstones studied. The range of the models fitted to the variograms provide a potentially important index of spatial variability in permeability for different depositional settings including aeolian, fluvial, shallow marine, and marine/mass- flow turbidite.     

Implementation aspects of sequential Gaussian simulation on irregular points

The increasing use of unstructured grids for reservoir modeling motivates the development of geostatistical techniques to populate them with properties such as facies proportions, porosity and permeability. Unstructured grids are often populated by upscaling high-resolution regular grid models, but the size of the regular grid becomes unreasonably large to ensure that there is sufficient resolution for small unstructured grid elements. The properties could be modeled directly on the unstructured grid, which leads to an irregular configuration of points in the three-dimensional reservoir volume. Current implementations of Gaussian simulation for geostatistics are for regular grids. This paper addresses important implementation details involved in adapting sequential Gaussian simulation to populate irregular point configurations including general storage and computation issues, generating random paths for improved long range variogram reproduction, and search strategies including the superblock search and the k-dimensional tree. An efficient algorithm for computing the variogram of very large irregular point sets is developed for model checking.     

黑龙江森林沼泽区超低密度 地球化学调查采样介质对比

黑龙江省中部的嘉荫地区为典型的森林沼泽景观,在该区进行了超低密度地球化学调查采样介质对比研究。通过河漫滩沉积物、活性水系沉积物进行对比研究,发现利用河漫滩表层沉积物所圈定的异常不仅能较好地反映矿化信息,而且能较好地反映该地区的地质背景。水系沉积物测量所圈定的异常很弱,只能反映出露矿信息。研究成果表明,河漫滩表层沉积物是森林超低密度地球化学调查的最佳采样介质。     

A simple sufficient condition for a variogram model to yield positive variances under restrictions

Estimation of linear combinations is accomplished by using the observed (available) data. Accordingly, to require the negative of a modeled variogram function to be positive definite for all possible data combinations is unnecessary when only the observed data are used in estimation. The requirement that the negative of a variogram model be conditionally positive semidefinite is then relaxed to apply at the observed spatial locations only. In this setting a simple, yet crude, sufficient condition is developed to ensure that a variogram model will yield nonnegative variances for the available data. It is seen that the condition is independent of the dimensionality of the data and applies to both isotropic and anisotropic models. An example of the application of the condition is also presented. The condition is harder to satisfy as the amount of data increases and must be adjusted as the variogram changes to accommodate new data.     

半变异函数及取样间距对克里金法在海洋地层分析中的影响研究

以浅剖数据为源数据,钻孔实测数据为验证数据,利用普通克里金法对海底地层厚度进行空间插值得到地层分布特征,采用3种半变异函数模型和不同取样间距对某井场3组地层厚度进行普通克里金插值并验证其插值效果。结果表明:普通克里金是一种有效的海底地层厚度预测方法;结构分析最佳的模型不一定是误差最小的模型,应对不同模型下的插值结果进行综合分析来选择最合适的模型,并提出球状模型在该井场厚度估计中最优,高斯模型次之;对于球状模型,增大取样间距对地层厚度变化剧烈的地层回归效果影响较小,对地层厚度变化不大的地层回归效果影响较大;同时,SE预测值变化率分析表明对于地层厚度变化剧烈的地层,减小取样间距可以大幅度地减少插值误差,而对于地层厚度变化不大的地层,减小取样间距对插值精度提高的意义不大。     

安徽铜陵狮子山铜矿床探采对比研究

本文选取矿体形态变化、厚度变化、底板位移、品位变化和储量误差作为铜陵狮子山铜矿床矿山探采对比项目开展研究,分析了地质勘探期间对勘探类型划分及勘探手段、勘探方法、勘探网度选择的合理性;明确了原地质勘探对矿体形态和规模的认识和控制程度及资源/储量估算的误差率,为今后地质条件类似矿床的勘查、设计及开采提供了很好借鉴作用。     

Nonstationarity of the Mean and Unbiased Variogram Estimation: Extension of the Weighted Least-Squares Method

When concerned with spatial data, it is not unusual to observe a nonstationarity of the mean. This nonstationarity may be modeled through linear models and the fitting of variograms or covariance functions performed on residuals. Although it usually is accepted by authors that a bias is present if residuals are used, its importance is rarely assessed. In this paper, an expression of the variogram and the covariance function is developed to determine the expected bias. It is shown that the magnitude of the bias depends on the sampling configuration, the importance of the dependence between observations, the number of parameters used to model the mean, and the number of data. The applications of the expression are twofold. The first one is to evaluate a priori the importance of the bias which is expected when a residuals-based variogram model is used for a given configuration and a hypothetical data dependence. The second one is to extend the weighted least-squares method to fit the variogram and to obtain an unbiased estimate of the variogram. Two case studies show that the bias can be negligible or larger than 20%. The residual-based sample variogram underestimates the total variance of the process but the nugget variance may be overestimated.