中国铁矿床品位-吨位模型

文章以2005年国家有关部门公布的铁矿储量数据为依据,在拥有翔实和权威的铁矿数据材料的基础上,经过分析整理和统计计算,以铁矿的6种矿床类型(沉积变质型、沉积型、矽卡岩型、岩浆岩型、火山岩型和风化型)为划分依据,建立了中国铁矿及其伴生矿种的品位和吨位模型,并对其品位-吨位联合分布的模型进行了研究。通过研究得出结论,中国铁矿的资源量不服从正态分布,经过对数变换后服从正态分布,品位除了矽卡岩之外都直接服从正态分布。由于铁矿的品位和吨位之间相关性较差,因而不具备分形特征。最后通过建立吨位和品位的直方图来构造正态分布函数,从而实现累计概率曲线的拟合。在取得了远景区的矿床类型、矿床数分布以及矿床的品位和吨位的累计概率曲线后,可以使用计算机通过蒙特卡洛模拟的方法来实现远景区资源量的预测和模拟。     

The Grade-Tonnage Model for China’s Molybdenum Deposits

The authors have studied the statistical characteristics of China’s molybdenum deposits to establish the grade-tonnage model based on the data updated to?the end of 2010. The results showed that each type of China’s molybdenum deposits complied with Lasky’s law approximately and the characteristics of grade-tonnage model obey the lognormal distribution. However, there are poor correlations between grade and tonnage respectively. Ultimately,?we aimed to fit the grade-tonnage model through the known distribution function, draw the cumulative?probability curves, and evaluated undiscovered mineral resources of China’s?molybdenum deposits by means of Monte Carlo simulation integrated in MRAS.     

Using non-Gaussian distributions in geostatistical simulations

Parametric geostatistical simulations such as LU decomposition and sequential algorithms do not need Gaussian distributions. It is shown that variogram model reproduction is obtained when Uniform or Dipole distributions are used instead of Gaussian distributions for drawing i. i.d. random values in LU simulation, or for modeling the local conditional probability distributions in sequential simulation. Both algorithms yield simulated values with a marginal normal distribution no matter if Gaussian, Uniform, or Dipole distributions are used. The range of simulated values decreases as the entropy of the probability distribution decreases. Using Gaussian distributions provides a larger range of simulated normal score values than using Uniform or Dipole distributions. This feature has a negligible effect for reproduction of the normal scores variogram model but have a larger impact on the reproduction of the original values variogram. The Uniform or Dipole distributions also produce lesser fluctuations among the variograms of the simulated realizations.     

Revisiting Multi-Gaussian Kriging with the Nataf Transformation or the Bayes’ Rule for the Estimation of Spatial Distributions

A multivariate probability transformation between random variables, known as the Nataf transformation, is shown to be the appropriate transformation for multi-Gaussian kriging. It assumes a diagonal Jacobian matrix for the transformation of the random variables between the original space and the Gaussian space. This allows writing the probability transformation between the local conditional probability density function in the original space and the local conditional Gaussian probability density function in the Gaussian space as a ratio equal to the ratio of their respective marginal distributions. Under stationarity, the marginal distribution in the original space is modeled from the data histogram. The stationary marginal standard Gaussian distribution is obtained from the normal scores of the data and the local conditional Gaussian distribution is modeled from the kriging mean and kriging variance of the normal scores of the data. The equality of ratios of distributions has the same form as the Bayes’ rule and the assumption of stationarity of the data histogram can be re-interpreted as the gathering of the prior distribution. Multi-Gaussian kriging can be re-interpreted as an updating of the data histogram by a Gaussian likelihood. The Bayes’ rule allows for an even more general interpretation of spatial estimation in terms of equality for the ratio of the conditional distribution over the marginal distribution in the original data uncertainty space with the same ratio for a model of uncertainty with a distribution that can be modeled using the mean and variance from direct kriging of the original data values. It is based on the principle of conservation of probability ratio and no transformation is required. The local conditional distribution has a variance that is data dependent. When used in sequential simulation mode, it reproduces histogram and variogram of the data, thus providing a new approach for direct simulation in the original value space.     

A Fractal Interpolatory Approach to Geochemical Exploration Data Processing

Traditional interpolation procedures used for processing geochemical data treat the data as a continuous smooth surface. In this paper, we proposed a fractal spatial interpolatory procedure based on the concept of the fractional delta variance. The method is suitable for processing the geochemical data that are measured at irregularly spaced discrete points, without resort to gridding procedures. The value at each interpolatory point is a function of the fractal disturbance that is related to the fractal dimension determined from the original data set, thus enhancing reconstruction of the natural spatial distribution of element concentration. The proposed procedure has been applied to the copper geochemical data measured from irregularly-spaced floodplain sediment samples in China. We compared geochemical maps created using different interpolatory procedures, including the new fractal method, the Kriging, and the weighted average, against the actual spatial distribution of copper mineral deposits in China, and found that the new fractal method detected more of known Cu-deposits from the test area than the other two methods.     

Uncertainty in Coal Property Valuation in West Virginia: A Case Study

Interpolated grids of coal bed thickness are being considered for use in a proposed method for taxation of coal in the state of West Virginia (United States). To assess the origin and magnitude of possible inaccuracies in calculated coal tonnage, we used conditional simulation to generate equiprobable realizations of net coal thickness for two coals on a 7 min topographic quadrangle, and a third coal in a second quadrangle. Coals differed in average thickness and proportion of original coal that had been removed by erosion; all three coals crop out in the study area. Coal tonnage was calculated for each realization and for each interpolated grid for actual and artificial property parcels, and differences were summarized as graphs of percent difference between tonnage calculated from the grid and average tonnage from simulations. Coal in individual parcels was considered minable for valuation purposes if average thickness in each parcel exceeded 30 inches. Results of this study show that over 75% of the parcels are classified correctly as minable or unminable based on interpolation grids of coal bed thickness. Although between 80 and 90% of the tonnages differ by less than 20% between interpolated values and simulated values, a nonlinear conditional bias might exist in estimation of coal tonnage from interpolated thickness, such that tonnage is underestimated where coal is thin, and overestimated where coal is thick. The largest percent differences occur for parcels that are small in area, although because of the small quantities of coal in question, bias is small on an absolute scale for these parcels. For a given parcel size, maximum apparent overestimation of coal tonnage occurs in parcels with an average coal bed thickness near the minable cutoff of 30 in. Conditional bias in tonnage for parcels having a coal thickness exceeding the cutoff by 10 in. or more is constant for two of the three coals studied, and increases slightly with average thickness for the third coal.     

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.     

Gaussian Process Emulators for Computer Experiments with Inequality Constraints

Physical phenomena are observed in many fields (science and engineering) and are often studied by time-consuming computer codes. These codes are analyzed with statistical models, often called emulators. In many situations, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all input variables. The aim is to build a model capable of incorporating both data interpolation and inequality constraints into a Gaussian process emulator. By using a functional decomposition, a finite-dimensional approximation of Gaussian processes such that all conditional simulations satisfy the inequality constraints in the entire domain is proposed. To show the performance of the proposed model, some conditional simulations with inequality constraints such as boundedness, monotonicity or convexity conditions in one and two dimensions are given. A simulation study to investigate the efficiency of the method in terms of prediction is included.     

A comparative study between sequential Gaussian simulation and kriging method grade modeling in open-pit mining

Grade estimation is very important in designing open pits. In the process of grade estimation, underestimation can result in loss of economic ore, whereas overestimation would unnecessarily increase stripping ratio. Normally, kriging method, which suffers from underestimation and/or overestimation due to smoothing effect, is used for grade estimation. To overcome drawbacks of the kriging method, more efficient techniques such as conditional simulation can be applied. In this paper, utilizing sequential Gaussian conditional simulation, grade models were constructed for Sungun copper deposit situated in the North West of Iran. According to the obtained results, it was observed that conditional simulation can effectively cope with the weakness of kriging method. Also, it was observed that as compared to the kriging method, grade distribution, resulted from the conditional simulation, is almost identical to that of the real exploration data. Accordingly, using conditional simulation, the amount of mineable ore was significantly increased, and also, average net present value as the mines’ most important economic indicator was improved by 40%.     

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

Assessing the accuracy of sequential Gaussian simulation and cosimulation

Sequential Gaussian simulation is widespread in Earth Science applications to quantify the uncertainty about regionalized properties. Its practical implementation relies on the screen effect approximation in order to determine the successive conditional distributions by considering only the information available in the neighborhood of the target location. A methodology is presented to assess the accuracy of sequential Gaussian simulation, by calculating the theoretical moments (expectation and variance–covariance matrix) of the simulated random vector and comparing them with the moments of the underlying model. The methodology can be applied in both the conditional and non-conditional contexts, as well as for univariate or multivariate simulation. It is helpful to determine appropriate implementation parameters, in particular about the visiting sequence and the design of the moving neighborhood for selecting relevant conditioning information, prior to performing simulation.     

Stochastic Distance Based Geological Boundary Modeling with Curvilinear Features

Obtaining accurate geological boundaries and assessing the uncertainty in these limits are critical for effective ore resource and reserve estimation. The uncertainty in the extent of an ore body can be the largest source of uncertainty in ore resource estimation when drilling is sparse. These limits are traditionally interpreted deterministically and it can be difficult to quantify uncertainty in the boundary and its impact on ore tonnage. The proposed methodology is to consider stochastic modeling of the ore boundary with a distance function recoding of the available data. This technique is modified to incorporate non-stationarities in the form of a locally varying anisotropy field used in kriging and sequential Gaussian simulation. Implementing locally varying anisotropy kriging retains the geologically realistic features of a deterministic model while allowing for a stochastic assessment of uncertainty. A case study of a gold deposit in Northern Canada is used to demonstrate the methodology. The proposed technique generates realistic, curvilinear geological boundary models and allows for an assessment of the uncertainty in the model.     

Geostatistical approach for identification of transmissivity structure at Dulliu area in Taiwan

 A thorough understanding of the characteristics of transmissivity makes groundwater deterministic models more accurate. These transmissivity data characteristics occasionally possess a complicated spatial variation over an investigated site. This study presents both geostatistical estimation and conditional simulation methods to generate spatial transmissivity maps. The measured transmissivity data from the Dulliu area in Yun-Lin county, Taiwan, is used as the case study. The spatial transmissivity maps are simulated by using sequential Gaussian simulation (SGS), and estimated by using natural log ordinary kriging and ordinary kriging. Estimation and simulation results indicate that SGS can reproduce the spatial structure of the investigated data. Furthermore, displaying a low spatial variability does not allow the ordinary kriging and natural log kriging estimates to fit the spatial structure and small-scale variation for the investigated data. The maps of kriging estimates are smoother than those of other simulations. A SGS with multiple realizations has significant advantages over ordinary kriging and even natural log kriging techniques at a site with a high variation in investigated data. These results are displayed in geographic information systems (GIS) as basic information for further groundwater study. Received: 27 August 1999 · Accepted: 22 February 2000     

Co-simulating Total and Soluble Copper Grades in an Oxide Ore Deposit

In oxide copper deposits, the acid soluble copper represents the fraction of total copper recoverable by heap leaching. Two difficulties often complicate the joint modeling and simulation of total and soluble copper grades: the inequality constraint linking both grade variables and the sampling design for soluble copper grade, which may be preferential and cause biases in sample statistics. A methodology is presented in order to accurately estimate the total and soluble copper grade bivariate distribution, based on an explicit modeling of the conditional distributions of soluble copper grade. Co-simulation is then realized by converting the copper grades into Gaussian random fields, through stepwise conditional transformation, and by fitting a coregionalization model while accounting for the preferential sampling design. The proposed approach is illustrated through an application to an ore deposit located in northern Chile.     

Bivariate simulation using copula and its application to probabilistic pile settlement analysis

This paper aims to propose a procedure for modeling the joint probability distribution of bivariate uncertain data with a nonlinear dependence structure. First, the concept of dependence measures is briefly introduced. Then, both the Akaike Information Criterion and the Bayesian Information Criterion are adopted for identifying the best‐fit copula. Thereafter, simulation of copulas and bivariate distributions based on Monte Carlo simulation are presented. Practical application for serviceability limit state reliability analysis of piles is conducted. Finally, four load–test datasets of load–displacement curves of piles are used to illustrate the proposed procedure. The results indicate that the proposed copula‐based procedure can model and simulate the bivariate probability distribution of two curve‐fitting parameters underlying the load–displacement models of piles in a more general way. The simulated load–displacement curves using the proposed procedure are found to be in good agreement with the measured results. In most cases, the Gaussian copula, often adopted out of expedience without proper validation, is not the best‐fit copula for modeling the dependence structure underlying two curve‐fitting parameters. The conditional probability density functions obtained from the Gaussian copula differ considerably from those obtained from the best‐fit copula. The probabilities of failure associated with the Gaussian copula are significantly smaller than the reference solutions, which are very unconservative for pile safety assessment. If the strong negative correlation between the two curve‐fitting parameters is ignored, the scatter in the measured load–displacement curves cannot be simulated properly, and the probabilities of failure will be highly overestimated. Copyright © 2011 John Wiley & Sons, Ltd.     

基于不同地质统计方法的渗透系数场对污染物运移的影响

渗透系数场的空间变异性是影响污染物运移结果的决定因素,而地质统计方法是解决渗透系数空间变异性的主要技术手段。本文利用野外场地实测数据,采用普通克里格法和指示克里格法、顺序高斯模拟法和顺序指示模拟法四种地质统计方法,插值估测和模拟再现随机渗透系数场,进而对比研究四种渗透系数场对大尺度污染物运移的影响。研究结果表明,污染羽的质心位置（一阶矩）主要由渗透系数的平均值来决定;污染羽在空间上的展布范围（二阶矩）主要受渗透系数空间变异方差的影响;条件模拟克服了估计法的平滑效果,较好地再现真实曲线的波动性,渗透系数（ lnK）估计方差与污染羽空间二阶矩随着条件模拟次数的增加而减小,并且顺序指示模拟程度更加明显。     

Stochastic Simulation of Nonstationary Rainfall Fields, Accounting for Seasonality and Atmospheric Circulation Pattern Evolution

A model for generating daily spatial correlated rainfall fields suitable for evaluating the impacts of climate change on water resources is presented. The model, termed Stochastic Rainfall Generating Process, is designed to incorporate two major nonstationarities: changes in the frequencies of different precipitation generating mechanisms (frontal and convective), and spatial nonstationarities caused by interactions of mesoscale atmospheric patterns with topography (orographic effects). These nonstationarities are approximated as discrete sets of the time-stationary Stochastic Rainfall Generating Process, each of which represents the different spatial patterns of rainfall (including its variation with topography) associated with different atmospheric circulation patterns and times of the year (seasons). Each discrete Stochastic Rainfall Generating Process generates daily correlated rainfall fields as the product of two random fields. First, the amount of rainfall is generated by a transformed Gaussian process applying sequential Gaussian simulation. Second, the delimitation of rain and no-rain areas (intermittence process) is defined by a binary random function simulated by sequential indicator simulations. To explore its applicability, the model is tested in the Upper Guadiana Basin in Spain. The result suggests that the model provides accurate reproduction of the major spatiotemporal features of rainfall needed for hydrological modeling and water resource evaluations. The results were significantly improved by incorporating spatial drift related to orographic precipitation into the model.     

History matching of approximated lithofacies models under uncertainty

In history matching of lithofacies reservoir model, we attempt to find multiple realizations of lithofacies configuration that are conditional to dynamic data and representative of the model uncertainty space. This problem can be formalized in the Bayesian framework. Given a truncated Gaussian model as a prior and the dynamic data with its associated measurement error, we want to sample from the conditional distribution of the facies given the data. A relevant way to generate conditioned realizations is to use Markov chains Monte Carlo (MCMC). However, the dimensions of the model and the computational cost of each iteration are two important pitfalls for the use of MCMC. Furthermore, classical MCMC algorithms mix slowly, that is, they will not explore the whole support of the posterior in the time of the simulation. In this paper, we extend the methodology already described in a previous work to the problem of history matching of a Gaussian-related lithofacies reservoir model. We first show how to drastically reduce the dimension of the problem by using a truncated Karhunen-Loève expansion of the Gaussian random field underlying the lithofacies model. Moreover, we propose an innovative criterion of the choice of the number of components based on the connexity function. Then, we show how we improve the mixing properties of classical single MCMC, without increasing the global computational cost, by the use of parallel interacting Markov chains. Applying the dimension reduction and this innovative sampling method drastically lowers the number of iterations needed to sample efficiently from the posterior. We show the encouraging results obtained when applying the methodology to a synthetic history-matching case.