Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics

Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial connectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes.     

High-order Stochastic Simulation of Complex Spatially Distributed Natural Phenomena

Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data. Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity in the data used, in addition to high order.     

High-Order Spatial Simulation Using Legendre-Like Orthogonal Splines

High-order sequential simulation techniques for complex non-Gaussian spatially distributed variables have been developed over the last few years. The high-order simulation approach does not require any transformation of initial data and makes no assumptions about any probability distribution function, while it introduces complex spatial relations to the simulated realizations via high-order spatial statistics. This paper presents a new extension where a conditional probability density function (cpdf) is approximated using Legendre-like orthogonal splines. The coefficients of spline approximation are estimated using high-order spatial statistics inferred from the available sample data, additionally complemented by a training image. The advantages of using orthogonal splines with respect to the previously used Legendre polynomials include their ability to better approximate a multidimensional probability density function, reproduce the high-order spatial statistics, and provide a generalization of high-order simulations using Legendre polynomials. The performance of the new method is first tested with a completely known image and compared to both the high-order simulation approach using Legendre polynomials and the conventional sequential Gaussian simulation method. Then, an application in a gold deposit demonstrates the advantages of the proposed method in terms of the reproduction of histograms, variograms, and high-order spatial statistics, including connectivity measures. The C++ course code of the high-order simulation implementation presented herein, along with an example demonstrating its utilization, are provided online as supplementary material.     

Validation Techniques for Geological Patterns Simulations Based on Variogram and Multiple-Point Statistics

Traditional simulation methods that are based on some form of kriging are not sensitive to the presence of strings of connectivity of low or high values. They are particularly inappropriate in many earth sciences applications, where the geological structures to be simulated are curvilinear. In such cases, techniques allowing the reproduction of multiple-point statistics are required. The aim of this paper is to point out the advantages of integrating such multiple-statistics in a model in order to allow shape reproduction, as well as heterogeneity structures, of complex geological patterns to emerge. A comparison between a traditional variogram-based simulation algorithm, such as the sequential indicator simulation, and a multiple-point statistics algorithm (e.g., the single normal equation simulation) is presented. In particular, it is shown that the spatial distribution of limestone with meandering channels in Lecce, Italy is better reproduced by using the latter algorithm. The strengths of this study are, first, the use of a training image that is not a fluvial system and, more importantly, the quantitative comparison between the two algorithms. The paper focuses on different metrics that facilitate the comparison of the methods used for limestone spatial distribution simulation: both objective measures of similarity of facies realizations and high-order spatial cumulants based on different third- and fourth-order spatial templates are considered.     

A Multiple Training Image Approach for Spatial Modeling of Geologic Domains

Characterization of complex geological features and patterns remains one of the most challenging tasks in geostatistics. Multiple point statistics (MPS) simulation offers an alternative to accomplish this aim by going beyond classical two-point statistics. Reproduction of features in the final realizations is achieved by borrowing high-order spatial statistics from a training image. Most MPS algorithms use one training image at a time chosen by the geomodeler. This paper proposes the use of multiple training images simultaneously for spatial modeling through a scheme of data integration for conditional probabilities known as a linear opinion pool. The training images (TIs) are based on the available information and not on conceptual geological models; one image comes from modeling the categories by a deterministic approach and another comes from the application of conventional sequential indicator simulation. The first is too continuous and the second too random. The mixing of TIs requires weights for each of them. A methodology for calibrating the weights based on the available drillholes is proposed. A measure of multipoint entropy along the drillholes is matched by the combination of the two TIs. The proposed methodology reproduces geologic features from both TIs with the correct amount of continuity and variability. There is no need for a conceptual training image from another modeling technique; the data-driven TIs permit a robust inference of spatial structure from reasonably spaced drillhole data.     

Kernel Principal Component Analysis for Efficient,Differentiable Parameterization of Multipoint Geostatistics

This paper describes a novel approach for creating an efficient, general, and differentiable parameterization of large-scale non-Gaussian, non-stationary random fields (represented by multipoint geostatistics) that is capable of reproducing complex geological structures such as channels. Such parameterizations are appropriate for use with gradient-based algorithms applied to, for example, history-matching or uncertainty propagation. It is known that the standard Karhunen–Loeve (K–L) expansion, also called linear principal component analysis or PCA, can be used as a differentiable parameterization of input random fields defining the geological model. The standard K–L model is, however, limited in two respects. It requires an eigen-decomposition of the covariance matrix of the random field, which is prohibitively expensive for large models. In addition, it preserves only the two-point statistics of a random field, which is insufficient for reproducing complex structures. In this work, kernel PCA is applied to address the limitations associated with the standard K–L expansion. Although widely used in machine learning applications, it does not appear to have found any application for geological model parameterization. With kernel PCA, an eigen-decomposition of a small matrix called the kernel matrix is performed instead of the full covariance matrix. The method is much more efficient than the standard K–L procedure. Through use of higher order polynomial kernels, which implicitly define a high-dimensionality feature space, kernel PCA further enables the preservation of high-order statistics of the random field, instead of just two-point statistics as in the K–L method. The kernel PCA eigen-decomposition proceeds using a set of realizations created by geostatistical simulation (honoring two-point or multipoint statistics) rather than the analytical covariance function. We demonstrate that kernel PCA is capable of generating differentiable parameterizations that reproduce the essential features of complex geological structures represented by multipoint geostatistics. The kernel PCA representation is then applied to history match a water flooding problem. This example demonstrates that kernel PCA can be used with gradient-based history matching to provide models that match production history while maintaining multipoint geostatistics consistent with the underlying training image.     

基于地质空间数据挖掘的区域成矿预测方法

以多源地质空间数据库和空间数据挖掘技术为基础,顾及地质数据的空间特征和不确定性,提出一种基于地质空间数据挖掘的区域成矿预测方法,主要包括:连续型地质空间数据离散化、成矿空间关系提取及属性化、成矿关联规则提取及质量评价、成矿关联规则综合评价与潜力制图.最后,以青海省东昆仑地区铁矿资源潜力预测为例进行实验,并将地质空间数据...     

Using the Entropy of Curves to Segment a Time or Spatial Series

The concept of the thermodynamics of curves is used to analyze change in the variability of time or spatial series. More specifically, the entropy of a curve makes it possible to divide a nonstationary random field, because of a change in variance, into subdomains where data are said to be stationary. It is demonstrated that for time and spatial series the entropy of a curve is the slope of the cumulative sum of absolute differences. Numerical simulations show the efficiency of this tool. It can be shown that the presence of a linear or quadratic trend is without effect on the localization of the stationary subdomains. A practical case is studied with data collected from a tunnel boring machine where several parameters are recorded. This analysis can bring more information on the mechanical behavior of the different geological formations and explain or justify unplanned delays.     

A Comparison of Random Field Models Beyond Bivariate Distributions

In order to determine to what extent a spatial random field can be characterized by its low-order distributions, we consider four models (specifically, random spatial tessellations) with exactly the same univariate and bivariate distributions and we compare the statistics associated with various multiple-point configurations and the responses to specific transfer functions. The three- and four-point statistics are found to be the same or experimentally hardly distinguishable because of ergodic fluctuations, whereas change of support and flow simulation produce very different outcomes. This example indicates that low-order distributions may not discriminate between contending random field models, that simulation algorithms based on such distributions may not reproduce the spatial properties of a given model or training image, and that the inference of high-order distribution may require very large training images.     

Simulation of Non-Gaussian Transmissivity Fields Honoring Piezometric Data and Integrating Soft and Secondary Information

The conditional probabilities (CP) method implements a new procedure for the generation of transmissivity fields conditional to piezometric head data capable to sample nonmulti-Gaussian random functions and to integrate soft and secondary information. The CP method combines the advantages of the self-calibrated (SC) method with probability fields to circumvent some of the drawbacks of the SC method—namely, its difficulty to integrate soft and secondary information or to generate non-Gaussian fields. The SC method is based on the perturbation of a seed transmissivity field already conditional to transmissivity and secondary data, with the perturbation being function of the transmissivity variogram. The CP method is also based on the perturbation of a seed field; however, the perturbation is made function of the full transmissivity bivariate distribution and of the correlation to the secondary data. The two methods are applied to a sample of an exhaustive non-Gaussian data set of natural origin to demonstrate the interest of using a simulation method that is capable to model the spatial patterns of transmissivity variability beyond the variogram. A comparison of the probabilistic predictions of convective transport derived from a Monte Carlo exercise using both methods demonstrates the superiority of the CP method when the underlying spatial variability is non-Gaussian.     

CDFSIM: Efficient Stochastic Simulation Through Decomposition of Cumulative Distribution Functions of Transformed Spatial Patterns

Simulation of categorical and continuous variables is performed using a new pattern-based simulation method founded upon coding spatial patterns in one dimension. The method consists of, first, using a spatial template to extract information in the form of patterns from a training image. Patterns are grouped into a pattern database and, then, mapped to one dimension. Cumulative distribution functions of the one-dimensional patterns are built. Patterns are then classified by decomposing the cumulative distribution functions, and calculating class or cluster prototypes. During the simulation process, a conditioning data event is compared to the class prototype, and a pattern is randomly drawn from the best matched class. Several examples are presented so as to assess the performance of the proposed method, including conditional and unconditional simulations of categorical and continuous data sets. Results show that the proposed method is efficient and very well performing in both two and three dimensions. Comparison of the proposed method to the filtersim algorithm suggests that it is better at reproducing the multi-point configurations and main characteristics of the reference images, while less sensitive to the number of classes and spatial templates used in the simulations.     

基于GIS的矿产预测地质解译空间信息集成

利用我国海量地质标准基础数据库中的数字地质图和矿产图,通过基于GIS的地质解译空间集成地质信息,将其用于综合信息矿产预测。以地质解译系统对内蒙大兴安岭南段1∶20万成矿预测的应用为案例,阐述地质信息的空间提取与集成过程：首先在建立地质字典库实现地质空间信息共享的基础上,通过矿化密集区对地质模型的分类图层进行空间分析,建立地质成矿空间信息库和图库;然后,基于典型矿床圈定模型单元,通过模型单元与地质成矿空间信息库和图库的空间分析,建立地质找矿模型;最后,基于地质单元对地质成矿空间信息库和图库的二次空间集成,完成预测模型的地质空间信息提取与集成。将本方法应用在银矿案例的综合信息矿产预测靶区评价上,得到可供进一步查证的新增靶区比已知靶区增加了近5倍。     

基于连通性恢复的潜流带生态修复研究进展

潜流带是流域生态修复的关键区域之一, 潜流带修复的根本目标是恢复水系间的能量流通、物质传递和信息流动, 即恢复潜流带的连通性。对于潜流带连通性恢复而言, 应统筹考虑水文连通性、生态连通性和功能连通性等多层次的内容。潜流带生态修复相关研究主要基于流体动力学、地质学和生态学等基础理论, 剖析潜流驱动的生物地球化学耦合机制, 研发可促进潜流交换和恢复生物多样性的生态修复技术, 实现潜流带水文条件的改善与生物物种的恢复, 进而达到潜流带生态系统结构和功能综合性修复的目的。本文从潜流带水文连通性、生态连通性和功能连通性等多层次出发, 从潜流带流体动力学性能、介质性能、生物群落组成、食物网结构及环境生态功能等方面, 综述基于生态修复目标的潜流带连通性恢复理论与技术进展, 以实现潜流带生态系统整体稳定性的提升。在未来潜流带生态修复理论与应用研究中, 需发挥多学科交叉的优势, 耦合多组学方法对潜流带生态过程进行微观探索, 系统探究时间和空间尺度上潜流带生态修复过程的演替规律, 进一步构建多因素作用下的潜流带生态修复框架体系。     

中国北方荒漠化地质环境特征及其空间表达

通过开展面向决策者的荒漠化地质环境特征研究和空间表达 (编图与信息系统建设 ) ,反映影响荒漠化的主要地质环境要素及其空间分布 ,为研究荒漠化发生机制、发展趋势、治理对策提供地质依据。空间信息系统建设和编图突出地反映地质环境背景条件的演变对荒漠化发展的内在制约作用 ,以及地下水开发利用对荒漠化的影响。空间信息系统包括8个逻辑图层集 ,16个图层类型 ,百余个图元属性类型 ,在此基础上编制了5张成果图件。对荒漠化分布现状进行了信息化统计 ,中国北方地区荒漠化总面积为 197.24× 104km2 ,占区域面     

Genetic Modeling of GIS-Based Cell Clusters and Its Application in Mineral Resources Prediction

This paper presents a synthetic analysis method for multi-sourced geological data from geo-graphic information system (GIS). In the previous practices of mineral resources prediction, a usually adopted methodology has been statistical analysis of cells delimitated based on thoughts of random sam-pling. That might lead to insufficient utilization of local spatial information, for a cell is treated as a point without internal structure. We now take “cell dusters“, L e. , spatial associations of cells, as basic units of statistics, thus the spatial configuration information of geological variables is easier to be detected and utilized, and the accuracy and reliability of prediction are improved. We build a linear multi-discriminating model for the dusters via genetic algorithm. Both the right-judgment rates and the in-class vs. betweewclass distance ratios are considered to form the evolutional adaptive values of the population. An application of the method in gold mineral resoerces prediction in east Xinjiang, China is presented.     

A reconstruction method for three‐dimensional pore space using multiple‐point geology statistic based on statistical pattern recognition and microstructure characterization

To predict the macroscopic properties (e.g., transport, electromagnetic, and mechanical properties) of porous media, it is necessary to have a three‐dimensional (3D) representation of porous media. We reconstruct the geologically realistic 3D structure of Fontainebleau sandstone based on the two‐dimensional (2D) thin sections by using the multiple‐point statistics method. For this method, the size of template is an important parameter that reflects the perceived scale of spatial structure of porous media. In this paper, we take advantage of entropy method to obtain the appropriate size of the template, which is proven to be correct and feasible. The reconstruction method proposed by us combines successive 2D MPS simulations as well as 3D MPS simulation, which takes account into the pore structure information (e.g., heterogeneity and connectivity) both intralayer and interlayer. This reconstruction method is tested on Fontainebleau sandstone for which 3D images from micro‐CT scanning are available. Applying local percolation theory analysis, this new approach can depict the expected patterns of geological heterogeneities. In addition, it also can well reproduce a high degree of connectivity of the pore space better than other reconstruction methods based on lower‐order statistics. Copyright © 2011 John Wiley & Sons, Ltd.     

空间模式的广义自相似性分析与矿产资源评价

尺度不变性(scale invariance)包括自相似性(各向同性)、自仿射性(成层结构)、广义自相似性(各向异性标度不变性),是由各种地质过程和地质事件所产生的地质特征和模式的本质属性．尺度不变性可用分形和多重分形模型来表征．这些尺度特征的定量化可为刻画地质空问模式和模式识别提供有力的工具．例如。热液矿床的群聚现象可以用局部分形特征(局部奇异性)来刻画．通过在特征空问中(如频率空问)识别空问模式的广义自相似性．可以将空间混合模式进行分解或异常的识别．介绍了几种相关的分形模型和方法。包括度量空问模式广义尺度独立性(GSI)的线性模型;基于广义尺度独立性的异常分解S—A方法;度量空问模式的局部奇异性方法;以及如何利用分形特征预测未发现矿床的2种方法．有些方法已应用于许多矿产资源评价实例中．给出了对加拿大Nova Scotia省西南部湖泊沉积物样品中的4种元素As、Pb、Zn和Cu的地球化学数据处理分析结果。证明了局部奇异性分析和S—A异常分解方法对地球化学异常的增强和分离的有效性．研究表明：由S—A方法分解的异常往往具有多重分形的特点,而且普遍具有局部奇异性．研究区内具有明显奇异性的地区(元素含量富集区)是金矿异常区域。它们与金矿成矿作用和已知矿床的赋存密切相关．     

基于饱和渗透系数空间变异结构的斜坡渗流及失稳特征

以往研究一般采用单随机变量方法（SRV）或基于水平或垂直方向波动范围生成的空间变异随机场来模拟岩土参数的空间变异性,对具有倾斜定向特征的空间变异随机场未有涉及.基于条件模拟相关理论和非侵入式随机有限元的理论框架,提出了利用序贯高斯模拟方法进行斜坡参数条件随机场模拟并运用有限元方法进行斜坡渗流和稳定性分析的方法.针对理想边坡,对各向同性和几何各向异性的共7种空间变异结构的饱和渗透系数（K_s）各进行了200次条件随机场模拟,基于条件随机场模拟结果进行了有限元渗流和稳定性计算,对每种空间变异结构多次计算结果进行了统计分析.结果表明:本文所提出的方法不仅再现了研究区域参数的空间二阶统计特性,通过设定变异函数参数进行不同空间变异类型、变异程度、变异定向性的随机场模拟,同时利用现场观测数据对随机场模拟结果进行条件限制,从而提高了随机场的赋值精度;K_s的空间变异结构对孔隙水压力的分布规律、地下水位线变化范围、稳定性系数和最危险滑动面分布特征均有一定程度的影响.本研究为库岸斜坡稳定性评价提供方法支撑.      

地学模拟相关技术的研究与进展

地学模拟技术的一个发展方向是与地学过程分析密切结合,另一个发展方向是与数据可视化技术相结合。前者试图通过使用各种数学方法模拟地学随机现象,并对这些不良结构化或半结构化的地学问题进行定量化描述;后者运用计算机的三维可视化功能,在三维环境下将空间信息管理、地学解译、空间分析、地学统计与预测、三维图形可视化等技术相结合,实现计算可视化、分析可视化、过程可视化、结果可视化和决策可视化,并用于地学分析。回顾了地学中计算机三维地学建模技术、地质统计学和地学非线性现象模拟方法,并对该领域的发展进行了展望,认为加强地学模拟的理论体系、方法体系、技术体系的研究和实践既有着重要的理论意义,又有着重要的现实意义。