High-Order Data-Driven Spatial Simulation of Categorical Variables

Mathematical Geosciences - Modern approaches for the spatial simulation of categorical variables are largely based on multi-point statistical methods, where a training image is used to derive...     

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.     

Block Simulation of Multiple Correlated Variables

Numerical representations of multivariate natural phenomena, including characteristics of mineral deposits, petroleum reservoirs and geo-environmental attributes, need to consider and reproduce the spatial relationships between correlated attributes of interest. There are, however, only a few methods that can practically jointly simulate large size multivariate fields. This paper presents a method for the conditional simulation of a non-Gaussian vector random field directly on block support. The method is derived from the group sequential simulation paradigm and the direct block simulation algorithm which leads to the efficient joint simulation of large multivariate datasets jointly and directly on the block support. This method is a multistage process. First, a vector random function is orthogonalized with minimum/maximum autocorrelation factors (MAF). Blocks are then simulated by performing LU simulation on their discretized points, which are later back-rotated and averaged to yield the block value. The internal points are then discarded and only the block value is stored in memory to be used for further conditioning through a joint LU, resulting in the reduction of memory requirements. The method is termed direct block simulation with MAF or DBMAFSIM. A proof of the concept using an exhaustive data set demonstrates the intricacies and the performance of the proposed approach.     

Training Image Free High-Order Stochastic Simulation Based on Aggregated Kernel Statistics

Mathematical Geosciences - A training image free, high-order sequential simulation method is proposed herein, which is based on the efficient inference of high-order spatial statistics from the...     

高阶统计量方法在地震信号分析中的应用

高阶统计量分析是近20年来国内外信号处理领域的一个前沿课题,广泛应用于所有需要考虑非高斯、非最小相位、有色噪声、非线性或循环平稳性的各类问题中。在地震信号处理方面,应用高阶统计可以消除高斯有色噪声的影响和提取与识别非最小相位子波;基于高阶统计特征的独立分量分析己成为信号处理领域的一个研究热点并应用于地震信号分析中。对高阶统计量的定义和性质以及高阶谱时频分析方法作了介绍,并以此为理论依据,进行了理论模型和实际地震数据的实验。     

High-Order Block Support Spatial Simulation Method and Its Application at a Gold Deposit

Mathematical Geosciences - High-order sequential simulation methods have been developed as an alternative to existing frameworks to facilitate the modeling of the spatial complexity of non-Gaussian...     

Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics

In many earth sciences applications, the geological objects or structures to be reproduced are curvilinear, e.g., sand channels in a clastic reservoir. Their modeling requires multiple-point statistics involving jointly three or more points at a time, much beyond the traditional two-point variogram statistics. Actual data from the field being modeled, particularly if it is subsurface, are rarely enough to allow inference of such multiple-point statistics. The approach proposed in this paper consists of borrowing the required multiple-point statistics from training images depicting the expected patterns of geological heterogeneities. Several training images can be used, reflecting different scales of variability and styles of heterogeneities. The multiple-point statistics inferred from these training image(s) are exported to the geostatistical numerical model where they are anchored to the actual data, both hard and soft, in a sequential simulation mode. The algorithm and code developed are tested for the simulation of a fluvial hydrocarbon reservoir with meandering channels. The methodology proposed appears to be simple (multiple-point statistics are scanned directly from training images), general (any type of random geometry can be considered), and fast enough to handle large 3D simulation grids.     

A Fixed-Path Markov Chain Algorithm for Conditional Simulation of Discrete Spatial Variables

The Markov chain random field (MCRF) theory provided the theoretical foundation for a nonlinear Markov chain geostatistics. In a MCRF, the single Markov chain is also called a “spatial Markov chain” (SMC). This paper introduces an efficient fixed-path SMC algorithm for conditional simulation of discrete spatial variables (i.e., multinomial classes) on point samples with incorporation of interclass dependencies. The algorithm considers four nearest known neighbors in orthogonal directions. Transiograms are estimated from samples and are model-fitted to provide parameter input to the simulation algorithm. Results from a simulation example show that this efficient method can effectively capture the spatial patterns of the target variable and fairly generate all classes. Because of the incorporation of interclass dependencies in the simulation algorithm, simulated realizations are relatively imitative of each other in patterns. Large-scale patterns are well produced in realizations. Spatial uncertainty is visualized as occurrence probability maps, and transition zones between classes are demonstrated by maximum occurrence probability maps. Transiogram analysis shows that the algorithm can reproduce the spatial structure of multinomial classes described by transiograms with some ergodic fluctuations. A special characteristic of the method is that when simulation is conditioned on a number of sample points, simulated transiograms have the tendency to follow the experimental ones, which implies that conditioning sample data play a crucial role in determining spatial patterns of multinomial classes. The efficient algorithm may provide a powerful tool for large-scale structure simulation and spatial uncertainty analysis of discrete spatial variables.     

Simulation of Geological Contacts from Interpreted Geological Model Using Multiple-Point Statistics

Applications of multiple-point statistics (mps) algorithms to large non-repetitive geological objects such as those found in mining deposits are difficult because most mps algorithms rely on pattern repetition for simulation. In many cases, an interpreted geological model built from a computer-aided design system is readily available but suffers as a training image due to the lack of patterns repetitiveness. Porphyry copper deposits and iron ore formations are good examples of such mining deposits with non-repetitive patterns. This paper presents an algorithm called contactsim that focuses on reproducing the patterns of the contacts between geological types. The algorithm learns the shapes of the lithotype contacts as interpreted by the geologist, and simulates their patterns at a later stage. Defining a zone of uncertainty around the lithological contact is a critical step in contactsim, because it defines both the zones where the simulation is performed and where the algorithm should focus to learn the transitional patterns between lithotypes. A larger zone of uncertainty results in greater variation between realizations. The definition of the uncertainty zone must take into consideration the geological understanding of the deposit, and the reliability of the contact zones. The contactsim algorithm is demonstrated on an iron ore formation.     

基于地下水多变量空间聚类分析的变异性评价

在现存地下水监测网站中,观测站点分布的任意性、随意性和层次不清以及观测数据的冗余性等问题普遍存在,应用空间聚类原理,对所选研究区域廊坊地下水的监测点位及监测指标分别进行了空间聚类分析,对原始数据和经聚类处理后的数据分别进行了空间变异性评价,结果显示空间聚类分析是有效合理的。试图将空间变异性和空间聚类方法结合起来,为环境监测点的重新布置提供了理论依据,使提高监测效率与监测点的代表性、优化监测网格成为了可能;了解监测指标及监测点位在空间上的相关程度,为环境监测指标的确定提供理论依据,进而为环境管理、污染物控制以及环境资源的综合利用提供基础依据。     

基于Copulas函数的二维干旱变量联合分布

通过构建干旱变量的联合分布揭示干旱演变规律,可作为干旱分析的重要手段。基于8种单参数族的Copulas函数进行新疆乌鲁木齐和石河子气象站二维干旱变量的联合分布。经拟合优度评价:Frank Copula对干旱历时和干旱烈度、干旱历时和烈度峰值的拟合度最好;Clayton Copula对于干旱烈度和烈度峰值的拟合效果最好。二维变量联合超越概率值随单变量值的减小而增大;单变量的重现期介于二维变量联合重现期与同现重现期之间。表明Copulas函数能够描述二维干旱特征变量的联合分布。     

节理等密图的计算机化

用VB6语言进行节理等密图绘制编程，可以提高构造研究的效率和数据的精度。程序首先对节理产状进行转换得到施密特网中的直角坐标，再投图得到节理极密图；其次，对转换的坐标数据进行统计后可作出节理等值线图；最后，程序可自动确定出节理分布的多个优势中心。     

Stepwise Conditional Transformation for Simulation of Multiple Variables

Most geostatistical studies consider multiple-related variables. These relationships often show complex features such as nonlinearity, heteroscedasticity, and mineralogical or other constraints. These features are not handled by the well-established Gaussian simulation techniques. Earth science variables are rarely Gaussian. Transformation or anamorphosis techniques make each variable univariate Gaussian, but do not enforce bivariate or higher order Gaussianity. The stepwise conditional transformation technique is proposed to transform multiple variables to be univariate Gaussian and multivariate Gaussian with no cross correlation. This makes it remarkably easy to simulate multiple variables with arbitrarily complex relationships: (1) transform the multiple variables, (2) perform independent Gaussian simulation on the transformed variables, and (3) back transform to the original variables. The back transformation enforces reproduction of the original complex features. The methodology and underlying assumptions are explained. Several petroleum and mining examples are used to show features of the transformation and implementation details.     

High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

The spatial distributions of earth science and engineering phenomena under study are currently predicted from finite measurements and second-order geostatistical models. The latter models can be limiting, as geological systems are highly complex, non-Gaussian, and exhibit non-linear patterns of spatial connectivity. Non-linear and non-Gaussian high-order geostatistics based on spatial connectivity measures, namely spatial cumulants, are proposed as a new alternative modeling framework for spatial data. This framework has two parts. The first part is the definition, properties, and inference of spatial cumulants—including understanding the interrelation of cumulant characteristics with the in-situ behavior of geological entities or processes, as examined in this paper. The second part is the research on a random field model for simulation based on its high-order spatial cumulants. Mathematical definitions of non-Gaussian spatial random functions and their high-order spatial statistics are presented herein, stressing the notion of spatial cumulants. The calculation of spatial cumulants with spatial templates follows, including anisotropic experimental cumulants. Several examples of two- and three-dimensional images, including a diamond bearing kimberlite pipe from the Ekati Mine in Canada, are analyzed to assess the relations between cumulants and the spatial behavior of geological processes. Spatial cumulants of orders three to five are shown to capture directional multiple-point periodicity, connectivity including connectivity of extreme values, and spatial architecture. In addition, they provide substantial information on geometric characteristics and anisotropy of geological patterns. It is further shown that effects of complex spatial patterns are seen even if only subsets of all cumulant templates are computed. Compared to second-order statistics, cumulant maps are found to include a wealth of additional information from underlying geological patterns. Further work seeks to integrate this information in the predictive capabilities of a random field model.     

Determination of Joint Roughness Coefficients Using Roughness Parameters

This study used precisely digitized standard roughness profiles to determine roughness parameters such as statistical and 2D discontinuity roughness, and fractal dimensions. Our methods were based on the relationship between the joint roughness coefficient (JRC) values and roughness parameters calculated using power law equations. Statistical and 2D roughness parameters, and fractal dimensions correlated well with JRC values, and had correlation coefficients of over 0.96. However, all of these relationships have a 4th profile (JRC 6–8) that deviates by more than ±5 % from the JRC values given in the standard roughness profiles. This indicates that this profile is statistically different than the others. We suggest that fractal dimensions should be measured within the entire range of the divider, instead of merely measuring values within a suitable range. Normalized intercept values also correlated with the JRC values, similarly to the fractal dimension values discussed above. The root mean square first derivative values, roughness profile indexes, 2D roughness parameter, and fractal dimension values decreased as the sampling interval increased. However, the structure function values increased very rapidly with increasing sampling intervals. This indicates that the roughness parameters are not independent of the sampling interval, and that the different relationships between the JRC values and these roughness parameters are dependent on the sampling interval.     

Evaluation of Indirect Tensile Strength of Track Ballast Using Weibull Statistics

Iraq is planning to expand and rehabilitate its current railway network, thus demanding huge amounts of proper ballast material. There are three common quarries in Iraq, Sinjaar in the northern sector, Al-Qaim in western desert and Najaf-sea in the middle and southern parts of Iraq. The selection of proper track ballast material is of prime importance in design and construction of railway projects. The single particle-crushing test is a simple indirect tensile test that provides useful data for the selection of the material. Two types of railway ballast materials from (Najaf-sea and Al-Qaim) quarries are commonly used in the construction of railway in Iraq. The single particle-crushing test was performed for various ballast sizes, and the obtained data were analyzed using Weibull analysis. Najaf-sea crushed stones indicate higher tensile strength for all particle sizes compared to corresponding particle sizes of Al-Qaim crushed stone. In addition, an inverse linear relationship had obtained, between tensile strength and particle size with R ² (0.837 and 0.979) for Al-Qaim and Najaf-sea crushed stone, respectively, such relationships are important in evaluating the suitability of the materials in terms of the survival probability     

Extrapolating the Fractal Characteristics of an Image Using Scale-Invariant Multiple-Point Statistics

The resolution of measurement devices can be insufficient for certain purposes. We propose to stochastically simulate spatial features at scales smaller than the measurement resolution. This is accomplished using multiple-point geostatistical simulation (direct sampling in the present case) to interpolate values at the target scale. These structures are inferred using hypothesis of scale invariance and stationarity on the spatial patterns found at the coarse scale. The proposed multiple-point super-resolution mapping method is able to deal with “both continuous and categorical variables,” and can be extended to multivariate problems. The advantages and limitations of the approach are illustrated with examples from satellite imaging.