岩土参数的空间变异性分析

岩土参数具有空间变异性和不确定性,本文将岩土参数视为具有随机性和结构性的区域化变量,利用区域化变量理论和地质统计方法分析上海地基土有代表性的抗剪强度指标的空间变异特性。结果表明该方法是可行的。     

单斜构造区岩溶倾斜洞穴发育趋势与岩层产状相关性识别

通过构建岩溶倾斜洞穴发育趋势面和岩层倾向面,探讨在单斜构造区岩溶倾斜洞发育趋势与岩层产状的相关性。首先,根据洞穴探测导线长度、方向、倾角等要素,以测量初始点为原点建立空间直角坐标系,将测量空间方位角转换为平面直角坐标系中的方向角,将测量数据投影在平面直角坐标系上,运用向量加法原理求出各测点平面向量及坐标,结合各测点的相对高差,运用趋势面分析方法进行一次拟合,构建洞道空间展布平面方程;其次,根据岩层产状信息在空间坐标系中构建平面方程;最后,通过计算两平面法向量的夹角,来判断洞穴空间展布平面和岩层产状平面的重合度,同时求出两平面与同一水平面的交线所成角来判断洞穴空间展布趋势面与岩层面的倾向关系。通过实例证明,该方法用于探讨单斜构造区岩层产状对倾斜洞穴发育趋势的影响识别具有一定的实用性。      

Geostatistical Space–Time Models: A Review

Geostatistical space–time models are used increasingly for addressing environmental problems, such as monitoring acid deposition or global warming, and forecasting precipitation or stream flow. Each discipline approaches the problem of joint space–time modeling from its own perspective, a fact leading to a significant amount of overlapping models and, possibly, confusion. This paper attempts an annotated survey of models proposed in the literature, stating contributions and pinpointing shortcomings. Stochastic models that extend spatial statistics (geostatistics) to include the additional time dimension are presented with a common notation to facilitate comparison. Two conceptual viewpoints are distinguished: (1) approaches involving a single spatiotemporal random function model, and (2) approaches involving vectors of space random functions or vectors of time series. Links between these two viewpoints are then revealed; advantages and shortcomings are highlighted. Inference from space–time data is revisited, and assessment of joint space–time uncertainty via stochastic imaging is suggested.     

An Embedded Model Estimator for Non-Stationary Random Functions Using Multiple Secondary Variables

Mathematical Geosciences - An algorithm for non-stationary spatial modelling using multiple secondary variables is developed herein, which combines geostatistics with quantile random forests to...     

Calculation of the Inverse of the Covariance

In reservoir characterization, the covariance is often used to describe the spatial correlation and variation in rock properties or the uncertainty in rock properties. The inverse of the covariance, on the other hand, is seldom discussed in geostatistics. In this paper, I show that the inverse is required for simulation and estimation of Gaussian random fields, and that it can be identified with the differential operator in regularized inverse theory. Unfortunately, because the covariance matrix for parameters in reservoir models can be extremely large, calculation of the inverse can be a problem. In this paper, I discuss four methods of calculating the inverse of the covariance, two of which are analytical, and two of which are purely numerical. By taking advantage of the assumed stationarity of the covariance, none of the methods require inversion of the full covariance matrix.     

Indicator kriging for unit vectors: Rock joint orientations

Indicator kriging (IK) is extended to analyze three-dimensional random unit vectors and evaluate the local probability distribution of rock joint orientations in geological formations. The pole vector representing joint orientations is regionalized and projected on a plane normal to the mean attitude of the joint family and centered at the mean. A two-dimensional cutoff system is developed to define the indicator variable, and corresponding indicator variograms and indicator kriging. The cutoff system defines probability regions similar to those of a bivariate distribution, concentric rings sliced into radial sectors. A case study made on an open pit mine proved positively the efficiency of IK and encourages its applications to localized probabilistic structural modeling for geotechnical or geohydrological analysis and oil and gas reservoir analysis.     

工程荷载作用下缓倾角反倾似层状岩质边坡变形稳定性分析

在反倾层状边坡的破坏机制和稳定性分析方面,由于其稳定性一般较好,系统深入地研究其破坏机制和稳定性的反而不多。以火山岩地层多次火山喷发旋回形成的缓倾角反倾似层状岩质边坡为研究对象,利用有限元分析其位移矢量图,揭示边坡的变形破坏机制和稳定性控制要素,利用极限平衡法、矢量和法,评价其稳定性,利用强度折减法分析塑性区的扩展规律,揭示滑动面发展的时间空间规律,结果表明,（1）类似层状反倾岩质边坡的潜在破坏模式为剪切–张拉破坏,滑面形态表现为近似折线状：顺坡向穿过凝灰岩夹层和强风化岩体,后缘通过陡倾结构面;（2）极限平衡法、矢量和法的边坡稳定性综合评价表明边坡浅层稳定性不满足工程稳定性要求,深层稳定性满足工程稳定性要求,且三维矢量和安全系数大于二维结果,是三维效应造成;（3）塑性区扩展揭示滑面的空间发展序列为滑面4、5、3。文中的技术路线和分析方案可用于缓倾角反倾层状岩质边坡稳定性评价。     

Geostatistics for Power Models of Gaussian Fields

This paper introduces geostatistical approaches (i.e., kriging estimation and simulation) for a group of non-Gaussian random fields that are power algebraic transformations of Gaussian and lognormal random fields. These are power random fields (PRFs) that allow the construction of stochastic polynomial series. They were derived from the exponential random field, which is expressed as Taylor series expansion with PRF terms. The equations developed from computation of moments for conditional random variables allow the correction of Gaussian kriging estimates for the non-Gaussian space. The introduced PRF geostatistics shall provide tools for integration of data that requires simple algebraic transformations, such as regression polynomials that are commonly encountered in the practical applications of estimation. The approach also allows for simulations drawn from skewed distributions.     

Bayesian Maximum Entropy Analysis and Mapping: A Farewell to Kriging Estimators?

The Bayesian Maximum Entropy (BME) method of spatial analysis and mapping provides definite rules for incorporating prior information, hard and soft data into the mapping process. It has certain unique features that make it a loyal guardian of plausible reasoning under conditions of uncertainty. BME is a general approach that does not make any assumptions regarding the linearity of the estimator, the normality of the underlying probability laws, or the homogeneity of the spatial distribution. By capitalizing on various sources of information and data, BME introduces an epistemological framework that produces predictive maps that are more accurate and in many cases computationally more efficient than those derived by traditional techniques. In fact, kriging techniques can be derived as special cases of the BME approach, under restrictive assumptions regarding the prior information and the data available. BME is a more rigorous approach than indicator kriging for incorporating soft data. The BME formulation, in fact, applies in a spatial or a spatiotemporal domain and its extension to the case of block and vector random fields is straightforward. New theoretical results are presented and numerical examples are discussed, which use the BME approach to account for important sources of knowledge in a systematic manner. BME can be useful in practical situations in which prior information can be used to compensate for the limited amount of measurements available (e.g., preliminary or feasibility study levels) or soft data are available that can be combined with hard data to improve mapping significantly. BME may be then viewed as an effort towards the development of a more general framework of spatial/temporal analysis and mapping, which includes traditional geostatistics as its limiting case, and it also provides the means to derive novel results that could not be obtained by traditional geostatistics.     

Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables

This paper proposes a stochastic response surface method for reliability analysis involving correlated non-normal random variables, in which the Nataf transformation is adopted to effectively transform the correlated non-normal variables into independent standard normal variables. Transformations of random variables that are often used in reliability analyses in terms of standard normal variables are summarized. The closed-form expressions for fourth to sixth order Hermite polynomial chaos expansions involving any number of random variables are formulated. The proposed method will substantially extend the application of stochastic response surface method for reliability problems. An example of reliability analysis of rock slope stability with plane failure is presented to demonstrate the validity and capability of the proposed stochastic response surface method. The results indicate that the proposed stochastic response surface method can evaluate the reliability of rock slope stability involving correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that for the first-order reliability method, and it is much more efficient than direct Monte-Carlo simulation. The results also show that the number of collocation points selected should ensure that the Hermite polynomial matrix has a full rank so that different order SRSMs can produce a robust estimation of probability of failure for a specified performance function. Generally, the accuracy of SRSM increases as the order of SRSM increases.     

变异函数在个旧锡矿X号矿体中的应用

变异函数是地质统计学的核心和基本工具。它既能描述区域化变量的空间结构性变化，又能描述其随机性变化，而且它的计算还是许多其它地质统计学计算的基础。在阐述了地质统计学原理和变异函数理论研究的基础上，根据个旧锡矿X号矿体的地质特征，对该矿体Sn品位进行了变异函数的模拟和结构分析，并在此基础上进行了地质解释。     

Interpretation of variability of rock mass rating by geostatistical analysis: a case study in Western Turkey

Geotechnical interpretation of drillholes for rock mass characterization is commonly based on the engineering judgment. This might result in misrepresentation of the rock mass at unsampled locations. This paper investigates the reproducibility of the spatial variability of rock mass quality (rock mass rating, RMR) using geostatistics. Geotechnical data from an exploratory phase mining project in Western Turkey that covers 2.7 km depth with a total of 14 drillholes composed of 700 samples was used to perform a geostatistical analysis. RMR values of positions, where geotechnical logging was missing, were estimated by applying inverse distance weighting (IDW) and the ordinary kriging (OK) methods. The validation process showed that the OK method is consistently estimating the RMR values better than IDW. The spatial variability of RMR values by geostatistical analysis could provide valuable information in the design and development of underground or surface structures.     

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.     

Implementation and evaluation of multivariate analysis for groundwater hydrochemistry assessment in arid environments: a case study of Hajeb Elyoun–Jelma, Central Tunisia

In most arid zones, groundwater (GW) is the major source of domestic, agriculture, drinking, and industrial water. Accordingly, the monitoring of its quality by different techniques and tools is a vital issue. The purpose of this paper is the evaluation of the combination of principal components analysis (PCA) and geostatistics as a technique for (1) identifying the processes affecting the groundwater chemistry of the detrital unconfined Middle Miocene Aquifer (MMA) of the Hajeb elyoun Jelma (HJ) aquifer (Tunisia) and (2) mapping the controlling variables for groundwater quality. This work is based on a limited database recorded in 22 wells tapping the aquifer and unequally distributed in the field. The proposed approach is carried out in two steps. In the first step, the application of PCA revealed that rock–water interaction, agriculture irrigation and domestic effluents could explain 85 % of the observed variability of the chemical GW quality of the MMA. As a result, two new variables are defined: V1 (rock–water interaction influence) and V2 (irrigation and domestic effluent influence). In the second step, the spatial variability of these variables over the extent of the MMA is mapped by applying a kriging interpolation technique. The results of this study suggest that, while both natural and anthropogenic processes contribute to the GW quality of the MMA, natural impacts can be considered as the most important.     

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.     

Applicability of Statistical Learning Algorithms for Spatial Variability of Rock Depth

Two algorithms are outlined, each of which has interesting features for modeling of spatial variability of rock depth. In this paper, reduced level of rock at Bangalore, India, is arrived from the 652 boreholes data in the area covering 220 sq⋅km. Support vector machine (SVM) and relevance vector machine (RVM) have been utilized to predict the reduced level of rock in the subsurface of Bangalore and to study the spatial variability of the rock depth. The support vector machine (SVM) that is firmly based on the theory of statistical learning theory uses regression technique by introducing ε-insensitive loss function has been adopted. RVM is a probabilistic model similar to the widespread SVM, but where the training takes place in a Bayesian framework. Prediction results show the ability of learning machine to build accurate models for spatial variability of rock depth with strong predictive capabilities. The paper also highlights the capability of RVM over the SVM model.     

岩体结构面强度的可靠度分析

基于岩体结构面破坏的Mohr-Coulomb准则和可靠度理论,视结构面和岩块强度参数以及结构面倾角为随机变量,分析了结构面强度参数的变异性对结构面破坏概率的影响,探讨了结构面破坏概率随结构面倾角的变化规律,提出了结构面和岩块破坏概率的计算公式,找出了影响结构面破坏概率的主要因素。结果表明,岩体结构面强度的可靠度分析比传统的定值分析方法更为客观合理。     

A Mathematical View of Weights-of-Evidence,Conditional Independence,and Logistic Regression in Terms of Markov Random Fields

New light is shed on mathematical methods of potential modeling from the point of view of Markov random fields. In particular, weights-of-evidence and logistic regression models are discussed in terms of graphical models possessing Markov properties, where the notion of conditional independence is essential, and will be related to log-linear models. While weights-of-evidence with respect to indicator predictor variables and logistic regression with unrestricted predictor variables model conditional probabilities of an indicator random target variable, the subject of log-linear models is the joint probability of random variables. The relationship to log-linear models leads to a likelihood ratio test of conditional independence, rendering an omnibus test of conditional independence restricted by a normality assumption obsolete. Moreover, it reveals a hierarchy of methods comprising weights-of-evidence, logistic regression without interaction terms, and logistic regression including interaction terms, where each former method is a special case of the consecutive latter method. The assumptions of conditional independence of all predictor variables given the target variable lead to logistic regression without interaction terms. Violations of conditional independence are compensated exactly by corresponding interaction terms, no cumbersome approximate corrections are needed. Thus, including interaction terms into logistic regression models is an appropriate means to account for lacking conditional independence. Logistic regression exempts from the burden to worry about lack of conditional independence. Eventually, the relationship to log-linear models renders logistic regression with indicator predictor variables optimum for discrete predictor variables. Weights-of-evidence applies for indicator predictor variables only, logistic regression applies without restrictions of the type of predictor variables and approximates the proper distribution in the general case.     

Risk analysis of fractured rock mass underground structures

Recent accidents in underground structures have raised the risk awareness of the geotechnical engineering community. Geotechnical design is subject to significant uncertainties in load and strength parameters as well as in engineering models. However, engineering models which objectively address such uncertainties in design are still scarce. This paper presents an objective framework for the quantification of the risks involved in underground structures excavated in fractured rock masses, where structural failures may occur due to block falls. The framework considers the structure as a distributed system, where falling block probabilities are integrated over the main structural dimension. Random block size and geometry, arising from random joint orientation, are taken into account, as well as uncertainties in joint strength and geometrical parameters. A cost function is used to quantify failure consequences in terms of the block size. The framework is demonstrated in an application to a case study involving a real structure: the Paulo Afonso IV power station cavern. Results of the case study show that the studied cavern presents high reliability and very low risk. The framework proposed herein is shown to be a practical tool for the risk evaluation of underground structures constructed in rock masses, such as caverns and tunnels.