Characterization and reconstruction of a rock fracture surface by geostatistics

It is well understood that, in studying the mechanical and hydromechanical behaviour of rock joints, their morphology must be taken into account. A geostatistical approach has been developed for characterizing the morphology of fracture surfaces at a decimetre scale. This allows the analysis of the spatial variability of elevations, and their first and second derivatives, with the intention of producing a model that gives a numerical three‐dimensional (3D) representation of the lower and upper surfaces of the fracture. Two samples (I and II) located close together were cored across a natural fracture. The experimental data are the elevations recorded along profiles (using recording steps of 0.5 and 0.02 mm, respectively, for the samples I and II). The goal of this study is to model the surface topography of sample I, so getting estimates for elevations at each node of a square grid whose mesh size will be, for mechanical purposes, no larger than the recording step. Since the fracture surface within the sample core is not strictly horizontal, geostatistical methods are applied to residuals of elevations of sample I. Further, since structural information is necessary at very low scale, theoretical models of variograms of elevations, first and second derivatives are fitted using data of both that sample I and sample II. The geostatistical reconstructions are computed using kriging and conditional simulation methods. In order to validate these reconstructions, variograms and distributions of experimental data are compared with variograms and distributions of the fitted data. Copyright © 2002 John Wiley & Sons, Ltd.     

Application of spatial filter theory to kriging

Data-processing requirements for remotely sensed, digital images include spatial filtering to suppress image noise, enhance edges/contacts, and improve image clarity. Spatial filter theory demonstrates that the addition of a high-pass filtered image to a low-pass filtered image yields the original digital image. Application of this principle in kriging can be accomplished by using the same covariance matrix to solve for two weighting vectors to yield a result analogous to low- and high-pass filtering. The addition of kriged estimates calculated using both weighting vectors is analogous to summing high-, and low-pass filtered digital images. This modified method of kriging yields estimates associated with less smoothing compared to ordinary kriging. Statistical moments of original sample data are better preserved through estimation by this method.     

Spatial continuity measures for probabilistic and deterministic geostatistics

Geostatistics has traditionally used a probabilistic framework, one in which expected values or ensemble averages are of primary importance. The less familiar deterministic framework views geostatistical problems in terms of spatial integrals. This paper outlines the two frameworks and examines the issue of which spatial continuity measure, the covarianceC (h) or the variogram (h), is appropriate for each framework. AlthoughC (h) and (h) were defined originally in terms of spatial integrals, the convenience of probabilistic notation made the expected value definitions more common. These now classical expected value definitions entail a linear relationship betweenC (h) and (h); the spatial integral definitions do not. In a probabilistic framework, where available sample information is extrapolated to domains other than the one which was sampled, the expected value definitions are appropriate; furthermore, within a probabilistic framework, reasons exist for preferring the variogram to the covariance function. In a deterministic framework, where available sample information is interpolated within the same domain, the spatial integral definitions are appropriate and no reasons are known for preferring the variogram. A case study on a Wiener-Levy process demonstrates differences between the two frameworks and shows that, for most estimation problems, the deterministic viewpoint is more appropriate. Several case studies on real data sets reveal that the sample covariance function reflects the character of spatial continuity better than the sample variogram. From both theoretical and practical considerations, clearly for most geostatistical problems, direct estimation of the covariance is better than the traditional variogram approach.This paper was presented at MGUS 87 Conference, Redwood City, California, 14 April 1987.     

Use of indicator variograms for an enhanced spatial analysis

Flat variograms often are interpreted as representing a lack of spatial autocorrelation. Recent research in earthquake engineering shows that nearby field noise can substantially mask a prominent spatial autocorrelation and result in what appears to be a purely random spatial process. A careful selection of threshold in assigning an indicator function can yield an indicator variogram which reveals underlying spatial autocorrelation. Although this application involves use of seismic data, the results are relevant to geostatistical applications in general.     

The integral of the semivariogram: A powerful method for adjusting the semivariogram in geostatistics

A good fining of the structural junction that describes the variability of a spatial phenomenon is an essential stage in the building of an accurate estimator by kriging. The technique of the integral of the semivariogram (ISV) makes it possible to find this structural function while overcoming the problem of grouping together the pairs of experimental points into classes of distances when the data are not sampled on a regular grid. The ISV is particularly useful when the dispersion of the values of the classical Semivariogram (SV) makes it difficult to fit a model. Since the ISV is composed of a large number of values, it is more continuous than a SV and therefore easier to fit analytically. In fact, when the general shape of the SV is known, the ISV method proves its worth in finding the parameters that best fit a given variogram model. The analytical models of ISV which will be used, are the integral expressions of the traditional analytical SV. In this paper and on the basis of hydrogeological examples, we propose a method to adjust all the parameters of each model. The first derivative of a filled ISV, used in the kriging equations, appears to be systematically the best SV for a cross-validation on the data. This is why we think that the ISV technique should be used when the strong spatial variability of a parameter spreads out the values of the experimental SV.     

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.     

Spatial analysis of extension fracture systems: A process modeling approach

Little consensus exists on how best to analyze natural fracture spacings and their sequences. Field measurements and analyses published in geotechnical literature imply fracture processes radically different from those assumed by theoretical structural geologists. The approach adopted in this paper recognizes that disruption of rock layers by layer-parallel extension results in two spacing distributions, one representing layer-fragment lengths and another separation distances between fragments. These two distributions and their sequences reflect mechanics and history of fracture and separation. Such distributions and sequences, represented by a 2 × nmatrix of lengths L,can be analyzed using a method that is history sensitive and which yields also a scalar estimate of bulk extension, e(L).The method is illustrated by a series of Monte Carlo experiments representing a variety of fracture-and-separation processes, each with distinct implications for extension history. Resulting distributions of e(L) are process-specific, suggesting that the inverse problem of deducing fracture-and-separation history from final structure may be tractable.     

基于GIS空间分析的虚拟钻孔确定与实现方法

在三维地质建模过程中,通过引入一定数量的虚拟钻孔,对地层分界面几何形态更加精确的控制,不仅能够明显改善地质分界面网格的质量,而且能够提高模型精度和可视化效果。现有虚拟钻孔的引入途径主要通过手工添加实现,受人为主观因素影响较大,且自动化水平低,耗时、费力,针对这一问题,提出了一种基于GIS空间分析的虚拟钻孔确定与实现方法。以地质勘查获取的钻孔数据和地质剖面图为数据源,在ArcGIS平台的支持下,综合运用自动矢量化（Arc Scan）、要素转点（Feature To Point）、创建TIN （Create Tin）、TIN转三角形（Tinto Triangle）、泰森多边形（Create Thiessen Polygons）和相交（Intersect）等空间分析和文件转换工具,分别生成断层构造和建模区域内的二维（2D）虚拟钻孔点;而后,在利用插值工具生成各地层栅格表面的基础上,结合图形插值（Interpolate Shape）工具将2D虚拟钻孔点投影为三维（3D）虚拟钻孔;然后,利用Model Builder可视化建模工具设计了生成虚拟钻孔的模型,实现虚拟钻孔创建的自动化;最后,将获得的3D虚拟钻孔应用于三维地质建模,并进行了建模对比实验,结果表明,该方法具有一定有效性、实用性和可操作性。     

土性参数的随机场模型及桩体沉降变异特性分析

考虑土体参数的空间变异性,建立了土体变形模量的随机场模型并推导了方差拆减系数计算公式,研究了竖向和径向自相关距离对桩体沉降变异性的影响。     

A Bayesian framework for updating model parameters while considering spatial variability

The study presents a recent slope failure in India which resulted in the burial of a village and claimed large number of lives. Current methods of probabilistic back analysis incorporate uncertainty in the analysis but do not consider spatial variability. In this study, back analysis is performed using Bayesian analysis in conjunction with random field theory. The probabilistic method is shown to be efficient in back-analysing a slope failure. It also provides confidence in parameter values to be used for post-failure slope design. The back analysis method which does not consider spatial variability overestimates the uncertainty in analysis, which can lead to uneconomical slope remediation design and measures.     

节理产状分组的k均值聚类分析及其分组结果的费歇尔分布验证法

裂隙岩体内的节理往往是变化的,有时甚至变化很大。测得大量节理产状以后,对其分组是一项基础性的工作,但是如何分组目前还没有很好的理论和方法。采用快速聚类分析的方法,将节理产状的样本数据划分为不同的簇,利用极大似然估计的原理,通过数值方法求解费歇尔概率分布模型的参数,并用皮尔逊检验说明了费歇尔逊概率模型的有效性。通过不同簇的概率模型计算及其簇心的分布特性比较说明了簇数分类的适度性。利用所提出的快速聚类分析方法及其推导的计算公式,可以方便地求解费歇尔分布的参数。结合皮尔逊检验,形成了解决这类问题的系统实用的方法。     

改进遗传算法和支持向量机的岩体结构面聚类分析

岩体结构面控制着岩质边坡和地下洞室等岩体工程的稳定性,在岩体力学及水力学分析中起到关键作用。为对岩体结构面进行合理分组,精确地模拟岩体结构面网络的分布,提出一种融合改进遗传算法和支持向量机的聚类方法。首先,根据岩体结构面产状信息建立结构面分组的数学模型,采用改进的遗传算法计算结构面样本的全局最优聚类中心,再以聚类中心为训练样本,利用支持向量机方法将结构面样本进行完全划分。通过随机产生的结构面数据以及实际工程的运用表明,遗传-支持向量机聚类算法对岩体结构面的分组合理,获得的优势结构面结果可靠。     

Probabilistic analysis of reinforced slopes using RFEM and considering spatial variability of frictional soil properties due to compaction

Probabilistic stability analyses of constructed wrapped-face reinforced slopes (or embankments) using frictional soils were carried out using the random finite element method (RFEM). Soil properties reported in the literature for unsaturated frictional fills compacted to different densities were used in the simulations. Bar elements were added to the RFEM code to simulate extensible geosynthetic reinforcement layers and the Davis approach was used to improve numerical stability for purely frictional soil slopes at collapse. The influence of isotropic and anisotropic spatially variable soil strength was investigated and shown to have a large influence on the variation of maximum mobilised tensile forces in reinforcement layers for the steep 5 m-high slopes in the study. The influence of fill placed at different layer thickness and compacted to different levels was simulated by adjusting the soil strength and unit weight, and the vertical strength correlation length in the anisotropic spatially variable strength field used in each slope realisation. Numerical results showed that vertical strength correlation lengths approaching the magnitude of fill lift heights can control the probability of failure for reinforced slopes constructed with weak fills placed in lift heights close to but less than the wrapped reinforcement spacing used in the study.     

土工有限元分析前后处理的一种可视化方法

讨论了利用 Super SA P前后处理软件 ViziC AD前处理模块来实现有限元分析前处理的方法 ,重点介绍了后处理模块 SV IEW所使用的有限元数值分析结果的二进制数据文件 . DO和 . N SO的格式 ,并给出相应的子例程序段。 实例表明 ,可以利用 ViziC AD快捷地实现土工有限元计算结果的可视化 。     

A spatial analysis method for geochemical anomaly separation

One purpose of using statistical methods in exploration geochemistry is to assist exploration geologists in separating anomalies from background. This always involves two types of negatively associated errors of misclassification: type I errors occur when samples with background levels are rejected as background; and type II errors occur when samples with anomalous values are accepted as background. A new spatial statistical approach is proposed to minimize errors of total misclassification using a moving average technique with variable window radius. This method has been applied for geochemical anomaly enhancement and recognition as demonstrated by a case study of Au and Au-associated data for 698 stream sediment samples in the Iskut River area, northwestern British Columbia. Similar results were obtained using the fractal concentration-area method on the same data. By employing spatial information in the analysis, the process of selecting anomalies becomes less subjective than in more traditional approaches.     

考虑空间变异性条件下的边坡稳定可靠度高效敏感性分析

在有限数据条件下,可靠度敏感性分析是研究各种不确定性因素对边坡失稳概率影响规律的重要途径。基于直接蒙特卡洛模拟和概率密度加权分析方法提出了一种高效边坡稳定可靠度敏感性分析方法。所提出的方法通过随机场表征岩土体参数的空间变异性,并采用局部平均理论建立岩土体参数的缩维概率密度函数,用于概率密度加权分析中高效、准确地计算不同敏感性分析方案对应的边坡失稳概率。最后,通过一个工程案例--詹姆斯湾堤坝说明了所提出方法的有效性和准确性。结果表明：在敏感性分析过程中,所提出的方法只需要执行一次直接蒙特卡洛模拟,避免了针对不同敏感性分析方案重新产生随机样本和执行边坡稳定分析,节约了大量的计算时间和计算资源,显著提高了基于蒙特卡洛模拟的敏感性分析计算效率;在概率密度加权分析中采用岩土体参数的缩维概率密度函数能够准确地计算边坡失稳概率,避免了有偏估计,使概率密度加权分析方法适用于考虑空间变异性条件下的边坡稳定可靠度敏感性分析问题。     

结构方程模型及其在地学数据建模中的回顾与展望

结构方程模型是一种建立、 估计和检验因果关系的方法.它可以替代多重回归、 路径分析、 因子分析、 协方差分析等方法,清晰分析单项指标对总体的作用和单项指标间的相互关系,是一种主要应用于验证性模型分析的多元统计建模技术.由于能够通过可观测变量来度量潜变量得分以及分析不同子模型下潜变量之间的协同效应等优点,结构方程模型被广...     

Combinatorial analysis on spatial information statistics for the karst water environment in Guiyang, China

The karst groundwater system is extremely vulnerable and easily contaminated by human activities.To understand the spatial distribution of contaminants in the groundwater of karst urban areas and contributors to the contamination,this paper employs the spatial information statistics analysis theory and method to analyze the karst groundwater environment in Guiyang City.Based on the karst ground water quality data detected in 61 detection points of the research area in the last three years,we made Kriging evaluation isoline map with some ions in the karst groundwater,such as SO4 2-,Fe 3+,Mn 2+and F -,analyzed and evaluated the spatial distribution,extension and variation of four types of ions on the basis of this isoline map.The results of the analysis show that the anomaly areas of SO4 2-,Fe 3+,Mn 2+,Fand other ions are mainly located in Baba’ao,Mawangmiao and Sanqiao in northwestern Gui- yang City as well as in its downtown area by reasons of the original non-point source pollution and the contamination caused by human activities(industrial and domestic pollution).     

Estimation of rock mass strength and deformability in 3-D for a 30 m cube at a depth of 485 m at Äspö Hard Rock Laboratory

The rock fracture data provided by Swedish Nuclear Fuel and Waste Management Company were used to develop a 3-D stochastic fracture network model for a 30 m cube of Äspö diorite located at a depth of 485 m at Äspö Hard Rock Laboratory, Sweden. This fracture network model was validated. A new procedure is developed to estimate rock block strength and deformability in three-dimensions allowing for the anisotropy and incorporating the inherently statistical fracture geometry for the selected cube. The mean rock mass strength was found to be 47% of the mean intact rock strength of 297 MPa at 485 m depth. The mean rock mass modulus was found to be 51% of the intact rock Young's modulus of 73 GPa. The rock mass Poisson's ratio was found to be 21% higher than the intact rock Poisson's ratio of 0.28. These percentages indicate the level of weakening of the rock mass due to the presence of fractures. The ratio of mean major principal rock mass strength/mean minor principal rock mass strength turned out to be 1.28. The ratio of mean major principal rock mass modulus/mean minor principal rock mass modulus turned out to be 1.21.     

面向构造分析的赤平投影的计算机实现与应用

地壳中的矿产分布受一定的地质构造控制,正确认识区域构造与矿床构造特征,有助于对矿产资源进行科学预测与评价。在地质构造分析中,赤平投影是经常使用的分析方法,能定量计算节理的优势方位,直观、形象地展示节理的几何形态。运用构造分析理论,系统介绍通过吴氏网绘制节理赤平投影弧、通过施氏网计算节理优势方位的方法。使用C++语言实现了绘制赤平投影的程序,运用该程序处理胶西北多个金矿床的地质体产状数据,计算得到的优势方位走向符合实际区域地质构造。与Dips软件的处理结果进行比较,优势方位的误差在±2°以内,符合实际精度需求,验证了程序的正确性和实用性。