MODES-2D—Method of detecting strike-shift shear zones

One of the goals of using the Global Positioning System (GPS) and other geodetic survey techniques in tectonics has been to detect boundaries such as faults or shear zones between rigid or mildly deforming crustal masses. The calculation of infinitesimal strains and rotations with GPS data has been widely used to detect shear zones but it has been largely unsuccessful because infinitesimal strain and rotation, although useful in many other ways, is non-diagnostic of shear zones or faults. Our approach is to work with components of deformation, not strain, and to design specifically a diagnostic method of detecting shear zones. This paper introduces the first part of our method, the detection of two-dimensional, strike-shift shear zones (MODES-2D). The MODES-2D method has three elements: (1) determination of the orientation of a suspected strike-shift shear zone by analyzing components of a deformation tensor derived from a data set of displacements in an arbitrary coordinate system; (2) resolution of the deformation tensor into the coordinate system parallel and normal to the detected shear zone; and (3) exploration of the resolved data set for evidence for a belt of inhomogeneous deformation, which is an essential characteristic of a shear zone. The operation of MODES-2D is illustrated herein with a theoretical survey network across an ideal shear zone developed with a buried dislocation-fault and with a survey network afforded by the crossing of the Kaynaşlı viaduct by the 1999 Düzce–Bolu earthquake rupture in Turkey.     

Determination of the strain ellipsoid from measurements on any three sections

An ellipsoid is defined by, and may be re-constructed from, any three sections through it. In the field, calculation of the strain ellipsoid from general sections (two-dimensional strain ellipses determined from measured strain markers) is complicated by the fact that, due to experimental error and/or strain inhomogeneity, the three ellipses may not come from the same ellipsoid. The ellipses must first be adjusted to make them compatible. A method is suggested by which an adjustment ellipse is determined analytically for each of the three sections. Application of these adjustment ellipses makes the three sections compatible, and the strain ellipsoid may be determined. The principal axes of the ellipsoid are derived from the ellipsoid matrix by eigenvector analysis. Examples are given of practical applications of this method.     

基于矩张量的深埋隧洞岩爆机制分析

以锦屏二级水电站微震监测数据为基础,将矩张量引入到深埋隧洞岩爆机制分析研究中。对分析过程中遇到的2个关键问题提出了针对性的建议：① 隧洞工程为线性工程,传感器布置空间有限,难以形成良性传感器阵列,矩张量分析结果的可靠性受到很大的影响,为此,提出了数值计算过程中的坐标系空间旋转法。该方法按最优路径旋转隧道坐标系,使对于同一个事件的每个传感器,源-传感器射线到新坐标系的3个方向余弦差值在10倍数值范围以内,在数值计算过程中,可以得到较理想的矩张量结果。②针对Ohtsu使用矩张量分解判断岩石破裂类型的方法中,提出的各分量比重计算公式,仅适用于各分量均为受拉的情况。考虑快速掘进条件下隧洞工程受力方向的不确定性,扩展其分量比重计算公式到更一般适用状态。在此基础上,建立了一套完整的岩爆孕育过程矩张量分析方法。利用该方法对锦屏二级水电站典型岩爆微震数据进行分析,结果表明,该方法可以较好地解释岩爆孕育过程岩石破裂演化机制。     

航空重力梯度测量飞行姿态影响及误差校正

在石油和金属矿勘探中,相对于重力异常数据,航空重力梯度数据含有更多的高频信号成分,能更好的描述小的异常特征。为了得到地下异常体的真实重力梯度数据,需要全张量梯度测量数据中有效地去除飞机自身梯度值的影响。针对这一影响,提出线性的校正方法。考虑飞行姿态的改变引起的空间坐标系的变化,在校正过程中给出旋转矩阵和张量转换矩阵来实现坐标系的统一。通过模型实验分析了自身梯度值随姿态角变化情况,用实际梯度数据计算证明了该方法能够减小几个E的姿态误差,有效提高数据的精度。     

Equivalent continuum strain calculations based on 3D particle kinematic measurements of sand

Constitutive modeling of granular materials has been a subject of extensive research for many years. While the calculation of the Cauchy stress tensor using the discrete element method has been well established in the literature, the formulation and interpretation of the strain tensor are not as well documented. According to Bagi, 1 researchers mostly adopt well‐known continuum or discrete microstructural approaches to calculate strains within granular materials. However, neither of the 2 approaches can fully capture the behavior of granular materials. They are considered complementary to each other where each has its own strengths and limitations in solving granular‐mechanics problems. Zhang and Regueiro 2 proposed an equivalent continuum approach to calculating finite strain measures at the local level in granular materials subjected to large deformations. They used three‐dimensional discrete element method results to compare the proposed strains measures. This paper presents an experimental application of the Zhang and Regueiro 2 approach using three‐dimensional synchrotron microcomputed tomography images of a sheared Ottawa sand specimen. Invariant Eulerian finite strain measures were calculated for representative element volumes within the specimen. The spatial maps of Eulerian octahedral shear and volumetric strain were used to identify zones of intense shearing within the specimen and compared well with maps of incremental particle translation and rotation for the same specimen. The local Eulerian volumetric strain was compared to the global volumetric strains, which also can be considered as an averaging of all local Eulerian volumetric strains.     

仿射坐标系下的因子分析—仿射因子分析

传统的因子分析以正交坐标系为其基础坐标系,因而存在某种固有缺陷。文中提出以仿射坐标系为因子分析的基础坐标系,并进而给出了仿射因子分析的方法。新方法不仅消除了原方法所存在的缺陷,同时还实现了样品和变量空间的统一。     

The stress state of the Earth’s lithosphere: Results of statistical processing of the world stress-map data

A method for the statistical processing of the input data on the stress state of the Earth’s lithosphere that takes the initial 3-D position of the principle stress axes into account is elaborated. This approach is based on the calculation of the arithmetic mean value for every six independent tensor components during determination of the average stress in any sampling. When determining the sampling for calculation of the average stress for the current cell of the calculated grid, it is proposed to insert the measurements into the sampling that are spatially located in such a manner where the distance from the measurement point to the cell center is less than some value that is named as a search radius. The latter either was specified as a constant for all cells of the calculated grid or was determined assuming that the dispersion of the average tensor was less than some preset value. The results of the application of this approach are presented based on the example of the processing of the measurements from the World Stress Map (Heidbach et al., 2008). The resultant set of the mean stress-field maps reflects the generalized pattern of the stress distribution in the Earth’s lithosphere.     

Analysis of the Variability of a Two-Dimensional Finite Strain Estimate

Finite strain estimation is a widely used technique for the study of rock deformation in structural geology. One particular algorithm proposed by Shimamoto and Ikeda uses the ‘average shape matrix’ of deformed markers. This paper provides a detailed error analysis for resulting strain estimates in two dimensions. When the number of markers exceeds 100, estimators of components of the strain tensor are shown to have an approximately Gaussian distribution with variances that increase with their mean. Equal variance estimators are obtained by applying a log transform for the elongation and an arcsin transformation for the orientation estimates. Confidence interval formulae for strain tensor components are proposed. Lithology specific constants arising in these formulae are estimated from undeformed samples. The results are validated by application to simulated data as well as observational data from thin sections of sandstone sampled from SE Ireland.     

Density distribution techniques and strained length methods for determination of finite strains

For a homogeneously deformed rock composed initially of an isotropic distribution of object shapes, finite strain may be determined from the correlation between the orientations of either two-dimensional or one-dimensional sample cuts and the frequencies with which they intersect marker objects. Mimran previously published an incorrect method for planar samples under the heading ‘density distribution technique’. Methods are described by which the three-dimensional strain may be directly determined from six general samples, either linear or planar. Construction of two-dimensional ellipses as an intermediate step is unnecessary and enforces practical difficulties.These methods may be simplified by use of samples parallel to known principal axes or planes of the finite strain. In this case the same large errors may arise from slight misorientation of samples as with other methods of strain measurement. A new quick method is proposed, combining linear and planar measurements of frequencies of intersected objects, which is thought to be the first method to circumvent a large part of the error from this error source. For example, if true X:Z ratio is 9 : 1, and orientations in the XZ plane are misjudged by 8°, normal methods give 38% error where the new method gives, with care, an error of 1.9%. For methods of strain measurement such as are described here the concept of strain ellipsoid is unnecessarily limiting, and should be abandoned.     

岩土工程中位移监测方法及精度分析

在岩土工程位移监测中,选择哪种方法取决于监测时的条件;本文对监测中常用的测小角法、角度前方交会方法、距离前方交会方法、全站仪后方交会及极坐标综合方法等各种方法进行了研究。在目前各种文献中,一般都只是给出了监测点坐标的计算方法,充其量还给出监测点沿坐标轴方向的中误差以及点位中误差,而监测点其它方向的误差情况则无从计算。基于此,本文对以上各种监测方法的误差椭圆进行了推导,从而可以计算监测点沿各个方向误差并估算精度。     

空心包体应变计地应力分量计算方法及应用

为进一步改进和完善空心包体应变计地应力计算方法,基于地应力现场实测过程中空心包体应变计的安装方式,采用线性参数的最小二乘拟合方法对地应力分量计算公式进行了推导,得出了6个地应力分量的改进算法及其标准误差的计算公式。采用实现完全温度补偿并考虑岩体非线性的地应力测量技术,对弓长岭井下矿-160、-220、-280 m 3个水平3个测点进行了地应力现场实测,得到了3组孔壁应变数据。分别使用常规算法和改进算法对3个测点的地应力分量进行计算并分析其标准误差。结果表明：采用改进算法计算得到的地应力分量其标准误差普遍小于常规算法计算地应力分量产生的标准误差,说明改进算法比常规算法具有更高的可靠度。在此基础上,对弓长岭井下矿地应力赋存规律进行了研究。结果显示：矿区地应力场以水平构造应力为主导,最大水平主应力的走向总体上为南东东-北西西方向,最大水平主应力、最小水平主应力和垂直主应力均随深度呈增长关系。     

Bayesian Spatial Prediction for Discrete Closed Skew Gaussian Random Field

Most approaches in statistical spatial prediction assume that the spatial data are realizations of a Gaussian random field. However, this assumption is hard to justify for most applications. When the distribution of data is skewed but otherwise has similar properties to the normal distribution, a closed skew normal distribution can be used for modeling their skewness. Closed skew normal distribution is an extension of the multivariate skew normal distribution and has the advantage of being closed under marginalization and conditioning. In this paper, we generalize Bayesian prediction methods using closed skew normal distributions. A simulation study is performed to check the validity of the model and performance of the Bayesian spatial predictor. Finally, our prediction method is applied to Bayesian spatial prediction on the strain data near Semnan, Iran. The mean-square error of cross-validation is improved by the closed skew Gaussian model on the strain data.     

土钉墙内部稳定性分析方法改进

现有土钉墙滑动面的确定方法大多假定圆弧滑动面通过坡脚,但事实并非如此。文中总结了实际中可能出现的圆弧滑动面的4种类型,利用圆弧滑动面圆心坐标与圆弧上3个位置参数的关系建立了不通过坡脚的土钉墙最危险滑动面的搜索方法。根据土钉墙滑动面的类型,讨论和完善了建筑基坑支护技术规程中土钉墙整体稳定安全系数计算公式。在3个算例中将不同方法得到的滑动面和整体稳定安全系数进行对比分析,结果表明文中方法得到的滑动面与其他方法结果接近,说明文中方法的可行性;部分情况下文中得到的滑动面不通过坡脚,而传统方法假定滑动面通过坡脚,二者计算偏差较大;对于除传统方法得到的滑动面之外的3种滑动面类型,在相同滑动面条件下完善后的公式由于考虑了部分土体重力沿滑动面切向的抗滑力作用,计算出的整体稳定安全系数稍大。     

Error propagation by the Monte Carlo method in geochemical calculations

The Monte Carlo method of error propagation consists of repeated calculation of a quantity, each time varying the input data randomly within their stated limits of precision. The distribution of the calculated quantity then shows the effects of the imprecision of the data. It is an inefficient but very simple method of error propagation. Three examples are given—calculation of the free energy of a compound from phase equilibrium data; calculation of a reaction equilibrium temperature from free energy data; and the calculation of sulfide solubilities. Fifty iterations were generally found to be sufficient in these cases. The method is also useful where precision or accuracy of the data is not known, since the effect of various assumed possible errors is easily calculated.     

基于空间插值的工程岩体RQD预测方法

作为工程实际中评价岩体性质的重要指标,RQD的应用已相当广泛。但工程岩体中RQD值的分布存在一定的结构性和随机性,因此,通过对特定位置钻探获取的RQD值无法很到位地表达整个空间区域的实际岩石质量。为了弥补这方面的不足,统计学中的空间插值法为此提供了有力的手段。考虑到空间RQD分布存在各向异性,以及在工程坐标系中很难描述RQD分布的结构性,首先计算工程坐标系下已获取RQD值的主要黑塞方向（pHd）;然后构建空间旋转与比例缩放矩阵对其进行坐标标准化处理。在标准化的空间中,引入Kriging插值法,采用变异函数描述RQD在标准空间结构上的变化,建立其空间变异规律的数学模型,并对其进行插值计算;最后将标准空间进行坐标逆变换,实现了对工程坐标系下RQD分布的预测。利用R语言编程将该方法应用于工程实例,结果表明,这种空间插值法能够有效地预测工程岩体中RQD值的分布规律,弥补了局部钻探的局限性。     

Quantifying Three-Dimensional Silicate Fabrics in Cumulates Using Cumulative Distribution Functions

We present a new method for quantifying three-dimensional silicatefabrics and the associated uncertainties from grain orientationdata on three orthogonal sections. Our technique is appliedto the orientation of crystallographic features and, hence,yields a fabric related to the lattice-preferred orientation,although the method could be applied to shape-preferred orientationsor strain analysis based on passive linear markers. The orientationdata for each section are represented by their cumulative distributionfunction, and an iterative procedure is used to find the symmetricsecond-rank strain tensor that will simultaneously satisfy thecumulative distribution functions observed on each section.For samples with well-developed fabrics, this technique providesa much closer match to the sectional data than do previous techniquesbased on eigenparameter analysis of two-dimensional orientationdata. Robust uncertainty estimates are derived from a non-parametricbootstrap resampling scheme. The method is applied to two cumulates:one with a well-developed fabric and the other with a weak fabric,from the Stillwater complex, Montana. The silicate petrofabricorientations obtained for these samples compare favorably withindependent direct estimates of the volume fabric from electronbackscatter diffraction and magnetic techniques. KEY WORDS: cumulates; fabrics; quantitative; Stillwater complex; textural analysis     

Combining Areal and Point Data in Geostatistical Interpolation: Applications to Soil Science and Medical Geography

A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.     

花岗岩巴西劈裂试验的矩张量反演研究

为了研究岩石张拉破坏的过程和机制,首先通过模拟计算,验证了加权反演法在进行巴西劈裂试验矩张量求解方面的优越性;其次基于花岗岩的巴西劈裂试验,进行声发射监测、事件定位及矩张量反演,并分析标准反演法和加权反演法对应矩张量结果的差别;使用聚类分析K均值算法对矩张量结果进行分簇研究。研究结果表明：模拟计算中加权反演法可以降低巴西劈裂试验矩张量反演误差;室内花岗岩巴西劈裂试验中,加权反演法较大程度地优化了声发射事件的剪切成分以及压/拉应力轴的分布,使得反演结果更加合理。针对加权反演结果,所有事件可分成3簇,同簇内事件的破裂类型、应力状态、破裂机制、辐射模式等均相近;不同簇中的事件在时空上交叉分布,试验后期在圆盘东?西方向上具有较为明显张拉特性的事件骤增,于圆盘中心附近聚集并占据主导,最终导致试样的宏观破裂。研究结果解释了花岗岩巴西劈裂试验的破裂过程和破裂机制,为岩石力学行为研究提供进一步指导。     

A geometric approach to two-dimensional finite strain compatibility: interpretation and review

The purpose of this paper is to collect, clarify, augment and modify the authors' previous work on the subject of finite strain compatibility. The derivations of the fundamental equations are reviewed so that the geometric meaning of each step can be explained. Besides providing a basis for the geological interpretations of the equations, these derivations also lead to a useful new form of the strain compatibility equations.We begin by showing that compatibility is a geometric property of continuous and smooth coordinate grids, and we derive and explain the coordinate grid compatibility equations. We then use the fact that every finite deformation may be described by two coordinate grids to derive finite strain compatibility equations in principal coordinates and Cartesian coordinates. The resulting strain compatibility equations are not easily solved for general strain fields in any coordinate system. Nonetheless, we show that many common geological strain patterns have simple geometries for which the compatibility equations can be interpreted. For example, if a deformation has constant strain in one direction, as most shear zones do, then compatibility provides an iterative method for determining the strain throughout the deformed region if the strain is initially known at any one point. Some of the other strain geometries to which we apply compatibility in this paper include simple shear, inhomogeneous pure shear, parallel and similar folding.     

MT阻抗张量的正则分解和参数重建

对大地电磁（ＭＴ）阻抗张量正则分解的基本理论进行了阐述,重建了物理意义明确的参数体系。通过理论推导和等价性讨论,说明新参数与传统的坐标轴旋转法参数之间的对应关系,使这一数据处理方法能够成为用于大地电磁测深的实用技术。