Geometry of stress function surfaces for an asymmetric continuum

A two-dimensional stress field of dislocation or fault is geometrically studied for an asymmetric continuum. For geometric surfaces of the stress and couple-stress functions, the mean and Gaussian curvatures are derived. The mean curvature of couple-stress function surface is connected with the asymmetric of stress tensor. Moreover, the Gaussian curvature of stress function surface is characterized by both the stress and couple-stress. On the other hand, the mean curvature of stress function surface is not affected by the asymmetry of stress. Based on these geometric expressions, the Coulomb’s failure criterion and the friction coefficient are expressed by the curvatures of couple-stress function surface. Moreover, geometric structures of stress and couple stress function surfaces are shown for edge and wedge dislocations as faults. The curvatures of these surfaces show that the effect of couple-stress is constrained around the dislocations only.     

Spatially non‐local model of inelastic deformations: applications for rock failure problem

A spatially non‐local model for inelastic deformation of solids is proposed and studied. The non‐locality of deformation is taken into account by the additional parameter of state beyond the classical parameters such as stress and strain tensors. This additional parameter is the curvature tensor expressed in terms of the metric strain tensor, and it is called the failure parameter. In the case of small deformation, it is equivalent to the Saint‐Venant incompatibility tensor. Thermodynamic properties of the model are studied, and governing differential equations for spatially non‐local model are formulated, which are composed by the elasticity equations and parabolic equation for the failure parameter. The model can be applied to the study of the rock failure problem, and as an example, the one‐dimensional problem for the deformation of half‐plane loaded by the normal surface stress is studied. Stationary and non‐stationary formulations of the problem are considered, and qualitative agreement with available experimental data is observed.     

The Transition from Three-Dimensional Embedding to Two-Dimensional Euler-Lagrange Deformation Tensor of the Second Kind: Variation of Curvature Measures

Based on the Stein formulation of changes of curvature parameters, we describe the changes of Riemann two-dimensional manifolds of surface geometries. The three-dimensional left and right Euclidean manifolds with the curvature parameters “geodetic curvature, normal curvature, geodetic torsion” characterize the embedding “Riemann to Euclid” or 2d into 3d. The variation in time of the Euler-Lagrange deformation tensor of the second kind, in short “curvature variation” is studied.     

Asymptotic Elastodynamic Green Function in the Kiss Singularity in Homogeneous Anisotropic Solids

The far-field asymptotic formula is derived for the elastodynamic Green function in the kiss singularity in homogeneous anisotropic solids. In contrast to standard asymptotics in regular directions the derived formula is more complex and expressed in the form of a 1-D integral. This integral is specified for the kiss singularity along the symmetry axis in transverse isotropy and along the fourfold symmetry axes in tetragonal and cubic symmetries. The shape of the slowness surface in the singularity is regular in transverse isotropy and the amplitude of the Green function is expressed by means of the Gaussian curvature of this surface in the singularity. However, the shape of the slowness surface is irregular and the Gaussian curvature is not defined in the singularity in tetragonal or cubic symmetries. In this case, the amplitude of the Green function is expressed by means of the generalized Gaussian curvature.     

Multivariate statistical analysis of deformation tensors: independent vs. correlated tensor observations

In the presence of errors in measuring a random displacement field (under the normal distribution assumption of displacement field), stochastic behaviors of principal components of deformation tensors (strain tensor and tensor of change of curvature (TCC)), based on the intrinsic assumption of geometrical modeling of surface deformation analysis, are discussed. We divided the contents into two parts: In the first, we considered independent random vectors of repeated tensor measurements. In the second step, we considered correlations among repeated measurements. Then, covariance components between tensor elements by Helmert estimator, based on prior information of variance components, are estimated. As a case study, both assumptions are applied to the estimation of principal components of deformation rate tensor observations in Zagros region (Western Iran). The results of numerical analysis showed that greatest shortening is accommodated in oblique orientation (NS) with respect to the Main Recent Fault (MRF), northwest part of North Zagros, Central Iran block and MRF, respectively. Most of the extensions occurred in the east part of the belt. The pattern of eigenspace components of TCC shows highest positive values across the NW region, nearly in orthogonal direction to the MRF and Main Zagros Fault (MZF). The pattern has insignificant values in the Central Zagros. It takes the significant negative values across the SW part, especially along the SPF and Persian Gulf shore. The effect of non-independent observations on the estimation of eigenspace components of deformation tensors (strain tensor and TCC) shows that the estimation of covariance components has influence on the confidence intervals of eigenspace components, especially in seismically active regions of the belt (along the Persian Gulf shore, NW of the belt and region between the Central Iran block and MRF). The results demonstrate the importance of considering the correlation structure among the observations on statistical behavior of principal components of deformation tensors in seismically active regions.     

The magnetotelluric impedance tensor through Clifford algebras: part II — a constrained stochastic heuristic method for recovering the magnetotelluric regional impedance tensor in 2D/3D case

A method for obtaining the galvanic distortion matrix is presented so that the regional impedance tensor (free of distortion) is recovered. The method is a constrained stochastic heuristic method, which consists in randomly exploring the space of the distortion parameters. Constraints are imposed on the shortest periods of the regional impedance tensor that, at these short periods, tends to be two dimensional (or one dimensional). Depending on the constraints used, two different methods to recover the regional impedance tensor in this 2D/3D case are presented. Method 1 needs to find the strike of the short periods and Method 2 applies to the measurement directions. Twist, shear and anisotropy parameters are obtained. Thus, the regional impedance tensor is recovered with the only exception being the vertical shift due to the gain, which is equal for all the components of the tensor. Examples with synthetic impedance tensors from 2D/3D models perturbed with galvanic distortion are presented to illustrate how the algorithm works. The presence of noise in data is considered and rules for proceeding are provided. The same examples perturbed by Gaussian noise together with experimental data illustrate the capabilities of the algorithm.     

The fractal geometry of the soil—covered landscape

The soil-covered landscape surface can be idealized from two viewpoints. The intuitive view is of a smooth, absolutely continuous surface with continuous contour lines and measurable in integral dimensions. The alternative view emphasizes the roughness, a surface of little regularity and at the limit of no contours, the appropriate measure being that of fractional Hausdorff dimension. Regularity is a local property and both idealizations need to stop far short of the limit to avoid awkward consequences. The dichotomy of viewpoint can be matched in the theory of Gaussian random fields. These, if they are smooth, are very smooth but if they are irregular they are highly irregular (erratic); there is no middle ground. This Belayev dichotomy is defined and both modes applied to the soil-covered landscape. On the one hand, if the landscape is subject to a general diffusive type degradation or more generally a Davisian downwasting regime then the curvature of the landscape surface is progressively straightened and the distribution of gradient (increments) along a typical traverse will eventually adopt a Gaussian form. Then from the irregular viewpoint the surface is ultimately well represented by a fractional Brownian surface of low Hausdorff dimension (2·0 < dim < 2·3). The Hausdorff dimension is directly related to the entropy of the landscape and as degradation proceeds both quantities decrease in value. On the other hand, if the surface is regarded as smooth and well represented by an absolutely continuous Gaussian field then the mean value of the number of upcrossings of a level or the extent of an excursion set will also be Gaussian. This analysis is restricted to one dimension; the number of times a profile curve crosses or the amount of time it spends above any given level. Predictions from both viewpoints are substantially corroborated in a map analysis of 15 sites on varied terrains in Southern England and the map analysis checked against one based upon digital tape data for one of the sites.     

复杂地表条件下高斯波束叠前深度偏移(英文)

在复杂地表条件的区域,地震数据的采集和处理是一项极大的挑战。虽然可以通过静校正来消除起伏地表的影响,然而当地表高程以及近地表速度剧烈变化时,简单的垂直时移对地震波场造成的畸变会严重降低偏移成像的质量。基于射线的偏移方法可以直接在起伏地表面进行波场的延拓成像,是解决上述问题的有效手段。本文针对复杂地表条件下的高斯波束叠前深度偏移进行研究,对倾斜叠加公式进行修改,使之包含地表高程以及速度的信息,通过直接在复杂地表面进行平面波的合成,得到了一种具有更高成像精度的改进方法。首先简单介绍常规高斯波束偏移的基本原理和计算流程,并以此为基础,给出复杂地表条件下高斯波束偏移原有的实现方法以及本文的改进方法,最后通过模型和实际资料的试算验证本文方法的有效性。     

近地表地震地质复杂性的一种定量分析方法

复杂近地表散射衰减所有的地面观测波场,形成半随机半相干的近地表强散射噪音背景,弥漫整个炮集,淹没深层反射信号,是导致地震资料极低信噪比的主要原因.如何研究和评价近地表散射强度一直是石油勘探未解决的问题,这与起伏地表的粗糙度、近地表速度横向变化和结构倾角分布密切相关.基于前期复杂近地表边界元法波动方程数值模拟研究,本文提出一种复杂近地表散射振幅矩阵方法来分析近地表散射强度.首先对复杂近地表结构进行边界元配置方法离散,根据边界积分方程生成矩阵方程.我们不求解该矩阵方程(涉及海量计算),只是利用矩阵分析技术来解析矩阵方程中的散射振幅系数矩阵,研究复杂近地表结构对不同频率波场的散射强度.该方法利用边界元对近地表结构几何特征的精确表征,研究起伏地表和非规则地质分界面对地震波传播的影响,由基本解及其在边界上的法向导数经过高斯数值积分计算得到的散射振幅系数矩阵,不仅描述了任意两点之间的相互影响,同时还刻画了边界形状特征的影响,为评价不同地质结构的散射强度提供了可能性.作为初步评价手段,我们采用矩阵元素总和与矩阵维数之比作为表征散射振幅系数矩阵散射特征的标量复杂系数,通过理论和实际模型测试,形成了一套行之有效、计算快速的近地表复杂性分析方法.     

Electromagnetic induction of a three-dimensional conductivity inhomogeneity in the two-layered earth

Summary The calculation of the elements of Green's tensor function is presented for solving the problem of the electromagnetic induction by means of a vector integral equation. A two-layered Earth is considered as the medium, the surface layer including a three-dimensional conductivity inhomogeneity. Use is made of the boundary condition requiring the vertical component of the electric current to be zero at the Earth's surface which partly simplifies the theoretical computation. Long-period asymptotics of the individual complicated functions, occurring in Green's tensor function as well as in the tensor function required to calculate the components of the anomalous magnetic field at the surface of the halfspace, were effected. With the aid of these asymptotics one can obtain estimates of the functions occurring in the theoretical analysis of the problem.     

On the limits of linear moment tensor inversion of surface wave spectra

Some theoretical and practical limits to linear moment tensor inversion of surface waves are analyzed in detail, in particular when one or few stations are used for rapid determination of source parameters. The theory is briefly outlined and steps of preprocessing, especially corrections for source history and propagation, are discussed in order to guarantee the validity of the moment tensor inversion. The inverse problem is first studied from a theoretical point of view. Then the feasibility of the moment tensor inversion is tested from synthetic computations and the formalism is improved in order to obtain a better system conditioning. Finally, some cases of practical nonuniqueness of the solution are shown and possible restrictions to the applicability of the method are discussed. All steps are illustrated with the example of theM _s =6.9 Erzincan (Turkey) earthquake of March, 13, 1992.     

The propagation of surface waves in elastic mediums with slightly curved boundaries of sinusoidal type

Summary The effect of slightly curved boundaries (free surfaces and interfaces) of the elastic mediums on the components of displacement of a particle in a medium due to the propagation of the surface waves has been investigated in this paper. It has been found that along with the usual displacement components for stratified boundaries (designated here as primary components), there exist secondary displacement components arising from the presence of curvature in the boundaries. They are constituted of different harmonic components with their amplitudes proportional to the parameters which measure the extent of curvature of the boundaries. The wave numbers of these harmonic constituents are related to those of the primary components in a definite way decided by the shapes of the boundaries.     

基于高斯曲率模态相关系数的梁桥支座损伤识别研究

为方便地检测梁桥支座损伤,提出了利用运营桥梁实测模态位移结合其无损状态的模态位移判断支座损伤的高斯曲率模态相关系数法。通过简支梁桥的室内试验,验证了利用高斯曲率模态相关系数判定支座损伤的合理性以及该方法中无损状态下的模态位移可以通过模态试验和有限元模拟两种方法获得。利用该方法对实际简支梁桥和连续梁桥进行的支座损伤识别结果表明:高斯曲率模态相关系数法可准确识别出单支座和多支座损伤的支座损伤位置,具有较强的鲁棒性,可将此方法应用于实际工程中的支座损伤识别。     

重力全张量数据联合欧拉反褶积法研究及应用

全张量测量技术是在空中或海上用加载了多个加速度计的移动平台技术测量位场的五个独立分量.各张量分量包含不同方向的地下地质体信息,水平张量分量T_(xx)、T_(yy)、T_(xy)、T_(xz)、T_(yz)通常用于识别和映射与地质构造或地层变化有关的测量区域中的目标,垂直张量分量Tzz用于估计深度.然而,这些分量传统上是彼此分开解释,经常遇到错失关键信息的风险.本文所用全张量欧拉反褶积是在单独z方向的欧拉反演基础上发展而来的,它融合了重力异常垂直分量以及其三个方向导数、水平分量以及其三个方向导数.全张量数据信息得以有效应用的同时,欧拉反褶积结果也比常规欧拉反褶积结果更加收敛.最后,结合美国墨西哥湾地区实测航空FTG数据,用重力梯度张量数据进行联合欧拉三维反演研究,有效的识别岩盖的边界信息,划分岩盖范围,为进一步研究盖层底下深部复杂地质情况提供可靠的解释结果.     

利用高斯求积和连分式展开计算电磁张量格林函数积分

针对于利用快速汉克尔变换求解电磁张量格林函数中含有贝塞尔函数积分项时，积分核函数不总是随贝塞尔函数（Jv(λr)）参数λ的增大而趋于收敛，从而产生错误结果的问题，我们利用一种直接数值积分——结合连分式展开的高斯求积方法进行计算，通过对不同模型的试算表明这种方法总能够保证电磁张量格林函数积分的正确计算.     

体积分方程法模拟电偶源三维电磁响应

利用体积分方程法模拟电偶源激发时均匀导电半空间频率域三维电磁响应.讨论了张量格林函数的计算,对于需要计算三重积分的张量格林函数电荷项一次部分,应用一种差分近似的方法求解,这种方法在保证计算精度的同时更加便于计算机实现;采用二次剖分的算法解决了计算张量格林函数时的奇异值问题;计算含有贝塞尔函数的积分项时,利用一种结合连分式展开的高斯求积代替常规的快速汉克尔变换方法,在确保正确计算的同时提高了计算精度.     

一种磁张量探测系统载体的磁张量补偿方法

针对磁张量系统载体产生的磁张量值对系统测量精度产生很大影响的问题,以及现有磁补偿模型存在非线性、分体式和参数多的问题,提出一种磁张量系统载体的一体化线性磁张量补偿方法.分析了载体硬磁材料产生固有磁张量值和软磁材料产生感应磁张量值的微观机理,并推导了相应的数学表达式,结合固有磁场影响和感应磁场影响建立了载体磁张量补偿模型.模型中含有20个载体磁张量补偿系数,对模型求解得到补偿系数,结合三分量磁场测量值即可达到对载体磁张量的补偿.实测实验表明,磁张量补偿方法计算得到的载体磁张量值与载体实际产生的磁张量值仅差32nT/m,可以有效完成对磁张量系统的载体磁张量补偿.     

网式大地电磁阻抗张量及其影响因素分析

网式大地电磁（Network-MT,N-MT）法采用长数公里至数十公里的电话线为电极线测量电场,很难形成两条笔直且相互垂直的电极线,因此阻抗张量的计算不如大地电磁法中直接.本文依据阻抗张量的旋转规则提出了一种计算N-MT阻抗张量的简便算法.依据该算法计算了中国东北地区5个N-MT测站的阻抗张量,获得了基于阻抗张量的视电阻率、相位曲线和最佳主轴方位角分布图像,为我国东北N-MT资料的进一步处理和解释提供了基础性数据.此外,本文对比分析了朝阳测站中6条N-MT观测电极线上阻抗张量旋转值与观测值之间的差异,重点讨论了产生这种系统性偏差的各种因素,提出其主要因素可能来自“电场等效各向异性”效应,即测站附近的地壳内部存在与观测电极线尺度相比拟的横向非均匀构造,而测站各电极线沿不同方向跨越不均匀构造,此时各电极线上的电场分量不遵循同一电场矢量的分解准则,导致地表观测三角形内阻抗分量不满足统一的阻抗张量旋转规则.     

Coherence algorithm with a high-resolution time–time transform and feature matrix for seismic data

Traditional coherence algorithms are often based on the assumption that seismic traces are stationary and Gaussian. However, seismic traces are actually non-stationary and non-Gaussian. A constant time window and the canonical correlation analysis in traditional coherence algorithms are not optimal for non-stationary seismic traces and cannot describe the similarity between adjacent seismic traces in detail. To overcome this problem, a new coherence algorithm using the high-resolution time–time transform and the feature matrix is designed. The high-resolution time–time transform used to replace the constant time window can produce a frequency-dependent time local series to analyse non-stationary seismic traces. The feature matrix, constructed by the frequency-dependent time local series and the related local gradients, defines a new correlation metric that enhances more details of the geological discontinuities in seismic images than does the canonical correlation analysis. Additionally, the Riemannian metric is introduced for related calculations because the feature matrices are not defined in a Euclidean space but rather in a manifold space. Application to field data illustrates that the proposed method reveals more details of structural and stratigraphic features.     

Doubly asymptotic multi-directional transmitting boundary for dynamic unbounded medium-structure-interaction analysis

The novel doubly asymptotic multi-directional transmitting boundary combines the advantages of the doubly asymptotic and multi-directional formulations. It is rigorous for the low-frequency limit (static case) and the high-frequency limit in the wave-propagation direction perpendicular to the artificial boundary and at all the preselected angles corresponding to various apparent velocities. It is implemented straightforwardly in the finite-element method. The doubly asymptotic multi-directional transmitting boundary is temporally local and spatially either global or local depending on the implementation of the static-stiffness matrix. It leads to much higher accuracy than other transmitting boundaries with the same finite-element mesh.