Direct current electric potential computation in an inhomogeneous and arbitrarily anisotropic layered earthl

We consider the calculation of the electrical field quantities, electric potential and the vertical component of the total volume density of electric current, in a horizontally layered, piecewise homogeneous and arbitrarily anisotropic earth due to a system of direct current point sources. By applying Fourier transformation with respect to the horizontal space coordinates to the static field equations, the field quantities are obtained as the solutions of the system of transform-domain differential equations in the vertical (depth) coordinates. A recurrence scheme has been given to compute the tranform-domain field quantities at any depth. The corresponding space-domain quantities are then obtained by inverse Fast Fourier Transformation (FFT). A complete computer program has been developed for computing the electric potentials at any depth of the layered earth, which is composed of an arbitrary number of anisotropic layers with arbitrary conductivity tensors. By considering the point sources at different depths from the surface, equipotential contours on the surface of arbitrarily anisotropic layered earth models are given.     

Dispersion splitting of Rayleigh waves in layered azimuthally anisotropic media

Rayleigh wave dispersion can be induced in an anisotropic medium or a layered isotropic medium. For a layered azimuthally anisotropic structure, traditional wave equation of layered structure can be modified to describe the dispersion behavior of Rayleigh waves. Numerical stimulation results show that for layered azimuthal anisotropy both the dispersion velocities and anisotropic parameters depend principally on anisotropic S-wave velocities. The splitting S-wave velocities may produce dispersion splitting of Rayleigh waves. Such dispersion splitting appears noticeable at azimuthal angle 45°. This feature was confirmed by the measured results of a field test. The fundamental mode splits into two branches at azimuthal angle 45° to the symmetry axis for some frequencies, and along the same direction the difference of splitting-phase velocities of the fundamental model reaches the maximum. Dispersion splitting of Rayleigh waves was firstly displayed for anisotropy study in dispersion image by means of multichannel analysis of surface waves, the image of which provides a new window for studying the anisotropic property of media.     

层状TI介质中微地震定位和各向异性速度结构同时反演

微地震监测被广泛应用于非常规油气资源的水力压裂作业、油藏描绘和水驱前缘监测工程中.微地震定位采用的初始速度模型一般是基于地震测井记录和射孔数据建立,该速度模型的不准确性易引起定位误差.为降低这种定位误差,本文发展了一种微地震定位和各向异性速度结构同时反演的方法.研究对象为1-D的层状TI介质,其中对称轴方向任意.利用改进的分区多步最短路径算法计算qP、qSV和qSH波的到达时间和射线路径,结合共轭梯度法求解带约束的阻尼最小二乘问题.数值模拟结果表明,该算法能同时进行各向异性速度结构模型（每层的Thomsen参数和界面深度）和微震震源参数（空间坐标和发震时刻）的反演,并且对随机噪声不敏感,有利于实际工程应用.

     

A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tilt anisotropic media

Introduction The real Earth usually presents anisotropy. Therefore, it is of theoretical and practical sig- nificance for many fields as oil and gas, seismic exploration and production, earthquake prediction, detection of deep structure and so on to study on seismic wave theory, numerical simulation method and its applications in the anisotropic media (Crampin, 1981, 1984; Crampin et al, 1986; Hudson et al, 1996; Liu et al, 1997; Thomsen, 1986, 1995; TENG et al, 1992; HE and ZHANG, 1996)…     

A precise integration approach for dynamic impedance of rigid strip footing on arbitrary anisotropic layered half-space

The precise integration method (PIM) is proposed for the dynamic response analysis of rigid strip footing resting on arbitrary anisotropic multi-layered half-space. In the frequency domain, the governing equation of wave motion is converted into dual vector form of first-order ordinary differential equations which is solved by PIM. Each layer is divided into a large number (say, 2^N) of mini-layers of equal thickness, within which characteristic matrices are assumed to vary following the Taylor series expansion to the fourth order. As a result, any desired accuracy of the displacements and stresses can be achieved by PIM. In addition, dual vector form equation makes it quite easily to combine two adjacent mini-layers into a new one. Each pass of combination reduces the total number of mini-layers by a half. The computational effort for the evaluation of the dynamic impedance of rigid strip footing can be reduced to a great extent. Numerical examples are provided to validate the efficiency and accuracy of the proposed approach.     

电导率任意各向异性海洋可控源电磁三维矢量有限元数值模拟

现有海洋可控源电磁三维数值模拟方法大多基于电导率各向同性介质理论,不能模拟海底地层电导率各向异性的实际情况.本文给出了电导率各向异性三维介质中电性源海洋可控源电磁二次电场的边值问题以及相应的变分问题,采用长方体单元对研究区域剖分,将场分量定义在剖分单元的边上,利用矢量有限单元法求解变分问题,实现了电导率任意各向异性海洋可控源电磁三维矢量有限元数值模拟.这个新的正演方法可以计算电导率任意各向异性三维地电模型的海洋可控源电磁响应,基于二次场矢量有限元法直接求解电磁场,避免了传统有限元方法可能遇到的伪解问题和难于处理电场法向分量不连续的问题,提高了数值模拟计算精度.一维电导率各向异性模型电磁场数值解与解析解吻合得相当好,无论在源附近还是远离源处相对误差均不超过1%.电导率各向异性二维模型的计算结果与已有文献采用的非结构有限元模拟结果十分吻合.三维地电模型数值模拟结果显示,电导率各向异性张量电导率主轴分量和欧拉角对不同装置海洋可控源电磁响应均有着明显的影响.

     

双轴各向异性介质多分量感应测井响应快速计算

为研究双轴各向异性介质多分量感应测井响应特征,本文基于三重傅里叶变换,推导任意方向偶极子源的谱域电磁场解析式;采用围线积分方法,自适应截断积分区间,结合谱域电磁场周期特性,实现三重傅里叶变换的精确快速积分;进而,针对双轴各向异性倾斜地层,模拟研究不同纵横向各向异性条件多分量感应测井响应特征.结果表明：利用谱域内电磁场在周向的周期特性简化解析式,可将计算速度提高4倍;自适应截断积分区间方法保证了计算精度,并极大地减少了积分节点数.对于倾斜双轴各向异性介质,倾角较大时,共面分量可反映地层横向各向异性,同轴分量可反映地层纵向各向异性;倾角较小时,同轴分量可反映地层横向各向异性,共面分量可反映地层纵向各向异性.

     

Any P-wave kinematic algorithm for vertical transversely isotropic media can be ADAPTed to moderately anisotropic media of arbitrary symmetry type

Most of P-wave anisotropic kinematic algorithms (modeling, processing, and inversion) have been developed for the case of Transverse Isotropy (TI). Does it mean that when dealing with more complex symmetry types (Arbitrarily tilted TI, orthorhombic, monoclinic or even triclinic), all these algorithms are irrelevant? In fact, not at all. It has recently been demonstrated that in 2D geometry any qP-wave TI kinematic algorithm can be simply generalized to the case of monoclinic symmetry using the so-called Azimuthally Dependent Anisotropy Parameter Transformation (ADAPT), assuming moderate anisotropy. The extension of the technique to the case of arbitrary anisotropy type (triclinic) is achieved in this paper. The method is successfully checked for seismic modeling in a full 2D model with layers of contrasted anisotropy types and with arbitrary vertical and horizontal velocity variations (non-constant gradient). Typically, the approximate travel times using ADAPT differ from the exact travel times by a few milliseconds for total travel times of the order of a few seconds. Applications to seismic processing are also described. The simplicity of the procedure and the generality of the applicability of the ADAPT recipe are striking and very convenient for practical applications. They certainly deserve further analysis.     

层状各向异性介质海洋可控源电磁场灵敏度计算及特征分析

海洋可控源电磁（CSEM）方法已广泛应用于地质构造研究以及海底资源探测,但其在各向异性地层中的分辨能力依然不明确.灵敏度分析是一种分析电磁场对探测目标分辨能力的有效方法,传统的地球物理反演方法也需要精确计算电磁场关于地下介质电阻率的灵敏度.在模拟和解释海洋CSEM资料时,地球物理数值模拟常在笛卡尔直角坐标系下进行,且通常假定发射源为理想的水平电偶极源.然而,在实际的海洋可控源电磁勘探作业中,由于海水运动等影响,发射源可能会发生旋转和倾斜等.复杂姿态的电偶极源可通过计算并矢量叠加三个正交方向发射源分量的电磁场以获得总电磁场,因此需三个正交方向电偶源电磁场的计算方法.本文推导了笛卡尔直角坐标系下,电阻率垂直各向异性介质中三个正交方向电偶极源电磁场表达式,并详细导出了电磁场分量关于各向异性电导率的灵敏度解析表达式.通过与各向同性算法对比,验证了本文所提出灵敏度计算方法的正确性;模拟了不同方向电偶源情况下地电模型的灵敏度并分析其特征.计算结果表明,薄层将显著影响地下介质各向异性电阻率的灵敏度分布,垂直电偶源对海底地层各向异性电阻率的分辨能力高于水平电偶源,通过反演各向异性率间接恢复低灵敏度的各向异性电阻率值是一个可行的反演策略.

     

利用探地雷达频谱反演层状介质几何与电性参数

通过对地下层状介质探地雷达(GPR)回波广义反射系数的奇偶分解,建立了联系GPR反射系数序列频谱与介质几何参数、电性参数的代价函数,这些介质参数包括地下反射面的深度、层厚度以及各层的介电常数和电导率,从而提出了一种由GPR频谱同时估算地下介质多参数的全局优化反演方法.为了对多参数全局优化算法给出一个合理的参数初值,研究了不同参数对反射系数序列频谱属性的影响规律,提出了利用不同频谱属性分别估算不同参数的分步反演方法.以分步反演方法得到的结果作为多参数全局优化反演的初值,可以极大地提高反演计算的效率和反演结果的可靠性.用理论模型合成数据和GPR公路检测数据对本文方法进行了测试,结果表明本文方法效果良好,具有较高的分辨率,能较好的给出厚度小于调谐厚度的薄层的深度、厚度和介电常数等参数.     

Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

Staggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.     

Looking for layered anisotropic structures in the mantle beneath the northern Apennines

Competing geodynamic scenarios proposed for northern Apennines (Italy) make very different predictions for the orientation of strain in the upper mantle. Constraints on the pattern are offered by observations of seismic anisotropy. Previous study of the anisotropy beneath the northern Apennines used birefringence of core-refracted shear waves (SKS phases), and demonstrated the presence of two domains: Tuscan and Adria. In the transition between the two domains, across the Apennines orogen, anisotropy measurements reflect a complex deep structure. To define better the upper-mantle structure beneath this area we analyze seismological data recorded by a set of seismic stations that operated for 3 years, between 2003 and 2006, located in the outer part of the Apennines belt, in the Adria terrane, collected by the RETREAT Project. Directionally distributed sets of SKS records were inverted for layered anisotropic structures with a well-tested method, adding new results to previous hypotheses for this area. New data analysis argues for two-layer anisotropy for sites located on the Apennines wedge and also one site in the Tuscan terrane. Beneath the wedge an upper layer with nearly north-south fast polarization pervades the lithospheric mantle, while at depth a nearly NW–SE Apennines-parallel direction is present in the lower layer. Beneath Tuscany a shallower NW–SE direction and a deeper E–W one suggest the deeper strain from active slab retreat, with a mantle-wedge circulation (i.e. an east–west corner flow), overlain by an Apennines-parallel fast polarization that could be a remnant of lower-crust deformation.     

Multi‐block finite‐difference method for 3D elastodynamic simulations in anisotropic subhorizontally layered media

Prediction of elastic full wavefields is required for reverse time migration, full waveform inversion, borehole seismology, seismic modelling, etc. We propose a novel algorithm to solve the Navier wave equation, which is based on multi‐block methodology for high‐order finite‐difference schemes on curvilinear grids. In the current implementation, the blocks are subhorizontal layers. Smooth anisotropic heterogeneous media in each layer can have strong discontinuities at the interfaces. A curvilinear adaptive hexahedral grid in blocks is generated by mapping the original 3D physical domain onto a parametric cube with horizontal layers and interfaces. These interfaces correspond to the main curvilinear physical contrast interfaces of a subhorizontally layered formation. The top boundary of the parametric cube handles the land surface with smooth topography. Free‐surface and solid–solid transmission boundary conditions at interfaces are approximated with the second‐order accuracy. Smooth media in the layers are approximated up to sixth‐order spatial schemes. All expected properties of the developed algorithm are demonstrated in numerical tests using corresponding parallel message passing interface code.     

层状各向异性介质转换波克希霍夫叠前时间偏移

在克希霍夫叠前时间偏移处理中,地震波走时的计算方法是决定大偏移距地震资料成像品质的重要因素.在常规的三维转换波各向异性叠前时间偏移公式中,走时的计算是基于等效单层各向异性介质的非双曲线方法.用这种方法处理的成像道集,在偏移/深度比超过一定阈值后,成像道集中的反射同相轴将出现过偏现象,这种偏移不平的同相轴将影响偏移叠加的最佳响应,使得偏移成像波组呈低频化特征,最终降低三维转换波偏移成像质量.我们采用层状介质的走时计算方法代替常规算法,并且利用了常规方法的转换波各向异性偏移速度模型.基于层状介质的算法能够提高大偏移距转换波走时计算精度,克服中浅地层大偏移距远道成像道集中反射同相轴逐渐上翘的问题.两个地区的三维转换波资料处理结果证实,基于层状各向异性介质的转换波克希霍夫叠前时间偏移方法,明显改善了反射成像剖面的连续性和分辨率,提高成像剖面构造的可解释性.     

Validity of the long-wave approximation in periodically layered media

In seismic modelling, a stack of thin layers is often replaced by an effective equivalent anisotropic homogeneous slab. For waves with finite wavelength, this is an approximation, and the error thus introduced can be quantified by considering the relative error in the phase velocity between the layer stack and the effective medium. For periodic layering, the relative phase-velocity error can be expressed in closed form as a function of wavelength, reflection coefficients and layer thicknesses. By comparing the relative phase-velocity error with laboratory measurements and numerical simulations, we find that the difference in seismic response between a periodic layer stack and an equivalent effective medium depends not only on wavelength, but it also depends significantly on reflection coefficients and the ratio between layer thicknesses. For a 1% relative error in the phase velocity, and if all layers have the same thickness measured in vertical traveltime, we find that the wavelength must be larger than approximately three times the layer period for a reflection coefficient of 0.1, but this increases to 13 times the layer period for a reflection coefficient of 0.9, which is highly unrealistic in a geological setting.     

Transformation relations for second-order derivatives of travel time in anisotropic media

In the computation of paraxial travel times and Gaussian beams, the basic role is played by the second-order derivatives of the travel-time field at the reference ray. These derivatives can be determined by dynamic ray tracing (DRT) along the ray. Two basic DRT systems have been broadly used in applications: the DRT system in Cartesian coordinates and the DRT system in ray-centred coordinates. In this paper, the transformation relations between the second-order derivatives of the travel-time field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used both in isotropic and anisotropic media, including computations of complex-valued travel times necessary for the evaluation of Gaussian beams.     

三维各向异性介质弹性波模拟的透射边界条件

在弹性波数值模拟中,为了压制人工边界反射,需要使用边界条件.本文从三维各向异性介质中弹性渡方程出发,在边界入射波为平面波的假设前提下,利用特征值分解以及坐标系旋转,导出人工边界处质点位移矢量计算公式,实现了在边界处为任意倾斜入射时的透射边界条件.并显示了边界条件的使用效果.     

Analytical description of electric currents in the magnetosheath region

Volume currents in the magnetosheath region are calculated within the framework of a new analytical model. Magnetic field structure in the region is found, satisfying boundary conditions on the bow shock and the magnetopause, and then volume currents are calculated using the Maxwell equation. Surface bow shock and magnetopause currents are calculated, too. Free parameters of the model are interplanetary magnetic field, Mach number of the solar wind flow, distances to the bow shock and to the magnetopause, and field compression at the magnetopause.     

Weak-contrast approximation of the elastic scattering matrix in anisotropic media

The plane-wave reflection and transmission coefficients at a plane interface between two anisotropic media constitute the elements of the elastic scattering matrix. For a 1-D anisotropic medium the eigenvector decomposition of the system matrix of the transformed elasto-dynamic equations is used to derive a general expression for the scattering matrix. Depending on the normalization of the eigenvectors, the expressions give scattering coefficients for amplitudes or for vertical energy flux.Computing the vertical slownesses and the corresponding polarizations, the eigenvector matrix and its inverse can be found. We give a simple formula for the inverse, regardless of the normalization of the eigenvectors. When the eigenvectors are normalized with respect to amplitudes of displacement (or velocity), the calculation of the scattering matrix for amplitudes is simplified.When the relative changes in all parameters are small, a weak-contrast approximation of the scattering matrix, based on the exactly determined polarization vectors in an average medium, is obtained. The same approximation is also derived directly from the transformed elasto-dynamic equations for a smooth vertically inhomogeneous medium, proving the consistency of the approximation.For monoclinic media, with the mirror symmetry plane parallel to the interface, the approximative scattering matrix is given in terms of analytic expressions for the non-normalized eigenvectors and vertical slownesses. For transversely isotropic media with a vertical axis of symmetry (VTI) and isotropic media, explicit solutions for the weak-contrast approximations of the scattering matrices have been obtained. The scattering matrix for amplitudes for isotropic media is well known. The scattering matrix for vertical energy flux may have applications in AVO analysis and inversion due to the reciprocity of the reflection coefficients for converted waves.Numerical examples for monoclinic and VTI media provide good agreement between the approximative and the exact reflection matrices. It is, however, expected that the approximations cannot be used when the symmetry properties of the two media are very different. This is because the approximation relies on a small relative contrast between the eigenvectors in the two media.Presented at the Workshop Meeting on Seismic Waves in Laterally Inhomogeneous Media, Castle of Trest, Czech Republic, May 22–27, 1995.