电性任意各向异性且分块连续变化CSAMT三维有限元数值模拟

考虑地球介质电导率任意各向异性且随空间位置连续变化的情况,本文实现了直接求解电磁场的可控源音频大地电磁测深（CSAMT）三维有限元数值模拟.首先给出了电导率任意各向异性介质中CSAMT二次电场满足的控制方程及其相应变分问题,然后采用任意六面体单元对研究区域进行剖分,在网格单元中对任意各向异性电导率进行线性插值,解决了实际工作中岩矿石电导率各向异性且连续变化的情况,将变分问题转化为线性代数方程组的求解.电导率各向异性且连续变化一维模型三维有限元数值模拟结果与电导率各向异性且分层均匀渐进模型解析解结果对比验证了方法的有效性;三维地电模型电导率随位置线性变化且各向同性、主轴各向异性、方位各向异性和倾斜各向异性的数值模拟结果表明,电导率各向异性且连续变化对CSAMT视电阻率和相位数据均有明显的影响.     

The electromagnetic response of a horizontal electric dipole buried in a multi‐layered earth

The electromagnetic response of a horizontal electric dipole transmitter in the presence of a conductive, layered earth is important in a number of geophysical applications, ranging from controlled‐source audio‐frequency magnetotellurics to borehole geophysics to marine electromagnetics. The problem has been thoroughly studied for more than a century, starting from a dipole resting on the surface of a half‐space and subsequently advancing all the way to a transmitter buried within a stack of anisotropic layers. The solution is still relevant today. For example, it is useful for one‐dimensional modelling and interpretation, as well as to provide background fields for two‐ and three‐dimensional modelling methods such as integral equation or primary–secondary field formulations. This tutorial borrows elements from the many texts and papers on the topic and combines them into what we believe is a helpful guide to performing layered earth electromagnetic field calculations. It is not intended to replace any of the existing work on the subject. However, we have found that this combination of elements is particularly effective in teaching electromagnetic theory and providing a basis for algorithmic development. Readers will be able to calculate electric and magnetic fields at any point in or above the earth, produced by a transmitter at any location. As an illustrative example, we calculate the fields of a dipole buried in a multi‐layered anisotropic earth to demonstrate how the theory that developed in this tutorial can be implemented in practice; we then use the example to examine the diffusion of volume charge density within anisotropic media—a rarely visualised process. The algorithm is internally validated by comparing the response of many thin layers with alternating high and low conductivity values to the theoretically equivalent (yet algorithmically simpler) anisotropic solution, as well as externally validated against an independent algorithm.     

The effect of the electrical anisotropy on the response of helicopter-borne frequency-domain electromagnetic systems

Helicopter electromagnetic (HEM) systems are commonly used for conductivity mapping and the data are often interpreted using an isotropic horizontally layered earth model. However, in regions with distinct dipping stratification, it is useful to extend the model to a layered earth with general anisotropy by assigning each layer a symmetrical 3 × 3 resistivity tensor. The electromagnetic (EM) field is represented by two scalar potentials, which describe the poloidal and toroidal parts of the magnetic field. Via a 2D Fourier transform, we obtain two coupled ordinary differential equations in the vertical coordinate. To stabilize the numerical calculation, the wavenumber domain is divided into two parts associated with small and large wavenumbers. The EM field for small wavenumbers is continued from layer to layer with the continuity conditions. For large wavenumbers, the EM field behaves like a DC field and therefore cannot be sensed by airborne EM systems. Thus, the contribution from the large wavenumbers is simply ignored. The magnetic fields are calculated for the vertical coaxial (VCX), horizontal coplanar (HCP) and vertical coplanar (VCP) coil configurations for a helicopter EM system. The apparent resistivities defined from the VCX, VCP and HCP coil responses, when plotted in polar coordinates, clearly identify the principal anisotropic axes of an anisotropic earth. The field example from the Edwards Aquifer recharge area in Texas confirms that the polar plots of the apparent resistivities identify the principal anisotropic axes that coincide well with the direction of the underground structures.     

AN INTERPRETATION THEORY FOR INDUCED POLARIZATION VERTICAL SOUNDINGS (TIME-DOMAIN)*

In this paper it is shown how one may obtain a generalized Ohm's law which relates the induced polarization electric field to the steady-state current density through the introduction of a fictitious resistivity defined as the product of the chargeability and the resistivity of a given medium. The potential generated by the induced polarization is calculated at any point in a layered earth by the same procedure as used for calculating the potential due to a point source of direct current. On the basis of the definition of the apparent chargeability m_a, the expressions of m_a for different stratigraphie situations are obtained, provided the IP measurements are carried out on surface with an appropriate AMNB array. These expressions may be used to plot master curves for IP vertical soundings. Finally some field experiments over sedimentary formations and the quantitative interpretation procedure are reported.     

Analytical solution for the electric potential in arbitrary anisotropic layered media applying the set of Hankel transforms of integer order

The analytical solution and algorithm for simulating the electric potential in an arbitrarily anisotropic multilayered medium produced by a point DC source is here proposed. The solution is presented as a combination of Hankel transforms of integer order and Fourier transforms based on the analytical recurrent equations obtained for the potential spectrum. For the conversion of the potential spectrum into the space domain, we have applied the algorithm of the Fast Fourier Transform for logarithmically spaced points. A comparison of the modelling results with the power‐series solution for two‐layered anisotropic structures demonstrated the high accuracy and computing‐time efficiency of the method proposed. The results of the apparent‐resistivity calculation for both traditional pole‐pole and tensor arrays above three‐layered sequence with an azimuthally anisotropic second layer are presented. The numerical simulations show that both arrays have the same sensitivity to the anisotropy parameters. This sensitivity depends significantly on the resistivity ratio between anisotropic and adjacent layers and increases for the models with a conductive second layer.     

横观各向同性层状半空间中的弹性位错

本文在柱向量函数系下,利用传播矩阵法求解了层状横观各向同性半空间由内部点源位锚引起的变形;对六个基本点源位错,以等价体力法推出了横观各向同性情形下的点源函数,并且给出了内部任意剪切位错源引起的地表位移的积分表达式。为研究地球的层状结构,特别是其上部的横观各向同性对地表的地震位移、应变以及倾斜场的影响提供了计算公式。     

各向异性层状介质中视电阻率与磁场响应研究

针对任意各向异性地层,利用极向型和环向型标量位函数,导出相应的直流电视电阻率和磁电阻率的磁场响应关系.计算了各向异性地层的直流电视电阻率和磁电阻率响应特征,重点分析了电阻率测深方法与磁电阻率探测方法对地下各向异性介质的探测能力.文中采用状态矩阵的分析方法,首先采用极向型和环向型标量位构造了各向异性层状介质电场与磁场的通解,利用各层界面电场、磁场的连续性及地面激励源的耦合条件,推导了不同层之间电磁场的状态矩阵,建立了空间电场与磁场的递归计算关系.其次,针对递归计算中指数项数值计算的不稳定性,借用状态矩阵的性质,导出了将不稳定指数计算项转化为稳定的指数项的转换关系.针对横向各向同性(TI)介质中极向位与环向位解耦的特点,导出了电磁场的直接积分解.最后,采用解析解验证了算法的正确性,给出了多层各向异性地层模型的视电阻率和磁场响应曲线,分析了直流电法探测裂缝性地层、估计裂缝分布性状的可能性.     

Vertical electric dipole over an inhomogeneous and anisotropic earth

Summary The propagation of electromagnetic waves in a two-layer anisotropic earth from an oscillating vertical electric dipole placed over it has been studied. Formal expression for the vector potential in the top layer has been determined for the general case when both the layers are electrically anisotropic. The conductivity and the permittivity tensors of the layers are assumed to have simple diagonal forms. Three special cases of practical interest have been considered in detail, for which the results are given in terms of tabulated functions. It has been shown that at the surface of the earth, only the electric field components are influenced by the anisotropy.     

THE SOLUTION OF THE STATIONARY ELECTRIC FIELD STRENGTH AND POTENTIAL OF A POINT CURRENT SOURCE IN A 2 1/2- DIMENSIONAL ENVIRONMENT*

A method is given for solving the dc electric field problem of a point current source in an anisotropic 2 1/2-dimensional structural model. The surface integral equation of the field strength is given. Parallel to the strike the field strength is represented by a Fourier series. On the plane perpendicular to the strike each term of the field strength series is solved by means of the method of sub-sections, where linear behaviour of field strength over the sub-sections is assumed. Some numerical examples for different galvanic effects are given.     

有限元直接迭代算法及其在线源频率域电磁响应计算中的应用

采用有限元直接迭代算法实现了线源频率域测深电磁响应的二维正演计算. 首先给出了线源正演问题的有限元直接迭代格式,然后由迭代法进行求解. 在处理奇异源问题上,采用向内递推的组合网格技巧,在源点附近可进行局部加密,并实现粗细网格的对接,从而较好地解决了奇异源附近的计算问题. 还提出一种迭代求取全区视电阻率的方法,避免了远近区的划分. 通过对均匀半空间、层状介质和二维模型电磁响应的计算,获得了与大地电磁测深相似的视电阻率曲线,验证了算法的正确性;通过对计算结果的分析,在理论上说明了线源频率域近区测深的可行性.     

Sensitivity and inversion of marine electromagnetic data in a vertically anisotropic stratified earth

The controlled‐source electromagnetic (CSEM) and magnetotelluric method (MT) are two techniques that can be jointly used to explore the resistivity structure of the earth. Such methods have, in recent years, been applied in marine environments to the exploration and appraisal of hydrocarbons. In many situations the electric properties of the earth are anisotropic, with differences between resistivity in the vertical direction typically much higher than those in the horizontal direction. In cases such as this, the two modes of the time‐harmonic electromagnetic field are altered in different ways, implying that the sensitivity to the earth resistivity may vary significantly from one particular resistivity component (scalar, horizontal or vertical) to another, depending on the measurement configuration (range, azimuth, frequency or water depth). In this paper, we examine the sensitivity of the electromagnetic field to a vertically anisotropic earth for a typical set of configurations, compare inversion results of synthetic data characterizing a vertically anisotropic earth obtained using the isotropic and anisotropic assumptions and show that correctly accounting for anisotropy can prevent artefacts in inversion results.     

Electromagnetic induction in spherical cap current layers under lunar and terrestrial conditions

We present the theory of electromagnetic induction in spherical cap current sheets of arbitrary angular size, with arbitrary axisymmetric integrated electrical conductivity variations and located at any radial position with respect to the surface of observation. The external time-varying magnetic field may be arbitrarily oriented with respect to the current layer cap and the induced fields are derived for vacuum boundary conditions appropriate to terrestrial induction and plasma confinement boundary conditions relevant to lunar induction in the solar wind or magnetosheath plasmas. Numerical evaluations show the induced magnetic field as a function of position over the current sheet cap, depth to the current layer, size of the cap, integrated electrical conductivity of the current sheet, and frequency of the fluctuating external field. The local vertical magnetic field component and the horizontal field component which is normal to the periphery of the cap exhibit peak inductive responses above the edge of the current sheet for external magnetic fields perpendicular to the axis of the cap. Thus, induced magnetic field fluctuations observed over the edge of a conductivity anomaly may exhibit a highly directional, or polarized behavior. This may provide an explanation for the asymmetric character of induced magnetic field fluctuations observed on the lunar surface.     

TIME-DOMAIN COMPUTATION OF NON-NORMAL INCIDENCE WAVEFIELDS IN PLANE LAYERED MEDIA1

A new time-domain method is introduced for the calculation of theoretical seismograms which include frequency dependent effects like absorption. To incorporate these effects the reflection and transmission coefficients become convolutionary operators. The method is based on the communication theory approach and is applicable to non-normal incidence plane waves in flat layered elastic media. Wave propagation is simulated by tracking the wave amplitudes through a storage vector inside the computer memory representing a Goupillaud earth model discretized by equal vertical transit times. Arbitrary numbers of sources and receivers can be placed at arbitrary depth positions, while the computational effort is independent of that number. Therefore, the computation of a whole plane-wave vertical seismic profile is possible with no extra effort compared to the computation of the surface seismogram. The new method can be used as an aid to the interpretation of plane-wave decomposed reflection data where the whole synthetic vertical seismic profile readily gives the interpreter the correct depth position of reflection events.     

Regional induction studies: A review of methods and results

The geomagnetic skin-effect is specified by setting three length scales in relation to each other: L₁ for the overhead source. L₂ for the lateral non-uniformity of the subsurface conductor, L₃ for the depth of penetration of a quasi-uniform transient field into this conductor. Relations for the skin-effect of a quasi-uniform source in layered conductors are generalized to include sources of any given geometry by introducing response kernels as functions of frequency and distance. They show that only those non-uniformities of the source which occur within a distance comparable to L₃ from the point of observation are significant. The skin-effect of a quasi-uniform source in a laterally non-uniform earth is expressed by linear transfer functions for the surface impedance and the surface ratio of vertical/horizontal magnetic variations. In the case of elongated structures and E-polarisation of the source, a modified apparent resistivity is defined which as a function of depth and distance gives a first orientation about the internal distribution of conductivity. The skin-effect of a non-uniform source in a non-uniform earth is considered for stationary and “running” sources. Recent observations on the sea floor and on islands indicate a deep-seated change of conductivity at the continent—ocean transition, bringing high conductivity close to the surface, a feature which may not prevail, however, over the full width of the ocean. There is increasingly reliable evidence for high conductivities (0.02 to 0.1 micro ^?1 m^?1) at subcrustal or even at crustal depth beneath certain parts of the continents, in some cases without obvious correlation to geological structure.     

1-D DC Resistivity Modeling and Interpretation in Anisotropic Media Using Particle Swarm Optimization

We examine the one-dimensional direct current method in anisotropic earth formation. We derive an analytic expression of a simple, two-layered anisotropic earth model. Further, we also consider a horizontally layered anisotropic earth response with respect to the digital filter method, which yields a quasi-analytic solution over anisotropic media. These analytic and quasi-analytic solutions are useful tests for numerical codes. A two-dimensional finite difference earth model in anisotropic media is presented in order to generate a synthetic data set for a simple one-dimensional earth. Further, we propose a particle swarm optimization method for estimating the model parameters of a layered anisotropic earth model such as horizontal and vertical resistivities, and thickness. The particle swarm optimization is a naturally inspired meta-heuristic algorithm. The proposed method finds model parameters quite successfully based on synthetic and field data. However, adding 5 % Gaussian noise to the synthetic data increases the ambiguity of the value of the model parameters. For this reason, the results should be controlled by a number of statistical tests. In this study, we use probability density function within 95 % confidence interval, parameter variation of each iteration and frequency distribution of the model parameters to reduce the ambiguity. The result is promising and the proposed method can be used for evaluating one-dimensional direct current data in anisotropic media.     

球体内点位错产生的球型位移场：II．位错Love数

本文进一步研究了ＳＮＲＥＩ地球模型内位错所产生的球型位移场问题．作为特例，讨论了位错位于地表时的解和计算技巧．对于１０６６Ａ地球模型进行了具体计算．给出了深度分别为５，１０，３２，１００，３００和６３７ｋｍ的４种类型位错的位错Ｌｏｖｅ数的结果．     

THE NON-LINEAR INVERSION OF SEISMIC WAVEFORMS CAN BE PERFORMED EITHER BY TIME EXTRAPOLATION OR BY DEPTH EXTRAPOLATION1

The classical aim of non-linear inversion of seismograms is to obtain the earth model which, for null initial conditions and given sources, best predicts the observed seismograms. This problem is currently solved by an iterative method: each iteration involves the resolution of the wave equation with the actual sources in the current medium, the resolution of the wave equation, backwards in time, with the current residuals as sources; and the correlation, at each point of space, of the two wavefields thus obtained. Our view of inversion is more general: we want to obtain a whole set of earth model, initial conditions, source functions, and predicted seismograms, which are the closest to some a priori values, and which are related through the wave equation. It allows us to justify the previous method, but it also allows us to set the same inverse problem in a different way: what is now searched for is the best fit between calculated and a priori initial conditions, for given sources and observed surface displacements. This leads to a completely different iterative method, in which each iteration involves the downward extrapolation of given surface displacements and tractions, down to a given depth (the‘bottom’), the upward extrapolation of null displacements and tractions at the bottom, using as sources the initial time conditions of the previous field, and a correlation, at each point of the space, of the two wavefields thus obtained. Besides the theoretical interest of the result, it opens the way to alternative numerical methods of resolution of the inverse problem. If the non-linear inversion using forward-backward time propagations now works, this non-linear inversion using downward-upward extrapolations will give the same results but more economically, because of some tricks which may be used in depth extrapolation (calculation frequency by frequency, inversion of the top layers before the bottom layers, etc.).     

任意各向异性介质中三维可控源音频大地电磁正演模拟

在一些地层层理发育的地区,地下介质存在显著的电各向异性,此时基于各向同性模型解释含各向异性效应的可控源音频大地电磁（CSAMT）测深观测数据会导致错误的结果.本文通过引入3×3的对称正定张量表征电导率各向异性,采用非结构四面体网格和矢量有限元方法离散电场满足的矢量Helmholtz方程,并将电磁场源等效为系列电偶极子,实现任意各向异性介质中CSAMT高效数值模拟.本文首先通过层状各向异性模型检验三维有限元算法的精度和有效性,进一步建立三维地电模型研究异常体各向异性和围岩各向异性对CSAMT响应的影响,最后使用视电阻率极性图来识别各向异性电导率主轴方向.数值模拟结果表明,各向异性电导率对CSAMT视电阻率幅值及分布规律都有很大影响,视电阻率极性图能够很好地识别各向异性主轴方向.     

Elimination of the water-layer response from multi-component source and receiver marine electromagnetic data

This paper presents the theory to eliminate from the recorded multi‐component source, multi‐component receiver marine electromagnetic measurements the effect of the physical source radiation pattern and the scattering response of the water‐layer. The multi‐component sources are assumed to be orthogonally aligned above the receivers at the seabottom. Other than the position of the sources, no source characteristics are required. The integral equation method, which for short is denoted by Lorentz water‐layer elimination, follows from Lorentz' reciprocity theorem. It requires information only of the electromagnetic parameters at the receiver level to decompose the electromagnetic measurements into upgoing and downgoing constituents. Lorentz water‐layer elimination replaces the water layer with a homogeneous half‐space with properties equal to those of the sea‐bed. The source is redatumed to the receiver depth. When the subsurface is arbitrary anisotropic but horizontally layered, the Lorentz water‐layer elimination scheme greatly simplifies and can be implemented as deterministic multi‐component source, multi‐component receiver multidimensional deconvolution of common source gathers. The Lorentz deconvolved data can be further decomposed into scattering responses that would be recorded from idealized transverse electric and transverse magnetic mode sources and receivers. This combined electromagnetic field decomposition on the source and receiver side gives data equivalent to data from a hypothetical survey with the water‐layer absent, with idealized single component transverse electric and transverse magnetic mode sources and idealized single component transverse electric and transverse magnetic mode receivers. When the subsurface is isotropic or transverse isotropic and horizontally layered, the Lorentz deconvolution decouples into pure transverse electric and transverse magnetic mode data processing problems, where a scalar field formulation of the multidimensional Lorentz deconvolution is sufficient. In this case single‐component source data are sufficient to eliminate the water‐layer effect. We demonstrate the Lorentz deconvolution by using numerically modeled data over a simple isotropic layered model illustrating controlled‐source electromagnetic hydrocarbon exploration. In shallow water there is a decrease in controlled‐source electromagnetic sensitivity to thin resistors at depth. The Lorentz deconvolution scheme is designed to overcome this effect by eliminating the water‐layer scattering, including the field's interaction with air.