Forward plane-wave electromagnetic model in three dimensions using hybrid finite volume–integral equation scheme

We present a concept of the hybrid finite volume–integral equation technique for solving Maxwell's equation in a quasi-static form. The divergence correction was incorporated to improve the convergence and stability of the governing linear system equations which pose a challenge on the discretization of the curl–curl Helmholtz equation. A staggered finite volume approach is applied for discretizing the system of equations on a structured mesh and solved in a secondary field technique. The bi-conjugate gradient stabilizer was utilized with block incomplete lower-upper factorization preconditioner to solve the system of equation. To obtain the electric and magnetic fields at the receivers, we use the integral Green tensor scheme. We verify the strength of our hybrid technique with benchmark models relative to other numerical algorithms. Importantly, from the tested models, our scheme was in close agreement with the semi-analytical solution. It also revealed that the use of a quasi-analytical boundary condition helps to minimize the runtime for the linear system equation. Furthermore, the integral Green tensor approach to compute at the receivers demonstrates better accuracy compared with the conventional interpolation method. This adopted technique can be applied efficiently to the inversion procedure.     

Coulomb规范下地电磁场的自适应有限元模拟的理论分析

地电磁场的直接求解法存在伪解现象,且电磁场分量在界面上的不连续性与节点型有限元的基本要求矛盾. 本文将Coulomb 规范下磁矢量势-电标量势与自适应有限元相结合,提出了地球物理电磁场计算的快速、高精度方法. 首先从地电磁场一般边值问题出发,给出了Coulomb 规范下磁矢量势-电标量势的公式系统,分析了求解域内势的连续性. 采用Galerkin 加权余值法推导出积分弱解形式和Delaunay非结构化四面体单元时Hierarchal 基函数的有限元方程. 基于超收敛恢复技术,提出了适用于电磁场的后验误差估计方法,阐述了地电磁场自适应计算的策略及迭代算法,分析了计算时间消耗和误差收敛性质,表明本文方法可以用最优的计算资源得到呈拟指数收敛到准确解的数值结果,从而为后续的数值计算奠定了理论基础.     

用交错网格有限差分法计算三维频率域电磁响应

用交错网格有限差分法(SFD)，实现了三维频率域电磁场响应 的数值模拟. 该方法适用于任何方向的磁偶极子源. 经与解析方法、积分方程等 其他方法的计算结果对比表明，交错网格有限差分法结合散度校正和不完全乔累斯基分解预 处理的双共轭梯度迭代方法进行正演计算，速度快、精度高、结果稳定，能适应三维复杂介 质的数值模拟，为三维电磁反演奠定了基础.     

Potential field of a stationary electric current in a stratified medium with a three-dimensional perturbing body

Summary The potential of the electric field of a stationary current in a two-layered Earth is calculated by applying Green's formula in the case where a three-dimensional inhomogeneity of different conductivity is located in the basement of the layer. It is proved that the potential outside and inside the perturbing body can be calculated from the potential of an electric double-layer distributed on the surface of this body. An integral equation of the Fredholm type is derived for the surface density of the double-layer, together with some of its integral properties. A similar procedure can be applied to computing the magnetic anomalies of three-dimensional magnetized bodies, geothermal anomalies due to three-dimensional inhomogeneities of different heat conductivity, as well as to potential problems of theoretical electrical engineering.     

On regularized time varying gravity field models based on grace data and their comparison with hydrological models

Determination of spherical harmonic coefficients of the Earth’s gravity field is often an ill-posed problem and leads to solving an ill-conditioned system of equations. Inversion of such a system is critical, as small errors of data will yield large variations in the result. Regularization is a method to solve such an unstable system of equations. In this study, direct methods of Tikhonov, truncated and damped singular value decomposition and iterative methods of ν, algebraic reconstruction technique, range restricted generalized minimum residual and conjugate gradient are used to solve the normal equations constructed based on range rate data of the gravity field and climate experiment (GRACE) for specific periods. Numerical studies show that the Tikhonov regularization and damped singular value decomposition methods for which the regularization parameter is estimated using quasioptimal criterion deliver the smoothest solutions. Each regularized solution is compared to the global land data assimilation system (GLDAS) hydrological model. The Tikhonov regularization with L-curve delivers a solution with high correlation with this model and a relatively small standard deviation over oceans. Among iterative methods, conjugate gradient is the most suited one for the same reasons and it has the shortest computation time.     

Penetration of Nonstationary Ionospheric Electric Fields into Lower Atmospheric Layers in the Global Electric Circuit Model

The problem of the penetration of nonstationary ionospheric electric fields into the lower atmospheric layers is considered based on the model of the global electric circuit in the Earth’s atmosphere. For the equation of the electric field potential, a solution that takes into account exponential variation in the electrical conductivity with height has been obtained. Analysis of the solution made it possible to reveal three cases of the dependence of the solution on height. The first case (the case of high frequencies) corresponds to the Coulomb approximation, when the electrical conductivity of the atmosphere can be neglected. In the case of low frequencies (when the frequency of changes in the ionosphere potential is less than the quantity reciprocal to the time of electric relaxation of the atmosphere), a quasi-stationary regime, in which the variation in the electric potential of the atmosphere is determined by the electric conduction currents, occurs. In the third case, due to the increase in the electrical conductivity of the atmosphere, two spherical regions appear: with the Coulomb approximation in the lower region and conduction currents in the upper one. For these three cases, formulas for estimating the electric field strength near the Earth’s surface have been obtained.     

基于有限体积法的二维大地电磁各向异性数值模拟

为了计算带任意地形的各向异性介质中二维大地电磁响应,本文在非结构化网格的基础上,采用有限体积法,开发了二维大地电磁各向异性正演模拟的新算法.首先,从Maxwell方程出发,推导二维各向异性介质中大地电磁场的边值问题;然后,采用三角网格自动生成技术对求解区域进行非结构化网格剖分,进而构建节点中心控制体积单元,利用有限体积方法,得到求解边值问题的大型稀疏线性方程组;最后,利用Pardiso精确地计算了大地电磁响应值.三个各向异性模型的计算结果表明,本文开发的有限体积算法,不仅能够高精度求解带任意地形的大地电磁电导率各向异性问题,而且对于同一模型,该方法的计算消耗和精度都与有限单元法相当.因此,有限体积法是处理电磁法各向异性问题的一种有效方法.     

一种新的三维大地电磁积分方程正演方法

采用规则六面体单元和并矢Green函数奇异积分等效积分技术,已有的大地电磁积分正演方法具有不能有效模拟地下复杂地质体和计算精度偏低的缺点.本文提出了一种新的三维大地电磁积分方程正演技术,即采用四面体单元、解析的并矢Green函数奇异积分表达式,达到既能模拟地下复杂异常体,又能有效提高已有积分方程法计算精度的目的.首先,采用四面体网格技术离散地下复杂异常体,获得四面体单元上的大地电磁积分方程.然后,利用针对四面体单元开发的新的奇异值积分的解析表达式,准确计算线性方程中的并矢Green函数的奇异积分,从而获得精确的线性方程.借助于PARDISO高性能并行直接求解器,实现了三维大地电磁问题的高精度求解.最后,基于国际标准3D-1模型和六棱柱模型,通过与其他方法结果的对比分析,验证了本文方法的正确性、处理高电导率对比度的能力(1000:1)和处理复杂模型的能力.     

Calculation of atmospheric electric fields penetrating from the ionosphere

The spatial distributions of electric fields and currents in the Earth’s atmosphere are calculated. Electric potential distributions typical of substorms and quiet geomagnetic conditions are specified in the ionosphere. The Earth is treated as a perfect conductor. The atmosphere is considered as a spherical layer with a given height dependence of electrical conductivity. With the chosen conductivity model and an ionospheric potential of 300 kV with respect to the Earth, the electric field near the ground is vertical and reaches 110 Vm⁻¹. With the 60-kV potential difference in the polar cap of the ionosphere, the electric field disturbances with a vertical component of up to 13 V m⁻¹ can occur in the atmosphere. These disturbances are maximal near the ground. If the horizontal scales of field nonuniformity are over 100 km, the vertical component of the electric field near the ground can be calculated with the one-dimensional model. The field and current distributions in the upper atmosphere can be obtained only from the three-dimensional model. The numerical method for solving electrical conductivity problems makes it possible to take into account conductivity inhomogeneities and the ground relief.     

三维陆地可控源电磁法有限元模型降阶快速正演

三维陆地可控源电磁法有限元快速正演的主要瓶颈在于多频率大型稀疏方程组求解问题.本文引入一种基于模型降阶的Krylov子空间投影算法,推导了有限元刚度矩阵的模型降阶形式,构建了频率域传递函数;采用标准正交向量序列,构建一个远远小于有限元刚度矩阵维度的矩阵,该矩阵与频率无关,通过一次模型降阶即可实现多频点有限元方程快速求解.采用基于电场的变分方程,加入散度校正条件,以消除伪解;引入伪δ函数,消除了源点的奇异性,可适用于复杂背景模型三维有限元数值模拟,并为多源的求解奠定了基础;以层状介质模型解析解为标准,通过和基于Pardiso直接求解器的有限元算法（3DFEM）进行比较,模型降阶法计算时间小于前者的1/10,平均相对误差在1.72%,在满足精度要求下,实现了高效率三维有限元数值求解;分别设计了横向高低阻模型和纵向高低阻模型,分析了从近区到远区电场和卡尼亚视电阻率的变化规律,假极值的表现特征,阴影效应的影响等,从而也验证了该算法的正确性.最后,建立了一个地层陷落柱模型,通过模型降阶有限元正演模拟,发现视电阻率断面图在陷落柱上方出现"凹陷",与模型设计吻合,表明该算法对复杂地层模拟具有同样的适用性.     

POTENTIAL FIELD OF A STATIONARY ELECTRIC CURRENT USING FREDHOLM'S INTEGRAL EQUATIONS OF THE SECOND KIND*

An integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the ‘second kind’ developed for the potential of the electric field. In this method the discontinuity-surfaces of electrical conductivity are divided into ‘sub-areas’ that are so small that the value of their potential can be regarded as constant. The equations are applied to 3-D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise-a-la-masse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.     

基于二次场电导率分块连续变化的三维可控源电磁有限元数值模拟

从电偶源三维地电断面可控源电磁法的二次电场边值问题及其变分问题出发,采用任意六面体单元对研究区域进行剖分,并且在单元分析中同时对电导率及二次电场进行三线性插值,实现电导率分块连续变化情况下,基于二次场的可控源电磁三维有限元数值模拟.这个新的可控源电磁三维正演方法可以模拟实际勘探中地下任意形状及电性参数连续变化的复杂模型.理论模型的计算结果表明,均匀大地计算的视电阻率误差和相位误差分别为0.002%和0.0005°.分层连续变化模型的有限元计算结果表明,其与对应的分层均匀模型解析结果有明显差异.三维异常体组合模型以及倾斜异常体等复杂模型的有限元计算结果也有效地反映了异常形态.     

An approximate analytical approach to compute geoelectric dipole–dipole responses due to a small buried cube

A simple analytical solution is presented for computing direct current (DC) electric field distortion due to a small cube in a homogeneous half-space, measured with a dipole–dipole array on the surface. Both the transmitter and the receiver may have any orientation; furthermore their position on the horizontal surface and the depth of the cube can be freely selected. It is shown that a simple approximate analytical method may replace more complicated 3D numerical modelling algorithms.
The approximation lies in the linearization of the problem: the secondary source (i.e. the cube) is considered as a system of three perpendicular electric dipoles. In spite of this first-order approximation, in the case of realistic depths z ( z R ≈0.1–0.5, where R is the transmitter–receiver distance), this approximate solution fits very well with true 3D numerical modelling results, and with analogue modelling results if a R ≤0.1, where a is the length of the side of the cube. Due to its simplicity, this method could be used for computing DC field distortion effects, estimating parameter-sensitivities, or even determining some initial models for further inversions.     

Penetration of electric field from the near ground atmospheric layer to the ionosphere

A mathematical model has been proposed for describing quasi-stationary atmospheric electric fields with approximate, but fairly accurate allowance for ionospheric conductivity. It is shown that some well-known models of electric field penetration from the Earth into the ionosphere have been deemed inadequate, though they work well in the atmosphere below 50 km. In these models, the arbitrarily specified boundary condition in the upper boundary of the atmosphere omits the existing good conductor or adds not existent conductor. The maximum possible field in our model is far less than in models where ionospheric conductivity is not taken into account, but vastly larger than in models based on the approximation with infinite Pedersen conductivity in the upper ionosphere.     

频率域2维电磁测深资料的RRI反演

本文将大地电磁反演中的RRI方法应用于线源频率测深模型资料的反演中.当背景电导率变化很小时,可用变化前的电场来近似代替变化后的电场,从而在反演方程的推导过程中将有源电磁场中的源项消掉,得到和大地电磁场相同的反演方程,使有源电磁波的反演也可以应用RRI方法.反演过程中所需要的模型资料通过有限元方法得到,该资料不需做近场校正,直接用适合于有源电磁场的RRI方法反演,避免了近场校正带来的误差.数值模型结果证明该方法是可行的.最后,用RRI方法讨论了当源和目标区间存在低阻异常体时只对目标区反演的可行性,对实际工作的解释有一定的指导意义.     

基于高斯牛顿法的频率域可控源电磁三维反演研究

三维反演解释是电磁法勘探发展的重要趋势,而如何提高三维反演的可靠性、稳定性和计算效率是算法开发者们目前的研究重点.本文实现了一种频率域可控源电磁(CSEM)三维反演算法.其中正演基于拟态有限体积法离散化,利用直接矩阵分解技术来求解大型线性系统方程,不仅准确、稳定,而且特别有利于含有大量发射场源位置的CSEM勘探情况;对目标函数的最优化采用高斯牛顿法(GN),具有近似二次的收敛性;使用预条件共轭梯度法(PCG)求解每次GN迭代所得到的法方程,避免了显式求解和存储灵敏度矩阵,减小了计算量.以上这些方法的结合应用,使得本文的三维反演算法准确、稳定且高效.通过陆地和海洋CSEM勘探场景中的典型理论模型的反演测试,验证了本文算法的有效性.     

Three‐dimensional modelling of controlled‐source electromagnetic surveys using an edge finite‐element method with a direct solver

A fully three‐dimensional finite‐element algorithm has been developed for simulating controlled‐source electromagnetic surveys. To exploit the advantages of geometric flexibility, frequency‐domain Maxwell's equations of the secondary electric field were discretised using edge‐based finite elements while the primary field was calculated analytically for a horizontally layered‐earth model. The resulting system of equations for the secondary field was solved using a parallel version of direct solvers. The accuracy of the algorithm was successfully verified by comparisons with integral‐equations and iterative solutions, and the applicability to models containing large conductivity contrasts was verified against published data. The advantages of geometry‐conforming meshes have been demonstrated by comparing different mesh systems to simulate an inclined sheet model. A comparison of the performance between direct and iterative solvers demonstrated the superior efficiency of direct solvers, particularly for multisource problems.     

Ground penetrating radar inversion in 1-D: an approach for the estimation of electrical conductivity, dielectric permittivity and magnetic permeability

This paper presents a method for inverting ground penetrating radargrams in terms of one-dimensional profiles. We resort to a special type of linearization of the damped E-field wave equation to solve the inverse problem. The numerical algorithm for the inversion is iterative and requires the solution of several forward problems, which we evaluate using the matrix propagation approach. Analytical expressions for the derivatives with respect to physical properties are obtained using the self-adjoint Green's function method. We consider three physical properties of materials; namely dielectrical permittivity, magnetic permeability and electrical conductivity. The inverse problem is solved minimizing the quadratic norm of the residuals using quadratic programming optimization. In the iterative process to speed up convergence we use the Levenberg–Mardquardt method. The special type of linearization is based on an integral equation that involves derivatives of the electric field with respect to magnetic permeability, electrical conductivity and dielectric permittivity; this equation is the result of analyzing the implication of the scaling properties of the electromagnetic field. The ground is modeled using thin horizontal layers to approximate general variations of the physical properties. We show that standard synthetic radargrams due to dielectric permittivity contrasts can be matched using electrical conductivity or magnetic permeability variations. The results indicate that it is impossible to differentiate one property from the other using GPR data.     

电导率各向异性的海洋电磁三维有限单元法正演

本文提出了一种基于非结构化网格的海洋电磁有限单元正演算法.为了回避场源奇异性,文中选用二次场算法,将背景电阻率设置为水平层状且各向异性,场源在水平层状各向异性介质中所激发的一次场通过汉克尔积分得到.基于Coulomb规范得到二次矢量位和标量位所满足的Maxwell方程组,通过Galerkin加权余量法形成大型稀疏有限元方程,采用不完全LU分解(ILU)预条件因子的quasi-minimum residual(QMR)迭代解法对有限元方程进行求解得到二次矢量位和标量位;进而,利用滑动平均方法得到二次矢量位和标量位在空间的导数,由此得到二次电磁场;通过一维模型对算法的可靠性进行验证,与此同时,针对实际复杂海洋电磁模型,比较有限元模拟结果与积分方程模拟结果,进一步验证算法精度.若干计算结果均表明,文中算法具有良好的通用性,适用于井中电磁、航空电磁,环境地球物理等非均匀且各向异性介质中的电磁感应基础研究.     

Fast and accurate three‐dimensional controlled source electromagnetic modelling†

We present a fast approximate method for three‐dimensional low frequency controlled source electro‐magnetic modeling. We apply the method to a synthetic model in a typical marine controlled source electromagnetic scenario, where conductivity and permittivity are different from the known background medium. For 3D configurations, fast computational methods are relevant for both forward and inverse modelling studies. Since this problem involves a large number of unknowns, it has to be solved efficiently to obtain results in a timely manner, without compromising accuracy. For this reason, the Born approximation, extended Born approximation and iterative extended Born approximation are implemented and compared with the full solution of the conjugate gradient fast Fourier transformation method. These methods are based on an electric field domain integral equation formulation. It is shown here how well the iterative extended Born approximation method performs in terms of both accuracy and speed with different configurations and different source positions. The improved accuracy comes at virtually no additional computational cost. With the help of this method, it is now possible to perform sensitivity analysis using 3D modelling in a timely manner, which is vital for controlled source electromagnetic applications. For forward modeling the solution at the sea‐bottom is of interest, because that is where the receivers are usually located. For inverse modeling, the accuracy of the solution in the target zone is important to obtain reasonably accurate conductivity values from the inversion using this approximate solution method. Our modelling studies show that the iterative extended Born approximation method is fast and accurate for both forward and inverse modelling. Sensitivity analysis as a function of the source position and different reservoir sizes validate the accuracy of the iterative extended Born approximation.