Quasi‐analytical approximations and series in electromagnetic modeling

Geophysics ◽  
2000 ◽  
Vol 65 (6) ◽  
pp. 1746-1757 ◽  
Author(s):  
Michael S. Zhdanov ◽  
Vladimir I. Dmitriev ◽  
Sheng Fang ◽  
Gábor Hursán
Keyword(s):  
Integral Equation ◽  
Linear Approximation ◽  
Linear Equations ◽  
Integral Equation Method ◽  
Forward Modeling ◽  
Computational Effort ◽  
Correct Result ◽  
Electromagnetic Modeling ◽  
Analytical Approximations ◽  
Analytical Series

The quasi‐linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background (normal) field through an electrical reflectivity tensor λ⁁. In the original formulation of the quasi‐linear approximation, λ⁁ was determined by solving a minimization problem based on an integral equation for the scattering currents. This approach is much less time‐consuming than the full integral equation method; however, it still requires solution of the corresponding system of linear equations. In this paper, we present a new approach to the approximate solution of the integral equation using λ⁁ through construction of quasi‐analytical expressions for the anomalous electromagnetic field for 3-D and 2-D models. Quasi‐analytical solutions reduce dramatically the computational effort related to forward electromagnetic modeling of inhomogeneous geoelectrical structures. In the last sections of this paper, we extend the quasi‐analytical method using iterations and develop higher order approximations resulting in quasi‐analytical series which provide improved accuracy. Computation of these series is based on repetitive application of the given integral contraction operator, which insures rapid convergence to the correct result. Numerical studies demonstrate that quasi‐analytical series can be treated as a new powerful method of fast but rigorous forward modeling solution.

A hybrid solver based on the integral equation method and vector finite-element method for 3D controlled-source electromagnetic method modeling

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. E319-E333 ◽  
Author(s):  
Rong Liu ◽  
Rongwen Guo ◽  
Jianxin Liu ◽  
Changying Ma ◽  
Zhenwei Guo
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Integral Equation ◽  
Finite Element ◽  
Differential Equations ◽  
Electric Fields ◽  
Hybrid Method ◽  
Linear Equations ◽  
Integral Equation Method ◽  
Controlled Source

The integral equation method (IEM) and differential equation methods have been widely applied to provide numerical solutions of the electromagnetic (EM) fields caused by inhomogeneity for the controlled-source EM method. IEM has a bounded computational domain and has been well-known for its efficiency, whereas differential equation methods are commonly used for complex geologic models. To use the advantages of the two types of approaches, a hybrid method is developed based on the combination of IEM and the edge-based finite-element method (vector FEM). In the hybrid scheme, Maxwell’s differential equations of the secondary electric fields in the frequency domain are derived for a volume with boundary placed slightly away from the inhomogeneity. The vector FEM is applied to solve Maxwell’s differential equations, and a system of linear equations for the secondary electric fields can be derived by the minimum theorem. The secondary electric fields on the boundary are represented by IEM in terms of the secondary electric fields inside the inhomogeneity. The linear equations from substituting the boundary values into the vector FEM linear equations then can be solved to obtain the secondary electric fields inside the inhomogeneity. The secondary electric fields at receivers are calculated by IEM based on the secondary electric field solutions inside the inhomogeneity. The hybrid algorithm is verified by comparison of simulated results with earlier works on canonical 3D disc models with a high accuracy. Numerical comparisons with two conventional IEMs demonstrate that the hybrid method is more accurate and efficient for high-conductivity contrast media.

Introducing new global electromagnetic modeling solver

2020 ◽  
Author(s):  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Linear Equations ◽  
Numerical Algorithms ◽  
Iterative Solution ◽  
Parallel Computations ◽  
Electromagnetic Modeling ◽  
System Of Linear Equations

<p> In this contribution, we present novel global 3-D electromagnetic forward solver based on a numerical solution of integral equation (IE) with contracting kernel. Compared to widely used x3dg code which is also based on IE approach, new solver exploits alternative (more efficient and accurate) numerical algorithms to calculate Green’s tensors, as well as an alternative (Galerkin) method to construct the system of linear equations (SLE). The latter provides guaranteed convergence of the iterative solution of SLE. The solver outperforms x3dg in terms of accuracy, and, in contrast to (sequential) x3dg, it allows for efficient parallel computations, meaning that the code has practically linear scalability up to the hundreds of processors.</p>

scholarly journals Linear Approximation of Volume Integral Equations for the Problem of Magnetostatics

2018 ◽  
Vol 173 ◽  
pp. 03001
Author(s):  
Pavel Akishin ◽  
Andrey Sapozhnikov
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Linear Approximation ◽  
Integral Equation Method ◽  
Equation Method ◽  
Volume Integral ◽  
Magnetic Systems ◽  
Volume Integral Equation ◽  
Volume Integral Equation Method

The volume integral equation method is considered for magnetic systems. New modeling results are reported.

A hybrid finite-element and integral-equation method for forward modeling of 3D controlled-source electromagnetic induction

2018 ◽  
Vol 15 (3-4) ◽  
pp. 536-544
Author(s):  
Feng Zhou ◽  
Jing-Tian Tang ◽  
Zheng-Yong Ren ◽  
Zhi-Yong Zhang ◽  
Huang Chen ◽  
...  
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Electromagnetic Induction ◽  
Integral Equation Method ◽  
Forward Modeling ◽  
Equation Method ◽  
Controlled Source ◽  
Hybrid Finite Element

Symmetry properties of the scattering matrix in 3-D electromagnetic modeling using the integral equation method

Geophysics ◽  
1992 ◽  
Vol 57 (9) ◽  
pp. 1199-1202 ◽  
Author(s):  
Zonghou Xiong
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Computation Time ◽  
Integral Equation Method ◽  
Scattering Matrix ◽  
Electromagnetic Modeling ◽  
Symmetry Properties ◽  
Computer Storage ◽  
Earth Conductivity ◽  
Number Of Cells

Modeling large three‐dimensional (3-D) earth conductivity structures continues to pose challenges. Although the theories of electromagnetic modeling are well understood, the basic computational problems are practical, involving the quadratically growing requirements on computer storage and cubically growing computation time with the number of cells required to discretize the modeling body.

Induced‐polarization and electromagnetic modeling of a three‐dimensional body buried in a two‐layer anisotropic earth

Geophysics ◽  
1986 ◽  
Vol 51 (12) ◽  
pp. 2235-2246 ◽  
Author(s):  
Zonghou Xiong ◽  
Yanzhong Luo ◽  
Shoutan Wang ◽  
Guangyao Wu
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Lower Layer ◽  
Integral Equation Method ◽  
Induced Polarization ◽  
Equation Method ◽  
Frequency Effect ◽  
Electromagnetic Modeling ◽  
The Earth

The integral equation method is used for induced‐polarization (IP) and electromagnetic (EM) modeling of a finite inhomogeneity in a two‐layer anisotropic earth. An integral equation relates the exciting electric field and the scattering currents in the homogeneity through the electric tensor Green’s function deduced from the vector potentials in the lower layer of the earth. Digital linear filtering and three‐point parabolic Lagrangian interpolation with two variables speed up the numerical evaluation of the Hankel transforms in the tensor Green’s function. The results of this integral equation method for isotropic media are checked by direct comparisons with results by other workers. The results for anisotropic media are indirectly verified, mainly by checking the tensor Green’s function. The calculated results show that the effects of anisotropy on apparent resistivity and percent frequency effect are to reduce the size of the anomalies, shift the anomaly region downward toward the lower centers of the pseudosections, and enhance the effect of overburden; in other words, to shade the target from detection. This is due to the increase of currents flowing horizontally through the earth over the target. The effects of anisotropy on horizontal‐loop EM responses are to reduce the amplitude and lower the critical frequency of the maximum of the quadrature component.

scholarly journals A hybrid finite-difference and integral-equation method for modeling and inversion of marine controlled-source electromagnetic data

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. E323-E336 ◽  
Author(s):  
Daeung Yoon ◽  
Michael S. Zhdanov ◽  
Johan Mattsson ◽  
Hongzhu Cai ◽  
Alexander Gribenko
Keyword(s):  
Integral Equation ◽  
Finite Difference ◽  
Electric Fields ◽  
Integral Equation Method ◽  
Numerical Differentiation ◽  
Forward Modeling ◽  
Staggered Grid ◽  
3D Inversion ◽  
Controlled Source ◽  
And Inversion

One of the major problems in the modeling and inversion of marine controlled-source electromagnetic (CSEM) data is related to the need for accurate representation of very complex geoelectrical models typical for marine environment. At the same time, the corresponding forward-modeling algorithms should be powerful and fast enough to be suitable for repeated use in hundreds of iterations of the inversion and for multiple transmitter/receiver positions. To this end, we have developed a novel 3D modeling and inversion approach, which combines the advantages of the finite-difference (FD) and integral-equation (IE) methods. In the framework of this approach, we have solved Maxwell’s equations for anomalous electric fields using the FD approximation on a staggered grid. Once the unknown electric fields in the computation domain of the FD method are computed, the electric and magnetic fields at the receivers are calculated using the IE method with the corresponding Green’s tensor for the background conductivity model. This approach makes it possible to compute the fields at the receivers accurately without the need of very fine FD discretization in the vicinity of the receivers and sources and without the need for numerical differentiation and interpolation. We have also developed an algorithm for 3D inversion based on the hybrid FD-IE method. In the case of the marine CSEM problem with multiple transmitters and receivers, the forward modeling and the Fréchet derivative calculations are very time consuming and require using large memory to store the intermediate results. To overcome those problems, we have applied the moving sensitivity domain approach to our inversion. A case study for the 3D inversion of towed streamer EM data collected by PGS over the Troll field in the North Sea demonstrated the effectiveness of the developed hybrid method.

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