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.

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.

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

Hybrid modeling of elastic‐wave propagation in two‐dimensional laterally inhomogeneous media

Geophysics ◽  
1987 ◽  
Vol 52 (6) ◽  
pp. 765-771 ◽  
Author(s):  
B. Kummer ◽  
A. Behle ◽  
F. Dorau
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Hybrid Scheme ◽  
Finite Difference Techniques

We have constructed a hybrid scheme for elastic‐wave propagation in two‐dimensional laterally inhomogeneous media. The algorithm is based on a combination of finite‐difference techniques and the boundary integral equation method. It involves a dedicated application of each of the two methods to specific domains of the model structure; finite‐difference techniques are applied to calculate a set of boundary values (wave field and stress field) in the vicinity of the heterogeneous domain. The continuation of the near‐field response is then calculated by means of the boundary integral equation method. In a numerical example, the hybrid method has been applied to calculate a plane‐wave response for an elastic lens embedded in a homogeneous environment. The example shows that the hybrid scheme enables more efficient modeling, with the same accuracy, than with pure finite‐difference calculations.

scholarly journals A parallel finite‐difference approach for 3D transient electromagnetic modeling with galvanic sources

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1192-1202 ◽  
Author(s):  
Michael Commer ◽  
Gregory Newman
Keyword(s):  
Magnetic Field ◽  
Finite Difference ◽  
Electric Field ◽  
Electric Fields ◽  
Time Derivative ◽  
Stable Solution ◽  
Parallel Computer ◽  
Staggered Grid ◽  
Electromagnetic Modeling ◽  
Transient Electromagnetic

A parallel finite‐difference algorithm for the solution of diffusive, three‐dimensional (3D) transient electromagnetic field simulations is presented. The purpose of the scheme is the simulation of both electric fields and the time derivative of magnetic fields generated by galvanic sources (grounded wires) over arbitrarily complicated distributions of conductivity and magnetic permeability. Using a staggered grid and a modified DuFort‐Frankel method, the scheme steps Maxwell's equations in time. Electric field initialization is done by a conjugate‐gradient solution of a 3D Poisson problem, as is common in 3D resistivity modeling. Instead of calculating the initial magnetic field directly, its time derivative and curl are employed in order to advance the electric field in time. A divergence‐free condition is enforced for both the magnetic‐field time derivative and the total conduction‐current density, providing accurate results at late times. In order to simulate large realistic earth models, the algorithm has been designed to run on parallel computer platforms. The upward continuation boundary condition for a stable solution in the infinitely resistive air layer involves a two‐dimensional parallel fast Fourier transform. Example simulations are compared with analytical, integral‐equation and spectral Lanczos decomposition solutions and demonstrate the accuracy of the scheme.

Irregular surface seismic forward modeling by a body-fitted rotated–staggered-grid finite-difference method

2018 ◽  
Vol 15 (3-4) ◽  
pp. 420-431
Author(s):  
Jing-Wang Cheng ◽  
Na Fan ◽  
You-Yuan Zhang ◽  
Xiao-Chun Lü
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Forward Modeling ◽  
Staggered Grid ◽  
Irregular Surface

scholarly journals On the applicability of a renormalized Born series for seismic wavefield modelling in strongly scattering media

2019 ◽  
Vol 17 (2) ◽  
pp. 277-299 ◽  
Author(s):  
Xingguo Huang ◽  
Morten Jakobsen ◽  
Ru-Shan Wu
Keyword(s):  
Integral Equation ◽  
Frequency Domain ◽  
Large Scale ◽  
Integral Equation Method ◽  
Equation Method ◽  
Scattering Media ◽  
Matrix Vector Multiplication ◽  
Born Series ◽  
And Inversion ◽  
Matrix Vector

Abstract Scattering theory is the basis for various seismic modeling and inversion methods. Conventionally, the Born series suffers from an assumption of a weak scattering and may face a convergence problem. We present an application of a modified Born series, referred to as the convergent Born series (CBS), to frequency-domain seismic wave modeling. The renormalization interpretation of the CBS from the renormalization group prospective is described. Further, we present comparisons of frequency-domain wavefields using the reference full integral equation method with that using the convergent Born series, proving that both of the convergent Born series can converge absolutely in strongly scattering media. Another attractive feature is that the Fast Fourier Transform is employed for efficient implementations of matrix–vector multiplication, which is practical for large-scale seismic problems. By comparing it with the full integral equation method, we have verified that the CBS can provide reliable and accurate results in strongly scattering media.

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.

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