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.

A block iterative algorithm for 3-D electromagnetic modeling using integral equations with symmetrized substructures

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 291-295 ◽  
Author(s):  
Zonghou Xiong ◽  
Alan C. Tripp
Keyword(s):  
Integral Equation ◽  
Iterative Algorithm ◽  
Computation Time ◽  
Cost Effective ◽  
Integral Equation Method ◽  
Scattering Matrix ◽  
Electromagnetic Response ◽  
Electromagnetic Modeling ◽  
Large Structures ◽  
Computer Storage

The integral equation method is in many cases a cost effective way of modeling the electromagnetic response of 3-D conductivity structures. Yet the requirements on computer storage and on computation time in forming and inverting the scattering matrix limit its applications for large structures.

Electromagnetic modeling of 3-D structures by the method of system iteration using integral equations

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1556-1561 ◽  
Author(s):  
Zonghou Xiong
Keyword(s):  
Integral Equations ◽  
Three Dimensional ◽  
Computation Time ◽  
Iterative Solution ◽  
Matrix Inversion ◽  
Electromagnetic Modeling ◽  
Direct Inversion ◽  
Conductivity Structure ◽  
The Matrix ◽  
Earth Conductivity

A new approach for electromagnetic modeling of three‐dimensional (3-D) earth conductivity structures using integral equations is introduced. A conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion. The influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively. This is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method. This method reduces computer memory requirements since only one submatrix at a time needs to be stored. The diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure. The block iterative solution requires much less computation time than direct matrix inversion or conventional point iterative methods as the convergence depends on the number of the submatrices, not on the total number of unknowns in the solution. As the submatrices are independent of each other, this method is suitable for parallel processing.

Advanced three-dimensional electromagnetic modeling using a nested integral equation approach

2021 ◽  
Author(s):  
Chaojian Chen ◽  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Region Of Interest ◽  
Electromagnetic Modeling ◽  
Multi Scale ◽  
Scale Modeling ◽  
Uniform Grids ◽  
The Matrix ◽  
Integral Equation Approach ◽  
Matrix Vector

Summary Most of the existing three-dimensional (3-D) electromagnetic (EM) modeling solvers based on the integral equation (IE) method exploit fast Fourier transform (FFT) to accelerate the matrix-vector multiplications. This in turn requires a laterally-uniform discretization of the modeling domain. However, there is often a need for multi-scale modeling and inversion, for instance, to properly account for the effects of non-uniform distant structures, and at the same time, to accurately model the effects from local anomalies. In such scenarios, the usage of laterally-uniform grids leads to excessive computational loads, both in terms of memory and time. To alleviate this problem, we developed an efficient 3-D EM modeling tool based on a multi-nested IE approach. Within this approach, the IE modeling is first performed at a large domain and on a (laterally-uniform) coarse grid, and then the results are refined in the region of interest by performing modeling at a smaller domain and on a (laterally-uniform) denser grid. At the latter stage, the modeling results obtained at the previous stage are exploited. The lateral uniformity of the grids at each stage allows us to keep using the FFT for the matrix-vector multiplications. An important novelty of the paper is a development of a “rim domain” concept which further improves the performance of the multi-nested IE approach. We verify the developed tool on both idealized and realistic 3-D conductivity models, and demonstrate its efficiency and accuracy.

THREE‐DIMENSIONAL INDUCED POLARIZATION AND ELECTROMAGNETIC MODELING

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 309-324 ◽  
Author(s):  
Gerald W. Hohmann
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Induced Polarization ◽  
The Body ◽  
Scale Model ◽  
Two Dimensional ◽  
Electromagnetic Modeling ◽  
The Earth

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response varies significantly with conductivity contrast; the peak response occurs at higher contrasts for two‐dimensional bodies than for bodies of limited length. Very conductive bodies can produce negative IP response due to EM induction. An electrically polarizable body produces a small magnetic field, so that it is possible to measure IP with a sensitive magnetometer. Calculations show that horizontal loop EM response is enhanced when the background resistivity in the earth is reduced, thus confirming scale model results.

A Hybrid Integral Equation Method for Wave Forces on Three-Dimensional Offshore Structures

1987 ◽  
Vol 109 (3) ◽  
pp. 229-236 ◽  
Author(s):  
M. M. F. Yuen ◽  
F. P. Chau
Keyword(s):  
Integral Equation ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Offshore Structures ◽  
Equation Method ◽  
Wave Forces ◽  
Test Cases ◽  
Outer Domain ◽  
Hybrid Integral

Wave forces on arbitrary-shaped three-dimensional offshore structures are analyzed by the proposed hybrid integral equation method. Eigenfunction expansion is used to represent velocity potential in the outer domain, while a set of integral equations is developed for the inner domain encompassing the structure. A floating dock and a step cylinder are used as test cases showing the validity of the method.

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