scholarly journals Feasibility of simplified integral equation modeling of low-frequency marine CSEM with a resistive target

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. F107-F117 ◽  
Author(s):  
Shaaban A. Bakr ◽  
Trond Mannseth
Keyword(s):  
Integral Equation ◽  
Sparse Matrix ◽  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Background Field ◽  
Petroleum Exploration ◽  
Equation Modeling ◽  
Dense Matrix ◽  
Matrix Methods ◽  
Marine Csem

We have assessed the accuracy of a simplified integral equation (SIE) modeling approach for marine controlled-source electromagnetics (CSEM) with low applied frequencies and a resistive target. The most computationally intensive part of rigorous integral equation (IE) modeling is the computation of the anomalous electric field within the target itself. This leads to a matrix problem with a dense coefficient matrix. It is well known that, in general, the presence of many grid cells creates a computational disadvantage for dense-matrix methods compared to sparse-matrix methods. The SIE approach replaces the dense-matrix part of rigorous IE modeling by sparse-matrix calculations based on an approximation of Maxwell’s equations. The approximation is justified theoretically if a certain dimensionless parameter [Formula: see text] is small. As opposed to approximations of the Born type, the validity of the SIE approach does not rely on the anomalous field to be small in comparison with the background field in the target region. We have calculated [Formula: see text] for a range of parameter values typical for marine CSEM, and compared the SIE approach numerically to the rigorous IE method and to the quasi-linear (QL) and quasi-analytic (QA) approximate solutions. It is found that the SIE approach is very accurate for small [Formula: see text], corresponding to frequencies in the lower range of those typical for marine CSEM for petroleum exploration. In addition, the SIE approach is found to be significantly more accurate than the QL and QA approximations for small [Formula: see text].

Rapid construction of equivalent sources using wavelets

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. L51-L59 ◽  
Author(s):  
Yaoguo Li ◽  
Douglas W. Oldenburg
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Sparse Matrix ◽  
Coefficient Matrix ◽  
Magnetic Data ◽  
Dense Matrix ◽  
Wavelet Domain ◽  
Space Domain ◽  
Equivalent Source ◽  
Equivalent Sources

We have developed a fast algorithm for generating an equivalent source by using fast wavelet transforms based on orthonormal, compactly supported wavelets. We apply a 2D wavelet transform to each row and column of the coefficient matrix and subsequently threshold the transformed matrix to generate a sparse representation in the wavelet domain. The algorithm then uses this sparse matrix to construct the the equivalent source directly in the wavelet domain. Performing an inverse wavelet transform then yields the equivalent source in the space domain. Using upward continuation of total-field magnetic data between uneven surfaces as examples, we have compared this approach with the direct solution using the dense matrix in the space domain. We have shown that the wavelet approach can reduce the CPU time by as many as two orders of magnitude.

Conjugate direction relaxation solutions for 3-D magnetotelluric modeling

Geophysics ◽  
1993 ◽  
Vol 58 (7) ◽  
pp. 1052-1057 ◽  
Author(s):  
Randall L. Mackie ◽  
Theodore R. Madden
Keyword(s):  
Integral Equation ◽  
Sparse Matrix ◽  
Three Dimensional ◽  
Approximate Solutions ◽  
Systems Of Equations ◽  
Integral Equation Methods ◽  
Tremendous Amount ◽  
Sparse Systems ◽  
Differential Methods ◽  
Made In

In recent years, there has been a tremendous amount of progress made in three‐dimensional (3-D) magnetotelluric modeling algorithms. Much of this work has been devoted to the integral equation technique (e.g., Hohmann, 1975; Weidelt, 1975; Wannamaker et al., 1984; Wannamaker, 1991). This method has contributed significantly to our understanding of electromagnetic field behavior in 3-D models. However, some of the very earliest work in 3-D modeling concentrated on differential methods (e.g., Jones and Pascoe, 1972; Reddy et al., 1977). It is generally recognized that differential methods are better suited than integral equation methods to model arbitrarily complex geometries, and consequently this area has recently been receiving a great deal of attention (e.g., Madden and Mackie, 1989; Xinghua et al., 1991; Mackie et al., 1993; Smith, 1992, personnal communication). Differential methods lead to large sparse systems of equations to be solved for the unknown field values. It is possible to use relaxation algorithms to quickly obtain approximate solutions to these systems of equations without resorting to standard matrix inversion routines or sparse matrix solvers.

scholarly journals Analysis of the Propagation in High-Speed Interconnects for MIMICs by Means of the Method of Analytical Preconditioning: A New Highly Efficient Evaluation of the Coefficient Matrix

2021 ◽  
Vol 11 (3) ◽  
pp. 933
Author(s):  
Mario Lucido
Keyword(s):  
Integral Equation ◽  
Integrated Circuits ◽  
High Speed ◽  
Coefficient Matrix ◽  
Single Step ◽  
Integral Theorem ◽  
Acceleration Technique ◽  
Cauchy Integral Theorem ◽  
Integral Equation Formulation ◽  
High Speed Interconnects

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

Analysis of Rotating Blade Forced Vibration Due to Torsional Excitation Using the Method of Harmonic Balance

2002 ◽  
Author(s):  
B. O. Al-Bedoor ◽  
A. A. Al-Qaisia
Keyword(s):  
Forced Vibration ◽  
Harmonic Balance ◽  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Blade Vibration ◽  
Varying Coefficients ◽  
Rotating Blade ◽  
Method Of Harmonic Balance ◽  
Torsional Excitation ◽  
Truncated Fourier Series

This paper presents an analysis of the forced vibration of rotating blade due to torsional excitation. The model analyzed is a multi-modal forced second order ordinary differential equation with multiple harmonically varying coefficients. The method of Harmonic Balance (HB) is employed to find approximate solutions for each of the blade modes in the form of truncated Fourier series. The solutions have shown multi resonance response for the first blade vibration mode. The examination of the determinant of the harmonic balance solution coefficient matrix for stability purposes has shown that the region between the two resonance points is an unstable vibration region. Numerical integration of the equations is conducted at different frequency ratio points and the results are discussed. This solution provides a very critical operation and design guidance for rotating blade with torsional vibration excitation.

Sparse Matrix Methods

2006 ◽  
pp. 40-1-40-20
Author(s):  
Esmond G. Ng
Keyword(s):  
Sparse Matrix ◽  
Matrix Methods
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