Hybrid finite-difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. F123-F137 ◽  
Author(s):  
M. Zaslavsky ◽  
V. Druskin ◽  
S. Davydycheva ◽  
L. Knizhnerman ◽  
A. Abubakar ◽  
...  
Keyword(s):  
Integral Equation ◽  
Finite Difference ◽  
Condition Number ◽  
Stiffness Matrix ◽  
Computational Cost ◽  
Integral Equation Method ◽  
Condition Numbers ◽  
Computational Domain ◽  
Electromagnetic Problems ◽  
Single Well

The modeling of the controlled-source electromagnetic (CSEM) and single-well and crosswell electromagnetic (EM) configurations requires fine gridding to take into account the 3D nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such as the seafloor bathymetry, the land topography, and targets with complex geometries and large contrasts in conductivities. Such problems significantly increase the computational cost of the conventional finite-difference (FD) approaches mainly due to the large condition numbers of the corresponding linear systems. To handle these problems, we employ a volume integral equation (IE) approach to arrive at an effective preconditioning operator for our FD solver. We refer to this new hybrid algorithm as the finite-difference integral equation method (FDIE). This FDIE preconditioning operator is divergence free and is based on a magnetic field formulation. Similar to the Lippman-Schwinger IE method, this scheme allows us to use a background elimination approach to reduce the computational domain, resulting in a smaller size stiffness matrix. Furthermore, it yields a linear system whose condition number is close to that of the conventional Lippman-Schwinger IE approach, significantly reducing the condition number of the stiffness matrix of the FD solver. Moreover, the FD framework allows us to substitute convolution operations by the inversion of banded matrices, which significantly reduces the computational cost per iteration of the hybrid method compared to the standard IE approaches. Also, well-established FD homogenization and optimal gridding algorithms make the FDIE more appropriate for the discretization of strongly inhomogeneous media. Some numerical studies are presented to illustrate the accuracy and effectiveness of the presented solver for CSEM, single-well, and crosswell EM applications.

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 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.

scholarly journals Improving condition number and convergence of the surface integral-equation method of moments for penetrable bodies

2012 ◽  
Vol 20 (15) ◽  
pp. 17237 ◽  
Author(s):  
Luis Landesa ◽  
Marta Gómez Araújo ◽  
José Manuel Taboada ◽  
Luis Bote ◽  
Fernando Obelleiro
Keyword(s):  
Integral Equation ◽  
Method Of Moments ◽  
Condition Number ◽  
Integral Equation Method ◽  
Equation Method ◽  
Surface Integral ◽  
Surface Integral Equation

scholarly journals Preconditioning in a wavelet basis and its application to some boundary value problems

2003 ◽  
pp. 86-93
Author(s):  
Marija Rasajski
Keyword(s):  
Finite Difference ◽  
Condition Number ◽  
Iterative Procedure ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Condition Numbers ◽  
Wavelet Basis ◽  
Numerical Examples ◽  
The Matrix ◽  
Difference Methods

Standard finite difference methods applied to the boundary value problem a(x)u" (x) + b(x)u'(x) + c(x)u(x) = f (x), u(0) = 0, u(1) = 0, lead to linear systems with large condition numbers. Solving a system, i.e. finding the inverse of a matrix with a large condition number can be achieved by some iterative procedure in a large number of iteration steps. By projecting the matrix of the system into the wavelet basis, and applying a diagonal pre-conditioner, we obtain a matrix with a small condition number. Computing the inverse of such a matrix requires fewer iteration steps, and that number does not grow significantly with the size of the system. Numerical examples, with various operators, are presented to illustrate the effect preconditioners have on the condition number, and the number of iteration steps.

Finite‐difference and frequency‐wavenumber modeling of seismic monopole sources and receivers in fluid‐filled boreholes

Geophysics ◽  
1994 ◽  
Vol 59 (7) ◽  
pp. 1053-1064 ◽  
Author(s):  
A. L. Kurkjian ◽  
R. T. Coates ◽  
J. E. White ◽  
H. Schmidt
Keyword(s):  
Finite Difference ◽  
Guided Waves ◽  
Computational Cost ◽  
Seismic Source ◽  
Low Velocity ◽  
Imaging Geometry ◽  
Receiver Array ◽  
Single Well ◽  
Tube Wave ◽  
Effective Source

In borehole seismic experiments the presence of the borehole has a significant effect on observations. Unfortunately, including boreholes explicitly in modeling schemes excludes the use of some methods (e.g., frequency‐wavenumber) and adds prohibitively to the cost of others (e.g., finite difference). To overcome this problem, we use the concept of an effective source/receiver array to replace the explicit representation of the borehole by a distributed seismic source/receiver. This method mimics the presence of the borehole at seismic frequencies under a wide variety of conditions without adding a significant computational cost. It includes the effects of dispersive and attenuative tube wave propagation, the generation of secondary sources at interfaces and caliper changes, and the generation of conical waves in low‐velocity layers. Comparison with a finite‐difference scheme with an explicit borehole representation validates the approach. The modeling method applied to a continuity logging geometry demonstrates that the presence of guided waves does not uniquely imply bed connectivity. Results for a single‐well imaging geometry emphasize the dominance of the tube wave in the hydrophone synthetics and demonstrates the necessity of using clamped geophones for single‐well experiments. The concept of an effective source/receiver array is an efficient way of including borehole phenomena in seismic modeling methods at minimal extra computational cost.

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