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A 3-D finite-difference algorithm for DC resistivity modelling using conjugate gradient methods

A 3-D finite-difference algorithm for DC resistivity modelling using conjugate gradient methods Summary An accurate and efficient 3-D finite-difference forward algorithm for DC resistivity modelling is developed. The governing differential equations of the resistivity problem are discretized using central finite differences that are derived by a second-order Taylor series expansion. Electrical conductivity values may be arbitrarily distributed within the half-space. Conductivities at the grid points are calculated by a volume-weighted arithmetic average from conductivities assigned to grid cells. Variable grid spacing is incorporated. The algorithm does not limit the number and configuration of the sources, although all illustrative examples are computed using two current electrodes at the surface. In general, the linear set of equations resulting from this kind of discretization is non-symmetric and requires generalized numerical equation solvers. However, after symmetrizing the matrix equations, the ordinary conjugate gradient method becomes applicable. It takes advantage of the matrix symmetry and, thus, is superior to the generalized methods. An efficient SSOR-preconditioner (SSOR symmetric successive overrelaxation) provides fast convergence by decreasing the spectral condition number of the matrix without using additional memory. Furthermore, a compact storage scheme reduces memory requirements and accelerates mathematical matrix operations. The performance of five different equation solvers is investigated in terms of cpu time. The preconditioned conjugate gradient method (CGPC) is shown to be the most efficient matrix solver and is able to solve large equation systems in moderate times (approximately 21/2 minutes on a DEC alpha workstation for a grid with 50 000 nodes, and 48 minutes for 200000 nodes). The importance of the tolerance value in the stopping criterion for the iteration process is pointed out. In order to investigate the accuracy, the numerical results are compared with analytical or other solutions for three different model classes, yielding maximum deviations of 3.5 per cent or much less for most of the computed values of the apparent resistivity. In conclusion, the presented algorithm provides a powerful and flexible tool for practical application in resistivity modelling. conjugate gradient methods, DC geoelectrics, finite differences, 3-D resistivity modelling References Axelsson O. , 1980 . Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations , Lin. Algebra Appl ., 29 , 293 – 322 . Google Scholar Crossref Search ADS WorldCat Axelsson O. , 1980 . A generalized conjugate direction method and its application on a singular perturbation problem , Lecture Notes in Mathematics , 773 , 1 – 11 . OpenURL Placeholder Text WorldCat Brewitt-Taylor C.R. Weaver J.T. , 1976 . On the finite difference solution of two-dimensional induction problems , Geophys. J. R. astr. Soc ., 47 , 375 – 396 . Google Scholar Crossref Search ADS WorldCat Coggon J.H. , 1971 . Electromagnetic and electrical modeling by the Finite Element Method , Geophysics , 36 , 132 – 155 . Google Scholar Crossref Search ADS WorldCat Dey A. Morrison H.F. , 1979 . Resistivity modelling for arbitrarily shaped two-dimenional structures , Geophys. Prospect ., 27 , 106 – 136 . Google Scholar Crossref Search ADS WorldCat Dey A. Morrison H.F. , 1979 . Resistivity modelling for arbitrarily shaped three-dimensional structures , Geophysics , 44 , 753 – 780 . Google Scholar Crossref Search ADS WorldCat Elman H.C. Saad Y. Saylor P.E. , 1986 . A Hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations , SIAM J. Sci. Statist. Comput ., 7 , 840 – 855 . Google Scholar Crossref Search ADS WorldCat Fletcher R. , 1976 . Conjugate gradient methods for indefinite systems , Lecture Notes in Mathematics , 506 , 73 – 89 . OpenURL Placeholder Text WorldCat Freund R.W. , 1992 . Transpose-Free Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems , AT&T Bell Laboratories Numerical Analysis Manuscripts , 92 – 07 . Hestenes M.R. Stiefel E. , 1952 . Method of conjugate gradients for solving linear systems , J. Res. Nat. Bur. Standards , 49 , 409 – 436 . Google Scholar Crossref Search ADS WorldCat Hvoidara M. , 1994 . The boundary integral calculation of the D.C. geoelectric field due to a point current source on the surface of 2-layered Earth with a 3-D outcropping perturbing body , Contr. Geophys. Inst. Slov. Acad. Sci ., 25 . OpenURL Placeholder Text WorldCat Koefoed O. , 1979 . Geosounding principles 1, Elsevier, Amsterdam. Li, Y. & Oldenburg, D.W., 1994. Inversion of 3-D DC resistivity data using an approximate inverse mapping , Geophys. J. Int ., 116 , 527 – 537 . OpenURL Placeholder Text WorldCat Madden T.R. , 1972 . Transmission systems and network analogies to geophysical forward and inverse problems, Report 72–3 , Dept. of Earth and Planetary Siences, MIT , Cambridge, MA . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Manteuffel T.A. , 1977 . The Tchebychev iteration for nonsymmetric linear systems , Numerische Mathematik ., 28 , 307 – 327 . Google Scholar Crossref Search ADS WorldCat Manteuffel T.A. , 1978 . Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration , Numerische Mathematik , 31 , 183 – 208 . Google Scholar Crossref Search ADS WorldCat Mufti I.R. , 1976 . Finite-difference resistivity modeling for arbitrarily shaped two-dimensional structures , Geophysics , 41 , 62 – 78 . Google Scholar Crossref Search ADS WorldCat Mundry E. , 1984 . Geoelectrical model calculations for two-dimensional resistivity distributions , Geophys. Prospect ., 32 , 124 – 131 . Google Scholar Crossref Search ADS WorldCat O'Leary D.P. , 1980 . The block conjugate gradient algorithm and related methods , Lin. Algebra Appl ., 29 , 293 – 322 . Google Scholar Crossref Search ADS WorldCat Paige C.C. Saunders M.A. , 1982 . LSQR An algorithm for sparse linear equations and sparse least squares , ACM Trans. Math. Software , 8 , 43 – 71 . Google Scholar Crossref Search ADS WorldCat Park S.K. Van G.P. , 1991 . Inversion of pole-pole data for 3-D resistivity structure beneath arrays of electrodes , Geophysics , 56 , 951 – 960 . Google Scholar Crossref Search ADS WorldCat Reid J.K. , 1977 . Solution of linear systems of equations: direct methods (general) , Lecture Notes in Mathematics , 572 , 102 – 129 . OpenURL Placeholder Text WorldCat Saad Y. Schultz M.H. , 1986 . GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems , SIAM J. Sci. Statist. Comput ., 7 , 856 – 869 . Google Scholar Crossref Search ADS WorldCat Schulz R. , 1983 . Potentialberechnungen zur Interpretation von gleichstromgeoelektrischen Messungen über dreidimensionalen Störkörpern , PhD thesis , Mathematisch-naturwissenschaftliche Fakultät der Technischen Universität Clausthal , Germany . Schulz R. , 1985 . The method of integral equation in the direct current resistivity method and its accuracy , J. Geophys ., 56 , 192 – 200 . OpenURL Placeholder Text WorldCat Schwarz H.R. , 1991 . Methode der finiten Elemente , Teubner , Stuttgart . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Scriba H. , 1981 . Computation of the electric potential in three-dimensional structures , Geophys. Prospect ., 29 , 790 – 802 . Google Scholar Crossref Search ADS WorldCat Sonneveld P. , 1989 . CGS: a fast Lanczos-type solver for nonsymmetric linear systems , SIAM J. Sci. Statist. Comput ., 10 , 36 – 52 . Google Scholar Crossref Search ADS WorldCat Telford W.M. Geldart L.P. Sheriff R.E. , 1990 . Applied Geophysics , 2nd edn, Cambridge University Press , Cambridge . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Van der Vorst H.A. , 1981 . Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems , J. Comp. Phys ., 44 , 1 – 19 . Google Scholar Crossref Search ADS WorldCat Van der Vorst H.A. , 1992 . BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems , SIAM J. Sci. Stat. Comput ., 13 , 631 – 644 . Google Scholar Crossref Search ADS WorldCat Vinsome P.K.W. , 1976 . Orthomin, an iterative method for solving sparse sets of simultaneous linear equations , Soc. Petroleum Engin. of AIME, Paper SPE 5729 , 149 – 159 . OpenURL Placeholder Text WorldCat Weiss R. Schönauer W. , 1990 . Data reduction (DARE) preconditioning for generalized conjugate gradient methods , Lecture Notes in Mathematics , 1457 , 137 – 153 . OpenURL Placeholder Text WorldCat Widlund O. , 1978 . A Lanczos method for a class of nonsymmetric systems of linear equations , SIAM J. Numer. Anal ., 15 , 801 – 812 . Google Scholar Crossref Search ADS WorldCat Wurmstich B. Morgan F.D. , 1994 . Modeling of streaming potential responses caused by oil well pumping , Geophysics , 59 , 46 – 56 . Google Scholar Crossref Search ADS WorldCat Young D.M. Jea K.C. , 1980 . Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods , Lin. Algebra Appl ., 34 , 159 – 194 . Google Scholar Crossref Search ADS WorldCat Zhang J. Mackie R.L. Madden T.R. , 1995 . Three-dimensional resistivity forward modeling and inversion using conjugate gradients , Geophysics (submitted). This content is only available as a PDF. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

A 3-D finite-difference algorithm for DC resistivity modelling using conjugate gradient methods

Spitzer,, K.
Geophysical Journal International , Volume 123 (3) – Dec 1, 1995
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References (39)

Publisher
Oxford University Press
ISSN
0956-540X
eISSN
1365-246X
DOI
10.1111/j.1365-246X.1995.tb06897.x
Publisher site
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Abstract

Summary An accurate and efficient 3-D finite-difference forward algorithm for DC resistivity modelling is developed. The governing differential equations of the resistivity problem are discretized using central finite differences that are derived by a second-order Taylor series expansion. Electrical conductivity values may be arbitrarily distributed within the half-space. Conductivities at the grid points are calculated by a volume-weighted arithmetic average from conductivities assigned to grid cells. Variable grid spacing is incorporated. The algorithm does not limit the number and configuration of the sources, although all illustrative examples are computed using two current electrodes at the surface. In general, the linear set of equations resulting from this kind of discretization is non-symmetric and requires generalized numerical equation solvers. However, after symmetrizing the matrix equations, the ordinary conjugate gradient method becomes applicable. It takes advantage of the matrix symmetry and, thus, is superior to the generalized methods. An efficient SSOR-preconditioner (SSOR symmetric successive overrelaxation) provides fast convergence by decreasing the spectral condition number of the matrix without using additional memory. Furthermore, a compact storage scheme reduces memory requirements and accelerates mathematical matrix operations. The performance of five different equation solvers is investigated in terms of cpu time. The preconditioned conjugate gradient method (CGPC) is shown to be the most efficient matrix solver and is able to solve large equation systems in moderate times (approximately 21/2 minutes on a DEC alpha workstation for a grid with 50 000 nodes, and 48 minutes for 200000 nodes). The importance of the tolerance value in the stopping criterion for the iteration process is pointed out. In order to investigate the accuracy, the numerical results are compared with analytical or other solutions for three different model classes, yielding maximum deviations of 3.5 per cent or much less for most of the computed values of the apparent resistivity. In conclusion, the presented algorithm provides a powerful and flexible tool for practical application in resistivity modelling. conjugate gradient methods, DC geoelectrics, finite differences, 3-D resistivity modelling References Axelsson O. , 1980 . Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations , Lin. Algebra Appl ., 29 , 293 – 322 . Google Scholar Crossref Search ADS WorldCat Axelsson O. , 1980 . A generalized conjugate direction method and its application on a singular perturbation problem , Lecture Notes in Mathematics , 773 , 1 – 11 . OpenURL Placeholder Text WorldCat Brewitt-Taylor C.R. Weaver J.T. , 1976 . On the finite difference solution of two-dimensional induction problems , Geophys. J. R. astr. Soc ., 47 , 375 – 396 . Google Scholar Crossref Search ADS WorldCat Coggon J.H. , 1971 . Electromagnetic and electrical modeling by the Finite Element Method , Geophysics , 36 , 132 – 155 . Google Scholar Crossref Search ADS WorldCat Dey A. Morrison H.F. , 1979 . Resistivity modelling for arbitrarily shaped two-dimenional structures , Geophys. Prospect ., 27 , 106 – 136 . Google Scholar Crossref Search ADS WorldCat Dey A. Morrison H.F. , 1979 . Resistivity modelling for arbitrarily shaped three-dimensional structures , Geophysics , 44 , 753 – 780 . Google Scholar Crossref Search ADS WorldCat Elman H.C. Saad Y. Saylor P.E. , 1986 . A Hybrid Chebyshev Krylov subspace algorithm for solving nonsymmetric systems of linear equations , SIAM J. Sci. Statist. Comput ., 7 , 840 – 855 . Google Scholar Crossref Search ADS WorldCat Fletcher R. , 1976 . Conjugate gradient methods for indefinite systems , Lecture Notes in Mathematics , 506 , 73 – 89 . OpenURL Placeholder Text WorldCat Freund R.W. , 1992 . Transpose-Free Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems , AT&T Bell Laboratories Numerical Analysis Manuscripts , 92 – 07 . Hestenes M.R. Stiefel E. , 1952 . Method of conjugate gradients for solving linear systems , J. Res. Nat. Bur. Standards , 49 , 409 – 436 . Google Scholar Crossref Search ADS WorldCat Hvoidara M. , 1994 . The boundary integral calculation of the D.C. geoelectric field due to a point current source on the surface of 2-layered Earth with a 3-D outcropping perturbing body , Contr. Geophys. Inst. Slov. Acad. Sci ., 25 . OpenURL Placeholder Text WorldCat Koefoed O. , 1979 . Geosounding principles 1, Elsevier, Amsterdam. Li, Y. & Oldenburg, D.W., 1994. Inversion of 3-D DC resistivity data using an approximate inverse mapping , Geophys. J. Int ., 116 , 527 – 537 . OpenURL Placeholder Text WorldCat Madden T.R. , 1972 . Transmission systems and network analogies to geophysical forward and inverse problems, Report 72–3 , Dept. of Earth and Planetary Siences, MIT , Cambridge, MA . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Manteuffel T.A. , 1977 . The Tchebychev iteration for nonsymmetric linear systems , Numerische Mathematik ., 28 , 307 – 327 . Google Scholar Crossref Search ADS WorldCat Manteuffel T.A. , 1978 . Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration , Numerische Mathematik , 31 , 183 – 208 . Google Scholar Crossref Search ADS WorldCat Mufti I.R. , 1976 . Finite-difference resistivity modeling for arbitrarily shaped two-dimensional structures , Geophysics , 41 , 62 – 78 . Google Scholar Crossref Search ADS WorldCat Mundry E. , 1984 . Geoelectrical model calculations for two-dimensional resistivity distributions , Geophys. Prospect ., 32 , 124 – 131 . Google Scholar Crossref Search ADS WorldCat O'Leary D.P. , 1980 . The block conjugate gradient algorithm and related methods , Lin. Algebra Appl ., 29 , 293 – 322 . Google Scholar Crossref Search ADS WorldCat Paige C.C. Saunders M.A. , 1982 . LSQR An algorithm for sparse linear equations and sparse least squares , ACM Trans. Math. Software , 8 , 43 – 71 . Google Scholar Crossref Search ADS WorldCat Park S.K. Van G.P. , 1991 . Inversion of pole-pole data for 3-D resistivity structure beneath arrays of electrodes , Geophysics , 56 , 951 – 960 . Google Scholar Crossref Search ADS WorldCat Reid J.K. , 1977 . Solution of linear systems of equations: direct methods (general) , Lecture Notes in Mathematics , 572 , 102 – 129 . OpenURL Placeholder Text WorldCat Saad Y. Schultz M.H. , 1986 . GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems , SIAM J. Sci. Statist. Comput ., 7 , 856 – 869 . Google Scholar Crossref Search ADS WorldCat Schulz R. , 1983 . Potentialberechnungen zur Interpretation von gleichstromgeoelektrischen Messungen über dreidimensionalen Störkörpern , PhD thesis , Mathematisch-naturwissenschaftliche Fakultät der Technischen Universität Clausthal , Germany . Schulz R. , 1985 . The method of integral equation in the direct current resistivity method and its accuracy , J. Geophys ., 56 , 192 – 200 . OpenURL Placeholder Text WorldCat Schwarz H.R. , 1991 . Methode der finiten Elemente , Teubner , Stuttgart . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Scriba H. , 1981 . Computation of the electric potential in three-dimensional structures , Geophys. Prospect ., 29 , 790 – 802 . Google Scholar Crossref Search ADS WorldCat Sonneveld P. , 1989 . CGS: a fast Lanczos-type solver for nonsymmetric linear systems , SIAM J. Sci. Statist. Comput ., 10 , 36 – 52 . Google Scholar Crossref Search ADS WorldCat Telford W.M. Geldart L.P. Sheriff R.E. , 1990 . Applied Geophysics , 2nd edn, Cambridge University Press , Cambridge . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Van der Vorst H.A. , 1981 . Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems , J. Comp. Phys ., 44 , 1 – 19 . Google Scholar Crossref Search ADS WorldCat Van der Vorst H.A. , 1992 . BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems , SIAM J. Sci. Stat. Comput ., 13 , 631 – 644 . Google Scholar Crossref Search ADS WorldCat Vinsome P.K.W. , 1976 . Orthomin, an iterative method for solving sparse sets of simultaneous linear equations , Soc. Petroleum Engin. of AIME, Paper SPE 5729 , 149 – 159 . OpenURL Placeholder Text WorldCat Weiss R. Schönauer W. , 1990 . Data reduction (DARE) preconditioning for generalized conjugate gradient methods , Lecture Notes in Mathematics , 1457 , 137 – 153 . OpenURL Placeholder Text WorldCat Widlund O. , 1978 . A Lanczos method for a class of nonsymmetric systems of linear equations , SIAM J. Numer. Anal ., 15 , 801 – 812 . Google Scholar Crossref Search ADS WorldCat Wurmstich B. Morgan F.D. , 1994 . Modeling of streaming potential responses caused by oil well pumping , Geophysics , 59 , 46 – 56 . Google Scholar Crossref Search ADS WorldCat Young D.M. Jea K.C. , 1980 . Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods , Lin. Algebra Appl ., 34 , 159 – 194 . Google Scholar Crossref Search ADS WorldCat Zhang J. Mackie R.L. Madden T.R. , 1995 . Three-dimensional resistivity forward modeling and inversion using conjugate gradients , Geophysics (submitted). This content is only available as a PDF.

Journal

Geophysical Journal International – Oxford University Press

Published: Dec 1, 1995

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