Surveys in Geophysics

, Volume 26, Issue 6, pp 767–799 | Cite as

Three-Dimensional Electromagnetic Modelling and Inversion from Theory to Application

Article

Abstract

The whole subject of three-dimensional (3-D) electromagnetic (EM) modelling and inversion has experienced a tremendous progress in the last decade. Accordingly there is an increased need for reviewing the recent, and not so recent, achievements in the field. In the first part of this review paper I consider the finite-difference, finite-element and integral equation approaches that are presently applied for the rigorous numerical solution of fully 3-D EM forward problems. I mention the merits and drawbacks of these approaches, and focus on the most essential aspects of numerical implementations, such as preconditioning and solving the resulting systems of linear equations. I refer to some of the most advanced, state-of-the-art, solvers that are today available for such important geophysical applications as induction logging, airborne and controlled-source EM, magnetotellurics, and global induction studies. Then, in the second part of the paper, I review some of the methods that are commonly used to solve 3-D EM inverse problems and analyse current implementations of the methods available. In particular, I also address the important aspects of nonlinear Newton-type optimisation techniques and computation of gradients and sensitivities associated with these problems.

Keywords

three-dimensional modelling and inversion electromagnetic fields optimisation 

Abbreviations

EM

electromagnetic

3-D

three-dimensional

FD

finite-difference

FE

finite-element

IE

integral equation

NLCG

nonlinear conjugate gradients

QN

Quasi–Newton

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abubakar, A., Berg, P. 2000‘Non-Linear Three-Dimensional Inversion of Cross-Well Electrical Measurements’Geophys. Prosp.48109134Google Scholar
  2. Abubakar, A., Berg, P. 2001‘Nonlinear Inversion of the Electrode Logging Measurements in a Deviated Well’Geophysics66110124CrossRefGoogle Scholar
  3. Alumbaugh, D. L., Newman, G. A., Prevost, L., Shadid, J. N. 1996‘Three-Dimensional Wide Band Electromagnetic Modeling on Massively Parallel Computers’Radio Sci.31123CrossRefGoogle Scholar
  4. Aruliah, D. A., Ascher, U. M. 2003‘Multigrid Preconditioning for Krylov Methods for Time-Harmonic Maxwell’s Equations in 3D’SIAM J. Scient. Comput.24702718Google Scholar
  5. Avdeev, D. B., Kuvshinov, A. V., Pankratov, O. V., Newman, G.A. 1997‘High-Performance Three-Dimensional Electromagnetic Modeling Using Modified Neumann Series. Wide-band Numerical Solution and Examples’J. Geomagn. Geoelectr.4915191539Google Scholar
  6. Avdeev, D. B., Kuvshinov, A. V., Pankratov, O. V., Newman, G. A. 1998‘Three-Dimensional Frequency-Domain Modelling of Airborne Electromagnetic Responses’Explor. Geophy.29111119Google Scholar
  7. Avdeev, D. B., Kuvshinov, A. V., Pankratov, O. V., and Newman, G. A.: 2000, ‘3D EM Modelling Using Fast Integral Equation Approach with Krylov Subspace Accelerator, in Expanded abstracts of the 62nd EAGE Conference, Glasgow, Scotland, pp. 195–198Google Scholar
  8. Avdeev, D. B., Kuvshinov, A. V., Pankratov, O. V., Newman, G. A. 2002a‘Three-Dimensional Induction Logging Problems. Part I. An Integral Equation Solution and Model Comparisons’Geophysics67413426CrossRefGoogle Scholar
  9. Avdeev, D. B., Kuvshinov, A. V., Epova, X. A. 2002b‘Three-Dimensional Modeling of Electromagnetic Logs From Inclined-Horizontal Wells, Izvestiya’Phys. Solid Earth38975980Google Scholar
  10. Badea, E. A., Everett, M. E., Newman, G. A., Biro, O. 2001‘Finite-Element Analysis of Controlled-Source Electromagnetic Induction Using Coulomb-Gauged Potentials’Geophysics66786799CrossRefGoogle Scholar
  11. Boyce, W., Lynch, D., Paulsen, K., Minerbot, G. 1992‘Nodal Based Finite Element Modeling Maxwell’s Equations’IEEE Trans. Antennas Propagat.40642651Google Scholar
  12. Broyden, C. G. 1969‘A New Double-Rank Minimization Algorithm’Notices Am. Math. Soc.16670Google Scholar
  13. Cerv, V. 1990‘Modelling and Analysis of Electromagnetic Fields in 3D Inhomogeneous Media’Surv. Geophys.11205230CrossRefGoogle Scholar
  14. Champagne, N. J., Berryman, J. G., Buettner, H. M., Grant, J. B., and Sharpe, R. M.: 1999, ‘A Finite-Difference Frequency-Domain Code for Electromagnetic Induction Tomography’, in Proc. SAGEEP, Oakland, CA, pp. 931–940Google Scholar
  15. Chew, W. C. 1999Waves and Fields in Inhomogeneous MediaWiley-IEEE PressPiscataway, NJGoogle Scholar
  16. Commer, M., Newman, G. 2004‘A Parallel Finite-Difference Approach for 3D Transient Electromagnetic Modeling with Galvanic Sources’Geophysics6911921202CrossRefGoogle Scholar
  17. Davydycheva, S., Druskin, V., Habashy, T. 2003‘An Efficient Finite Difference Scheme for Electromagnetic Logging in 3D Anisotropic Inhomogeneous Media’Geophysics6815251536CrossRefGoogle Scholar
  18. Dawson, T. W., Weaver, J. T. 1979‘Three-Dimensional Electromagnetic Induction in a Non-Uniform Thin Sheet at the Surface of Uniformly Conducting Earth’Geophys. J. Roy. Astr. Soc.59445462Google Scholar
  19. Dennis, J. E., Schnabel, R. B. 1996Numerical Methods for Unconstrained Optimization and Nonlinear EquationsSIAMPhiladelphiaGoogle Scholar
  20. Dey, A., Morrison, H. F. 1979‘Resistivity Modelling for Arbitrary Shaped Three-Dimensional Structures’Geophysics44753780CrossRefGoogle Scholar
  21. Dmitriev, V. I. 1969Electromagnetic Fields in Inhomogeneous MediaMoscow State UniversityMoscow (in Russian)Google Scholar
  22. Dmitriev, V. I., Nesmeyanova, N. I. 1992‘Integral Equation Method in Three-Dimensional Problems of Low-Frequency Electrodynamics’Comput. Math. Model.3313317Google Scholar
  23. Dorn, O., Bertete-Aguirre, H., Berryman, J. G., Papanicolaou, G. C. 1999‘A Nonlinear Inversion Method for 3D Electromagnetic Imaging Using Adjoint Fields’Inv. Prob.1515231558CrossRefGoogle Scholar
  24. Druskin, V., Knizhnerman, L. 1994‘Spectral Approach to Solving Three-Dimensional Maxwell’s Equations in the Time and Frequency Domains’Radio Sci.29937953CrossRefGoogle Scholar
  25. Druskin, V., Knizhnerman, L., Lee, P. 1999‘A New Spectral Lanczos Decomposition Method for Induction Modeling in Arbitrary 3D Geometry’Geophysics64701706CrossRefGoogle Scholar
  26. Eaton, P. A. 1989‘3D Electromagnetic Inversion Using Integral Equations’Geophys. Prosp.37407426Google Scholar
  27. Ellis, R. G.: 1999, ‘Joint 3-D Electromagnetic Inversion’, in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7, pp. 179–192Google Scholar
  28. Ellis, R. G.: 2002, Electromagnetic Inversion Using the QMR-FFT Fast Integral Equation Method, in 72st Ann. Internat. Mtg., Soc. Expl. Geophys., pp. 21–25Google Scholar
  29. Everett, M., Schultz, A. 1996‘Geomagnetic Induction in a Heterogeneous Sphere, Azimuthally Symmetric Test Computations and the Response of an Undulating 660-km Discontinuity’J. Geophys. Res.10127652783CrossRefGoogle Scholar
  30. Jones, F. W., Pascoe, L. J. 1972‘The Perturbation of Alternating Geomagnetic Fields by Three-Dimensional Conductivity Inhomogeneities’Geophys. J. Roy. Astr. Soc.27479484Google Scholar
  31. Judin, M. N. 1980Magnetotelluric Field Calculation in Three-Dimensional Media Using a Grid Method, in Problems of the Sea Electromagnetic InvestigationsIZMIRANMoscow96101(in Russian)Google Scholar
  32. Faber, V., Manteuffel, T. 1984‘Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method’SIAM J. Numer. Anal.24352362Google Scholar
  33. Farquharson, C. G., Oldenburg, D. W. 1996‘Approximate Sensitivities for the Electromagnetic Inverse Problem’Geophys. J. Int.126235252Google Scholar
  34. Farquharson, C. G., Oldenburg, D. W. 1998‘Non-Linear Inversion Using General Measures of Data Misfit and Model Structure’Geophys. J. Int.134213233CrossRefGoogle Scholar
  35. Farquharson, C. G., Oldenburg, D. W., Haber, E., and Shekhtman, R.: 2002, ‘An Algotithm for The Three-Dimensional Inversion of Magnetotelluric Data,’ in 72st Ann. Internat. Mtg., Soc. Expl. Geophys., pp.649–652Google Scholar
  36. Farquharson, C. G., Oldenburg, D. W. 2004‘A Comparison of Automatic Techniques for Estimating the Regularization Parameter in Non-Linear Inverse Problems’Geophys. J. Int.156411425CrossRefGoogle Scholar
  37. Fletcher, R., Reeves, C. M. 1964‘Function Minimization by Conjugate Gradients’Comput. J.7149154CrossRefGoogle Scholar
  38. Fomenko, E. Y., Mogi, T. 2002‘A New Computation Method for a Staggered Grid of 3D EM Field Conservative Modeling’Earth Planets Space54499509Google Scholar
  39. Golub, G. H., Loan, C. F. 1996Matrix ComputationsThirdThe Johns Hopkins University PressBaltimore and LondonGoogle Scholar
  40. Grammatica, N., Tarits, P. 2002‘Contribution at Satellite Altitude of Electromagnetically Induced Anomalies Arising from a Three-Dimensional Heterogeneously Conducting Earth, using Sq as an Inducing Source Field’Geophys. J. Int.151913923CrossRefGoogle Scholar
  41. Greenbaum, A. 1997Iterative Methods for Solving Linear SystemsSIAMPhiladelphiaGoogle Scholar
  42. Habashy, T. M., Groom, R. W., Spies, B. R. 1993‘Beyond the Born and Rytov Approximations: A Nonlinear Approach to Electromagnetic Scattering’J. Geophys. Res.9817591775Google Scholar
  43. Haber, E.: 1999, ‘Modeling 3D EM Using Potentials and Mixed Finite Elements,’ in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7, pp. 12–15Google Scholar
  44. Haber, E., Ascher, U. M., Aruliah, D. A., Oldenburg, D. W. 2000a‘Fast Simulation of 3D Electromagnetic Problems Using Potentials’J. Comp. Phys.163150171Google Scholar
  45. Haber, E., Ascher, U. M., Aruliah, D. A., Oldenburg, D. W. 2000b‘On Optimisation Techniques for Solving Nonlinear Inverse Problems’Inv. Prob.1612631280CrossRefGoogle Scholar
  46. Haber, E., Ascher, U. M., Oldenburg, D. W., Shekhtman R. and Chen J.: 2002a, ‘3-D Frequency Domain CSEM Inversion Using Unconstrained Optimization,’ in 72st Ann. Internat. Mtg., Soc. Expl. Geophys., pp.653–656Google Scholar
  47. Haber, E., Ascher, U. M., Oldenburg, D. W. 2004‘Inversion of 3D Electromagnetic Data in Frequency and Time Domain Using an Inexact All-at-Once Approach’Geophysics6912161228CrossRefGoogle Scholar
  48. Haber, E. 2005‘Quasi–Newton Methods for Large-Scale Electromagnetic Inverse Problems’Inv. Prob.21305323CrossRefGoogle Scholar
  49. Hamano, Y. 2002‘A New Time-Domain Approach for the Electromagnetic Induction Problem in a Three-Dimensional Heterogeneous Earth’Geophys. J. Int.150753169CrossRefGoogle Scholar
  50. Hestenes, M. R., Stiefel, E. 1952‘Methods of Conjugate Gradients for Solving Linear Systems’J. Res. Nat. Bur. Stand.49409436Google Scholar
  51. Hohmann, G. W. 1975‘Three-Dimensional Induced-Polarization and Electromagnetic Modeling’Geophysics40309324CrossRefGoogle Scholar
  52. Hohmann G. W.: 1988, ‘Numerical Modelling of Electromagnetic Methods of Geophysics’, in M. N. Nabighian (ed.), Electromagnetic methods in applied geophysics, Vol. 1, S.E.G. Investigations in geophysics 3, pp. 314–364Google Scholar
  53. Hursan, G., Zhdanov, M. S. 2002‘Contraction Integral Equation Method in Three-Dimensional Modeling’Radio Sci.371089doi, 10.1029/2001RS002513Google Scholar
  54. Kaufman, A. A., Eaton, P. A. 2001The Theory of Inductive Prospectings, Methods in Geochemistry and Geophysics 35ElsevierAmsterdam–NewYork–TokyoGoogle Scholar
  55. Kelly, C. T. 1999Iterative Methods for OptimizationSIAMPhiladelphiaGoogle Scholar
  56. Koyama, T., Shimizu, H., Utada, H. 2002‘Possible Effects of Lateral Heterogeneity in the D’ Layer on Electromagnetic Variations of Core Origin’Phys. Earth Planet. Interiors12999116CrossRefGoogle Scholar
  57. Kuvshinov, A. V., Avdeev, D. B., Pankratov, O. V., Golyshev, S. A., and Olsen, N.: 2002, ‘Modelling Electromagnetic Fields in 3-D Spherical Earth Using Fast Integral Equation Approach’, in M. S. Zhdanov and P. E. Wannamaker (eds.), Three Dimensional Electromagnetics, Methods in Geochemistry and Geophysics 35: Elsevier, pp. 43–54Google Scholar
  58. Kuvshinov, A. V., Utada, H., Avdeev, D. B., Koyama, T. 2005‘3-D Modelling and Analysis of Dst C-Responses in the North Pacific Ocean Region, Revisited’Geophys. J. Int.160505526CrossRefGoogle Scholar
  59. Kuvshinov, A. V., Olsen, N. 2004

    ‘Modelling the Coast Effect of Geomagnetic Storms at Ground and Satellite Altitude’

    Reigber, C.Luhr, H.Schwintzer, P.Wickert, J. eds. Earth Observation with CHAMP. Results from Three Years in OrbitSpringer-VerlagBerlin353359
    Google Scholar
  60. LaBrecque, D.: 1999, ‘Finite Difference Modeling of 3-D EM Fields with Scalar and Vector Potentials’, in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7, pp. 148–160Google Scholar
  61. Lager, I. E., Mur, G. 1998‘Generalized Cartesian finite elements’IEEE Trans. Magn.3422202227Google Scholar
  62. Lanczos, C. 1952‘Solution of Systems of Linear Equations by Minimized Iterations’J. Res. Nat. Bur. Stand.493353Google Scholar
  63. Lesselier, D. and Habashy, T. (eds.) 2000, ‘Special Section on Electromagnetic Imaging and Inversion of the Earth’s subsurface’, Inv. Prob. 16(5) 1083–1376Google Scholar
  64. Lesselier, D. and Chew, W. C. (eds.): 2004, ‘Special Section on Electromagnetic Characterization of Buried Obstacles’, Inv. Prob. 20(6), S1–S256Google Scholar
  65. Li, Y., Oldenburg, D. W. 2000‘3-D Inversion of Induced Polarization Data’Geophysics6519311945Google Scholar
  66. Li, Y., Spitzer, K. 2002‘Three-Dimensional DC Resistivity Forward Modeling Using Finite Elements in Comparison With Finite-Difference Solutions’Geophys. J. Int.151924934CrossRefGoogle Scholar
  67. Livelybrooks, D. 1993‘Program 3Dfeem, A Multidimensional Electromagnetic Finite Element Model’Geophys. J. Int.114443458Google Scholar
  68. Mackie, R. L., Madden, T. R. 1993‘Three-Dimensional Magnetotelluric Inversion Using Conjugate Gradients’Geophys. J. Int.115215229Google Scholar
  69. Mackie, R. L., Madden, T. R., Wannamaker, P. 1993‘3-D Magnetotelluric Modeling Using Difference Equations – Theory and Comparisons to Integral Equation Solutions’Geophysics58215226Google Scholar
  70. Mackie, R. L., Smith, T. J., Madden, T. R. 1994‘3-D Electromagnetic Modeling Using Difference Equations, The Magnetotelluric Example’Radio Sci.29923935CrossRefGoogle Scholar
  71. Mackie, R.L., Rodi, W., and Watts, M.D.: 2001, ‘3-D Magnetotelluric Inversion for Resource Exploration, in 71st Ann. Internat. Mtg., Soc. Expl. Geophys., pp. 1501–1504Google Scholar
  72. Mackie, R.L.: 2004, Private communicationGoogle Scholar
  73. Macnae, J. and Liu, G. (eds.): 2003, Three Dimensional Electromagnetics III, Austr. Soc. Expl. GeophysGoogle Scholar
  74. Madden, T. R., Mackie, R. L. 1989‘Three-Dimensional Magnetotelluric Modeling and Inversion’Proc. IEEE77318333CrossRefGoogle Scholar
  75. Martinec, Z. 1999‘Spectral-Finite Element Approach to Three-Dimensional Electromagnetic Induction in a Spherical Earth’Geophys. J. Int.136229250CrossRefGoogle Scholar
  76. McGillivray, P. R., Oldenburg, D. W. 1990‘Methods for Calculating Frechet Derivatives and Sensitivities for the Non-Linear Inverse Problems’Geophysics60899911Google Scholar
  77. McKirdy, D., Weaver, J. T., Dawson, T. W. 1985‘Induction in a Thin Sheet of Variable Conductance at the Surface of a Stratified Earth- II. Three-dimensional theory’Geophys. Roy. Astr. Soc.80177194Google Scholar
  78. Mitsuhata, Y., Uchida, T., Amano, H. 2002‘2.5-D Inversion of Frequency-Domain Electromagnetic Data Generated by a Grounded-Wire Source’Geophysics6717531768CrossRefGoogle Scholar
  79. Mitsuhata, Y., Uchida, T. 2004‘3D Magnetotelluric Modeling Using the T-Ω Document Finite-Element Method’Geophysics69108119CrossRefGoogle Scholar
  80. Newman, G. A., Hohmann, G. W. 1988‘Transient Electromagnetic Response of High-Contrast Prisms in a Layered Earth’Geophysics53691706CrossRefGoogle Scholar
  81. Newman, G. A., Alumbaugh, D. L. 1995‘Frequency-Domain Modeling of Airborne Electromagnetic Responses Using Staggered Finite Differences’Geophys. Prosp.4310211042Google Scholar
  82. Newman, G. A., Alumbaugh, D. L. 1997‘Three-Dimensional Massively Parallel Electromagnetic Inversion- I.’Theory, Geophys. J. Int.128345354Google Scholar
  83. Newman, G. A., Alumbaugh, D. L. 2000‘Three-Dimensional Magnetotelluric Inversion Using Non-Linear Conjugate Gradients’Geophys. J. Int.140410424CrossRefGoogle Scholar
  84. Newman, G. A., Hoversten, G. M. 2000‘Solution Strategies for 2D and 3D EM Inverse Problem’Inv. Prob.1613571375CrossRefGoogle Scholar
  85. Newman, G. A., Hoversten, G. M., and Alumbaugh, D. L.: 2002, ‘3D Magnetotelluric Modeling and Inversion, Applications to Sub-Salt Imaging’, in M.S. Zhdanov and P.E. Wannamaker (eds.), Three Dimensional Electromagnetics, Methods in Geochemistry and Geophysics 35, Elsevier, pp. 127–152Google Scholar
  86. Newman, G. A., Alumbaugh, D. L. 2002‘Three-Dimensional Induction Logging Problems. Part I. An Integral Equation Solution and Model Comparisons’Geophysics67484491CrossRefGoogle Scholar
  87. Newman, G. A., Recher, S., Tezkan, B., Neubauer, F. M. 2003‘3D Inversion of a Scalar Radio Magnetotelluric Field Data Set’Geophysics68791802CrossRefGoogle Scholar
  88. Newman, G. A., Boggs, P. T. 2004‘Solution Accelerators for Large-Scale Three-Dimensional Electromagnetic Inverse Problem’Inv. Prob.20s151s170CrossRefGoogle Scholar
  89. Newman, G. A., Commer, M. 2005‘New Advances in Three-Dimensional Transient Electromagnetic Inversion’Geophys. J. Int.160532CrossRefGoogle Scholar
  90. Nocedal, J., Wright, S. 1999Numerical OptimizationSpringer-VerlagNew YorkGoogle Scholar
  91. O’Leary, D. P.: 1996, ‘Conjugate Gradients and Related KMP Algorithms, the Beginnings’, in L. Adams and J.L. Nazareth (eds.), Linear and Nonlinear Conjugate Gradient-Related Methods, SIAM, pp. 1–9Google Scholar
  92. Oristaglio, M. J. and Spies, B.R.: 1999, ‘Three Dimensional Electromagnetics’, in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7Google Scholar
  93. Paige, C. C., Sounders, M. A. 1974‘Solution of Sparse Indefinite Systems of Linear Equations’SIAM J. Numer. Anal.11197209CrossRefGoogle Scholar
  94. Pankratov, O. V., Avdeev, D. B., Kuvshinov, A. V. 1995‘Electromagnetic Field Scattering in a Heterogeneous Earth, A Solution to the Forward Problem’Phys. Solid Earth31201209Google Scholar
  95. Pankratov, O. V., Kuvshinov, A. V., Avdeev, D. B. 1997‘High-Performance Three-Dimensional Electromagnetic Modeling Using Modified Neumann series. Anisotropic case’J. Geomagn. Geoelectr.4915411547Google Scholar
  96. Paulsen, K. D., Linch, D. R., Strohbehn, J. W. 1988‘Three-dimensional finite, boundary, and hybrid element solutions of the Maxwell equations for lossy dielectric media’IEEE Trans. Microwave Theory Tech.36682693Google Scholar
  97. Polak, E., Ribiere, G. 1969‘Note sur la convergence de methode de directions conjuguees’Revue Francaise d’Informatique et de Recherche Operationnelle163543Google Scholar
  98. Portniaguine, O., Zhdanov, M. S. 1999‘Focusing Geophysical Inversion Images’Geophysics64874887CrossRefGoogle Scholar
  99. Pridmore, D. F., Hohmann, G. W., Ward, S. H., Still, W. R. 1981‘An Investigation of Finite-Element Modeling For Electrical and Electromagnetic Data in Three Dimensions’Geophysics4610091024CrossRefGoogle Scholar
  100. Raiche, A. 1974‘An Integral Equation Approach to Three-Dimensional Modeling’Geophys. J.36363376Google Scholar
  101. Ratz, S.: 1999, ‘A 3D Finite Element Code for Modeling of Electromagnetic Responses, in Expanded abstracts of the 2nd International Symposium on 3D Electromagnetics, Salt Lake City, Utah, pp.33–36Google Scholar
  102. Reddy, I. K., Rankin, D., Phillips, R. J. 1977‘Three-dimensional modelling in magnetotelluric and magnetic variational sounding’Geophys. J. Roy. Astr. Soc.51313325Google Scholar
  103. Rodi, W., Mackie, R. L. 2000‘Nonlinear conjugate gradients algorithm for 2-D magnetotelluric inversion’Geophysics66174187Google Scholar
  104. Romanov, V. G, Kabanikhin, S. I. 1994Inverse Problems for Maxwell’s EquationsVSPUtrechtGoogle Scholar
  105. Saad, Y., Schultz, M. H. 1986‘GMRES, A Generalized Minimal Residual Algorithm for Nonsymmetric Linear Systems’SIAM J. Sci. Stat. Comput.7856869CrossRefGoogle Scholar
  106. Sasaki, Y. 2001‘Full 3-D Inversion of Electromagnetic Data on PC’J. Appl. Geophys.464554CrossRefGoogle Scholar
  107. Sasaki, Y. 2004‘Three-Dimensional Inversion of Static-Shifted Magnetotelluric Data’Earth Planets Space56239248Google Scholar
  108. Schultz, A. and Pritchard, G.: 1999, Three-Dimensional Inversion for Large-Scale Structure in a Spherical Domain’, in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7, pp. 451–476Google Scholar
  109. Singer, B. Sh., Fainberg, E. B. 1985Electromagnetic Induction in Non-uniform Thin LayersIZMIRANMoscow (in Russian)Google Scholar
  110. Singer, B. Sh. 1995‘Method for Solution of Maxwell’s Equations in Non-Uniform Media’Geophys. J. Int.120590598Google Scholar
  111. Singer, B. Sh., Fainberg, E. B. 1995‘Generalization of the Iterative-Dissipative Method for Modeling Electromagnetic Fields in Nonuniform Media with Displacement Currents’J. Appl. Geophys.344146CrossRefGoogle Scholar
  112. Singer, Sh. B., Fainberg, E. B. 1997‘Fast and Stable Method for 3-D Modelling of Electromagnetic Field’Explor. Geophys.28130135Google Scholar
  113. Singer, B. Sh., Mezzatesta, A., and Wang, T.: 2003, Integral equation approach based on contraction operators and Krylov subspace optimisation, in J. Macnae and G. Liu (eds.), Three Dimensional Electromagnetics III, Austr. Soc. Expl. GeophysGoogle Scholar
  114. Siripunvaraporn, W., Uyeshima, M., Egbert, G. 2004a‘Three-Dimensional Inversion for Network-Magnetotelluric Data’Earth Planets Space56893902Google Scholar
  115. Siripunvaraporn, W., Egbert, G., Lenbury, Y., Uyeshima, M. 2004b‘Three-Dimensional Magnetotelluric Inversion: Data Space Method’Phys. Earth Planet. Inter.150314Google Scholar
  116. Smith, J. T., Booker, J. R. 1991‘Rapid Inversion of Two- and Three-Dimensional Magnetotelluric Data’J. Geophys. Res.9639053922CrossRefGoogle Scholar
  117. Smith, J. T. 1996a‘Conservative Modeling of 3-D Electromagnetic Fields, Part I, Properties and Error Analysis’Geophysics6113081318Google Scholar
  118. Smith, J. T. 1996b‘Conservative Modeling of 3-D Electromagnetic Fields, Part II, Biconjugate Gradient Solution and an Accelertor’Geophysics6113191324Google Scholar
  119. Song, L.-P., Liu, Q. H. 2004‘Fast Three-Dimensional Electromagnetic Nonlinear Inversion in Layered Media with a Novel Scattering Approximation’Inv. Prob.20S171S194CrossRefGoogle Scholar
  120. Spichak, V. V.: 1983, Numerically modeling the electromagnetic fields in three-dimensional media, Ph. D. thesis, Moscow, 215 p. (in Russian)Google Scholar
  121. Spichak, V., Popova, I. 2000‘Artificial Neural Network Inversion of Magnetotelluric Data in Terms of Three-Dimensional Earth Macroparameters’Geophys. J. Int.1421526CrossRefGoogle Scholar
  122. Sugeng, F., Raiche, A. and Xiong, Z.: 1999, ‘An Edge-Element Approach to Model the 3D EM Response of Complex Structures with High Contrasts’, in Expanded abstracts of the 2nd International Symposium on 3D Electromagnetics, Salt Lake City, Utah, pp. 25–28Google Scholar
  123. Tabarovsky, L. A. 1975Application of Integral Equation Method to Geoelectrical ProblemsNovosibirskNauka (in Russian)Google Scholar
  124. Tamarchenko, T., Frenkel, M., and Mezzatesta, A.: 1999, Three-dimensional modeling of microresistivity devices, in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7, pp. 77–83Google Scholar
  125. Tarantola, A. 1987Inverse Problem TheoryElsevierAmsterdam–Oxford–New York–TokyoGoogle Scholar
  126. Tarits, P. 1994‘Electromagnetic Studies of Global Geodynamic Processes’Surv. Geophys.15209238CrossRefGoogle Scholar
  127. Tikhonov, A. N., Arsenin, V. Y. 1977Solutions of Ill-posed ProblemsWileyNew YorkGoogle Scholar
  128. Ting, S. C., Hohmann, G. W. 1981‘Integral Equation Modeling of Three-Dimensional Magnetotelluric Response’Geophysics46182197CrossRefGoogle Scholar
  129. Torres-Verdin, C., Habashy, T. M. 1994‘Rapid 2.5-D Forward Modeling and Inversion Via a New Nonlinear Scattering Approximation’Radio Sci.2910511079Google Scholar
  130. Torres-Verdin, C., Habashy, T. M. 2002‘Rapid Numerical Simulations of Axisymmetric Single-well Induction Data Using the Extended Born Approximation’Radio Sci.3612871306Google Scholar
  131. Tseng, H. -W., Lee, K. H., Becker, A. 2003‘3-D Interpretation of Electromagnetic Data Using a Modified Extended Born Approximation’Geophysics68127137CrossRefGoogle Scholar
  132. Uyeshima, M., Schultz, A. 2000‘Geoelectromagnetic Induction in a Heterogeneous Sphere, A New 3-D Forward Solver Using a Staggered-Grid Integral Formulation’Geophys. J. Int.140636650CrossRefGoogle Scholar
  133. Uchida, T. and Sasaki, Y.: 2003, Stable 3-D Inversion of MT Data and Its Application for Geothermal Exploration, in J. Macnae and G. Liu (eds.), Three Dimensional Electromagnetics III, Austr. Soc. Expl. GeophysGoogle Scholar
  134. Varentsov, Iv. M.: 1999, ‘The Selection of Effective Finite Difference Solvers in 3D Electromagnetic Modeling’, in Expanded Abstracts of 2nd International Symposium on 3D Electromagnetics, Salt Lake City, UtahGoogle Scholar
  135. Varentsov, Iv. M.: 2002, ‘A General Approach to the Magnetotelluric Data Inversion in a Piece-Continuous Medium’ Fizika Zemli 11 (in Russian)Google Scholar
  136. Vasseur, G., Weidelt, P. 1977‘Bimodal Electromagnetic Induction in Non-Uniform Thin Sheets with Application to the Northern Pyrenean Induction Anomaly’Geophys. J. R. Astr. Soc.51669690Google Scholar
  137. Velimsky, J., Everett, M. E., and Martinec, Z.: 2003, ‘The Transient Dst Electromagnetic Induction Signal at Satellite Altitudes for a Realistic 3-D Electrical Conductivity in the Crust and Mantle, Geophys. Res. Letts. 30(7), doi,10.1029/2002GL016671Google Scholar
  138. Wang, T., Hohmann, G. W. 1993‘A Finite-Difference Time-Domain Solution for Three-Dimensional Electromagnetic Modeling’Geophysics58797809CrossRefGoogle Scholar
  139. Wang, T., Oristaglio, M., Tripp, A., Hohmann, G. W. 1994‘Inversion of Diffusive Transient Electromagnetic Data by a Conjugate Gradient Method’Radio Sci.2911431156Google Scholar
  140. Wang, T., Tripp, A. 1996‘FDTD Simulation of EM Wave Propagation in 3-D Media’Geophysics61110120Google Scholar
  141. Wang, T., Fang, S. 2001‘3D Electromagnetic Anisotropy Modeling Using Finite Differences’Geophysics6613861398Google Scholar
  142. Wannamaker, P. E., Hohmann, G. W., San Filipo, W. A. 1984‘Electromagnetic Modeling of Three-Dimensional Bodies in Layered Earth Using Integral Equations’Geophysics496074Google Scholar
  143. Wannamaker, P. E. 1991‘Advances in Three-Dimensional Magnetotelluric Modeling Using Integral Equations’Geophysics5617161728Google Scholar
  144. Weaver, J. T. 1994Mathematical Methods for Geo-electromagnetic InductionJohn Wiley and SonsTaunton, UKGoogle Scholar
  145. Weaver, J. T., Agarwal, A. K. and Pu, X. H.: 1999, ‘Three-Dimensional Finite-Difference Modeling of the Magnetic Field in Geo-Electromagnetic Induction’, in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7, pp. 426–443Google Scholar
  146. Weidelt, P. 1975‘Electromagnetic Induction in 3D Structures’J. Geophys.4185109Google Scholar
  147. Weidelt, P.: 1999, 3D conductivity models, Implications of electrical anisotropy, in M. J. Oristaglio and B. R. Spies (eds.), Three Dimensional Electromagnetics, S.E.G. Geophysical Developments Series 7, pp. 119–137Google Scholar
  148. Weiss, Ch. J., Everett, M. E. 1998‘Geomagnetic Induction in a Heterogeneous Sphere, Fully Three-Dimensional Test Computation and the Response of a Realistic Distribution of Oceans and Continents’Geophys. J. Int.135650662CrossRefGoogle Scholar
  149. Weiss, Ch. J., Newman, G. A. 2002‘Electromagnetic Induction in a Fully 3-D Anisotropic Earth’Geophysics6711041114CrossRefGoogle Scholar
  150. Weiss, Ch. J., Newman, G. A. 2003‘Electromagnetic Induction in a Fully 3-D Anisotropic Earth, Part 2, The LIN Preconditioner’Geophysics68922930CrossRefGoogle Scholar
  151. Xiong, Z. 1992‘EM Modeling Three-Dimensional Structures by the Method of System Iteration Using Integral Equations’Geophysics5715561561Google Scholar
  152. Xiong, Z., Tripp, A. C. 1995‘Electromagnetic Scattering of Large Structures in Layered Earth Using Integral Equations’Radio Sci.30921929CrossRefGoogle Scholar
  153. Xiong, Z., Raiche, A., Sugeng, F. 2000‘Efficient Solution of Full Domain 3D Electromagnetic Modeling Problems’Explor. Geophys.31158161Google Scholar
  154. Yamane, K., Kim, H. J., Ashida, Y. 2000‘Three-Dimensional Magnetotelluric Inversion Using a Generalized RRI Method and Its Applications’Butsuri-Tansa (Geophys. Explor.)5315011513Google Scholar
  155. Yee, K. S. 1966‘Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media’IEEE Trans. Ant. Prop.AP-14302307Google Scholar
  156. Yoshimura, R. and Oshiman, N.: 2002, Edge-based Finite Element Approach to the Simulation of Geoelectromagnetic Induction in a 3-D sphere, Geophys. Res. Letts. 29(3), doi,10.1029/2001GL014121Google Scholar
  157. Zhang, Z. 2003‘3D Resistivity Mapping of Airborne EM Data’Geophysics6818961905Google Scholar
  158. Zhdanov, M. S., Fang, S. 1996‘Quasi-Linear Approximation in 3-D EM Modeling’Geophysics61646665Google Scholar
  159. Zhdanov, M. S., Fang, S. 1997‘Quasi-Linear Series in Three-Dimensional Electromagnetic Modeling’Radio Sci.3221672188CrossRefGoogle Scholar
  160. Zhdanov, M. S., Portniaguine, O. 1997‘Time-Domain Electromagnetic Migration in the Solution of Inverse Problems’Geophys. J. Int.131293309Google Scholar
  161. Zhdanov, M. S., Varentsov, I. M., Weaver, J. T., Golubev, N. G., and Krylov, V. A.: 1997, ‘Methods for Modelling Electromagnetic Fields; Results from COMMEMI – the International Project On the Comparison of Modelling Methods for Electromagnetic Induction’, in J. T. Weaver (ed.), J. Appl. Geophys. 37, 133–271Google Scholar
  162. Zhdanov, M. S., Fang, S., Hursan, G. 2000‘Electromagnetic Inversion using Quasi-Linear Approximation’Geophysics6515011513Google Scholar
  163. Zhdanov, M. S. 2002Geophysical Inverse Theory and Regularization problemsElsevierAmsterdam-New York-TokyoGoogle Scholar
  164. Zhdanov, M. S. and Wannamaker, P. E.: 2002, ‘Three Dimensional Electromagnetics’, in M. S. Zhdanov and P. E. Wannamaker (eds.), Three Dimensional Electromagnetics, Methods in Geochemistry and Geophysics 35, ElsevierGoogle Scholar
  165. Zhdanov, M. S. and Golubev, N. G.: 2003, ‘Three-Dimensional Inversion of Magnetotelluric Data in Complex Geological Structures’, in J. Macnae and G. Liu (eds.), Three Dimensional Electromagnetics III, Austr. Soc. Expl. GeophysGoogle Scholar
  166. Zhdanov, M. S., Tolstaya, E. 2004‘Minimum Support Nonlinear Parametrization in the Solution of a 3D Magnetotelluric Inverse Problem’Inv. Prob.20937952CrossRefGoogle Scholar
  167. Zunoubi, M. R., Jin, J. -M., Donepudi, K. C., Chew, W. C. 1999‘A Spectral Lanczos Decomposition Method for Solving 3-D Low-Frequency Electromagnetic Diffusion by the Finite-Element Method’IEEE Trans. Antennas Propogat.47242248Google Scholar
  168. Zyserman, F. I., Santos, J. E. 2000‘Parallel Finite Element Algorithm with Domain Decomposition for Three-Dimensional Magnetotelluric Modeling’J. Appl. Geophys.44337351CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Russian Academy of SciencesInstitute of Terrestrial Magnetism, Ionosphere and Radiowave PropagationTroitskRussia
  2. 2.Dublin Institute for Advanced Studies, School of Cosmic PhysicsDublin 2Ireland

Personalised recommendations