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Solution of 3D time-domain electromagnetic problems using optimal subspace projection


Time-domain problems for controlled-source electromagnetic exploration require accurate discretization of the solution for multiple spacial and temporal scales. Therefore, forward simulation using conventional computational methods becomes computationally expensive, even without accounting for induced-polarization (IP) effects. These effects create another complication caused by the presence of a convolution integral in the time-domain Maxwell system. We suggested a novel, fast, and robust algorithm to solve the 3D time-domain electromagnetic (EM) problems that can be considered as a generalization of the spectral Lanczos decomposition method. The new method also allowed us to incorporate the IP effects without significant cost increase. The discretized large-scale Maxwell system was projected onto a small subspace consisting of the Laplace-domain solutions (the so-called parameter-dependent Krylov subspace) for an optimally chosen set of Laplace parameters. The projected system preserved stability and passivity of the original problem. Moreover, our approach (even without the IP effects) yielded an optimal solution within a wide class of computational algorithms that included the conventional time-domain finite-difference, discrete Fourier transform and spectral Lanczos decomposition methods. Numerical examples for the controlled-source EM problem showed that the new algorithm produces accurate solutions on time intervals spanning from milliseconds to hundreds of seconds with the cost of (at most) 60 time steps of the implicit finite-difference time domain scheme. This showed significant improvement even compared with results for nonpolarized media reported in recent literature. Additionally, the new algorithm had the unique capability to accurately handle large-scale 3D models, including the IP effects.


  • Abubakar, A., T. Habashy, V. Druskin, L. Knizhnerman, and D. Alumbaugh, 2008, Two-and-half dimensional forward and inverse modeling for the interpretation of low-frequency electromagnetic measurements: Geophysics, 73, no. 4, F165–F177, doi: 10.1190/1.2937466.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Bai, Z., and Y. Su, 2005, Dimension reduction of second-order dynamical systems via a second-order Arnoldi method: SIAM Journal on Scientific Computing, 26, 1692–1709, doi: 10.1137/040605552.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Barsukov, P., E. Fainberg, and B. Singer, 2007, A method for hydrocarbon reservoir mapping and apparatus for use when performing the method: International patent WO 2007/053025.Google Scholar
  • Beattie, C., and S. Gugercin, 2009, Interpolatory projection methods for structure-preserving model reduction: Systems and Control Letters, 58, 225–232, doi: 10.1016/j.sysconle.2008.10.016.SCLEDC0167-6911CrossrefWeb of ScienceGoogle Scholar
  • Beckermann, B., and L. Reichel, 2009, Error estimation and evaluation of matrix functions via the Faber transform: SIAM Journal on Numerical Analysis, 47, 3849–3883, doi: 10.1137/080741744.SJNAEQ0036-1429CrossrefWeb of ScienceGoogle Scholar
  • Boerner, R.-U., O. Ernst, and K. Spitzer, 2008, Fast 3D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection: Geophysical Journal International, 173, 766–780, doi: 10.1111/gji.2008.173.issue-3.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Chandrasekharan, K., 1968, Introduction to analytic number theory: Springer-Verlag.CrossrefGoogle Scholar
  • Cole, K., and R. Cole, 1941, Dispersion and absorption in dielectrics. I. Alternating current characteristics: Journal of Chemical Physics, 9, 341–351, doi: 10.1063/1.1750906.JCPSA60021-9606CrossrefGoogle Scholar
  • Constable, S., and C. Cox, 1996, Marine controlled-source electromagnetic sounding, 2, The PEGASUS experiment: Journal of Geophysical Research, 101, 5519–5530, doi: 10.1029/95JB03738.JGREA20148-0227CrossrefWeb of ScienceGoogle Scholar
  • Davydycheva, S., V. Druskin, and T. Habashy, 2003, An efficient finite-difference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media: Geophysics, 68, 1525–1536, doi: 10.1190/1.1620626.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Davydycheva, S., N. Rykhlinski, and P. Legeido, 2006, Electrical-prospecting method for hydrocarbon search using the induced-polarization effect: Geophysics, 71, no. 4, G179–G189, doi: 10.1190/1.2217367.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Dias, C., 2000, Development in a model to describe low-frequency electrical polarization of rocks: Geophysics, 65, 437–451, doi: 10.1190/1.1444738.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Druskin, V., and L. Knizhnerman, 1988, A spectral semi-discrete method for the numerical solution of three-dimensional non-stationary electrical prospecting problems: Physics of solid earth, 8, 63–74, (in Russian, translated into English).Google Scholar
  • Druskin, V., and L. Knizhnerman, 1989, Two polynomial methods of calculating functions of symmetric matrices: U.S.S.R. Journal of Computational Mathematics and Mathematical Physics, 29, 112–121.Google Scholar
  • Druskin, V., and L. Knizhnerman, 1994, Spectral approach to solving three-dimensional Maxwell’s equations in the time and frequency domains: Radio Science, 29, 937–953, doi: 10.1029/94RS00747.RASCAD0048-6604CrossrefWeb of ScienceGoogle Scholar
  • Druskin, V., L. Knizhnerman, and M. Zaslavsky, 2009, Solution of large scale evolutionary problems using rational Krylov subspaces with optimized shifts: SIAM Journal on Scientific Computing, 31, 3760–3780, doi: 10.1137/080742403.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Druskin, V., C. Lieberman, and M. Zaslavsky, 2010, On adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problems: SIAM Journal on Scientific Computing, 32, 2485–2496, doi: 10.1137/090774082.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Druskin, V., and M. Zaslavsky, 2010, On convergence of Krylov subspace approximations of time-invariant self-adjoint dynamical systems: Linear algebra and its applications, in press, corrected proof, doi: 10.1016/j.laa.2011.02.039.CrossrefGoogle Scholar
  • Flis, M., G. Newman, and G. Hohmann, 1989, Induced-polarization effects in time-domain electromagnetic measurements: Geophysics, 54, 514–523, doi: 10.1190/1.1442678.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Flosadottir, A., and S. Constable, 1996, Marine controlled-source electromagnetic sounding, 1, Modeling and experimental design: Journal of Geophysical Research, 101, 5507–5517, doi: 10.1029/95JB03739.JGREA20148-0227CrossrefWeb of ScienceGoogle Scholar
  • Freund, R., 2005, Padé-type model reduction of second order and higher order dynamical systems: Benner, P.V. MehrmannD. C. Sorensen, eds., Dimension reduction of large scale systems (Lecture notes in computational science and engineering): Springer-Verlag, 45, 191–223.CrossrefGoogle Scholar
  • Gallopulos, E., and Y. Saad, 1992, Efficient solution of parabolic equations by Krylov approximation method: SIAM Journal on Scientific and Statistical Computing, 13, 1236–1264, doi: 10.1137/0913071.SIJCD40196-5204CrossrefWeb of ScienceGoogle Scholar
  • Gohberg, I., P. Lancaster, and L. Rodman, 2009, Matrix polynomials: Classics in applied mathematics: SIAM.CrossrefGoogle Scholar
  • Goldman, Y., 1989, The electric field in the conductive half space as a model in mining and petroleum prospecting: Mathematical Methods in the Applied Sciences, 11, 373–401, doi: 10.1002/(ISSN)1099-1476.MMSCDB0170-4214CrossrefWeb of ScienceGoogle Scholar
  • Hochbruck, M., and C. Lubich, 1997, On Krylov subspace approximations to the matrix exponential operator: SIAM Journal on Numerical Analysis, 34, 1911–1925, doi: 10.1137/S0036142995280572.SJNAEQ0036-1429CrossrefWeb of ScienceGoogle Scholar
  • Knizhnerman, L., V. Druskin, and M. Zaslavsky, 2009, On optimal convergence rate of the rational Krylov subspace reduction for electromagnetic problems in unbounded domains: SIAM Journal on Numerical Analysis, 47, 953–971, doi: 10.1137/080715159.SJNAEQ0036-1429CrossrefWeb of ScienceGoogle Scholar
  • Maao, F., 2007, Fast finite-difference time-domain modeling for marine-subsurface electromagnetic problems: Geophysics, 72, no. 2, A19–A23, doi: 10.1190/1.2434781.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Moro, G., and J. Freed, 1981, Calculation of ESR spectra and related Fokker-Planck forms by the use of the Lanczos algorithm: Journal of Chemical Physics, 74, 3757–3773, doi: 10.1063/1.441604.JCPSA60021-9606CrossrefWeb of ScienceGoogle Scholar
  • Mulder, W., M. Wirianto, and E. Slob, 2008, Time-domain modeling of electromagnetic diffusion with a frequency-domain code: Geophysics, 73, no. 1, F1–F8, doi: 10.1190/1.2799093.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Newman, G., G. Hohmann, and W. Anderson, 1986, Transient electromagnetic response of a three-dimensional body in a layered earth: Geophysics, 51, 1608–1627, doi: 10.1190/1.1442212.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Nour-Omid, B., 1987, Lanczos method for heat conduction analysis: International Journal for Numerical Methods in Engineering, 24, 251–262, doi: 10.1002/(ISSN)1097-0207.IJNMBH0029-5981CrossrefWeb of ScienceGoogle Scholar
  • Pelton, W., 1977, Interpretation of induced polarization and resistivity data: Ph.D. thesis, University of Utah.Google Scholar
  • Petropoulos, P., 1994, Stability and phase error analysis of FD-TD in dispersive dielectrics: IEEE Transactions on Antennas and Propagation, 42, 62–69, doi: 10.1109/8.272302.IETPAK0018-926XCrossrefWeb of ScienceGoogle Scholar
  • Ramati da Rocha, B., and T. Habashy, 1997a, Fractal geometry, porosity and complex resistivity. I: From rough pore interfaces to hand specimens: Developments in petrophysics: Geological Society Publishing House, 122, 277–286.Google Scholar
  • Ramati da Rocha, B., and T. Habashy, 1997b, Fractal geometry, porosity and complex resistivity. II: From hand specimens to field data: Developments in petrophysics: Geological Society Publishing House, 122, 287–298.Google Scholar
  • Ruhe, A., 1994, The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices: Behaviour and Information Technology, 34, 165–176.BEITD50144-929XGoogle Scholar
  • Saad, Y., 1992, Analysis of some Krylov subspace approximations to the matrix exponential operator: SIAM Journal on Numerical Analysis, 29, 209–228, doi: 10.1137/0729014.SJNAEQ0036-1429CrossrefWeb of ScienceGoogle Scholar
  • Saff, E., and V. Totik, 1997, Logarithmic potentials with external fields: Springer-Verlag.CrossrefGoogle Scholar
  • Schadle, A., M. Lopez-Fernandez, and C. Lubich, 2006, Fast and oblivious convolution quadrature: SIAM Journal on Scientific Computing, 28, 421–438, doi: 10.1137/050623139.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Schlumberger, C., 1920, Etude sur la prospection électrique du sous-sol: Gauthier-Villars et Cie.Google Scholar
  • Tal-Ezer, H., J. Carcione, and D. Kosloff, 1990, An accurate and efficient scheme for wave propagation in linear viscoelastic media: Geophysics, 55, 1366–1379, doi: 10.1190/1.1442784.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Um, E., J. Harris, and D. Alumbaugh, 2010, 3-D time-domain simulation of electromagnetic diffusion phenomena: A finite-element electric-field approach: Geophysics, 75, no. 4, F115–F126, doi: 10.1190/1.3473694.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Van der Vorst, M., 1987, An iterative solution method for solving f(A)x=b using Krylov subspace information obtained for the symmetric positive definite matrix: Journal of Computational and Applied Mathematics, 18, 249–263, doi: 10.1016/0377-0427(87)90020-3.JCAMDI0377-0427CrossrefWeb of ScienceGoogle Scholar
  • Varga, R. S., 1984, Functional analysis and approximation theory in numerical analysis: CBMS-NSF regional conference series in applied mathematics, 3.CRCMENGoogle Scholar
  • Wang, T., and G. Hohmann, 1993, A finite-difference time-domain solution for three-dimensional electromagnetic modeling: Geophysics, 58, 797–813, doi: 10.1190/1.1443465.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Weedon, W., and C. Rappaport, 1997, A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media: IEEE Antennas and Propagation Magazine, 45, 401–410.IAPMEZ1045-9243Web of ScienceGoogle Scholar
  • Weideman, J. A.C., and L. N. Trefethen, 2007, Parabolic and hyperbolic contours for computing the Bromwich integral: Mathematics of Computation, 76, 1341–1357, doi: 10.1090/S0025-5718-07-01945-X.MCMPAF0025-5718CrossrefWeb of ScienceGoogle Scholar
  • West, B., M. Bologna, and P. Grigolini, 2003, Physics of fractal operators: Springer.CrossrefGoogle Scholar
  • Zaslavsky, M., S. Davydycheva, V. Druskin, A. Abubakar, T. Habashy, and L. Knizhnerman, 2006, Finite-difference solution of the three-dimensional electromagnetic problem using divergence-free preconditioners: 76th Annual International Meeting, SEG, Expanded Abstracts, 775–778.AbstractGoogle Scholar
  • Zaslavsky, M., S. Davydycheva, V. Druskin, A. Abubakar, T. Habashy, and L. Knizhnerman, 2011, Hybrid finite-difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems: Geophysics, 76, no. 2, F123–F137, doi: 10.1190/1.3552595.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Zaslavsky, M., and V. Druskin, 2010, Solution of time-convolutionary Maxwell’s equations using parameter-dependent Krylov subspace reduction: Journal of Computational Physics, 229, 4831–4839, doi: 10.1016/ of ScienceGoogle Scholar
  • Zhdanov, M., 2008, Generalized effective-medium theory of induced polarization: Geophysics, 73, no. 5, F197–F211, doi: 10.1190/1.2973462.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar