Surveys in Geophysics

, Volume 37, Issue 1, pp 109–147 | Cite as

Applied Mathematics in EM Studies with Special Emphasis on an Uncertainty Quantification and 3-D Integral Equation Modelling

  • Oleg Pankratov
  • Alexey Kuvshinov


Despite impressive progress in the development and application of electromagnetic (EM) deterministic inverse schemes to map the 3-D distribution of electrical conductivity within the Earth, there is one question which remains poorly addressed—uncertainty quantification of the recovered conductivity models. Apparently, only an inversion based on a statistical approach provides a systematic framework to quantify such uncertainties. The Metropolis–Hastings (M–H) algorithm is the most popular technique for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. However, all statistical inverse schemes require an enormous amount of forward simulations and thus appear to be extremely demanding computationally, if not prohibitive, if a 3-D set up is invoked. This urges development of fast and scalable 3-D modelling codes which can run large-scale 3-D models of practical interest for fractions of a second on high-performance multi-core platforms. But, even with these codes, the challenge for M–H methods is to construct proposal functions that simultaneously provide a good approximation of the target density function while being inexpensive to be sampled. In this paper we address both of these issues. First we introduce a variant of the M–H method which uses information about the local gradient and Hessian of the penalty function. This, in particular, allows us to exploit adjoint-based machinery that has been instrumental for the fast solution of deterministic inverse problems. We explain why this modification of M–H significantly accelerates sampling of the posterior probability distribution. In addition we show how Hessian handling (inverse, square root) can be made practicable by a low-rank approximation using the Lanczos algorithm. Ultimately we discuss uncertainty analysis based on stochastic inversion results. In addition, we demonstrate how this analysis can be performed within a deterministic approach. In the second part, we summarize modern trends in the development of efficient 3-D EM forward modelling schemes with special emphasis on recent advances in the integral equation approach.


Quantification of uncertainties Bayesian approach  Metropolis–Hastings Gradient and Hessian Adjoint sources approach Finite elements Contracting integral equations 



The authors would like to thank Alexander Grayver for many seminal discussions over the course of this work, and Alexey Geraskin and Alexander Grayver for their input to Sect. 11. We wish to thank William Lowrie who helped us to improve the English presentation of this paper. We extend our gratitude to Chester Weiss and an anonymous reviewer for constructive comments on the manuscript. This work has been supported by the European Space Agency through ESTEC contract No. 4000102140/10/NL/JA and in part by the Russian Foundation for Basic Research under Grant No. 13-05-12111. Oleg Pankratov acknowledges the support of ETH during his stay in Zurich as a visiting professor.


  1. Abubakar A, Habashy T (2013) Three-dimensional visco-acoustic modeling using a renormalized integral equation iterative solver. J Comput Phys 249:1–12CrossRefGoogle Scholar
  2. Avdeev D, Knizhnik S (2009) 3D integral equation modeling with a linear dependence on dimensions. Geophysics 74:89–94CrossRefGoogle Scholar
  3. Avdeev D, Kuvshinov A, Epova K (2002) Three-dimensional modelling of electromagnetic modelling of electromagnetic logs from inclined-horizontal wells. Phys Solid Earth 38:975–980Google Scholar
  4. Avdeev D, Kuvshinov A, Pankratov O, Newman G (1997) High-performance three-dimensional electromagnetic modeling using modified Neumann series. Wide-band numerical solution and examples. J Geomagn Geoelectr 49:1519–1539CrossRefGoogle Scholar
  5. Avdeev D, Kuvshinov A, Pankratov O, Newman G (2000) 3-D EM modelling using fast integral equation approach with Krylov subspaces accelerator. In: 2nd EAGE conference and technical exhibition, vol 2, Scotland, ScotlandGoogle Scholar
  6. Avdeev D, Kuvshinov A, Pankratov O, Newman G (2002) Three-dimensional induction logging problems, part I: an integral equation solution and model comparisons. Geophysics 67(2):413–426CrossRefGoogle Scholar
  7. Bayes T (1763) An essay towards solving a problem in the doctrine of chances. Philos Trans R Soc Lond 53:370–418Google Scholar
  8. Beck R, Hiptmair R, Hoppe RH, Wohlmuth B (2000) Residual based a posteriori error estimators for eddy current computation. Math Model Numer Anal 34:159–182CrossRefGoogle Scholar
  9. Bodin T, Sambridge M, Rawlinson N, Arroucau P (2012) Transdimensional tomography with unknown data noise. Geophys J Int 189:1536–1556CrossRefGoogle Scholar
  10. Börner RU (2010) Numerical modelling in geo-electromagnetics: advances and challenges. Surv Geophys 31:225–245CrossRefGoogle Scholar
  11. Brown V, Hoversten M, Key K, Chen J (2012) Resolution of reservoir scale electrical anisotropy from marine CSEM data. Water Resour Res 77(2):E147–E158Google Scholar
  12. Bürg M (2000) A residual-based a posteriori error estimator for the hp-finite element method for Maxwells equations. Appl Numer Math 62:922–940CrossRefGoogle Scholar
  13. Bürg M (2013) Convergence of an automatic hp-adaptive finite element strategy for Maxwells equations. Appl Numer Math 72:188–206CrossRefGoogle Scholar
  14. Chen J, Hoversten GM, Key K, Nordquist G, Cumming W (2011) Stochastic inversion of magnetotelluric data using a sharp boundary parameterization and application to a geothermal site. Geophysics 77(4):E265–E279CrossRefGoogle Scholar
  15. Chen J, Hoversten GM, Vasco D, Rubin Y, Hou Z (2007) A Bayesian model for gas saturation estimation using marine seismic AVA and CSEM data. Geophysics 72(2):WA85–WA95Google Scholar
  16. Chew W, Jin J, Lu C, Michielssen E, Song J (2014) Fast solution methods in electromagnetics. IEEE Trans Antenna Propag 45:533–543CrossRefGoogle Scholar
  17. Christensen N (1990) Optimized fast Hankel transform filters. Geophys Prospect 38:545–568CrossRefGoogle Scholar
  18. Epanomeritakis I, Akcelik V, Ghattas O, Bielak J (2008) A Newton-CG method for large-scale three-dimensional elastic full waveform seismic inversion. Inverse Probl 24:1–26CrossRefGoogle Scholar
  19. Ernst OG, Gander MJ (2011) Why it is difficult to solve Helmholtz problems with classical iterative methods. Numer Anal Multiscale Probl 83:325–361CrossRefGoogle Scholar
  20. Fainberg E, Zinger B (1980) Electromagnetic induction in a nonuniform spherical model of the Earth. Ann Geophys 36:127–134Google Scholar
  21. Farquharson CG, Miensopust MP (2011) Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction. J Appl Geophys 75:699–710CrossRefGoogle Scholar
  22. Fichtner A, Trampert J (2011) Hessian kernels of seismic data functionals based upon adjoint techniques. Geophys J Int 185:775–798CrossRefGoogle Scholar
  23. Frayss V, Giraud L, Gratton S, Langou J (2003) A set of GMRES routines for real and complex arithmetics on high performance computers. CERFACS technical report TR/PA/03/3Google Scholar
  24. Geraskin A, Kruglyakov M, Kuvshinov A (2015) Novel robust and scalable 3-D forward solver based on contracting integral equation method and modern programming technologies. Comput Geosci (submitted)Google Scholar
  25. Gilks WR, Richardson S, Spiegelhalter D (eds) (1996) Markov chain Monte Carlo in practice. Chapman and Hall, London, pp 1–485Google Scholar
  26. Grandis H, Menvielle M, Roussignol M (1999) Bayesian inversion with Markov chains-I. The magnetotelluric one-dimensional case. GJI 138:757–768Google Scholar
  27. Grandis H, Sumintaredja P, Irawan D (2012) A template for 1-D inversion of geo-electromagnetic data using MCMC method. EMSEV 2012, Gotemba Kogen Resort, Gotemba, Japan, October 14, 2012, pp 1–4Google Scholar
  28. Grayver A, Burg M (2014) Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method. Geophys J Int. doi: 10.1093/gji/ggu119
  29. Grayver A, Kolev T (2015) Large-scale 3D geo-electromagnetic modeling using parallel adaptive high-order finite element method. Geophysics 80(6):277–291Google Scholar
  30. Greenbaum A (1997) Iterative methods for solving linear systems. SIAM, Philadelphia, USAGoogle Scholar
  31. Haber E, Ascher UM (2001) Fast finite volume simulation of 3D electromagnetic problems with highly discontinuous coefficients. SIAM J Sci Comput 22(6):1943–1961CrossRefGoogle Scholar
  32. Hastings WK (1970) Monte carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109CrossRefGoogle Scholar
  33. Hiptmair R (2002) Finite elements in computational electromagnetism. Acta Numer 11:237–339Google Scholar
  34. Hohmann G (1975) Three-dimensional induced polarization and electromagnetic modeling. J Geophys 40:309–324CrossRefGoogle Scholar
  35. Hursan G, Zhdanov M (2002) Contraction integral equation method in three-dimensional electromagnetic modeling. Radio Sci 37:2001R. doi: 10.1029/S002513 CrossRefGoogle Scholar
  36. Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, New YorkGoogle Scholar
  37. Kamm J, Pedersen L (2014) Inversion of airborne tensor VLF data using integral equations. Geophys J Int. doi: 10.1093/gji/ggu161
  38. Kelbert A, Kuvshinov A, Velimsky J, Koyama T, Ribaudo J, Sun J, Martinec Z, Weiss C (2014) Global 3-D electromagnetic forward modelling: a benchmark study. Geophys J Int 197:785–814CrossRefGoogle Scholar
  39. Kirk BS, Peterson JW, Stogner RH, Carey GF (2006) libmesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng Comput 22:237–254CrossRefGoogle Scholar
  40. Kolmogorov AN (1956) Foundations of the theory of probability, 2nd edn. Chelsea, New YorkGoogle Scholar
  41. Koyama T, Shimizu H, Utada H, Ichiki M, Ohtani E, Hae R (2006) Water content in the mantle transition zone beneath the North Pacific derived from the electrical conductivity anomaly. AGU Geophys Monogr Ser 168:171–179Google Scholar
  42. Koyama T, Utada H, Avdeev D (2008) Fast and memory-saved 3-D forward modeling code for MT by using integral equation method. In: Abstract book. 19th workshop on electromagnetic induction in the Earth, ChinaGoogle Scholar
  43. Kuvshinov A (2008) 3-D global induction in the oceans and solid Earth: recent progress in modeling magnetic and electric fields from sources of magnetospheric, ionospheric and oceanic origin. Surv Geophys. doi: 10.1007/s10712-008-9045-z Google Scholar
  44. Kuvshinov AV, Utada H, Avdeev D, Koyama T (2005) 3-D modelling and analysis of Dst C-responses in the North Pacific Ocean region, revisited. Geophys J Int 160:505–526CrossRefGoogle Scholar
  45. Logg A, Wells GN (2010) Dolfin: automated finite element computing. ACM Trans Math Softw 37:20:1–20:28CrossRefGoogle Scholar
  46. Lu CC, Chew W (1994) A multilevel algorithm for solving boundary-value scattering. Microwave Opt Technol Lett 7:466–470CrossRefGoogle Scholar
  47. Mackie R, Smith J, Madden T (1994) 3-Dimensional electromagnetic modeling using finite-difference equation—the magnetotelluric example. Radio Sci 29(4):923–935CrossRefGoogle Scholar
  48. Martin J, Wilcox LC, Burstedde C, Ghattas O (2012) A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM J Sci Comput 34(3):A1460–A1487CrossRefGoogle Scholar
  49. Metivier L, Brossier R, Virieux J, Operto S (2013) Full waveform inversion and the truncated Newton method. SIAM J Sci Comput 35:401–437CrossRefGoogle Scholar
  50. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092CrossRefGoogle Scholar
  51. Michielssen E, Boag A (1996) A multilevel matrix decomposition algorithm for analyzing scattering from large structures. IEEE Trans Antennas Propag 44:1086–1093CrossRefGoogle Scholar
  52. Millard X, Liu QH (2003) A fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered media. IEEE Trans Antennas Propag 51:2393–2401CrossRefGoogle Scholar
  53. Minsley BJ (2011) A trans-dimensional Bayesian Markov chain Monte Carlo algorithm for model assessment using frequency-domain electromagnetic data. GJI 187:252–272Google Scholar
  54. Monk P (2003) Finite element methods for Maxwells equations. Oxford University Press, OxfordCrossRefGoogle Scholar
  55. Mulder W (2006) A multigrid solver for 3D electromagnetic diffusion. Geophys Prospect 54:633–649CrossRefGoogle Scholar
  56. Newman G, Alumbaugh D (2002) Three-dimensional induction logging problems, part 2: a finite-difference solution. Geophysics 61:484–491CrossRefGoogle Scholar
  57. Newman GA, Hoversten GM (2000) Solution strategies for two- and three-dimensional electromagnetic inverse problems. Inverse Probl 16:1357–1375CrossRefGoogle Scholar
  58. Nie X, Li LW, Yuan N, Yeo TS (2013) A fast integral equation solver for 3D induction well logging in formations with large conductivity contrasts. J Comput Phys 61:645–657Google Scholar
  59. Pankratov O, Avdeev D, Kuvshinov A (1995) Electromagnetic field scattering in a homogeneous Earth: a solution to the forward problem. Phys Solid Earth 31:201–209Google Scholar
  60. Pankratov O, Kuvshinov A (2010) General formalism for the efficient calculation of derivatives of EM frequency domain responses and derivatives of the misfit. Geophys J Int 181:229–249Google Scholar
  61. Pankratov O, Kuvshinov A (2015) General formalism for the efficient calculation of the Hessian matrix of EM data misfit and Hessian-vector products based upon adjoint sources approach. Geophys J Int 200:1449–1465Google Scholar
  62. Pankratov O, Kuvshinov A, Avdeev D (1997) High-performance three-dimensional electromagnetic modeling using modified Neumann series. Anisotropic case. J Geomag Geoelectr 49:1541–1547CrossRefGoogle Scholar
  63. Phillips JR, White J (1996) A precorrected-FFT method for electrostatic analysis of complicated 3-D structures. IEEE Trans Comput Aided Des Integr Circuits Syst 16:1059–1071CrossRefGoogle Scholar
  64. Prasolov VV (2004) Lobachevsky geometry. MCCME Publishing House, Moscow (in Russian) Google Scholar
  65. Puzyrev V, Koldan J, de la Puente J, Houzeaux G, Vazquez M, Cela JM (2013) Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the earth: finite-element frequency-domain approach. Geophys J Int 193:678–693CrossRefGoogle Scholar
  66. Raiche A (1974) An integral equation approach to three-dimensional modelling. Geophys J R Astr Soc 36:363–376CrossRefGoogle Scholar
  67. Ren Z, Kalscheuer T, Greenhalgh S, Maurer H (2013) A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling. Geophys J Int 194:700–718CrossRefGoogle Scholar
  68. Rius JM, Parron J, Heldring A, Tamayo J, Ubeda E (2008) Fast iterative solution of integral equations with method of moments and matrix decomposition algorithm - singular value decomposition. IEEE Trans Anteenas Propag 56:2314–2324CrossRefGoogle Scholar
  69. Rokhlin V (1990) Rapid solution of integral equations of scattering theory in two dimensions. J. Comp. Phys. 36:414–439CrossRefGoogle Scholar
  70. Rosas-Carbajal M, Linde N, Kalscheuer T, Vrugt JA (2013) Two-dimensional probabilistic inversion of plane-wave electromagnetic data: methodology, model constraints and joint inversion with electrical resistivity data. GJI 196:1–17Google Scholar
  71. Santosa F, Symes WW (1988) Computation of the Hessian for least-squares solutions of inverse problems of reflection seismology. Inverse Probl 4:211–213CrossRefGoogle Scholar
  72. Schwarzbach C, Börner RU, Spitzer K (2011) Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics—a marine CSEM example. Geophys J Int 187:63–74CrossRefGoogle Scholar
  73. Singer B (1995) Method for solution of Maxwell’s equations in non-uniform media. Geophys J Int 120:590–598CrossRefGoogle Scholar
  74. Singer B (2008) Electromagnetic integral equation approach based on contraction operator and solution optimization in Krylov subspace. Geophys J Int 175:857–884CrossRefGoogle Scholar
  75. Singer B, Fainberg E (1995) Generalization of the iterative dissipative method for modeling electromagnetic fields in nonuniform media with displacement currents. J Appl Geophys 34:41–46CrossRefGoogle Scholar
  76. Singer B, Fainberg E (1997) Fast and stable method for 3D modeling of electromagnetic field. Explor Geophys 34:130–135CrossRefGoogle Scholar
  77. Smith TJ (1996) Conservative modeling of 3-D electromagnetic fields, part II: biconjugate gradient solution as an accelerator. Geophysics 61:1319–1324CrossRefGoogle Scholar
  78. Sun J, Egbert G (2012) A thin-sheet model for global electromagnetic induction. Geophys J Int 189:343–356CrossRefGoogle Scholar
  79. Sun J, Kuvshinov A (2015) Accelerating EM integral equation forward solver for global geomagnetic induction using SVD based matrix compression method. Geophys J Int 1200:1003–1009Google Scholar
  80. Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
  81. Um ES, Commer M, Newman GA (2013) Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the earth: finite-element frequency-domain approach. Geophys J Int 193:1460–1473CrossRefGoogle Scholar
  82. Weidelt P (1975) Electromagnetic induction in three-dimensional structures. J Geophys 41:85–109Google Scholar

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave PropagationRussian Academy of SciencesTroitsk, MoscowRussia
  2. 2.Institute of GeophysicsETH ZurichZurichSwitzerland

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