Some Computational Aspects of the Time and Frequency Domain Formulations of Seismic Waveform Inversion

  • René-Édouard PlessixEmail author
Part of the Geosystems Mathematics book series (GSMA)


Seismic waveform inversion relies on efficient solutions of the elasto-dynamic wave equations. The associated inverse problem can be formulated either in the time domain or in the frequency domain. The choice between these two approaches mainly depends on their numerical efficiency. Here, I discuss some of the computational aspects of the frequency-domain solution of the visco-acoustic vertical transverse isotropic wave equations based on a Krylov subspace iterative solver and a complex shifted Laplace preconditioner. In the context of least-square migration or non-linear impedance inversion, the frequency domain approaches are currently not attractive because a complete frequency band response is required. However, in the context of waveform tomography when a small number of frequency responses are inverted, the frequency-domain approaches become relevant, especially when viscous effects are modeled, depending on the geological context.


Iterative Solver Waveform Inversion Direct Solver Full Waveform Inversion Amplitude Versus Offset 
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  1. 1.
    Achenbach, J.: Wave Propagation in Elastic Solids, North-Holland (1973).Google Scholar
  2. 2.
    Aki, K, Richards, P.: Quantitative Seismology, Vol. I, Freeman & Co (1980).Google Scholar
  3. 3.
    Alkhalifah, T.: An acoustic wave equation for anisotropic media: Geophysics, 65, 1239–1250 (2000).Google Scholar
  4. 4.
    Aminzadeh, F., Brac, J., Kunz, T.: 3-D salt and overthrust models, SEG/EAGE 3-D Modeling Series no.1, SEG (1997).Google Scholar
  5. 5.
    Anderson, J.E., Tan, L., Wang, D.: Time-reversal checkpointing methods for RTM and FWI, Geophysics, 77, S93–S103 (2012).CrossRefGoogle Scholar
  6. 6.
    Aruliah, D. A., Ascher, U. A.: Multigrid preconditioning for Krylov methods for time-harmonic Maxwells equations in 3D, SIAM J. Sci. Comput., 24, 702–18 (2003).CrossRefzbMATHGoogle Scholar
  7. 7.
    Bamberger, A., Chavent, G., Hemon, C., Lailly, P.: Inversion of normal incidence seismograms, Geophysics, 47, 757–770 (1982).CrossRefGoogle Scholar
  8. 8.
    Bérenger, J.-P.: A perfectly matched layer for absorption of electromagnetic waves, J. Comput. Phys., 114, 185–200 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Baumann, M., van Gijzen, M.B.: Nested Krylov Methods for shifted linear systems, SIAM Journal on Scientific Computing, 37, S90–S112 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Blanch, J., Robertson, J.O.A., Symes, W.W.: Modeling of a constant Q: methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique, Geophysics, 60, 176–184 (1995).CrossRefGoogle Scholar
  11. 11.
    Brenders A.J., Pratt, R.G.: Full waveform tomography for lithospheric imaging: results from a blind test in a realistic crustal model, Geophysical Journal International, 168, 133–151 (2007).CrossRefGoogle Scholar
  12. 12.
    Briggs, W.L., Henson, V.E., McCormick, S.F.: A multigrid tutorial, 2nd ed., SIAM (2000).Google Scholar
  13. 13.
    Carcione, J.M.: Wave fields in real media:Wave propagation in anisotropic, anelastic and porous media: Pergamon Press (2011).Google Scholar
  14. 14.
    Chavent G.: Nonlinear least squares for inverse problems: theoretical foundations and step-by-step guide for applications, Springer (2009).Google Scholar
  15. 15.
    Christensen, R.M.: Theory of viscoelasticity - An introduction, Academic Press Inc (1982).Google Scholar
  16. 16.
    Clapp, R. G.: Reverse time migration with random boundaries, 79th Annual International Meeting, SEG, Expanded Abstracts, 2809–2813 (2009).Google Scholar
  17. 17.
    Duveneck, E., Milcik, P., Bakker, P.M., Perkins, C.: Acoustic VTI wave equations and their applications for anisotropic reverse-time migration: 78th Annual International Meeting, SEG, Expanded Abstract, 2186–2189 (2008).CrossRefGoogle Scholar
  18. 18.
    Elman, H., Ernst, O., O’ Leary, D.: A multigrid based preconditioner for heterogeneous Helmholtz equation, SIAM Journal on Scientific Computing, 23, 1291–1315 (2001).MathSciNetCrossRefGoogle Scholar
  19. 19.
    Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation; Hierarchical matrix representation, Communications on Pure and Applied Mathematics, LXIV, 0697–0735 (2011).Google Scholar
  20. 20.
    Epanomeritakis, I., Akçelik V., Ghatta,s O., Bielak, J.: A Newton-CG method for large-scale three- dimensional elastic full waveform seismic inversion, Inverse Problems, 24, 1–26 (2008).Google Scholar
  21. 21.
    Erlangga, Y.A., Vuik, C., Oosterlee, C.: On a class of preconditioners for the Helmholtz equation, Applied Numerical Mathematics, 50, 409–425 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Erlangga, Y.A., Vuik, C., Oosterlee, C.: A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM Journal on Scientific Computing, 27, 1471–1492 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Erlangga, Y.A., Nabben, R.: On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted laplacian, Electronic transactions on Numerical Analysis, 31, 408–424 (2008).MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gauthier O., Virieux J., Tarantola A.: Two-dimensional nonlinear inversion of seismic waveform: numerical results. Geophysics 51, 1387–1403 (1986).CrossRefGoogle Scholar
  25. 25.
    Gordon, D., Gordon, R.: Robust and highly scalable parallel solution of the Helmholtz equation with large wave numbers, Journal of computation and applied mathematics, 237, 182–196 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Griewank, A., Walther, A.: Algorithm 799: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation: ACM Transactions on Mathematical Software, 26 (1), 19–45 (2000).Google Scholar
  27. 27.
    Lailly, P.: The seismic inverse problem as a sequence of before stack migrations: Conference on Inverse Scattering, Theory and Application, Society of Industrial and Applied Mathematics, Expanded Abstracts, 206–220 (1983).Google Scholar
  28. 28.
    Marfurt, K.J.: Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics, 49, 533–549 (1984).CrossRefGoogle Scholar
  29. 29.
    Métivier L., Brossier, R., Virieux, J., Operto, S.: Full Waveform Inversion and the Truncated Newton Method, SIAM J. Sci. Comput., 35, B401–437 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mulder, W. A.: A multigrid solver for 3D electromagnetic diffusion Geophys. Prosp., 54, 633–649 (2006).Google Scholar
  31. 31.
    Mulder, W.A., Plessix, R.-É.: Exploring some issues in acoustic full waveform inversion, Geophysical Prospecting, 56, 827–841 (2008).CrossRefGoogle Scholar
  32. 32.
    Nihei, K.T., Li, X.: Frequency response modelling of seismic waves using finite difference time domain with phase sensitive detection (TDPSD), Geophysical Journal International, 169, 1069–1078 (2006).CrossRefGoogle Scholar
  33. 33.
    Operto, S., Virieux, J., Amestoy, P., L’Excellent, J.Y., Giraud, L.: 3D finite-difference frequency-domain modeling of viscoacoustic wave propagation using a massively parallel direct solver: a feasibility study, Geophysics, 72, SM195–SM211 (2007).CrossRefGoogle Scholar
  34. 34.
    Operto, S., Brossier, R., Combe, L., Métivier, L.,Ribodetti, A., Virieux,J., 2014. Computationally-efficient three-dimensional visco-acoustic finite difference frequency-domain seismic modeling in vertical transversely isotropic media with sparse direct solver, Geophysics, 79,T257–T275 (2014).Google Scholar
  35. 35.
    Plessix, R.-É, Mulder, W.A.: Separation of variables as a preconditioner for an iterative Helmholtz solver, Applied Numerical Mathematics, 44, 385–400 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Plessix, R.-É.: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophys. J. Int., 167, 495–503 (2006).CrossRefGoogle Scholar
  37. 37.
    Plessix, R.-É., Darnet, M,Mulder, W. A.: An approach for 3D multi-source, multi-frequency CSEM modeling Geophysics 72 SM177–84 (2007).Google Scholar
  38. 38.
    Plessix, R.-É.: A Helmholtz iterative solver for 3D seismic-imaging problems, Geophysics, 72, SM185–SM194 (2007).CrossRefGoogle Scholar
  39. 39.
    Plessix, R.-É.: Three-dimensional frequency-domain full-waveform inversion with an iterative solver, Geophysics, 74,WCC149–WCC157 (2009).CrossRefGoogle Scholar
  40. 40.
    Plessix, R.-É., Perkins, C.: Full waveform inversion of a deep water ocean bottom dataset. First Break 28, 71–78 (2010).CrossRefGoogle Scholar
  41. 41.
    Pratt, R. G., Song, Z.M., Williamson, P.R., Warner, M.: Two-dimensional velocity model from wide-angle seismic data by wavefield inversion: Geophysical Journal International, 124, 323–340 (1996).Google Scholar
  42. 42.
    Saad, Y.: Iterative methods for linear systems, Second edition, SIAM (2003).CrossRefzbMATHGoogle Scholar
  43. 43.
    Shin, C., Cha, Y.H.: Waveform inversion in the Laplace-Fourier domain, Geophysical Journal International, 171, 1067–1079 (2009).CrossRefGoogle Scholar
  44. 44.
    Sonneveld, P., van Gijzen, M.B.: IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM Journal on Scientific Computing, 31, 1035–1062 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Tarantola, A.: Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, 1259–1266 (1984).CrossRefGoogle Scholar
  46. 46.
    Tarantola A.: Inverse Problem Theory, Elsevier (1987).Google Scholar
  47. 47.
    Tarantola, A.: Theoretical background for the inversion of seismic waveforms, including elasticity and attenuation, Pure Appl. Geophys., 128, 365–399 (1988).CrossRefGoogle Scholar
  48. 48.
    van der Vorst, H.A.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13, 631–644 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Virieux, J., Operto, S.: An overview of full waveform inversion in exploration geophysics, Geophysics, 74, WCC1–WCC2 (2009).CrossRefGoogle Scholar
  50. 50.
    Virieux, J, Calandra A., Plessix, R.-É.: A review of the spectral, pseudo-spectral, finite-difference and finite-element modelling techniques for geophysical imaging, Geophysical Prospecting, 59, 794–813 (2011).CrossRefGoogle Scholar
  51. 51.
    Wang, S., de Hoop, M.V., Xia, J.: On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver, Geophys. Prospect., 59, 857–873 (2011).CrossRefGoogle Scholar
  52. 52.
    Yilmaz, O., 2001. Seismic Data Analysis, SEG.CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Shell Global Solutions InternationalRijswijkThe Netherlands

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