Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Modern Solvers for Helmholtz Problems

Part of the book series: Geosystems Mathematics ((GSMA))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Achenbach, J.: Wave Propagation in Elastic Solids, North-Holland (1973).

    Google Scholar 

  2. Aki, K, Richards, P.: Quantitative Seismology, Vol. I, Freeman & Co (1980).

    Google Scholar 

  3. Alkhalifah, T.: An acoustic wave equation for anisotropic media: Geophysics, 65, 1239–1250 (2000).

    Google Scholar 

  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. Anderson, J.E., Tan, L., Wang, D.: Time-reversal checkpointing methods for RTM and FWI, Geophysics, 77, S93–S103 (2012).

    Article  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  7. Bamberger, A., Chavent, G., Hemon, C., Lailly, P.: Inversion of normal incidence seismograms, Geophysics, 47, 757–770 (1982).

    Article  Google Scholar 

  8. Bérenger, J.-P.: A perfectly matched layer for absorption of electromagnetic waves, J. Comput. Phys., 114, 185–200 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  9. Baumann, M., van Gijzen, M.B.: Nested Krylov Methods for shifted linear systems, SIAM Journal on Scientific Computing, 37, S90–S112 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  12. Briggs, W.L., Henson, V.E., McCormick, S.F.: A multigrid tutorial, 2nd ed., SIAM (2000).

    Google Scholar 

  13. Carcione, J.M.: Wave fields in real media:Wave propagation in anisotropic, anelastic and porous media: Pergamon Press (2011).

    Google Scholar 

  14. Chavent G.: Nonlinear least squares for inverse problems: theoretical foundations and step-by-step guide for applications, Springer (2009).

    Google Scholar 

  15. Christensen, R.M.: Theory of viscoelasticity - An introduction, Academic Press Inc (1982).

    Google Scholar 

  16. Clapp, R. G.: Reverse time migration with random boundaries, 79th Annual International Meeting, SEG, Expanded Abstracts, 2809–2813 (2009).

    Google Scholar 

  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).

    Book  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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. 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. Erlangga, Y.A., Vuik, C., Oosterlee, C.: On a class of preconditioners for the Helmholtz equation, Applied Numerical Mathematics, 50, 409–425 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  24. Gauthier O., Virieux J., Tarantola A.: Two-dimensional nonlinear inversion of seismic waveform: numerical results. Geophysics 51, 1387–1403 (1986).

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Marfurt, K.J.: Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics, 49, 533–549 (1984).

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  30. Mulder, W. A.: A multigrid solver for 3D electromagnetic diffusion Geophys. Prosp., 54, 633–649 (2006).

    Google Scholar 

  31. Mulder, W.A., Plessix, R.-É.: Exploring some issues in acoustic full waveform inversion, Geophysical Prospecting, 56, 827–841 (2008).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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. Plessix, R.-É, Mulder, W.A.: Separation of variables as a preconditioner for an iterative Helmholtz solver, Applied Numerical Mathematics, 44, 385–400 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  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. Plessix, R.-É.: A Helmholtz iterative solver for 3D seismic-imaging problems, Geophysics, 72, SM185–SM194 (2007).

    Article  Google Scholar 

  39. Plessix, R.-É.: Three-dimensional frequency-domain full-waveform inversion with an iterative solver, Geophysics, 74,WCC149–WCC157 (2009).

    Article  Google Scholar 

  40. Plessix, R.-É., Perkins, C.: Full waveform inversion of a deep water ocean bottom dataset. First Break 28, 71–78 (2010).

    Article  Google Scholar 

  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. Saad, Y.: Iterative methods for linear systems, Second edition, SIAM (2003).

    Book  MATH  Google Scholar 

  43. Shin, C., Cha, Y.H.: Waveform inversion in the Laplace-Fourier domain, Geophysical Journal International, 171, 1067–1079 (2009).

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  45. Tarantola, A.: Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, 1259–1266 (1984).

    Article  Google Scholar 

  46. Tarantola A.: Inverse Problem Theory, Elsevier (1987).

    Google Scholar 

  47. Tarantola, A.: Theoretical background for the inversion of seismic waveforms, including elasticity and attenuation, Pure Appl. Geophys., 128, 365–399 (1988).

    Article  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  49. Virieux, J., Operto, S.: An overview of full waveform inversion in exploration geophysics, Geophysics, 74, WCC1–WCC2 (2009).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  52. Yilmaz, O., 2001. Seismic Data Analysis, SEG.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shell Global Solutions International, Kesslerpark 1, 2288 GS, Rijswijk, The Netherlands

    René-Édouard Plessix

Authors
  1. René-Édouard Plessix
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to René-Édouard Plessix .

Editor information

Editors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Domenico Lahaye

  2. Delft Institute of Applied Mathematics, Delft University of Technology VORtech B.V., Delft, The Netherlands

    Jok Tang

  3. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Kees Vuik

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Plessix, RÉ. (2017). Some Computational Aspects of the Time and Frequency Domain Formulations of Seismic Waveform Inversion. In: Lahaye, D., Tang, J., Vuik, K. (eds) Modern Solvers for Helmholtz Problems. Geosystems Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28832-1_7

Download citation

Publish with us

Policies and ethics

Search

Navigation