On the Use of Optimal Transport Distances for a PDE-Constrained Optimization Problem in Seismic Imaging

  • L. MétivierEmail author
  • A. Allain
  • R. Brossier
  • Q. Mérigot
  • E. Oudet
  • J. Virieux
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 163)


Full waveform inversion is a PDE-constrained nonlinear least-squares problem dedicated to the estimation of the mechanical subsurface properties with high resolution. Since its introduction in the early 80s, a limitation of this method is related to the non-convexity of the misfit function which is minimized when the method is applied to the estimation of the subsurface wave velocities. Recently, the definition of an alternative misfit function based on an optimal transport distance has been proposed to mitigate this difficulty. In this study, we review the difficulties for exploiting standard optimal transport techniques for the comparison of seismic data. The main difficulty is related to the oscillatory nature of the seismic data, which requires to extend optimal transport to the transport of signed measures. We review three standard possible extensions relying on a decomposition of the data into its positive and negative part. We show how the two first might not be adapted for full waveform inversion and focus on the third one. We present a numerical strategy based on the dual formulation of a particular optimal transport distance yielding an efficient implementation. The interest of this approach is illustrated on the 2D benchmark Marmousi model.



This study was partially funded by the SEISCOPE consortium ( ), sponsored by CGG, CHEVRON, EXXON-MOBIL, JGI, SHELL, SINOPEC, STATOIL, TOTAL, and WOODSIDE. This study was granted access to the HPC resources of the Froggy platform of the CIMENT infrastructure (, which is supported by the Rhône-Alpes region (GRANT CPER07_13 CIRA), the OSUG@2020 labex (reference ANR10 LABX56), and the Equip@Meso project (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche, and the HPC resources of CINES/IDRIS/TGCC under the allocation 046091 made by GENCI.


  1. 1.
    Adams, J. C. (1989). MUDPACK: Multigrid portable FORTRAN software for the efficient solution of linear elliptic partial differential equations. Applied Mathematics and Computation, 34(2):113–146.CrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Gigli, N., and Savaré, G. (2008). Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media.Google Scholar
  3. 3.
    Benamou, J.-D. and Brenier, Y. (2000). A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik.Google Scholar
  4. 4.
    Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L., and Peyré, G. (2015). Iterative Bregman Projections for Regularized Transportation Problems. SIAM Journal on Scientific Computing, 37(2):A1111–A1138.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bogachev, V. I. (2007). Measure Theory. Number vol. I,II in Measure Theory. Springer Berlin Heidelberg.zbMATHGoogle Scholar
  6. 6.
    Bozdağ, E., Trampert, J., and Tromp, J. (2011). Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements. Geophysical Journal International, 185(2):845–870.CrossRefGoogle Scholar
  7. 7.
    Brandt, A. (1977). Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation, 31:333–390.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bunks, C., Salek, F. M., Zaleski, S., and Chavent, G. (1995). Multiscale seismic waveform inversion. Geophysics, 60(5):1457–1473.CrossRefGoogle Scholar
  9. 9.
    Chavent, G. (1971). Analyse fonctionnelle et identification de coefficients répartis dans les équations aux dérivées partielles. PhD thesis, Université de Paris.Google Scholar
  10. 10.
    Combettes, P. L. and Pesquet, J.-C. (2011). Proximal splitting methods in signal processing. In Bauschke, H. H., Burachik, R. S., Combettes, P. L., Elser, V., Luke, D. R., and Wolkowicz, H., editors, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, volume 49 of Springer Optimization and Its Applications, pages 185–212. Springer New York.Google Scholar
  11. 11.
    Cuturi, M. (2013). Sinkhorn distances: lightspeed computation of optimal transportation distances. Advances in Neural Information Processing Systems.Google Scholar
  12. 12.
    Devaney, A. (1984). Geophysical diffraction tomography. Geoscience and Remote Sensing, IEEE Transactions on, GE-22(1):3–13.CrossRefGoogle Scholar
  13. 13.
    Engquist, B. and Froese, B. D. (2014). Application of the Wasserstein metric to seismic signals. Communications in Mathematical Science, 12(5):979–988.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fichtner, A., Kennett, B. L. N., Igel, H., and Bunge, H. P. (2008). Theoretical background for continental- and global-scale full-waveform inversion in the time-frequency domain. Geophysical Journal International, 175:665–685.CrossRefGoogle Scholar
  15. 15.
    Hale, D. (2013). Dynamic warping of seismic images. Geophysics, 78(2):S105–S115.CrossRefGoogle Scholar
  16. 16.
    Jannane, M., Beydoun, W., Crase, E., Cao, D., Koren, Z., Landa, E., Mendes, M., Pica, A., Noble, M., Roeth, G., Singh, S., Snieder, R., Tarantola, A., and Trezeguet, D. (1989). Wavelengths of Earth structures that can be resolved from seismic reflection data. Geophysics, 54(7):906–910.CrossRefGoogle Scholar
  17. 17.
    Kantorovich, L. (1942). On the transfer of masses. Dokl. Acad. Nauk. USSR, 37:7–8.Google Scholar
  18. 18.
    Lailly, P. (1983). The seismic inverse problem as a sequence of before stack migrations. In Bednar, R. and Weglein, editors, Conference on Inverse Scattering, Theory and application, Society for Industrial and Applied Mathematics, Philadelphia, pages 206–220.Google Scholar
  19. 19.
    Le Dimet, F. and Talagrand, O. (1986). Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A, 38A(2):97–110.CrossRefGoogle Scholar
  20. 20.
    Lellmann, J., Lorenz, D., Schönlieb, C., and Valkonen, T. (2014). Imaging with Kantorovich–Rubinstein discrepancy. SIAM Journal on Imaging Sciences, 7(4):2833–2859.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lions, J. L. (1968). Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod, Paris.zbMATHGoogle Scholar
  22. 22.
    Luo, S. and Sava, P. (2011). A deconvolution-based objective function for wave-equation inversion. SEG Technical Program Expanded Abstracts, 30(1):2788–2792.CrossRefGoogle Scholar
  23. 23.
    Luo, Y. and Schuster, G. T. (1991). Wave-equation traveltime inversion. Geophysics, 56(5):645–653.CrossRefGoogle Scholar
  24. 24.
    Mainini, E. (2012). A description of transport cost for signed measures. Journal of Mathematical Sciences, 181(6):837–855.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Métivier, L. and Brossier, R. (2016). The SEISCOPE optimization toolbox: A large-scale nonlinear optimization library based on reverse communication. Geophysics, 81(2):F11–F25.CrossRefGoogle Scholar
  26. 26.
    Métivier, L., Brossier, R., Mérigot, Q., Oudet, E., and Virieux, J. (2016). Increasing the robustness and applicability of full waveform inversion: an optimal transport distance strategy. The Leading Edge, 35(12):1060–1067.CrossRefGoogle Scholar
  27. 27.
    Métivier, L., Brossier, R., Mérigot, Q., Oudet, E., and Virieux, J. (2016). Measuring the misfit between seismograms using an optimal transport distance: Application to full waveform inversion. Geophysical Journal International, 205:345–377.CrossRefGoogle Scholar
  28. 28.
    Métivier, L., Brossier, R., Mérigot, Q., Oudet, E., and Virieux, J. (2016c). An optimal transport approach for seismic tomography: Application to 3D full waveform inversion. Inverse Problems, 32(11):115008.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nash, S. G. (2000). A survey of truncated Newton methods. Journal of Computational and Applied Mathematics, 124:45–59.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Nocedal, J. (1980). Updating Quasi-Newton Matrices With Limited Storage. Mathematics of Computation, 35(151):773–782.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nocedal, J. and Wright, S. J. (2006). Numerical Optimization. Springer, 2nd edition.Google Scholar
  32. 32.
    Operto, S., Brossier, R., Gholami, Y., Métivier, L., Prieux, V., Ribodetti, A., and Virieux, J. (2013). A guided tour of multiparameter full waveform inversion for multicomponent data: from theory to practice. The Leading Edge, Special section Full Waveform Inversion(September):1040–1054.CrossRefGoogle Scholar
  33. 33.
    Philippis, G. D. and Figalli, A. (2014). The Monge-Ampère equation and its link to optimal transportation. BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY.Google Scholar
  34. 34.
    Plessix, R. E. (2006). A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophysical Journal International, 167(2):495–503.CrossRefGoogle Scholar
  35. 35.
    Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, part I : theory and verification in a physical scale model. Geophysics, 64:888–901.CrossRefGoogle Scholar
  36. 36.
    Santambrogio, F. (2015). Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling. Progress in Nonlinear Differential Equations and Their Applications. Springer International Publishing.CrossRefGoogle Scholar
  37. 37.
    Shipp, R. M. and Singh, S. C. (2002). Two-dimensional full wavefield inversion of wide-aperture marine seismic streamer data. Geophysical Journal International, 151:325–344.CrossRefGoogle Scholar
  38. 38.
    Swarztrauber, P. N. (1974). A Direct Method for the Discrete Solution of Separable Elliptic Equations. SIAM Journal on Numerical Analysis, 11(6):1136–1150.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Symes, W. W. (2008). Migration velocity analysis and waveform inversion. Geophysical Prospecting, 56:765–790.CrossRefGoogle Scholar
  40. 40.
    Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49(8):1259–1266.CrossRefGoogle Scholar
  41. 41.
    Villani, C. (2003). Topics in optimal transportation. Graduate Studies In Mathematics, Vol. 50, AMS.Google Scholar
  42. 42.
    Villani, C. (2008). Optimal transport : old and new. Grundlehren der mathematischen Wissenschaften. Springer, Berlin.Google Scholar
  43. 43.
    Virieux, J., Asnaashari, A., Brossier, R., Métivier, L., Ribodetti, A., and Zhou, W. (2017). An introduction to Full Waveform Inversion. In Grechka, V. and Wapenaar, K., editors, Encyclopedia of Exploration Geophysics, page R1–1–R1–40. Society of Exploration Geophysics.Google Scholar
  44. 44.
    Virieux, J. and Operto, S. (2009). An overview of full waveform inversion in exploration geophysics. Geophysics, 74(6):WCC1–WCC26.CrossRefGoogle Scholar
  45. 45.
    Warner, M. and Guasch, L. (2014). Adaptative waveform inversion - FWI without cycle skipping - theory. In 76th EAGE Conference and Exhibition 2014, page We E106 13.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • L. Métivier
    • 1
    Email author
  • A. Allain
    • 2
  • R. Brossier
    • 3
  • Q. Mérigot
    • 4
  • E. Oudet
    • 2
  • J. Virieux
    • 3
  1. 1.ISTerre/LJK, CNRSUniv. Grenoble AlpesSaint-Martin-d’HèresFrance
  2. 2.LJKUniv. Grenoble AlpesSaint-Martin-d’HèresFrance
  3. 3.Univ. Grenoble AlpesISTerreGrenobleFrance
  4. 4.LMOUniv. Paris SudOrsayFrance

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