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Full-Wavefield Inversion: An Extreme-Scale PDE-Constrained Optimization Problem

  • Martin-D. Lacasse
  • Laurent White
  • Huseyin Denli
  • Lingyun Qiu
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 163)

Abstract

Full-wavefield inversion is a geophysical method aimed at estimating the mechanical properties of the earth subsurface. This parameter estimation problem is solved iteratively using optimization techniques aimed at minimizing some measure of misfit between computer-simulated data and real data measured in a seismic survey. This PDE-constrained optimization problem poses many challenges due to the extreme size of the surveys considered. Practical issues related to the physical fidelity and numerical accuracy of the forward problem are presented. Also, issues related to the inverse problem such as the limitations of the optimization methods employed, and the many heuristic strategies used to obtain a solution are discussed. The goal of this paper is to demonstrate some of the progress achieved over the last decades while highlighting the many areas where further investigation could bring this method to full technical maturity. It is our hope that this paper, together with other contributions in this book, will motivate a new generation of researchers to contribute to this broad and challenging research area.

Notes

Acknowledgements

The authors thank ExxonMobil Research and Engineering Company for permission to publish this work. The authors would also like to thank Fadil Santosa and the Institute of Mathematics and its Applications for hosting the workshop where this work was presented. We would also like to thank Jeremy Brandman, Jerry Krebs, Anatoly Baumstein, and Dimitar Trenev for their insightful suggestions and comments on the original manuscript.

Appendix

We start the mathematical description of attenuation by considering a rheological model composed of springs and dashpots as shown in Figure 2. The effective modulus c(ω) of this mechanical model can be expressed as a function of auxiliary variables representing relaxation angular frequencies ωl = 2πfl and nondimensional anelastic coefficients al,
$$\displaystyle \begin{aligned} c(\omega) = c_u \left(1 - \sum_{l=1}^{n} \frac{a_l\omega_l}{\omega_l + \mathrm{i}\omega} \right),\end{aligned} $$
(49)
where ωl = Δclηl, al = Δclcu where the unrelaxed modulus \(c_u = c(\omega \rightarrow \infty ) = c_r + \sum _{l=1}^{n} \varDelta c_l\), in contrast to the relaxed modulus cr = c(ω → 0). This only says that if one moves the system in Figure 2 very slowly, only spring cr is felt as dashpots are relaxing and not transmitting force, while if one moves it very quickly all springs are fully active. Anything in between depends on the frequency according to Equation (49). This model will have an attenuation quality factor following the ratio of real and imaginary parts of the modulus [54, 71], leading to the following self-consistent relation
$$\displaystyle \begin{aligned} Q^{-1}(\omega) = \frac{\Im\left[c(\omega)\right]} {\Re\left[c(\omega)\right]} = \sum_{l=1}^{n} a_l \frac{\omega_l \omega + \omega_l^2 Q^{-1}(\omega)} {\omega_l^2 + \omega^2}.\end{aligned} $$
(50)
The frequency dependence of Q(ω) is set by carefully picking values for ωl and al. This task is usually achieved by sampling frequencies ωl logarithmically in the band of interest and fitting the anelastic coefficients al using a least-squares method [28, 37, 86]. In order to obtain a constant-Q attenuation, i.e., Q(ω) = Qo, we have shown [25] that at least three such relaxation mechanisms are required to obtain a response close to the desired behavior. Figure 9 shows the effect of using a different number of relaxation mechanisms on the frequency response Q(f) of a generalized Maxwell solid. The parameters of the relaxation mechanisms are optimally tuned over a frequency band ranging from 3 Hz to 40 Hz in view of obtaining a constant target quality factor of Qo = 50.
Fig. 9

Frequency response of quality factor Q(f) for generalized Maxwell solids with 1, 2, 3, and 5 relaxation mechanisms over a frequency band ranging from 3 Hz to 40 Hz. The parameters of the relaxation mechanisms are optimized (least-squares) to mimic a constant target quality factor Q = 50 in the frequency band

Each spring and dashpot added to the relaxation model introduce an additional anelastic function ζl(t) (sometimes called memory variable) that has to be solved as part of the governing equations. Each equation in (1) is then replaced by
$$\displaystyle \begin{aligned} \tau(e, t) = c_u e(t) - \sum_{l=1}^{n} a_l\zeta_l(t),\end{aligned} $$
(51)
which are coupled to n additional equations for the anelastic functions,
$$\displaystyle \begin{aligned} \frac{d\zeta_l(t)}{dt} + \omega_l\zeta_l(t) = \omega_l e(t). \end{aligned} $$
(52)
These equations are obtained after integrating the frequency-dependent moduli while maintaining causality (Boltzmann superposition principle). See [71] for details.

Because of this additional complexity, viscoelastic simulations are more costly by up to an order of magnitude, which can be reduced if special algorithms are used [26].

References

  1. 1.
    M. Ainsworth and H. A. Wajid, Dispersive and dissipative behavior of the spectral element method, SIAM Journal on Numerical Analysis, 47 (2009), pp. 3910–3937.MathSciNetCrossRefGoogle Scholar
  2. 2.
    V. Akçelik, H. Denli, A. Kanevsky, K. K. Patel, L. White, and M.-D. Lacasse, Multiparameter material model and source signature full waveform inversion, in SEG Technical Program Expanded Abstracts, San Antonio, 2011, Society of Exploration Geophysics, p. 2406.Google Scholar
  3. 3.
    K. Aki and P. G. Richards, Quantitative Seismology, Theory and Methods, Freeman, San Francisco, 1980.Google Scholar
  4. 4.
    T. Alkalifah and R.-É. Plessix, A recipe for practical full-waveform inversion in anisotropic media: An analytical parameter resolution study, Geophysics, 79 (2014), p. R91.CrossRefGoogle Scholar
  5. 5.
    J. E. Anderson, L. Tan, and D. Wang, Time-reversal checkpointing methods for RTM and FWI, Geophysics, 77 (2012), p. S93.CrossRefGoogle Scholar
  6. 6.
    G. E. Backus, Long-wave elastic anisotropy produced by horizontal layering, J. Geophys. Res., 11 (1962), p. 4427.CrossRefGoogle Scholar
  7. 7.
    R. Bansal, J. R. Krebs, P. Routh, S. Lee, J. E. Anderson, A. Baumstein, A. Mullur, S. Lazaratos, I. Chikichev, and D. McAdow, Simultaneous-source full-wavefield inversion, The Leading Edge, 32 (2013), p. 1100.CrossRefGoogle Scholar
  8. 8.
    R. A. Bartlett, D. M. Gay, and E. T. Phipps, Automatic differentiation of C++ codes for large-scale scientific computing, in Computational Science – ICCS 2006, V. N. Alexandrov, G. D. van Albada, P. M. A. Sloot, and J. Dongarra, eds., Springer, 2006, pp. 525–532.Google Scholar
  9. 9.
    C. C. Bates, T. F. Gaskell, and R. B. Rice, Geophysics in the Affair of Man: A Personalized History of exploration geophysics and its allied sciences of seismology and oceanography, Pergamon Press, Oxford, 1982.Google Scholar
  10. 10.
    J. T. Betts and S. L. Campbell, Discretize then optimize, in Mathematics for industry: Challenger and Frontiers — A Process Review: Practice and Theory, D. R. Fergusson and T. J. Peters, eds., Society of Industrial and Applied Mathematics, Toronto, 2003, p. 140.Google Scholar
  11. 11.
    R. E. Bixby, A brief history of linear and mixed-integer programming computation, in Documenta Mathematica – Extra Volume ISMP, Berlin, 2012, 21st International Symposium on Mathematical Programming, pp. 107–121.Google Scholar
  12. 12.
    A. Bourgeois, P. Lailly, and R. Vesteeg, The Marmousi model, in The Marmousi experience, R. Versteeg and G. Grau, eds., Paris, 1991, IFP/Technip.Google Scholar
  13. 13.
    J. Brandman, H. Denli, and D. Trenev, Introduction to PDE-constrained optimization in the oil and gas industry, in Frontiers in PDE-Constrained Optimization, H. Antil, M.-D. Lacasse, D. Ridzal, and D. P. Kouri, eds., Berlin, 2017, Springer.Google Scholar
  14. 14.
    R. Brossier, L. Métivier, S. Operto, A. Ribodetti, and J. Vireux, VTI acoustic equations: a first-order symmetrical PDE, tech. report, 2013.Google Scholar
  15. 15.
    C. Bunks, F. M. Salek, S. Zaleski, and G. Chavent, Multiscale seismic waveform inversion, Geophysics, 60 (1995), p. 1457.CrossRefGoogle Scholar
  16. 16.
    C. Burstedde and O. Ghattas, Algorithmic strategies for full waveform inversion: 1D experiments, Geophysics, 74 (2009), pp. WCC37–WCC46.CrossRefGoogle Scholar
  17. 17.
    V. Cerveny, Seismic Ray Theory, Cambridge University Press, Cambridge, 2001.CrossRefGoogle Scholar
  18. 18.
    G. Chavent, Identification of functional parameters in partial differential equations, in Identification of functional parameters in distributed systems, R. E. Goodson and M. Polis, eds., American Society of Mechanical Engineers, 1974, p. 31.Google Scholar
  19. 19.
    G. Chavent, Nonlinear Least Squares for Inverse Problems, Springer, Berlin, 2006.zbMATHGoogle Scholar
  20. 20.
    J. Claerbout and D. Nichols, Spectral preconditioning, Stanford Exploration Project Report, 82 (1994), pp. 183–186.Google Scholar
  21. 21.
    R. Clapp, Reverse-time migration: Saving the boundaries, in SEP – 138, 2009, p. 29.Google Scholar
  22. 22.
    S. S. Collis, C. C. Ober, and B. G. van Bloemen Waanders, Unstructured discontinuous Galerkin for seismic inversion, in SEG Technical Program Expanded Abstracts, Denver, 2010, Society of Exploration Geophysics, p. 2767.Google Scholar
  23. 23.
    D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer, New York, 3 ed., 2013.CrossRefGoogle Scholar
  24. 24.
    D. Dagnino, V. Sallarès, and C. R. Ranero, Scale- and parameter-adaptive model-based gradient pre-conditioner for elastic full-waveform inversion, Geophysical Journal International, 198 (2014), p. 1130.CrossRefGoogle Scholar
  25. 25.
    H. Denli, V. Akçelik, A. Kanevsky, D. Trenev, L. White, and M.-D. Lacasse, Full-wavefield inversion of acoustic wave velocity and attenuation, in SEG Technical Program Expanded Abstracts, Houston, 2013, Society of Exploration Geophysics, p. 980.Google Scholar
  26. 26.
    H. Denli and A. Kanevsky, Fast viscoacoustic and viscoelastic full wavefield inversion, Dec 2015, http://www.google.com/patents/US20150362622. US Patent App. 14/693,464.
  27. 27.
    M. Dumbser and M. Käser, An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes — ii. the three-dimensional isotropic case, Geophys. J. Int., 167 (2006), p. 319.CrossRefGoogle Scholar
  28. 28.
    H. Emmerich and M. Korn, Incorporation of attenuation into time-domain computations of seismic wave fields, Geophysics, 52 (1987), p. 1252.CrossRefGoogle Scholar
  29. 29.
    B. Engquist and B. D. Frosse, Application of the Wasserstein metric to seismic signals, 2013. arXiv 1311.4581 [math-ph].Google Scholar
  30. 30.
    B. Engquist, B. D. Frosse, and Y. Yang, Optimal transport for seismic full waveform inversion, 2016. arXiv:1602.01540 [physics.geo-ph].MathSciNetCrossRefGoogle Scholar
  31. 31.
    V. Étienne, E. Chaljub, J. Virieux, and N. Glinsky, An h-p adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modeling, Geophys. J. Int., 183 (2010), p. 941.CrossRefGoogle Scholar
  32. 32.
    P. M. Farrell, D. A. Ham, S. W. Funke, and M. E. Runkes, Automated derivation of the adjoint of high-level transient finite element programs, SIAM Journal of Scientific Computing, 35 (2013), p. C369.MathSciNetCrossRefGoogle Scholar
  33. 33.
    M. Fehler and P. J. Keliher, SEAM Phase I: Challenges of Subsalt Imaging in Tertiary Basins, with Emphasis on Deepwater Gulf of Mexico, Society of Exploration Geophysicists, Tulsa, 2011.Google Scholar
  34. 34.
    A. Fichtner, Full Seismic Waveform Modelling and Inversion, Springer, Berlin, 2011.CrossRefGoogle Scholar
  35. 35.
    W. I. Futterman, Dispersive body waves, J. Geophys. Res., 67 (1962), pp. 5279–5291.MathSciNetCrossRefGoogle Scholar
  36. 36.
    O. Gauthier, J. Virieux, and A. Tarantola, Two-dimensional nonlinear inversion of seismic waveforms: Numerical results, Geophysics, 5 (1986), p. 1387.CrossRefGoogle Scholar
  37. 37.
    R. W. Graves and S. M. Day, Stability and accuracy analysis of coarse-grain viscoelastic simulations, Bulletin Seismological Society of America, 93 (2003), p. 283.CrossRefGoogle Scholar
  38. 38.
    A. Griewank and A. Walther, Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation, Trans. Math. Software, 26 (2000), p. 19.CrossRefGoogle Scholar
  39. 39.
    A. Griewank and A. Walther, Evaluating Derivatives — Principles and Techniques of Algorithmic Differentiation, Society of Industrial and Applied Mathematics, Philadelphia, second ed., 2008.Google Scholar
  40. 40.
    P. C. Hansen and D. P. O’Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), p. 1487.MathSciNetCrossRefGoogle Scholar
  41. 41.
    J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods, Springer, Berlin, 2008.CrossRefGoogle Scholar
  42. 42.
    B. Hofmann and O. Scherzer, Factors influencing the ill-posedness on nonlinear problems, Inverse Problems, 10 (1994), p. 1277.MathSciNetCrossRefGoogle Scholar
  43. 43.
    B. Hofmann and M. Yamamoto, On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems, Applicable Analysis, 89 (2010), p. 1705.MathSciNetCrossRefGoogle Scholar
  44. 44.
    H. Igel, Computational Seismology: A Practical Introduction, Oxford University Press, Oxford, 2017.Google Scholar
  45. 45.
    M. Jakobsen and B. Ursin, Full waveform inversion in the frequency domain using direct iterative t-matrix methods, J. Geophys. Engineer., 12 (2015), p. 400.CrossRefGoogle Scholar
  46. 46.
    W. Kaplan, Advanced Calculus, Addison Wesley, Reading, Massachusetts, second ed., 1973.Google Scholar
  47. 47.
    M. Käser, J. de la Puente, A.-A. Gabriel, and other contributors, seisol. http://www.seissol.org/, Retrieved March 1, 2018.
  48. 48.
    E. Kjartansson, Constant Q-wave propagation and attenuation, Journal of Geophysical Research, 84 (1979), p. 4737.CrossRefGoogle Scholar
  49. 49.
    L. Knopoff, Q, Rev. Geophysics, 2 (1964), p. 625.CrossRefGoogle Scholar
  50. 50.
    H. Kolsky, The propagation of stress pulses in viscoelastic solids, Phys. Mag., 1 (1956), pp. 693–710.CrossRefGoogle Scholar
  51. 51.
    D. Komatitsch, Méthodes spectrales et éléments spectraux pour l’équation de l’élastodynamique 2D et 3D en milieu hétérogènes, PhD thesis, Institut de Physique du Globe de Paris, France, 1997.Google Scholar
  52. 52.
    D. Komatitsch, J. Tromp, and other contributors, specfem3d. http://geodynamics.org/cig/software/specfem3d, Retrieved March 1, 2018.
  53. 53.
    J. R. Krebs, J. E. Anderson, D. Hinkley, R. Neelamani, S. Lee, A. Baumstein, and M.-D. Lacasse, Fast full-wavefield seismic inversion using encoded sources, Geophysics, 74 (2009), p. WCC177.CrossRefGoogle Scholar
  54. 54.
    J. Kristek and P. Moczo, Seismic wave propagation in viscoelastic media with material discontinuities — a 3D 4th -order staggered-grid finite-difference modeling, Bulletin Seismological Society of America, 93 (2003), p. 2273.CrossRefGoogle Scholar
  55. 55.
    P. Lailly, The seismic inverse problem as a sequence of before-stack migrations, in Conference on Inverse Scattering: Theory and Applications, J. B. Bednar, R. Redner, E. Robinson, and A. Weglein, eds., Philadelphia, 1983, Society of Industrial and Applied Mathematics, p. 206.Google Scholar
  56. 56.
    L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Pergamon, Oxford, 1959.zbMATHGoogle Scholar
  57. 57.
    S. Lazaratos, I. Chikichev, and Y. Wang, Improving convergence rate of full wavefield inversion using spectral shaping, in SEG Technical Program Expanded Abstracts, San Antonio, 2011, Society of Exploration Geophysics, p. 2428.Google Scholar
  58. 58.
    R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, New York, 2002.CrossRefGoogle Scholar
  59. 59.
    A. Logg, K. A. Mardal, and G. N. Wells, eds., The Fenics project, Lecture notes in computational science and engineering, Springer, Berlin, 2012.Google Scholar
  60. 60.
    C. C. Lopez, Accélération et régularisation de la méthode d’inversion des formes d’ondes complètes en exploration sismique, PhD thesis, Université de Nice-Sophia Antipolis, 2014.Google Scholar
  61. 61.
    R. Madariaga, Seismic source: Theory, in Encyclopedia of Earth Sciences Series – Geophysics, C. W. Finkl, ed., Springer, Boston, MA, 1989, pp. 1129–1133.Google Scholar
  62. 62.
    G. Marchuk, V. Shutyaev, and G. Bocharov, Adjoint equations and analysis of complex systems: Application to virus infection modelling, J. Computational and Applied Mathematics, 184 (2005), pp. 177–204.MathSciNetCrossRefGoogle Scholar
  63. 63.
    G. I. Marchuk, Adjoint Equations and Analysis of Complex Systems, Springer, Netherlands, 1995.CrossRefGoogle Scholar
  64. 64.
    G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint equations and perturbations algorithms in nonlinear problems, CRC Press, Boca Raton, 1996.Google Scholar
  65. 65.
    G. S. Martin, R. Wiley, and K. J. Marfurt, Marnousi2: An elastic upgrade to Marmousi, The Leading Edge, 25 (2006), p. 156.CrossRefGoogle Scholar
  66. 66.
    Mavko, Quantitative seismic interpretation, Springer, 2006.Google Scholar
  67. 67.
    G. Mavko, T. Mukerji, and J. Dvorkin, The Rock Physics Handbook, Cambridge University Press, Cambridge, 1998.Google Scholar
  68. 68.
    L. Métivier, F. Bretaudeau, R. Brossier, S. Operto, and J. Virieux, Full waveform inversion and the truncated Newton method: quantitative imaging of complex subsurface structures, Geophysical Prospecting, 62 (2014), p. 1353.CrossRefGoogle Scholar
  69. 69.
    L. Metivier and J. Virieux, Optimal transport theory, in Frontiers in PDE-Constrained Optimization, H. Antil, M.-D. Lacasse, D. Ridzal, and D. P. Kouri, eds., Berlin, 2017, Springer.Google Scholar
  70. 70.
    P. Moczo and J. Kristek, On the rheological models in the time-domain methods for seismic wave propagation, Geophysical Review Letters, 32 (2005), p. L01306.Google Scholar
  71. 71.
    P. Moczo, J. Kristek, and P. Franek, Lectures notes on rheological models. http://www.fyzikazeme.sk/mainpage/stud_mat/Moczo_Kristek_Franek_Rheological_Models.pdf, 2006. retrieved March 1, 2018.
  72. 72.
    R. Modrak and J. Tromp, Seismic waveform inversion best practices, Geophysical Journal International, 206 (2016), p. 1864.CrossRefGoogle Scholar
  73. 73.
    P. R. Mora, Non-linear two-dimensional elastic inversion of multi-offset seismic data, Geophysics, 52 (1987), p. 1211.CrossRefGoogle Scholar
  74. 74.
    J. Nocedal and S. J. Wright, Numerical optimization, Springer Series in Operations Research and Financial Engineering, Springer, Berlin, 2006.Google Scholar
  75. 75.
    G. Noh and S. H. ans Klaus-Jürgen Bathe, Performance of an implicit time integration scheme in the analysis of wave propagations, Computers and Structures, 123 (2013), pp. 93–105.CrossRefGoogle Scholar
  76. 76.
    C. C. Ober, T. M. Smith, J. R. Overfelt, S. S. Collis, G. J. von Winckel, B. G. van Bloemen Waanders, N. J. Downey, S. A. Mitchell, S. D. Bond, D. F. Aldridge, and J. R. Krebs, Visco-TTI-elastic FWI using discontinuous Galerkin, in SEG Technical Program Expanded Abstracts, Dallas, 2016, Society of Exploration Geophysics, p. 5654.Google Scholar
  77. 77.
    S. Operto, Y. Gholami, V. Prieux, A. Ribodetti, R. Brossier, L. Metivier, and J. Virieux, A guided tour of multiparameter full waveform inversion with multicomponent data: from theory to practice, The Leading Edge, 32 (2013), p. 936.CrossRefGoogle Scholar
  78. 78.
    S. Operto, J. Virieux, P. Amestoy, J.-Y. L’Excellent, L. Giraud, and H. Ben Hadj Ali, 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study, Geophysics, 72 (2007), p. SM195.CrossRefGoogle Scholar
  79. 79.
    W. J. Parnell and I. D. Abrahams, New integral equation approach to elastodynamic homogenization, Proceedings of the Royal Society A, 464 (2008), p. 1461.MathSciNetCrossRefGoogle Scholar
  80. 80.
    R.-É. Plessix and Q. Cao, A parametrization study for surface seismic full waveform inversion in an acoustic vertical transversely isotropic medium, Geophys J Int, 185 (2011), p. 539.CrossRefGoogle Scholar
  81. 81.
    R. G. Pratt, C. Shin, and G. J. Hicks, Gauss-newton and full newton methods in frequency-space seismic waveform inversion, Geophys. J. Int, 133 (1998), p. 341.CrossRefGoogle Scholar
  82. 82.
    R. G. Pratt and M. H. Worthington, Inverse theory applied to multi-source cross-hole tomography. Part I: acoustic wave-equation method, Geophys. Prospect., 38 (1990), p. 287.Google Scholar
  83. 83.
    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, New York, third ed., 2007.zbMATHGoogle Scholar
  84. 84.
    L. Qiu and M.-D. Lacasse, Effects of parameterization on non-linear parameter estimation problems, to be submitted.Google Scholar
  85. 85.
    C. D. Riyanti, Y. A. Erlangga, R.-É. Plessix, W. A. Mulder, C. Vuik, and C. Oosterlee, New iterative solver for the time-harmonic wave equation, Geophysics, 71 (2006), p. E57.CrossRefGoogle Scholar
  86. 86.
    J. O. A. Robertsson, J. O. Blanch, and W. W. Symes, Viscoelastic finite-difference modeling, Geophysics, 59 (1994), p. 1444.CrossRefGoogle Scholar
  87. 87.
    P. S. Routh, J. R. Krebs, S. Lazaratos, and J. E. Anderson, Encoded simultaneous-source full-wavefield inversion for spectrally shaped marine streamer data, in SEG Technical Program Expanded Abstracts, San Antonio, 2011, Society of Exploration Geophysics, p. 2433.Google Scholar
  88. 88.
    R. Sargent, Progress in modelling and simulation, in Verification and Validation of Simulation Models, F. Celier, ed., Academic Press, London, 1982, p. 159.Google Scholar
  89. 89.
    S. Scheslinger, R. E. Crosby, R. E. Gagné, G. S. Innis, C. S. Lalwani, J. Loch, R. J. Sylvester, R. D. Wright, N. Kheir, and D. Bartos, Terminology for model credibility, Simulation, (1979), pp. 103–104.Google Scholar
  90. 90.
    J. H. Schön, Physical properties of rocks — Fundamentals and principles of petrophysics, in Handbook of Geophysical Exploration, K. Helbig and S. Treitel, eds., vol. 18, Elsevier, 2004, p. 583.Google Scholar
  91. 91.
    P. M. Shearer, Introduction to Seismology, Cambridge University Press, Cambridge, 1999.Google Scholar
  92. 92.
    SIAM Working Group on CSE Education, Graduate education in computational science and engineering, SIAM Review, 43 (2001), p. 163.CrossRefGoogle Scholar
  93. 93.
    I. S. Sokolnikov, Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956.Google Scholar
  94. 94.
    W. W. Symes, I. S. Terentyev, and T. W. Vdovina, Gridding requirements for accurate finite difference simulation, in SEG Technical Program Expanded Abstracts, Las Vegas, 2008, Society of Exploration Geophysics, pp. 2077–2081.Google Scholar
  95. 95.
    A. Tarantola, Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49 (1984), p. 1259.CrossRefGoogle Scholar
  96. 96.
    A. Tarantola, Inverse Problem Theory And Methods For Model Parameter Estimation, Society of Applied and Industrial Mathematics, Philadelphia, 2005.CrossRefGoogle Scholar
  97. 97.
    L. Thomsen, Weak elastic anisotropy, Geophysics, 51 (1986), p. 1954.CrossRefGoogle Scholar
  98. 98.
    V. A. Titarev and E. F. Toro, ADER: Arbitrary high-order Godunov approach, J. Sci. Comput., 17 (2002), pp. 609–18.MathSciNetCrossRefGoogle Scholar
  99. 99.
    M. N. Toksoz, D. H. Johnston, and A. Timur, Attenuation of seismic waves in dry and saturated rocks: I. Laboratory measurements, Geophysics, 44 (1979), p. 681.CrossRefGoogle Scholar
  100. 100.
    S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, vol. 16 of Interdisciplinary applied mathematics, Springer-Verlag, New York, 2002.CrossRefGoogle Scholar
  101. 101.
    J. Tromp, D. Komatitsch, and Q. Liu, Spectral elements and adjoint methods in seismology, Communications in Computational Physics, 3 (2008), p. 1.zbMATHGoogle Scholar
  102. 102.
    B. Ursin and T. Toverud, Comparison of seismic dispersion and attenuation models, Stud. Geophys. Geod., 46 (2002), p. 293.CrossRefGoogle Scholar
  103. 103.
    J. Virieux, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method, Geophysics, 51 (1986), p. 889.CrossRefGoogle Scholar
  104. 104.
    J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74 (2009), p. WCC127.CrossRefGoogle Scholar
  105. 105.
    J. Virieux, S. Operto, H. Ben Hadj Ali, R. Brossier, V. Etienne, F. S. amd L. Giraud, and A. Haidar, Seismic wave modeling for seismic imaging, The Leading Edge, 28 (2009), p. 538.CrossRefGoogle Scholar
  106. 106.
    C. Vogel, Computational methods for inverse problems, Society for Industrial and Applied Mathematics, Philadelphia, 2002.CrossRefGoogle Scholar
  107. 107.
    S. Wang, M. V. de Hoop, and J. Xia, On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver, Geophysical Prospecting, 59 (2011), p. 857.CrossRefGoogle Scholar
  108. 108.
    Y. Wang, Seismic Inverse Q Filtering, John Wiley and Sons, New York, 2009.Google Scholar
  109. 109.
    M. Warner and L. Guasch, Adaptive waveform inversion: Theory, Geophysics, 81 (2016), pp. R429–R445.CrossRefGoogle Scholar
  110. 110.
    R. Wu and K. Aki, Scattering characteristics of elastic waves by an elastic heterogeneity, Geophysics, 50 (1985), p. 582.CrossRefGoogle Scholar
  111. 111.
    P. Yang, R. Brossier, L. Métivier, and J. Virieux, Wavefield reconstruction in attenuating media: A checkpointing-assisted reverse-forward simulation method, Geophysics, 81 (2016), pp. R349–R362.CrossRefGoogle Scholar
  112. 112.
    Y. O. Yuan, F. J. Simons, and J. Tromp, Double-difference adjoint seismic tomography, Geophys. J. Int., 206 (2017), pp. 1599–1618.CrossRefGoogle Scholar

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Authors and Affiliations

  • Martin-D. Lacasse
    • 1
  • Laurent White
    • 1
  • Huseyin Denli
    • 1
  • Lingyun Qiu
    • 2
  1. 1.Corporate Strategic ResearchExxonMobil Research and Engineering CompanyAnnandaleUSA
  2. 2.Petroleum Geo-ServicesHoustonUSA

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