Seismic wavefield imaging of Earth’s interior across scales


Seismic full-waveform inversion (FWI) for imaging Earth’s interior was introduced in the late 1970s. Its ultimate goal is to use all of the information in a seismogram to understand the structure and dynamics of Earth, such as hydrocarbon reservoirs, the nature of hotspots and the forces behind plate motions and earthquakes. Thanks to developments in high-performance computing and advances in modern numerical methods in the past 10 years, 3D FWI has become feasible for a wide range of applications and is currently used across nine orders of magnitude in frequency and wavelength. A typical FWI workflow includes selecting seismic sources and a starting model, conducting forward simulations, calculating and evaluating the misfit, and optimizing the simulated model until the observed and modelled seismograms converge on a single model. This method has revealed Pleistocene ice scrapes beneath a gas cloud in the Valhall oil field, overthrusted Iberian crust in the western Pyrenees mountains, deep slabs in subduction zones throughout the world and the shape of the African superplume. The increased use of multi-parameter inversions, improved computational and algorithmic efficiency, and the inclusion of Bayesian statistics in the optimization process all stand to substantially improve FWI, overcoming current computational or data-quality constraints. In this Technical Review, FWI methods and applications in controlled-source and earthquake seismology are discussed, followed by a perspective on the future of FWI, which will ultimately result in increased insight into the physics and chemistry of Earth’s interior.

Key points

  • Modern numerical methods and high-performance computers have facilitated the characterization of Earth’s interior constrained by the physics of seismic-wave propagation.

  • Seismic full-waveform inversion (FWI) has enabled unprecedented imaging across nine orders of magnitude in frequency and wavelength, with applications ranging from medical imaging and nondestructive testing to global seismology.

  • FWI continues to be developed and improved, with opportunities for a more complete description of the physics of seismic-wave propagation (for example, anisotropy, attenuation and poroelasticity), as well as better and more effective optimization algorithms (such as source encoding, uncertainty quantification and Hamiltonian Monte Carlo methods).

  • Computers in the exascale era (~2020–2021) will enable global FWI at frequencies of up to ~1 Hz, potentially facilitating sub-10-km-scale imaging of Earth’s mantle.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Timeline of major developments in seismic FWI and notable applications.
Fig. 2: FWI workflow.
Fig. 3: Multi-parameter visco-acoustic FWI of the Valhall oil field.
Fig. 4: Teleseismic FWI experiment using the MAUPASACQ array in the western Pyrenees.
Fig. 5: Vertical cross sections of compressional wave-speed perturbations in various subduction zones.
Fig. 6: 3D morphology of the African superplume from two angles.


  1. 1.

    Aki, K., Christoffersson, A. & Husebye, E. S. Determination of the three-dimensional seismic structure of the lithosphere. J. Geophys. Res. 82, 277–296 (1977).

    Google Scholar 

  2. 2.

    Dziewoński, A. M., Hager, B. H. & O’Connell, R. J. Large-scale heterogeneities in the lower mantle. J. Geophys. Res. 82, 239–255 (1977).

    Google Scholar 

  3. 3.

    Woodhouse, J. H. & Dziewoński, A. M. Mapping the upper mantle: three-dimensional modeling of Earth structure by inversion of seismic waveforms. J. Geophys. Res. 89, 5953–5986 (1984). The first application of global waveform tomography in earthquake seismology using mantle waves.

    Google Scholar 

  4. 4.

    Masters, T. G., Johnson, S., Laske, G. & Bolton, H. A shear-velocity model of the mantle. Phil. Trans. R. Soc. Lond. A 354, 1385–1411 (1996).

    Google Scholar 

  5. 5.

    Van der Hilst, R. D., Widiyantoro, S. & Engdahl, R. Evidence for deep mantle circulation from global tomography. Nature 386, 578–584 (1997).

    Google Scholar 

  6. 6.

    Grand, S. P., Van der Hilst, R. D. & Widiyantoro, S. High resolution global tomography: a snapshot of convection in the Earth. Geol. Soc. Am. Today 7 (1997).

  7. 7.

    Bassin, C., Laske, G. & Masters, G. The current limits of resolution for surface wave tomography in North America. EOS Trans. AGU 81 (2000).

  8. 8.

    Laske, G., Masters, G., Ma, Z. & Pasyanos, M. Update on CRUST1.0 – A 1-degree global model of Earth’s crust. Geophys. Res. Abstr. 15, 2658 (2013).

  9. 9.

    Ratcliff, D. W., Gray, S. H. & Whitmore, N. D. Jr. Seismic imaging of salt structures in the Gulf of Mexico. Lead. Edge 11, 15–31 (1992).

    Google Scholar 

  10. 10.

    Schreiman, J., Gisvold, J., Greenleaf, J. F. & Bahn, R. Ultrasound transmission computed tomography of the breast. Radiology 150, 523–530 (1984).

    Google Scholar 

  11. 11.

    Duric, N. et al. in Proc. Medical Imaging 2015: Ultrasonic Imaging and Tomography Vol. 9419 (International Society for Optics and Photonics, 2015).

  12. 12.

    Li, C., Sandhu, G. Y., Boone, M. & Duric, N. in Proc. Medical Imaging 2017: Ultrasonic Imaging and Tomography Vol. 10139 (International Society for Optics and Photonics, 2017).

  13. 13.

    Boehm, C., Martiartu, N. K., Vinard, N., Balic, I. J. & Fichtner, A. in Proc. Medical Imaging 2018: Ultrasonic Imaging and Tomography Vol. 10580 (International Society for Optics and Photonics, 2017).

  14. 14.

    Wiskin, J. et al. Full wave 3D inverse scattering: 21st century technology for whole body imaging. J. Acoust. Soc. Am. 145, 1857–1857 (2019).

    Google Scholar 

  15. 15.

    Huthwaite, P. & Simonetti, F. High-resolution guided wave tomography. Wave Motion 50, 979–993 (2013).

    Google Scholar 

  16. 16.

    Huthwaite, P. Guided wave tomography with an improved scattering model. Proc. R. Soc. A 472, 20160643 (2016).

    Google Scholar 

  17. 17.

    Rao, J., Ratassepp, M. & Fan, Z. Guided wave tomography based on full waveform inversion. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63, 737–745 (2016).

    Google Scholar 

  18. 18.

    Seidl, R. & Rank, E. Iterative time reversal based flaw identification. Comput. Math. Appl. 72, 879–892 (2016).

    Google Scholar 

  19. 19.

    Rao, J., Ratassepp, M. & Fan, Z. Limited-view ultrasonic guided wave tomography using an adaptive regularization method. J. Appl. Phys. 120, 194902 (2016).

    Google Scholar 

  20. 20.

    Jalinoos, F., Tran, K. T., Nguyen, T. D. & Agrawal, A. K. Evaluation of bridge abutments and bounded wall type structures with ultraseismic waveform tomography. J. Bridge Eng. 22, 04017104 (2017).

    Google Scholar 

  21. 21.

    Rao, J., Ratassepp, M. & Fan, Z. Investigation of the reconstruction accuracy of guided wave tomography using full waveform inversion. J. Sound Vib. 400, 317–328 (2017).

    Google Scholar 

  22. 22.

    Lamert, A., Nguyen, L. T., Friederich, W. & Nestorović, T. Imaging disturbance zones ahead of a tunnel by elastic full-waveform inversion: adjoint gradient based inversion vs. parameter space reduction using a level-set method. Undergr. Space 3, 21–33 (2018).

    Google Scholar 

  23. 23.

    Nguyen, L. T. & Modrak, R. T. Ultrasonic wavefield inversion and migration in complex heterogeneous structures: 2D numerical imaging and nondestructive testing experiments. Ultrasonics 82, 357–370 (2018).

    Google Scholar 

  24. 24.

    He, J., Rocha, D. C., Leser, P. E., Sava, P. & Leser, W. P. Least-squares reverse time migration (LSRTM) for damage imaging using Lamb waves. Smart Mater. Struct. 28, 065010 (2019).

    Google Scholar 

  25. 25.

    Gao, F., Levander, A. R., Pratt, R. G., Zelt, C. A. & Fradelizio, G. L. Waveform tomography at a groundwater contamination site: VSP-surface data set. Geophysics 71, H1–H11 (2006).

    Google Scholar 

  26. 26.

    Chen, J., Zelt, C. A. & Jaiswal, P. Detecting a known near-surface target through application of frequency-dependent traveltime tomography and full-waveform inversion to P- and SH-wave seismic refraction data. Geophysics 82, R1–R17 (2017).

    Google Scholar 

  27. 27.

    Alam, M. I. & Jaiswal, P. Near surface characterization using VP/VS and Poisson’s ratio from seismic refractions. J. Environ. Eng. Geophysics 22, 101–109 (2017).

    Google Scholar 

  28. 28.

    Alam, M. I. Near-surface characterization using traveltime and full-waveform inversion with vertical and horizontal component seismic data. Interpretation 7, T141–T154 (2019).

    Google Scholar 

  29. 29.

    Wang, Y. et al. Tunnel detection at Yuma Proving Ground, Arizona, USA – Part 1: 2D full-waveform inversion experiment. Geophysics 84, B95–B105 (2019).

    Google Scholar 

  30. 30.

    Smith, J. A. et al. Tunnel detection at Yuma Proving Ground, Arizona, USA – Part 2: 3D full-waveform inversion experiments. Geophysics 84, B107–B120 (2019).

    Google Scholar 

  31. 31.

    Gauthier, O., Virieux, J. & Tarantola, A. Two-dimensional non-linear inversion of seismic waveforms: numerical results. Geophysics 51, 1387–1403 (1986).

    Google Scholar 

  32. 32.

    Mora, P. Nonlinear two-dimensional elastic inversion of multi-offset seismic data. Geophysics 52, 1211–1228 (1987).

    Google Scholar 

  33. 33.

    Mora, P. Elastic wavefield inversion of reflection and transmission data. Geophysics 53, 750–759 (1987).

    Google Scholar 

  34. 34.

    Pratt, R. G. & Worthington, M. H. Inverse theory applied to multi-source cross-hole tomography. Part 1: Acoustic wave-equation method. Geophys. Prospecting 38, 287–310 (1990). Successful application of frequency-domain waveform inversion using crosshole transmitted waves.

    Google Scholar 

  35. 35.

    Igel, H., Djikpréssé, H. & Tarantola, A. Waveform inversion of marine reflection seismograms for P impedance and Poisson’s ratio. Geophys. J. Int. 124, 363–371 (1996).

    Google Scholar 

  36. 36.

    Pratt, R. G., Song, Z. M., Williamson, P. R. & Warner, M. Two-dimensional velocity models from wide-angle seismic data by wavefield inversion. Geophys. J. Int. 124, 323–340 (1996).

    Google Scholar 

  37. 37.

    Pratt, R. G. Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model. Geophysics 64, 888–901 (1999).

    Google Scholar 

  38. 38.

    Pratt, R. G. & Shipp, R. M. Seismic waveform inversion in the frequency domain, Part 2: Fault delineation in sediments using crosshole data. Geophysics 64, 902–914 (1999).

    Google Scholar 

  39. 39.

    Brenders, A. J. & Pratt, G. Full waveform tomography for lithospheric imaging: results from a blind test in a realistic crustal model. Geophys. J. Int. 168, 133–151 (2007). Successful blind test inversion based on seismic waveform inversion in exploration seismology.

    Google Scholar 

  40. 40.

    Dessa, J.-X. et al. Multiscale seismic imaging of the eastern Nankai trough by full waveform inversion. Geophys. Res. Lett. 31 (2004).

  41. 41.

    Ravaut, C. et al. Multiscale imaging of complex structures from multifold wide-aperture seismic data by frequency-domain full-waveform tomography: Application to a thrust belt. Geophys. J. Int. 159, 1032–1056 (2004).

    Google Scholar 

  42. 42.

    Operto, S., Virieux, J., Dessa, J.-X. & Pascal, G. Crustal seismic imaging from multifold ocean bottom seismometer data by frequency domain full waveform tomography: Application to the eastern Nankai trough. J. Geophys. Res. Solid Earth 111 (2006).

  43. 43.

    Kamei, R., Pratt, R. G. & Tsuji, T. Waveform tomography imaging of a megasplay fault system in the seismogenic Nankai subduction zone. Earth Planet. Sci. Lett. 317–318, 343–353 (2012).

    Google Scholar 

  44. 44.

    Jian, H., Singh, S. C., Chen, Y. J. & Li, J. Evidence of an axial magma chamber beneath the ultraslow-spreading Southwest Indian Ridge. Geology 45, 143–146 (2017).

    Google Scholar 

  45. 45.

    Górszczyk, A., Operto, S. & Malinowski, M. Toward a robust workflow for deep crustal imaging by FWI of OBS data: The eastern Nankai Trough revisited. J. Geophys. Res. Solid Earth 122, 4601–4630 (2017).

    Google Scholar 

  46. 46.

    Huot, G. & Singh, S. C. Seismic evidence for fluid/gas beneath the Mentawai Fore-Arc Basin, Central Sumatra. J. Geophys. Res. Solid Earth 123, 957–976 (2018).

    Google Scholar 

  47. 47.

    Gorszczyk, A., Operto, S., Schenini, L. & Yamada, Y. Crustal-scale depth imaging via joint full-waveform inversion of ocean-bottom seismometer data and pre-stack depth migration of multichannel seismic data: a case study from the eastern Nankai Trough. Solid Earth 10, 765–784 (2019).

    Google Scholar 

  48. 48.

    Chen, P., Zhao, L. & Jordan, T. H. Full 3D tomography for the crustal structure of the Los Angeles region. Bull. Seism. Soc. Am. 97, 1094–1120 (2007). Earthquake seismology FWI of the Los Angeles region.

    Google Scholar 

  49. 49.

    Tape, C., Liu, Q., Maggi, A. & Tromp, J. Adjoint tomography of the southern California crust. Science 325, 988–992 (2009). Earthquake seismology FWI of the southern California crust.

    Google Scholar 

  50. 50.

    Tape, C., Liu, Q., Maggi, A. & Tromp, J. Seismic tomography of the southern California crust based on spectral-element and adjoint methods. Geophys. J. Int. 180, 433–462 (2010).

    Google Scholar 

  51. 51.

    Fichtner, A., Kennett, B. L. N., Igel, H. & Bunge, H. P. Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods. Geophys. J. Int. 179, 1703–1725 (2009).

    Google Scholar 

  52. 52.

    Fichtner, A., Kennett, B. L. N., Igel, H. & Bunge, H.-P. Full waveform tomography for radially anisotropic structure: New insights into present and past states of the Australasian upper mantle. Earth Planet. Sci. Lett. 290, 270–280 (2010).

    Google Scholar 

  53. 53.

    French, S. W. & Romanowicz, B. Broad plumes rooted at the base of the Earth’s mantle beneath major hotspots. Nature 525, 95–99 (2015). The first application of global waveform inversion based on a hybrid method, combining forward simulations in 3D models with inverse simulations based on a perturbation method.

    Google Scholar 

  54. 54.

    Bozdag˘, E. et al. Global adjoint tomography: first-generation model. Geophys. J. Int. 207, 1739–1766 (2016). The first application of global FWI.

    Google Scholar 

  55. 55.

    Tromp, J., Luo, Y., Hanasoge, S. & Peter, D. Noise cross-correlation sensitivity kernels. Geophys. J. Int. 183, 791–819 (2010).

    Google Scholar 

  56. 56.

    Sager, K., Ermert, L., Boehm, C. & Fichtner, A. Towards full waveform ambient noise inversion. Geophys. J. Int. 212, 566–590 (2018).

    Google Scholar 

  57. 57.

    Virieux, J. & Operto, S. An overview of full-waveform inversion in exploration geophysics. Geophysics 74, WCC1–WCC26 (2009).

    Google Scholar 

  58. 58.

    Fichtner, A. Full Seismic Waveform Modelling and Inversion (Springer, 2010).

  59. 59.

    Liu, Q. & Gu, Y. Seismic imaging: from classical to adjoint tomography. Tectonophysics 566–567, 31–66 (2012).

    Google Scholar 

  60. 60.

    Bamberger, A., Chavent, G. & Lailly, P. Une application de la théorie du contrôle à un problème inverse de sismique. Ann. Geophys. 33, 183–200 (1977).

    Google Scholar 

  61. 61.

    Lailly, P. in Conf. on Inverse Scattering: Theory and Application (ed Bednar, J.) 206–220 (Society for Industrial and Applied Mathematics, 1983).

  62. 62.

    Tarantola, A. Inversion of seismic reflection data in the acoustic approximation. Geophysics 49, 1259–1266 (1984). The magnificent work at the root of FWI.

    Google Scholar 

  63. 63.

    Lions, J. L. & Magenes, E. Non-Homogeneous Boundary Value Problems and Applications (Springer, 1972).

  64. 64.

    Chavent, G. in Identification of Parameter Distributed Systems (eds Goodson, R. E. & Polis, M. P.) 65–74 (American Society of Mechanical Engineers, 1974).

  65. 65.

    Le Dimet, F.-X. & Talagrand, O. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A 38, 97–110 (1986).

    Google Scholar 

  66. 66.

    Talagrand, O. & Courtier, P. Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. Q. J. R. Meteorol. Soc. 113, 1311–1328 (1987).

    Google Scholar 

  67. 67.

    Nolet, G. in Seismic Tomography (ed. Nolet, G.) 301–322 (D. Reidel, 1987).

  68. 68.

    Nolet, G. Partitioned waveform inversion and two-dimensional structure under the Network of Autonomously Recording Seismograph. J. Geophys. Res. 95, 8499–8512 (1990). Introduction of partitioned waveform inversion in earthquake seismology.

    Google Scholar 

  69. 69.

    Zielhuis, A. & Nolet, G. Deep seismic expression of an ancient plate boundary in Europe. Science 265, 79–81 (1994).

    Google Scholar 

  70. 70.

    Li, X. D. & Tanimoto, T. Waveforms of long-period body waves in a slightly aspherical Earth model. Geophys. J. Int. 112, 92–102 (1993).

    Google Scholar 

  71. 71.

    Li, X. D. & Romanowicz, B. Comparison of global waveform inversions with and without considering cross-branch modal coupling. Geophys. J. Int. 121, 695–709 (1995).

    Google Scholar 

  72. 72.

    Li, X.-D. & Romanowicz, B. Global mantle shear velocity model developed using nonlinear asymptotic coupling theory. J. Geophys. Res. 101, 22245–22272 (1996). Construction of a global shear wave speed model based on NACT.

    Google Scholar 

  73. 73.

    Marquering, H., Dahlen, F. A. & Nolet, G. Three-dimensional sensitivity kernels for finite-frequency traveltimes: the banana-doughnut paradox. Geophys. J. Int. 137, 805–815 (1999). Introduction of finite-frequency sensitivity kernels, affectionately known as ‘banana-doughnut’ kernels.

    Google Scholar 

  74. 74.

    Dahlen, F. A., Hung, S.-H. & Nolet, G. Fréchet kernels for finite-frequency traveltimes - I. Theory. Geophys. J. Int. 141, 157–174 (2000).

    Google Scholar 

  75. 75.

    Dahlen, F. A. & Baig, A. M. Fréchet kernels for body-wave amplitudes. Geophys. J. Int. 150, 440–466 (2002).

    Google Scholar 

  76. 76.

    Montelli, R., Nolet, G., Dahlen, F. A. & Masters, G. A catalogue of deep mantle plumes: new results from finite-frequency tomography. Geochem. Geophys. Geosyst. 7, Q11007 (2006).

    Google Scholar 

  77. 77.

    Tromp, J., Tape, C. & Liu, Q. Y. Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys. J. Int. 160, 195–216 (2005). This article draws connections between finite-frequency sensitivity kernels, adjoint-state methods and time-reversal imaging.

    Google Scholar 

  78. 78.

    Tape, C., Liu, Q. & Tromp, J. Finite-frequency tomography using adjoint methods — methodology and examples using membrane surface waves. Geophys. J. Int. 168, 1105–1129 (2007).

    Google Scholar 

  79. 79.

    Liu, Q. & Tromp, J. Finite-frequency kernels based on adjoint methods. Bull. Seism. Soc. Am. 96, 2383–2397 (2006).

    Google Scholar 

  80. 80.

    Liu, Q. & Tromp, J. Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods. Geophys. J. Int. 174, 265–286 (2008).

    Google Scholar 

  81. 81.

    Plessix, R. E. A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys. J. Int. 167, 495–503 (2006).

    Google Scholar 

  82. 82.

    Nocedal, J. & Wright, S. Numerical Optimization 2nd edn (Springer, 2006).

  83. 83.

    Biegler, L., Ghattas, O., Heinkenschloss, M. & van Bloemen Waanders, B. in Large-Scale PDE-Constrained Optimization Vol. 30 (eds Biegler, L. T., Heinkenschloss, M., Ghattas, O. & van Bloemen Waanders, B.) 3–13 (Springer, 2003).

  84. 84.

    Dziewoński, A. & Anderson, D. Preliminary reference Earth model. Phys. Earth Planet. Inter. 25, 297–356 (1981).

    Google Scholar 

  85. 85.

    Ritzwoller, M. H. & Lavely, E. M. Three-dimensional models of the Earth’s mantle. Rev. Geophys. 33, 1–66 (1995).

    Google Scholar 

  86. 86.

    Trampert, J. & Woodhouse, J. H. Assessment of global phase velocity models. Geophys. J. Int. 144, 165–174 (2001).

    Google Scholar 

  87. 87.

    Becker, T. W. & Boschi, L. A comparison of tomographic and geodynamic mantle models. Geochem. Geophys. Geosyst 3, 1003 (2002).

    Google Scholar 

  88. 88.

    Lekic, V., Cottaar, S., Dziewonski, A. & Romanowicz, B. Cluster analysis of global lower mantle tomography: a new class of structure and implications for chemical heterogeneity. Earth Planet. Sci. Lett. 357–358, 68–77 (2012).

    Google Scholar 

  89. 89.

    Bunks, C., Saleck, F. M., Zaleski, S. & Chavent, G. Multiscale seismic waveform inversion. Geophysics 60, 1457–1473 (1995). Introduction of the important concept of multiscale waveform inversion.

    Google Scholar 

  90. 90.

    Dahlen, F. A. & Tromp, J. Theoretical Global Seismology (Princeton Univ. Press, 1998).

  91. 91.

    Zhu, H., Bozdağ, E., Peter, D. & Tromp, J. Structure of the European upper mantle revealed by adjoint tomography. Nat. Geosci. 5, 493–498 (2012). Continental-scale VTI FWI of the European crust and upper mantle.

    Google Scholar 

  92. 92.

    Plessix, R.-E., Baeten, G., de Maag, J. & ten Kroode, F. Full waveform inversion and distance separated simultaneous sweeping: a study with a land seismic data set. Geophys. Prospecting 60, 733–747 (2012).

    Google Scholar 

  93. 93.

    Igel, H. Computational Seismology (Oxford Univ. Press, 2016).

  94. 94.

    Virieux, J. SH-wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 49, 1933–1942 (1984).

    Google Scholar 

  95. 95.

    Virieux, J. P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 51, 889–901 (1986).

    Google Scholar 

  96. 96.

    Levander, A. R. Fourth-order finite-difference P-SV seismograms. Geophysics 53, 1425–1436 (1988).

    Google Scholar 

  97. 97.

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

    Google Scholar 

  98. 98.

    Crase, E., Pica, A., Noble, M., McDonald, J. & Tarantola, A. Robust elastic non-linear waveform inversion: application to real data. Geophys. J. Int. 55, 527–538 (1990).

    Google Scholar 

  99. 99.

    Pratt, R. G. Inverse theory applied to multi-source cross-hole tomography. Part II: Elastic wave-equation method. Geophys. Prospecting 38, 311–330 (1990). Application and evaluation of frequency-domain FWI in exploration seismology.

    Google Scholar 

  100. 100.

    Komatitsch, D. & Vilotte, J. P. The spectral-element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull. Seism. Soc. Am. 88, 368–392 (1998).

    Google Scholar 

  101. 101.

    Komatitsch, D. & Tromp, J. Introduction to the spectral-element method for 3-D seismic wave propagation. Geophys. J. Int. 139, 806–822 (1999).

    Google Scholar 

  102. 102.

    Komatitsch, D. & Tromp, J. Spectral-element simulations of global seismic wave propagation-I. Validation. Geophys. J. Int. 149, 390–412 (2002).

    Google Scholar 

  103. 103.

    Komatitsch, D. & Tromp, J. Spectral-element simulations of global seismic wave propagation-II. 3-D models, oceans, rotation, and self-gravitation. Geophys. J. Int. 150, 303–318 (2002).

    Google Scholar 

  104. 104.

    Afanasiev, M. et al. Modular and flexible spectral-element waveform modelling in two and three dimensions. Geophys. J. Int. 216, 1675–1692 (2019).

    Google Scholar 

  105. 105.

    Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, 2005).

  106. 106.

    Métivier, L., Brossier, R., Mérigot, Q., Oudet, E. & Virieux, J. Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion. Geophys. J. Int. 205, 345–377 (2016).

    Google Scholar 

  107. 107.

    Park, J., Lindberg, C. R. & Vernon III, F. L. Multitaper spectral analysis of high-frequency seismograms. J. Geophys. Res. 92, 12675–12684 (1987).

    Google Scholar 

  108. 108.

    Laske, G. & Masters, G. Constraints on global phase velocity maps from long-period polarization data. J. Geophys. Res. 101, 16059–16075 (1996).

    Google Scholar 

  109. 109.

    Ekström, G., Tromp, J. & Larson, E. Measurements and global models of surface wave propagation. J. Geophys. Res. 102, 8137–8157 (1997).

    Google Scholar 

  110. 110.

    Fichtner, A., Kennett, B. L. N., Igel, H. & Bunge, H. P. Theoretical background for continental- and global-scale full-waveform inversion in the time-frequency domain. Geophys. J. Int. 175, 665–685 (2008).

    Google Scholar 

  111. 111.

    Bozdağ, E., Trampert, J. & Tromp, J. Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements. Geophys. J. Int. 185, 845–870 (2011).

    Google Scholar 

  112. 112.

    Yuan, Y., Simons, F. & Tromp, J. Double-difference adjoint seismic tomography. Geophys. J. Int. 206, 1599–1618 (2016).

    Google Scholar 

  113. 113.

    Shin, C. & Min, D.-J. Waveform inversion using a logarithmic wavefield. Geophysics 71, R31–R42 (2006).

    Google Scholar 

  114. 114.

    Shin, C., Pyun, S. & Bednar, J. B. Waveform inversion using a logarithmic wavefield. Geophys. Prospecting 55, 449–464 (2007).

    Google Scholar 

  115. 115.

    Shin, C. & Cha, Y. H. Waveform inversion in the Laplace domain. Geophys. J. Int. 173, 922–931 (2008).

    Google Scholar 

  116. 116.

    Shin, C. & Cha, Y. H. Waveform inversion in the Laplace–Fourier domain. Geophys. J. Int. 177, 1067–1079 (2009).

    Google Scholar 

  117. 117.

    Warner, M. & Guasch, L. Adaptive waveform inversion: theory. Geophysics 81, R429–R445 (2018).

    Google Scholar 

  118. 118.

    Ramos-Martínez, J., Qiu, L., Valenciano, A. A., Jiang, X. & Chemingui, N. Long-wavelength FWI updates in the presence of cycle skipping. Lead. Edge 38, 193–196 (2019).

    Google Scholar 

  119. 119.

    Huang, G., Nammour, R. & Symes, W. Full-waveform inversion via source-receiver extension. Geophysics 82, R153–R171 (2017).

    Google Scholar 

  120. 120.

    Biondi, B. & Almomin, A. Simultaneous inversion of full data bandwidth by tomographic full-waveform inversion. Geophysics 79, WA129–WA140 (2014).

    Google Scholar 

  121. 121.

    Engquist, B. & Froese, B. Application of the Wasserstein metric to seismic signals. Commun. Math. Science 12, 979–988 (2014).

    Google Scholar 

  122. 122.

    Yang, Y. & Engquist, B. Analysis of optimal transport and related misfit functions in full-waveform inversion. Geophysics 83, A7–A12 (2018).

    Google Scholar 

  123. 123.

    Métivier, L., Brossier, R., Mérigot, Q. & Oudet, E. A graph space optimal transport distance as a generalization of L p distances: application to a seismic imaging inverse problem. Inverse Probl. 35, 085001 (2019).

    Google Scholar 

  124. 124.

    van Leeuwen, T. & Herrmann, F. Mitigating local minima in full-waveform inversion by expanding the search space. Geophys. J. Int. 195, 661–667 (2013).

    Google Scholar 

  125. 125.

    Wang, C., Yingst, D., Farmer, P. & Leveille, J. Full-waveform inversion with the reconstructed wavefield method. Geophysics 81, 1237–1241 (2016).

    Google Scholar 

  126. 126.

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

    Google Scholar 

  127. 127.

    Komatitsch, D. et al. Anelastic sensitivity kernels with parsimonious storage for adjoint tomography and full waveform inversion. Geophys. J. Int. 206, 1467–1478 (2016).

    Google Scholar 

  128. 128.

    Akçelik, V. Multiscale Newton-Krylov Methods for Inverse Acoustic Wave Propagation. Thesis, Carnegy-Mellon Univ. (2002).

  129. 129.

    Plessix, R.-E. Three-dimensional frequency-domain full-waveform inversion with an iterative solver. Geophysics 74, WCC53–WCC61 (2009).

    Google Scholar 

  130. 130.

    Operto, S. et al. Efficient 3-D frequency-domain mono-parameter full-waveform inversion of ocean-bottom cable data: application to Valhall in the visco-acoustic vertical transverse isotropic approximation. Geophys. J. Int. 202, 1362–1391 (2015).

    Google Scholar 

  131. 131.

    Operto, S. & Miniussi, A. On the role of density and attenuation in three-dimensional multiparameter viscoacoustic VTI frequency-domain FWI: an OBC case study from the North Sea. Geophys. J. Int. 213, 2037–2059 (2018).

    Google Scholar 

  132. 132.

    Komatitsch, D., Tsuboi, S., Ji, C. & Tromp, J. A 14.6 billion degrees of freedom, 5 teraflops, 2.5 terabyte earthquake simulation on the Earth Simulator. Proc. 2003 ACM/IEEE Conf. Supercomputing 1, 4–11 (2003).

  133. 133.

    Peter, D. et al. Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes. Geophys. J. Int. 186, 721–739 (2011).

    Google Scholar 

  134. 134.

    Gunzburger, M. Perspectives in Flow Control and Optimization (SIAM, 2000).

  135. 135.

    Pratt, R. G., Shin, C. & Hicks, G. J. Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion. Geophys. J. Int. 133, 341–362 (1998).

    Google Scholar 

  136. 136.

    Akçelik, V., Biros, G. & Ghattas, O. Parallel multiscale Gauss–Newton–Krylov methods for inverse wave propagation. Proc. 2002 ACM/IEEE Conf. Supercomputing 1–15 (2002).

  137. 137.

    Akçelik, V. et al. in Proceedings of the 2003 ACM/IEEE Conference on Supercomputing 52 (2003).

  138. 138.

    Burstedde, C. & Ghattas, O. Algorithmic strategies for full waveform inversion: 1D experiments. Geophysics 74, WCC37–W3346 (2009).

    Google Scholar 

  139. 139.

    Modrak, R. & Tromp, J. Seismic waveform inversion best practices: regional, global and exploration test cases. Geophys. J. Int. 206, 1864–1889 (2016).

    Google Scholar 

  140. 140.

    Liu, D. & Nocedal, J. On the limited memory BFGS method for large scale optimization. Math. Program. 45, 504–528 (1989).

    Google Scholar 

  141. 141.

    Nash, S. & Nocedal, J. A numerical study of the limited memory BFGS method and the truncated-Newton method for large scale optimization. SIAM J. Optim. 1, 358–372 (1991).

    Google Scholar 

  142. 142.

    Zou, X. et al. Numerical experience with limited-memory quasi-Newton and truncated Newton methods. SIAM J. Optim. 3, 582–608 (1993).

    Google Scholar 

  143. 143.

    Nocedal, J. Theory of algorithms for unconstrained optimization. Acta Numerica 1, 199–242 (1992).

    Google Scholar 

  144. 144.

    Koren, Z., Mosegaard, K., Landa, E., Thore, P. & Tarantola, A. Monte Carlo estimation and resolution analysis of seismic background velocities. J. Geophys. Res. 96, 20289–20299 (1991).

    Google Scholar 

  145. 145.

    Mosegaard, K. & Tarantola, A. Monte Carlo sampling of solutions to inverse problems. J. Geophys. Res. 100, 12431–12447 (1995).

    Google Scholar 

  146. 146.

    Sambridge, M. & Mosegaard, K. Monte Carlo methods in geophysical inverse problems. Rev. Geophys. 40, 1–29 (2002).

    Google Scholar 

  147. 147.

    Fichtner, A., Zunino, A. & Gebraad, L. Hamiltonian Monte Carlo solution of tomographic inverse problems. Geophys. J. Int. 216, 1344–1363 (2019).

    Google Scholar 

  148. 148.

    Sengupta, M. & Toksöz, N. Three-dimensional model of seismic velocity variation in the Earth’s mantle. Geophys. Res. Lett. 3, 84–86 (1977).

    Google Scholar 

  149. 149.

    Claerbout, J. F. Toward a unified theory of reflector mapping. Geophysics 36, 467–481 (1971).

    Google Scholar 

  150. 150.

    Claerbout, J. & Doherty, S. Downward continuation of moveout-corrected seismograms. Geophysics 37, 741–768 (1972).

    Google Scholar 

  151. 151.

    Aki, K. & Richards, P. G. Quantitative Seismology, Theory and Methods (W. H. Freeman, 1980).

  152. 152.

    Montagner, J.-P. & Jobert, N. Vectorial tomography; II. Application to the Indian Ocean. Geophys. J. 94, 309–344 (1988).

    Google Scholar 

  153. 153.

    Marone, F. & Romanowicz, B. Non-linear crustal corrections in high-resolution regional waveform seismic tomography. Geophys. J. Int. 170, 460–467 (2007).

    Google Scholar 

  154. 154.

    Schneider, W. A. Integral formulation for migration in two and three dimensions. Geophysics 43, 49–76 (1978).

    Google Scholar 

  155. 155.

    Baysal, E., Kosloff, D. & Sherwood, J. Reverse time migration. Geophysics 48, 1514–1524 (1983).

    Google Scholar 

  156. 156.

    Hill, N. R. Gaussian beam migration. Geophysics 55, 1416–1428 (1990).

    Google Scholar 

  157. 157.

    Stolt, R. H. Migration by Fourier transform. Geophysics 43, 23–48 (1978).

    Google Scholar 

  158. 158.

    Gazdag, J. Wave equation migration with the phase-shift method. Geophysics 43, 1342–1351 (1978).

    Google Scholar 

  159. 159.

    Aki, K. Space and time spectra of stationary stochastic waves, with special reference to microtremors. Bull. Earthq. Res. Inst. 35, 415–456 (1957).

    Google Scholar 

  160. 160.

    Claerbout, J. F. Synthesis of a layered medium from its acoustic transmission response. Geophysics 33, 264–269 (1968).

    Google Scholar 

  161. 161.

    Fichtner, A. & Tsai, V. C. Theoretical foundations of noise interferometry. in Seismic Ambient Noise (eds Nakata, N., Gualtieri, L. & Fichtner, A.) 109–143 (Cambridge Univ. Press, 2019).

  162. 162.

    Sirgue, L. et al. Full waveform inversion: the next leap forward in imaging at Valhall. First Break 28, 65–70 (2010).

    Google Scholar 

  163. 163.

    Barkved, O. et al. in Expanded Abstracts, 91st Annual SEG Meeting and Exposition (October 17–22, Denver) 925–929 (Society of Exploration Geophysics, 2010).

  164. 164.

    Amestoy, P. et al. Fast 3D frequency-domain full waveform inversion with a parallel block low-rank multifrontal direct solver: application to OBC data from the North Sea. Geophysics 81, R363–R383 (2016).

    Google Scholar 

  165. 165.

    Operto, S. et al. 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 

  166. 166.

    Kurzmann, A., Przebindowska, A., Kohn, D. & Bohlen, T. Acoustic full waveform tomography in the presence of attenuation: a sensitivity analysis. Geophys. J. Int. 195, 985–1000 (2013).

    Google Scholar 

  167. 167.

    Operto, S. et al. A guided tour of multiparameter full-waveform inversion with multicomponent data: from theory to practice. Lead. Edge 32, 1040–1054 (2013).

    Google Scholar 

  168. 168.

    Luo, Y., Modrak, R. & Tromp, J. in Handbook of Geomathematics 2nd edn (eds Freeden, W., Nahed, Z. & Sonar, T.) 1–52 (Springer, 2014).

  169. 169.

    Roecker, S., Baker, B. & McLaughlin, J. A finite-difference algorithm for full waveform teleseismic tomography. Geophys. J. Int. 181, 1017–1040 (2010).

    Google Scholar 

  170. 170.

    Monteiller, V., Chevrot, S., Komatitsch, D. & Fuji, N. A hybrid method to compute short-period synthetic seismograms of teleseismic body waves in a 3-D regional model. Geophys. J. Int. 192, 230–247 (2013).

    Google Scholar 

  171. 171.

    Monteiller, V., Chevrot, S., Komatitsch, D. & Wang, Y. Three-dimensional full waveform inversion of short-period teleseismic wavefields based upon the SEM-DSM hybrid method. Geophys. J. Int. 202, 811–827 (2015).

    Google Scholar 

  172. 172.

    Tong, P., Chen, C. W., Komatitsch, D., Basini, P. & Liu, Q. High-resolution seismic array imaging based on an SEM-FK hybrid method. Geophys. J. Int. 197, 369–395 (2014).

    Google Scholar 

  173. 173.

    Tong, P. et al. A 3-D spectral-element and frequency-wave number hybrid method for high-resolution seismic array imaging. Geophys. Res. Lett. 41, 7025–7034 (2014).

    Google Scholar 

  174. 174.

    Masson, Y. & Romanowicz, B. Box tomography: Localised imaging of remote targets buried in an unknown medium, a step forward for understanding key structures in the deep Earth. Geophys. J. Int. 211, 141–163 (2017).

    Google Scholar 

  175. 175.

    Wang, Y. et al. The deep roots of the western Pyrenees revealed by full waveform inversion of teleseismic P waves. Geology 44, 475–478 (2016).

    Google Scholar 

  176. 176.

    Beller, S. et al. Lithospheric architecture of the South-Western Alps revealed by multiparameter teleseismic full-waveform inversion. Geophys. J. Int. 212, 1369–1388 (2018).

    Google Scholar 

  177. 177.

    Clouzet, P., Masson, Y. & Romanowicz, B. Box Tomography: first application to the imaging of upper-mantle shear velocity and radial anisotropy structure beneath the North American continent. Geophys. J. Int. 213, 1849–1875 (2018).

    Google Scholar 

  178. 178.

    Chevrot, S. & Sylvander, M. Maupasacq. International Federation of Digital Seismograph Networks. Dataset/Seismic Network. 10.7914/SN/XD_2017 (2017).

  179. 179.

    Polychronopoulou, K. et al. Broadband, short-period or geophone nodes? Quality assessment of passive seismic signals acquired during the Maupasacq experiment. First Break 36, 71–75 (2018).

    Google Scholar 

  180. 180.

    Fichtner, A. et al. The deep structure of the North Anatolian Fault Zone. Earth Planet. Sci. Lett. 373, 109–117 (2013).

    Google Scholar 

  181. 181.

    Colli, L., Fichtner, A. & Bunge, H.-P. Full waveform tomography of the upper mantle in the South Atlantic region: imaging a westward fluxing shallow asthenosphere? Tectonophysics 604, 26–40 (2013).

    Google Scholar 

  182. 182.

    Zhu, H. & Tromp, J. Mapping tectonic deformation in the crust and upper mantle beneath Europe and the North Atlantic Ocean. Science 341, 871–875 (2013). Continental-scale horizontal transverse isotropy FWI of the European crust and upper mantle.

    Google Scholar 

  183. 183.

    Zhu, H., Bozdağ, E., Duffy, T. & Tromp, J. Seismic attenuation beneath Europe and the North Atlantic: Implications for water in the mantle. Earth Planet. Sci. Lett. 381, 1–11 (2013).

    Google Scholar 

  184. 184.

    Zhu, H., Bozdağ, E. & Tromp, J. Seismic structure of the European upper mantle based on adjoint tomography. Geophys. J. Int. 201, 18–52 (2015).

    Google Scholar 

  185. 185.

    Rickers, F., Fichtner, A. & Trampert, J. The Iceland–Jan Mayen plume system and its impact on mantle dynamics in the North Atlantic region: Evidence from full-waveform inversion. Earth Planet. Sci. Lett. 367, 39–51 (2013).

    Google Scholar 

  186. 186.

    Fichtner, A. & Villaseñor, A. Crust and upper mantle of the western Mediterranean – Constraints from full-waveform inversion. Earth Planet. Sci. Lett. 428, 52–62 (2015).

    Google Scholar 

  187. 187.

    Çubuk Sabuncu, Y., Taymaz, T. & Fichtner, A. 3-D crustal velocity structure of western Turkey: Constraints from full-waveform tomography. Phys. Earth Planet. Inter. 270, 90–112 (2017).

    Google Scholar 

  188. 188.

    Zhu, H., Komatitsch, D. & Tromp, J. Radial anisotropy of the North American upper mantle based on adjoint tomography with USArray. Geophys. J. Int. 211, 349–377 (2017).

    Google Scholar 

  189. 189.

    Krischer, L., Fichtner, A., Boehm, C. & Igel, H. Automated large-scale full seismic waveform inversion for North America and the North Atlantic. J. Geophys. Res. 123, 5902–5928 (2018).

    Google Scholar 

  190. 190.

    Chen, M., Niu, F., Liu, Q., Tromp, J. & Zheng, X. Multiparameter adjoint tomography of the crust and upper mantle beneath East Asia: 1. Model construction and comparisons. J. Geophys. Res. 120, 1762–1786 (2015).

    Google Scholar 

  191. 191.

    Simuté, S., Steptoe, H., Cobden, L. J., Gokhberg, A. & Fichtner, A. Full-waveform inversion of the Japanese Islands region. J. Geophys. Res. 121, 3722–3741 (2016).

    Google Scholar 

  192. 192.

    Tao, K., Grand, S. & Niu, F. Seismic structure of the upper mantle beneath eastern Asia from full waveform seismic tomography. Geochem. Geophys. Geosyst. 19, 2732–2763 (2018).

    Google Scholar 

  193. 193.

    Lloyd, A. et al. Radially anisotropic seismic structure of the Antarctic upper mantle based on full-waveform adjoint tomography. Geophys. J. Int. (in the press).

  194. 194.

    Capdeville, Y., Chaljub, E. & Montagner, J. P. Coupling the spectral element method with a modal solution for elastic wave propagation in global earth models. Geophys. J. Int. 152, 34–67 (2003).

    Google Scholar 

  195. 195.

    Li, X.-D. & Romanowicz, B. Global mantle shear velocity model developed using nonlinear asymptotic coupling theory. J. Geophys. Res. 101, 22245–22272 (1996).

    Google Scholar 

  196. 196.

    French, S. W. & Romanowicz, B. Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography. Geophys. J. Int. 199, 1303–1327 (2014).

    Google Scholar 

  197. 197.

    Valentine, A. & Trampert, J. The impact of approximations and arbitrary choices on geophysical images. Geophys. J. Int. 204, 59–73 (2016).

    Google Scholar 

  198. 198.

    Fichtner, A. et al. Multi-scale full waveform inversion. Geophys. J. Int. 194, 534–556 (2013).

    Google Scholar 

  199. 199.

    Afanasiev, M. et al. Foundations for a multiscale collaborative global Earth model. Geophys. J. Int. 204, 39–58 (2016).

    Google Scholar 

  200. 200.

    Fichtner, A. et al. The collaborative seismic earth model: generation 1. Geophys. Res. Lett. 45, 4007–4016 (2019).

    Google Scholar 

  201. 201.

    Fukao, Y. & Obayashi, M. Subducted slabs stagnant above, penetrating through, and trapped below the 660 km discontinuity. J. Geophys. Res. 118, 5920–5938 (2013).

    Google Scholar 

  202. 202.

    Van der Meer, D. G., Van Hinsbergen, D. J. & Spakman, W. Atlas of the underworld: Slab remnants in the mantle, their sinking history, and a new outlook on lower mantle viscosity. Tectonophysics 723, 309–448 (2018).

    Google Scholar 

  203. 203.

    Grand, S. P. Mantle shear structure beneath the Americas and surrounding oceans. J. Geophys. Res. 99, 11591–11621 (1994).

    Google Scholar 

  204. 204.

    Fukao, Y., Obayashi, M., Inoue, H. & Nenbai, M. Subducting slabs stagnant in the mantle transition zone. J. Geophys. Res. 97, 4809–4822 (1992).

    Google Scholar 

  205. 205.

    Su, W., Woodward, R. & Dziewonski, A. Degree 12 model of shear velocity heterogeneity in the mantle. J. Geophys. Res. 99, 6945–6980 (1994).

    Google Scholar 

  206. 206.

    Ruan, Y. et al. Balancing unevenly distributed data in seismic tomography: a global adjoint tomography example. Geophys. J. Int. 219, 1225–1236 (2019).

    Google Scholar 

  207. 207.

    Pratt, R. G., Plessix, R. E. & Mulder, W. A. in 63rd EAGE Conf. Exhibition P092 (SEAGE, 2001).

  208. 208.

    Pratt, R. G., Sirgue, L., Hornby, B. & Wolfe, J. in 70th EAGE Conf. Exhibition incorporating SPE EUROPEC 2008 F020 (2008).

  209. 209.

    Gholami, Y., Brossier, R., Operto, S., Ribodetti, A. & Virieux, J. Which parametrization is suitable for acoustic VTI full waveform inversion? Geophysics 78, R81–R105 (2013).

    Google Scholar 

  210. 210.

    Smith, M. & Dahlen, F. The azimuthal dependence of Love and Rayleigh wave propagation in a slightly anisotropic medium. J. Geophys. Res. 78, 3321–3333 (1973).

    Google Scholar 

  211. 211.

    Montagner, J.-P. & Nataf, H. A simple method for inverting the azimuthal anisotropy of surface waves. J. Geophys. Res. 91, 511–520 (1986).

    Google Scholar 

  212. 212.

    Duveneck, E. & Bakker, P. M. Stable P-wave modeling for reverse-time migration in tilted TI media. Geophysics 76, S65–S75 (2011).

    Google Scholar 

  213. 213.

    Oropeza, E. & McMechan, G. A. Common-reflection-point migration velocity analysis of 2D P-wave data from TTI media. Geophysics 79, C65–C79 (2014).

    Google Scholar 

  214. 214.

    Rusmanugroho, H., Modrak, R. & Tromp, J. Anisotropic full-waveform inversion with tilt-angle recovery. Geophysics 82, R135–R151 (2017).

    Google Scholar 

  215. 215.

    Liao, Q. & McMechan, G. A. 2.5D full-wavefield viscoacoustic inversion. Geophys. Prospecting 43, 1043–1059 (1995).

    Google Scholar 

  216. 216.

    Song, Z., Williamson, P. & Pratt, G. Frequency-domain acoustic-wave modeling and inversion of crosshole data, Part 2: Inversion method, synthetic experiments and real-data results. Geophysics 60, 786–809 (1995).

    Google Scholar 

  217. 217.

    Hicks, G. J. & Pratt, R. G. Reflection waveform inversion using local descent methods: Estimating attenuation and velocity over a gas-sand deposit. Geophysics 66, 598–612 (2001).

    Google Scholar 

  218. 218.

    Prieux, V., Brossier, R., Operto, S. & Virieux, J. Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: Imaging compressional wave speed, density and attenuation. Geophys. J. Int. 194, 1640–1664 (2013).

    Google Scholar 

  219. 219.

    Yuan, Y. O., Simons, F. J. & Bozdağ, E. Multiscale adjoint waveform tomography for surface and body waves. Geophysics 80, R281–R302 (2015).

    Google Scholar 

  220. 220.

    Blom, N., Boehm, C. & Fichtner, A. Synthetic inversions for density using seismic and gravity data. Geophys. J. Int. 209, 1204–1220 (2017).

    Google Scholar 

  221. 221.

    Bernauer, M., Fichtner, A. & Igel, H. Optimal observables for multiparameter seismic tomography. Geophys. J. Int. 198, 1241–1254 (2014).

    Google Scholar 

  222. 222.

    Modrak, R. T., Borisov, D., Lefebvre, M. & Tromp, J. Seisflows – flexible waveform inversion software. Comput. Geosci. 115, 88–95 (2018).

    Google Scholar 

  223. 223.

    Balasubramanian, V. et al. in 2018 IEEE International Parallel and Distributed Processing Symposium (IPDPS) 536–545 (IEEE, 2018).

  224. 224.

    Lefebvre, M. et al. in Exascale Scientific Applications — Scalability and Performance Portability (eds Straatsma, T., Antypas, K. & Williams, T.) (CRC, 2018).

  225. 225.

    Liu, Q. et al. Hello ADIOS: the challenges and lessons of developing leadership class I/O frameworks. Concurr. Comput. Pract. Exp. 26, 1453–1473 (2014).

    Google Scholar 

  226. 226.

    Boehm, C., Hanzich, M., de la Puente, J. & Fichtner, A. Wavefield compression for adjoint methods in full-waveform inversion. Geophysics 81, R385–R397 (2016).

    Google Scholar 

  227. 227.

    Krischer, L. et al. An adaptable seismic data format. Geophys. J. Int. 207, 1003–1011 (2016).

    Google Scholar 

  228. 228.

    Maggi, A., Tape, C., Chen, M., Chao, D. & Tromp, J. An automated time-window selection algorithm for seismic tomography. Geophys. J. Int. 178, 257–281 (2009).

    Google Scholar 

  229. 229.

    Chen, Y. et al. Automated time-window selection based on machine learning for full-waveform inversion. SEG Technical Program Expanded Abstracts 1604–1609 (2017).

  230. 230.

    Rawlinson, N., Fichtner, A., Sambridge, M. & Young, M. K. Seismic tomography and the assessment of uncertainty. Adv. Geophysics 55, 1–76 (2014).

    Google Scholar 

  231. 231.

    Fichtner, A. & Trampert, J. Hessian kernels of seismic data functionals based upon adjoint techniques. Geophys. J. Int. 185, 775–798 (2011).

    Google Scholar 

  232. 232.

    Fichtner, A. & Trampert, J. Resolution analysis in full waveform inversion. Geophys. J. Int. 187, 1604–1624 (2011).

    Google Scholar 

  233. 233.

    Zhu, H., Li, S., Fomel, S., Städler, G. & Ghattas, O. A Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration. Geophysics 81, R307–R323 (2016).

    Google Scholar 

  234. 234.

    Fichtner, A. & van Leeuwen, T. Resolution analysis by random probing. J. Geophys. Res. 120, 5549–5573 (2015).

    Google Scholar 

  235. 235.

    Fang, Z., Silva, C., Kuske, R. & Herrmann, F. Uncertainty quantification for inverse problems with weak partial-differential-equation constraints. Geophysics 83, R629–R647 (2018).

    Google Scholar 

  236. 236.

    Thurin, J., Brossier, R. & Métivier, L. Ensemble-based uncertainty estimation in full waveform inversion. Geophys. J. Int. 219, 1613–1635 (2019).

    Google Scholar 

  237. 237.

    Eikrem, K. S., Nævdal, G. & Jacobsen, M. Iterated extended Kalman filter method for time-lapse seismic full-waveform inversion. Geophys. Prospecting 67, 379–394 (2019).

    Google Scholar 

  238. 238.

    Liu, Q., Peter, D. & Tape, C. Square-root variable metric based elastic full-waveform inversion – Part 1: theory and validation. Geophys. J. Int. 218, 1121–1135 (2019).

    Google Scholar 

  239. 239.

    Liu, Q. & Peter, D. Square-root variable metric based elastic full-waveform inversion – Part 2: uncertainty estimation. Geophys. J. Int. 218, 1100–1120 (2019).

    Google Scholar 

  240. 240.

    Martin, G., Wiley, R. & Marfurt, K. Marmousi2: an elastic upgrade for Marmousi. Lead. Edge 25, 156–166 (2006).

    Google Scholar 

  241. 241.

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

    Google Scholar 

  242. 242.

    Krebs, J. et al. Fast full-wavefield seismic inversion using encoded sources. Geophysics 74, WCC177–WCC188 (2009).

    Google Scholar 

  243. 243.

    Ben-Hadj-Ali, H., Operto, S. & Virieux, J. An efficient frequency-domain full waveform inversion method using simultaneous encoded sources. Geophysics 76, R109–R124 (2009).

    Google Scholar 

  244. 244.

    Choi, Y. & Alkhalifah, T. Source-independent time-domain wave-form inversion using convolved wavefields. Geophysics 76, R125–R134 (2011).

    Google Scholar 

  245. 245.

    Schuster, G., Wang, X., Huang, Y., Dai, W. & Boonyasiriwat, C. Theory of multisource crosstalk reduction by phase-encoded statics. Geophys. J. Int. 184, 1289–1303 (2011).

    Google Scholar 

  246. 246.

    Schiemenz, A. & Igel, H. Accelerated 3-D full-waveform inversion using simultaneously encoded sources in the time domain: application to Valhall ocean-bottom cable data. Geophys. J. Int. 195, 1970–1988 (2013).

    Google Scholar 

  247. 247.

    Castellanos, C., Métivier, L., Operto, S., Brossier, R. & Virieux, J. Fast full waveform inversion with source encoding and second-order optimization methods. Geophys. J. Int. 200, 718–742 (2015).

    Google Scholar 

  248. 248.

    Zhao, Z., Sen, M. & Stoffa, P. Double-plane-wave reverse time migration in the frequency domain. Geophysics 81, S367–S382 (2016).

    Google Scholar 

  249. 249.

    Romero, L., Ghiglia, D., Ober, C. & Morton, S. Phase encoding of shot records in prestack migration. Geophysics 65, 426–436 (2000).

    Google Scholar 

  250. 250.

    Krebs, J. R. et al. Orthogonal source and receiver encoding. US Patent 10,012,745) (2013).

  251. 251.

    Huang, Y. & Schuster, G. in 75th EAGE Conf. Exhibition incorporating SPE EUROPEC 2013 (2013).

  252. 252.

    Huang, Y. & Schuster, G. Full-waveform inversion with multisource frequency selection of marine streamer data. Geophys. Prospecting 66, 1243–1257 (2018).

    Google Scholar 

  253. 253.

    Zhang, Q., Mao, W., Zhou, H., Zhang, H. & Chen, Y. Hybrid-domain simultaneous-source full waveform inversion without crosstalk noise. Geophys. J. Int. 215, 1659–1681 (2018).

    Google Scholar 

  254. 254.

    Tromp, J. & Bachmann, E. Source encoding for adjoint tomography. Geophys. J. Int. 218, 2019–2044 (2019).

    Google Scholar 

  255. 255.

    Herrmann, F. J. Randomized sampling and sparsity: Getting more information from fewer samples. Geophysics 75, WB173–WB187 (2009).

    Google Scholar 

  256. 256.

    Herrmann, F. J. & Li, X. Efficient least-squares imaging with sparsity promotion and compressive sensing. Geophys. Prospecting 60, 696–712 (2012).

    Google Scholar 

  257. 257.

    Li, X., Aravkin, A. Y., van Leeuwen, T. & Herrmann, F. J. Fast randomized full-waveform inversion with compressive sensing. Geophysics 77, A13–A17 (2012).

    Google Scholar 

  258. 258.

    van Leeuwen, T. & Herrmann, F. J. Fast waveform inversion without source-encoding. Geophys. Prospecting 61, 10–19 (2013).

    Google Scholar 

  259. 259.

    Silva, C. D., Zhang, Y., Kumar, R. & Herrmann, F. J. Applications of low-rank compressed seismic data to full-waveform inversion and extended image volumes. Geophysics 84, R371–R383 (2019).

  260. 260.

    Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953).

    Google Scholar 

  261. 261.

    Hastings, W. K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970).

    Google Scholar 

  262. 262.

    Mosegaard, K. & Tarantola, A. Monte Carlo sampling of solutions to inverse problems. J. Geophys. Res. 100, 431–447 (1995).

    Google Scholar 

  263. 263.

    Wolpert, D. & Macready, W. G. No free lunch theorems for optimization. IEEE Trans. Evolut. Comput. 1, 67–82 (1997).

    Google Scholar 

  264. 264.

    Mosegaard, K. Limits to Nonlinear Inversion (Springer, 2012).

  265. 265.

    Bellman, R. E. Dynamic Programming (Rand Corporation, 1957).

  266. 266.

    Curtis, A. & Lomax, A. Prior information, sampling distributions, and the curse of dimensionality. Geophysics 66, 372–378 (2001).

    Google Scholar 

  267. 267.

    Kennedy, S. D. A. D., Pendleton, B. J. & Roweth, D. Hybrid Monte Carlo. Phys. Lett. B 195, 216–222 (1987).

    Google Scholar 

  268. 268.

    Neal, R. M. MCMC using Hamiltonian dynamics. in Handbook of Markov Chain Monte Carlo (eds Brooks, S., Gelman, A., Jones, G. & Meng, X.-L.) 113–162 (Chapman and Hall, 2011).

  269. 269.

    Betancourt, M. A conceptual introduction to Hamiltonian Monte Carlo. Preprint at arXiv (2017).

  270. 270.

    Fichtner, A. & Zunino, A. Hamiltonian nullspace shuttles. Geophys. Res. Lett. 46, 644–651 (2019).

    Google Scholar 

Download references


The author is grateful to S. Operto and S. Beller for their feedback and input on controlled-source seismology applications of full-waveform inversion and for contributing figures. W. Lei and Y. Ruan also contributed figures to this article. Comments and suggestions by the reviewers helped improve an earlier version of the manuscript. This research used the resources of the Oak Ridge Leadership Computing Facility, which is a US Department of Energy Office of Science User Facility supported under contract DE-AC05-00OR22725. Additional computational resources were provided by the Princeton Institute for Computational Science and Engineering (PICSciE).

Author information



Corresponding author

Correspondence to Jeroen Tromp.

Ethics declarations

Competing interests

The author declares no competing interests.

Additional information

Peer review information

Nature Reviews Earth & Environment thanks A. Fichtner, B. Romanowicz and the other, anonymous, reviewer for their contribution to the peer review of this work.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Related links








Body waves

Seismic waves that travel through Earth’s interior as compressional or shear waves.

Surface waves

Seismic waves that rumble along Earth’s surface in the form of a Love wave with transverse linear particle motion or a Rayleigh wave with vertical and radial retrograde elliptical particle motion.

Misfit function

A multivariate function of a set of model parameters indicative of the fit between observed and simulated data.

Mantle waves

Very-long-period (>~120 s) surface waves.

Forward simulations

Numerical modelling of seismic-wave propagation given a set of source parameters and an Earth model.

Adjoint simulations

Numerical modelling based on an Earth model and a fictitious set of sources that inject measurements simultaneously from all receivers.

Fréchet derivatives

The derivatives of a misfit function with respect to model parameters, such as seismic wave speeds or source parameters.

Banana-doughnut kernels

A finite-frequency version of an infinite-frequency seismic ray, which, in a spherical Earth model, looks like a banana in the vertical plane between the source and receiver and like a doughnut in a cross section perpendicular to this plane.

Checkboard tests

Inversion experiments in which synthetic data are generated for a checkboard model parameter pattern. These data are then inverted to assess how well the checkboard pattern can be recovered.

Marmousi model

A fictitious model created by a consortium led by the Institut Français du Pétrole. The initial model was 2D acoustic but there is an elastic version called Marmousi2.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tromp, J. Seismic wavefield imaging of Earth’s interior across scales. Nat Rev Earth Environ 1, 40–53 (2020).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing