no. 1
A review of the use of optimal transport distances for high resolution seismic imaging based on the full waveform
Ludovic Métivier1 ; Romain Brossier2 ; Félix Kpadonou3 ; Jérémie Messud3 ; Arnaud Pladys2
1 Laboratoire Jean Kuntzmann, Université Grenoble Alpes, France
2 ISTerre, Université Grenoble Alpes, France
3 CGG, Massy, France
MathematicS In Action, Tome 11 (2022) no. 1, pp. 3-42.

This study is a review of recent applications to seismic imaging of optimal transport based numerical tools. Modern seismic imaging methods used in the industry rely on the interpretation of the full signal. The characterization of the subsurface mechanical properties is formulated as a PDE-constrained optimization problem, solved through local optimization strategies. The choice of the misfit function used to measure the distance between actual seismic recordings and those synthetized by the solution of wave propagation PDE is crucial. Indeed, the conventional least-squares distance function leads to a non-convex optimization problem whose solution through local optimization then strongly depends on the initial guess. Using an optimal transport distance is an interesting alternative from its convexity properties with respect to translation and dilation. Specific strategies need however to be implemented as seismic data are oscillatory, while the optimal transport theory has been developed for the comparison of positive measures. In this study we review two optimal transport based misfit functions, from their mathematical formulation to their application to field data through their numerical implementation. Advantages and drawbacks of both strategies are discussed. Numerical experiments show that they represent two interesting and complementary alternative to the classical least-squares misfit function, mitigating the dependency to the choice of the initial guess.

Publié le :
DOI : 10.5802/msia.15
Classification : 35R30, 86A22, 86A15
Mots clés : Optimal transport, convexity, optimization, seismic imaging
Ludovic Métivier 1 ; Romain Brossier 2 ; Félix Kpadonou 3 ; Jérémie Messud 3 ; Arnaud Pladys 2

1 Laboratoire Jean Kuntzmann, Université Grenoble Alpes, France
2 ISTerre, Université Grenoble Alpes, France
3 CGG, Massy, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Ludovic Métivier; Romain Brossier; Félix Kpadonou; Jérémie Messud; Arnaud Pladys. A review of the use of optimal transport distances for high resolution seismic imaging based on the full waveform. MathematicS In Action, Tome 11 (2022) no. 1, pp. 3-42. doi : 10.5802/msia.15. https://msia.centre-mersenne.org/articles/10.5802/msia.15/

[1] H. Aghamiry; A. Gholami; S. Operto Improving full-waveform inversion by wavefield reconstruction with alternating direction method of multipliers, Geophysics, Volume 84 (2019) no. 1, p. R139-R162 | DOI

[2] M. Akgül A genuinely polynomial primal simplex algorithm for the assignment problem, Discrete Appl. Math., Volume 45 (1993) no. 2, pp. 93-115 | DOI | MR | Zbl

[3] K. Aki; P. Richards Quantitative Seismology: Theory and Methods, W. H. Freeman and Co, 1980

[4] L. Ambrosio Lecture Notes on Optimal Transport Problems, Mathematical Aspects of Evolving Interfaces (Lecture Notes in Mathematics), Volume 1812, Springer, 2003, pp. 1-52 | DOI | MR | Zbl

[5] L. Ambrosio; E. Mainini; S. Serfaty Gradient flow of the Chapman Rubinstein Schatzman model for signed vortices, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 2, pp. 217-246 | Numdam | MR | Zbl

[6] M. L. Balinski Signature Methods for the Assignment Problem, Oper. Res., Volume 33 (1985) no. 3, pp. 527-536 | DOI | MR | Zbl

[7] O. I. Barkved; A. G. Bærheim; D. J. Howe; J. H. Kommedal; G. Nicol Life of Field Seismic Implementation - Another “First at Valhal”, 65th EAGE Workshop, Stavanger (2003)

[8] D. P. Bertsekas Network Optimization: Continuous and Discrete Models, Athena Scientific, 1998

[9] D. P. Bertsekas; D. A. Castanon The auction algorithm for the transportation problem, Ann. Oper. Res., Volume 20 (1989) no. 1, pp. 67-96 | DOI | MR | Zbl

[10] N. Bleistein On the imaging of reflectors in the Earth, Geophysics, Volume 52 (1987) no. 7, pp. 931-942 | DOI

[11] V. I. Bogachev Measure Theory. Vol. I and II, Springer, 2007 | DOI

[12] E. Bozdağ; D. Peter; M. Lefebvre; D. Komatitsch; J. Tromp; J. Hill; N. Podhorszki; D. Pugmire Global adjoint tomography: first-generation model, Geophys. J. Int., Volume 207 (2016) no. 3, pp. 1739-1766 | DOI

[13] E. Bozdağ; J. Trampert; J. Tromp Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements, Geophys. J. Int., Volume 185 (2011) no. 2, pp. 845-870 | DOI

[14] A. Brandt Multi-level adaptive solutions to boundary-value problems, Math. Comput., Volume 31 (1977), pp. 333-390 | DOI | MR | Zbl

[15] F. Bretaudeau; R. Brossier; D. Leparoux; O. Abraham; J. Virieux 2D elastic full waveform imaging of the near surface: Application to synthetic and a physical modelling data sets, Near Surface Geophysics, Volume 11 (2013), pp. 307-316 | DOI

[16] R. Brossier; S. Operto; J. Virieux Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion, Geophysics, Volume 74 (2009) no. 6, p. WCC105-WCC118 | DOI

[17] R. Brossier; S. Operto; J. Virieux Velocity model building from seismic reflection data by full waveform inversion, Geophysical Prospecting, Volume 63 (2015), pp. 354-367 | DOI

[18] C. Bunks; F. M. Salek; S. Zaleski; G. Chavent Multiscale seismic waveform inversion, Geophysics, Volume 60 (1995) no. 5, pp. 1457-1473 | DOI

[19] R. Burkard; M. Dell’Amico; S. Martello Assignment Problems, Society for Industrial and Applied Mathematics, 2012 | DOI

[20] D. Carotti; O. Hermant; S. Masclet; M. Reinier; J. Messud; A. Sedova; G. Lambaré Optimal Transport Full-Waveform Inversion - Applications, 82nd EAGE Conference and Exhibition, Expanded Abstracts, European Association of Geoscientists and Engineers, 2020, p. Th Dome1 17

[21] J. F. Claerbout Imaging the Earth’s interior, Blackwell Scientific Publication, 1985

[22] P. L. Combettes; J-C. Pesquet Proximal Splitting Methods in Signal Processing, Fixed-Point Algorithms for Inverse Problems in Science and Engineering (H. H. Bauschke; R. S. Burachik; L. ttes P. Combe; V. Elser; D. R. Luke; H. Wolkowicz, eds.) (Springer Optimization and Its Applications), Volume 49, Springer, 2011, pp. 185-212 | DOI | MR | Zbl

[23] J. Delon Movie and video scale-time equalization application to flicker reduction, IEEE Trans. Image Process., Volume 15 (2006) no. 1, pp. 241-248 | DOI

[24] A. Dominitz; A. Tannenbaum Texture Mapping via Optimal Mass Transport, IEEE Trans. Visual. Comput. Graph., Volume 16 (2010) no. 3, pp. 419-433 | DOI

[25] B. Engquist; B. D. Froese Application of the Wasserstein metric to seismic signals, Commun. Math. Sci., Volume 12 (2014) no. 5, pp. 979-988 | DOI | MR | Zbl

[26] B. Engquist; B. D. Froese; Y. Yang. Optimal transport for seismic full waveform inversion, Commun. Math. Sci., Volume 14 (2016) no. 8, pp. 2309-2330 | DOI | MR | Zbl

[27] A. Fichtner; B. L. N. Kennett; H. Igel; H.-P. Bunge Theoretical background for continental- and global-scale full-waveform inversion in the time-frequency domain, Geophys. J. Int., Volume 175 (2008), pp. 665-685 | DOI

[28] A. Fichtner; B. L. N. Kennett; H. Igel; H.-P. Bunge Full waveform tomography for radially anisotropic structure: New insights into present and past states of the Australasian upper mantle, Earth and Planetary Science Letters, Volume 290 (2010) no. 3-4, pp. 270-280 | DOI

[29] O. Gauthier; J. Virieux; A. Tarantola Two-dimensional nonlinear inversion of seismic waveforms: numerical results, Geophysics, Volume 51 (1986) no. 7, pp. 1387-1403 | DOI

[30] L. Groos; M. Schäfer; T. Forbriger; T. Bohlen The role of attenuation in 2D full-waveform inversion of shallow-seismic body and Rayleigh waves, Geophysics, Volume 79 (2014) no. 6, p. R247-R261 | DOI

[31] O. Hermant; A. Aziz; S. Warzocha; M. Al Jahdhami Imaging Complex Fault Structures On-shore Oman Using Optimal Transport Full-Waveform Inversion, 82nd EAGE Conference and Exhibition, Expanded Abstracts, European Association of Geoscientists and Engineers, 2020, p. We Dome1 19

[32] O. Hermant; A. Sedova; G. Royle; M. Retailleau; J. Messud; G. Lambaré; S. Al Abri; M. Al Jahdhami Broadband FAZ land data: an opportunity for FWI, 81st EAGE Conference and Exhibition, Workshop Programme, European Association of Geoscientists and Engineers, 2019, p. WS08 11

[33] G. Huang; R. Nammour; W. W. Symes; M. Dolliazal Waveform inversion via source extension (2019), pp. 4761-4766 | DOI

[34] T. M. Irnaka; R. Brossier; L. Métivier; T. Bohlen; Y. Pan Towards 3D 9C Elastic Full Waveform Inversion of Shallow Seismic Wavefields - Case Study Ettlingen Line, 81st EAGE Conference and Exhibition, Expanded Abstracts, European Association of Geoscientists and Engineers, 2019, p. We P01 04 | DOI

[35] M. Jannane; W. Beydoun; E. Crase; D. Cao; Z. Koren; E. Landa; M. Mendes; A. Pica; M. Noble; G. Roeth; S. Singh; R. Snieder; A. Tarantola; D. Trezeguet Wavelengths of Earth structures that can be resolved from seismic reflection data, Geophysics, Volume 54 (1989) no. 7, pp. 906-910 | DOI

[36] N. Kamath; R. Brossier; L. Métivier; A. Pladys; P. Yang Multiparameter full-waveform inversion of 3D ocean-bottom cable data from the Valhall field, Geophysics, Volume 86 (2021) no. 1, p. B15-B35 | DOI

[37] L. Kantorovich On the transfer of masses, Dokl. Akad. Nauk SSSR, Volume 37 (1942), pp. 7-8

[38] F. Kpadonou; J. Messud; A. Sedova; M. Reinier Optimal transport FWI with graph transform: Analysis and proposal of a partial shift strategy, 82nd EAGE Annual Conference and Exhibition, European Association of Geoscientists & Engineers, 2021 | DOI

[39] H. W. Kuhn The Hungarian method for the assignment problem, Nav. Res. Logist. Q., Volume 2 (1955) no. 1-2, pp. 83-97 | DOI | MR | Zbl

[40] P. Lailly The seismic inverse problem as a sequence of before stack migrations, Conference on Inverse Scattering, Theory and application, Society for Industrial and Applied Mathematics, Philadelphia (1983), pp. 206-220 | MR

[41] G. Lambaré Stereotomography, Geophysics, Volume 73 (2008) no. 5, p. VE25-VE34 | DOI

[42] J. Lellmann; D. A. Lorenz; C. Schönlieb; T. Valkonen Imaging with Kantorovich–Rubinstein Discrepancy, SIAM J. Imaging Sci., Volume 7 (2014) no. 4, pp. 2833-2859 | DOI | MR | Zbl

[43] S. Luo; P. Sava A deconvolution-based objective function for wave-equation inversion, SEG Technical Program Expanded Abstracts 2011, Society of Exploration Geophysicists, 2011 | DOI

[44] Y. Luo; G. T. Schuster Wave-equation traveltime inversion, Geophysics, Volume 56 (1991) no. 5, pp. 645-653 | DOI

[45] E. Mainini A description of transport cost for signed measures, J. Math. Sci., New York, Volume 181 (2012) no. 6, pp. 837-855 | DOI | MR | Zbl

[46] G. S. Martin; R. Wiley; K. J. Marfurt Marmousi2: An elastic upgrade for Marmousi, The Leading Edge, Volume 25 (2006) no. 2, pp. 156-166 | DOI

[47] J. Messud; R. Poncet; G. Lambaré Optimal transport in full-waveform inversion: Analysis and practice of the multidimensional Kantorovich-Rubinstein norm, Inverse Probl., Volume 37 (2021) no. 065012, pp. 1-42 | MR | Zbl

[48] J. Messud; A. Sedova Multidimensional Optimal Transport for 3D FWI: Demonstration on Field Data, 81st EAGE Conference and Exhibition, Expanded Abstracts, European Association of Geoscientists and Engineers, 2019, p. Tu R08 02

[49] L. Métivier; A. Allain; R. Brossier; Q. Mérigot; E. Oudet; J. Virieux On the Use of Optimal Transport Distances for a PDE-Constrained Optimization Problem in Seismic Imaging, Frontiers in PDE-Constrained Optimization (Harbir Antil; Drew P. Kouri; Martin-D. Lacasse; Denis Ridzal, eds.), Springer, 2018, pp. 377-397 | DOI | Zbl

[50] L. Métivier; A. Allain; R. Brossier; Q. Mérigot; E. Oudet; J. Virieux Optimal transport for mitigating cycle skipping in full waveform inversion: a graph space transform approach, Geophysics, Volume 83 (2018) no. 5, p. R515-R540 | DOI

[51] L. Métivier; R. Brossier New insights on the graph space optimal transport distance for full waveform inversion, SEG Technical Program Expanded Abstracts 2021 (2021)

[52] L. Métivier; R. Brossier Receiver-extension strategy for time-domain full-waveform inversion using a relocalization approach, Geophysics, Volume 87 (2022) no. 1, p. R13-R33 | DOI

[53] L. Métivier; R. Brossier; Q. Mérigot; E. Oudet A graph space optimal transport distance as a generalization of L p distances: application to a seismic imaging inverse problem, Inverse Probl., Volume 35 (2019) no. 8, 085001, 49 pages | DOI | MR | Zbl

[54] L. Métivier; R. Brossier; Q. Mérigot; E. Oudet; J. Virieux Increasing the robustness and applicability of full waveform inversion: an optimal transport distance strategy, The Leading Edge, Volume 35 (2016) no. 12, pp. 1060-1067 | DOI | Zbl

[55] L. Métivier; R. Brossier; Q. Mérigot; E. Oudet; J. Virieux Measuring the misfit between seismograms using an optimal transport distance: Application to full waveform inversion, Geophys. J. Int., Volume 205 (2016), pp. 345-377 | DOI

[56] L. Métivier; R. Brossier; Q. Mérigot; E. Oudet; J. Virieux An optimal transport approach for seismic tomography: Application to 3D full waveform inversion, Inverse Probl., Volume 32 (2016) no. 11, p. 115008 | DOI | MR | Zbl

[57] G. Monge Mémoire sur la théorie des déblais et des remblais, Histoire de l’Académie Royale des Sciences de Paris (1781)

[58] J. Nocedal Updating Quasi-Newton Matrices With Limited Storage, Math. Comput., Volume 35 (1980) no. 151, pp. 773-782 | DOI | MR | Zbl

[59] J. Nocedal; S. J. Wright Numerical Optimization, Springer, 2006

[60] G. Nolet A Breviary of Seismic Tomography, Cambridge University Press, 2008 | DOI

[61] S. Operto; R. Brossier; Y. Gholami; L. Métivier; V. Prieux; A. Ribodetti; J. Virieux A guided tour of multiparameter full waveform inversion for multicomponent data: from theory to practice, The Leading Edge, Volume 32 (2013) no. 9, pp. 1040-1054 (Special section Full Waveform Inversion) | DOI

[62] S. Operto; A. Miniussi; R. Brossier; L. Combe; L. Métivier; V. Monteiller; A. Ribodetti; J. Virieux 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., Volume 202 (2015) no. 2, pp. 1362-1391 | DOI

[63] F. Pitié; A. C. Kokaram; R. Dahyot Automated colour grading using colour distribution transfer, Computer Vision and Image Understanding, Volume 107 (2007) no. 1, pp. 123-137 (Special issue on color image processing) | DOI

[64] A. Pladys; R. Brossier; N. Kamath; L. Métivier Robust FWI with graph space optimal transport: application to 3D OBC Valhall data, Geophysics, Volume 87 (2022) no. 3, pp. 1-76

[65] A. Pladys; R. Brossier; Y. Li; L. Métivier On cycle-skipping and misfit function modification for full-wave inversion: Comparison of five recent approaches, Geophysics, Volume 86 (2021) no. 4, p. R563-R587 | DOI

[66] R.-E. Plessix A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophys. J. Int., Volume 167 (2006) no. 2, pp. 495-503 | DOI

[67] R.-E. Plessix; C. Perkins Full waveform inversion of a deep water ocean bottom seismometer dataset, First Break, Volume 28 (2010), pp. 71-78

[68] R. Poncet; J. Messud; M. Bader; G. Lambaré; G. Viguier; C. Hidalgo FWI with Optimal Transport: A 3D Implementation and an Application on a Field Dataset, 80th EAGE Conference and Exhibition, European Association of Geoscientists and Engineers, 2018, p. We A12 02

[69] A. Pratelli On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 43 (2007) no. 1, pp. 1-13 | DOI | Numdam | MR | Zbl

[70] V. Prieux; R. Brossier; Y. Gholami; S. Operto; J. Virieux; O. I. Barkved; J. H. Kommedal On the footprint of anisotropy on isotropic full waveform inversion: the Valhall case study, Geophys. J. Int., Volume 187 (2011), pp. 1495-1515 | DOI

[71] V. Prieux; R. Brossier; S. Operto; J. Virieux Multiparameter full waveform inversion of multicomponent OBC data from Valhall. Part 1: imaging compressional wavespeed, density and attenuation, Geophys. J. Int., Volume 194 (2013) no. 3, pp. 1640-1664 | DOI

[72] G. Provenzano; R. Brossier; L. Métivier; Y. Li Joint FWI of diving and reflected waves using a graph space optimal transport distance: Synthetic tests on limited-offset surface seismic data (2020), pp. 780-784 | DOI

[73] L. Qiu; J. Ramos-Martìnez; A. Valenciano; Y. Yang; B. Engquist Full-waveform inversion with an exponentially encoded optimal-transport norm, SEG Technical Program Expanded Abstracts 2017 (2017), pp. 1286-1290 | DOI

[74] J. Rabin; G. Peyré; L. D. Cohen Geodesic Shape Retrieval via Optimal Mass Transport, Computer Vision – ECCV 2010 (2010), pp. 771-784 | DOI

[75] J. Rabin; G. Peyré; J. Delon; M. Bernot Wasserstein Barycenter and Its Application to Texture Mixing, Scale Space and Variational Methods in Computer Vision (2012), pp. 435-446 | DOI

[76] Y. Rubner; C. Tomasi; L. J. Guibas The Earth Mover’s Distance as a Metric for Image Retrieval, Int. J. Comput. Vision, Volume 40 (2000) no. 2, pp. 99-121 | DOI | Zbl

[77] F. Santambrogio Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and their Applications, Springer, 2015 | DOI

[78] M. Schäfer; L. Groos; T. Forbriger; T. Bohlen 2D Full Waveform Inversion of Recorded Shallow Seismic Rayleigh Waves on a Significantly 2D Structure, Proceedings of 19th European Meeting of Environmental and Engineering Geophysics, Expanded Abstracts, Bochum, Germany (2013)

[79] A. Sedova; J. Messud; H. Prigent; S. Masclet; G. Royle; G. Lambaré Acoustic Land Full-Waveform Inversion on a Broadband Land Dataset: The Impact of Optimal Transport, 81st EAGE Conference and Exhibition, Expanded Abstracts, European Association of Geoscientists and Engineers, 2019, p. Th R08 07

[80] R. M. Shipp; S. C. Singh Two-dimensional full wavefield inversion of wide-aperture marine seismic streamer data, Geophys. J. Int., Volume 151 (2002), pp. 325-344 | DOI

[81] L. Sirgue; O. I. Barkved; J. Dellinger; J. Etgen; U. Albertin; J. H. Kommedal Full waveform inversion: the next leap forward in imaging at Valhall, First Break, Volume 28 (2010), pp. 65-70 | DOI

[82] A. Stopin; R.-E. Plessix; S. Al Abri Multiparameter waveform inversion of a large wide-azimuth low-frequency land data set in Oman, Geophysics, Volume 79 (2014) no. 3, p. WA69-WA77 | DOI

[83] P. N. Swarztrauber A Direct Method for the Discrete Solution of Separable Elliptic Equations, SIAM J. Numer. Anal., Volume 11 (1974) no. 6, pp. 1136-1150 | DOI | MR | Zbl

[84] W. W. Symes Migration velocity analysis and waveform inversion, Geophysical Prospecting, Volume 56 (2008), pp. 765-790 | DOI

[85] C. Tape; Q. Liu; A. Maggi; J. Tromp Seismic tomography of the southern California crust based on spectral-element and adjoint methods, Geophys. J. Int., Volume 180 (2010), pp. 433-462 | DOI

[86] A. Tarantola Inversion of seismic reflection data in the acoustic approximation, Geophysics, Volume 49 (1984) no. 8, pp. 1259-1266 | DOI

[87] T. van Leeuwen; F. J. Herrmann Mitigating local minima in full-waveform inversion by expanding the search space, Geophys. J. Int., Volume 195 (2013) no. 1, pp. 661-667 | DOI

[88] T. van Leeuwen; F. J. Herrmann A penalty method for PDE-constrained optimization in inverse problems, Inverse Probl., Volume 32 (2016) no. 1, 015007, 26 pages | MR | Zbl

[89] T. van Leeuwen; W. A. Mulder A correlation-based misfit criterion for wave-equation traveltime tomography, Geophys. J. Int., Volume 182 (2010) no. 3, pp. 1383-1394 | DOI

[90] C. Villani Topics in optimal transportation, Graduate Studies in Mathematics, 50, American Mathematical Society, 2003

[91] C. Villani Optimal transport: old and new, Grundlehren der Mathematischen Wissenschaften, Springer, 2008

[92] J. Virieux; A. Asnaashari; R. Brossier; L. Métivier; A. Ribodetti; W. Zhou An introduction to Full Waveform Inversion, Encyclopedia of Exploration Geophysics, Society of Exploration Geophysicists, 2017, p. R1-1–R1-40 | DOI

[93] Y. Wang; Y. Rao Reflection seismic waveform tomography, J. Geophys. Res., Volume 114 (2009) no. B3, pp. 1978-2012

[94] M. Warner; L. Guasch Adaptive waveform inversion: Theory, Geophysics, Volume 81 (2016) no. 6, p. R429-R445 | DOI

[95] R.-S. Wu; J. Luo; B. Wu Seismic envelope inversion and modulation signal model, Geophysics, Volume 79 (2014) no. 3, p. WA13-WA24

[96] S. Xu; D. Wang; F. Chen; G. Lambaré; Y. Zhang Inversion on Reflected Seismic Wave, 2012 | DOI

[97] P. Yang; R. Brossier; L. Métivier; J. Virieux; W. Zhou A Time-Domain Preconditioned Truncated Newton Approach to Multiparameter Visco-acoustic Full Waveform Inversion, SIAM J. Sci. Comput., Volume 40 (2018) no. 4, p. B1101-B1130 | DOI | Zbl

[98] Y. Yang; B. Engquist Analysis of optimal transport and related misfit functions in full-waveform inversion, Geophysics, Volume 83 (2018) no. 1, p. A7-A12 | DOI

[99] Y. Yang; B. Engquist Model recovery below reflectors by optimal-transport FWI (2018), pp. 1178-1182 | DOI

[100] Y. Yang; B. Engquist Improving optimal transport based FWI through data normalization (2019), pp. 1315-1319 | DOI

[101] Y. Yang; B. Engquist; J. Sun; B. F. Hamfeldt Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion, Geophysics, Volume 83 (2018) no. 1, p. R43-R62 | DOI

[102] W. Zhou; R. Brossier; S. Operto; J. Virieux Full Waveform Inversion of Diving and Reflected Waves for Velocity Model Building with Impedance Inversion Based on Scale Separation, Geophys. J. Int., Volume 202 (2015) no. 3, pp. 1535-1554 | DOI

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