none 10.1190/geo2022-0604.1 Society of Exploration Geophysicists Society of Exploration Geophysicists 186 151520838 5981 2025121211490472100 10.1190 2025-12-12T16:49:17Z 2023-07-21T12:38:28Z 22 Geophysics 0016-8033 1942-2156 11 01 2023 88 6 Xingguo Huang Jilin University 1 , College of Instrumentation and Electrical Engineering, Changchun, China. xingguohuang@jlu.edu.cn (corresponding author) http://orcid.org/0000-0001-9719-6297 ABSTRACT Imaging earth’s structures is fundamental to the earth’s interior, yet how to reconstruct earth’s heterogeneities using full-waveform inversion of full wavefield data that includes primary reflections and multiples remains enigmatic. I develop a nonlinear Bayesian approach for full-waveform inversion method with multiple scattering. Instead of using single scattering Born approximation to formulate the sensitivity kernels, I develop multiple scattering sensitivity kernels using multiple scattering-based Green’s functions. It is based on the Lippmann-Schwinger integral and Marchenko methods, for which the Green’s functions are retrieved from reflection data by solving a Marchenko equation. To estimate the uncertainty of velocities, I apply a Bayesian framework to the inverse problem. Our results indicate that the method does not depend on the earth’s scattering potential and the full-waveform inversion method with multiple scattering is a good alternative approach to image the earth’s structures. 09 22 2023 11 01 2023 07 21 2023 T289 T303 1 10.1190/crossmark-policy library.seg.org true 2022-09-29 2023-05-28 2023-07-16 2023-09-22 10.1190/geo2022-0604.1 https://pubs.geoscienceworld.org/geophysics/article/88/6/T289/628277/Full-wavefield-inversion-with-multiples-Nonlinear https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/88/6/T289/6009019/geo-2022-0604.1.pdf https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/88/6/T289/6009019/geo-2022-0604.1.pdf https://pubs.geoscienceworld.org/geophysics/article/88/6/t289/628277/full-wavefield-inversion-with-multiples-nonlinear Quantitative seismology Aki 2002 0018-9286 IEEE Transactions on Automatic Control Bell 38 294 1993 10.1109/9.250476 The iterated Kalman filter update as a Gauss-Newton method Journal of Geophysical Research: Solid Earth Brackenhoff 124 11802 2019 10.1029/2019JB018485 Virtual sources and receivers in the real Earth: Considerations for practical applications 0143-0807 European Journal of Physics Broggini 33 593 2012 10.1088/0143-0807/33/3/593 Connection of scattering principles: A visual and mathematical tour 0016-8033 Geophysics Broggini 77 5 A25 2012 10.1190/geo2012-0060.1 Focusing the wavefield inside an unknown 1D medium: Beyond seismic interferometry 0016-8033 Geophysics Brossier 74 6 WCC105 2009 10.1190/1.3215771 Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion 0165-2125 Wave Motion Burridge 2 305 1980 10.1016/0165-2125(80)90011-6 The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems Integral equation methods for electromagnetic and elastic waves Chew 1 2008 Journal of Geophysical Research: Solid Earth Coen 86 6052 1981 10.1029/JB086iB07p06052 The inverse problem of the shear modulus and density profiles of a layered earth Inverse acoustic and electromagnetic scattering theory Colton 1998 10.1007/978-3-662-03537-5 10.5194/egusphere-egu2020-10804 Cummings D. Curtis A. , 2020, Defining the Green’s function derivative for imaging & inversion without the Born approximation: EGU General Assembly 2020. 10.1190/segam2020-3411883.1 Diekmann L. Vasconcelos I. , 2020, Imaging with the exact linearised Lippmann Schwinger integral by means of redatumed in-volume wavefields: 90th Annual International Meeting, SEG, Expanded Abstracts, 3598–3602, doi: http://dx.doi.org/10.1190/segam2020-3411883.1. Eikrem K. S. Jakobsen M. Nævdal G. , 2017, Bayesian inversion of time-lapse seismic waveform data using an integral equation method: 19th European Symposium on Improved Oil Recovery. 0016-8025 Geophysical Prospecting Eikrem 67 379 2018 10.1111/1365-2478.12730 Iterated extended Kalman filter method for time-lapse seismic full-waveform inversion Journal of Geophysical Research: Solid Earth Fichtner 123 2984 2018 10.1002/2017JB015249 Hamiltonian Monte Carlo inversion of seismic sources in complex media 0956-540X Geophysical Journal International Fichtner 185 775 2011 10.1111/j.1365-246X.2011.04966.x Hessian kernels of seismic data functionals based upon adjoint techniques 0956-540X Geophysical Journal International Fichtner 187 1604 2011 10.1111/j.1365-246X.2011.05218.x Resolution analysis in full waveform inversion 0956-540X Geophysical Journal International French 199 1303 2014 10.1093/gji/ggu334 Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography Multiple scattering in solids Gonis 1999 Journal of Geophysical Research: Solid Earth Gouveia 103 2759 1998 10.1029/97JB02933 Bayesian seismic waveform inversion: Parameter estimation and uncertainty analysis The Leading Edge Huang 40 324 2021 10.1190/tle40050324.1 Full-waveform inversion for full-wavefield imaging: Decades in the making 0016-8033 Geophysics Huang 85 4 C125 2020 Bayesian full-waveform inversion in anisotropic elastic media using the iterated extended Kalman filter 1815-2406 Communications in Computational Physics Huang 28 249 2020 10.4208/cicp.OA-2018-0143 Target-oriented inversion of time-lapse seismic waveform data Journal of Geophysics and Engineering Huang 17 277 2020 10.1093/jge/gxz105 On the applicability of a renormalized Born series for seismic wavefield modelling in strongly scattering media 1815-2406 Communications in Computational Physics Jakobsen 4 1 2020 Transition operator approach to seismic full-waveform inversion in arbitrary anisotropic elastic media Journal of Geophysics and Engineering Jakobsen 12 400 2015 10.1088/1742-2132/12/3/400 Full waveform inversion in the frequency domain using direct iterative T-matrix methods 0956-540X Geophysical Journal International Jakobsen 206 880 2016 10.1093/gji/ggw169 Renormalized scattering series for frequency-domain waveform modelling of strong velocity contrasts 0016-8025 Geophysical Prospecting Jakobsen 66 1039 2018 10.1111/1365-2478.12619 Accelerating the T-matrix approach to seismic full-waveform inversion by domain decomposition 10.1190/segam2018-2992001.1 Jakobsen M. Wu R.-S. Huang X. , 2018, Seismic waveform modeling in strongly scattering media using renormalization group theory: 88th Annual International Meeting, SEG, Expanded Abstracts, 5007–5011, doi: http://dx.doi.org/10.1190/segam2018-2992001.1. 0021-9991 Journal of Computational Physics Jakobsen 409 2020 10.1016/j.jcp.2020.109343 Convergent scattering series solution of the inhomogeneous Helmholtz equation via renormalization group and homotopy continuation approaches 0016-8033 Geophysics Jia 83 5 S409 2018 10.1190/geo2017-0646.1 A practical implementation of subsalt Marchenko imaging with a Gulf of Mexico data set 0031-899X Physical Review Jost 87 1952 10.1103/PhysRev.87.977 Construction of a potential from a phase shift 1539-3755 Physical Review E Kouri 67 2003 10.1103/PhysRevE.67.046614 Inverse scattering theory: Renormalization of the Lippmann-Schwinger equation for acoustic scattering in one dimension Computers & Mathematics with Applications Kwon 59 3377 2010 10.1016/j.camwa.2009.07.088 Born expansion and Frechet derivatives in nonlinear diffuse optical tomography The seismic inverse problem as a sequence of before stack migrations Lailly 206 1983 0016-8033 Geophysics Liu 76 5 WB209 2011 10.1190/geo2010-0312.1 Reverse time migration of multiples for subsalt imaging 0016-8033 Geophysics Lomas 84 2 F35 2019 10.1190/geo2018-0068.1 An introduction to Marchenko methods for imaging methods 0021-9991 Journal of Computational Physics Malovichko 346 318 2017 10.1016/j.jcp.2017.06.021 Approximate solutions of acoustic 3D integral equation and their application to seismic modeling and full-waveform inversion Doklady Akademii Nauk SSSR Marchenko 104 695 1955 On reconstruction of the potential energy from phases of the scattered waves 0016-8025 Geophysical Prospecting Mcgillivray 38 499 1990 10.1111/j.1365-2478.1990.tb01859.x Methods for calculating Frechet derivatives and sensitivities for the non-linear inverse problem: A comparative study 1 0002-9505 American Journal of Physics Morse 22 410 1954 10.1119/1.1933765 Methods of theoretical physics 0956-540X Geophysical Journal International Newton 65 191 1981 10.1111/j.1365-246X.1981.tb02708.x Inversion of reflection data for layered media: A review of exact methods Scattering theory of waves and particles Newton 2013 0021-9991 Journal of Computational Physics Osnabrugge 322 113 2016 10.1016/j.jcp.2016.06.034 A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media 0956-540X Geophysical Journal International Panning 190 463 2012 10.1111/j.1365-246X.2012.05489.x Non-linear 3-D Born shear waveform tomography in Southeast Asia 0956-540X Geophysical Journal International Panning 177 161 2009 10.1111/j.1365-246X.2008.04050.x Seismic waveform modelling in a 3-D earth using the Born approximation: Potential shortcomings and a remedy Scattering, two-volume set: Scattering and inverse scattering in pure and applied science Pike 2001 0956-540X Geophysical Journal International Pratt 133 341 1998 10.1046/j.1365-246X.1998.00498.x Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion Geophysical Supplements to the Monthly Notices of the Royal Astronomical Society Ravasi 205 99 2016 10.1093/gji/ggv528 Target-oriented Marchenko imaging of a North Sea field Advances in geophysics Rawlinson 1 2014 0956-540X Geophysical Journal International Ray 212 522 2017 10.1093/gji/ggx428 Low frequency full waveform seismic inversion within a tree based Bayesian framework The Journal of the Acoustical Society of America Razavy 58 956 1975 10.1121/1.380756 Determination of the wave velocity in an inhomogeneous medium from the reflection coefficient 0956-540X Geophysical Journal International Romanowicz 175 520 2008 10.1111/j.1365-246X.2008.03914.x On the computation of long period seismograms in a 3-D earth using normal mode based approximations Physical Review A Rose 65 2001 10.1103/PhysRevA.65.012707 “Single-sided” focusing of the time-dependent Schrodinger equation 0266-5611 Inverse Problems Rose 18 2002 10.1088/0266-5611/18/6/329 ‘Single-sided’ autofocusing of sound in layered materials First Break Sirgue 28 65 2010 10.3997/1365-2397.2010012 Thematic set: Full waveform inversion: The next leap forward in imaging at Valhall Skoglund M. A. Hendeby G. Axehill D. , 2015, Extended Kalman filter modifications based on an optimization view point: 18th International Conference on Information Fusion (Fusion), IEEE, 1856–1861. Inverse problem theory and methods for model parameter estimation Tarantola 2005 10.1137/1.9780898717921 8755-1209 Reviews of Geophysics Tarantola 20 219 1982 10.1029/RG020i002p00219 Generalized nonlinear inverse problems solved using the least squares criterion Scattering theory: The quantum theory of non-relativistic collisions Taylor 2006 0010-3616 Communications in Mathematical Physics Teschl 196 175 1998 10.1007/s002200050419 Trace formulas and inverse spectral theory for Jacobi operators 0016-8033 Geophysics Thorbecke 82 6 WB29 2017 10.1190/geo2017-0108.1 Implementation of the Marchenko method 10.1190/segam2017-17733053.1 Thurin J. Brossier R. Metivier L. , 2017, An ensemble-transform Kalman filter: Full waveform inversion scheme for uncertainty estimation: 87th Annual International Meeting, SEG, Expanded Abstracts, 1307–1313, doi: http://dx.doi.org/10.1190/segam2017-17733053.1. 10.1190/segam2016-13944013.1 Tveit S. Mannseth T. Jakobsen M. , 2016, Discriminating time-lapse saturation and pressure changes in CO2 monitoring from seismic waveform and CSEM data using ensemble-based Bayesian inversion: 86th Annual International Meeting, SEG, Expanded Abstracts, 5485–5489, doi: http://dx.doi.org/10.1190/segam2016-13944013.1. 10.3997/2214-4609.201412939 van der Neut J. Thorbecke J. Wapenaar K. Slob E. , 2015a, Inversion of the multidimensional Marchenko equation: 77th Annual International Conference and Exhibition, EAGE, Extended Abstracts, doi: http://dx.doi.org/10.3997/2214-4609.201412939. 0956-540X Geophysical Journal International van der Neut 203 792 2015 10.1093/gji/ggv330 On Green’s function retrieval by iterative substitution of the coupled Marchenko equations 0016-8025 Geophysical Prospecting van Leeuwen 61 10 2013 10.1111/j.1365-2478.2012.01096.x Fast waveform inversion without source-encoding 0016-8033 Geophysics Verschuur 57 1166 1992 10.1190/1.1443330 Adaptive surface-related multiple elimination 0016-8033 Geophysics Virieux 74 6 WCC1 2009 10.1190/1.3238367 An overview of full-waveform inversion in exploration geophysics 0016-8033 Geophysics Wapenaar 86 5 WC117 2021 10.1190/geo2020-0854.1 Marchenko redatuming, imaging, and multiple elimination and their mutual relations 0031-9007 Physical Review Letters Wapenaar 110 2013 10.1103/PhysRevLett.110.084301 Three-dimensional single-sided Marchenko inverse scattering, data-driven focusing, Green’s function retrieval, and their mutual relations 0016-8033 Geophysics Wapenaar 79 3 WA39 2014 10.1190/geo2013-0302.1 Marchenko imaging The Journal of the Acoustical Society of America Ware 45 911 1969 10.1121/1.1911568 Continuous and discrete inverse-scattering problems in a stratified elastic medium. I. Plane waves at normal incidence 0016-8033 Geophysics Warner 78 2 R59 2013 10.1190/geo2012-0338.1 Anisotropic 3D full-waveform inversion 0266-5611 Inverse Problems Weglein 19 2003 10.1088/0266-5611/19/6/R01 Inverse scattering series and seismic exploration 0016-8033 Geophysics Weglein 62 1975 1997 10.1190/1.1444298 An inverse-scattering series method for attenuating multiples in seismic reflection data 0016-8033 Geophysics Weglein 74 6 WCD1 2009 10.1190/1.3256286 Clarifying the underlying and fundamental meaning of the approximate linear inversion of seismic data The Leading Edge Wong 33 970 2014 10.1190/tle33090970.1 Imaging with multiples using least-squares reverse time migration 10.1190/segam2016-13962956.1 Wu R.-S. Luo J. Chen G. , 2016, Seismic envelope inversion and renormalization group theory: Nonlinear scale separation and slow dynamics: 86th Annual International Meeting, SEG, Expanded Abstracts, 1346–1351, doi: http://dx.doi.org/10.1190/segam2016-13962956.1. 0266-5611 Inverse Problems Wu 31 2015 10.1088/0266-5611/31/11/115004 Renormalized nonlinear sensitivity kernel and inverse thin-slab propagator in T-matrix formalism for wave-equation tomography 0956-540X Geophysical Journal International Wu 196 1827 2014 10.1093/gji/ggt496 Non-linear partial derivative and its De Wolf approximation for non-linear seismic inversion The Leading Edge Yang 34 780 2015 10.1190/tle34070780.1 Reverse time migration of multiples: Applications and challenges 0016-8033 Geophysics Zhang 74 6 WCD29 2009 10.1190/1.3256283 Direct nonlinear inversion of 1D acoustic media using inverse scattering subseries 10.1190/segam2018-2997711.1 Zhang Z. Mei J. Lin F. Huang R. Wang P. , 2018, Correcting for salt misinterpretation with full-waveform inversion: 88th Annual International Meeting, SEG, Expanded Abstracts, 1143–1147, doi: http://dx.doi.org/10.1190/segam2018-2997711.1.