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A novel wavefield reconstruction inversion method using an approximated model-domain Hessian

Huaishan Liu1,2,3 ,  Yuzhao Lin3* ,  Lei Xing3 ,  Jinghao Li3 ,  Kun Huang3 ,  Hehao Tang3
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1 State Key Laboratory of Shale Oil and Gas Enrichment Mechanisms and Efficient Development, Beijing, China
2 Key Laboratory of Oil & Gas Reservoir Geophysics, Sinopec, Beijing, China
3 Department of Marine Geophysics, College of Marine Geosciences, Ocean University of China, Qingdao, Shandong, China
JSE, 025250018 https://doi.org/10.36922/JSE025250018
Submitted: 16 June 2025 | Revised: 15 August 2025 | Accepted: 3 September 2025 | Published: 3 November 2025
© 2025 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License ( https://creativecommons.org/licenses/by/4.0/ )
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Abstract

The Hessian matrix, though computationally expensive, plays a critical role in ensuring inversion accuracy and mitigating cross-talk in multi-parameter inversion. The well-known wavefield reconstruction inversion (WRI) or extended space full-waveform inversion can reduce nonlinearity and mitigate cycle skipping in traditional FWI. However, most implementations omit the Hessian. In this study, the Hessian—formulated as a function of measurement and theoretical covariance matrices—is incorporated into WRI within a Bayesian inference framework. Furthermore, the connections between the data- and model-domain Hessian equations are discussed, leading to a simplified calculation method for the extended source. Based on this approach, a new definition for the theoretical covariance matrix is proposed and validated through numerical tests, demonstrating its accuracy.

Keywords
Inversion
Bayesian inference
Theory covariance matrix
Funding
The study is supported by the Qingdao New Energy Shandong Laboratory Open Project under Grant QNESL OP202304.
Conflict of interest
Huaishan Liu is employed by the Key Laboratory of Oil & Gas Reservoir Geophysics, Sinopec, Beijing, China. The remaining authors declare they have no competing interests.
References
  1. Tarantola A, Valette B. Generalized nonlinear inverse problems solved using the least squares criterion. Rev Geophys. 1982;20(2):219‐232. doi: 10.1029/RG020i002P00219

 

  1. Tarantola A. Inversion of seismic reflection data in the acoustic approximation. Geophysics. 1984;49(8):1259‐1266. doi: 10.1190/1.1441754

 

  1. Tarantola A. Inverse Problem Theory and Methods for Model Parameter Estimation. Indian: SIAM; 2005.

 

  1. Virieux J, Operto S. An overview of full-waveform inversion in exploration geophysics. Geophysics. 2009;74(6): WCC1‐WCC26. doi: 10.1190/1.3238367

 

  1. Gauthier O, Virieux J, Tarantola A. Two-dimensional nonlinear inversion of seismic waveforms: Numerical results. Geophysics. 1986;51(7):1387‐1403. doi: 10.1190/1.1442188

 

  1. Warner M, Guasch L. Adaptive waveform inversion: Theory. Geophysics. 2016;81(6):R429‐R445. doi: 10.1190/geo2015-0387.1

 

  1. Guasch L, Warner M, Ravaut C. Adaptive waveform inversion: Practice. Geophysics. 2019;84(1):R447‐R461. doi: 10.1190/geo2018-0377.1

 

  1. Wu Z, Alkhalifah T. Selective data extension for full-waveform inversion: An efficient solution for cycle skipping. Geophysics. 2018;83(2):R201-R211. doi: 10.1190/geo2016-0649.1

 

  1. Métivier L, Brossier R, Merigot Q, Oudet E, Virieux J. An optimal transport approach for seismic tomography: Application to 3D full waveform inversion. Inverse Probl. 2016;32(11):115008. doi: 10.1088/0266-5611/32/11/115008

 

  1. Métivier L, Brossier R, Merigot Q, Oudet E, Virieux J. Measuring the misfit between seismograms using an optimal transport distance: Application to full waveform inversion. Geophys J Int. 2017;205(1):332-364. doi: 10.1093/gji/ggw027

 

  1. Pladys A, Brossier R, Irnaka M, Kamath N, Métivier L. Graph Space Optimal Transport Based FWI: 3D OBC Valhall Case Study. SEG Tech Program Expanded Abstracts. United States: Society of Exploration Geophysicists; 2020. p. 696-700.

 

  1. Lin Y, Liu H, Sun J, Xing L. Time-domain wavefield reconstruction inversion based on unbalanced optimal transport. IEEE Trans Geosci Remote Sens. 2022;60:5920612. doi: 10.1109/TGRS.2022.3210576

 

  1. Bunks C, Saleck FM, Zaleski S, Chavent G. Multiscale seismic waveform inversion. Geophysics. 1995;60(5): 1457-1473. doi: 10.1190/1.1443880

 

  1. Fu L, Guo B, Schuster GT. Multiscale phase inversion of seismic data. Geophysics. 2018;83(3):R159-R171.

 

  1. Symes WW. Objective Functions for Full Waveform Inversion. In: 74th EAGE Conference and Exhibition- Workshops; 2012.

 

  1. Fu L, Symes WW. A discrepancy-based penalty method for extended waveform inversion. Geophysics. 2017;82(4):R287-R298.

 

  1. Fu L, Symes WW. An adaptive multiscale algorithm for efficient extended waveform inversion. Geophysics. 2017;82(3):R183-R197. doi: 10.1190/geo2016-0426.1

 

  1. Huang G, Nammour R, Symes WW. Full waveform inversion via source-receiver extension. Geophysics. 2017;82(3):R153-R171.doi: 10.1190/geo2016-0301.1

 

  1. Huang G, Nammour R, Symes WW. Source-independent extended waveform inversion based on spacetime source extension: Frequency-domain implementation. Geophysics. 2018;83(5):R449-R461. doi: 10.1190/geo2017-0333.1

 

  1. Symes WW. Full Waveform Inversion by Source Extension: Why it Works. Preprint arXiv; 2020. doi: 10.48550/arXiv.2003.12538

 

  1. Leeuwen TV, Herrmann FJ. Mitigating local minima in full-waveform inversion by expanding the search space. Geophys J Int. 2013;195(1):661-667. doi: 10.1093/gji/ggt277

 

  1. Leeuwen T, Herrmann FJ. A penalty method for PDE-constrained optimization in inverse problems. Inverse Probl. 2015;32(1):015007. doi: 10.1088/0266-5611/32/1/015007

 

  1. Wang C, Yingst D, Farmer P, Leveille J. Full-Waveform Inversion with the Reconstructed Wavefield Method. In: Conference SEG Tech Program Expanded Abstracts; 2016. p. 1237-1241.

 

  1. Li ZC, Lin YZ, Zhang K, Li YY, Yu ZN. Time-domain wavefield reconstruction inversion. Appl Geophys. 2017;14(4):523-528. doi: 10.1007/s11770-017-0629-6

 

  1. Rizzuti G, Louboutin M, Wang R, Daskalakis E, Herrmann F. A dual formulation for time-domain wavefield reconstruction inversion. In: SEG Tech Program Expanded Abstracts. United States: Georgia Institute of Technology; 2019. p. 1480-1485.

 

  1. Rizzuti G, Louboutin M, Wang R, Herrmann F. A dual formulation of wavefield reconstruction inversion for large-scale seismic inversion. Geophysics. 2021;86(6):R879-R893. doi: 10.1190/geo2020-0746.1

 

  1. Aghamiry HS, Gholami A, Operto S. Improving full waveform inversion by wavefield reconstruction with the alternating direction method of multipliers. Geophysics. 2019;84(1):R139-R162. doi: 10.1190/geo2018-0093.1

 

  1. Aghamiry HS, Gholami A, Operto S. Implementing bound constraints and total-variation regularization in extended full waveform inversion with the alternating direction method of multiplier: Application to large contrast media. Geophys J Int. 2019;212(2):855-872. doi: 10.1093/gji/ggz189

 

  1. Aghamiry HS, Gholami A, Operto S. ADMM-based multi-parameter wavefield reconstruction inversion in VTI acoustic media with TV regularization. Geophys J Int. 2019;219(2):1316-1333. doi: 10.1093/gji/ggz369

 

  1. Lin Y, Liu H, Xing L, Lin H. Time-domain wavefield reconstruction inversion solutions in the weighted full waveform inversion form. IEEE Trans Geosci Remote Sens. 2022;60(1):5923514. doi: 10.1109/TGRS.2022.3224383

 

  1. Gholami A, Aghamiry HS, Operto S. Multiplier Waveform Inversion. Preprint arXiv; 2021.

 

  1. Gholami A, Aghamiry HS, Operto S. Clarifying Some Issues on Extended FWI: Scattered-Field Equation, Time Reversal and Source Reconstruction. In: Conference SEG Annual Meeting Expanded Abstracts; 2021. p. 802-806.

 

  1. Lin Y, Liu H, Xing L, Lin H. Time-domain wavefield reconstruction inversion solutions in the weighted full waveform inversion form. IEEE Trans Geosci Remote Sens. 2022;60(1):5923514. doi: 10.1109/TGRS.2022.3224383

 

  1. Operto S, Gholami A, Aghamiry HS, Guo G, Mamfoumbi F, Beller S. Full waveform inversion beyond the Born approximation: A tutorial review. Geophysics. 2023;88(6):WCC87-WC127. doi: 10.1190/geo2022-0758.1

 

  1. Stuart AM. Inverse problems: A Bayesian perspective. Acta Numer. 2010;19:451-559. doi: 10.1017/S0962492910000031

 

  1. Figueiredo LP, Grana D, Bordignon FL, Santos M, Roisenberg M, Rodrigues BB. Joint Bayesian inversion based on rock-physics prior modeling for the estimation of spatially correlated reservoir properties. Geophysics. 2018;83(1):M49-M61. doi: 10.1190/geo2017-0363.1

 

  1. Huang X, Eikrem KS, Jakobsen M, Nævdal G. Bayesian full-waveform inversion in anisotropic elastic media using the iterated extended Kalman filter. Geophysics. 2020;85(4):C125-C139. doi: 10.1190/geo2019-0329.1

 

  1. Leeuwen T. A Note on Extended Full Waveform Inversion. Preprint arXiv; 2019. doi: 10.48550/arXiv.1904.00363

 

  1. Lin Y, Leeuwen T, Liu H, Sun J, Xing L. A fast wavefield reconstruction inversion solution in the frequency domain. Geophysics. 2023;88(1):1-62. doi: 10.1190/geo2022-0385.1

 

  1. Gholami A, Aghamiry HS, Operto S. Extended Full Waveform Inversion in the Time Domain by the Augmented Lagrangian Method. arXiv Preprint; 2021. doi: 10.48550/arXiv.2011.14102

 

  1. Plessix RE. A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys J Int. 2006;167(2):495-503. doi: 10.1111/j.1365-246X.2006.02978.x
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