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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 2025, 34(4), 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.
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