AccScience Publishing / JSE / Volume 35 / Issue 2 / DOI: 10.36922/JSE025500124
ARTICLE

Two-way acoustic wave-equation-based wavefield depth extrapolation for imaging in vertically transversely isotropic media

Shanzheng Hu1,2 Pengyuan Sun1,2 Shuqin Li1,2 Feng Hu1,2 Min Guan1,2 Jiachun You3*
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1 BGP Inc., China National Petroleum Corporation, Zhuozhou, Hebei, China
2 National Engineering Research Center of Oil and Gas Exploration Computer Software, China National Petroleum Corporation, Zhuozhou, Hebei, China
3 Department of Geophysics, Chengdu University of Technology, Chengdu, Sichuan, China
JSE 2026, 35(2), 025500124 https://doi.org/10.36922/JSE025500124
Received: 9 December 2025 | Revised: 7 February 2026 | Accepted: 7 February 2026 | Published online: 27 April 2026
© 2026 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/ )
Abstract

Anisotropy is a prevalent characteristic of the Earth’s subsurface, giving rise to more complex wavefield behavior than in purely acoustic media. Seismic data acquired in such anisotropic environments pose significant challenges for conventional acoustic imaging methods that rely on isotropic assumptions and acoustic approximations, often leading to inaccurate imaging of complex geologic structures. To address these challenges, we propose a wavefield depth extrapolation method grounded in the two-way acoustic anisotropic wave equation, specifically tailored for imaging in vertically transversely isotropic (VTI) media. Theoretically, the proposed method extends the two-way wave-equation-based wavefield extrapolation framework from acoustic media to anisotropic VTI media. To further improve image quality, we used a frequency–wavenumber-domain approach to remove pseudo-S-wave energy, enhancing the clarity of P-wave descriptions and reducing imaging artifacts. We evaluated the effectiveness of the proposed approach through several numerical tests, including experiments on a VTI three-layer model, the Hess VTI model, and the Marmousi VTI model. These benchmarks demonstrated that our method consistently outperformed conventional acoustic schemes designed for isotropic media, yielding clearer, higher-resolution structural images. In addition, we applied both the acoustic and the VTI-based migration methods to a real seismic dataset. The VTI-based migration provided a clearer, more continuous image of the deep structures than the acoustic migration method. The numerical experiments and real data application demonstrated that the proposed VTI imaging strategy is both practical and effective. More importantly, the findings underscore the need to explicitly account for anisotropic parameters in seismic imaging workflows to achieve geologically consistent and reliable interpretations in VTI-dominated regions.

Keywords
Vertically transversely isotropic media
Two-way acoustic wave equation
Wavefield depth extrapolation
Depth migration
Funding
This work was supported by the open research project of the National Engineering Research Center for Oil and Gas Exploration Computer Software (DFWT-ZYRJ- 2025-JS-56) and Sichuan Science and Technology Program (2026NSFSC0234, 2024NSFSC0810).
Conflict of interest
Jiachun You is a Guest Editor of this special issue, but was not in any way involved in the editorial and peer-review process conducted for this paper, directly or indirectly. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References
  1. Alkhalifah T. Acoustic approximations for processing in transversely isotropic media. Geophysics. 1998;63(2):623- 631. doi: 10.1190/1.1444361

 

  1. Alkhalifah T. An acoustic wave equation for anisotropic media. Geophysics. 2000;65(4):1239-1250. doi: 10.1190/1.1444815

 

  1. Li ZC, Qu YM. Research progress on seismic imaging technology. Pet Sci. 2022;19(1):128-146. doi: 10.1016/j.petsci.2022.01.015

 

  1. Duveneck E, Milcik P, Bakker PM, Perkins C. Acoustic VTI wave equations and their application for anisotropic reverse‐time migration. In: SEG Technical Program Expanded Abstracts 2008; Society of Exploration Geophysicists; 2008:2186-2190. doi: 10.1190/1.3059320

 

  1. Bube KP, Nemeth T, Stefani JP, et al. On the instability in second-order systems for acoustic VTI and TTI media. Geophysics. 2012;77(5):T171-T186. doi: 10.1190/geo2011-0250.1

 

  1. Zhou H, Zhang G, Bloor R. An Anisotropic Acoustic Wave Equation for VTI Media. In: Proceedings of the 68th EAGE Conference and Exhibition Incorporating SPE EUROPEC 2006; June 12-15, 2006; Vienna, Austria. European Association of Geoscientists & Engineers; 2006:cp-2-00318. doi: 10.3997/2214-4609.201402310

 

  1. Yan J, Sava P. Elastic wave-mode separation for VTI media. Geophysics. 2009;74(5):WB19-WB32. doi: 10.1190/1.3184014

 

  1. Zhang Q, McMechan GA. 2D and 3D elastic wavefield vector decomposition in the wavenumber domain for VTI media. Geophysics. 2010;75(3):D13-D26. doi: 10.1190/1.3431045

 

  1. Yao G, Fang X, Zheng Q, et al. Pseudo-Helmholtz decomposition for an elastic VTI wavefield based on wavefront phase direction. Geophysics. 2024;89(3):T151-T162. doi: 10.1190/geo2023-0574.1

 

  1. Sun S, Mao W, Zhang Q, et al. Reverse-time migration for pure qP-wave based on elliptical decomposition vector equation. IEEE Trans Geosci Remote Sens. 2024;62:1-12. doi: 10.1109/TGRS.2024.3406763

 

  1. Shan T, Yang J, Huang J, Wang W, Qin S. Least-Squares Gaussian Beam Migration in the VTI Media With a Cauchy Constrain. IEEE Trans Geosci Remote Sens. 2025;63:1-9. doi: 10.1109/tgrs.2025.3554996

 

  1. Li A, Li C, Lai F, Liao J, Liu T. An Acoustic Wave-Equation Depth Migration Method Using Generalized Two-Way Phase Shift Operator for Anisotropic Media. IEEE Geosci Remote Sens Lett. 2025;22:1-5. doi: 10.1109/lgrs.2025.3549919

 

  1. Qin S, Yang J, Wang P, Huang J, Qin N. Seismic imaging of pure quasi-P-wave in the VTI media by using the optical flow to calculate phase-velocity direction. Geophys J Int. 2025;242(3):ggaf270. doi: 10.1093/gji/ggaf270

 

  1. Zhang K, Guo LJ, Li ZC, et al. Reverse time migration of VTI media in undulating surface based on the first-order velocity-stress equation. Geophys Prospect Petr. 2026;65(1):77-87. [In Chinese]. doi: 10.12431/issn.1000-1441.2024.0135

 

  1. Fomel S, Ying L, Song X. Seismic wave extrapolation using lowrank symbol approximation. Geophys Prospect. 2013;61(3):526-536. doi: 10.1111/j.1365-2478.2012.01064.x

 

  1. Sun H, Sun J. A VTI medium prestack migration method based on the De Wolf approximation. Comput Geosci. 2025;196:105835. doi: 10.1016/j.cageo.2024.105835

 

  1. Mao Q, Huang JP, Mu XR, et al. Accurate simulations of pure-viscoacoustic wave propagation in tilted transversely isotropic media. Pet Sci. 2024;21(2):866-884. doi: 10.1016/j.petsci.2023.11.005

 

  1. Liang K, Cao D, Sun S, et al. Decoupled wave equation and forward modeling of qP wave in VTI media with the new acoustic approximation. Geophysics. 2023;88(1):WA335-WA344. doi: 10.1190/geo2022-0292.1

 

  1. Bai T, Zhu T, Tsvankin I. Attenuation compensation for time-reversal imaging in VTI media. Geophysics. 2019;84(4):C205-C216. doi: 10.1190/geo2018-0532.1

 

  1. Qu Y, Zhu J, Chen Z, et al. Q-compensated least-squares reverse time migration with velocity-anisotropy correction based on the first-order velocity-pressure equations. Geophysics. 2022;87(6):S335-S350. doi: 10.1190/geo2021-0689.1

 

  1. Wang N, Xing G, Zhu T, et al. Propagating seismic waves in VTI attenuating media using fractional viscoelastic wave equation. J Geophys Res Solid Earth. 2022;127(4):e2021JB023280. doi: 10.1029/2021jb023280.

 

  1. Wu S, Wang T, Cheng J. Second-order optimization for multiparameter reflection waveform inversion in acoustic VTI media. Geophys J Int. 2024;236(1):249-269. doi: 10.1093/gji/ggad406

 

  1. Singh S, Tsvankin I, Zabihi Naeini E. Full-waveform inversion with borehole constraints for elastic VTI media. Geophysics. 2020;85(6):R553-R563. doi: 10.1190/geo2019-0816.1

 

  1. Lang K, Yin X, Zong Z, et al. Anisotropic nonlinear inversion based on a novel PP wave reflection coefficient for VTI media. IEEE Trans Geosci Remote Sens. 2023;61:1-13. doi: 10.1109/TGRS.2023.3295800

 

  1. Wang B, Zhang F, Dai FC, et al. Study on seismic inversion method of SH-SH wave in VTI media. Chin J Geophys. 2023;66(5):2112-2122. [In Chinese]. doi: 10.6038/cjg2022P0750

 

  1. Kamath N, Tsvankin I. Elastic full-waveform inversion for VTI media: Methodology and sensitivity analysis. Geophysics. 2016;81(2):C53-C68. doi: 10.1190/geo2014-0586.1

 

  1. Li V, Tsvankin I, Alkhalifah T. Analysis of RTM extended images for VTI media. Geophysics. 2016;81(3):S139-S150. doi: 10.1190/geo2015-0384.1

 

  1. Luo C, Ba J, Carcione JM. A hierarchical prestack seismic inversion scheme for VTI media based on the exact reflection coefficient. IEEE Trans Geosci Remote Sens. 2022;60:1-16. doi: 10.1109/TGRS.2021.3140133

 

  1. Song C, Alkhalifah T, Waheed UB. Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks. Geophys J Int. 2021;225(2):846-859. doi: 10.1093/gji/ggab010

 

  1. Ristow D. Migration in transversely isotropic media using implicit finite-difference operators. J Seism Explor. 1999;8:39-56.

 

  1. Han Q, Wu RS. A one-way dual-domain propagator for scalar qP-waves in VTI media. Geophysics. 2005;70(2):D9- D17. doi: 10.1190/1.1884826

 

  1. Bale RA. Phase-Shift Migration and the Anisotropic Acoustic Wave Equation. In: Proceedings of the 69th EAGE Conference and Exhibition Incorporating SPE EUROPEC 2007; June 11-14, 2007; London, UK. European Association of Geoscientists & Engineers; 2007:cp-27-00104. doi: 10.3997/2214-4609.201401522

 

  1. Fei TW, Liner CL. Hybrid Fourier finite-difference 3D depth migration for anisotropic media. Geophysics. 2008;73(2):S27- S34. doi: 10.1190/1.2828704

 

  1. Bakker PM. A stable one-way wave propagator for VTI media. Geophysics. 2009;74(5):WB3-WB17. doi: 10.1190/1.3196818

 

  1. Amazonas D, Aleixo R, Schleicher J, et al. Anisotropic complex Padé hybrid finite-difference depth migration. Geophysics. 2010;75(2):S51-S59. doi: 10.1190/1.3337317

 

  1. Salcedo M, Novais A, Schleicher J, et al. Optimization of the parameters in complex Padé Fourier finite-difference migration. Geophysics. 2017;82(3):S259-S269. doi: 10.1190/geo2016-0324.1

 

  1. Liu LN, Zhang JF. Wave equation prestack depth migration method in 3D VTI media. Chin J Geophys. 2011;54(6):844- 855. doi: 10.1002/cjg2.1667

 

  1. Vyas M, Wang X, Etgen J. One-way TTI propagation: Is it still relevant? In: SEG Technical Program Expanded Abstracts 2015; Society of Exploration Geophysicists; 2015:3986-3990. doi: 10.1190/segam2015-5915894.1

 

  1. Alshuhail AA, Verschuur DJ. Robust estimation of vertical symmetry axis models via joint migration inversion: Including multiples in anisotropic parameter estimation. Geophysics. 2019;84(1):C57-C74. doi: 10.1190/geo2017-0856.1

 

  1. Wu B, Wu RS, Gao J. Preliminary Investigation of Wavefield Depth Extrapolation by Two‐Way Wave Equations. Int J Geophys. 2012;2012:968090. doi: 10.1155/2012/968090

 

  1. You J, Wu RS, Liu X, et al. Two-way wave equation-based depth migration using one-way propagators on a bilayer sensor seismic acquisition system. Geophysics. 2018;83(3):S271-S278. doi: 10.1190/geo2017-0172.1

 

  1. You J, Pan N, Liu W, et al. Efficient wavefield separation by reformulation of two-way wave-equation depth-extrapolation scheme. Geophysics. 2022;87(4):S209-S222. doi: 10.1190/geo2021-0629.1

 

  1. You JC, Zhang GL, Huang XG, et al. Internal-multiple-elimination with application to migration using two-way wave equation depth-extrapolation scheme. Pet Sci. 2025;22(1):178-192. doi: 10.1016/j.petsci.2024.06.021

 

  1. Li A, Zhu X, Liu T, et al. One-step sparse-matrix method for full wave equation depth migration. Geophysics. 2024;89(6):S405-S413. doi: 10.1190/geo2023-0372.1

 

  1. Sandberg K, Beylkin G. Full-wave-equation depth extrapolation for migration. Geophysics. 2009;74(6):WCA121-WCA128. doi: 10.1190/1.3202535

 

  1. You J, Cao J. Full-wave-equation depth extrapolation for migration using matrix multiplication. Geophysics. 2020;85(6):S395-S403. doi: 10.1190/geo2019-0323.1

 

  1. You J, Sun R, Huang X, Huang J. DeepImaging: Deep neural networks driven full waveform migration. IEEE Trans Geosci Remote Sens. 2026;64:5900719. doi: 10.1109/TGRS.2025.3648341

 

  1. Sun R, You J, Li Z, Hu H. Viscoacoustic Full Wavefield Migration and Its Application. IEEE Trans Geosci Remote Sens. 2025;63:5914517. doi: 10.1109/TGRS.2025.3574991

 

  1. Alkhalifah T. Traveltime computation with the linearized eikonal equation for anisotropic Media. Geophys Prospect. 2002;50(4):373–382. doi: 10.1046/j.1365-2478.2002.00322.x

 

  1. Liu YZ, Wang GY, Dong LG, et al. Joint inversion of VTI parameters using nonlinear traveltime tomography. Chin J Geophys. 2014;57(10):3402-3410. [In Chinese]. doi: 10.6038/cjg20141026
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