1887
Volume 73, Issue 9
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

Vertically transverse isotropic (VTI) acoustic wave equations are widely used to simulate wave propagation in VTI media. A commonly used acoustic VTI wave equation can be derived by setting the vertical shear‐wave velocity to zero in the elastic VTI wave equation. However, the resulting acoustic VTI wave equation has a non‐symmetric propagation operator, which leads to the operator of the adjoint equation in full‐waveform inversion (FWI) being different from that of the forward equation. Consequently, two separate sets of code are required for simulating the forward and adjoint wavefields. To simplify code implementation, we propose a symmetric‐form acoustic VTI equation. This new formulation allows both the forward and adjoint equations in FWI to share the same operator, enabling a unified code for both the forward and adjoint wavefield simulation and streamlined implementation. In addition, although both the symmetric and non‐symmetric formulations yield the same gradient, the adjoint wavefield from the non‐symmetric equation shows weaker amplitudes in deeper regions compared to that from the symmetric equation. As a result, FWI using the non‐symmetric formulation may suffer from insufficient compensation when employing a spatial preconditioner based on an approximated diagonal pseudo‐Hessian, leading to slower convergence. Numerical examples using the Marmousi2 and BP anisotropic models, as well as an ocean‐bottom cable (OBC) field data set, demonstrate that the proposed symmetric‐form acoustic VTI FWI achieves better inversion results and faster convergence than its non‐symmetric counterpart.

© 2025 European Association of Geoscientists & Engineers. © 2025 European Association of Geoscientists & Engineers.
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2025-12-16
2026-01-19

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References

  1. Alkhalifah, T.1998. “Acoustic Approximations for Processing in Transversely Isotropic Media.” Geophysics63, no. 2: 623–631. https://doi.org/10.1190/1.1444361.
     [Google Scholar]
  2. Alkhalifah, T.2000. “An Acoustic Wave Equation for Anisotropic Media.” Geophysics65, no. 4: 1239–1250. https://doi.org/10.1190/1.1444815.
     [Google Scholar]
  3. Alkhalifah, T., and R. É.Plessix. 2014. “A Recipe for Practical Full‐Waveform Inversion in Anisotropic Media: An Analytical Parameter Resolution Study.” Geophysics79, no. 3: R91–R101. https://doi.org/10.1190/geo2013‐0366.1.
     [Google Scholar]
  4. Bishop, T. N., K. P.Bube, R. T.Cutler, et al. 1985. “Tomographic Determination of Velocity and Depth in Laterally Varying Media.” Geophysics50, no. 6: 903–923. https://doi.org/10.1190/1.1441970.
     [Google Scholar]
  5. Chen, K., L.Liu, L. N.Xu, et al. 2024. “Linearized Waveform Inversion for Vertical Transversely Isotropic Elastic Media: Methodology and Multi‐Parameter Crosstalk Analysis.” Petroleum Science21, no. 1: 252–271. https://doi.org/10.1016/j.petsci.2024.01.002.
     [Google Scholar]
  6. Cheng, X., J.Xu, D.Vigh, and M.Hegazy. 2023. “Enhanced Template Matching Full‐Waveform Inversion.” In Proceedings of the 84th EAGE Annual Conference & Exhibition. European Association of Geoscientists & Engineers. https://doi.org/10.3997/2214‐4609.202310765.
  7. da Silva, N. V., G.Yao, and M.Warner. 2019. “Semiglobal Viscoacoustic Full‐Waveform Inversion.” Geophysics84, no. 2: R271–R293. https://doi.org/10.1190/geo2017‐0773.1.
     [Google Scholar]
  8. Ding, P. B., F.Gong, F.Zhang, and X. Y.Li. 2021. “A Physical Model Study of Shale Seismic Responses and Anisotropic Inversion.” Petroleum Science18, no. 4: 1059–1068. https://doi.org/10.1016/j.petsci.2021.01.001.
     [Google Scholar]
  9. Duveneck, E., P.Milcik, P. M.Bakker, and C.Perkins. 2008. “Acoustic VTI Wave Equations and Their Application for Anisotropic Reverse‐Time Migration.” In SEG Technical Program Expanded Abstracts. SEG. https://doi.org/10.1190/1.3059320.
  10. Duveneck, E., and P. M.Bakker. 2011. “Stable P‐Wave Modeling for Reverse‐Time Migration in Tilted TI Media.” Geophysics76, no. 2: S65–S75. https://doi.org/10.1190/1.3533964.
     [Google Scholar]
  11. Fletcher, R. P., X.Du, and P. J.Fowler. 2009. “Reverse Time Migration in Tilted Transversely Isotropic (TTI) Media.” Geophysics74, no. 6: WCA179–WCA187. https://doi.org/10.1190/1.3269902.
     [Google Scholar]
  12. Fu, L. Y.2022. “Interpretative Seismic Imaging With the Wavenumber‐Structure Monitoring of Velocity Models.” IEEE Transactions on Geoscience and Remote Sensing60: 1–14. https://doi.org/10.1109/TGRS.2022.3225454.
     [Google Scholar]
  13. Gardner, G. H. F., L. W.Gardner, and A.Gregory. 1974. “Formation Velocity and Density—The Diagnostic Basics for Stratigraphic Traps.” Geophysics39, no. 6: 770–780. https://doi.org/10.1190/1.1440465.
     [Google Scholar]
  14. Green, D. H., and H. F.Wang. 1986. “Fluid Pressure Response to Undrained Compression in Saturated Sedimentary Rock.” Geophysics51, no. 4: 948–956. https://doi.org/10.1190/1.1442152.
     [Google Scholar]
  15. Hadden, S., R.Gerhard Pratt, and B.Smithyman. 2019. “Anisotropic Full‐Waveform Inversion of Crosshole Seismic Data: A Vertical Symmetry Axis Field Data Application.” Geophysics84, no. 1: B15–B32. https://doi.org/10.1190/geo2017‐0790.1.
     [Google Scholar]
  16. He, W., R. E.Plessix, and S.Singh. 2018. “Parametrization Study of the Land Multiparameter VTI Elastic Waveform Inversion.” Geophysical Journal International213, no. 3: 1660–1672. https://doi.org/10.1093/gji/ggy099.
     [Google Scholar]
  17. Huang, J., Q.Mao, X.Mu, et al. 2024. “Least‐Squares Reverse Time Migration Based on an Efficient Pure qP‐Wave Equation.” Geophysical Prospecting72, no. 4: 1290–1311. https://doi.org/10.1111/1365‐2478.13326.
     [Google Scholar]
  18. Kamath, N., and I.Tsvankin. 2016. “Elastic Full‐Waveform Inversion for VTI Media: Methodology and Sensitivity Analysis.” Geophysics81, no. 2: C53–C68. https://doi.org/10.1190/geo2014‐0586.1.
     [Google Scholar]
  19. Kamath, N., I.Tsvankin, and E.Díaz. 2017. “Elastic Full‐Waveform Inversion for VTI Media: A Synthetic Parameterization Study.” Geophysics82, no. 5: C163–C174. https://doi.org/10.1190/geo2016‐0375.1.
     [Google Scholar]
  20. Lailly, P., and J.Bednar. 1983. “The Seismic Inverse Problem as a Sequence of Before Stack Migrations.” Presented at the Conference on Inverse Scattering: Theory and Application.
  21. Lambaré, G.2008. “Stereotomography.” Geophysics73, no. 5: VE25–VE34. https://doi.org/10.1190/1.2952039.
     [Google Scholar]
  22. Li, J., K.Xin, and F. S.Dzulkefli. 2022. “A New qP‐Wave Approximation in Tilted Transversely Isotropic Media and Its Reverse Time Migration for Areas With Complex Overburdens.” Geophysics87, no. 4: S237–S248. https://doi.org/10.1190/geo2021‐0433.1.
     [Google Scholar]
  23. Liu, F., S. A.Morton, S.Jiang, L.Ni, and J. P.Leveille. 2009. “Decoupled Wave Equations for P and SV Waves in an Acoustic VTI Media.” In SEG International Exposition and Annual Meeting. Society of Exploration Geophysicists.
  24. Liu, Y., and M. K.Sen. 2009. “Advanced Finite‐Difference Methods for Seismic Modeling.” Geohorizons14, no. 2: 5–16.
     [Google Scholar]
  25. Mora, P.1987. “Nonlinear Two‐Dimensional Elastic Inversion of Multioffset Seismic Data.” Geophysics52, no. 9: 1211–1228. https://doi.org/10.1190/1.1442384.
     [Google Scholar]
  26. Pan, W., K. A.Innanen, Y.Geng, and J.Li. 2019. “Interparameter Trade‐Off Quantification for Isotropic‐Elastic Full‐Waveform Inversion With Various Model Parameterizations.” Geophysics84, no. 2: R185–R206. https://doi.org/10.1190/geo2017‐0832.1.
     [Google Scholar]
  27. Pan, W., K. A.Innanen, and Y.Wang. 2020. “Parameterization Analysis and Field Validation of VTI‐Elastic Full‐Waveform Inversion in a Walk‐Away Vertical Seismic Profile Configuration.” Geophysics85, no. 3: B87–B107. https://doi.org/10.1190/geo2019‐0089.1.
     [Google Scholar]
  28. Plessix, R. E.2006. “A Review of the Adjoint‐State Method for Computing the Gradient of a Functional With Geophysical Applications.” Geophysical Journal International167, no. 2: 495–503. https://doi.org/10.1111/j.1365‐246X.2006.02978.x.
     [Google Scholar]
  29. Plessix, R. E., and Q.Cao. 2011. “A Parametrization Study for Surface Seismic Full Waveform Inversion in an Acoustic Vertical Transversely Isotropic Medium.” Geophysical Journal International185, no. 1: 539–556. https://doi.org/10.1111/j.1365‐246X.2011.04957.x.
     [Google Scholar]
  30. Pratt, R. G.1999. “Seismic Waveform Inversion in the Frequency Domain, Part 1: Theory and Verification in a Physical Scale Model.” Geophysics64, no. 3: 888–901. https://doi.org/10.1190/1.1444597.
     [Google Scholar]
  31. Pratt, R. G., and R. M.Shipp. 1999. “Seismic Waveform Inversion in the Frequency Domain, Part 2: Fault Delineation in Sediments Using Crosshole Data.” Geophysics64, no. 3: 902–914. https://doi.org/10.1190/1.1444598.
     [Google Scholar]
  32. Prieux, V., R.Brossier, Y.Gholami, et al. 2011. “On the Footprint of Anisotropy on Isotropic Full Waveform Inversion: The Valhall Case Study.” Geophysical Journal International187, no. 3: 1495–1515. https://doi.org/10.1111/j.1365‐246X.2011.05209.x.
     [Google Scholar]
  33. Prieux, V., R.Brossier, S.Operto, and J.Virieux. 2013. “Multiparameter Full Waveform Inversion of Multicomponent Ocean‐Bottom‐Cable Data From the Valhall Field. Part 1: Imaging Compressional Wave Speed, Density and Attenuation.” Geophysical Journal International194, no. 3: 1640–1664. https://doi.org/10.1093/gji/ggt177.
     [Google Scholar]
  34. Ren, Z., X.Dai, and L.Wang. 2024. “Full‐Waveform Inversion of Pure Quasi‐P Waves in Titled Transversely Isotropic Media Based on Different Parameterizations.” Geophysics89, no. 2: C57–C73. https://doi.org/10.1190/geo2023‐0367.1.
     [Google Scholar]
  35. Ren, Z., and Y.Liu. 2016. “A Hierarchical Elastic Full‐Waveform Inversion Scheme Based on Wavefield Separation and the Multistep‐Length Approach.” Geophysics81, no. 3: R99–R123. https://doi.org/10.1190/geo2015‐0431.1.
     [Google Scholar]
  36. Robertsson, J. O. A.1996. “A Numerical Free‐Surface Condition for Elastic/Viscoelastic Finite‐Difference Modeling in the Presence of Topography.” Geophysics61, no. 6: 1921–1934. https://doi.org/10.1190/1.1444107.
     [Google Scholar]
  37. Shen, X., I.Ahmed, A.Brenders, J.Dellinger, J.Etgen, and S.Michell. 2017. “Full‐Waveform Inversion: The Next Leap Forward in Subsalt Imaging.” Leading Edge37, no. 1: 67b61–67b66. https://doi.org/10.1190/tle37010067b1.1.
     [Google Scholar]
  38. Shipp, R. M., and S. C.Singh. 2002. “Two‐Dimensional Full Wavefield Inversion of Wide‐Aperture Marine Seismic Streamer Data.” Geophysical Journal International151, no. 2: 325–344. https://doi.org/10.1046/j.1365‐246X.2002.01645.x.
     [Google Scholar]
  39. Shin, C., S.Jang, and D. J.Min. 2001. “Improved Amplitude Preservation for Prestack Depth Migration by Inverse Scattering Theory.” Geophysical Prospecting49, no. 5: 592–606. https://doi.org/10.1046/j.1365‐2478.2001.00279.x.
     [Google Scholar]
  40. Singh, S., I.Tsvankin, and E. Z.Naeini. 2021. “Facies‐Based Full‐Waveform Inversion for Anisotropic Media: A North Sea Case Study.” Geophysical Prospecting69, no. 8–9: 1650–1663. https://doi.org/10.1111/1365‐2478.13139.
     [Google Scholar]
  41. Sirgue, L., O. I.Barkved, J. P.Van Gestel, O. J.Askim, and J. H.Kommedal. 2009. “3D Waveform Inversion on Valhall Wide‐Azimuth OBC.” In 71st EAGE Conference and Exhibition Incorporating SPE EUROPEC 2009 (cp‐127). European Association of Geoscientists & Engineers.
  42. Stockwell, J. W.2025. SeisUnix. GitHub. https://github.com/JohnWStockwellJr/SeisUnix.
     [Google Scholar]
  43. Tarantola, A.1984. “Inversion of Seismic Reflection Data in the Acoustic Approximation.” Geophysics49, no. 8: 1259–1266. https://doi.org/10.1190/1.1441754.
     [Google Scholar]
  44. Tavakoli, F. B., S.Operto, A.Ribodetti, and J.Virieux. 2019. “Matrix‐Free Anisotropic Slope Tomography: Theory and Application.” Geophysics84, no. 1: R35–R57. https://doi.org/10.1190/geo2018‐0112.1.
     [Google Scholar]
  45. Thomsen, L.1986. “Weak Elastic Anisotropy.” Geophysics51, no. 10: 1954–1966. https://doi.org/10.1190/1.1442051.
     [Google Scholar]
  46. Virieux, J., and S.Operto. 2009. “An Overview of Full‐Waveform Inversion in Exploration Geophysics.” Geophysics74, no. 6: WCC1–WCC26. https://doi.org/10.1190/1.3238367.
     [Google Scholar]
  47. Wang, N., G.Xing, T.Zhu, H.Zhou, and Y.Shi. 2022. “Propagating Seismic Waves in VTI Attenuating Media Using Fractional Viscoelastic Wave Equation.” Journal of Geophysical Research: Solid Earth127, no. 4: e2021JB023280. https://doi.org/10.1029/2021JB023280.
     [Google Scholar]
  48. Wang, T. F., and J. B.Cheng. 2017. “Elastic Full Waveform Inversion Based on Mode Decomposition: the Approach and Mechanism.” Geophysical Journal International209, no. 2: 606–622. https://doi.org/10.1093/gji/ggx038.
     [Google Scholar]
  49. Warner, M., A.Ratcliffe, T.Nangoo, et al. 2013. “Anisotropic 3D Full‐Waveform Inversion.” Geophysics78, no. 2: R59–R80. https://doi.org/10.1190/geo2012‐0338.1.
     [Google Scholar]
  50. wavetomo . 2025. wavetomo/VTIAcousticModeling: VTIAcousticModelingTestV1.0 (Modeling). Zenodo. https://doi.org/10.5281/zenodo.16749741.
  51. Winner, V., P.Edme, and H.Maurer. 2022. “Model‐Based Optimization of Source Locations for 3D Acoustic Seismic Full‐Waveform Inversion.” Geophysical Prospecting71, no. 1: 3–16. https://doi.org/10.1111/1365‐2478.13264.
     [Google Scholar]
  52. Xie, C., Z. L.Qin, J. H.Wang, et al. 2024. “Full Waveform Inversion Based on Hybrid Gradient.” Petroleum Science21, no. 3: 1660–1670. https://doi.org/10.1016/j.petsci.2024.01.013.
     [Google Scholar]
  53. Xu, S., and H.Zhou. 2014. “Accurate Simulations of Pure Quasi‐P‐Waves in Complex Anisotropic Media.” Geophysics79, no. 6: T341–T348. https://doi.org/10.1190/geo2014‐0242.1.
     [Google Scholar]
  54. Yao, G., N.da Silva, and D.Wu. 2018. “An Effective Absorbing Layer for the Boundary Condition in Acoustic Seismic Wave Simulation.” Journal of Geophysics and Engineering15, no. 2: 495–511. https://doi.org/10.1088/1742‐2140/aaa4da.
     [Google Scholar]
  55. Yao, G., N. V.da Silva, M.Warner, and T.Kalinicheva. 2018. “Separation of Migration and Tomography Modes of Full‐Waveform Inversion in the Plane Wave Domain.” Journal of Geophysical Research: Solid Earth123, no. 2: 1486–1501. https://doi.org/10.1002/2017JB015207.
     [Google Scholar]
  56. Yao, G., N. V.da Silva, M.Warner, D.Wu, and C.Yang. 2019. “Tackling Cycle Skipping in Full‐Waveform Inversion With Intermediate Data.” Geophysics84, no. 3: R411–R427. https://doi.org/10.1190/geo2018‐0096.1.
     [Google Scholar]
  57. Yao, G., B.Wu, P.Zhang, N.Liu, and D.Wu. 2024. “FWI With a Symmetric‐Form Acoustic VTI Wave Equation” [Data Set]. Zenodo. https://doi.org/10.5281/zenodo.14581171.
  58. Zhang, F., and X. Y.Li. 2020. “Inversion of the Reflected SV‐Wave for Density and S‐Wave Velocity Structures.” Geophysical Journal International221, no. 3: 1635–1639. https://doi.org/10.1093/gji/ggaa096.
     [Google Scholar]
  59. Zhang, P., and G.Yao. 2024. “Numerical Simulation of Seismic Wavefield Based on Modified EAL Boundary Condition.” Chinese Journal of Geophysics (In Chinese)67, no. 1: 261–276. https://doi.org/10.6038/cjg2023Q0746.
     [Google Scholar]
  60. Zhang, P., G.Yao, Q.Zheng, X.Fang, and D.Wu. 2024. “An Improved Pure Quasi‐P‐Wave Equation for Complex Anisotropic Media.” Journal of Geophysics and Engineering21, no. 2: 517–533. https://doi.org/10.1093/jge/gxae020.
     [Google Scholar]
  61. Zhang, Y., H.Zhang, and G.Zhang. 2011. “A Stable TTI Reverse Time Migration and Its Implementation.” Geophysics76, no. 3: WA3–WA11. https://doi.org/10.1190/1.3554411.
     [Google Scholar]
  62. Zhang, Z. D., and T.Alkhalifah. 2019. “Local‐Crosscorrelation Elastic Full‐Waveform Inversion.” Geophysics84, no. 6: R897–R908. https://doi.org/10.1190/geo2018‐0660.1.
     [Google Scholar]
  63. Zhang, Z., J.Mei, F.Lin, R.Huang, and P.Wang. 2018. “Correcting for Salt Misinterpretation With Full‐Waveform Inversion.” In SEG Technical Program Expanded Abstracts. Society of Exploration Geophysicists. https://doi.org/10.1190/segam2018‐2997711.1.
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Full‐Waveform Inversion With a Symmetric‐Form Acoustic VTI Wave Equation
Geophysical Prospecting 73, e70116 (2025); https://doi.org/10.1111/1365-2478.70116
/content/journals/10.1111/1365-2478.70116
/content/journals/10.1111/1365-2478.70116
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  • Article Type: Research Article
Keyword(s): full‐waveform inversion; Lagrange multiplier; matrix formulation; symmetric acoustic vertically transverse isotropic (VTI) equation

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