1887
Volume 73, Issue 6
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

Accurate simulation of seismic waves is essential for achieving high‐precision full‐waveform inversion (FWI). Within the Cartesian coordinate system‐based frequency‐domain finite‐difference (FDFD) framework, we propose a one‐direction composition average‐derivative optimal method for the 3D heterogeneous isotropic elastic‐wave equation, referred to as the 45‐point scheme. The results of dispersion analysis and weighted coefficient optimization demonstrate that the 45‐point scheme achieves higher dispersion accuracy than the existing 27‐point average‐derivative scheme. More importantly, by constructing the impedance matrix along the ‘composition’ direction, the bandwidth of the sparse impedance matrix increases only slightly, with nonzero elements compactly distributed in strips. On the basis of the multifrontal massively parallel sparse direct solver (MUMPS) on a supercomputer platform, the 45‐point scheme does not significantly increase computational complexity compared to the 27‐point scheme. To further test the performance of the 45‐point scheme, we provide several numerical experiments, including simple homogeneous and complex SEG/EAGE overthrust models. In comparison with the 27‐point scheme, the 45‐point scheme yields a notable improvement in computational accuracy, particularly for large grid ratios, while imposing only a modest increase in computational cost. These findings thus strongly suggest that the 45‐point scheme holds promise as a viable option for the forward part of frequency‐domain FWI in practical high‐accuracy seismic imaging applications.

© 2025 European Association of Geoscientists & Engineers. © 2025 European Association of Geoscientists & Engineers.
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2025-08-22
2026-02-11

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References

  1. Aghamiry, H. S., A.Gholami, L.Combe, and S.Operto. 2022. “Accurate 3D Frequency‐Domain Seismic Wave Modeling With the Wavelength‐Adaptive 27‐Point Finite‐Difference Stencil: A Tool for Full‐Waveform Inversion.” Geophysics87: R305–R324. https://doi.org/10.1190/geo2021‐0606.1.
     [Google Scholar]
  2. Aki, K., and P. G.Richards. 2002. Quantitative Seismology. 2nd ed.University Science Books.
     [Google Scholar]
  3. Amestoy, P. R., A.Buttari, J.‐Y.L'excellent, and T.Mary. 2019. “Performance and Scalability of the Block Low‐Rank Multifrontal Factorization on Multicore Architectures.” ACM Transactions on Mathematical Software45: 1–23. https://doi.org/10.1145/3242094.
     [Google Scholar]
  4. Amestoy, P. R., I. S.Duff, J.‐Y.L'Excellent, and J.Koster. 2001. “A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling.” SIAM Journal on Matrix Analysis and Applications23: 15–41. https://doi.org/10.1137/S0895479899358194.
     [Google Scholar]
  5. Berenger, J.‐P.1994. “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves.” Journal of Computational Physics114: 185–200. https://doi.org/10.1006/jcph.1994.1159.
     [Google Scholar]
  6. Brenders, A., and R.Pratt. 2007. “Efficient Waveform Tomography for Lithospheric Imaging: Implications for Realistic, Two‐Dimensional Acquisition Geometries and Low‐Frequency Data.” Geophysical Journal International168: 152–170. https://doi.org/10.1111/j.1365‐246X.2006.03096.x.
     [Google Scholar]
  7. Cai, X., Q.Hu, K. A.Innanen, et al. 2025. “Rapid‐Repeat Time‐Lapse Vertical Seismic Profile Imaging of CO2 Injection.” Geophysics90: B91–B99. https://doi.org/10.1190/geo2024‐0202.1.
     [Google Scholar]
  8. Cao, S.‐H., and J.‐B.Chen. 2012. “A 17‐Point Scheme and Its Numerical Implementation for High‐Accuracy Modeling of Frequency‐Domain Acoustic Equation.” Chinese Journal of Geophysics55: 3440–3449. https://doi.org/10.6038/j.issn.0001‐5733.2012.10.027.
     [Google Scholar]
  9. Chen, J.‐B.2011. “A Stability Formula for Lax‐Wendroff Methods With Fourth‐Order in Time and General‐Order in Space for the Scalar Wave Equation.” Geophysics76: T37–T42. https://doi.org/10.1190/1.3554626.
     [Google Scholar]
  10. Chen, J.‐B.2012. “An Average‐Derivative Optimal Scheme for Frequency‐Domain Scalar Wave Equation.” Geophysics77: T201–T210. https://doi.org/10.1190/geo2011‐0389.1.
     [Google Scholar]
  11. Chen, J. B.2014. “A 27‐Point Scheme for a 3D Frequency‐Domain Scalar Wave Equation Based on an Average‐Derivative Method.” Geophysical Prospecting62: 258–277. https://doi.org/10.1111/1365‐2478.12090.
     [Google Scholar]
  12. Chen, J. B., and J.Cao. 2016. “Modeling of Frequency‐Domain Elastic‐Wave Equation With an Average‐Derivative Optimal Method.” Geophysics81: T339–T356. https://doi.org/10.1190/geo2016‐0041.1.
     [Google Scholar]
  13. Chen, J. B., and J.Cao. 2018. “An Average‐Derivative Optimal Scheme for Modeling of the Frequency‐Domain 3D Elastic Wave Equation.” Geophysics83: T209–T234. https://doi.org/10.1190/geo2017‐0641.1.
     [Google Scholar]
  14. Dong, S.‐L., and J.‐B.Chen. 2022. “An Affine Generalized Optimal Scheme With Improved Free‐surface Expression Using Adaptive Strategy for Frequency‐Domain Elastic Wave Equation.” Geophysics87: T183–T204. https://doi.org/10.1190/geo2021‐0345.1.
     [Google Scholar]
  15. Dong, S.‐L., and J.‐B.Chen. 2023. “Finite‐Difference Modeling of 3D Frequency‐Domain Elastic Wave Equation Using an Affine Mixed‐Grid Method.” Geophysics88: T45–T63. https://doi.org/10.1190/geo2022‐0074.1.
     [Google Scholar]
  16. Dong, S. L., and J. B.Chen. 2024. “Affine 25‐Point Scheme for High‐Accuracy Numerical Simulation of Frequency‐Domain Acoustic Wave Equation.” Chinese Journal of Geophysics67: 3859–3873. https://doi.org/10.6038/cjg2024R0660.
     [Google Scholar]
  17. Fan, N., L.‐F.Zhao, X.‐B.Xie, X.‐G.Tang, and Z.‐X.Yao. 2017. “A General Optimal Method for a 2D Frequency‐Domain Finite‐Difference Solution of Scalar Wave Equation.” Geophysics82: T121–T132. https://doi.org/10.1190/geo2016‐0457.1.
     [Google Scholar]
  18. Gosselin‐Cliche, B., and B.Giroux. 2014. “3D Frequency‐Domain Finite‐Difference Viscoelastic‐Wave Modeling Using Weighted Average 27‐Point Operators With Optimal Coefficients.” Geophysics79: T169–T188. https://doi.org/10.1190/geo2013‐0368.1.
     [Google Scholar]
  19. Graves, R. W.1996. “Simulating Seismic Wave Propagation in 3D Elastic Media Using Staggered‐Grid Finite Differences.” Bulletin of the Seismological Society of America86: 1091–1106. https://doi.org/10.1785/BSSA0860041091.
     [Google Scholar]
  20. Gu, B., G.Liang, and Z.Li. 2013. “A 21‐Point Finite Difference Scheme for 2D Frequency‐Domain Elastic Wave Modeling.” Exploration Geophysics44: 156–166. https://doi.org/10.1071/EG12064.
     [Google Scholar]
  21. Guan, J., Y.Li, R.Ji, G.Liu, and Y.Yan. 2022. “Love Wave Full‐Waveform Inversion for Archaeogeophysics: From Synthesis Tests to a Field Case.” Journal of Applied Geophysics202: 104653. https://doi.org/10.1016/j.jappgeo.2022.104653.
     [Google Scholar]
  22. Guasch, L., O.Calderón Agudo, M. X.Tang, P.Nachev, and M.Warner. 2020. “Full‐Waveform Inversion Imaging of the Human Brain.” NPJ Digital Medicine3: 28. https://doi.org/10.1038/s41746‐020‐0240‐8.
     [Google Scholar]
  23. He, C., and J. B.Chen. 2024. “A 17‐Point Composite Average‐Derivative Optimal Method for High‐Accuracy Frequency‐Domain Scalar Wave Modeling.” Journal of Applied Geophysics220: 105269. https://doi.org/10.1016/j.jappgeo.2023.105269.
     [Google Scholar]
  24. Huang, C., T.Zhu, and G.Xing. 2023. “Data‐Assimilated Time‐Lapse Visco‐Acoustic Full‐Waveform Inversion: Theory and Application for Injected CO2 Plume Monitoring.” Geophysics88: R105–R120. https://doi.org/10.1190/geo2021‐0804.1.
     [Google Scholar]
  25. Hustedt, B., S.Operto, and J.Virieux. 2004. “Mixed‐Grid and Staggered‐Grid Finite‐Difference Methods for Frequency‐Domain Acoustic Wave Modelling.” Geophysical Journal International157: 1269–1296. https://doi.org/10.1111/j.1365‐246X.2004.02289.x.
     [Google Scholar]
  26. Jo, C.‐H., C.Shin, and J. H.Suh. 1996. “An Optimal 9‐Point, Finite‐Difference, Frequency‐Space, 2‐D Scalar Wave Extrapolator.” Geophysics61: 529–537. https://doi.org/10.1190/1.1443979.
     [Google Scholar]
  27. Johnson, L. R.1974. “Green's Function for Lamb's Problem.” Geophysical Journal International37: 99–131. https://doi.org/10.1111/j.1365‐246X.1974.tb02446.x.
     [Google Scholar]
  28. Kawasaki, Y., S.Minato, and R.Ghose. 2022. “Reducing Parameter Coupling Effect for Subsoil Density Estimation Using 3D Seismic Full‐waveform Inversion With Horizontal‐force Source.” in SEG Technical Program Expanded Abstracts, 2138–2142. SEG Publications. https://doi.org/10.1190/image2022‐3750995.1.
  29. Li, S. Z., and C. Y.Sun. 2022. “An Improved 25‐Point Finite‐Difference Scheme for Frequency‐Domain Elastic Wave Modelling.” Geophysical Prospecting70: 702–724. https://doi.org/10.1111/1365‐2478.13188.
     [Google Scholar]
  30. Li, S. Z., C. Y.Sun, H.Wu, R. Q.Cai, and N.Wu. 2021. “An Optimal Finite‐Difference Method Based on the Elongated Stencil for 2D Frequency‐Domain Acoustic‐Wave Modeling.” Geophysics86: T523–T541. https://doi.org/10.1190/geo2021‐0024.1.
     [Google Scholar]
  31. Li, Y., R.Brossier, and L.Métivier. 2020. “3D Frequency‐Domain Elastic Wave Modeling With the Spectral Element Method Using a Massively Parallel Direct Solver.” Geophysics85: T71–T88. https://doi.org/10.1190/geo2019‐0172.1.
     [Google Scholar]
  32. Li, Y., B.Han, L.Métivier, and R.Brossier. 2016. “Optimal Fourth‐Order Staggered‐Grid Finite‐Difference Scheme for 3D Frequency‐Domain Viscoelastic Wave Modeling.” Journal of Computational Physics321: 1055–1078. https://doi.org/10.1016/j.jcp.2016.06.018.
     [Google Scholar]
  33. Li, Y., L.Métivier, R.Brossier, B.Han, and J.Virieux. 2015. “2D and 3D Frequency‐Domain Elastic Wave Modeling in Complex Media With a Parallel Iterative Solver.” Geophysics80: T101–T118. https://doi.org/10.1190/geo2014‐0480.1.
     [Google Scholar]
  34. Liu, L., H.Liu, and H.Liu. 2013. “Optimal 15‐Point Finite Difference Forward Modeling in Frequency‐Space Domain.” Chinese Journal of Geophysics56: 644–652. https://doi.org/10.6038/cjg20130228.
     [Google Scholar]
  35. Marfurt, K. J.1984. “Accuracy of Finite‐Difference and Finite‐Element Modeling of the Scalar and Elastic Wave Equations.” Geophysics49: 533–549. https://doi.org/10.1190/1.1441689.
     [Google Scholar]
  36. Marty, P., C.Boehm, and A.Fichtner. 2022. “Elastic Full‐Waveform Inversion for Transcranial Ultrasound Computed Tomography Using Optimal Transport.” In IEEE International Ultrasonics Symposium (IUS), 1–4. IEEE. https://doi.org/10.1109/IUS54386.2022.9957394.
  37. Min, D.‐J., C.Shin, B.‐D.Kwon, and S.Chung. 2000. “Improved Frequency‐domain Elastic Wave Modeling Using Weighted‐Averaging Difference Operators.” Geophysics65: 884–895. https://doi.org/10.1190/1.1444785.
     [Google Scholar]
  38. Nocedal, J., and S. J.Wright. 1999. Numerical Optimization. Springer New York.
     [Google Scholar]
  39. Operto, S., J.Virieux, P.Amestoy, J.‐Y.L'Excellent, L.Giraud, and H. B. H.Ali. 2007. “3D Finite‐Difference Frequency‐Domain Modeling of Visco‐Acoustic Wave Propagation Using a Massively Parallel Direct Solver: A Feasibility Study.” Geophysics72: SM195–SM211. https://doi.org/10.1190/1.2759835.
     [Google Scholar]
  40. Operto, S., J.Virieux, A.Ribodetti, and J. E.Anderson. 2009. “Finite‐Difference Frequency‐Domain Modeling of Visco‐Acoustic Wave Propagation in Two‐Dimensional TTI media.” Geophysics74: T75–T95. https://doi.org/10.1190/1.2759835.
     [Google Scholar]
  41. Pan, W., K. A.Innanen, and Y.Geng. 2018. “Elastic Full‐Waveform Inversion and Parametrization Analysis Applied to Walk‐away Vertical Seismic Profile Data for Unconventional (Heavy Oil) Reservoir Characterization.” Geophysical Journal International213: 1934–1968. https://doi.org/10.1093/gji/ggy087.
     [Google Scholar]
  42. Pratt, R. G.1990. “Frequency‐Domain Elastic Wave Modeling by Finite Differences: A Tool for Crosshole Seismic Imaging.” Geophysics55: 626–632. https://doi.org/10.1190/1.1442874.
     [Google Scholar]
  43. Pratt, R. G., C.Shin, and G.Hick. 1998. “Gauss‐Newton and Full Newton Methods in Frequency‐Space Seismic Waveform Inversion.” Geophysical Journal International133: 341–362. https://doi.org/10.1046/j.1365‐246X.1998.00498.x.
     [Google Scholar]
  44. Pratt, R. G., and M.‐H.Worthington. 1990. “Inverse Theory Applied to Multisource Cross‐Hole Tomography, Part I: Acoustic Wave‐Equation Method.” Geophysical Prospecting38: 287–310. https://doi.org/10.1111/j.1365‐2478.1990.tb01846.x.
     [Google Scholar]
  45. Qu, Y., Z.Xu, J.Zhu, L.Xie, and J.Li. 2025. “Average‐Derivative Optimized 21‐Point and Improved 25‐Point Forward Modelling and Full Waveform Inversion in Frequency Domain.” Geophysical Prospecting73: 874–894. https://doi.org/10.1111/1365‐2478.13587.
     [Google Scholar]
  46. Rusch, K., D.Köhn, H.Stümpel, W.Gauß, and W.Rabbel. 2022. “Prehistoric Chamber Tombs or Geological Pitfall? A Multimethod Case Study From Ancient Aigeira With a Focus on Seismic Full‐Waveform Inversion.” Archaeological Prospection29: 45–68. https://doi.org/10.1002/arp.1835.
     [Google Scholar]
  47. Shin, C., and H.Sohn. 1998. “A Frequency‐Space 2D Scalar Wave Extrapolator Using Extended 25‐Point Finite‐Difference Operator.” Geophysics63: 289–296. https://doi.org/10.1190/1.1444323.
     [Google Scholar]
  48. Štekl, I., and R. G.Pratt. 1998. “Accurate Viscoelastic Modeling by Frequency‐Domain Finite Difference Using Rotated Operators.” Geophysics63: 1779–1794. https://doi.org/10.1190/1.1444472.
     [Google Scholar]
  49. Takekawa, J., and H.Mikada. 2019. “An Elongated Finite‐Difference Scheme for 2D Acoustic and Elastic Wave Simulations in the Frequency Domain.” Geophysics84: T29–T45. https://doi.org/10.1190/geo2018‐0014.1.
     [Google Scholar]
  50. Tang, X., H.Liu, H.Zhang, L.Liu, and Z.Wang. 2015. “An Adaptable 17‐Point Scheme for High‐Accuracy Frequency‐Domain Acoustic Wave Modeling in 2D Constant Density Media.” Geophysics80: T211–T221. https://doi.org/10.1190/geo2014‐0124.1.
     [Google Scholar]
  51. Virieux, J.1986. “P‐SV Wave Propagation in Heterogeneous Media: Velocity‐Stress Finite‐Difference Method.” Geophysics51: 889–901. https://doi.org/10.1190/1.1442147.
     [Google Scholar]
  52. Virieux, J., and S.Operto. 2009. “An Overview of Full‐Waveform Inversion in Exploration Geophysics.” Geophysics74: WCC1–WCC26. https://doi.org/10.1190/1.3238367.
     [Google Scholar]
  53. Wang, H., and J.‐B.Chen. 2023. “Frequency‐Domain Elastic Wave Modeling for Vertical Transversely Isotropic Media With an Average‐Derivative Optimal Method.” Geophysics88: T237–T258. https://doi.org/10.1190/geo2022‐0178.1.
     [Google Scholar]
  54. Wang, H., and J.‐B.Chen. 2024. “An Improved 27‐Point Frequency‐Domain Elastic‐Wave Average‐Derivative Method With Applications to a Moving Point Source.” Geophysics89: T95–T110. https://doi.org/10.1190/geo2023‐0257.1.
     [Google Scholar]
  55. Wang, M., Y.Li, Y.Chen, Z.Fu, D.Wang, and F.Tang. 2025. “An Average‐Derivative Based Second‐Order Staggered‐Grid Finite‐difference Scheme for 3D Frequency‐Domain Elastic Wave Modeling.” Geophysics90: T79–T97. https://doi.org/10.1190/geo2024‐0574.1.
     [Google Scholar]
  56. Xu, W., B.Wu, Y.Zhong, J.Gao, and Q. H.Liu. 2021. “Adaptive 27‐Point Finite‐Difference Frequency‐Domain Method for Wave Simulation of 3D Acoustic Wave Equation.” Geophysics86: T439–T449. https://doi.org/10.1190/geo2021‐0050.1.
     [Google Scholar]
  57. Yang, L., G.Wu, Y.Wang, Q.Li, and H.Zhang. 2022. “An Optimal 125‐Point Scheme for 3D Frequency‐Domain Scalar Wave Equation.” Acta Geophysica70: 71–88. https://doi.org/10.1007/s11600‐021‐00681‐8.
     [Google Scholar]
  58. Yao, G., D.Wu, and S. X.Wang. 2020. “A Review on Reflection‐Waveform Inversion.” Petroleum Science17: 334–351. https://doi.org/10.1007/s12182‐020‐00431‐3.
     [Google Scholar]
  59. Zhang, H., H.Liu, L.Liu, W.Jin, and X.Shi. 2014. “Frequency Domain Acoustic Equation High‐Order Modeling Based on an Average‐Derivative Method.” Chinese Journal of Geophysics57: 1599–1611. https://doi.org/10.6038/cjg20140523.
     [Google Scholar]
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High‐Accuracy Modelling of 3D Frequency‐Domain Elastic‐Wave Equation Based on One‐Direction Composition of the Average‐Derivative Optimal Method
Geophysical Prospecting 73, e70070 (2025); https://doi.org/10.1111/1365-2478.70070
/content/journals/10.1111/1365-2478.70070
/content/journals/10.1111/1365-2478.70070
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  • Article Type: Research Article
Keyword(s): 3D; elastic wave; finite‐difference modelling; frequency domain

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