1887
Volume 72, Issue 9
  • E-ISSN: 1365-2478

Abstract

Abstract

Accurate seismic models with anisotropy and attenuation characteristics are crucial to accurately imaging subsurface structures. However, the anisotropic viscoelastic equations are complex and require significant computational resources. In addition, the single‐mode waves have been sufficient for most practical exploration needs. However, separating the qP‐ and qSV‐waves in anisotropic viscoelastic wavefields is challenging. Thus, we propose a new method to approximate and efficiently separate the qP‐ and qSV‐waves in attenuated transversely isotropic media. First, we obtain the decoupled approximate phase velocities of qP‐ and qSV‐waves by a curve‐fitting method. Consequently, based on the average and maximum relative error analysis, our approximate qP‐ and qSV‐wave phase velocities are more accurate than the existing approximations. Additionally, our approximations have broader applicability, resulting in acceptable errors during their application. Second, based on the approximate qP‐ and qSV‐wave phase velocities, we derive the corresponding qP‐ and qSV‐wave equations for a complete decoupling of the qP‐ and qSV‐wave components in transversely isotropic media. Third, to combine the attenuation and anisotropy characteristics, we incorporate the Kelvin–Voigt attenuation model and obtain the decoupled qP‐ and qSV‐wave equations in attenuated transversely isotropic media. Then, we use an efficient and stable hybrid finite‐difference and pseudo‐spectral method to solve the new decoupled qP‐ and qSV‐wave equations. Finally, several numerical examples demonstrate the separability and high accuracy of the proposed qP‐ and qSV‐wave equations. We obtain a qP‐wave wavefield entirely devoid of SV‐wave artefacts. In addition, the decoupled approximate qP‐ and qSV‐wave equations are accurate and stable in heterogeneous media with different velocities and attenuation. The decoupled, approximated qP‐wave and qSV‐wave equations proposed in this paper can effectively separate the qP‐wave and qSV‐wave components, resulting in fully decoupled qP‐ and qSV‐wave wavefields in attenuated transversely isotropic media.

© 2024 European Association of Geoscientists & Engineers. © 2024 European Association of Geoscientists & Engineers.
Loading

Article metrics loading...

/content/journals/10.1111/1365-2478.13591
2024-10-11
2026-02-19

  • From This Site

    /content/journals/10.1111/1365-2478.13591
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. Abedi, M.M. & Stovas, A. (2020) A new acoustic assumption for orthorhombic media. Geophysical Journal International, 223, 1118–1129.
     [Google Scholar]
  2. Aki, K. & Richards, P. (2002) Quantitative Seismology: Second Edition. Sausalito, CA: University Science Books.
     [Google Scholar]
  3. Alkhalifah, T. (1998) Acoustic approximations for processing in transversely isotropic media. Geophysics, 63, 623–631.
     [Google Scholar]
  4. Alkhalifah, T. (2000) An acoustic wave equation for anisotropic media. Geophysics, 65, 1239–1250.
     [Google Scholar]
  5. Chen, K., Liu, L., Zhang, L. & Zhao, Y. (2022) Vertical transversely isotropic elastic least‐squares reverse time migration based on elastic wavefield vector decomposition. Geophysics, 88(1), S27–S45.
     [Google Scholar]
  6. Cheng, J., Alkhalifah, T., Wu, Z., Zou, P. & Wang, C. (2016) Simulating propagation of decoupled elastic waves using low‐rank approximate mixed‐domain integral operators for anisotropic media. Geophysics, 81(2), T63–T77.
     [Google Scholar]
  7. Chu, C., Macy, B.K. & Anno, P.D. (2011) Approximation of pure acoustic seismic wave propagation in TTI media. Geophysics, 76, WB97–WB107.
     [Google Scholar]
  8. Chu, C., Macy, B.K. & Anno, P.D. (2013) Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method. Geophysical Prospecting, 61, 556–567.
     [Google Scholar]
  9. da Silva, N.V., Yao, G. & Warner, M. (2019) Wave modeling in viscoacoustic media with transverse isotropy. Geophysics, 84, C41–C56.
     [Google Scholar]
  10. Dellinger, J. & Etgen, J. (1990) Wave‐field separation in 2‐dimensional anisotropic media. Geophysics, 55, 914–919.
     [Google Scholar]
  11. Dix, C.H. (1955) Seismic velocities from surface measurements. Geophysics, 20(1), 68–86.
     [Google Scholar]
  12. Fathalian, A., Trad, D.O. & Innanen, K.A. (2021) Q‐compensated reverse time migration in tilted transversely isotropic media. Geophysics, 86, S73–S89.
     [Google Scholar]
  13. Fomel, S., Ying, L. & Song, X. (2013) Seismic wave extrapolation using lowrank symbol approximation. Geophysical Prospecting, 61, 526–536.
     [Google Scholar]
  14. Fowler, P.J., Du, X. & Fletcher, R.P. (2010) Coupled equations for reverse time migration in transversely isotropic media. Geophysics, 75, S11–S22.
     [Google Scholar]
  15. Gazdag, J. & Sguazzero, P. (1984) Migration of seismic data by phase shift plus interpolation. Geophysics, 49(2), 124–131.
     [Google Scholar]
  16. Gray, S.H. (1986) Efficient traveltime calculations for Kirchhoff migration. Geophysics, 51(8), 1685–1688.
     [Google Scholar]
  17. Grechka, V., Zhang, L. & Rector, James W., I,. (2004) Shear waves in acoustic anisotropic media. Geophysics, 69, 576–582.
     [Google Scholar]
  18. Hao, Q. & Alkhalifah, T. (2017) An acoustic Eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis. Geophysics, 82(1), C9–C20.
     [Google Scholar]
  19. Hestholm, S. (2009) Acoustic VTI modeling using high‐order finite differences. Geophysics, 74, T67–T73.
     [Google Scholar]
  20. Jin, S. & Stovas, A. (2018) S‐wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis. Geophysical Prospecting, 66, 1123–1137.
     [Google Scholar]
  21. Jin, S. & Stovas, A. (2020) S‐wave in 2D acoustic transversely isotropic media with a tilted symmetry axis. Geophysical Prospecting, 68, 483–500.
     [Google Scholar]
  22. Li, B. & Stovas, A. (2021) Decoupled approximation and separate extrapolation of P‐ and SV‐waves in transversely isotropic media. Geophysics, 86, C133–C142.
     [Google Scholar]
  23. Li, X. & Zhu, H. (2018) A finite‐difference approach for solving pure quasi‐P‐wave equations in transversely isotropic and orthorhombic media. Geophysics, 83, C161–C172.
     [Google Scholar]
  24. Liang, K., Cao, D., Sun, S. & Yin, X. (2023) Decoupled wave equation and forward modeling of qP wave in VTI media with the new acoustic approximation. Geophysics, 88, WA335–WA344.
     [Google Scholar]
  25. Liang, K., Sun, S., Cao, D. & Yin, X. (2021) Approximate decoupled wave equations for elastic waves in TTI media. Chinese Journal of Geophysics, 64, 4607–4617 (in Chinese).
     [Google Scholar]
  26. Ma, X., Yang, D., Huang, X. & Zhou, Y. (2018) Nonsplit complex‐frequency shifted perfectly matched layer combined with symplectic methods for solving second‐order seismic wave equations ‐ Part 1: Method. Geophysics, 83(6), T301–T311.
     [Google Scholar]
  27. Mu, X., Huang, J., Yang, J., Zhang, J. & Wang, Z. (2022) Modeling of pure visco‐qP‐wave propagation in attenuating tilted transversely isotropic media based on decoupled fractional Laplacians. Geophysics, 87, T291–T313.
     [Google Scholar]
  28. Mu, X., Huang, J., Yong, P., Huang, J., Guo, X., Liu, D. & Hu, Z. (2020) Modeling of pure qP‐ and qSV‐waves in tilted transversely isotropic media with the optimal quadratic approximation. Geophysics, 85, C71–C89.
     [Google Scholar]
  29. Müller, T.M., Gurevich, B. & Lebedev, M. (2010) Seismic wave attenuation and dispersion resulting from wave‐induced flow in porous rocks – a review. Geophysics, 75(5), 75A147–75A164.
     [Google Scholar]
  30. Peters, L.E., Anandakrishnan, S., Alley, R.B. & Voigt, D.E. (2012) Seismic attenuation in glacial ice: a proxy for englacial temperature. Journal of Geophysical Research: Earth Surface, 117, 1–10.
     [Google Scholar]
  31. Qiao, Z., Sun, C. & Tang, J. (2020) Modelling of viscoacoustic wave propagation in transversely isotropic media using decoupled fractional Laplacians. Geophysical Prospecting, 68, 2400–2418.
     [Google Scholar]
  32. Qu, Y., Huang, J., Li, Z., Guan, Z. & Li, J. (2017) Attenuation compensation in anisotropic least‐squares reverse time migration. Geophysics, 82, S411–S423.
     [Google Scholar]
  33. Ravve, I. & Koren, Z. (2017) Traveltime approximation in vertical transversely isotropic layered media. Geophysical Prospecting, 65(6), 1559–1581.
     [Google Scholar]
  34. Robertsson, J.O., Blanch, J.O. & Symes, W.W. (1994) Viscoelastic finite‐difference modeling. Geophysics, 59, 1444–1456.
     [Google Scholar]
  35. Schleicher, J. & Costa, J.C. (2016) A separable strong‐anisotropy approximation for pure qP‐wave propagation in transversely isotropic media. Geophysics, 81, C337–C354.
     [Google Scholar]
  36. Schneider, W.A. (1978) Integral formulation for migration in 2 and 3 dimensions. Geophysics, 43(1), 49–76.
     [Google Scholar]
  37. Shen, Y., Biondi, B. & Clapp, R. (2017) Q‐model building using one‐way wave‐equation migration Q analysis – Part 1: theory and synthetic test. Geophysics, 83, S93–S109.
     [Google Scholar]
  38. Song, G., Huang, R., Tian, J., Chen, Y., Chen, P. & Yang, Y. (2016) A new qP‐wave equation for 2D VTI media. In: the 86th Annual International Meeting, SEG expanded abstracts. Houston, TX: Society of Exploration Geophysicists, pp. 3977–3981.
     [Google Scholar]
  39. Song, X., Fomel, S. & Ying, L. (2013) Lowrank finite‐differences and lowrank Fourier finite‐differences for seismic wave extrapolation in the acoustic approximation. Geophysical Journal International, 193, 960–969.
     [Google Scholar]
  40. Stoffa, P., Fokkema, J., de Luna Freire, R. & Kessinger, W. (1990) Split‐step Fourier migration. Geophysics, 55, 410–421.
     [Google Scholar]
  41. Stovas, A. (2015) Azimuthally dependent kinematic properties of orthorhombic media. Geophysics, 80(6), C107–C122.
     [Google Scholar]
  42. Stovas, A., Alkhalifah, T. & Waheed, U.b. (2020) Pure P‐ and S‐wave equations in transversely isotropic media. Geophysical Prospecting, 68, 2762–2769.
     [Google Scholar]
  43. Suh, S., Yoon, K., Cai, J. & Wang, B. (2012) Compensating visco‐acoustic effects in anisotropic resverse‐time migration. In: the 82nd Annual International Meeting, SEG expanded abstracts. Houston, TX: Society of Exploration Geophysicists, pp. 1–5.
     [Google Scholar]
  44. Sun, B. & Alkhalifah, T. (2021) Pseudoelastic pure p‐mode wave equation. Geophysics, 86(6), T469–T485.
     [Google Scholar]
  45. Sun, Y., Qin, F., Checkles, S. & Leveille, J.P. (2000) 3‐D prestack Kirchhoff beam migration for depth imaging. Geophysics, 65(5), 1592–1603.
     [Google Scholar]
  46. Thomsen, L. (1986) Weak elastic anisotropy. Geophysics, 51, 1954–1966.
     [Google Scholar]
  47. Tsvankin, I. (1996) P‐wave signatures and notation for transversely isotropic media: an overview. Geophysics, 61, 467–483.
     [Google Scholar]
  48. Tsvankin, I. (2012) Seismic signatures and analysis of reflection data in anisotropic media: Third Edition. Houston, TX: Society of Exploration Geophysicists.
     [Google Scholar]
  49. Tsvankin, I. & Thomsen, L. (1994) Nonhyperbolic reflection moveout in anisotropic media. Geophysics, 59(8), 1290–1304.
     [Google Scholar]
  50. Wang, C., Cheng, J. & Arntsen, B. (2016) Scalar and vector imaging based on wave mode decoupling for elastic reverse time migration in isotropic and transversely isotropic media. Geophysics, 81(5), S383–S398.
     [Google Scholar]
  51. Wiens, D.A., Conder, J.A. & Faul, U.H. (2008) The seismic structure and dynamics of the mantle wedge. Annual Review of Earth and Planetary Sciences, 36, 421–455.
     [Google Scholar]
  52. Wu, Y., Lan, H. & Huang, W. (2020) Discussion on the relationship between elastic anisotropy and mineral distribution of Longmaxi shale. Acta Geophysica Sinica, 63, 1856–1866.
     [Google Scholar]
  53. Wu, Z. & Alkhalifah, T. (2014) The optimized expansion based low‐rank method for wavefield extrapolation. Geophysics, 79, T51–T60.
     [Google Scholar]
  54. Xie, Y., Sun, J., Zhang, Y. & Zhou, J. (2015) Compensating for visco‐acoustic effects in TTI reverse time migration. In: the 85th Annual International Meeting, SEG expanded abstracts. Houston, TX: Society of Exploration Geophysicists, pp. 3996–4001.
     [Google Scholar]
  55. Xu, S. & Stovas, A. (2019) Estimation of the conversion point position in elastic orthorhombic media. Geophysics, 84(1), C15–C25.
     [Google Scholar]
  56. Xu, S., Stovas, A., Alkhalifah, T. & Mikada, H. (2019) New acoustic approximation for transversely isotropic media with a vertical symmetry axis. Geophysics, 85, C1–C12.
     [Google Scholar]
  57. Xu, S., Stovas, A. & Sripanich, Y. (2018) An anelliptic approximation for geometric spreading in transversely isotropic and orthorhombic media. Geophysics, 83(1), C37–C47.
     [Google Scholar]
  58. Xu, S. & Zhou, H. (2014) Accurate simulations of pure quasi‐P‐waves in complex anisotropic media. Geophysics, 79, T341–T348.
     [Google Scholar]
  59. Xu, W., Li, Z., Wang, J. & Zhang, Y. (2015) A pure viscoacoustic equation for VTI media applied in anisotropic RTM. Journal of Geophysics and Engineering, 12, 969–977.
     [Google Scholar]
  60. Yan, J. & Sava, P. (2008) Isotropic angle‐domain elastic reverse‐time migration. Geophysics, 73(6), S229–S239.
     [Google Scholar]
  61. Yang, J., Zhang, H., Zhao, Y. & Zhu, H. (2019) Elastic wavefield separation in anisotropic media based on eigenform analysis and its application in reverse‐time migration. Geophysical Journal International, 217(2), 1290–1313.
     [Google Scholar]
  62. Yang, P., Li, Z. & Gu, B. (2017) Pure quasi‐P wave forward modeling method in TI media and its application to RTM. Chinese Journal of Geophysics, 60(11), 4447–4467 (in Chinese).
     [Google Scholar]
  63. Zhan, G., Pestana, R.C. & Stoffa, P.L. (2013) An efficient hybrid pseudospectral/finite‐difference scheme for solving the TTI pure P‐wave equation. Journal of Geophysics and Engineering, 10, 25004–25011.
     [Google Scholar]
  64. Zhang, H.J. & Zhang, Y. (2008) Reverse time migration in 3D heterogeneous TTI media. In: the 78th Annual International Meeting, SEG expanded abstracts. Houston, TX: Society of Exploration Geophysicists, pp. 2196–2200.
     [Google Scholar]
  65. Zhang, J. & Yao, Z. (2012) Globally optimized finite‐difference extrapolator for strongly VTI media. Geophysics, 77, T125–T135.
     [Google Scholar]
  66. Zhang, Y., Liu, Y. & Xu, S. (2020a) Anisotropic viscoacoustic wave modelling in VTI media using frequency‐dependent complex velocity. Journal of Geophysics and Engineering, 17, 700–717.
     [Google Scholar]
  67. Zhang, Y., Liu, Y. & Xu, S. (2020b) Arbitrary‐order Taylor series expansion‐based viscoacoustic wavefield simulation in 3D vertical transversely isotropic media. Geophysical Prospecting, 68, 2379–2399.
     [Google Scholar]
  68. Zhang, Z., Alkhalifah, T. & Wu, Z. (2019) A hybrid finite‐difference/low‐rank solution to anisotropy acoustic wave equations. Geophysics, 84, T83–T91.
     [Google Scholar]
  69. Zhang, Z., Liu, Y., Alkhalifah, T. & Wu, Z. (2017) Efficient anisotropic quasi‐P wavefield extrapolation using an isotropic low‐rank approximation. Geophysical Journal International, 213, 48–57.
     [Google Scholar]
  70. Zhou, H., Zhang, G. & Bloor, R. (2006) An anisotropic acoustic wave equation for modeling and migration in 2D TTI media. In: the 76th Annual International Meeting, SEG expanded abstracts. Houston, TX: Society of Exploration Geophysicists, pp. 194–198.
     [Google Scholar]
  71. Zhou, T., Hu, W. & Ning, J. (2020) Broadband finite‐difference Q‐compensated reverse time migration algorithm for tilted transverse isotropic media. Geophysics, 85, S241–S253.
     [Google Scholar]
  72. Zhu, H. (2017) Elastic wavefield separation based on the Helmholtz decomposition. Geophysics, 82, S173–S183.
     [Google Scholar]
  73. Zuo, J., Niu, F., Liu, L., Zhang, H., Yang, J., Zhang, L. & Zhao, Y. (2022) 3D anisotropic P‐ and S‐mode wavefields separation in 3D elastic reverse‐time migration. Surveys in Geophysics, 43(3), 673–701.
     [Google Scholar]
/content/journals/10.1111/1365-2478.13591
Loading
Decoupled approximate qP‐ and qSV‐wave equations in attenuated transversely isotropic media
Geophysical Prospecting 72, 3495 (2024); https://doi.org/10.1111/1365-2478.13591
/content/journals/10.1111/1365-2478.13591
/content/journals/10.1111/1365-2478.13591
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): anisotropy; attenuation; seismic modelling; wave propagation; wave separation

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error