1887
Volume 71, Issue 3
  • E-ISSN: 1365-2478

Abstract

Abstract

For 3D seismic data processing, azimuth plays an important role, especially in complex subsurface regions. In such regions, elliptical or anelliptical orthorhombic models are commonly used to describe wave propagation. In these models, the behaviour of the slowness surface needs more detailed analysis. Umbilic points defined by the equal principal curvatures exist in a complex 3D model. In the vicinity of the umbilic points, the traveltime surface has the shifted hyperbola form that needs to be considered in processing operations like velocity analysis. Fractured media characterized by the orthorhombic model are more likely to have umbilic points, and it is important to address their positions. If exists, umbilic points can provide additional constraints in inverting for model parameters. Through a defined condition, we examine the position of the umbilic point and derive their explicit formulas. We analyse umbilic points for elliptical and anelliptical orthorhombic models in the numerical example. For the elliptical orthorhombic model, the formulas for the umbilic point position on different symmetry planes are derived and corresponding conditions are also identified. Further, we numerically examine umbilic point positions for anelliptical orthorhombic models and observe that the umbilic points are located out of two vertical symmetry planes. Moreover, caused by interference from two neighbouring umbilic points, a more significant deviation in traveltime is found in the anelliptical orthorhombic model.

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/content/journals/10.1111/1365-2478.13316
2023-02-17
2024-03-28
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References

  1. Abbena, E., Salamon, S. & Gray, A. (2017) Modern differential geometry of curves and surfaces with Mathematica. New York: Chapman and Hall/CRC.
    [Google Scholar]
  2. Abedi, M.M. (2020) Rational approximation of P‐wave kinematics. Part 2: Orhorhombic media. Geophysics, 85, C175–C186.
    [Google Scholar]
  3. Abedi, M.M., Pardo, D. & Stovas, A. (2021) Approximations for traveltime, slope, curvature, and geometric spreading of elastic waves in layered transversely isotropic media. Geophysics, 86, C173–C184.
    [Google Scholar]
  4. Alkhalifah, T. (1997) Kinematics of 3‐D DMO operators in transversely isotropic media. Geophysics, 62, 1214–1219.
    [Google Scholar]
  5. Alkhalifah, T. & Tsvankin, I. (1995) Velocity analysis for transversely isotropic media. Geophysics, 60, 1550–1566.
    [Google Scholar]
  6. Casey, J. (2012) Exploring curvature. New York/Berlin/Heidelberg: Springer Science & Business Media.
    [Google Scholar]
  7. Cazals, F. & Pouget, M. (2004) Smooth surfaces, umbilics, lines of curvatures, foliations, ridges and the medial axis: a concise overview. (Doctoral dissertation, INRIA).
  8. Červený, V. (2001) Seismic ray theory. Cambridge: Cambridge University Press.
    [Google Scholar]
  9. Chapman, C.H. & Shearer, P.M. (1989) Ray tracing in azimuthally anisotropic media—II. Quasi‐shear wave coupling. Geophysical Journal International, 96, 65–83.
    [Google Scholar]
  10. Chopra, S. & Marfurt, K. (2007) Curvature attribute applications to 3D surface seismic data. The Leading Edge, 26, 404–414.
    [Google Scholar]
  11. Farra, V. & Pšenčík, I. (2021) Weak‐anisotropy approximation of P‐wave geometrical spreading in horizontally layered anisotropic media of arbitrary symmetry: TTI specification. Geophysics, 86, C119–C132.
    [Google Scholar]
  12. Grechka, V. (2013) Ray‐direction velocities in VTI media. Geophysics, 78, F1–F5.
    [Google Scholar]
  13. Grechka, V. (2017) Algebraic degree of a general group‐velocity surface. Geophysics, 82, WA45–WA53.
    [Google Scholar]
  14. Ibanez‐Jacome, W., Alkhalifah, T. & Bin Waheed, U. (2013) Effective orthorhombic anisotropic models for wavefield extrapolation. Geophysical Journal International, 198, 1653–1661.
    [Google Scholar]
  15. Masmoudi, N. & Alkhalifah, T. (2016b) A new parametrization for waveform inversion in acoustic orthorhombic media. Geophysics, 81, R157–R171.
    [Google Scholar]
  16. Oh, J. & Alkhalifah, T. (2018) Optimal full waveform inversion strategy for marine data in azimuthally rotated elastic orthorhombic media. Geophysics, 83, R307–R320.
    [Google Scholar]
  17. Oh, J. & Alkhalifah, T. (2019) Study on the full‐waveform inversion strategy for 3D elastic orthorhombic anisotropic media: application to ocean bottom cable data. Geophysical Prospecting, 67, 1219–1242.
    [Google Scholar]
  18. Ravve, I. & Koren, Z. (2016) Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium. Geophysical Prospecting, 64, 1235–1258.
    [Google Scholar]
  19. Ravve, I. & Koren, Z. (2017) Traveltime approximation in vertical transversely isotropic layered media. Geophysical Prospecting, 65, 1559–1581.
    [Google Scholar]
  20. Ravve, I. & Koren, Z. (2019) Slowness domain kinematical characteristics for horizontally layered orthorhombic media. Part I: critical slowness match. Geophysical Prospecting, 67, 1097–1133.
    [Google Scholar]
  21. Roberts, A. (2001) Curvature attributes and their application to 3D interpreted horizons. First Break, 19, 85–100.
    [Google Scholar]
  22. Sava, P. & Alkhalifah, T. (2013) Wide‐azimuth angle gathers for anisotropic wave‐equation migration. Geophysical Prospecting, 61, 75–91.
    [Google Scholar]
  23. Schoenberg, M. & Helbig, K. (1997) Orthorhombic media: modeling elastic wave behavior in a vertically fractured earth. Geophysics, 62, 1683–2002.
    [Google Scholar]
  24. Song, L.P. & Every, A.G. (2000) Approximate formulae for acoustic wave group slownesses in weakly orthorhombic media. Journal of Physics D: Applied Physics, 33, L81.
    [Google Scholar]
  25. Stovas, A. (2017) Kinematic parameters of pure‐ and converted‐mode waves in elastic orthorhombic media. Geophysical Prospecting, 65, 426–452.
    [Google Scholar]
  26. Stovas, A. & Alkhalifah, T. (2013) A titled transversely isotropic slowness surface approximation. Geophysical Prospecting, 61, 568–573.
    [Google Scholar]
  27. Stovas, A. & Fomel, S. (2012) Shifted hyperbola moveout approximation revisited. Geophysical Prospecting, 60, 395–399.
    [Google Scholar]
  28. Stovas, A. & Roganov, Y. (2009) Slowness surface approximations for qSV‐waves in transversely isotropic media. Geophysical Prospecting, 57, 1–11.
    [Google Scholar]
  29. Stovas, A., Roganov, Y. & Roganov, V. (2021a) Geometrical characteristics of phase and group velocity surfaces in anisotropic media. Geophysical Prospecting, 69, 53–69.
    [Google Scholar]
  30. Stovas, A., Roganov, Y. & Roganov, V. (2021b) Wave characteristics in elliptical orthorhombic media. Geophysics, 86, C89–C99.
    [Google Scholar]
  31. Thomsen, L. (1986) Weak elastic anisotropy. Geophysics, 51, 1954–1966.
    [Google Scholar]
  32. Tsvankin, I. (1997) Anisotropic parameters and P‐wave velocity for orthorhombic media. Geophysics, 62, 1292–1309.
    [Google Scholar]
  33. Tsvankin, I. (2012) Seismic signatures and analysis of reflection data in anisotropic media. Houston, TX: Society of Exploration Geophysicists.
    [Google Scholar]
  34. Vavryčuk, V. (2001) Ray tracing in anisotropic media with singularities. Geophysical Journal International, 145, 265–276.
    [Google Scholar]
  35. Waheed, U. & Alkhalifah, T. (2015) Effective ellipsoidal models for wavefield extrapolation in tilted orthorhombic media. Studia Geophysica et Geodaetica, 1–21.
    [Google Scholar]
  36. Xu, S. & Stovas, A. (2017) Three‐dimensional generalized non‐hyperbolic approximation for relative geometrical spreading. Geophysical Journal International, 211, 1140–1153.
    [Google Scholar]
  37. Xu, S., Stovas, A. & Hao, Q. (2017) Perturbation‐based moveout approximations in anisotropic media. Geophysical Prospecting, 65, 1218–1230.
    [Google Scholar]
  38. Xu, S., Stovas, A. & Sripanich, Y. (2018) An anelliptic approximation for geometric spreading in transversely isotropic and orthorhombic media. Geophysics, 83, C37–C47.
    [Google Scholar]
  39. Yilmaz, O., (2001) Investigations in geophysics: Seismic data analysis – processing, inversion, and interpretation of seismic data. Houston, TX: SEG.
    [Google Scholar]
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