Computation of velocities and polarization vectors in weakly anisotropic media

Authors

  • Yu. V. Roganov Tesseral Technologies Inc., Ukraine
  • A. Stovas Norwegian University of Science and Technology, Norway
  • V. Yu. Roganov Glushkov Institute of Cybernetic of NAS of Ukraine, Ukraine

DOI:

https://doi.org/10.24028/gzh.v43i3.236381

Keywords:

phase velocity, group velocity, polarization vector, Christoffel equation, perturbation theory

Abstract

To compute the phase velocities in the weakly anisotropic media, we propose to transform the Christoffel matrix K into an adapted coordinate system, and, then, apply the perturbation theory to the resulting matrix X. For a weakly anisotropic medium, the off-diagonal elements of the matrix X are small compared to the diagonal ones, and two of them are equal to 0. The diagonal elements of the matrix X are initial approximations of the phase velocities squared. To refine them, it is proposed to use either iterative schemes or Taylor series expansions. The initial terms of the series and the formulas of iterative schemes are expressed through the elements of the matrix X and have a compact analytical representation. The odd-order terms in the series are equal to 0. To approximate the phase velocities of the S1 and S2 waves, a stable method is proposed based on solving a quadratic equation with the coefficients being expressed in terms of the matrix elements and the precomputed value of the qP wave phase velocity squared. For all iterative schemes and series, the convergence conditions are derived. The polarization vector of the wave with the square of the phase velocity  is defined as the column with maximum modulus of cofactor of the matrix K-I. The group velocities vectors are computed based on the known components of the polarization vector, the directional vector, and the density-normalized stiffness coefficients. The computational accuracy is demonstrated for the standard orthorhombic model. It is shown how the perturbation theory can be applied to media with strong anisotropy. To do this, first we need to apply several QR transforms or Jacobi rotations of the Christoffel matrix, and then use the perturbation theory. This method with four Jacobi rotations is applied to the calculation of the phase velocities squared for a triclinic medium with a maximum number (32) of singularity points. In this case, the phase velocities are computed with a relative error less than 0,004 %.

References

Bronstein, I. N., & Semendyaev, K. A. (1986). Handbook of mathematics. Moscow: Nauka, 545 p. (in Russian).

Kato, T. (1972). Perturbation theory for linear operators. Moscow: Mir, 740 p. (in Russian).

Madelung, E. (1968). Mathematical tools for the physicist. Moscow: Nauka, 604 p. (in Russian).

Parlett, B. (1983). The symmetric eigenvalue problem. Moscow: Mir, 384 p. (in Russian).

Petrashen, G. I. (1980). Wave propagation in anisotropic elastic media. Leningrad: Nauka, 280 p. (in Russian).

Roganov, Yu., Stovas, A., & Roganov, V. (2020). Dispersion of phase velocities in horizontally layered anisotropic weak contrast periodic media. Geofizicheskiy Zhurnal, 42(3), 109—126. https://doi.org//10.24028/gzh.0203-3100.v42i3.2020.204704 (in Russian).

Roganov, Yu., Stovas, A., & Roganov, V. (2019). Propeties of acoustic axes in triclinic media. Geofizicheskiy zhurnal, 41(3), 3—17. https://doi.org//10.24028/gzh.0203-3100.v41i3.2019.172417 (in Russian).

Samarskiy, A. A. (1987). Introduction to numerical methods. Moscow: Nauka, 271 p. (in Russian).

Wilkinson, J. H. (1970). The algebraic eigenvalue problem. Moscow: Nauka, 564 p. (in Russian).

Fedorov, F. I. (1965). Theory of elastic waves in crystals. Moscow: Nauka, 386 p. (in Russian).

Horn, R., Johnson, C. (1989). Matrix analysis. Moscow: Mir, 655 p. (in Russian).

Abedi, M. M. & Stovas, A. (2019). Extended gene¬ra¬lized non-hyperbolic moveout approxima¬ti¬on. Geophysical Journal International, 216(2), 1428—1440. https://doi.org/10.1093/gji/ ggy504.

Červeny, V. (2005). Seismic Ray Theory. Prague: Charles University, 724 p.

Farra, V. (2004). Improved first-order approximation of group velocities in weakly anisotro¬pic Media. Studia Geophysica et Geodeti¬ca, 48, 199—213. https://doi.org/10.1023/B:SGEG. 0000015592.36894.3b.

Farra, V. (2005a). High order expressions of the phase velocity and polarization of qP and qS waves in anisotropic media. Geophysical Journal International, 147(1), 93—105. https://doi.org/10.1046/j.1365-246X.2001.00510.x.

Farra, V. (2005b). First-order ray tracing for qS waves in inhomogeneous weakly anisotropic me¬dia. Geophysical Journal International, 161(2), 309—324. https://doi.org/10.1111/j. 1365-246X.2005.02570.x.

Farra, V. & Pšenčík, I. (2003). Properties of the zero-, first- and higher-order approximations of attributes of elastic waves in weakly anisotropic media. Journal of the Acoustical Society of America, 114, 1366—1378. https://doi.org/10.1121/1.1591772.

Farra, V., Pšenčík, I. & Jílek, P. (2016). Weak-anisotropy moveout approximations for P-waves in homogeneous layers of monoclinic or higher anisotropy symmetries. Geophysics, 81(2), C39—C59. https://doi.org/10.1190/geo2015-0223.1.

Jech, J. & Pšenčík, I. (1989). First-order perturbation method for anisotropic media. Geophysical Journal International, 99(2), 369—376. https://doi.org/10.1111/j.1365-246X.1989.tb01694.x.

Ohanian, V., Syder, T. M. & Carcione, J. (2006) Weak Elastic Anisotropy by Perturbation Theory. Geophysics, 71(3), D45D58. https://doi.org/10.1190/1.2194520.

Pšenčík, I. & Gajewski, D. (1998). Polarization, phase velocity and NMO velocity of qP wa¬ves in arbitrary weakly anisotropic media. Geo¬physics, 63(5), 1754—1766 https://doi.org/10. 1190/1.1444470.

Pšenčík, I. & Farra, V. (2005). First-order ray tracing for qP waves in inhomogeneous weakly anisotropic media. Geophysics, 70(10), D65—D75. https://doi.org/10.1190/1.2122411.

Roganov, Yu., & Stovas, A. (2014). Low-frequency normal wave propagation in a periodically layered medium with weak contrast in elastic properties. Geophysical Prospecting, 62(4), 1205—1210. https://doi.org/10.1111/1365-2478.12167.

Roganov, Yu., Stovas, A., & Roganov, V. (2019). Low-frequency layer-induced dispersion in a weak contrast vertically heterogeneous orthorhombic medium. Geophysical Prospecting, 67(9), 2269—2279. https:// doi.org/10.1111/1365-2478.12804.

Rommel, B. E. (1994). Approximate polarization of plane waves in a medium having weak transverse isotropy: Geophysics, 59, 1605—1612. https://doi.org/10.1190/1.1443549.

Schoenberg, M., & Helbig, K. (1997). Orthorhombic media: Modeling elastic wave behavior in a vertically fractured earth. Geophysics, 62(6), 19541974. https://doi.org/10.1190/1.1444297.

Smith, O. K. (1961). Eigenvalues of a Symmetric 3×3 Matrix. Communications ACM, 4, 168. http://dx.doi.org/10.1145/355578.366316.

Stovas, A. & Fomel, S. (2017). The generalized moveout approximation: a new parameter selection, Geophysical Prospecting, 65(3), 687695. https://doi.org/10.1111/1365-2478.12445.

Stovas, A. & Fomel, S. (2019). Generalized velocity approximation, Geophysics, 84(1), C27C40. https://doi.org/10.1190/geo2018-0401.1.

Stovas, A., Roganov, Yu. & Roganov, V. (2021). Waves in elliptical orthorhombic model. Geophysics (aссepted).

Thomsen, L. (1986). Weakly Elastic Anisotropy. Geophysics, 51, 19541966. https://doi.org/10.1190/1.1442051.

Vavryčuk, V. (2005). Acoustic axes in triclinic anisotropy. The Journal of the Acoustical Society of America, 118, 647653. http://dx.doi.org/10.1121/1.1954587.

Wang, Y., Nemeth, T. & Langan, R. (2006). An expanding-wavefront method for solving the eikonal equations in general anisotropic media. Geophysics, 51, T129T135. https://doi.org/10.1190/1.2235563.

Xu, S., Stovas, A. & Hao, Q. (2017). Perturbation-based moveout approximation in anisotropic media, Geophysical Prospecting, 65(5), 12181230. https://doi.org/10.1111/1365-2478. 12480.

Published

2021-10-05

How to Cite

Roganov, Y. V. ., Stovas, A. ., & Roganov, V. Y. . (2021). Computation of velocities and polarization vectors in weakly anisotropic media. Geofizicheskiy Zhurnal, 43(3), 64–81. https://doi.org/10.24028/gzh.v43i3.236381

Issue

Section

Articles