Abstract
Anisotropic common S-wave rays are traced using the averaged Hamiltonian of both S-wave polarizations. They represent very practical reference rays for calculating S waves by means of the coupling ray theory. They eliminate problems with anisotropic-ray-theory ray tracing through some S-wave slowness-surface singularities and also considerably simplify the numerical algorithm of the coupling ray theory for S waves.
The equations required for anisotropic-common-ray tracing for S waves in a smooth elastic anisotropic medium, and for corresponding dynamic ray tracing in Cartesian or ray-centred coordinates, are presented. The equations, for the most part generally known, are summarized in a form which represents a complete algorithm suitable for coding and numerical applications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bakker P.M., 2002. Coupled anisotropic shear wave raytracing in situations where associated slowness sheets are almost tangent. Pure Appl. Geophys., 159, 1403–1417.
Bucha V. and Bulant P. (Eds.), 2003. SW3D-CD-7 (CD-ROM). Seismic Waves in Complex 3-D Structures, Report 13, Dep. Geophys., Charles Univ., Prague, 251–251, online at “http://sw3d.mff.cuni.cz”.
Bucha V. and Bulant P. (Eds.), 2004. SW3D-CD-8 (CD-ROM). Seismic Waves in Complex 3-D Structures, Report 14, Dep. Geophys., Charles Univ., Prague, 229–229, online at “http://sw3d.mff.cuni.cz”.
Bulant P. and Klimeš L., 2002. Numerical algorithm of the coupling ray theory in weakly anisotropic media. Pure Appl. Geophys., 159, 1419–1435.
Bulant P. and Klimeš L., 2004. Comparison of quasi-isotropic approximations of the coupling ray theory with the exact solution in the 1-D anisotropic “oblique twisted crystal” model. Stud. Geophys. Geod., 48, 97–116.
Bulant P. and Klimeš L., 2006. Numerical comparison of the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory. Seismic Waves in Complex 3-D Structures, Report 16, Dep. Geophys., Charles Univ., Prague, 155–178, online at “http://sw3d.mff.cuni.cz”.
Bulant P., Klimeš L., Pšenčík I. and Vavryčuk V., 2004. Comparison of ray methods with the exact solution in the 1-D anisotropic “simplified twisted crystal” model. Stud. Geophys. Geod., 48, 675–688.
Červený V., 1972. Seismic rays and ray intensities in inhomogeneous anisotropic media. Geophys. J. R. Astr. Soc., 29, 1–13.
Červený V., 2001. Seismic Ray Theory. Cambridge Univ. Press, Cambridge.
Červený V., Klimeš L. and Pšenčík I., 1988. Complete seismic-ray tracing in three-dimensional structures. In: Doornbos D.J. (Ed.): Seismological Algorithms, Academic Press, New York, 89–168.
Coates R.T. and Chapman C.H., 1990. Quasi-shear wave coupling in weakly anisotropic 3-D media. Geophys. J. Int., 103, 301–320.
Hanyga A., 1982. Dynamic ray tracing in an anisotropic medium. Tectonophysics, 90, 243–251.
Kendall J-M., Guest W.S. and Thomson C.J., 1992. Ray-theory Green’s function reciprocity and ray-centred coordinates in anisotropic media. Geophys. J. Int., 108, 364–371.
Klimeš L., 1994. Transformations for dynamic ray tracing in anisotropic media. Wave Motion, 20, 261–272.
Klimeš L., 2002. Transformations for dynamic ray tracing in anisotropic media with a homogeneous Hamiltonian of an arbitrary degree. Seismic Waves in Complex 3-D Structures, Report 12, Dep. Geophys., Charles Univ., Prague, 67–78, online at “http://sw3d.mff.cuni.cz”.
Klimeš L., 2006. Ray-centred coordinate systems in anisotropic media. Stud. Geophys. Geod., 50, 431–447.
Klimeš L. and Bulant P., 2004. Errors due to the common ray approximations of the coupling ray theory. Stud. Geophys. Geod., 48, 117–142.
Klimeš L. and Bulant P., 2006. Errors due to the anisotropic-common-ray approximation of the coupling ray theory. Stud. Geophys. Geod., 50, 463–477.
Luneburg R.K., 1944. Mathematical Theory of Optics. Lecture notes, Brown University, Providence, Rhode Island. Reedition: University of California Press, Berkeley and Los Angeles, 1964.
Popov M.M. and Pšenčík I., 1978a. Ray amplitudes in inhomogeneous media with curved interfaces. Travaux Instit. Géophys. Acad. Tchécosl. Sci. No. 454, Geofys. Sborník, 24, 111–129, Academia, Praha.
Popov M.M. and Pšenčík I., 1978b. Computation of ray amplitudes in inhomogeneous media with curved interfaces. Stud. Geophys. Geod., 22, 248–258.
Pšenčík I., 1998. Green’s functions for inhomogeneous weakly anisotropic media. Geophys. J. Int., 135, 279–288.
Pšenčík I. and Dellinger J., 2001. Quasi-shear waves in inhomogeneous weakly anisotropic media by the quasi-isotropic approach: A model study. Geophysics, 66, 308–319.
Vavryčuk V., 2001. Ray tracing in anisotropic media with singularities. Geophys. J. Int., 145, 265–276.
Vavryčuk V., 2003. Behavior of rays near singularities in anisotropic media. Phys. Rev. B, 67, 054105-1-054105-8.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Klimeš, L. Common-ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium. Stud Geophys Geod 50, 449–461 (2006). https://doi.org/10.1007/s11200-006-0028-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11200-006-0028-6