Abstract
Most common parameterization of anisotropic media is by twenty one independent elements aijkl of the density-normalized stiffness tensor or by twenty one independent elements Aαβ of the density-normalized matrix of elastic parameters in the Voigt notation. These parameters are commonly of significantly different sizes, are dimensional, in (km/s)2, often appear in combinations. We are offering an alternative parameterization by twenty one A-parameters (anisotropic parameters), which removes the mentioned disadvantages and possesses some additional useful properties. For example, axes or planes of coordinate systems, in which A-parameters are defined, need not be related to symmetry axes or planes of the considered anisotropy symmetry as required in other similar parameterizations. In combination with the first-order weak-anisotropy approximation, in which anisotropy is considered as the first-order perturbation of reference isotropy, parameterization by A-parameters yields insight into the role of individual A-parameters in the wave propagation problems. For example, it turns out that in the first-order weak-anisotropy approximation, P- and S-wave velocities are each controlled by fifteen A-parameters. A set of six of them appears only in the expression for P-wave velocity, a set of other six A-parameters appears only in S-waves velocity expressions. Remaining set of nine A-parameters is common for both waves. We present transformation of A-parameters, analogue to Bond transformation, and useful formulae for the weak-anisotropy approximation for anisotropy of any symmetry and arbitrary tilt.
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References
Bakker P., 2002. Coupled anisotropic shear-wave ray tracing in situations where associated slowness sheets are almost tangent. Pure Appl. Geophys., 159, 1403–1417
Bond W., 1943. The mathematics of the physical properties of crystals. Bell System Technical Journal, 22, https://doi.org/10.1002/j.1538-7305.1943.tb01304.x
Carcione J.M., 2014. Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media. 3rd Edition. Elsevier Science, Oxford, U.K.
Červený V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge, U.K.
Chapman C.H., 2004. Fundamentals of Seismic Wave Propagation. Cambridge University Press, Cambridge, U.K.
Farra V. and Pšenčík I., 2003. Properties of the zero-, first- and higher-order approximations of attributes of elastic waves in weakly anisotropic media. J. Acoust. Soc. Am., 114, 1366–1378
Farra V. and Pšenčík I., 2008. First-order ray computations of coupled S waves in inhomogeneous weakly anisotropic media. Geophys. J. Int., 173, 979–989
Farra V. and Pšenčík I., 2016. Weak-anisotropy approximations of P-wave phase and ray velocities for anisotropy of arbitrary symmetry. Stud. Geophys. Geod., 60, 403–418
Farra V. and Pšenčík I., 2023. PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt. J. Appl. Geophys., 215, Art.No. 105062, https://doi.org/10.1016/j.jappgeo.2023.105062
Farra V., Pšenčík I. and Jílek P., 2016. Weak-anisotropy moveout approximations for P waves in homogeneous layers of monoclinic or higher anisotropy symmetries. Geophysics, 81, C39–C59
Fedorov F.I., 1968. Theory of Elastic Waves in Crystals. Plenum Publ., New York
Gomes E., Zheng X., Pšenčík I., Horne S. and Leaney S., 2004. Local determination of weak anisotropy parameters from a walkaway VSP qP-wave data in the Java Sea region. Stud. Geophys. Geod., 48, 215–230
Klimeš L., 2006. Common-ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium. Stud. Geophys. Geod., 50, 449–461
Mensch T. and Rasolofosaon P., 1997. Elastic-wave velocities in anisotropic media of arbitrary symmetry - generalization of Thomsens parameters ε, δ and γ. Geophys. J. Int., 128, 43–64
Pšenčík I. and Gajewski D., 1998. Polarization, phase velocity and NMO velocity of qP waves in arbitrary weakly anisotropic media. Geophysics, 63, 1754–1766
Pšenčík I., Růžek B. and Jílek 2020. Practical concept of arbitrary anisotropy applied in traveltime inversion of simulated P-wave VSP data. Geophysics, 85, C107–C123
Pšenčík I., Růžek B., Lokajíček T. and Svitek T., 2018. Determination of rock-sample anisotropy from P- and S-wave traveltime inversion. Geophys. J. Int., 214, 1088–1104
Sayers C.M., 1994. P-wave propagation in weakly anisotropic media. Geophys. J. Int., 116, 799–805
Thomsen L., 1986. Weak elastic anisotropy. Geophysics, 51, 1954–1966
Tsvankin I., 1997. Anisotropic parameters and P-wave velocities for orthorhombic media. Geophysics, 62, 1292–1309
Tsvankin I. and Grechka V., 2011. Seismology of azimuthally anisotropic media and seismic fracture characterization. Society of Exploration Geophysicists, Tulsa, OK
Zheng X. and Pšenčík I., 2002. Local determination of weak anisotropy parameters from the qP-wave slowness and particle motion measurements. Pure Appl. Geophys., 159, 1881–1905
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We are grateful to the Research Project 20-06887S of the Czech Science Foundation for support.
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Pšenčík, I., Farra, V. Parameterization of anisotropic media by A-parameters. Stud Geophys Geod 68, 41–60 (2024). https://doi.org/10.1007/s11200-023-1136-2
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DOI: https://doi.org/10.1007/s11200-023-1136-2