Abstract
We consider the partial derivatives of travel time with respect to both spatial coordinates and perturbation parameters. These derivatives are very important in studying wave propagation and have already found various applications in smooth media without interfaces. In order to extend the applications to media composed of layers and blocks, we derive the explicit equations for transforming these travel–time derivatives of arbitrary orders at a general smooth curved interface between two arbitrary media. The equations are applicable to both real–valued and complex–valued travel time. The equations are expressed in terms of a general Hamiltonian function and are applicable to the transformation of travel–time derivatives in both isotropic and anisotropic media. The interface is specified by an implicit equation. No local coordinates are needed for the transformation.
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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., 2008. Numerical comparison of the isotropic–common–ray and anisotropic–common–ray approximations of the coupling ray theory. Geophys. J. Int., 175, 357–374.
Cervený V., 1972. Seismic rays and ray intensities in inhomogeneous anisotropic media. Geophys. J. R. Astr. Soc., 29, 1–13.
Cervený V., 2001. Seismic Ray Theory. Cambridge Univ. Press, Cambridge.
Cervený V., Klimeš L. and Pšencík I., 1988. Complete seismic–ray tracing in three–dimensional structures. In: Doornbos D.J. (Ed.), Seismological Algorithms. Academic Press, New York, 89–168.
Cervený V., Klimeš L. and Pšencík I., 2008. Attenuation vector in heterogeneous, weakly dissipative, anisotropic media. Geophys. J. Int., 175, 346–355.
Cervený V. and Pšencík I., 2009. Perturbation Hamiltonians in heterogeneous anisotropic weakly dissipative media. Geophys. J. Int., 178, 939–949.
Duchkov A.A. and Goldin S.V., 2001. Analysis of seismic wave dynamics by means of integral representations and the method of discontinuities. Geophysics, 66, 413–418.
Goldin S.V. and Duchkov A.A., 2003. Seismic wave field in the vicinity of caustics and higher–order travel time derivatives. Stud. Geophys. Geod., 47, 521–544.
Hamilton W.R., 1837. Third supplement to an essay on the theory of systems of rays. Trans. Roy. Irish Acad., 17, 1–144.
Klimeš L., 2002a. Second–order and higher–order perturbations of travel time in isotropic and anisotropic media. Stud. Geophys. Geod., 46, 213–248.
Klimeš L., 2002b. Application of the medium covariance functions to travel–time tomography. Pure Appl. Geophys., 159, 1791–1810.
Klimeš L., 2006. Spatial derivatives and perturbation derivatives of amplitude in isotropic and anisotropic media. Stud. Geophys. Geod., 50, 417–430.
Klimeš L., 2013. Paraxial Super–Gaussian beams. Seismic Waves in Complex 3–D Structures, 23, 145–148 (http://sw3d.cz).
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.
Klimeš L. and Bulant P., 2012. Single–frequency approximation of the coupling ray theory. Seismic Waves in Complex 3–D Structures, 22, 143–167 (http://sw3d.cz).
Klimeš L. and Bulant P., 2014. Prevailing–frequency approximation of the coupling ray theory for S waves along the SH and SVreference rays in a transversely isotropic medium. Seismic Waves in Complex 3–D Structures, 24, 165–177 (http://sw3d.cz).
Klimeš L. and Bulant P., 2015. Ray tracing and geodesic deviation of the SH and SV reference rays in a heterogeneous generally anisotropic medium which is approximately transversely isotropic. Seismic Waves in Complex 3–D Structures, 25, 187–208 (http://sw3d.cz).
Klimeš L. and Bulant P., 2016. Prevailing–frequency approximation of the coupling ray theory for electromagnetic waves or elastic S waves. Stud. Geophys. Geod., 60, 419–450.
Klimeš M. and Klimeš L., 2011. Perturbation expansions of complex–valued traveltime along real–valued reference rays. Geophys. J. Int., 186, 751–759.
Shekar B. and Tsvankin I., 2014. Point–source radiation in attenuative anisotropic media. Geophysics, 79, WB25–WB34.
Zheng C., 2010. Gaussian beam approach for the boundary value problem of high frequency Helmholtz equation. Commun. Math. Sci., 8, 1041–1066.
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Klimeš, L. Transformation of spatial and perturbation derivatives of travel time at a curved interface between two arbitrary media. Stud Geophys Geod 60, 451–470 (2016). https://doi.org/10.1007/s11200-015-0479-8
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DOI: https://doi.org/10.1007/s11200-015-0479-8