Transformation relations for second derivatives of travel time in anisotropic media

Vlastislav Cerveny & Ludek Klimes


In the computation of paraxial travel times and Gaussian beams, the basic role is played by second derivatives of the travel-time field at the reference ray Ω. These derivatives can be determined by dynamic ray tracing (DRT) along the ray. Two basic DRT systems have been broadly used in applications: the DRT system in Cartesian coordinates and the DRT in ray-centred coordinates. In this paper, the transformation relations between the second derivatives of the travel time field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used in many applications in isotropic and anisotropic media, including computations of complex-valued travel times necessary for the evaluation of Gaussian beams.


Paraxial travel times, paraxial approximation of the displacement vector, Gaussian beams, dynamic ray tracing, second-order travel time derivatives.

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In: Seismic Waves in Complex 3-D Structures, Report 19, pp. 115-122, Dep. Geophys., Charles Univ., Prague, 2009.
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