Transformation relations for second-order 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 the second-order derivatives of the travel-time field at the reference ray. 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 system in ray-centred coordinates. In this paper, the transformation relations between the second-order derivatives of the travel-time field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used both 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 displacementvector, Gaussian beams, dynamic ray tracing, second-order travel-time derivatives.

Whole paper

The image of the paper in GIF 150dpi (417 kB) is designed for an instant screen preview.

The reprint is available in PDF (221 kB).

Stud. geophys. geod., 54 (2010), 257-267.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .