## A note on two-point paraxial travel times

**Vlastislav Cerveny**
### Summary

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or
anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point
*S'* to point *R'*, both these points being
situated close to a known reference ray Ω,
along which the ray-propagator matrix was calculated by dynamic ray tracing.
The reference ray and
the position of points *S'* and *R'* are specified in Cartesian coordinates. Two such
expressions for the two-point paraxial travel time play an important role. The first is based on the
4×4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred
coordinates. The second requires the knowledge of the 6×6 ray propagator matrix computed by
dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully
independently, using different methods, and are expressed in quite different forms. In this paper we prove that
the two expressions are fully equivalent and can be transformed into each other.

### Keywords

Seismic anisotropy, theoretical seismology, wave propagation,
paraxial ray methods.

### Whole paper

The reprint is available in
PDF (123 kB).

*Stud. geophys. geod.*, **57** (2013), 267-275.