## Approximate expression for the group velocity vector in weakly anisotropic media

Ivan Psencik & Vaclav Vavrycuk

### Summary

Determination of the direction of the group velocity vector or the ray vector (unit vector in the direction of the group velocity vector) corresponding to a given phase normal (unit vector in the direction of the phase velocity or slowness vector) in an arbitrary anisotropic medium can be performed using the well-known ray tracing formula. Determination of the phase normal from the direction of the group velocity vector (ray vector) is a more complicated task, which is usually solved iteratively. We review the derivation of an approximate, first-order ray perturbation formula solving this problem and test it on several examples. The formula can be used for the determination of the ray vector from a given phase normal and vice versa. The formula is applicable to qP as well as qS waves in arbitrary weakly anisotropic media, in regions, in which the waves can be dealt with separately (i.e. outside singular regions of qS waves). We show that the formula for the determination of the ray vector from phase normal can roughly describe even triplications in regions corresponding to concave sections of slowness surface. The reverse formula for the determination of the phase normal from the ray vector yields satisfactory results for waves with convex sections of slowness surfaces but fails for waves with concave slowness surface sections. In addition to the described formula approximate expressions for the phase and group velocities of qP and qS waves are also presented and tested. Although anisotropy of some test samples is rather high, the approximate formulae yield surprisingly good results.

### Whole paper

The paper is available in PostScript (1126 kB !, colour figures) and GZIPped PostScript (225 kB, colour figures).

In: Seismic Waves in Complex 3-D Structures, Report 11, pp. 279-300, Dep. Geophys., Charles Univ., Prague, 2001.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .