## 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.

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