The expressions previously derived for the elastic constants are a function of the phase velocities. However, only in a few cases phase velocities can be easily estimated from traveltime measurements that give, instead, group velocities. The correspondence between phase and group velocities is trivial only when the model is isotropic or elliptically anisotropic or when the ray travels along an axis of symmetry.

Equations (3), (6), (12), and (15) show that the phase velocities of
*P*- and *SV*-wave are
elliptical near the axes of symmetry. Those of *SH*-waves
are also elliptical [equation (1b)].
It has been shown that when the phase velocity
has an elliptical shape, the corresponding impulse response
is also elliptical (Levin, 1978; Byun, 1982).
Therefore, the group slowness expression that
corresponds to these equations has the general form

(25) |

(26) |

To estimate *S _{*}*, I use the expression
for the traveltime
of a ray that travels a distance
between two points:

(27) |

11/17/1997