OBTAINING THE PHASE VELOCITIES

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

$$S^2 (\phi) \ =\ S^{2}_{z} \cos^2 \phi + S^{2}_{x} \sin^2 \phi,$$ (25)

where $\phi$ is the ray angle measured from the vertical and S_* (the ray slowness) is

$$S_{*}^{2} \ =\ \frac{1}{W_*}.$$ (26)

To estimate S_*, I use the expression for the traveltime of a ray that travels a distance $l = \sqrt{ \Delta x^2 + \Delta z^2}$ between two points:

$$t^2 \ =\ { \Delta x^2 S_{x}^{2} + \Delta z^2 S_{z}^{2}}.$$ (27)

This equation, which has the same form as the isotropic moveout equation, is obtained after multiplying equation (25) by l². Velocities estimated from the moveout near one axis using this traveltime equation are called NMO velocities, and velocities estimated from arrival times along the same axis are called direct velocities. Hence the different names chosen for the phase velocities W_*.