**Elastic constants from
phase velocities near the vertical.-** When the phase
angles are close to zero, it is possible to estimate elastic constants
from the corresponding phase velocities
by using equations (4), (5), (7), and (8), a
system of four equations and four unknowns. The independent
term (that I haven't explained how to obtain yet)
is formed by *W*_{P,z}, *W*_{P,xnmo}, *W*_{SV,z}, and *W*_{SV,xnmo}.
The solution of this system of
equations is

(1) | ||

(2) | ||

(3) | ||

(4) |

Since this approximation simultaneously uses
the elliptical parameters of two
ellipses fitted near the vertical, I call it *vertical* double
elliptic approximation, analogous to Muir's double elliptic
approximation that uses horizontal and vertical ellipses
to approximate the slowness surface and impulse response for
all angles (Muir, 1990a; Dellinger et al., 1992).
It is important to point out the differences between these two
approximations.
On the one hand, the *vertical* double
elliptic approximation is used to estimate elastic constants from
phase velocities
near the vertical axis.
Slowness surfaces and impulse responses can be calculated from these elastic
constants
using (1a) and the exact relationships between phase and group velocities.
On the other hand, Muir's double elliptic approximation is used to
estimate directly slowness surfaces and impulse responses from
data near both axes
without having to know the elastic constants.

Fitting *P*- and *SV*-wave
phase velocities with ellipses near the vertical
is not the same as using the vertical double elliptic approximation.
However, the fitting is a necessary intermediate step in the estimation
of elastic
constants using equations (22). When the elliptical
fitting is done near the horizontal
axis the result is the *horizontal* double elliptic
approximation, as follows.

**Elastic constants from
phase velocities near the horizontal.-** When the phase angle
is close to 90 degrees, the expressions for the elastic constants
as a function of *P*- and *SV*-wave
phase velocities are obtained by solving
the system of equations
(13), (14), (16), and (17), with the following result:

(1) | ||

(2) | ||

(3) | ||

(4) |

The estimation of *W _{33}* from near-horizontal phase velocities (equation
(23d))
and

11/17/1997