Figure shows how the vertical double elliptic
approximation works for different angles around the vertical.
The parameters of the ellipses that approximate
the impulse response have been
calculated at the angles shown by the
straight lines.
The left column compares the given impulse responses for *P*- and
*SV*-waves (continuous lines)
with the elliptical approximations around the
vertical (dashed lines). With the four elliptical parameters obtained for
each aperture, I calculate the elastic constants of the medium
by using equation (22). From the estimated elastic
constants, I calculate the corresponding impulse responses for
both *P*- and *SV*-waves. The result is shown in the central
column (dashed lines) simultaneously with the
given impulse responses. In most cases (except at 40-degree
aperture) the agreement is excellent. At 40-degree aperture the
horizontal *P*-wave group velocity has been overestimated and the
shear wave triplication is larger than expected. The right column shows
the absolute value of the percentage error made in the estimation
of the elastic constants. For small angles ( 10 degrees), the error
is negligible. For angles between 10 and 30 degrees the error is
smaller in *W _{11}* (A) than in

Figure 1

In Figure , the elliptical approximations to the impulse
responses are calculated using two ray angles at a time, one zero
and one nonzero. In
Figure , I show what happens when all these angles
(or different VSP offsets)
are used simultaneously to calculate the elliptical approximations
and the elastic constants. The horizontal *P*-wave group velocity
is now slightly overestimated and the shear velocities are retrieved well.
The errors in the estimated elastic constants are 6%
in *W _{11}* (A) and 4% in

Figure 2

11/17/1997