Figure shows the result of diffracting, then migrating,
a model consisting of five point diffractors arranged in a diamond.
Scalar Stolt diffraction and time-migration with an anisotropic
dispersion relation were used to do the calculations; the dispersion
relation is for the *q*P (left column) and *q*SV (right column)
wavetypes of the TI medium Greenhorn Shale (Jones and Wang, 1981).
The result is not perfect because of the offset truncation of the
modeled hyperboloids.
The right-hand depth scale on the time migrations uses the true
vertical velocities to convert from time to depth.

Figure shows the results of time-migrating the modeled
*q*P and *q*SV hyperboloids from Figure using
isotropic dispersion relations.
The velocities used are the usual ones determined from moveout,
.Even for the less extreme *q*P wavetype (left column) the
isotropic migration
manages to properly focus only the nearest-offset part of the hyperboloids.

There is a more subtle error in the isotropic migration as well.
The true vertical scale cannot be determined from surface data alone,
but isotropy makes no allowance for this ambiguity and just assumes
the vertical velocity is the same as the moveout velocity.
As a result, the right-hand depth scale for the *q*SV waves is off
by nearly a factor of two.
(The depth scale for the *q*P waves also mislocates the diffractors,
but only by about 5%.)

Figure shows the results of using the first anelliptic
approximation to time-migrate the modeled TI hyperboloids in
Figure . The approximation is meant to be fit using only
surface data,
so as in Figure the unknowable vertical velocity has been
set equal to the moveout velocity.
Unlike the isotropic approximation in Figure , though,
the first anelliptic approximation
does a good job of focusing the hyperboloids, especially for the less
extreme *q*P wavetype.

In Figure we improve the fit to the
second anelliptic approximation. The main difference is that
the *q*SV diffractors are better focused.
Since fitting the second anelliptic approximation assumes access to
horizontal as well as vertical data, the correct vertical velocity
has been used and the right-hand depth scales are correct.

11/17/1997