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The analytical kinematic results can be verified by
numerical computations of impulse responses
by wavefield migration and transformation of the resulting
prestack image cubes
into the angle domain.
Figure shows
four zero subsurface-offset sections cut through
the impulse responses computed by
wavefield-continuation anisotropic migration
for three anisotropic rocks and for an isotropic rock.
The first anisotropic rock is the Taylor Sand defined above.
The second is the Mesa Clay Shale as defined by Tsvankin (2001)
using the three Thomsen parameters:
.The third is derived from the Green River Shale
as described by Tsvankin (2001)
by halving the anisotropic parameters
( and );
that is, it is defined by the three Thomsen parameters:
.The strong anelliptic nature
of the original Green River Shale () causes
the group-slowness approximation
in equation 6 to break down,
and makes the kinematic computations based on ray tracing,
and thus on group velocity and angles,
inconsistent with wavefield migrations
based on the dispersion relation in equation 7.
Hereupon I will refer to this rock, for obvious reasons,
as the GreenLight River Shale.
Notice that the GreenLight River Shale
is still strongly elliptical.
The other parameters defining the impulse responses
are the same as for Figure ;
that is, *t*_{D}=.9 seconds, *m*_{D}=0 kilometers,
and *h*_{D}=.4 kilometers, and vertical slowness .Figure a shows the isotropic case,
Figure b shows the Taylor Sand case,
Figure c shows the Mesa Clay Shale case,
and
Figure d shows the GreenLight River Shale case.
As in Figure ,
the line superimposed onto the images represent the
impulse response computed using the kinematic expressions
in equations 18-24.
The kinematic curves perfectly predict the shape of the images
even for very steep dips.

Figure
shows two-dimensional slices cut through the cube obtained by
the transformation to the angle domain of the impulse responses
shown in Figure .
The slices are cut at the midpoint and depth corresponding
to the expected location of the impulse responses;
that is, at the location tracked by the lines shown
in Figure .
There are three lines superimposed onto the angle-domain images.
The solid lines display the numerical computation
of
by applying equation 25.
They perfectly track, as expected,
the results of the transformation of the prestack images
to angle domain.
The dotted lines display the phase aperture angle .As expected, they overlap with the solid line around
the zero midpoint (i.e. flat reflector),
and depart from them at larger midpoints,
which correspond to steeper reflections.
However, the error introduced by ignoring
the difference between
and is small,
and likely to be negligible in most practical situations.
Finally, the dashed lines display the
group aperture angle .The differences between and are substantial,
up to 20% in some cases.
Ignoring them might
be detrimental to the application of ADCIGs.
Notice that in the isotropic case the three lines
perfectly overlap and all of them match the image.

**Quad_hxd_.4-overn
**

Figure 5
Impulse responses evaluated at zero subsurface offset for four
rock types:
a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale,
and
d) GreenLight River Shale.
Superimposed onto the images are the impulse responses computed by
the kinematic expressions presented in
equations 18-24.

**Quad_Mx-Ang_hxd_.4-overn
**

Figure 6
Slices of the impulse responses transformed into the angle-domain
for four rock types:
a) Isotropic, b) Taylor Sand, c) Mesa Clay Shale,
and
d) GreenLight River Shale.
Superimposed onto the images there are the curves
computed by applying the kinematic analysis:
(dashed line),
(dotted line),
and
(solid line).

** Next:** Gulf of Mexico data
** Up:** Kinematic analysis of ADCIGs
** Previous:** Analytical evaluation of the
Stanford Exploration Project

11/1/2005