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The relevance of obtaining an estimation of the several angledependent
reflection coefficients is the prospect of later using these information
in a Zoeppritzbased elastic inversion. However, since the Zoeppritz
equations represent the planewave response of a planar interface,
it is necessary that the reflectivity estimation also represent
the in situ planewave response of the interface.
From the previous criteria, only the Vstack represents a reasonable
approximation to this requirement. To obtain an estimation completely
consistent with the planewave response it necessary to perform a
complete planewave decomposition of the upward and downward propagating
wavefields at each point of the model, for all time steps. Although
this criterion can be applied to general RTM schemes, it is particularly
appropriate for the oneexperiment approach described in
Cunha (1992), where a single wavefield (with combined information
form the modeled and recorded wavefields) is backward propagated
in time using a discontinuous background model.
The implementation of the planewave decomposition criterion for the
oneexperiment RTM scheme is outlined below

First, it is necessary to separate the downgoing and
upcoming components of the wavefield potential, using
the propagationdirection unit vector
Appendix A describes how to obtain the propagation direction
from the field potentials. The downgoing field (within the forward time
reference system) includes the incident and the transmitted wavefields,
while the upcoming field contains only the reflections^{}. Figure shows the
estimated propagation directions overlain on the wavefield and
Figure shows the resulting upcoming and downgoing
separated wavefields.

The second step involves the computation of the stacking power of
the two wavefields ( and ) along the directions
represented by the angle around each point of the model
As in the Vstack case, the stacking power here is computed outward
from the image point (stacking trajectories are like a clock's hand),
instead of across it. This implies that
. As discussed in appendix B
the point of a given interface that intersects the downgoing wavefield
at time t (reflection point) will have two directions (almost
opposite) for which the has a local maximum, corresponding
to the incident and transmitted waves, but only one direction for
which the is maximum.
wdir
Figure 7 Wavefield partition at an interface. The incident and transmitted wavefields
propagate downward while the reflected wavefield propagates upward. The
overlaid arrows indicate the local propagation direction estimated from
the wavefield as described in appendix A.
wavesep
Figure 8 Separation of the upwardpropagating and downwardpropagating parts of
a wavefield using the local propagation direction. (a) The downwardpropagating
part corresponding to the incident and transmitted wavefields. (b) The
upwardpropagating part corresponding to the reflected wavefields.

Evaluate the angle for which is maximum. For
reasons discussed in appendix B this angle determines
the subdomain in which the correlation
between the two stacking power functions is to be evaluated.

Next, the crosscorrelation of the two stacking powers, and
the autocorrelation of the stacking power of the downgoing
wavefield are evaluated, in the interval
for points where
and
for points where where is the reflection angle.

The estimated attribute is given by
 

 (12) 
where
mig1time
Figure 9 PP reflectivity for a small (6 time steps) time window of the wavefield
in Figure , for different values of the angle of incidence
.
Figure shows the twelve images obtained over a small time
window centered at the timeframe of Figure . Each of these
images correspond to at a different local angle of
incidence . The raytheoretical values for the angle of incidence
of the wavefronts that reach the two interfaces at that particular time
are: 48 degrees for upper interface and 30.5 degrees for the bottom
interface. From the figure it is clear that the point of the
upper interface that is imaged at this time has maximum energy at
the frames corresponding to 45 and 52.5 degrees, while for
the bottom interface the maximum occurs at 30 and 37.5 degrees.
Though special care will be required to avoid the undesirable focusing
at small angles that we observe in the figure, it is encouraging to see
that the location of the main maxima are consistent with the expected values.
Next: SUMMARY
Up: DEFINING AN IMAGING CRITERION
Previous: The Vstack criterion
Stanford Exploration Project
11/18/1997