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(18) |
Because of the linearity
of equations 12
and 13,
the data computed by modeling the subset Aj
can be expressed as the sum of the data sets
obtained by modeling each independently;
that is,
![]() |
(19) | |
(20) |
The result of migrating this combined data set can be written as follows:
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||
(21) |
The first term in equation 22 is the desired result; that is, the image that we would obtain if we had independently modeled and imaged each SODCIG belonging to Aj, and summed the results. The second term in equation 22 represents the ``cross-talk'' between the SODCIGs; these artifacts are the unwanted consequence of combining SODCIGs before modeling in order to save computations.
The second term in equation 22
becomes easier to analyze
in the special case when migration
velocity is the same as the modeling velocity.
The ``residual propagation'' operator
thus
approximates a delta function
and equation 22 simplifies into:
![]() |
(22) |
In this case, the the cross-talks terms are given by the product
of each SODCIG in Aj,
shifted by the subsurface offset ,
with all the other SODCIG in Aj, shifted by
.If we assume that the SODCIGs have limited subsurface offset range
because they are partially focused by migration,
we can easily eliminate the cross-talks interference with
the desired image in a window around zero subsurface offset
by ensuring that the SODCIG belonging to Aj
are sufficiently separated in space.
The numerical examples in the next section demonstrates
this point.
However, if the migration and modeling velocities are dissimilar,
the shifted versions of the SODCIGs
contributing to the cross-talk are distorted and shifted by
the ``residual propagation'' operator
(equation 22).
This additional shift may increase, or decrease, the amount
of interference of the cross talks with the desired image.
The last example in the next section illustrates this point.