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``Mathematical derivations of integral operators assume that physical
quantities such as time, source-receiver midpoint, azimuth and offset
are continuous. In practice, the resulting algorithms are simply then
applied to the discretely sampled seismic field data'' Beasley and Klotz (1992).
However, a fundamental problem exists in the implementation of
integrals as discrete summations;
the integrands must be regularly and adequately sampled.
A way of thinking that can be very
helpful takes us back to the definition of definite 3D integrals, and their
volume approximations. This suggests that we should
consider a multiplicity concept
to avoid summing over redundant traces within
bin elements.
A straightforward extension of the NMO-Stack compensation for variable
fold, based on dividing the summed data by a function of the multiplicity,
cannot be used for our application since traces
within local bins may have different azimuths and offsets.
The summed trace will then suffer severe amplitude and phase
distortions.
What we propose as a new development in our imaging sequence is
a normalization procedure which includes
all the traces
within each CMP bin in the discrete summation while it normalizes
each input trace
by the local fold of its corresponding bin.
This solution is accurate for the case in which all the traces within the
CMP bin share the same offset and azimuth
and provides a good approximation for the case of small
variations in offset and azimuth.
Application of this normalization procedure leads to an approximation
of the survey subdivision by a single-fold 3D subset
that is better suited for
prestack processing when the wave equation operator is implemented as a
discrete integral summation.

**model
**

Figure 4 Synthetic reflectivity model of a flat reflector. The white circles are zones
of high reflection coefficient of 2.5 in a constant background of unit reflectivity.

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Stanford Exploration Project

11/11/1997