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Equation (5) contains all the velocity information in the data
and can be precomputed, at least in part. Notice that, although we described
the algorithm for , that is not necessary because the range of
velocities is limited and independent of the spatial dimensions of the data
(although the velocities themselves vary spatially).
Start by binning the velocities in small bins, for
example at 10 m/s (which would imply a maximum velocity error of 5 m/s,
well below the likely error in the estimation of the velocities themselves)
such that the vertical wavenumber k_{zl} (that is, the dispersion
relation),
needs to be computed only a few hundred times and can thus be stored as a
function of the horizontal wavenumber and the velocity. From the standpoint of
the theoretical algorithm, all that changes is the selection process to
choose the trace from the extrapolated wavefield that corresponds to the
binned velocity at each spatial location. That is, instead of the selection
being simply a multiplication by a Kronecker delta to choose l=j as it was
before, it is now a multiplication with a Kronecker delta, to select
l=p(j), that is, the wavefield that was migrated with the binned velocity
corresponding to the bin of V(j).
Equation (2) can then be rewritten as:
The equation for the wavefield extrapolation then becomes:
 
(6) 
Notice that summation over l involves summing over all the binned velocities
whereas summation over p involves selecting the different wavefield
components that correspond to a given velocity. That is, p ranges over
the spatial locations whose binned velocity is V_{l} for each l.
Figure shows the velocity selection. This time, since
we don't have a wavefield migrated with each velocity, at each spatial
location, it is likely that several locations correspond to the same wavefield,
since they correspond to the same velocity bin. There is, obviously, just
one possible velocity at each spatial location, but many spatial locations
may share the same velocity. Also, it is possible for a particular velocity
not to be required at a specific depth step.
bin_vels2
Figure 2 Diagram illustrating velocity selection
when there are fewer velocities than spatial locations.

 
Next: Computation of the horizontal
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Stanford Exploration Project
5/3/2005