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For data I began with a CMP gather used by Dave Hale. Denoting missing by m and known by k I interleaved and extended the data offsets by
The movie illustrated that a concept familiar in Fourier analysis also applies to the velocity transform. This is the idea that roughness in the interior of the data corresponds to energy ``off ends'' in velocity space whereas energy ``off ends'' in offset space, controls resolution in the interior in velocity space.
The movie also illustrated that a subjective choice of the
best value of was a function of location in the (x,t)-plane.
The best choices of
for internal regions
and for external regions were quite different.
Internally the best choice was one that would
make interleaved traces look most like their neighbors
which simply meant that they would be the same polarity and amplitude.
Within the data the best fitting
was about
near the top and
near the bottom.
Off the sides of the data,
a subjective analysis was not so simple,
but it seemed clear to me that the best value of
needed to be much larger,
say about
.
I was prepared to see a spectral difference between
and
, but none was apparent,
indicating the rho filters
in the definition of
had generally done their job.
The internal results can be explained by
less than perfect quality of the operator .Clearly a more stationary value of
in the internal
region could be achieved by a definition of the velocity transformation
that put more effort to yielding a more unitary operator.
The external behavior is more troublesome,
and potentially far more interesting
because it controls the resolution in velocity space.
From the results of this movie study I decided to separate the
issues of trace interpolation from those of trace extrapolation
though obviously there is some interaction near the ends of the cable.
Of the two problems I chose the data extrapolation problem to have
the most interest because it affects resolution of the geologically
interesting part of our data.
Since the operator chosen for this study works best at wide offsets,
I chose to extend the far end of the cable.