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We should prefer a transform of a single source gather
because these gathers correspond to a physical experiment
that can be modeled easily by wave-equation methods.
Unfortunately,
reflections from dipping layers and point scatters may have
a very complicated expression in a source gather.
We may be obliged to use many dips to capture their coherence.
Worse, many reflections will have minimum times at finite offset,
and a slant stack will alias some of their energy.
If the data are first sorted by midpoint y and half-offset h,
then reflections from dipping lines and from points will
still remain symmetric about zero offset. A slant
stack of a midpoint gather will better capture the
coherence of the reflections:
| ![\begin{displaymath}
Y(y,p_y, \tau_y ) \equiv \int d ( s=y-h/2, r=y+h/2, t=\tau_y + p_y h ) dh\end{displaymath}](img13.gif) |
(7) |
where
| ![\begin{displaymath}
p_y = \left. {\partial t \over \partial h } \right\vert _y .\end{displaymath}](img14.gif) |
(8) |
The Fourier version of a common-midpoint slant stack
can be derived exactly as before.
Let fy be the Fourier frequency of
:
| ![\begin{displaymath}
\tilde Y (y,p_y ,f_y ) =
\int \exp( i 2 \pi f_y p_y h ) \tilde d (s=y-h/2,r=y+h/2,f=f_y ) dh .\end{displaymath}](img16.gif) |
(9) |
Unfortunately, this slant stack does not correspond to any
single seismic experiment, and wave-equation modeling is much
more awkward.
Next: Conversion of midpoint to
Up: NOTES FROM TIEMAN's SEMINAR
Previous: The Fourier version
Stanford Exploration Project
11/12/1997