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For 3-D prestack time migration,
the reflectors' dips
and ,
and the wavelet-stretch factor ,can be analytically derived
as functions of the input and output trace geometry.
Starting from the prestack time-migration ellipsoid,
expressed as a parametric function of the angles and
| |
|
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| (7) |
where tD is the time of the input impulse
and tN is the time after application of NMO.
We differentiate the image coordinates
with respect to the angles and ;that is,
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|
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| (8) |
We then eliminate the differentials and from this set of equations
by setting respectively equal to zero when evaluating the
dip in the in-line direction,
and set equal to zero when evaluating the dip in the cross-line
direction .The second step is to eliminate the angles themselves
and express the image dips
as a function of the image coordinates
,
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|
| (9) |
The wavelet-stretch factor can be easily derived
by differentiating the summation surfaces of 3-D prestack time migration
expressed as the hyperboloids
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(10) |
where and are the source and receiver coordinates vector,
and
represents the horizontal components
of the image coordinates vector.
After a few simple algebra steps, we obtain
| |
(11) |
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Stanford Exploration Project
7/5/1998