Next: About this document ...
Up: Biondi: Kirchhoff imaging beyond
Previous: REFERENCES
For 3D prestack time migration,
the reflectors' dips
and ,
and the waveletstretch factor ,can be analytically derived
as functions of the input and output trace geometry.
Starting from the prestack timemigration ellipsoid,
expressed as a parametric function of the angles and
 

 
 (7) 
where t_{D} is the time of the input impulse
and t_{N} is the time after application of NMO.
We differentiate the image coordinates
with respect to the angles and ;that is,
 

 
 (8) 
We then eliminate the differentials and from this set of equations
by setting respectively equal to zero when evaluating the
dip in the inline direction,
and set equal to zero when evaluating the dip in the crossline
direction .The second step is to eliminate the angles themselves
and express the image dips
as a function of the image coordinates
,
 

 (9) 
The waveletstretch factor can be easily derived
by differentiating the summation surfaces of 3D prestack time migration
expressed as the hyperboloids
 
(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) 
Next: About this document ...
Up: Biondi: Kirchhoff imaging beyond
Previous: REFERENCES
Stanford Exploration Project
7/5/1998