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The waterbottom multiple from a dipping reflector has exactly the same
kinematics as a primary from a reflector with twice the dip
Alvarez (2005), that is,
 
(31) 
where is the dip of the reflector, is the perpendicular
depth to the equivalent reflector with twice the dip (at the CMP location)
and is the NMO velocity of the equivalent primary
.
Following the same procedure as for the flat waterbottom, we compute
the coordinates of the image point using
equations 24 and noting that in this case
,
 
(32) 
 (33) 
 (34) 
where, according to equations 710,
 
(35) 
 (36) 
 (37) 
 (38) 
The traveltimes of the individual ray segments are computed by
repeated application of the law of sines as shown in Appendix C:
 
(39) 
 (40) 
 (41) 
 (42) 
where is the perpendicular depth to the reflector at the CMP
location and is given by (Appendix C):
 
(43) 
Notice that this is not the same as in equation 31,
which corresponds to the perpendicular depth to the equivalent reflector whose
primary has the same kinematics as the waterbottom multiple.
The traveltime of the refracted ray segments are given by
equations 5 and 6 with
 
(44) 
In order for equation 3234
to be useful in practice, we need to
express them entirely in terms of the known data coordinates, which means that
we need to find an expression for in terms of
. In Appendix C it is shown that
 
(45) 
We now have all the pieces to compute the image space coordinates, since
and can be computed from
equations 5 and 6 using
equations 3545.
Appendix D shows that equations 3234 reduce to the
corresponding equations for the nondiffracted multiple from a flat water
bottom when , as they should.
Figure shows the zero subsurface offset section from a
migrated nondiffracted multiple from a dipping waterbottom. The overlaid
curve was computed with equations 3234.
The dip of the waterbottom is 15 degrees and intercepts the surface at
CMP location zero. The CMP range of the data is from 2000 to 3000 m and
the surface offsets from 0 to 2000 m. The multiple was migrated with the
same twolayer model described before. Notice how the multiple was migrated
as a primary. Since the migration velocity is faster than watervelocity, the
multiple is overmigrated and appears much steeper and shallower than it
should (recall that it would be migrated as a reflector with twice the
dip is the migration velocity were that of the water.)
image3
Figure 11 image section at zero
subsurface offset for a nondiffracted multiple from a dipping waterbottom.
The overlaid curve was computed with equation 34
and 33.

 
Figure shows the SODCIG at CMP 1500 m in i
Figure . Just as for the flat waterbottom, the multiple
energy is mapped to negative subsurface offsets since . The overlaid
curve is the moveout computed with equations 3234.
odcig3
Figure 12 SODCIG from a nondiffracted multiple
from a dipping waterbottom. The overlaid residual moveout curve was computed with equation 32 and 33.

 
The aperture angle is given by equation 11 with
 
(46) 
and given by equation 45. The image depth in the
ADCIG is given by equation 12 with and given
by equation 46 and and given by
equations 32 and 33. Figure shows the
ADCIG corresponding to the SODCIG in Figure . Notice that the
apex is at zero aperture angle.
adcig3
Figure 13 ADCIG corresponding to the SODCIG
shown in Figure . The overlaid residual moveout curve was
computed with equation 33, 11, 12,
and 46.

 
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Up: Dipping waterbottom
Previous: Dipping waterbottom
Stanford Exploration Project
11/1/2005