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Consider now a diffractor sitting at the waterbottom as illustrated in the
sketch in Figure . The source and receiverside multiples
are described by equations 24
as did the waterbottom multiple.
In this case, however, the takeoff angles from source and receiver
are different even if the surface offset is the same as that in
Figure . In fact, since the reflection is nonspecular
at the location of the diffractor, X_{diff} needs to be known
in order for the receiver takeoff angle to be computed. The traveltime
of the diffracted multiple is given by
 
(21) 
where Z_{wb}=Z_{diff} can be computed from the traveltime of the multiple
for the zero subsurface offset trace (t_{m}(0)) by solving the quadratic
equation in Z_{wb}^{2} that results from setting h_{D}=0 in
equation 21:

64Z_{wb}^{4}20V_{1}^{2}t_{m}^{2}(0)Z_{wb}^{2}+(V_{1}^{4}t_{m}^{4}(0)4V_{1}^{2}t_{m}^{2}(0)(m_{D}X_{diff})^{2})=0

(22) 
mul_sktch5
Figure 7 Imaging of receiverside diffracted
waterbottom multiple from a diffractor sitting on top of a flat waterbottom.
At the diffractor the reflection is nonspecular.
Notice that . 
 
The coordinates of the image point, according to
equations 24 are given by
 
(23) 
 (24) 
 (25) 
The traveltimes of the individual ray segments are given by
 
(26) 
whereas the traveltimes of the refracted rays can be computed
from equation 5:
 
(27) 
where, according to equations 9 and 10:
 
(28) 
In order to express , and entirely in terms of the
data space coordinates,
all we need to do is compute the sines and cosines of and
which can be easily done from the sketch of
Figure :
Notice that the diffraction multiple does not migrate as a primary even if
migrated with water velocity. In other words, even if , .The only exception is when X_{diff}=m_{D}+h_{D}/2 since then the
diffractor is in the right place to make a specular reflection and therefore is
indistinguishable from a nondiffracted waterbottom multiple. In that case,
(which in turn implies ) and from
equations 5 and 6,
and therefore
equations 2325 reduce to
equations 1416, respectively.
image2
Figure 8 image sections at 0 and 400 m
subsurface offset for a diffracted multiple from a flat waterbottom. The
depth of the waterbottom is 500 m and the diffractor is located at 2500 m.
The solid line represents image reflector computed with
equations 24 and 25.
Figure shows two subsurfaceoffset sections of a
migrated diffracted multiple from a diffractor sitting on top of a flat
reflector as in the schematic of Figure .
The diffractor
position is X_{diff}=2,500 m, the CMP range is from 2,000 m to 3,000 m, the
offsets range from 0 to 2,000 m and the water depth is 500 m. The data were
migrated with the same twolayer model described before.
Panel (a) corresponds to zero subsurface offset () whereas panel (b)
corresponds to subsurface offset of 400 m. Overlaid are the residual
moveout curves computed with equations 24 and 25.
Obviously, the zero subsurface offset section is not a good image of the
waterbottom or the diffractor.
Figure shows three SODCIGs taken at locations 2,300 m,
2,500 m and 2,700 m. Unlike the nondiffracted multiple, this time energy
maps to positive or negative subsurface offset depending on the relative
position of the CMP with respect to the diffractor.
In ADCIGs the aperture angle is given by equation 11 which, given
the geometry of Figure , reduces to
 
(29) 
The depth of the image is given by equation 12,
 
(30) 
odcig2
Figure 9 SODCIGs from a diffracted multiple from
a flat waterbottom at locations 2,300 m, 2,500 m and 2,700 m.
The diffractor is at 2,500 m. The overlaid residual moveout curves were
computed with equations 23 and 24.
Again, this equation shows that the diffracted multiple is not migrated as a
primary even if (except in the trivial case X_{diff}=m_{D}+h_{D}/2
discussed before for which, since , in agreement with equation 19 and so equation 30
reduces to equation 20).
adcig2
Figure 10 ADCIGs corresponding to the SODCIGs
in Figure . The overlaid curves are the residual moveout
curves computed with equations 24 and 30.
Figure shows the angle gathers corresponding to the SODCIGs
of Figure . Notice the shift in the apex of the
moveout curves.
Next: Dipping waterbottom
Up: Flat waterbottom
Previous: Nondiffracted multiple
Stanford Exploration Project
11/1/2005