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From Figure we can immediately compute the takeoff
angle of the diffracted receiver ray as
 
(78) 
mul_sktch13
Figure 26 Sketch showing the
geometry of the zero surface halfoffset diffracted multiple from a
dipping waterbottom.

 
In this equation the depth of the diffractor is not known, but it can be
calculated from the geometry of Figure :
 
(79) 
As we did for the diffracted multiple from the flat waterbottom, we can
use the traveltime of the multiple at the zero surfaceoffset trace to compute
, except that this time the computation is much more involved.
Figure shows the basic geometry. From triangle ABC we
have
 
(80) 
were t_{r1} is the traveltime of the diffracted segment that, according
to triangle DEF in Figure is given by

[V_{1}t_{r1}(0)]^{2}=Z_{diff}^{2}+(X_{diff}m_{D})^{2}.

(81) 
Replacing equations 79 and 81 into equation 80
gives a quartic equation for which can be solved numerically.
Once is known, we can easily compute Z_{diff} with
equation 79 and therefore with equation 78
in terms of the known quantities h_{D}, m_{D}, X_{diff} and t_{m}(0).
mul_sktch15
Figure 28 Sketch to compute
the takeoff angle of the source ray from a diffracted multiple.

 
In order to compute , we apply the law of sines to triangle ABC
in Figure to get
 
(82) 
where and V_{1}t_{r1} is the
length of the diffracted receiver ray and is given by
 
(83) 
Therefore, plugging equation 83 into equation 82 we get
equation 50:
 
(84) 
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Up: Alvarez: Multiples in image
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Stanford Exploration Project
11/1/2005