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PSCAM theoretical impulse response
This appendix derives the exact solution for the PS commonazimuth
downwardcontinuation operator using the
pointscatterer geometry
The equation for the total travel time is the sum
of a downgoing travel path with Pvelocity (v_{p}) and an upgoing
travel path with Svelocity (v_{s}),
 
(97) 
where and are the source and receiver vector locations,
respectively. The vector denotes the pointscatterer location.
The same expression can be represented in midpointoffset coordinates ()
 
(98) 
Figure shows the singlemode summation
surface in midpointoffset coordinates for
a point scatterer at a depth of 500 m, a Pvelocity
of 3000 m/s, and inlineoffset of 3000 m.
Figure
shows the equivalent surface for converted waves, with an
Svelocity of 1500 m/s.
summation
ppcheops
Figure 3 Singlemode (PP) totaltraveltime surface
(equation ) for a point scatterer at 500 m of depth in
a medium with constant Pvelocity=3000 m/s,
and for an inlineoffset=3000 m.
pscheops
Figure 4 Convertedmode (PS) totaltraveltime surface
(equation ) for a point scatterer at 500 m of depth in
a medium with constant Pvelocity=3000 m/s, and
constant Svelocity=1500m/s, for an
inlineoffset=3000 m.
The following procedure shows how to go from
to :
 
(99) 
where represents the PtoS velocities ratio.
If we make the following definitions,
 

 
 (100) 
() becomes
 
(101) 
We square both sides to get a new equation with only one square root:
 
(102) 
Squaring again to eliminate the square root, and combining elements, we obtain
 
(103) 
This expression is a 4^{th} degree polynomial in ; which is:
 

 (104) 
This can also be writen as follows
 

 
 
 (105) 
This polynomial equation has 4 solutions, which take the following well known form:
 
(106) 
where
 

 
 
 (107) 
Figure presents the four solutions described in the
previous equation. This plot, eventhough is a simple way to verify the
appropiate solution for our case, it's a simple way to select the
solution that is adequate to our problem. The plot suggests that the
fourth solution, that is the solution with the negative both inside
and outside the external square root is the appropiate solution.
ps4sol
Figure 5 Solutions for the analytical
solution of the PS commonazimuth migration operator. The final solution
is the first plot, that corresponds to the solution with both
negative signs.
Next: REFERENCES
Up: Imaging of convertedwave Oceanbottom
Previous: Derivation of the PSDMO
Stanford Exploration Project
12/14/2006