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A Kirchhoff perspective for PSADCIG transformation
In this appendix, I obtain the relation to transform subsurface offsetdomain
commonimage gathers into angledomain commonimage gathers for converted waves.
To perform this derivation, I use the geometry in Figure
in order to obtain the parametric equations for migration on a constant
velocity medium.
Following the derivation of Fomel (1996) and
Fomel and Prucha (1999),
and applying simple trigonometry and geometry to Figure ,
I obtain parametric equations for migrating an impulse recorded at time t_{D},
datamidpoint m_{D}, and data surfaceoffset h_{D} as follows:
angles2
Figure 1 Parametric formulation of the
impulse response.
 

 
 (79) 
where the total path length is:
 

 (80) 
From that system of equations, Biondi (2005) shows that the total path length is
 
(81) 
I can rewrite system as:
 

 
 (82) 
where and follow the same definition as in equation .
The total path length, L, in terms of the angles and is:
 
(83) 