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A Kirchhoff perspective for PS-ADCIG transformation
In this appendix, I obtain the relation to transform subsurface offset-domain
common-image gathers into angle-domain common-image 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 tD,
data-midpoint mD, and data surface-offset hD as follows:
angles2
Figure 1 Parametric formulation of the
impulse response.
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| (79) |
where the total path length is:
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| (80) |
From that system of equations, Biondi (2005) shows that the total path length is
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(81) |
I can rewrite system as:
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| (82) |
where and follow the same definition as in equation .
The total path length, L, in terms of the angles and is:
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(83) |