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Mapping of PS-ADCIGs
This appendix presents the derivation for the equations that allow to
map the existing PS-ADCIGs that are a function of the
half-aperture angle into two angle-domain common-image gathers.
The first angle gather is a function of the P-incidence angle,
the second angle gather depends on the S-reflection angle.
The first element to obtain these mapping equations
is Snell's law:
| |
(10) |
The angles and are the incidence and
reflection angles, respectively.
From the definition of the full-aperture angle, ,(equation ), I obtain the following:
| |
(11) |
| (12) |
Introducing equations A-11 and A-12
into equation A-10,
and using the P-to-S velocity ratio (), I obtain:
| |
(13) |
| (14) |
Using simple trigonometric relations and basic algebra,
from equations A-13 and A-14, I get, respectively,
| |
(15) |
| (16) |
Equations A-15 and A-16 translate into
equations and , in Chapter 3.
Next: A Kirchhoff perspective for
Up: Table of Contents
Previous: PS angle-domain transformation
Stanford Exploration Project
12/14/2006