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PS angledomain transformation
This appendix presents the transformation
from the subsurfaceoffset domain into the angledomain
for convertedwave data.
The derivation follows the wellknown equations for apparent slowness
in a constantvelocity medium in the neighborhood of the reflection/conversion
point. This derivation is consistent with the one presented by
Fomel (2001);Sava and Fomel (2003); and
Biondi (2005).
The expressions for the partial derivatives of the total
traveltime with respect to the image point coordinates () are
as follows Rosales and Rickett (2001):
 

 
 (63) 
Where S_{s} and S_{r} are the slowness (inverse of velocity) at the
source and receiver locations, respectively.
Figure illustrates all the angles in this discussion. The angle is the
direction of the wave propagation for the source, and the angle is the
direction of the wave propagation for the receiver.
Throughout these set of equations, we obtain:
 

 (64) 
At this step, I define two angles, and , to relate and as follows:
 
(65) 
as I discussed in Chapter 3, the angles and ,for the case of convertedwave data, are the halfaperture angle
and the pseudogeological dip, respectively.
Following the change of angles suggested on equation ,
and by following basic trigonometric identities,
we can rewrite equations as follows:
 

 (66) 
where,
 
(67) 
I introduce the following notation in equation :
 

 (68) 
In the notation , is the pseudoreflection angle,
and is local imagedips. The bases for this
definition resides in the conventional isotropic singlemode PP case. For this case,
the pseudoreflection angle is the
reflection angle, and the field represents the
geological dip Fomel (1996).
Using the notation into equation , I
present the equations
 
(69) 
 (70) 
Following basic algebra, equation reduces to
 
(71) 
Substituing equation into equation ,
and following basic algebraic manipulations, we obtain equation
in Chapter 3.
Next: Mapping of PSADCIGs
Up: Imaging of convertedwave Oceanbottom
Previous: Chapter 5: PS commonazimuth
Stanford Exploration Project
12/14/2006