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# PS angle-domain transformation

This appendix presents the transformation from the subsurface-offset domain into the angle-domain for converted-wave data. The derivation follows the well-known equations for apparent slowness in a constant-velocity 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 Ss and Sr 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 converted-wave data, are the half-aperture angle and the pseudo-geological 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 pseudo-reflection angle, and is local image-dips. The bases for this definition resides in the conventional isotropic single-mode PP case. For this case, the pseudo-reflection 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 PS-ADCIGs Up: Imaging of converted-wave Ocean-bottom Previous: Chapter 5: PS common-azimuth
Stanford Exploration Project
12/14/2006