The transformation to the angle domain of PS-SODCIGs follows an approach similar to the 2-D isotropic single-mode (PP) method Sava and Fomel (2003). Figure describes the angles I use in this section. For the converted-mode case, I define the following angles:
(31) |
In definition the angles , , and represent the incident, reflected, and geological dip angles, respectively. This definition is consistent with the single-mode case; notice that for the single-mode case the angles and are the same. Therefore, the angle represents the reflection angle, and the angle represents the geological dip Biondi and Symes (2004); Sava and Fomel (2003). For the converted-mode case, the angles and are not the same. Hence, the angle is the half-aperture angle, and the angle is the pseudo-geological dip.
Throughout this chapter, I present a relationship between the known quantities from our image, , and the half-aperture angle (). Appendix A presents the full derivation of this relationship. Here, I present only the final result, its explanation and its implications. The final relationship to obtain converted-mode angle-domain common-image gathers is the following (Appendix A):
(32) |
Equation consists of three main components. First is the P-to-S velocity ratio. Next, is the pseudo-opening angle. This pseudo-opening angle is the angle obtained throughout the conventional method to transform SODCIGs into isotropic ADCIGs as described by Sava and Fomel (2003). Finally, is the field of local image-dips. Equation describes the transformation from the subsurface-offset domain into the angle-domain for converted-wave data. This equation is valid under the assumption of constant velocity. However, it remains valid in a differential sense in an arbitrary velocity medium, by considering that is the subsurface half-offset. Therefore, the limitation of constant velocity applies in the neighborhood of the image. For , it is important to consider that every point of the image is related to a point on the velocity model with the same image coordinates. Notice that for the non-physical case of v_{p} = v_{s}, i.e. no converted waves, ,and the angles and are the same.