Next: Transformation into independent angles Up: Rosales et al.: PS-ADCIG Previous: Wave-equation imaging

Transformation to the angle domain

The transformation to the angle domain follows an approach similar to that of its single-mode (PP) counterpart Sava and Fomel (2003). Figure  describes the angles we use in this section. For the converted-mode case, we define the following angles:
 (3)
In definition 3 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 condition that the angles and are the same holds. 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 angle is the full-aperture angle, and the angle is the pseudo-geological dip.

 angles Figure 1 Definition of angles for the converted-mode reflection experiment. The angles , , , represent the full-aperture, the incident, the reflection, and the geological dip angles, respectively.

The main goal of this paper is to obtain a relationship between the known quantities from our image, and the full-aperture angle (). Appendix A presents the full derivation of this relationship. Here, we present only the final result, its explanation and its implications. The final relationship we use to obtain converted-mode angle-domain common-image gathers is the following (Appendix A):
 (4)

this equation consists of three main components: is the P-to-S velocity ratio, is the pseudo-opening angle, and is the field of local step-outs of the image. Equation 4 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.