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:

$$\begin{eqnarray} \theta &\equiv& \frac{\phi + \sigma}{2}, \nonumber\\ \alpha &\equiv& \frac{2 \alpha_x + \phi - \sigma}{2}.\end{eqnarray}$$ (3)

In definition 3 the angles $\phi$, $\sigma$, and $\alpha_x$ 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 $\phi$ and $\sigma$ are the same holds. Therefore, the angle $\theta$ represents the reflection angle, and the angle $\alpha$ represents the geological dip Biondi and Symes (2004); Sava and Fomel (2003). For the converted-mode case, the angle $\theta$ is the full-aperture angle, and the angle $\alpha$is the pseudo-geological dip.

 

angles Figure 1 Definition of angles for the converted-mode reflection experiment. The angles $\theta$, $\phi$, $\sigma$, $\alpha_x$ 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, $I({\bf m_\xi},{\bf h_\xi})$ and the full-aperture angle ($\theta$). 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):  

$$\tan{\theta} = \frac{4\gamma({\bf m_{\xi}})\tan{\theta_0}+ \... ...0}(\gamma({\bf m_{\xi}})- 1)^2 + (\gamma({\bf m_{\xi}})+ 1)^2},$$ (4)

this equation consists of three main components: $\gamma({\bf m_{\xi}})$ is the P-to-S velocity ratio, $\tan{\theta_0}$ is the pseudo-opening angle, and $\d$ 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 $h_\xi$ is the subsurface half-offset. Therefore, the limitation of constant velocity applies in the neighborhood of the image. For $\gamma({\bf m_{\xi}})$, 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.

 

• Transformation into independent angles
• Implementation