Transformation to the angle domain

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:

$$\begin{eqnarray} \theta &\equiv& \frac{\phi + \sigma}{2}, \nonumber\ \alpha &\equiv& \frac{2 \alpha_x + \phi - \sigma}{2}.\end{eqnarray}$$ (31)

In definition  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 angles $\phi$ and $\sigma$ are the same. 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 angles $\phi$ and $\sigma$ are not the same. Hence, the angle $\theta$ is the half-aperture angle, and the angle $\alpha$is the pseudo-geological dip.

 

angles_new
Figure 1 Definition of angles for the converted-mode reflection experiment. The angles $\theta$, $\phi$, $\sigma$, $\alpha_x$ represent the half-aperture, the incident, the reflection, and the geological dip angles, respectively.

Throughout this chapter, I present a relationship between the known quantities from our image, $I(m_{\xi},z_{\xi},h_{\xi})$, and the half-aperture angle ($\theta$). 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):

 

$$\tan{\theta} = \frac{4\gamma({\bf m_{\xi}})\tan{\theta_0}+ \... ...0}(\gamma({\bf m_{\xi}})- 1)^2 + (\gamma({\bf m_{\xi}})+ 1)^2},$$ (32)

where

$$\begin{eqnarray} \tan{\theta_0}&=& -\frac{\partial z_\xi}{\partial h_\xi}, \nonumber\ \d &=& -\frac{\partial z_\xi}{\partial m_\xi}. \nonumber\end{eqnarray}$$

Equation  consists of three main components. First $\gamma({\bf m_{\xi}})$ is the P-to-S velocity ratio. Next, $\theta_0$ 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, $\d$ 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 $h_\xi$ 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. Notice that for the non-physical case of v_p = v_s, i.e. no converted waves, $\gamma({\bf m_{\xi}})= 1$,and the angles $\theta_0$ and $\theta$ are the same.

 

• Mapping of PS-ADCIGs