Geometric interpretation

Figure 2 provides a geometric interpretation of the transformation of an image point in the subsurface offset domain $\left(z_\xi,h_\xi\right)$ to the corresponding image point in the angle-domain $\left(z_{\tilde{\gamma}},\widetilde{\gamma}\right)$. In this figure,

$$\begin{eqnarray} \beta_s=\alpha_x+\gamma, \\ \beta_r=\alpha_x-\gamma, \\ \wideti... ...amma}, \\ \widetilde{\beta_r}=\widetilde{\alpha_x}-\tilde{\gamma}.\end{eqnarray}$$ (11) (12) (13) (14)

From a ``plane-wave'' viewpoint, the image point in the angle-domain is determined by the intersection of the lines passing through the points $\left(z_{\xi},m_{\xi}\pm h_{\xi}\right)$ and tilted by $\widetilde{\beta_s}$ and $-\widetilde{\beta_r}$. This interpretation is consistent with the one for flat reflectors illustrated in Figure 3.

 

cig-2d-aniso-dipping-1 Figure 2 Geometry of the transformation to the angle-domain with a dipping reflector. The image point in the subsurface-offset domain $\left(z_\xi,h_\xi\right)$ moves to the image point in the angle-domain $\left(z_{\tilde{\gamma}},\widetilde{\gamma}\right)$.

 

cig-2d-aniso-flat-v1 Figure 3 Geometry of the transformation to the angle-domain with a flat reflector. The image point in the subsurface-offset domain $\left(z_\xi,h_\xi\right)$ moves to the image point in the angle-domain $\left(z_{\tilde{\gamma}},\widetilde{\gamma}\right)$.

The geometric interpretation of the angle $\widehat{\gamma}$ is illustrated in Figure 4. According to equation (10), the depth of the image point in the angle-domain $z_{\tilde{\gamma}}$ is given by the intersection of the two lines passing through the points $\left(z_{\xi},m_{\xi}\pm h_{\xi}\right)$ and tilted by $\pm \widehat{\gamma}$. For flat reflectors, $\widehat{\gamma}=\tilde{\gamma}$Biondi (2005a), which is consistent with Figure 3.

 

cig-2d-aniso-dipping-2 Figure 4 Geometric interpretation of the angle $\widehat{\gamma}$.