Theory of ADCIGs in anisotropic media from a Kirchhoff viewpoint

Figure 1 illustrates the generalization of the migration operator from a Kirchhoff viewpoint. Simple geometric relations allow us to derive the kinematics of the generalized migration operator. If we migrate an impulse recorded at time t_D, midpoint m_D and surface offset h_D, the migration impulse response can be expressed as follows:

$$\begin{eqnarray} z_\xi& = & L\left(\alpha_x,\gamma\right)\frac{\cos ^2 \alpha_x-... ...}- L\left(\alpha_x,\gamma\right)\frac{\sin \gamma}{\cos \alpha_x},\end{eqnarray}$$ (1) (2) (3)

where $\alpha_x$ is the group dip angle, $\gamma$ is the group average aperture angle, $z_\xi$, $m_\xi$ and $h_\xi$ are the depth, midpoint and subsurface offset of the imaging point as illustrated in Figure 1. $L\left(\alpha_x,\gamma\right)$ is the average half-path length and is given by

$$\begin{eqnarray} L\left(\alpha_x,\gamma\right)= \frac{L_s+ L_r}{2} = \frac {t_{D... ...ft(S_r+S_s\right) + \left(S_r-S_s\right)\tan \alpha_x\tan \gamma},\end{eqnarray}$$ (4)

where S_s and S_r are the group slowness along the source and receiver rays, respectively.

 

imp-resp Figure 1 Geometry used for evaluating the impulse response of integral migration, generalized to produce a prestack image function of the subsurface offset $h_\xi$.

The expression for the generalized impulse response of prestack anisotropic migration leads to the analytical evaluation of the offset dip ($\left.\frac{\partial z_\xi}{\partial h_\xi}\right\vert _{m_\xi=\widebar m_\xi}$) and midpoint dip ($\left.\frac{\partial z_\xi}{\partial m_\xi}\right\vert _{h_\xi=\widebar h_\xi}$) along the planes tangent to the impulse response. When we adopt the Kirchhoff viewpoint, the group aperture angles can then be related to the offset dips in the image, and the group dip angles can be similarly related to the midpoint dips in the image.