Angle-Domain Common Image Gathers and kinematic anisotropic migration

In Biondi (2005) I develop the theory of ADCIGs in anisotropic media from both a ``plane-wave'' viewpoint and a ``ray'' viewpoint. The two methods are equivalent and yield the same results, but the ray-theoretical approach is the natural starting point for analyzing RMO functions in ADCIGs. The kinematic approach is based on the conceptual generalization of integral (Kirchhoff) migration to the computation of a prestack image that include the sub-surface offset dimension. The image-space of integral migration are usually restricted to the zero subsurface-offset section; that is, with the integral operators (either summation surfaces or spreading surfaces) evaluated when source and receiver rays meet at the reflection point. The image space can be expanded to include non-zero subsurface offsets by integrating the data over surfaces evaluated with the end points of the source and receiver rays horizontally shifted with respect to each other, instead of being coincidental at the image point. Figure  illustrates this concept and provide the basis for computing the kinematics of the generalized migration operator.

Since the transformation to ADCIGs operates in the image space, I analyze the spreading surfaces (impulse responses) of the generalized prestack migration operator, which are defined in the image space. In homogeneous anisotropic medium the shape of the impulse responses of the generalized integral migration can be easily evaluated analytically as a function of the subsurface offset $h_\xi$, in addition to the usual image depth $z_\xi$ and midpoint $m_\xi$.Figure  illustrates the geometry used to evaluate this impulse response.

 

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 $h_\xi$.

Assuming an arbitrary homogeneous anisotropic medium, simple trigonometry applied to Figure  allows us to express the impulse response in parametric form, as a function of the group dip angle $\alpha_x$ and the group aperture angle $\gamma$.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)

with the average half-path length $L\left(\alpha_x,\gamma\right)$ given by:  

$$L\left(\alpha_x,\gamma\right)= \frac{L_s+ L_r}{2} = \frac {t... ...S_r+S_s\right) + \left(S_r-S_s\right)\tan \alpha_x\tan \gamma},$$ (4)

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

In 2-D, The ADCIGs are computed by applying a slant-stack decomposition on the prestack image along the subsurface offset axis, at constant midpoint. The kinematics of the transformation are defined by the following change of variables:

$$\begin{eqnarray} \widehat{\gamma} &=& \arctan \left. \frac{\partial z_\xi}{\part... ...artial z_\xi}{\partial h_\xi} \right\vert _{m_\xi=\widebar m_\xi},\end{eqnarray}$$ (5) (6)

where $z_\gamma$ is the depth of the image point after the transformation. In the general case, the angle $\widehat{\gamma}$ is related to the reflection aperture angle in a non-trivial way. However, in Biondi (2005) I demonstrate that for flat reflectors the slope of the impulse response along the subsurface offset axis, is equal to the tangent of the phase aperture angle $\widetilde{\gamma}$;that is,  

$$\left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{\... ...ial S}{\partial \gamma} \tan \gamma }= \tan \widetilde{\gamma}.$$ (7)

Notice that throughout this paper I use the tilde symbol to distinguish between phase quantities (with a tilde) and group quantities (without a tilde). Appendix A summarizes the relationships between group angles and velocities and phase angles and velocities. Equation 33 is directly used to derive the result in equation 7.

Substituting equation 7 in equations 5 and 6 we obtain

$$\begin{eqnarray} \widetilde{\gamma} &=& \arctan \left. \frac{\partial z_\xi}{\pa... ...idebar m_\xi}, \\ z_\gamma &=& z_\xi-h_\xi\tan \widetilde{\gamma}.\end{eqnarray}$$ (8) (9)

 

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

Figure  provides a geometrical interpretation of the transformation to angle domain of an image point with non-zero subsurface offset. The transformation to angle domain moves the image point in the subsurface-offset domain $\left(z_\xi,h_\xi\right)$to the image point in angle domain $\left(z_\gamma,\widetilde{\gamma}\right)$.The depth of the image point in 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 $\mp \widetilde{\gamma}$ with respect to the horizontal. When the migration velocity is correct, and the image is fully focused at zero subsurface offset, the transformation to angle domain does not change the depth of the image point and the reflections are imaged at the same depth for all aperture angles. On the contrary, when the reflections are not focused at zero offset, the transformation to angle domain maps the events at different depths for each different angle. The variability of the depth $z_\gamma$ with the aperture angle is described by the RMO function that we want to measure and quantify as a function of the perturbations in anisotropic parameters encountered along the propagation paths.