Angle Gathers by anisotropic downward-continuation migration

In this section I develop the theory for anisotropic ADCIGs from the ``plane-wave'' viewpoint. I assume that in the proximity of the reflection point the source wavefield and the receiver wavefield are plane waves and I derive the relationships between the propagation angles of these plane waves and the slopes computed in the prestack image. This assumption is not restrictive because the source and receiver wavefields can always be considered as the superposition of plane waves.

In anisotropic media, when the reflector is dipping with respect to the normal to the isotropic axis of symmetry (horizontal direction for VTI) the incident and reflected aperture angles differ. This difference is caused by the fact that, although the phase slowness is function of the propagation angle, Snell law requires that the components parallel to the reflector of the incident and reflected slowness vectors must match at the interface. However, we can still define an ``average'' aperture angle $\widetilde{\gamma}$and ``average'' dip angle $\widetilde{\alpha}_x$using the following relationships:  

$$\widetilde{\gamma}=\frac{\widetilde\beta_r-\widetilde\beta_s... ...etilde{\alpha}_x=\frac{\widetilde\beta_s+\widetilde\beta_r}{2},$$ (8)

where the $\widetilde\beta_s$ and $\widetilde\beta_r$ are the phase angles of the incident and reflected plane waves, respectively.

 

cig-aniso-v3
Figure 1 Sketch representing the reflection of a plane wave from a planar reflector in an anisotropic medium. The angles marked in the figure are all phase angles. They are defined as follows: $\widetilde\beta_s$ and $\widetilde\beta_r$ are the propagation angles of the incident and reflected plane waves, $\widetilde{\gamma}_s$ and $\widetilde{\gamma}_r$are the true aperture angles for the the incident and reflected plane waves, $\widebar{\alpha}_x$ is the true geological dip angle, $\widetilde{\gamma}$ are $\widetilde{\alpha}_x$ the average aperture angle and the average dip angle.

Figure  shows the geometric interpretation of these angles. Notice that the average dip angle $\widetilde{\alpha}_x$ is different from the true geological dip angle $\widebar{\alpha}_x$,and that the average aperture angle $\widetilde{\gamma}$ is obviously different from the true aperture angles $\widetilde{\gamma}_s$ and $\widetilde{\gamma}_r$.However, these five angles are related and, if needed, the true angles can be derived from the average angles, as shown in Appendix A.

Prestack images defined in the subsurface-offset $(h_\xi)$ domain are transformed into the angle domain by applying slant stacks. The transformation axis is thus the physical dip of the image along the subsurface offset; that is, ${\partial z_\xi}/{\partial h_\xi}$.The dip angles can be similarly related to the midpoint dips in the image; that is, ${\partial z_\xi}/{\partial m_\xi}$.Following the derivation of acoustic isotropic ADCIGs by Sava and Fomel (2003) and of converted-waves ADCIGs by Rosales and Rickett (2001), we can write the following relationships between the propagation angles and the derivative measured from the wavefield:

$$\begin{eqnarray} \left. \frac{\partial t}{\partial z_\xi} \right\vert _{\left(m... ...lde{S}_r\sin\left(\widetilde{\alpha}_x+ \widetilde{\gamma}\right),\end{eqnarray}$$ (9) (10) (11)

where $\widetilde{S}_s$ and $\widetilde{S}_r$ are the phase slownesses for the source and receiver wavefields, respectively. We obtain the expression for the offset dip by taking the ratio of equation 11 with equation 9, and similarly for the midpoint dips by taking the ratio of equation 10 with equation 9, and after some algebraic manipulations, we obtain the following expressions:

$$\begin{eqnarray} \left. \frac{\partial z_\xi}{\partial h_\xi} \right\vert _{\le... ...idetilde{S}_s} \tan \widetilde{\gamma}\tan \widetilde{\alpha}_x }.\end{eqnarray}$$ (12) (13)

In contrast with the equivalent relationships valid for isotropic media, these relationships depend on both the aperture angle $\widetilde{\gamma}$and the dip angle $\widetilde{\alpha}_x$.The expression for the offset dip (equation 9) simplifies into the known relationship valid in isotropic media when either the difference between the phase slownesses is zero, or the dip angle $\widetilde{\alpha}_x$ is zero. In VTI media this happens for flat geological dips. In a general TTI medium this condition is fulfilled when the geological dip is normal to the axis of symmetry.

Solving for $\tan \widetilde{\gamma}$ and $\tan \widetilde{\alpha}_x$we obtain the following:

$$\begin{eqnarray} \tan \widetilde{\gamma} &=& \frac { \frac{\partial z_\xi}{\part... ...}{\partial m_\xi} \Delta_\widetilde{S} \tan \widetilde{\gamma} },\end{eqnarray}$$ (14) (15)

where for convenience I substituted the symbol $\Delta_\widetilde{S}$for the ``normalized slowness difference'' $(\widetilde{S}_r-\widetilde{S}_s)/(\widetilde{S}_r+\widetilde{S}_s)$.

Substituting equation 15 in equation 14, and equation 14 into equation 15, we get the following two quadratic expressions that can be solved to estimate the angles as a function of the dips measured from the image:

$$\begin{eqnarray} \left[ \frac{\partial z_\xi}{\partial m_\xi} \Delta_\widetilde... ...Delta_\widetilde{S} - \frac{\partial z_\xi}{\partial m_\xi} &=&0.\end{eqnarray}$$ (16) (17)

These are two independent quadratic equations in $\tan \widetilde{\gamma}$ and $\tan \widetilde{\alpha}_x$that can be solved independently. If the ``normalized slowness difference'' $\Delta_\widetilde{S}$between the slowness along the propagation directions of the source and receiver wavefields are known, we can directly compute $\widetilde{\gamma}$ and $\widetilde{\alpha}_x$,and then the true $\widetilde\beta_s$ and $\widetilde\beta_r$.One important case in this category is when we image converted waves.

For anisotropic velocities, the slownesses depend on the propagation angles, and thus the normalized difference depends on the unknown $\widetilde{\gamma}$ and $\widetilde{\alpha}_x$.In practice, these equations can be solved by a simple iterative process that starts by assuming the ``normalized difference'' to be equal to zero. In all numerical test I conducted this iterative process converges to the correct solution in only a few iterations, and thus is not computationally demanding.

If the anisotropic slowness function were spatially homogeneous, equations 16 and 17 could be solved iteratively in the Fourier domain, and the transformation to the average angles $\widetilde{\gamma}$ and $\widetilde{\alpha}_x$could be computed exactly without the need of estimating the apparent reflector dip in the space domain. When the anisotropic parameters are a function of the spatial variables; that is, in the majority of the real situations, the solution of equations 16 and 17 requires the estimation of the local reflector dip in the space domain. If necessary, the reflectors' dip can be either extracted from the interpretation of the horizons of interest, or can be automatically estimated from the image by applying one of the several methods that have been presented in the literature (see for example Fomel (2002)). In practice, the estimation of the reflector dip is seldom necessary. The numerical and real-data examples shown below indicate that for practical values of the anisotropy parameters the dependency of the estimate from the dip angles can be safely ignored for small dips, and it is unlikely to constitute a problem for steep dips.