Angle-domain common-image gathers by reverse time migration

The standard imaging condition for prestack reverse time migration is based on the crosscorrelation in time of the source wavefield (S) with the receiver wavefield (R). The equivalent of the stacked image is the average over the sources (s) of the zero lag of this crosscorrelation, that is:  

$$I\left(z, {\bf x}\right) = \sum_s \sum_t S_s\left( t,z,{\bf x}\right) R_s\left( t,z,{\bf x}\right),$$ (1)

where z and ${\bf x}$ are respectively depth and the horizontal axes, and t is time. The image created using this imaging condition is the equivalent to the stack over offsets for Kirchhoff migration.

This imaging condition has the disadvantage of not allowing a prestack analysis of the image for either velocity updating or amplitude analysis. The conventional way of overcoming this limitation is to avoid the averaging over sources, and thus to create CIGs where the horizontal axis is related to a surface offset; that is, the distance between the source location and the image point. This kind of CIG is known to be prone to artifacts even when the migration velocity is correct because the non-specular reflections do not destructively interfere. Furthermore, in presence of migration velocity errors and structural dips, this kind of CIG does not provide useful information for improving the velocity function.

Rickett and Sava (2001) proposed a method for creating more useful angle-domain CIGs with shot profile migration using downward continuation. Their method can be easily extended to reverse time migration. Equation (1) can be generalized by crosscorrelating the wavefields shifted with respect to each other. The prestack image becomes function of the horizontal relative shift, that has the physical meaning of a subsurface offset (${\bf x}_h$). It can be computed as  

$$I\left(z,{\bf x},{\bf x}_h\right) = \sum_s \sum_t S_s\left... ...h}{2} \right) R_s\left( t,z,{\bf x}- \frac{{\bf x}_h}{2}\right)$$ (2)

This imaging condition generates CIGs in the offset domain that can be easily transformed to the more useful ADCIG applying the same methodology described in Sava et al. (2001).

Reverse time migration is more general than downward continuation migration because it allows events to propagate both upward and downward. Therefore the ADCIG computed from reverse time migration can be more general than the ones computed from downward-continuation migration. This more general imaging condition is actually needed when the source and receiver wavefields meet at the reflector when propagating along opposite vertical direction. This condition may occur either when we image overturned events or image prismatic reflections. I analyze these situations in more details in the following sections. To create useful ADCIGs in these situations we can introduce a vertical offset z_h into equation (2) and obtain  

$$I\left(z,{\bf x},{z}_h,{\bf x}_h\right) = \sum_s \sum_t S_... ...left( t,z - \frac{{z}_h}{2},{\bf x}- \frac{{\bf x}_h}{2}\right)$$ (3)

These offset-domain CIGs should be amenable to being transformed into angle-domain CIG, by generalizing the methodology described in Sava et al. (2001). However, I have not tested it yet, and in the examples I show only the application of the less general equation (2).