Offset to angle transformation in common-azimuth migration

We first present the simplest extension to 3-D: the common-azimuth case.

In 2-D, the transformation from offset-gather to angle-gather is accomplished by applying the relation Sava and Fomel (2000)  

$$\tan \gamma = \frac{k_{h_x}}{k_z},$$ (1)

where $\gamma$ is the aperture angle, k_hx the in-line offset wavenumber and k_z the vertical wavenumber.

This relation can be applied in 3-D, but it is then assumed that the two rays propagate vertically. In 3-D however, rays can propagate out of the vertical plane. In the common-azimuth assumption, the two rays propagate within a slanted plane. We call $\delta$ the dip angle of the slanted propagation plane. The transformation from offset to angle needs to take into account $\delta$. The new form of equation (1) is  

$$\tan \gamma = \frac{k_{h_x}}{k_z} \cos \delta = \frac{k_{h_x}}{k_z} \frac{1}{\sqrt{1+\frac{k_{m_y}^2}{k_z^2}}},$$ (2)

where k_my is the cross-line midpoint wavenumber. We give in Appendix A a geometrical and an analytical derivation of equation (2). This equation is valid for common-azimuth migration, that is to say, for an in-line orientation of the source-receiver axis and for the coplanarity of the source and receiver rays.