Multi-offset imaging

We produce angle gathers for shot-profile migrations by combining de Bruin et al.'s 1991 approach with that of Sava and Fomel (2000). Rather than extracting a single zero-offset/zero-time reflectivity image, we extract multiple zero-time images with a range of offsets. In 2-D this can be performed with the following sum over frequency:  

$$I(x,h,z) = \sum_\omega \; q_{-}(x-h,z,\omega) \; q_{+}(x+h,z,\omega)^\ast.$$ (1)

Gathers produced this way contain off-diagonal elements of Berkhout's reflectivity matrix 1985, and are equivalent to those produced by imaging multiple non-zero offsets in an offset-midpoint shot-geophone migration. Consequently, the offset axes can be mapped to angle with Sava and Fomel's 2000 transformation, which is based on the relationship

$$\tan \gamma = -\left. \frac{\partial z}{\partial h} \right\vert _{t,x} = \frac{\vert\vec{k_h}\vert}{k_z},$$ (2)

where $\gamma$ is the half-opening (incidence) angle. The imaging condition (t=0) and their common-midpoint nature (constant x) allow image gathers to be related to the partial derivative at constant t and x.