## Analytical derivation

Another way to find the relation between z and z' starts from the cascade of two dispersion relations Biondi and Palacharla (1996). The first one is for 2-D prestack downward-continuation along the in-line direction,
 (15)
which is equivalent to equation (11). The second one is for 2-D zero-offset downward continuation along the cross-line axis,
 (16)
which is equivalent to equation (12). Indeed, by taking the square and dividing by kz,
 (17)
and by introducing ,
 (18)
BThe coplanarity condition presented in Biondi and Palacharla (1996) is:
 (19)
Each square root in the previous equation can be substituted with the expression including the dip angle of the source or the receiver ray
 (20) (21)
where and can be expressed in terms of the aperture angle and the normal dip angle :
 (22)
Or, after applying some trigonometric relations
 (23)
We now introduce equation (2) as well as where is the projection of the angle on the vertical plane passing through the source-receiver axis. The two angles are linked by the relation .Equation 23 becomes
 (24)
CConsider one image point in the Fourier domain, i.e. at given kmx, kmy and kz. For one image point, we have all the values of the offset gather. Each value is referenced in the Fourier domain by the offset wavenumbers khx and khy. Given a reflection azimuth and an aperture angle ,we want to get the value of the sample associated with , hence what are the offset wavenumbers associated with .Given kmx, kmy, kz, and :
• Because of the reflection azimuth, we need to get into the new coordinates system. k'mx, k'my are computed using kmx, kmy and in equation (4).
• For a given and kz we can determine k'hx from equation (6).
• The second component of the offset wavenumber, k'hy, must satisfy the coplanarity condition (7).
• The two components of the offset wavenumber were obtained in the new system of coordinates. To return to the original system of coordinates, we use the inverse of transformations (4).
• We have determined which sample of the offset gather should be associated with the aperture angle .
The entire angle gather at the image point (kmx, kmy, kz) and at the reflection azimuth is obtained by looping over .