Apex-shifted Radon Transform

In order to account for the apex-shift of the diffracted multiples (h), we define the forward and adjoint Radon transforms as a modified version of the ``tangent-squared'' Radon transform introduced by Biondi and Symes 2003. We define the transformation from data space (ADCIGs) to model space (Radon-transformed domain) as:

$$m(h,q,z')=\sum_\gamma d(\gamma,z=z'+q\tan^2(\gamma-h)),$$

and from model space to data space as

$$d(\gamma,z)=\sum_q\sum_h m(h,q,z'=z-q\tan^2(\gamma-h)),$$

where z is depth in the data space, $\gamma$ is the aperture angle, z' is the depth in the model space, q is the moveout curvature and h is the lateral apex shift. In this way, we transform the two-dimensional data space of ADCIGs, $d(z,\gamma)$, into a three-dimensional model space, m(z',q,h).

In the ideal case, primaries would be perfectly horizontal in the ADCIGs and would thus map in the model space to the zero-curvature (q=0) plane, i.e., a plane of dimensions depth and apex-shift distance (h,z'). Specularly-reflected multiples would map to the zero apex-shift distance (h=0) plane, i.e., a plane of dimensions depth and curvature (q,z'). Diffracted multiples would map elsewhere in the cube depending on their curvature and apex-shift distance.