RTT implementation

A third source of inaccuracy is the actual implementation of the radial-trace transform. In theory, RTT can be implemented either as a push operator (loop over input) or as a pull operator (loop over output). Figure (6) shows an example with a push implementation (from the k_z-k_h domain to the $k_z-\gamma$ domain), and Figure (7) shows a pull implementation (in the $k_z-\gamma$ domain from the k_z-k_h domain).

As expected, the push implementation leaves empty regions in the $k_z-\gamma$ domain, while the pull implementation fills the entire domain. After inverse Fourier transform, the amplitude responses in the two cases are completely different (Figure 8): the pull implementation generates flat amplitudes, while the push implementation does not.

 

angpushA
Figure 6 Conversion from offset-domain to angle-domain image gathers. The push implementation of RTT leaves empty spaces that degrade the amplitudes. This effect can be partially corrected by weighting with the transformation Jacobian.

Under those circumstances, it would be tempting to consider just the pull implementation. However, inversion for angle-domain regularization, requires implementations for both the forward and adjoint operators. Therefore, if we use pull in one transformation, we need to use a corrected push in its adjoint Claerbout (1995). This correction is given by the Jacobian of the transformation from k_h to $\gamma$ or p_h.

 

angpullA
Figure 7 Conversion from offset-domain to angle-domain image gathers. The pull implementation of the RTT fills the entire usable space, therefore the amplitude response is correct.

 

ampA
Figure 8 Amplitude response for RTT implemented as unweighted push (left), and pull (right). Ideally, the amplitude should be constant at all angles.

Figure 9 shows the angle-domain representation for the ideal offset-gather in Figure 3. Since we use a wide temporal frequency band and a pull implementation of RTT, the amplitude response is flat for the whole usable angle range.