Mapping to the Radon domain

With ideal data, attenuating both specular and diffracted multiples could, in principle, be accomplished simply by zeroing out (with a suitable taper) all the $q$-planes except $q=0$ in the model cube $m(z',q,\Gamma)$ and taking the inverse apex-shifted Radon transform. In practice, however, the primaries may not be well-corrected and primary energy may map to a other nearby $q$-planes. Energy from the multiples may also map to those planes and so we have the usual trade-off of primary preservation versus multiple attenuation. The advantage of the apex-shift transform is that the diffracted multiples are well focused to their corresponding $\Gamma$-planes instead of being mapped as unfocused noise that interferes with the primaries.

To illustrate the mapping of the primaries, the specular multiples and the diffracted multiples, between the image space $(z,\gamma)$ and the apex-shifted Radon space $(z',q,\Gamma)$, I chose the ADCIG in Figure 15(a). Although this ADCIG shows no discernible primaries below the salt, it nicely shows the apex-shifted moveout of the diffracted multiples. This ADCIG was transformed to the Radon domain with the apex-shifted transform described by equations 28 and 29. The kernel of the Radon transform is given by equation  27 and I applied the Cauchy regularization given in equation  31. Figure 16 shows envelopes of the data in the Radon domain. Panel (a) shows the $\Gamma =0$ plane from the $(z',q,\Gamma)$ volume. This plane corresponds to zero apex-shift and therefore this is where the majority of the specular multiples should map. Figure 16(b) shows the zero-curvature $q=0$ plane, that is, the plane where the primaries should map. Notice that since the primaries are flat, they are independent of the apex-shift $\Gamma$ and therefore map as flat lines on this plane. Notice also that there are no significant primaries on the ADCIG below 2000 m. For comparison, Figure 16(c) shows the $\Gamma =8$ deg plane. This corresponds to the apex-shift of the most obvious diffracted multiple and we see its energy mapped on this plane at about 4000 m. Finally, Figure 16(d) shows a plane at a large curvature, $q=7200$ m/deg. Notice the energy from the diffracted multiple at approximately $\Gamma =8$ deg.

envelopes
Figure 16. Different views from the cube of the apex-shifted transform for the ADCIG at 6744 m. (a): zero apex-shift plane. (b) zero curvature plane. (c): plane at apex shift $\Gamma =8$ deg and (d): plane at curvature $q=7200$ m/deg.

It is important to emphasize the difference between the standard transform and the apex-shifted transform. While the $\Gamma =0$ plane of the apex-shifted transform is similar to the standard transform, they are not the same, as shown in Figure 17. Both panels in this figure are plotted with the exact same plotting parameters. Primaries are mapped near the $q=0$ line in both planes while specular multiples are mapped to other $q$ values. Notice how in the standard transform Figure 17(a), the diffracted-multiple energy is mapped as background noise, especially at the largest positive and negative $q$ values. In the $\Gamma =0$ plane of the apex-shifted transform (panel (b)), however, the diffracted multiples are not present since their moveout apex is not zero. These multiples, therefore, do not obscure the mapping of the specular multiples. Notice also that the primary energy is much lower than in Figure 17(a) since in the apex-shifted transform the primary energy is mapped not only to the $\Gamma =0$ plane but to other $\Gamma$ planes as well as illustrated previously in Figure 16(b).

radon-comp
Figure 17. Radon transforms of the ADCIG in Figure 15b. (a): standard 2D transform. (b): $h=0$ plane of the apex-shifted 3D transform. Both panels plotted at the exact same clip value.