I analytically demonstrated that when source or receiver wavefields are poorly sampled, subsurface-offset gathers are no longer linearly related to angle gathers. Slant-stack or radial-trace transform is prone to artifacts in such situations. This often happens when the underground velocity structure is very complex, since in complex geologies with limited recording geometries, it is difficult to have wavefields well sampled. To eliminate those artifacts, I presented a selective-stacking approach based on local smoothing of the envelope function. My test on the complex SEG/EAGE salt data set shows that by selectively stacking in the reflection angle and azimuth domain, a cleaner image with higher signal-to-noise ratio and less migration artifacts can be obtained.
Though the method discussed in this paper is effective, it is purely based on signal processing. If the artifacts in the ADCIGs are coherent and strong, this signal-processing approach may not work well, because it tends to heavily weight strong energy. Instead, we can use the illumination of the subsurface as a reference for designing the weighting functions. The illumination can be measured in the least-square sense by computing the Hessian matrix. As illustrated in Valenciano and Biondi (2004), the Hessian is diagonal-dominated in well-illuminated areas, but the energy smears in areas that are not well illuminated, resulting in strong off-diagonal components. Therefore, the focus of energy in the diagonal of the Hessian can be a measure of illumination of the subsurface. We can further transform the Hessian into the reflection angle and azimuth domain, where well-illuminated reflection angles and azimuths will receive high energy, while poorly illuminated angles and azimuths will receive low energy. Therefore, we could design weighting functions based on the diagonal of the Hessian to attenuate unwanted artifacts and selectively stack those well-illuminated angles and azimuths to get a high-quality image. This methodology will be a future research topic.