Discussion

Compared with standard interpolation methods, such as PEF interpolation or Fourier POCS interpolation, the proposed algorithm has different characteristics. First, the generalized velocity-dependent (VD)-seislet transform may have difficulties in handling diffractions, and the velocity scan occasionally produces errors, which would decrease its recovery ability. It is possible to preprocess by using dip moveout. However, the generalized VD-seislet can handle crossing events only up to a certain extent because crossing points only account for a low percentage of data along time-distance curves. It also avoids the problem of event stretch according to the prediction method (Fomel and Liu, 2010), thus producting less artifacts than simple interpolation strategies, such as NMO before data interpolation and subsequent reverting. Meanwhile, standard methods can also not provide accurate interpolation of curved diffractions. Without patching, the generalized VD-seislet transform provides relatively accurate representation of seismic events with aliasing and amplitude anomalies. Fourier POCS are prone to create artifacts, which occur in the frequency-domain method even if data are cut into overlapping windows (patching). Furthermore, data patching causes non-intuitive selection of parameters. The standard PEF interpolation is designed under the assumption of stationary data and becomes less effective when this assumption is violated. Patching in PEF interpolation still fails in the presence of variable dips and usually leads to the problem of insufficient data availability in some child windows, e.g., complete lack of data at near-offset locations and large gaps. Finally, the proposed method applies a modified Bregman iteration, which guarantees fast convergence using a large threshold value, whereas Abma's POCS iteration with linear threshold model produces a slow convergence. Although a different threshold model (Gao et al., 2010) can be used, the POCS method cannot provide fast convergence even with more iterations when random noise are present. Therefore, the proposed method provides an alternative tool for interpolating missing traces in CMP gathers for structurally simple areas even with random noise, amplitude variation, and large gaps.