Figure shows the acquisition geometry for one particular sail line. The first streamer (shown as 1) is extracted and used for this field data example. The ship is moving from right to left. Surface currents generate strong cable feathering around 642 km in the inline direction. The receiver positions are displayed for nine shots only every eight kilometers in Figure .
A near offset section for streamer 1 in Figure is shown in Figure . Salt intrusions make the S/N ratio very low below three seconds. Two orders of surface-related multiples (WBM1 and WBM2) are present in the data. The trace spacing is 75 m. The goal is to interpolate the shots every 25 m and to recover the near offset information before 2-D SRMP.
The shot gathers are first sorted into CMP gathers. The binning parameters are illustrated in Figure . The bin size is 2512.5 m. The goal of the interpolation is to fill-up the empty bins shown in Figure . From the CMP gathers, a masking operator is built. This masking operator is set to zero where traces are missing and to one where traces are present.
From the CMP gathers and the masking operator, a model is estimated by minimizing equations () and (). Figures c and d show when the sparseness constraint is or is not applied, respectively. Most of the aliasing artifacts caused by the missing traces in Figure are well attenuated when the Cauchy regularization is used. Note that the remaining artifacts in Figure c could be attenuated by increasing in equation () with the possible effect of damaging some useful signal, e.g., below 5 s in Figure a.
Once is estimated, the interpolated CMP gathers can be created with equation (). Figure illustrates the interpolation result for one CMP gather. The sparseness constraint (Figure b) gives a cleaner result and preserve the steep dips better than the radon transform without regularization (Figure c). Note that the reconstructed traces are less noisy than the known data and that adding some white noise might be needed Gulunay (2003). Because the shots are interpolated for multiple prediction only, no processing is applied to correct for this defect.
To better understand the effect of the Cauchy regularization on the steep dips, Figures a, b, and c show the F-K spectra of the CMP gathers in Figure for the input data, the reconstructed data with sparseness constraint and the reconstructed data without regularization, respectively. Most of the steep dips are attenuated when no regularization is applied during the inversion. This effect is clearly shown in the 15-25 Hz band where the sparse interpolation shows some aliased energy for events going slower than 1700 m/s (Figure b). The same events are attenuated when no regularization is applied in Figure c. Therefore, inversion with Cauchy regularization is preferred for data interpolation in the CMP domain. Note that the aliased energy could be attenuated in the CMP domain by sampling the offset axis on a thinner grid. However, because SRMP is performed in the shot domain, it is difficult to anticipate the effects of this aliasing on the multiple prediction result. In addition, by resampling the CMP offset axis, more CMP gathers would be needed to maintain a uniform grid in the shot domain for the multiple prediction. This extra-requirement would increase the total cost and would add more strain on the interpolation technique.
Finally, the interpolated CMP gathers are resorted into shot gathers. Figure displays four shot gathers after interpolation. Shots one and four are original shots from the survey while shots two and three are new. Note that the near offset traces have been interpolated for all the shots (known and interpolated). Similar to what we observed in the CMP domain, the Cauchy regularization helped to preserve the steep dips better in Figure a. In addition, the noise level is quite high for the interpolation result without regularization in Figure b. Below 5 s, the interpolation without sparseness constraint gives better results. However, for the multiple prediction, these events are of little interest. Overall, despite a fairly coarse acquisition geometry, the radon-based interpolation yields accurate results for a successful multiple prediction.