Theory and practice of interpolation in the pyramid domain

Interpolation of irregularly-sampled data

The interpolation of irregularly-sampled data in the pyramid domain requires a binning of the data onto a regular grid first. This binning is done with a simple nearest-neighbor scheme. Like what is done for the regular case, the weighting operator ${\bf W_d}$ in equation (15) is set to zero at the empty trace locations. For all following examples, 50$\%$ of the traces were set to zero randomly in order to emulate a realistic acquisition geometry.

Figure 8a shows the data to interpolate with its amplitude spectrum (Figure 8b). Many methods exploit the $FK$ domain directly in order to interpolate missing data [Abma and Kabir (2006); Xu et al. (2005); Zwartjes and Sacchi (2007)], with or without nonuniform Fourier transforms. The interpolation result in Figure 8c proves that the proposed algorithm works in this case as well. The $FK$ domain in Figure 8d validates these findings.

Figure 9a and 9c display the input data and the interpolation results, respectively. This dataset is similar to the one used in Figure 6a. The missing traces have been properly interpolated almost everywhere. Where gaps are big, however, the proposed algorithm might have some difficulties which could be overcome by using a multiscaling strategy. Note, in Figures 9b and 9d, the clean up of the $FK$ domain after interpolation.

Finally, we interpolate irregularly-spaced data for a field data example shown in Figure 10a. This dataset was also used in Figure 7a. Like what we observed in the previous example, the interpolation result in Figure 10c exemplifies how the proposed algorithm can interpolate missing data: the reconstructed traces look very similar to the original ones. The $FK$ spectra in Figures 10b and 10d show the attenuation of artifacts due to the random sampling in Figure 10a.

Theory and practice of interpolation in the pyramid domain