Seismic data interpolation without iteration using $t$ -$x$ -$y$ streaming prediction filter with varying smoothness

field data example

To evaluate the performance of the $t$ -$x$ -$y$ SPF interpolation method in 3D field conditions, we chose a set of marine shot gathers from a deep-water Gulf of Mexico survey (Liu and Fomel, 2011; Fomel, 2002). Figure 8a shows the complicated diffraction events caused by a salt body. We selected $35\%$ traces of the input data by subsampling in the shot direction and removing $30\%$ random traces (Figure 8b). For comparison, we used 3D Fourier POCS method and the conventional SPF to reconstruct the missing traces (Figure 9a and 9b, respectively). The Fourier POCS method also failed to interpolate the decimated traces and created some artificial events at the locations of the randomly-missing traces. The interpolated result could be partially improved by slicing data into patching windows. The conventional 3D $t$ -$x$ -$y$ SPF also failed to recover the decimated data. Figure 9c shows that the proposed $t$ -$x$ -$y$ SPF method produced better result, in which the missing gaps were recovered reasonable well, except for weaker amplitude in the common-offset sections. Figure 10 provides the $f$ -$k$ spectra corresponding to the original data and interpolated results with the Fourier POCS, the conventional 3D $t$ -$x$ -$y$ SPF, and the proposed 3D $t$ -$x$ -$y$ SPF, respectively. Figure 11 show the interpolation errors using these methods. The simultaneous occurrence of regular and irregular data missing is a challenge in the interpolation process. The proposed 3D $t$ -$x$ -$y$ SPF method shows more reasonable results than the Fourier POCS and the conventional streaming PEF. Meanwhile, the proposed algorithm is more efficient, and the CPU times for the 3D POCS with 500 iterations was 380.42 s whereas those of the 3D $t$ -$x$ -$y$ SPF was 12.27 s.

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Figure 8. (a) A 3D field dataset and (b) data after subsampling in the shot direction and $30\%$ randomly selected traces removed.

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Figure 9. Interpolated results using different methods. (a) The 3D Fourier POCS, (b) the conventional 3D $t$ -$x$ -$y$ SPF, and (c) the proposed 3D $t$ -$x$ -$y$ SPF.

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Figure 10. The $f$ -$k$ spectra for different data. (a) Original data (Figure 8a), (b) data in Figure 9a, (c) data in Figure 9b, and (d) data in Figure 9c.

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Figure 11. Interpolation errors using different methods. (a) The 3D Fourier POCS, (b) the conventional 3D $t$ -$x$ -$y$ SPF, and (c) the proposed 3D $t$ -$x$ -$y$ SPF.

Seismic data interpolation without iteration using $t$ -$x$ -$y$ streaming prediction filter with varying smoothness