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

3D synthetic data test

We created a 3D prestack dataset (Figure 6a) from a 2D slice out of the benchmark French model (French, 1974), and the data was subsampled by a factor of two in both offset and shot axes, which caused visible aliasing of dipping events. Furthermore, we removed $15\%$ of randomly selected traces from the decimated data. The data interleaved with zero traces along the offset and shot directions is shown in Figure 6b. The challenge of this test was to account for nonstationarity, aliasing, both decimated and irregular missing traces, and computational cost. Figure 7a and Figure 7b display the interpolated result using 3D Fourier POCS and the conventional 3D $t$ -$x$ -$y$ SPF, respectively. Notably, the Fourier POCS method can only recover randomly missing traces, and it fails in handling regularly missing traces. For the proposed 3D $t$ -$x$ -$y$ SPF, the choices of the filter length were seven samples in time axis, nine samples in the offset axis, and three samples in the shot axis. We designed the scale parameters, $\xi_t=0.4$ , $\xi_x=0.5$ , $\xi_y=0.4$ , and $\delta_t=\delta_x=\delta_y=0.01$ , to deal with the variability of events. The conventional 3D $t$ -$x$ -$y$ SPF did not recover the decimated data well. However, the proposed method succeeded in interpolating irregular and regular missing traces simultaneously (Figure 7c), which produced reasonable results for curved events. The CPU times of the 3D Fourier POCS with 500 iterations and the 3D $t$ -$x$ -$y$ SPF were 889.21 s and 33.72 s, respectively.

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Figure 6. (a) 3D synthetic prestack data and (b) missing data interleaved with zero traces.

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Figure 7. Reconstructed data volumes 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