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

2D nonstationary missing-trace interpolation test

We utilized a 2D synthetic model composed of nonstationary events with conflicting dips to test the effectiveness of the proposed method (Figure 2a). Two curved events and one dipping event show different amplitudes. We removed $40\%$ of randomly selected traces specially at the intersecting position (Figure 2b). For comparison, we used a 2D Fourier POCS with 19$\times$ 8 patches and a conventional streaming PEF (Fomel and Claerbout, 2016) to interpolate the missing traces (Figure 3a and Figure 3b). The size of each patch was set as 40$\times$ 50 and the iteration number was selected to be 150 in the Fourier POCS method. The interpolated results using the SPF with causal and non-causal filters are shown in Figure 3c and Figure 3d, respectively. The difference between the reconstructed traces and the original traces is shown in Figure 4. For space non-causal filters, we designed a 2D $t$ -$x$ SPF with 25 (time) $\times$ 23 (space) coefficients and scalar parameters of $\xi_t$ = 0.05, $\xi_x$ = 0.8, $\delta_t$ = 0.5, and $\delta_x$ = 0.5 for each sample. The 2D Fourier POCS method with patching windows and SPF with causal filters produced similar results, except that more spatial aliasing was generated for the POCS method. The conventional streaming PEF has evident spatial aliasing at the cross position of the events and the lower right corner due to the helix transform. However, the proposed method reconstructed a reasonable result, where the interpolation errors were substantially reduced. The CPU times, for a single 2.10 GHZ CPU used in this study, were 2.68 s for the 2D Fourier POCS and 0.90 s for the proposed method.

mc,gc
Figure 2. (a) Synthetic data with three events and (b) model with $40\%$ randomly selected traces removed.

pocsp,interh,adds,ac
Figure 3. Reconstructed results by using different methods. (a) The 2D Fourier POCS, (b) the 2D streaming PEF, the 2D $t$ -$x$ SPF with (c) causal filter, and (d) non-causal filter.

dcpp,difh,difs,dcs
Figure 4. Interpolation errors by using different methods. (a) The 2D Fourier POCS, (b) the 2D streaming PEF, the 2D $t$ -$x$ SPF with (c) causal filter, and (d) non-causal filter.

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