Seismic data interpolation using streaming prediction filter in the frequency domain

2D data interpolation using the $f$ -$x$ SPF

We generated a 2D synthetic model with one linear event and two curve events to evaluate the interpolation ability of the $f$ -$x$ SPF. The curve events bend in opposite directions (Fig. 3a and 3b), which challenges the adaptability of interpolation method. For a missing trace interpolation test (Fig. 3c), we removed $40\%$ of the randomly selected traces, which caused the appearance of aliasing (Fig. 3d). For comparison, we used the 2D Fourier Project Onto Convex Sets (POCS) to recover the missing traces (Fig. 4a). The interpolation result from the 2D Fourier POCS showed that the linear event was interpolated, but many discontinuities were present on the curve events. The interpolated error was slightly larger in the locations missing traces (Fig. 4b), which is caused by strongly variable slopes. The seislet POCS method (Gan et al., 2016) shows a clean interpolation result (Fig. 4c), but it produces interpolation errors where the events intersect (Fig. 4d). We designed the $f$ -$x$ SPF with $\lambda_{f}=0.2$ , $\lambda_{x}=0.5$ , and 30 (space) filter coefficients. The proposed method provided successful amplitude preservation, and the missing traces were interpolated reasonably well (Fig. 4e). The difference between the interpolated and original traces showed that most of the redundant fluctuations and artifacts were smaller than those created by the Fourier POCS (Fig. 4f). The $F$ -$K$ spectra (Fig. 5) showed that the $f$ -$x$ SPF recovered the missing data and successfully suppressed aliasing.

mod,fkmod,gap,fkgap
Figure 3. Synthetic model (a) and $F$ -$K$ spectrum (b). Model with $40\%$ of the data traces randomly removed (c) and $F$ -$K$ spectrum (d).

pocs,errpocs,stpocs,errstpocs,fxspf,errfxspf
Figure 4. Reconstructed result (a) and interpolation error (b) using the 2D Fourier POCS, reconstructed result (c) and interpolation error (d) using the 2D seislet POCS, reconstructed result (e) and interpolation error (f) using the 2D $f$ -$x$ SPF.

fkpocs,fkstpocs,fkfxspf
Figure 5. $F$ -$K$ spectrum of the interpolation result using the 2D Fourier POCS (a), $F$ -$K$ spectrum of the interpolation result using the 2D seislet POCS (b), $F$ -$K$ spectrum of the interpolation result using the 2D $f$ -$x$ SPF (c).

Fig. 6a shows a marine shot gather from a deepwater Gulf of Mexico survey with $40\%$ of the data traces randomly removed (Fig. 6c). The $F$ -$K$ spectra (Fig. 6b and 6d) reflect the impact caused by missing data. Fig. 7a shows the interpolated result using the 2D Fourier POCS, which fails to recover steeply dipping events at the far-offset positions (Fig. 7b). In the interpolation result of seislet POCS, discontinuities of seismic events are present (Fig. 7c), and some energy remains in the interpolation error profile (Fig. 7d). We designed the $f$ -$x$ SPF with $\lambda_{f}=0.05$ , $\lambda_{x}=0.1$ , and 20 (space) coefficients. Fig. 7e and 7f show the interpolated result using the proposed method and the difference between the interpolated and original traces plotted at the same clip value. The reconstructed data naturally filled the broken events; meanwhile, the steeply dipping events and diffraction events were reasonably interpolated. The $F$ -$K$ spectrum of the interpolation result using the $f$ -$x$ SPF (Fig. 8c) is similar to that of the original data (Fig. 6b); it suppresses the low-frequency interference compared to the seislet POCS (Fig. 8b), and gives a cleaner spectrum than the Fourier POCS (Fig. 8a).

sean,fksn,gapsn,fkgapsn
Figure 6. Field data (a) and $F$ -$K$ spectrum (b). Data with $40\%$ of the seismic traces randomly removed (c) and $F$ -$K$ spectrum (d).

pocssn,errpocssn,stpocssn,errstpocssn,fxspfsn,errfxspfsn
Figure 7. Reconstructed result (a) and interpolation error (b) using the 2D Fourier POCS, reconstructed result (c) and interpolation error (d) using the 2D seislet POCS, reconstructed result (e) and interpolation error (f) using the 2D $f$ -$x$ SPF.

fkpocssn,fkstpocssn,fkfxspfsn
Figure 8. $F$ -$K$ spectrum of the interpolation result using the 2D Fourier POCS (a), $F$ -$K$ spectrum of the interpolation result using the 2D seislet POCS (b), $F$ -$K$ spectrum of the interpolation result using the 2D $f$ -$x$ SPF (c).

Seismic data interpolation using streaming prediction filter in the frequency domain