3D data interpolation using the -

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3D data interpolation using the - - SPF

For the 3D data interpolation test, we selected a 3D synthetic model (Fig. 9a) containing curve events and faults, and the data cube randomly removed $70\%$ of the seismic traces, where the faults were hard to distinguish (Fig. 9c). The - $K_{x}$ - $K_{y}$ spectrum of the synthetic model and the missing data model are shown in Fig. 9b and 9d, respectively. Because currently seislet transform can only handle 2D datasets, here we evaluated the recovery ability of the - - SPF by comparing with the 2D seislet POCS method and the 3D Fourier POCS method. The parameters of the - - SPF are $\lambda_{f}=0.0005$ , $\lambda_{x}=0.001$ , $\lambda_{y}=0.0008$ , and 11 (space ) $\times$ 6 (space ) filter coefficients. The 2D seislet POCS does not reconstruct the missing data well (Fig. 10a), and generates large interpolation errors (Fig. 10b). The interpolated results (Fig. 10c and 10e) show that both the 3D Fourier POCS and the - - SPF can recover the missing traces even when faults are present. However, the interpolation errors (Fig. 10d and 10f) display that the - - SPF produces less error and preserves the amplitude of the events reasonably better than the 3D Fourier POCS. The comparison of the - $K_{x}$ - $K_{y}$ spectra is shown in Fig. 11, the - - SPF reduces the influence of aliasing. More importantly, the - - SPF significantly reduces the computational cost by avoiding the iterative algorithm especially in higher dimensions. Compared with the Fourier POCS, the proposed methods solve the problem without iterations, which reduces the computational cost (Table 1). Table 2 shows the time consumption of each method, and the computation platform uses 2.0GHz E5-2650 CPU.

To further evaluate the interpolation ability of the proposed methods, we defined the signal-to-noise ratio (SNR) as a measurement:

$\displaystyle SNR = 10 \log_{10} \left( \frac{\Arrowvert \mathbf{D} \Arrowvert_{2}^{2}} {\Arrowvert\mathbf{D} - \hat{\mathbf{D}}\Arrowvert_{2}^{2}} \right),$

(19)

where $\mathbf{D}$ denotes original data, and $\hat{\mathbf{D}}$ denotes interpolation result. The traces in the 3D model (Fig. 9a) have been randomly removed from $5\%$ to $95\%$ . The interpolation results of the

SPF are shown in Fig. 12. The proposed SPF interpolation method in the frequency domain reconstructs data even under the severely degraded circumstance. Fig. 13 shows that the SPF in higher dimensions can effectively improve the SNR.


qdome,fkqd,gapqd,fkgapqd Figure 9. Synthetic 3D model (a) and - $K_{x}$ - $K_{y}$ spectrum (b). Model with $70\%$ of the data traces randomly removed (c) and - $K_{x}$ - $K_{y}$ spectrum (d).


stpocsqd,errstpocsqd,pocsqd,errpocsqd,spfqd,errspfqd Figure 10. Reconstructed result (a) and interpolation error (b) using the 2D seislet POCS, reconstructed result (c) and interpolation error (d) using the 3D Fourier POCS, reconstructed result (e) and interpolation error (f) using the 3D - - SPF.


fkstpocsqd,fkpocsqd,fkspfqd Figure 11. - $K_{x}$ - $K_{y}$ spectrum of the interpolation result using the 2D seislet POCS (a), - $K_{x}$ - $K_{y}$ spectrum of the interpolation result using the 3D Fourier POCS (b), - $K_{x}$ - $K_{y}$ spectrum of the interpolation result using the 3D - - SPF (c).


gap3d-5,intp3d-5,gap3d-95,intp3d-95 Figure 12. Model with $5\%$ of the data traces randomly removed (a), interpolated data using the 3D - - SPF corresponding to Fig. 12a (b), model with $95\%$ of the data traces randomly removed (c), and interpolated data using the 3D - - SPF corresponding to Fig. 12c (d).


snr3d Figure 13. The SNR changing with different degree of trace missing (range from $5\%$ to $95\%$ ). Dash line is the SNR of the model with trace missing and solid line is the SNR of interpolation result.

We used 3D field common mid-point (CMP) gathers after normal moveout (NMO) to further test the proposed method (Fig. 14a). Fig. 14b shows data binning, where the blank space denotes approximately $75\%$ of the missing traces. A large amount of missing traces lead to the spatial aliasing artifacts as shown in Fig. 18a. The 2D seislet POCS works well with the horizontal events, but the data interpolation result is not reasonable where curve and horizontal events intersect (Fig. 15a). The 3D Fourier POCS can recover most horizontal events, however, some large gaps remain in Fig. 16a. We chose $\lambda_{f}=6$ , $\lambda_{x}=60$ , $\lambda_{y}=15$ , and 101 (space ) $\times$ 5 (space ) filter coefficients for the - - SPF. The design of the filter coefficients is large, which increases the computational time-consumption of the - - SPF. The interpolation result using the 3D - - SPF (Fig. 17a) shows that the missing traces are reasonably reconstructed, the broken events are well recovered, and the continuity of both linear and curve events are interpolated well. A close-up comparison at TraceY=8 (Fig. 15b, 16b, and 17b) shows that the - - SPF can handle more gaps and recover the amplitude of nonstationary events better than the 2D seislet POCS and the 3D Fourier POCS. The - $K_{x}$ - $K_{y}$ spectra (Fig. 18) demonstrate that the - - SPF can handle aliasing, and the energy is converged in the - $K_{x}$ - $k_{y}$ spectrum (Fig. 18d).

**Table 1:** Comparison of computational cost: the - SPF, the 2D Fourier POCS, the - - SPF, and the 3D Fourier POCS.
3.5pt $\begin{threeparttable} \begin{tabular}{\vert c\vert c\vert c\vert} \toprule &... ...{iter}$ is the number of iterations. \par \end{tablenotes} \end{threeparttable}$

**Table 2:** Comparison of time consumption.
3.5pt 20mm $\begin{threeparttable} \begin{tabular}{\vert c\vert c\vert c\vert c\vert c\vert... ...{1}$s & $3.63\times10^{2}$s \\ \bottomrule \end{tabular} \end{threeparttable}$


gapcmp,mask Figure 14. 3D field data with trace missing (a) and data binning of the field data (b).


stpocscmp,stpocszoom Figure 15. Reconstructed result using the 2D seislet POCS (a), and close-up of the interpolated result (b).


pocscmp,pocszoom Figure 16. Reconstructed result using the 3D Fourier POCS (a), and close-up of the interpolated result (b).


spfcmp,spfzoom Figure 17. Reconstructed result using the 3D - - SPF (a), and close-up of the interpolated result (b).


fkgapcmp,fkstpocscmp,fkpocscmp,fkspfcmp Figure 18. The - $K_{x}$ - $K_{y}$ spectrum of the 3D field data (a), the - $K_{x}$ - $K_{y}$ spectrum of the reconstructed result using the 2D seislet POCS (b), the - $K_{x}$ - $K_{y}$ spectrum of the reconstructed result using the 3D Fourier POCS (c), the - $K_{x}$ - $K_{y}$ spectrum of the reconstructed result using the 3D - - SPF (d).

Next: Discussion Up: Numerical examples Previous: 2D data interpolation using

2022-04-15