Examples

We use two synthetic examples and one field data set to compare the performance of the aforementioned percentile-half-thresholding approach with the conventional thresholding approaches. For simplicity, we only compare half thresholding with soft thresholding in two cases: percentile and constant-value threshold.

The first example is a synthetic example that is composed of four linear events, one of which crosses the other three events. The original data is shown in Figure 1a. After randomly removing 30% traces, we create the decimated section, as shown in Figure 1b. Figure 2 show the reconstruction results for the first synthetic example. From the result using percentile half thresholding (Figure 2d) and constant-value half thresholding (Figure 2b), and their corresponding estimation error sections (the difference between the true data and estimated data), we can observe that half thresholding nearly do a perfect job while the percentile thresholding strategy helps obtain a even better result. Both conventional constant-value and percentile soft-thresholding results seem to have significant estimation error. We can also observe that the percentile soft thresholding outperforms the constant-value soft thresholding by obtaining slightly less estimation error. In order to better compare the error sections using different approaches, we amplified the magnitude of each error section shown in Figures 2e, 2f, 2g and 2h by multiplying the magnitude by 10 times. The magnitude amplified error sections are shown in Figures 2i, 2j, 2k and 2l. It's much clearer that the percentile-half-thresholding strategy obtains a much better performance than any other approaches. The diagrams shown in Figure 3 further prove the observations in that the converged result for percentile half thresholding has the largest SNR, which is followed by constant-value half thresholding, and the constant-value soft thresholding obtains the smallest SNR.

The second example is a hyperbolic-events synthetic example. The original data and decimated data with 30 % traces randomly removed traces are shown in Figure 4. We apply four different thresholding approaches as mentioned above to this example and get the reconstructed results and their corresponding reconstruction error sections, which are shown in Figure 5. Similarly, the reconstructed result for the proposed percentile half thresholding (shown in Figure 5d) obtains the best results, causing negligible reconstruction error (shown in Figure 5h). The constant-value half thresholding also performs well because the error section (shown in Figure 5b) contains small amount of coherent signals. The reconstruction results for soft thresholding using either percentile or constant-value strategy (shown in Figures 5a and 5c )are not pleasant. Because they still contain some zero or weak-amplitude traces and cause a significant energy loss in the error sections as shown in Figures 5e and 5g, which indicates a failure in interpolating the missing traces. From the convergence diagram as shown in Figure 6, we can conclude that the half-thresholding approaches outperforms the conventional soft-thresholding approaches, while the percentile-thresholding strategy can helps obtain better results that the constant-value strategy.

The third example is a field marine data, which was investigated widely in the literature (Liu and Fomel, 2011; Fomel, 2002). We can obtain a similar conclusion as the previous two examples. However, from the reconstructed data and the estimation error sections, the conventional soft-thresholding (both constant-value and percentile) results (shown in Figures 8a and 8c) suffer from a heavy useful-energy loss as shown in Figures 8e and 8g and the reconstructed data as shown in Figures 8a and 8c still contain some obvious zero or weak-amplitude traces, which is no longer acceptable. From the convergence diagrams shown in Figure 9, the final SNRs of conventional soft-thresholding approaches are lower than the half-thresholding approaches.. Although the constant-value half thresholding can get slightly higher SNR than percentile half thresholding during the first several iterations, the percentile half thresholding converges to a higher final SNR than constant-value half thresholding.

linear,linear-zero
Figure 1. (a) Synthetic data (linear events). (b) Decimated synthetic data (created by randomly removing 30% traces).

data2,data3,data5,data6,diff2,diff3,diff5,diff6,diff2_10,diff3_10,diff5_10,diff6_10
Figure 2. Reconstructed results for synthetic data (linear events). (a) Using constant-value soft thresholding. (b) Using constant-value half thresholding. (c) Using percentile soft thresholding. (d) Using percentile half thresholding. (e)-(h) Reconstruction error sections corresponding to (a)-(d), respectively. (i)-(l) Amplitude amplified error sections by $\times 10$, corresponding to (e)-(h), respectively.

SNRs
Figure 3. Convergence diagrams for synthetic data (linear events). "p" corresponds to percentile half thresholding. "o" corresponds to constant-value half thresholding. "s" corresponds to percentile soft thresholding. "*" corresponds to constant-value soft thresholding.

hyper,hyper-zero
Figure 4. (a) Synthetic data (hyperbolic events). (b) Decimated synthetic data (created by randomly removing 30% traces).

hyper-data2,hyper-data3,hyper-data5,hyper-data6,hyper-diff2,hyper-diff3,hyper-diff5,hyper-diff6
Figure 5. Reconstructed results for synthetic data (hyperbolic events). (a) Using constant-value soft thresholding. (b) Using constant-value half thresholding. (c) Using percentile soft thresholding. (d) Using percentile half thresholding. (e)-(h) Reconstruction error sections corresponding to (a)-(d), respectively.

hyper-SNRs
Figure 6. Convergence diagrams for synthetic data (hyperbolic events). "p" corresponds to percentile half thresholding. "o" corresponds to constant-value half thresholding. "s" corresponds to percentile soft thresholding. "*" corresponds to constant-value soft thresholding.

sean,sean-zero
Figure 7. (a) Field data. (b) Decimated field data (created by randomly removing 30% traces).

sean-data2,sean-data3,sean-data5,sean-data6,sean-diff2,sean-diff3,sean-diff5,sean-diff6
Figure 8. Reconstructed results for field data. (a) Using constant-value soft thresholding. (b) Using constant-value half thresholding. (c) Using percentile soft thresholding. (d) Using percentile half thresholding. (e)-(h) Reconstruction error sections corresponding to (a)-(d), respectively.

sean-SNRs
Figure 9. Convergence diagrams for field data. "p" corresponds to percentile half thresholding. "o" corresponds to constant-value half thresholding. "s" corresponds to percentile soft thresholding. "*" corresponds to constant-value soft thresholding.