Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization

Numerically blended synthetic data - linear events

Our first synthetic example contains three plane-wave events, with one high-dip-angle event crossing two other events. The original unblended and numerically blended sections are shown in Figure 4 shows the deblended results using soft thresholding in the $f-k$ domain, soft thresholding in the seislet domain, and $f-x$ predictive filtering. All of these deblending results are generally acceptable despite some artifacts left in Figure 4a and some weak-level noise left in Figure 4c. By computing the difference between the deblended section and the blended section, we can obtain the blending noise section. Comparing the blending noise sections (see Figures 4d-4f), we find that there is some leakage of useful energy for $f-x$ predictive filtering. Computing the differences between the deblended sections and the unblended sections, we get the estimation error sections, shown in Figures 4g-4i. Compared with the other two shaping approaches, seislet-domain soft thresholding causes a nearly zero estimation error, which indicates a nearly perfect deblending result. The estimation error for $f-x$ predictive filtering is comparatively large, because of the large predictive error problem when several dipping components are taken into consideration (Chen and Ma, 2014). The small estimation errors in Figure 4g are caused by spectrum mixture in the $f-k$ domain because some useful energy reside in the low amplitude part that is easy to be removed during soft thresholding. The small estimation errors shown in Figure 4h are caused by dip estimation error because of the conflicting dips when using plane wave destruction (PWD) algorithm with a single slope (Fomel, 2002), which is a limitation of seislet-domain denoising approaches, unless a seislet frame is used instead of the seislet transform (Fomel and Liu, 2010). The diagrams for convergence rates are shown in Figure 5, which demonstrates a superior behavior of seislet-domain soft thresholding in comparison with the other two approaches. In order to make the comparisons fair, we try to find the best deblending performance through the parameter-selection process for corresponding shaping operators. In this case, the percentages we use for $f-k$ domain and seislet domain thresholding are both 8 %, the filter length we use for $f-x$ predictive filtering is 4 samples.

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Figure 3. Numerically blended synthetic data (linear case). (a) Unblended data. (b) Blended data.

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Figure 4. Deblending comparison for numerically blended synthetic data (linear case). (a) Deblended result using $f-k$ domain thresholding. (b) Deblended result using seislet-domain thresholding. (c) Deblended result using $f-x$ predictive filtering. (d) Blending noise corresponding to (a). (e) Blending noise corresponding to (b). (f) Blending noise corresponding to (c). (g) Estimation error corresponding to (a). (h) Estimation error corresponding to (b). (i) Estimation error corresponding to (c).

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Figure 5. Diagrams of SNR for synthetic example (linear case). The "+" line corresponds to seislet-domain thresholding. The "o" line corresponds to $f-k$ domain soft thresholding. The "*" line corresponds to $f-x$ predictive filtering.

Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization