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Field data examples

field1 field1n
field1,field1n
Figure 11.
(a) Clean data. (b) Noisy data.
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field1n-re0 field1n-ddtf0 field1n-ddtf-re00
field1n-re0,field1n-ddtf0,field1n-ddtf-re00
Figure 12.
Denoised sections using different approaches. (a) Seislet thresholding. (b) DDTF thresholding. (c) DSD thresholding. The best results are selected for each method.
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field1n-dif0 field1n-ddtf-dift field1n-ddtf-dif00
field1n-dif0,field1n-ddtf-dift,field1n-ddtf-dif00
Figure 13.
Noise sections using different approaches. (a) Seislet thresholding. (b) DDTF thresholding. (c) DSD thresholding. The best results are selected for each method.
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For field data tests, we first use a relatively simple 2-D seismic image, shown in Figure 11a. The simulated noisy data with Gaussian white noise is shown in Figure 11b. The denoised results from the three different approaches specified previously are shown in Figure 12. In this example, we use $ 7\times7$ patch size for the tight frames and run 30 iterations for updating the tight frames and the coefficients. The corresponding three noise sections are shown in Figure 13. From the comparison of denoised results, we conclude that, while all of the three methods can obtain acceptable results, the proposed DSD-based thresholding produces the cleanest image (Figure 12c) with the least amount of useful energy left in the noise section (Figure 13c). The seislet transform based thresholding causes some damage to signal events around time sample 300. The DDTF based thresholding leaves some residual random noise, which comes from the fact that DDTF might automatically learn the behavior of random noise and thus become sensitive to random noise. In this example, we carefully selected parameters for each method in order to obtain the best performance. Specifically, we use 4 % coefficients in the seislet domain, 8 % coefficients in the DDTF domain, and 3 % coefficients in the DSD domain. The comparison of SNRs is shown in the third row in Table 2. The SNR comparison confirms the previous observation that DSD-based thresholding can obtain the highest SNR.

field2-f field2n-f
field2-f,field2n-f
Figure 14.
(a) Clean data. (b) Noisy data. The frame boxes are zoomed for detailed comparisons in Figure 17.
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field2n-re0-f field2n-ddtf0-f field2n-ddtf-re00-f
field2n-re0-f,field2n-ddtf0-f,field2n-ddtf-re00-f
Figure 15.
Denoised sections using different approaches. (a) Seislet thresholding. (b) DDTF thresholding. (c) DSD thresholding. The best results are selected for each method. The frame boxes are zoomed for detailed comparisons in Figure 17.
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field2n-dif0 field2n-ddtf-dift field2n-ddtf-dif00
field2n-dif0,field2n-ddtf-dift,field2n-ddtf-dif00
Figure 16.
Noise sections using different approaches. (a) Seislet thresholding. (b) DDTF thresholding. (c) DSD thresholding. The best results are selected for each method.
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field2-f-zoom-a field2n-f-zoom-a field2n-re0-f-zoom-a field2n-ddtf0-f-zoom-a field2n-ddtf-re00-f-zoom-a
field2-f-zoom-a,field2n-f-zoom-a,field2n-re0-f-zoom-a,field2n-ddtf0-f-zoom-a,field2n-ddtf-re00-f-zoom-a
Figure 17.
Zoomed sections corresponding to the frame boxes shown in Figures 14 and 15. (a) Clean data. (b) Noisy data. (c) Seislet thresholding. (d) DDTF thresholding. (e) DSD thresholding.
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Our second field data example is a more complicated case. Figures 14a and 14b show the clean data and simulated noisy data, respectively. In this example, we use $ 7\times7$ patch size for the tight frames and run 30 iterations for updating the tight frames and the DSD coefficients. Because of the complex signal structure, seislet transform fails to sparsify the data well. On the other hand, because of strong noise, some noise gets treated as useful signal and gets incorporated into the dictionary when constructing DDTF. Therefore, DDTF also fails to obtain good denoising result for this dataset. The proposed DSD compensates for the weaknesses of both the seislet transform and DDTF and thus achieves a better result. Figure 15 shows the optimal denoised results after using thresholding in the seislet domain with 17 % coefficients, thresholding in the DDTF domain with 11 % coefficients, thresholding in the DSD domain with 7 % coefficients. The corresponding noise sections are shown in Figure 16. From observing the denoised results and noise sections, we conclude that thresholding in the DSD domain obtained the cleanest image and with the smallest amount of useful coherent signals lost in the noise sections. The calculated SNRs are shown in the Field 2 row in Table 2. The SNR comparison confirms the observation that thresholding in DSD domain can get the highest SNR. In this example, we also tried different parameters in order to obtain the best result for each approach. The denoising performance using DSD is noticeably better than the other two approaches, in terms of both SNR measures and visual observation.

To see the differences more clearly, we zoomed parts of the denoised sections and show them in Figure 17. The zoomed sections correspond to the portions pointed out by the frame boxes in Figures 14 and 15. We can observe more clearly a better performance of the proposed DSD thresholding in comparison with the other two approaches in that it gets a cleaner image and preserves more of the useful signal.

Our third field data example is a noisy seismic dataset, shown in Figure 18. The denoised sections using different denoising approaches are shown in Figure 19. In addition to the three thresholding-based approaches, we also demonstrate the denoising performance using the classic $ f-x$ deconvolution (Chen and Ma, 2014; Galbraith, 1991; Gulunay, 1986). For $ f-x$ deconvolution, we use $ 50\times50$ windows with 50% overlap between different local windows. The prediction length is 6 points. These parameters have been tuned for obtaining the best denoising performance. In this example, we use $ 7\times7$ patch size for the tight frames and 30 iterations for updating the tight frames and the coefficients in DDTF and DSD. Correspondingly, Figure 20 shows the noise sections using four approaches. Since for the real case, we do not know the true answer, we are not able to numerically compare the performance of each approach as we did in the previous examples. Instead, we can only judge the performance by observation. The denoised section using $ f-x$ deconvolution and its corresponding noise section are shown in Figures 19c and 20c, respectively. The denoising results show that the DSD thresholding removes the most noise while preserving the most significant details of the original seismic data.

real
real
Figure 18.
Real noisy seismic data.
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real-re real-ddtf-t0 real-fx0 real-sletddtf-re
real-re,real-ddtf-t0,real-fx0,real-sletddtf-re
Figure 19.
Denoised sections using different approaches. (a) Seislet thresholding. (b) DDTF thresholding. (c) $ f-x$ deconvolution. (d) DSD thresholding. The best results are selected for each method.
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real-dif real-ddtf-dif real-fx-dif0 real-sletddtf-dif
real-dif,real-ddtf-dif,real-fx-dif0,real-sletddtf-dif
Figure 20.
Noise sections using different approaches. (a) Seislet thresholding. (b) DDTF thresholding. (c) $ f-x$ deconvolution. (d) DSD thresholding. The best results are selected for each method.
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Percentage Seislet DDTF DSD
1 % 12.67 16.35 19.25
2 % 13.10 17.34 21.32
3 % 13.31 18.32 20.65
4 % 15.03 18.78 19.74
5 % 15.63 18.56 19.32
10 % 14.84 16.32 15.12
15 % 13.33 12.85 11.54

Table 1. Comparison of SNR in dB using different approaches for first example (the original SNR is -7.31 dB).

Models Original Seislet DDTF DSD
Synthetic 2 20.04 33.67 (20%) 29.43 (16%) 37.25 (10%)
Field 1 -3.015 23.12 (4%) 18.04 (8%) 26.34 (3%)
Field 2 -0.94 15.54 (17%) 18.23 (11%) 21.97 (7%)

Table 2. Comparison of SNR in dB using different approaches. The percentage corresponds to the optimum percentage of coefficients in the transform domain that can obtain the presented SNRs.


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Next: Discussion Up: Examples Previous: Synthetic examples

2016-02-27