Examples

We use one synthetic example and one field data example to demonstrate the performances of the traditional and the improved MSSA algorithms from the Matlab package. The first synthetic example is shown in Figures 1, 2, 3, and 4. Since this example is a synthetic example, we have the clean dataset in Figure 1a. We add some bandlimited random noise to the clean data to simulate the noisy data, as shown in Figure 1b. We then randomly remove 50% traces from the noisy synthetic data, and obtain the simulated observed noisy data, as shown in Figure 1c. Figures 1d and 1e show the reconstructed and denoised data using the traditional MSSA and the proposed modified MSSA algorithms. It is obvious that the proposed approach can obtain a very accurate recovery (Figure 1e) compared with the true clean data shown in Figure 1a. However, the traditional MSSA approach still causes some noticeable residual noise. Note that in Figure 1e, we use $N=3,K=2$. We also show the reconstructed and denoised data using $K=1$ and $K=3$ in Figure 2a and 2b, respectively. It is obvious that when $K=1$, as shown in Figure 2a, the proposed algorithm causes too much damages to the useful signals. When $K=3$, as shown in Figure 2b, there is a bit more residual noise left in the data compared with that when $K=2$. Figure 3 shows the comparison of 5th crossline section for all the figures in Figure 1. Figure 4 shows the comparison of 5th inline section for all the figures in Figure 1. From both Figures 3 and 4 we can confirm that the proposed improved MSSA algorithm can obtain a much better reconstruction and denoising performance than the traditional MSSA algorithm.

The field data example is shown in Figure 5. In this example, we do not know the true answer and thus we can not judge the reconstruction and denoising performance by comparing the final results with the true model as used in the synthetic example. Instead, we can only judge the performance by the spatial coherency of the reconstructed data. Figure 5a shows the observed noisy and incomplete data, which has been binned to regular grids. There are about 50% missing traces in this example. Figures 5b and 5c show the final results using the traditional and proposed improved MSSA algorithms. Figures 5d, 5e, and 5f show the 4th crossline slices of Figures 5a, 5b, and 5c, respectively. It can be seen that both methods can effectively reconstruct the missing data and attenuate the strong random noise, but the proposed improved MSSA algorithm can obtain even better performance since the events are more spatially coherent. It is worth mentioning that since there is no true answer for the field data example, it is hard to use quantitive measure to compare the performance of different approaches. The visual observation on the spatial coherency, instead, is the most effective and straightforward way.

Both examples are processed using the introduced Matlab packages. The binary files output from the Matlab packages are then put into the Madagascar platform to generate the final figures shown in Figures 1-5.

synth-clean synth-noisy synth-obs synth-mssa synth-dmssa2
synth-clean,synth-noisy,synth-obs,synth-mssa,synth-dmssa2
Figure 1.
(a) Clean data. (b) Noisy data. (c) Observed data with 50% missing traces. (d) Denoised and reconstructed using the MSSA method. (e) Denoised and reconstructed using the proposed approach ($K=2$).
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synth-dmssa1 synth-dmssa
synth-dmssa1,synth-dmssa
Figure 2.
(a) Denoised and reconstructed data using the proposed approach when $K=3$. (b) Denoised and reconstructed data using the proposed approach when $K=1$.
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synth-s-clean synth-s-noisy synth-s-obs synth-s-mssa synth-s-dmssa2
synth-s-clean,synth-s-noisy,synth-s-obs,synth-s-mssa,synth-s-dmssa2
Figure 3.
Single slice comparison (5th crossline section). (a) Clean data. (b) Noisy data. (c) Observed data with 50% missing traces. (d) Denoised and reconstructed using the MSSA method. (e) Denoised and reconstructed data using the proposed approach ($K=2$).
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synth-s-clean-i synth-s-noisy-i synth-s-obs-i synth-s-mssa-i synth-s-dmssa2-i
synth-s-clean-i,synth-s-noisy-i,synth-s-obs-i,synth-s-mssa-i,synth-s-dmssa2-i
Figure 4.
Single slice comparison (5th inline section). (a) Clean data. (b) Noisy data. (c) Observed data with 50% missing traces. (d) Denoised and reconstructed using the MSSA method. (e) Denoised and reconstructed using the proposed approach ($K=2$).
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field-obs field-mssa field-dmssa field-s-obs field-s-mssa field-s-dmssa
field-obs,field-mssa,field-dmssa,field-s-obs,field-s-mssa,field-s-dmssa
Figure 5.
(a) Observed field data (binned to the regular grid). (b) Reconstructed data using the MSSA method. (c) Reconstructed data using the DMSSA method. (d) 4th crossline slice of the observed data. (e) 4th crossline slice of the MSSA reconstructed data. (f) 4th crossline slice of the reconstructed data using the proposed approach.
[pdf] [pdf] [pdf] [pdf] [pdf] [pdf] [png] [png] [png] [png] [png] [png] [scons]


2020-03-10