Synthetic examples

The first example is shown in Figure 1 including both clean and noisy data. The noisy data has an SNR of -8.39 dB. In this test, we compare the denoising performance of four different methods: the frequency-wavenumber (FK) domain thresholding method (Mahdad, 2012; Abma et al., 2015), the rank-reduction method (RR), the damped rank-reduction method (DRR), and the proposed method (ODRR). In this example, we do not use local windows. We use the Ricker wavelet with a dominant frequency of 40 Hz to generate the clean data. We add Gaussian white noise with variance of 0.2 (as compared with the normalized clean data). Since the proposed method is an improved version of the DRR method, we compare the proposed method with two other rank-reduction methods. Since the FK method is one of most commonly used approaches to attenuate high-frequency high-wavenumber random noise, we also treat it as a competing method for this comparison. Figure 2 shows denoised data using different methods. The top row of Figure 2 shows the the dataset denoised by the four methods, while the middle row of Figure 2 presents the noise extracted by the four methods. In this test, we preserve the 10% largest coefficients for the FK method to obtain Figure 2a. We use a rank of $N=3$ for all three rank-reduction based methods. By observing Figure 2a, it is clear that the FK thresholding causes strong artifacts around the edges. Both FK thresholding and the rank-reduction methods leaves a large amount of residual noise in the denoised results. The calculated SNRs of the four methods are 6.03 dB for the FK thresholding method, 6.57 dB for the rank-reduction method, 10.29 dB for the damped rank-reduction method, and 11.27 dB for the proposed method. It is clear that the SNR comparison quantitatively confirms our initial observation that the proposed method obtains the best result among the four methods. In this test we use a very simple synthetic example that contains three planar components. In the rank-reduction method, it can be proved that the number of the planar/linear events is equal to the input rank parameter, i.e., $N=3$ in this test (Oropeza and Sacchi, 2011; Trickett and Burroughs, 2009). Because we know the ground truth in this test, we the same rank for all methods, and the proposed method obtains the best performance but is slightly better than the damped method. However, in the case of the more complex data, where the exact number of events with distinct dips is unknown, we prefer to choose a more conservative rank to avoid losing too many useful details. The bottom row of Figure 2 shows the comparison of local similarity, where we can find that the local similarity of the ODRR method is much smaller. The spectrum comparison when $N=3$ is plotted in Figure 3, which further verify the best performance of the proposed method.

Next, based on the same example, we use a larger rank $N=6$. We also use a more conservative threshold value for the FK thresholding method, e.g., we preserve the 20% largest coefficients in the transformed domain. The results of the four methods are shown in the top row of Figure 4. The middle row of Figure 4 shows the corresponding noise cubes. It is clear that the denoised results of FK thresholding method, rank-reduction method, damped rank-reduction method are all noisier than the corresponding results presented in Figure 2. However, the result from the proposed method, as shown in Figure 4d, is less affected. The calculated SNRs in this case are 4.31 dB for the FK method, 3.45 dB for the RR method, 8.35 dB for the DRR method, and 10.86 dB for the proposed method. The differences of SNR with respect to the previous example for the three methods are 1.72 dB for the FK thresholding method, 3.12 dB for the rank-reduction method, 1.94 dB for the damped rank-reduction method, and 0.41 dB for the proposed method. Comparison of SNRs demonstrates that while the other three methods are sensitive to the input parameter, the proposed method is much less sensitive, because it can still find the appropriate rank by using weights. The bottom row of Figure 4 shows the comparison of local similarity, where we find that the local similarity of the presented method is distinctly smaller. The spectrum comparison when $N=6$ is plotted in Figure 5, which further verify the best performance of the proposed method.

For a better comparison, we extract the 5th Xline slice from Figures 1 and 4 and show the slices in Figure 6. In this display, it is even more evident that the proposed method produces the cleanest result while minimizing signal damage. Figure 7 plots the SNR diagrams of the different methods with respect to the input parameters. The input parameters for the rank-reduction based methods and transform based methods are the selected rank and the percentage of sparse coefficients, respectively. When the rank is chosen large enough, e.g., larger than 8, the proposed method is less sensitive in all methods. The two transform based methods are also sensitive to the percentage of selected coefficients.

To test the performance of the proposed method in denoising high-frequency band. We create a slightly different example shown in Figure 8. We increase the dominant frequency of the Ricker wavelet to 60 Hz and then extract the frequency band of 60-100 Hz. We also extract the same frequency band of the Gaussian white noise, and add the high-frequency noise to the clean data to generate the noisy data. The denoising comparison of different methods for the high-frequency band is shown in Figure 9. The spectrum comparison is shown in Figure 10. This test demonstrates the lowrank methods also work well in high-frequency band.

The next example is a synthetic dataset with hyperbolic events. In this example, we use the Ricker wavelet with a dominant frequency of 10 Hz to generate the clean data. We add Gaussian white noise with variance of 0.2 (as compared with the normalized clean data). The clean and noisy data are shown in Figures 11a and 11b, respectively. The SNR of the noisy data is -2.17 dB. We apply FK thresholding method and the three rank-reduction based methods to this example and show the results in Figure 12. For the FK thresholding method, we preserve 20% largest coefficients in the transformed domain. For all the rank-reduction based methods, we use $N=10$. Since the hyperbolic events no longer satisfy the assumption, i.e., being lowrank, the hyperbolic events in this example are over-smoothed when $N=10$. It is also clear that both FK thresholding and the rank-reduction methods have significant residual noise. The damped rank-reduction method and the proposed method are both very clean, but the proposed method is slightly smoother. In this example, because of the hyperbolic events and their small rank, the removed noise cubes as shown in the bottom row of Figure 12 contain a small amount of spatially coherent energy, which indicates signal leakage (Chen and Fomel, 2015). The calculated SNRs in this example are 7.05 dB for the FK thresholding method, 8.27 dB for the rank-reduction method, 9.58 dB for the damped rank-reduction method, and 9.65 dB for the proposed method. In this example, we do not use local windows to locally pretend that hyperbolic events act as linear events. When applying local windows, additional parameters (e.g., the window size) need to be compared and considered. To avoid the this step, we can use a relatively large rank to avoid the signal damage. From the comparison of local similarity, significant damage is highlighted as high similarity anomalies. We increase the rank from $N=10$ to $N=20$ and show the results in Figure 14. For the FK thresholding method, we increase the threshold percentage from 20% to 40%. We find that in this test, both FK thresholding method and the rank-reduction method leave more residual noise while the results from the damped rank-reduction method and the proposed method are still very smooth. However, when $N=20$, the rank-reduction methods do not produce significant damage to useful signals but they also do not attenuate the noise very well. In addition, when $N=20$, the proposed method becomes obviously smoother than the damped rank-reduction method. The comparison is more noticeable when observing the local similarity maps. The calculated SNRs in this example are 5.85 dB for the FK thresholding method, 7.04 dB for the rank-reduction method, 10.08 dB for the damped rank-reduction method, and 11.00 dB for the proposed method. Figure 16 shows a single slice comparison of this example (5th Xline slice) when $N=20$, where it is more noticeable that the proposed method obtains the best result. The SNRs for all the aforementioned tests are given in Table 2 for a detailed comparison. Figure 17 plots the SNR diagrams of the different methods with respect to the input parameters for the hyperbolic example. When the rank is sufficiently large, e.g., larger than 18, the proposed approach is clearly more insensitive to the rank when compared to the other rank-reduction based methods due to the calculation of adaptive weights for the singular-values. Table 3 compares the computational costs for all these tests. All three methods are comprable but the proposed method is slightly more expensive.

syn3d-c,syn3d-n
Figure 1. Synthetic data examples with linear/planar events. (a) Clean data. (b) Noisy data. The events shown on the outside of the cube are situated at the blue lines within the cube.

syn3d-fk,syn3d-lr,syn3d-dlr,syn3d-olr,syn3d-n-fk,syn3d-n-lr,syn3d-n-dlr,syn3d-n-olr,syn3d-simi1,syn3d-simi2,syn3d-simi3,syn3d-simi4
Figure 2. Denoising comparison ($N=3$). Top row: denoised results using (a) FK method with 10% largest coefficients, (b) rank-reduction method, (c) damped rank-reduction method, and (d) the proposed method. Middle row: separated noise corresponding to the top row. Bottom row: local similarity corresponding to the top row.

syn3d-c-fkk,syn3d-n-fkk,syn3d-fk-fkk,syn3d-lr-fkk,syn3d-dlr-fkk,syn3d-olr-fkk
Figure 3. Spectrum comparison ($N=3$). FK spectrum of (a) clean data, (b) noisy data, (c) FK method, (d) rank-reduction method, (e) damped rank-reduction method, and (f) the proposed method.

syn3d-6-fk,syn3d-6-lr,syn3d-6-dlr,syn3d-6-olr,syn3d-6-n-fk,syn3d-6-n-lr,syn3d-6-n-dlr,syn3d-6-n-olr,syn3d-6-simi1,syn3d-6-simi2,syn3d-6-simi3,syn3d-6-simi4
Figure 4. Denoising comparison ($N=6$). Top row: denoised results using (a) FK method with 20% largest coefficients, (b) rank-reduction method, (c) damped rank-reduction method, and (d) the proposed method. Middle row: separated noise corresponding to the top row. Bottom row: local similarity corresponding to the top row.

syn3d-6-c-fkk,syn3d-6-n-fkk,syn3d-6-fk-fkk,syn3d-6-lr-fkk,syn3d-6-dlr-fkk,syn3d-6-olr-fkk
Figure 5. Spectrum comparison ($N=6$). FK spectrum of (a) clean data, (b) noisy data, (c) FK method, (d) rank-reduction method, (e) damped rank-reduction method, and (f) the proposed method.

syn3d-6-c-s,syn3d-6-fk-s,syn3d-6-lr-s,syn3d-6-dlr-s,syn3d-6-olr-s,syn3d-6-n-s,syn3d-6-n-fk-s,syn3d-6-n-lr-s,syn3d-6-n-dlr-s,syn3d-6-n-olr-s,syn3d-6-simi1-s,syn3d-6-simi2-s,syn3d-6-simi3-s,syn3d-6-simi4-s
Figure 6. 2D slice view of denoising comparison ($N=6$). (a) Clean data. (b)-(e): denoised results using FK method, (b) rank-reduction method, (c) damped rank-reduction method, and the proposed method, respectively. (f) Noisy data. (g)-(j) Separated noise corresponding to (b)-(e), respectively. (k)-(n) Local similarity corresponding to (b)-(e), respectively.

snrs
Figure 7. SNR diagrams of different lowrank approaches with respect to the selected rank parameters for the linear synthetic example.

syn3dhf-c,syn3dhf-n
Figure 8. Synthetic data examples with linear/planar events for the high-frequency denoising test (60-100 Hz). (a) Clean data. (b) Noisy data. The events shown on the outside of the cube are situated at the blue lines within the cube.

syn3dhf-fk,syn3dhf-lr,syn3dhf-dlr,syn3dhf-olr,syn3dhf-n-fk,syn3dhf-n-lr,syn3dhf-n-dlr,syn3dhf-n-olr,syn3dhf-simi1,syn3dhf-simi2,syn3dhf-simi3,syn3dhf-simi4
Figure 9. Denoising comparison ($N=6$) for the high-frequency denoising test (60-100 Hz). Top row: denoised results using (a) FK method with 10% largest coefficients, (b) rank-reduction method, (c) damped rank-reduction method, and (d) the proposed method. Middle row: separated noise corresponding to the top row. Bottom row: local similarity corresponding to the top row.

syn3dhf-c-fkk,syn3dhf-n-fkk,syn3dhf-fk-fkk,syn3dhf-lr-fkk,syn3dhf-dlr-fkk,syn3dhf-olr-fkk
Figure 10. Spectrum comparison ($N=6$) for the high-frequency denoising test (60-100 Hz). FK spectrum of (a) clean data, (b) noisy data, (c) FK method, (d) rank-reduction method, (e) damped rank-reduction method, and (f) the proposed method.

hyp3d-c,hyp3d-n
Figure 11. Synthetic data examples with hyperbolic events. (a) Clean data. (b) Noisy data. The gray scales for all of the images shown in Figures 11-16 are the same.

hyp3d-fk,hyp3d-lr,hyp3d-dlr,hyp3d-olr,hyp3d-n-fk,hyp3d-n-lr,hyp3d-n-dlr,hyp3d-n-olr,hyp3d-simi1,hyp3d-simi2,hyp3d-simi3,hyp3d-simi4
Figure 12. Denoising comparison ($N=10$). The top row: denoised results using (a) FK method, (b) rank-reduction method, (c) damped rank-reduction method, and (d) the proposed method. The middle row: separated noise corresponding to the top row. The bottom row: local similarity corresponding to the top row.

hyp3d-c-fkk,hyp3d-n-fkk,hyp3d-fk-fkk,hyp3d-lr-fkk,hyp3d-dlr-fkk,hyp3d-olr-fkk
Figure 13. Spectrum comparison ($N=10$). FK spectrum of (a) clean data, (b) noisy data, (c) FK method, (d) rank-reduction method, (e) damped rank-reduction method, and (f) the proposed method.

hyp3d-20-fk,hyp3d-20-lr,hyp3d-20-dlr,hyp3d-20-olr,hyp3d-20-n-fk,hyp3d-20-n-lr,hyp3d-20-n-dlr,hyp3d-20-n-olr,hyp3d-20-simi1,hyp3d-20-simi2,hyp3d-20-simi3,hyp3d-20-simi4
Figure 14. Denoising comparison ($N=20$). The top row: denoised results using (a) FK method, (b) rank-reduction method, (c) damped rank-reduction method, and (d) the proposed method. The middle row: separated noise corresponding to the top row. The bottom row: local similarity corresponding to the top row.

hyp3d-20-c-fkk,hyp3d-20-n-fkk,hyp3d-20-fk-fkk,hyp3d-20-lr-fkk,hyp3d-20-dlr-fkk,hyp3d-20-olr-fkk
Figure 15. Spectrum comparison ($N=20$). FK spectrum of (a) clean data, (b) noisy data, (c) FK method, (d) rank-reduction method, (e) damped rank-reduction method, and (f) the proposed method.

hyp3d-20-c-s,hyp3d-20-fk-s,hyp3d-20-lr-s,hyp3d-20-dlr-s,hyp3d-20-olr-s,hyp3d-20-n-s,hyp3d-20-n-fk-s,hyp3d-20-n-lr-s,hyp3d-20-n-dlr-s,hyp3d-20-n-olr-s,hyp3d-20-simi1-s,hyp3d-20-simi2-s,hyp3d-20-simi3-s,hyp3d-20-simi4-s
Figure 16. Denoising comparison ($N=20$). (a) Clean data. (b)-(e): denoised results using FK method, (b) rank-reduction method, (c) damped rank-reduction method, and the proposed method, respectively. (f) Noisy data. (g)-(j) Separated noise corresponding to (b)-(e), respectively. (k)-(n) Local similarity corresponding to (b)-(e), respectively.

snrs_hyp2
Figure 17. SNR diagrams of different lowrank approaches with respect to the selected rank parameters for the hyperbolic synthetic example.