## 3 Results
### Synthetic Data  
Figure 10 shows application of local SVD and SOSVD on the synthetic dataset from [17], while Figure 11 shows the application of SMD to a very similar data set. To create the synthetic data in Figure 11 we recreated the signal observed in Figure 10 and added random noise to it.  
Figure 9: Flowchart describing the process of applying SMD to seismic data. The fourth step from above is described by the flowchart presented in Figure 5.  
In Figure 10 we see that both local SVD and SOSVD are able to remove most of the noise. However, both local SVD and SOSVD struggle to properly reconstruct the signal in the area highlighted by the blue rectangle in Figure 10c,b. The highlighted area contains two waves of different slopes intersecting with each other.  
Figure 11 shows results of applying SMD to the synthetic data set twice, with 95% compression (Figure 11c) and 80% compression (Figure 11d). In both SMD applications the coherent waves were reconstructed perfectly. We can see that there are no numerical artifacts in the area highlighted by the blue rectangle in Figure 11c,b, which contains intersecting waves with different slopes.  
While we can see significant noise reduction, we could not find specific information in [17] describing the amount of noise present in synthetic data in Figure 10, However, we can estimate the signal-to-noise ratio in the synthetic data on which SMD is tested. By measuring the root mean square value from all receivers in time increment range from 300 to 320, where only noise is present, we can estimate the amplitude of the noise. To estimate the amplitude of the signal, we measure the root mean square value from all receivers in time increment range from 340 to 360, from the data presented in Figure 11a,  
Figure 10: (**a**) Synthetic data showing several wave arrivals without noise. (**b**) The noisy data, created by adding noise to the synthetic data. (**c**) The noisy data filtered using local SVD. (**d**) The noisy data filtered with SOSVD. The blue rectangles highlight areas of interest in which local SVD and SOSVD produce numerical artifacts. This figure was modified from [17].  
which contains only pure signal. The signal-to-noise (\(\rho\)) ratio is therefore calculated with the Equation (10):  
\[\rho=\frac{\sqrt{\sum_{i=340}^{i=360}\sum_{j}{M_{i,j}^{p}}^{2}}}{\sqrt{\sum_{i=3 00}^{i=320}\sum_{j}{M_{i,j}^{n}}^{2}}} \tag{10}\]  
where **M**P represents the data containing only pure signal presented in Figure 11a, and **M**n represents the data from Figure 11b,c, or d, which contains some amount of noise.  
The signal-to-noise ratios in the original data, data reconstructed from SMD results after 80% compression, and after 95% compression were 1.9, 4.7, and 12.3, respectively. Notice that the signal-to-noise ratio is much larger when SMD is applied with 95% compression than when it is applied with 80% compression. This is because there is only a few coherent waves present in the data. These coherent waves are represented with first several singular vectors and shift vectors, while the following singular vectors and shift vectors are describing noise. However, SMD results represent noise a lot less efficiently than coherent waves, and the majority of the noise is still not recorded when SMD is applied with 80% compression. In conclusion, the SMD application with 80% reduction in memory requirements picked up some of the noise, while SMD application with 95% data compression ignored the noise almost completely.  
Figure 11: (**a**) Synthetic data showing several wave arrivals without noise. (**b**) The noisy data, created by adding noise to the synthetic data. (**c**) Data obtained by applying SMD to the noisy data with 95% compression. (**d**) Data obtained by applying SMD to the noisy data with 80% compression. The blue rectangles highlight areas of interest in which local SVD and SOSVD produce numerical artifacts.