# Noise Attenuation Based on Wave Vector Characteristics
## 3 Numerical Data Test
### Two-dimensional Two-Component Synthetic Data  
We apply our method to the synthetic shot records from the theoretical model shown in Table 1. The near-surface ground roll is numerically generated using the methods introduced by [PERSON] and [PERSON] and _PP_- and _PS_-reflections from the deep interfaces are obtained by ray tracing [30; 31; 32]. The simulated ground roll is dispersive with a frequency band of 1-12 Hz (Figure 9). The frequency band of the added non-ground-roll related coherent noise is 10-20 Hz. The dominant frequencies of _PP_- and _PS_ valid waves are 30 Hz and 15 Hz, respectively. Compared with the valid waves in the synthetic shot records (Figure 10a,b), the added ground roll is of 350% level (Figure 10c,d), the added non-ground-roll related coherent noise is of 50% level (Figure 10e,f), and the added random noise is of 400% level. The sample rate of the synthetic shot records is 1 ms and the maximum apparent period of the valid wave is 60 ms.  
Figure 8: Hodogram plots of the ground-roll-attenuated signal. (**a**) Ground-roll-attenuated signal; (**b**) the unscaled valid signal after random noise attenuation; (**c**) the scaled valid signal after random noise attenuation and (**d**) the removed random noise. Figure 7: Results of random noise filtering. (**a**) Valid seismic signals after random noise attenuation and (**b**) the removed random noise. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Density (g/cm\({}^{3}\))** & **Thickness (m)** & **P-Wave Velocity (m/s)** & **S-Wave Velocity (m/s)** \\ \hline
1.47 & 5 & 500 & 200 \\
1.53 & 5 & 600 & 300 \\
1.74 & 30 & 1000 & 500 \\
2.19 & 760 & 2500 & 1443 \\
2.26 & 800 & 2800 & 1617 \\
2.33 & 800 & 3200 & 1848 \\
2.4 & 600 & 3600 & 2078 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Theoretical model parameters. Figure 10: _Cont._  
Figure 9: Dispersion curve of the simulated ground roll. First, we determine the moving window size for the attenuation of ground roll according to the map of wave vector modulus (Figure 11). As shown in Figure 11a, a 50-ms moving window corresponding to the apparent period of the _PP_-wavelet can well suppress _PP_-wave vectors but can hardly differentiate between the _PS_-waves and ground-roll vectors. A 60-ms moving window (Figure 11b) is a little smaller than the apparent period of the _PS_-wavelet, so parts of the _PS_-vectors were left. The 70-ms moving window (Figure 11c) is exactly equal to the apparent period of the _PS_-wavelet, and perfectly highlighted the wave vectors of the ground roll and suppressed both the _PP_- and _PS_-vectors. As shown in Figure 11d, too-large time windows (e.g., 80 ms) caused high-frequency ground roll to be less recognizable, although valid wave vectors are well suppressed. After testing, we choose the 70-ms moving window for the separation of the ground roll. Figure 10: Synthetic shot records. (**a**) \(Z\)-component with noise; (**b**) \(X\)-component with noise; (**c**) pure ground roll added to the \(Z\)-component; (**d**) pure ground roll added to the \(X\)-component; (**e**) non-ground-roll related coherent noise added to the \(Z\)-component; (**f**) and non-ground-roll related coherent noise added to the \(X\)-component. Second, we test different moving time and moving trace windows based on the ground-roll-attenuated data. As the cutoff frequency of a wavelet with 30 Hz dominant frequency is 75 Hz, the proper moving window size to suppress random noise is 7 ms. As shown in Figure 12, compared with other moving windows, a 5-trace and 7-ms moving window can better suppress both the random noise and the non-ground-roll related coherent noise. Larger moving windows (e.g., 9-trace and 9 ms) damage the valid wave to some extent (Figure 12d). Figure 11: Wave vector moduli calculated from the raw synthetic shot records. Moving time widow of (**a**) 50 ms, (**b**) 60 ms, (**c**) 70 ms, and (**d**) 80 ms.  
As displayed in Figure 13, the both valid _PP_- and _PS_-waves are effectively separated from their noisy backgrounds.