# Noise Attenuation Based on Wave Vector Characteristics
## 3 Numerical Data Test
### Comparison with Instantaneous Polarization Filtering  
We compare our approach with the instantaneous polarization filtering method based on complex trace analysis. Using the Hilbert transform, we get 2C analytical seismic signals \(X(t)\) and \(Z(t)\) as:  
\[\left\{\begin{array}{l}X(t)=x_{r}(t)+ix_{q}(t)\\ Z(t)=z_{r}(t)+iz_{q}(t)\end{array}\right. \tag{19}\]  
where \(x_{r}(t)\) and \(z_{r}(t)\) are the real \(X\)- and \(Z\)-component data, and \(x_{q}(t)\) and \(z_{q}(t)\) are the Hilbert transforms of \(x_{r}(t)\) and \(z_{r}(t)\), respectively. From [PERSON] et al., we obtain the squares of the semi-major and semi-minor axis lengths as, respectively [33],  
\[\left\{\begin{array}{l}a^{2}(t)=\left\{S_{0}(t)+\left[S_{1}^{2}(t)+S_{2}^{2} (t)\right]^{\frac{1}{2}}\right\}/2\\ b^{2}(t)=\left\{S_{0}(t)-\left[S_{1}^{2}(t)+S_{2}^{2}(t)\right]^{\frac{1}{2}} \right\}/2\end{array}\right. \tag{20}\]  
where  
Figure 14: Frequency spectra of the synthetic shot records. (**a**) The raw \(Z\)-component; (**b**) the raw \(R\)-component, (**c**) the final filtered \(Z\)-component; (**d**) the final filtered \(R\)-component; (**e**) noise removed from the \(Z\)-component; and (**f**) noise removed from the \(R\)-component.  
\[\left\{\begin{array}{l}S_{0}(t)=\left|Z(t)\right|^{2}+\left|X(t)\right|^{2}\\ S_{1}(t)=\left|Z(t)\right|^{2}-\left|X(t)\right|^{2}\\ S_{2}(t)=2\left|Z(t)\right||X(t)|\cos\Bigl{(}\arctan\frac{x_{r}(t)z_{q}(t)-z_{ r}(t)x_{q}(t)}{z_{r}(t)x_{r}(t)+z_{q}(t)x_{q}(t)}\Bigr{)}\end{array}\right. \tag{21}\]  
Then, the instantaneous reciprocal ellipticity and tilt angle are given by, respectively,  
\[\left\{\begin{array}{l}e(t)=\frac{b(t)}{a(t)}\\ \theta(t)=\left(\arctan\frac{S_{2}(t)}{S_{1}(t)}\right)/2\end{array}\right. \tag{22}\]  
Following the method proposed by [PERSON] and [PERSON] (2008), a separated valid signal may be defined as:  
\[R(t)=G_{1}(t)G_{2}(t)S(t) \tag{23}\]  
where \(S(t)\) is the input data for \(X(t)\) or \(Z(t)\), and \(G_{1}(t)\) and \(G_{2}(t)\) are the weighting functions based on the ellipticity and tilt angle, respectively, and  
\[G_{1}(t)=\exp\Bigl{(}-\left(1-e(t)\right)^{2}/2\sigma_{e}^{2}\Bigr{)} \tag{24}\]  
\[G_{2}(t)=\exp\Bigl{(}-\Delta\theta(t)^{2}/2\sigma_{\theta}^{2}\Bigr{)} \tag{25}\]  
where \(\Delta\theta\) is the angular difference between the actual tilt angle and desired tilt angle and \(\sigma_{e}\) and \(\sigma_{\theta}\) are the standard deviations of ellipticity and tilt angle, respectively.  
As shown in Figure 15, the instantaneous polarization filtering method can effectively filter out the ground roll. However, the instantaneous polarization filtering method relies on the polarization difference between body waves and noise. Therefore, if the polarization trajectory of a body wave is close to the ground roll, the performance of the instantaneous polarization filtering method declines. For example, the \(PP\)- and \(PS\)-reflections near 1800 ms (Figure 15c,d) are damaged to some extent because they are interfered together and exhibited similar polarization characteristics as the ground roll. Moreover, the non-ground-roll related coherent noise and random noise are not adequately suppressed (Figure 15a,b). The correlation coefficients between the filtered \(Z\)- and \(X\)-components (Figure 13a,b) derived by the proposed method and the noise-free \(Z\)- and \(X\)-components are 0.804 and 0.839, respectively. Nevertheless, those between the filtered \(Z\)- and \(X\)-components (Figure 15a,b) derived with the instantaneous polarization filtering and the noise-free \(Z\)- and \(X\)-components are 0.124 and 0.246, respectively. Compared with Figure 13, it is obvious that our noise attenuation method is more powerful in the discrimination of body waves from a noisy background.