## 3 Synthetic Examples  
### Synthetic Wavelet Example  
To test the noise attenuation performance of the presented method, a synthetic example based on Ricker wavelet is introduced to simulate a seismic wave:  
\[f(t)=(1-2\pi^{2}f_{p}^{2}t^{2})\text{e}^{-\pi^{2}f_{p}^{2}t^{2}} \tag{13}\]  
where _fp_ is the dominant frequency. In this example, the wavelet with a dominant frequency of 25 Hz, has a sampling frequency of 1000 Hz. Its waveform is contaminated by adding white gaussian noise. Figure 3 shows its waveform and spectrum before and after adding noise. It is apparent that the whole frequency domain has been polluted by random noise, so the low-pass filter is not a useful approach. For comparing the performance difference between the proposed method with other existing popular algorithms, EEMD, CEEMD, and VMD are also used to decompose the synthetic signal and to restore it into a clean signal. Although the mode number \(K\) of VMD needs to be preset, the reconstructed results under different \(K\) is approximately the same [40]. To control the variables, the number of VMD sub-signals is set to 10 just as EEMD. The amplitude coefficient and the ensemble size of EEMD and CEEMD are set to 0.2 and 100, respectively. Figures 4 and 5 illustrate the difference between ten component waveforms obtained by EEMD and VMD, respectively. It is shown from Figure 4 that the first two components of EEMD have obvious mode aliasing. The frequency of the first component is distributed nearly throughout the whole spectrum and the fluctuations with distinct frequency both exist in the second component, which causes two frequency peaks to be present in one spectrum. It is a sign of insufficient decomposition that waveforms in different frequency intervals are not clearly separated, which has a negative effect on noise reduction. However, the decomposition result of VMD has a rather different side. Their bandwidth is severely limited to a certain extent, unlike that of EEMD whose frequency bandwidth is either particularly large or extremely small. The distributions of different frequency bands in the spectrum show a certain regularity, which their dominant frequencies are relatively uniform in the whole interval. Taking the position of the component of mode mixing as the demarcation point, there are eight dominant frequencies of EEMD compactly concentrated below 100 Hz. It means that the high-frequency part of the same signal is decomposed more fully by VMD, which is favorable to suppress noise. Figure 6 displays the calculated \(CC\) and \(\triangle CC\) for the ten decomposed components by VMD. The \(CC\) values of the first eight components are less than 0.1, and the latter two are more than 0.7. Their \(\triangle CC\) values as a whole show a trend of rising first and then falling, reaching its maximum at the ninth mode. The change of the coefficient is consistent with the VMD decomposition result in Figure 5. It is clear that the last two modes are used to reconstruct the valuable signal. Figure 4: Waveforms and spectrograms of ten components obtained by ensemble empirical mode decomposition (EEMD). (**a**) Waveforms of components; (**b**) spectrograms of components. Figure 3: Waveforms and spectrums of a 25 Hz Ricker wavelet before and after adding white gaussian noise. (**a**) Noise-free wavelet waveform; (**b**) noise-free wavelet spectrum; (**c**) noisy wavelet waveform; (**d**) noisy wavelet spectrum. Figure 7 depicts the waveforms and spectrums of denoised signals using different methods. It can be seen from the figure that the noise whose frequency is above 300 Hz is almost filtered out by EEMD or CEEMD, but there is apparent noise residue in the frequency range between 100 Hz and 300 Hz. Not only does the proposed VMD-AIC method eliminate the noise above 100 Hz, but also it weakens the excess oscillation below 100 Hz by removing the non-signal data points.