## 3 Synthetic Examples
### Synthetic MS Examples with Different SNRs  
In order to further compare the denoising ability of VMD-AIC under different intensity of noise interface, a clean MS signal is added with Gaussian white noise of different degrees to form 11 synthetic  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**SNR/dB**} \\ \cline{2-3}  & **Before** & **After** \\ \hline VMD-AIC & 2.49 & 23.47 \\ VMD & 2.49 & 11.90 \\ EEMD & 2.49 & 6.73 \\ CEEMD & 2.49 & 6.92 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The \(SNR\) before and after denoising the wavelet by different methods.  
Figure 7: Noise reduction results of synthetic signals by different methods. **(a)** EEMD; **(b)** complete EEMD (CEEMD); **(c)** VMD and VMD-Akaike information criterion (AIC). The letter S means the start position of the MS signal segment, and the letter E indicates its end position. The portion in the solid box is an enlarged view of the portion in the dotted box.  
signal records with different \(SNR\)s. These synthetic MS signals are closer to the noisy signal of the construction site compared with a completely synthetic signal, and their denoised results are convenient for evaluation. Table 2 shows their noise reduction effect of VMD-AIC, VMD, EEMD, and CEEMD. It can be seen from the table that no matter what the initial noise level is, the \(SNR\) after VMD-AIC noise suppression can be increased by more than 10. These increases are generally larger than those of VMD by 2, and much larger than those of EEMD and CEEMD. When the \(SNR\) is equal to \(-4\) or even lower, the signal is completely overwhelmed by noise. The algorithm can also suppress the noise very well and extract the effective waveform, while the noise of denoised signals of EEMD and CEEMD still occupy more than half of the effective signal (\(10 log2\approx 3\)). The waveform and spectrum of the initial noiseless MS signal are shown in Figure 8a. Figure 8b-g display the noisy signals with different \(SNR\)s and their results after noise attenuation by VMD-AIC.  
It can be clearly seen from their spectrums that the surplus frequency components above 500 Hz are basically removed by the proposed method. When the tail wave of the MS signal is completely submerged by noise in Figure 8b-d, there is some error in the pickup of the end position of the signal. Although the position is somewhat lagging, removing the noise in the latter section also improves the signal quality. Moreover, when the \(SNR\) of the signal is not too low, the location of the post-noise  
Figure 8: Waveforms and spectrograms of noisy signals with different \(SNR\)s before and after noise attenuation by VMD-AIC. (**a**) The original noiseless MS signal; (**b**) noisy signal with \(SNR=-8\); (**c**) noisy signal with \(SNR=-4\); (**d**) noisy signal with \(SNR=0\); (**e**) noisy signal with \(SNR=4\); (**f**) noisy signal with \(SNR=8\); (**g**) noisy signal with \(SNR=10\). The red line in the figure represents the signal after denoising by VMD-AIC, and the blue line represents the waveform before noise reduction.  
section is still relatively accurate. The position of the first arrival of MS is well determined, regardless of whether the initial \(SNR\) is high or low. The above results all indicate that the VMD-AIC algorithm has the ability to adapt to signals with different \(SNR\)s, and can further improve signal quality on the basis of VMD.