## 3 Simulation Experiment  
### EMD_CS_ST Algorithm Application  
Using the digital simulation analysis in Section 2.4.1, the EMD adaptive decomposition of the noisy Ricker wavelet is performed to obtain 7 IMF components. The waveforms of each component and its spectrum are shown in Figure 5.  
The calculated cross-correlation coefficient is shown in Figure 6 with bar and dot charts. According to the demarcation point discrimination method of noise and signal, it can be known that \(\textit{imf}_{6}\) is a boundary. Then, the first six IMF components are respectively subjected to sparse transformation to obtain coefficients to be reconstructed \(\theta\) = {\(\theta_{1}\), \(\theta_{2}\),..., \(\theta_{6}\)}. The threshold \(T\) is calculated according to the algorithm of Section 2.3, and the calculation result is shown in Figure 7, and soft-thresholding filtering is performed on the reconstruction coefficient \(\theta\). Finally, the denoised IMF\({}_{new}\) component is reconstructed by discrete wavelet inverse transform, and the remaining IMF component is combined with IMF\({}_{new}\) to obtain the denoised MS signal. Figure 4: The EMD_CS_ST algorithm flow chart. First, Empirical Mode Decomposition (EMD) is performed on the MS signal. Then, CS_ST is used to denoise the Intrinsic Mode Functions (IMF) components to be denoised. Finally, the denoised components and the remaining IMF components are combined to obtain the final effective signal. Figure 5: EMD decomposed the results of the noise Ricker waveform and the IMF’s corresponding spectrum. Figure 6: Correlation coefficients of signal and IMF. To further verify the effectiveness and feasibility of the proposed method, this paper compares with traditional discrete wavelet transform (DWT) denoising, EEMD denoising, and VMD_DWT [37] denoising methods. The simulation experiments of these denoising methods are carried out on the MATLAB platform. For the DWT denoising method of MS signals, the perform function is _wdenoise_, _sym_ is used as the wavelet base, the number of filters is 8, and the number of decomposition layers is 5. The threshold rule is soft-threshold; estimate noise using only finest-scale wavelet coefficients. According to the signal-to-noise ratio (\(SNR\)), the signal standard deviation (\(SD\)), the correlation coefficient (\(CC\)) of the pure signal, and the denoising signal, these three indicators quantitatively illustrate the denoising effect of this paper. The waveforms and their spectra after denoising by these four methods are shown in Figure 8.  
The \(SNR\) is defined as:  
\[SNR=10\log\left\{\frac{\sum\limits_{t=0}^{N-1}x(t)^{2}}{\sum\limits_{t=0}^{N-1 }\left[x(t)-x(t)_{new}\right]^{2}}\right\} \tag{11}\]  
where \(x(t)\) is a pure signal, \(x(t)_{new}\) is a denoised signal, and \(N\) is the number of sampling points. If the signal has a high \(SNR\) after denoising, the denoising effect is better. Figure 8: Four methods for denoised of the Ricker waveform and their spectra. Figure 7: The T value of the coefficient to be reconstructed of the first 6 components IMF. The \(SD\) is defined as:  
\[SD=\sqrt{\sum_{t=0}^{N-1}[x(t)-x(t)_{new}]^{2}/N} \tag{12}\]  
If the \(SD\) is smaller, it means that the smaller the fluctuation between the denoising signal and the pure signal, the closer to the pure signal. The \(CC\) between the pure signal and the denoising signal is defined as:  
\[CC=\frac{\text{Cov}[x(t),x(t)_{new}]}{\sqrt{\text{Var}[x(t)]\cdot\text{Var}[x(t )_{new}]}} \tag{13}\]  
where Cov is the covariance function, Var is the variance function. If the \(CC\) value is closer to 1, the denoising signal is similar to the clean signal. Apply the Equations (11)-(13) to calculate the \(SNR\) before and after denoising of the noisy Picker wavelet, and the \(SD\) and \(CC\) after denoising. These results are shown in Figure 4:0  
It can be seen from Table 1 and Figure 8 that the EMD_CS_ST has a better denoising effect than the other three methods. Analysis from Table 1 shows that the \(SNR\) of the noisy Ricker wavelet is 4.651 dB.