## 4 Numerical Experiment
### Anti-Noise Performance Testing  
To evaluate the anti-noise performance of the MWT-MSSA method, a Gaussian noise with a SNR of 35 dB is added to the synthetic data. The noisy profile, as depicted in Figure 18, causes diffractions only at the vertex. However, the majority of energy around two wings is concealed by noise, making it rather difficult to discern, in addition, the diffraction on the right side is nearly undetectable. Even though the reflection can still be identified, the noise has an effect on its energy, leading to a significant decrease in resolution. Figure 19 shows the low-frequency approximation and high-frequency detailed coefficient profiles obtained by performing a three-level wavelet transform with the bior4.4 wavelet basis function. As can be observed, the distribution of the reflection and diffraction across the coefficient profiles at each level closely mirrors the patterns seen under noise-free conditions. It is noteworthy that after the three-level wavelet transform, the noise is mainly decomposed into the low-frequency approximation coefficients and high-frequency horizontal detailed coefficients at each level. Likewise, as the wavelet transformation level of increases, the reflection, diffraction, and noise are progressively decomposed into the high-frequency detailed coefficients at each level. This observation underscores the effectiveness of MWT in separating diffraction in noisy GPR simulation profiles. Figure 19: Coefficient profiles of the three levels wavelet transform of the GPR data from Figure 18. (**a,e,i**) are the profiles reconstructed from the low-frequency approximation coefficients of levels 1-3. (**b,f,j**) are the profiles reconstructed from the high-frequency horizontal detail coefficients of levels 1-3. (**c,g,k**) are the profiles reconstructed from the high-frequency vertical detail coefficients of levels 1-3. (**d,h,l**) are the profiles reconstructed from the high-frequency diagonal detail coefficients of levels 1-3. Figure 18: Noisy GPR profile by adding Gaussian noise (SNR = 35 dB) to the synthetic data shown in Figure 3b. As shown in Figure 19e, the profile contains a certain amount of noise, while the noise presented in Figure 19c,g is more prominent, making it difficult to identify the diffraction. Since the singular values that correspond to noise are relatively small and typically located in the latter half of the singular spectrum after SVD, the \(k\)-means clustering algorithm is employed to perform cluster analysis on the coefficient profiles. Figure 20 displays the reconstructed diffraction and reflection profiles created by the MSSA, and Table 4 lists the parameters related to cluster analysis. As can be seen, the noise energy is relatively weak in Figure 20a, and the diffraction is clearly visible in Figure 20b,c. The GPR diffraction and reflection profiles, which were reconstructed from the coefficient profiles after diffraction extraction, are presented in Figure 20b,c. It has been observed that the diffraction profiles obtained using the MWT-MSSA show a significant decrease in reflection and noise components. The separation performance is highly encouraging due to the presence of three clearly visible hyperbolic diffraction events. Figure 21e illustrates the diffraction profile obtained by using PWD filtering on the dip field result depicted in Figure 21d. It can be found that the diffraction profile still contains substantial noise, which potentially reduces the clarity of the three hyperbolic diffraction events and manifests a suboptimal separation performance. More importantly, as listed in Table 1, the time it took for PWD was approximately 39 min, while MWT-MSSA completed the process in 12 min, resulting in a 69.2% improvement in computational efficiency. A comparison between the diffraction profiles separated from the noisy data and those derived from the noise-free data revealed a noticeable difference in SNR.