## 4 Numerical Experiment
### Effectiveness Verification of MWT-MSSA Method
However, reflective energy is also presented in these coefficients, as indicated by the arrows in Figure 4k. This progression illustrates that, with increasing wavelet transform levels, wavefield energy can be incrementally transferred to higher-level of high-frequency detailed coefficients. Notably, diffraction energy tends to shift into the high-frequency detailed coefficients at earlier stages compared to reflection energy, which highlights the distinct decomposition characteristics of this work. Figure 3: (**a**) Model of relative dielectric permittivity and electric conductivity distribution and (**b**) its simulated GPR data. After the second level wavelet transform, a portion of the energy at the vertex of the diffraction remains within the second level of low-frequency approximation coefficient. In this sense, reconstructing the diffraction wavefield using the high-frequency vertical detailed coefficients \(d_{HL1}\) and \(d_{HL2}\) from the first and second levels could result in a partial energy loss at the vertex of the diffraction. Conversely, incorporating the high-frequency vertical detailed coefficient from the third level can reduce the vertex energy loss but introduce significant reflection energy into the reconstruction, ultimately compromising the separation accuracy. To avoid the energy loss at the vertex of diffraction for wavelet transform-based separation methods, the MSSA method is used coupled with \(k\)-means clustering. With this coupling strategy, the diffraction energy can effectively be separated from the original second level of low-frequency approximation coefficient \(d_{LL2}\), thereby elevating the precision of wave separation. Figure 5a,b present the diffraction and reflection profiles extracted from the second level of low-frequency approximation coefficient \(d_{LL2}\) (in Figure 4e) using the MSSA method coupled with \(k\)-means clustering. The results indicate that the majority of the diffraction vertex energy has been successfully separated, with minimal energy loss, and the reflection is practically invisible. The profile in Figure 5b shows a greater amount of complete reflection energy and a significant decrease in diffraction energy, compared to the low-frequency approximation coefficient profile derived from the third level wavelet transform. Figure 4: Three-level wavelet transform coefficient profiles of the GPR data present in Figure 3b. (**a,e,i**) are the reconstructed profiles of the low-frequency approximation coefficients for the first, second, and third levels, respectively. (**b,f,j**) are the reconstructed profiles of the high-frequency horizontal detail coefficients for the first, second, and third levels, respectively. (**c,g,k**) are the reconstructed profiles of the high-frequency vertical detail coefficients for the first, second, and third levels, respectively. (**d,h,l**) are the reconstructed profiles of the high-frequency diagonal detail coefficients for the first, second, and third levels, respectively. Figure 6b showcases the reconstructed diffraction profile that was derived from the high-frequency vertical detailed coefficient profiles \(d_{HL1}\) and \(d_{HL2}\) (using the first and second level wavelet transforms) and the diffraction profile (in Figure 5a), with the reconstructed reflection profile illustrated in Figure 6c. It can be observed that the diffraction profile eliminates a significant amount of reflection energy, with only minimal energy loss at the top of the diffraction, indicating encouraging separation performance. To further confirm the outperformance of the proposed method, we compare the reconstruction profile with the PWD method, as shown in Figure 6e,f. The comparison result suggests that the diffraction profile reconstructed by PWD using the dip field (Figure 6d) loses more energy at the vertex of the diffraction, while simultaneously maintaining a portion of reflective energy. As demonstrated, the reflection profile contains a greater quantity of residual diffraction energy throughout the pair-tail of the hyperbolic events.