## 4 Numerical Experiment
### Impact of Wavelet Basis Functions on Separation Performance  
To evaluate the influence of different wavelet basis functions on the performance of MWT-MSSA, three typical wavelet basis functions (i.e., haar, db4, and rbio4.4) were performed by a third level wavelet transformation on the synthetic data. The selection of an appropriate wavelet basis function is a critical step for the given problem, as it can affect the results to some extent. Specifying this effect necessitates a careful consideration of the specific problem and an evaluation of key wavelet properties such as vanishing moments, support width, regularity, symmetry, and other relevant characteristics. A comprehensive, multi-faceted assessment of these properties is essential to ensure optimal performance in wavelet-based analysis. As shown in Table 2, the bior4.4 wavelet has a large number of vanishing moments that enable it to capture local features of the signal effectively, while its extended support length enhances the accuracy of signal representation. Also, its favorable regularity ensures a smooth reconstruction, and its symmetry minimizes the chance of reconstruction errors [42]. Based on these properties, we prefer the bior4.4 wavelet in our study to separate GPR diffraction waves for subsequent analyses. The MWT coefficient profiles are presented in Figures 8-10. In comparison to the third level bior4.4 wavelet transform coefficient profiles shown in Figure 4, the high-frequency vertical detail coefficient profiles \(d_{HL1}\) and \(d_{HL2}\) obtained by using haar wavelet basis function exhibit significant linear reflective components at the positions marked by the black arrows (see Figure 8c,g), which could be detrimental for diffraction separation. The high-frequency vertical detailed coefficient profile \(d_{HL2}\) obtained by using rbio4.4 wavelet basis function still retains some portion of weak reflection (see Figure 9g). In contrast, the high-frequency vertical detailed coefficient profile \(d_{HL2}\) from the db4 wavelet basis function exhibits minimal reflection energy. However, compared to the high-frequency vertical detail coefficient profile \(d_{HL2}\) from the bior4.4 basis function, a slight loss of energy can be observed at the diffraction vertex, as indicated by black arrow in Figure 10g. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Calculated Time (min)** & **Synthetic Data** &
\begin{tabular}{c} **Synthetic Data** \\ **with Noise** \\ \end{tabular} & **In-Site Field Data** \\ \hline MWT-MSSA & 8 & 12 & 16 \\ \hline PWD & 38 & 39 & 45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computation time statistics for MWT-MSSA and PWD methods. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Properties** & **Haar Wavelet** & \begin{tabular}{c} **Daubechies** \\ **Wavelet (dbN)** \\ \end{tabular} & \begin{tabular}{c} **Reverse** \\ **Biorthogonal** \\ **Wavelet (rbioN\({}_{d}\),N\({}_{r}\))** \\ \end{tabular} & \begin{tabular}{c} **Biorthogonal** \\ **Wavelet (biorN\({}_{r}\)-N\({}_{d}\))** \\ \end{tabular} \\ \hline Orthogonality & Yes & Yes & No & No \\ \hline Biorthogonality & Yes & Yes & Yes \\ \hline Support Width & 1 & \(2N-1\) & \begin{tabular}{c} \(2N_{d}+1\) \\ (Reconstruction) \\ \end{tabular} & \begin{tabular}{c} \(2N_{r}+1\) \\ (Reconstruction) \\ \end{tabular} \\  & 1 & \
\begin{tabular}{c} \(2N_{r}+1\) \\ (Decomposition) \\ \end{tabular} &
\begin{tabular}{c} \(2N_{d}+1\) \\ (Decomposition) \\ \end{tabular} \\ \hline Regularity & Discontinuous & Approximately \(0.2N\) & \(N_{d}-1,N_{d}-2\) & \(N_{r}-1,N_{r}-2\) \\ \hline Symmetry & Yes & No & Yes & Yes \\ \hline Vanishing Moments & 1 & \(N\) & \(N_{d}\) & \(N_{r}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of Properties of Common Wavelet Basis Functions. Figure 9: Third level wavelet transform coefficient profiles of the GPR data shown in Figure 3b using the rbio4.4 wavelet. (**a**,**e**,**i**) represent the reconstructed profiles of the first to third level low-frequency approximation coefficients.