## 4 Numerical Experiment
### Impact of Clustering Parameter Selection on Separation Performance in the k-Means Algorithm  
As stated, the proposed MWT-MSSA leverages the distinct amplitude differences between direction, reflection, diffraction and noise in the GPR profile to achieve effective diffraction separation. In contrast to seismic wavefields, the GPR wavefield predominantly consists of these four components: the direct wave and reflection have the highest energy levels, diffraction possess moderate energy levels, and noise features the weakest energy [40]. According to the SVD principle, direct wave and reflection are exemplified by the largest singular values, diffraction is represented by moderate singular values, and noise is characterized by the smallest singular values [25; 27; 40]. To accurately determine the thresholds between different components in the singular spectrum, we employ a partitioning-based \(k\)-means clustering algorithm to classify the singular values in the singular spectrum into \(k\) clusters based on their similarity attribute. The diffraction is then reconstructed by selecting the singular values associated with the diffraction cluster. Typically, there is no definitive criterion for determining the number of clusters \(k\). Recent studies in seismic exploration have demonstrated that while \(k\) can be guided by the complexity of the wavefield. The more complex wavefields typically require a larger number of clusters to accurately represent their distinct components [27; 40]. Take the GPR wavefield shown in Figure 3b as an example. This wavefield consists of a direct wave, two reflections that originate from undulating interfaces, three diffractions that are caused by small-scale anomalous bodies, and ambient noise. In this regard, the number of clusters \(k\) may need to exceed seven to warrant an accurate representation of these distinct components. Also, the complexity of the wavefield necessitates an appropriate increase in \(k\), as discussed in [39]. To evaluate the impact of the total number of \(k\)-means clusters on separation efficiency, we employ the \(k\)-means clustering algorithm to analyze the GPR data depicted in Figure 3b. The clustering analysis is performed with the number of \(k\)-means clusters set to 9, 11, and 13, respectively, while the first six clusters are fixed at zero and all other parameters remain unchanged. As the results shown in Figure 13, the effectiveness of diffraction separation is significantly affected by the number of k-means clusters chosen. Specifically, when the  
Figure 12: Comparison of the single trace waveform of the diffraction wave’s vertex (**a**) and wings (**b**) extracted from Figure 11a, b, and c, respectively. cluster count is set to 13, the separated diffraction profile exhibits optimal results. Conversely, a decrease in cluster count to 9 leads to a notable decline in separation performance, highlighting the sensitivity of the process to the selected number of clusters. After performing MWT on the synthetic data (shown in Figure 3b), residual diffraction can be observed in the low-frequency approximation coefficient profile shown in Figure 4e. The singular spectrum of the coefficient profile at the dominant frequency in the profile illustrated in Figure 4e was analyzed using \(k\)-means clustering with a total number of 13 clusters. As shown in Figure 14, the singular spectrum at the dominant frequency is divided into 13 distinct clusters, each with a gray region. The distribution of singular values across the clusters is non-uniform, with distinct amounts of singular values in each cluster. By examining a total of 13 clusters listed in Table 3, we investigated the impact of selecting different numbers of clusters (singular values) representing diffraction on the separation of GPR diffraction. The reconstruction of the GPR diffraction is achieved by utilizing various numbers of singular values from the clusters.