## 4 Numerical Experiment
### Impact of Clustering Parameter Selection on Separation Performance in the k-Means Algorithm
A total of six cases are considered, where the following singular values are set to zero: 5 singular values from the first 4 clusters (Case 11), 6 singular values from the first 5 clusters (Case 12), 7 singular values from the first 6 clusters (Case 13), 9 singular values from the first 7 clusters (Case 14), 12 singular values from the first 8 clusters (Case 15), and 17 singular values from first 9 clusters (Case 16). The reconstruction performance can be evaluated in detail by utilizing the remaining singular values from the other clusters to reconstruct the diffraction, as demonstrated in Figure 15.  
Figure 14: \(k\)-means clustering result of the singular spectrum at the dominant frequency location of the coefficient profile \(d_{LL_{2}}\) (**a**) shows the singular value distribution of the first cluster in the singular spectrum, while (**b**) shows the singular value distribution of the second to thirteenth clusters in the singular spectrum. Figure 13: Diffraction and reflection profiles extracted from Figure 4a with \(k\)-means cluster numbers set to 9, 11, and 13. (**a–c**) represent the diffraction profiles extracted with \(k\)-means cluster numbers of 9, 11, and 13, respectively. (**d–f**) represent the reflection profiles extracted with \(k\)-means cluster numbers of 9, 11, and 13, respectively. It can be seen in Figure 15 that setting a smaller number of clusters (singular values) to zero results in less energy loss at the diffraction vertex. However, this also leads to a stronger residual reflection energy within the diffraction profile, as can be observed in Figure 15a,b. Conversely, by decreasing the number of clusters (singular values) to zero, the residual reflection energy in the reconstructed diffraction profile can be decreased. It is worth noting that the reduction in diffraction vertex energy remains significantly smaller than the reduction in reflection energy, thereby enhancing the overall separation performance. Also, it should be noted that as the number of clusters (singular values) set to zero continues to increase, the residual reflection energy in the reconstructed diffraction profile is reduced to \(\sim 10^{-12}\). Figure 15: Diffraction profiles reconstructed by selecting different numbers of clusters (singular values) in 6 Cases when the total number of clusters is 13. (**a**) Case 11, (**b**) Case 12, (**c**) Case 13, (**d**) Case 14, (**e**) Case 15, (**f**) Case 16. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cases** & **Total Clusters** & **Zeroed Clusters** & **Reconstructed Clusters** & **Zeroed Singular** & **Reconstructed Singular Values** \\ \hline Case 1 & & 1–3 & 4–9 & 1–5 & 6–78 \\ Case 2 & 9 & 1–4 & 5–9 & 1–6 & 7–78 \\
**Case 3** & & **1–5** & **6–9** & **1–9** & **10–78** \\ Case 4 & & 1–6 & 7–9 & 1–12 & 13–78 \\ \hline Case 5 & & 1–3 & 4–11 & 1–5 & 6–78 \\ Case 6 & & 1–4 & 5–11 & 1–6 & 7–78 \\ Case 7 & 11 & 1–5 & 6–11 & 1–7 & 8–78 \\
**Case 8** & & **1–6** & **7–11** & **1–9** & **10–78** \\ Case 9 & & 1–7 & 8–11 & 1–12 & 13–78 \\ Case 10 & & 1–8 & 9–11 & 1–17 & 18–78 \\ \hline Case 11 & & 1–4 & 5–13 & 1–5 & 6–78 \\ Case12 & & 1–5 & 6–13 & 1–6 & 7–78 \\ Case 13 & & 1–6 & 7–13 & 1–7 & 8–78 \\
**Case 14** & 13 & **1–7** & **8–13** & **1–9** & **10–78** \\ Case 15 & & 1–8 & 9–13 & 1–12 & 13–78 \\ Case 16 & & 1–9 & 10–13 & 1–17 & 18–78 \\ \hline Case 17 & & 1–5 & 6–15 & 1–5 & 6–78 \\ Case 18 & & 1–6 & 7–15 & 1–6 & 7–78 \\ Case 19 & & 1–7 & 8–15 & 1–7 profile is eliminated. This, however, comes with an increased energy loss in the diffraction itself. This finding can also be substantiated in the A-scan waveform comparison of the diffraction vertex and two wings, as shown in Figure 16. The optimal balance between separation results can be maintained by Case 14, which preserves the highest diffraction energy at the vertex and maintains superior energy retention in the wings indicated by the arrow. This selection suggests a model that utilizes the most efficient performance to separate GPR diffraction.