## 3 Model Validation and Engineering Examples
### Verification of the SSA-VMD Method  
The subsidence data of YG1-17 on the monitoring cross-section were obtained. The number of SSA populations was set to 10, the number of iterations was set to 30, the search range of \(K\) was set to [2; 10], the search range of \(\alpha\) was [100, 3000], and the parameter combination [\(K\), \(\alpha\)] of the VMD after SSA seeking was [6; 500], as shown in Table 2. The corresponding runtime software is Python 3.11.4, and the runtime libraries include NumPy, SciPy, Matplotlib, and the hardware is Intel i9 13900K and Nvdia RTX49090.  
To explore the performance of the proposed adaptive VMD-improved method, the center frequency method was used to verify the accuracy of \(K\). The theoretical basis of the center frequency method is that the modes differ in terms mainly in terms of the center frequency; hence, appropriate values of the modes can be selected by observing the distribution of the center frequency under different numbers of modes. When the center frequencies of the adjacent modes are close to each other, the sign  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline K & \multicolumn{5}{c}{**Center Frequency (Hz)**} \\ \hline
2 & 24 & 2876 & – & – & – & – & – \\
3 & 20 & 241 & 2877 & – & – & – & – \\
4 & 20 & 241 & 2874 & 3749 & – & – & – \\
5 & 20 & 241 & 1847 & 2878 & 4330 & – & – \\
6 & 20 & 241 & 1864 & 2877 & 3604 & 4395 & – \\
7 & 20 & 241 & 1123 & 2219 & 2880 & 3607 & 4341 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Center frequencies corresponding to different values of \(K\).  
overly decomposed. In this case, the best decomposition layer is the \(K\) value of the previous term when the center frequencies are close to each other. First, by predecomposing the time series of the settlement at YG1-17, the center frequencies of each component under different \(K\) values are shown in Table 2.  
When \(K=6\), there is a gap between the center frequencies of the components. When the \(K\) value is seven, the center frequencies of _IMF_4 and _IMF_5 are close to each other, which indicates frequency mixing between the signaling components; thus, the VMD of the signal is better for \(K=6\).  
Therefore, the most suitable modal number for the VMD of the YG1-17 settling time signal is six. In the VMD process, the number of _IMF_ components is determined by the preset \(K\) value, whereas the bandwidths of these components are controlled by the penalty coefficient. The smaller the penalty factor is, the larger the bandwidth of each IMF component; an excessively large bandwidth will cause some components to overlap with the other components of the signal. Conversely, the larger the value of \(\alpha\) is, the smaller the bandwidth of each _IMF_ component. An excessively small bandwidth results in signal loss during decomposition. The commonly used empirical value of 1.5 times the frequency of the signal was applied in this study. To more clearly demonstrate the effect of \(\alpha\) on the signal decomposition, a normalization process was conducted for each component signal. Taking \(\alpha\) as 1.5 and 500, the signal spectra of the corresponding \(\alpha\) values were obtained, as shown in Figure 9.  
Figure 9: Modal signal spectrum distributions for \(\alpha=1.5\) and 500.