## 2 Principle and Construction of the Settlement Prediction Model  
### Variational Mode Decomposition  
Considering that VMD can effectively prevent the occurrence of mode mixing phenomena, it has great potential in the field of surface settlement prediction. The decomposition effect of VMD depends on the parameter settings, especially the settings of two key parameters, namely the number of decomposition layers (\(K\)) and the penalty factor (\(a\)). The setting of this set of parameters is not easy, and bad decomposition results can easily be achieved through empirical methods. Some studies have achieved better results by combining optimization algorithms to determine these two parameters. In other words, VMD is a fully nonrecursive decomposition method with significant advantages in dealing with nonlinear data and noise [32]. It essentially constructs and solves bandwidth-constrained amplitude modulation functions with five steps and six required parameters, namely \(a\), tau, \(K\), init, \(DC\), and tol, among which the parameters with the greatest influence on the decomposition are \(K\) and \(a\).  
Assuming that the decomposed modal function is \(u_{k}(t)\) the unilateral spectrum of the analyzed signal derived via the Hilbert transform is [33]  
\[(\delta(t)+\frac{j}{\pi t})*u_{k}(t) \tag{1}\]where \(\delta(t)\) is the unit impulse function (Dirac function), \(\delta(t)\) is infinite when \(t\) is equal to 0, \(\delta(t)\) is 0 when \(t\) is not equal to 0, and \(j\) is an imaginary unit, * is the convolution operation.  
Then, the spectrum is mixed with the exponential term \(e^{-j\omega_{k}t}\) of the central frequency \(\omega_{k}\) so that it is leveled with the baseband, and the expression of the analytic signal is obtained as  
\[[(\delta(t)+\frac{j}{\pi t})*u_{k}(t)]e^{-j\omega_{k}t} \tag{2}\]  
The sum of the modal components should be equal to that of the original sequence to ensure that when the original sequence is decomposed into \(K\) modal components, the sum of the estimated bandwidths of each modal component with central frequencies is minimized. Thus, the constrained variational expression is  
\[\left\{\begin{aligned} \underset{\{u_{k}\},\{u_{k}\}}{\min} \left\{\begin{aligned} \sum\limits_{k=1}^{K}\left\|\partial_{t} \left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j\omega_{k}t} \right\|_{2}^{2}\right\}\\ s.t.\sum\limits_{k=1}^{K}u_{k}=f(t)\end{aligned}\right. \tag{3}\]  
where \(u_{k}\) is the decomposed \(K\) modal function, \(\omega_{k}\) is the central frequency corresponding to each of the \(K\) modal functions, \(\partial_{t}\) is the partial derivative of the solution with respect to \(t\), \(\mid\mid\mid\) is the parametric number, and \(f\) (\(t\)) is the original time series signal.  
The above equation is solved by transforming the constrained variational problem into an unconstrained variational problem by penalizing the factor \(\mathfrak{a}\) with the Lagrange multiplier operator \(\lambda\). The corresponding equation is  
\[\begin{aligned} L(\{u_{k}\},\{\omega_{k}\},\lambda)=& \  
where the upper right corner labels (\(n\)) and (\(n+1\)) indicate the number of iterations, the superscript \({}^{*}\) denotes the Fourier transform, the lower left corner \(k\) denotes the \(k\)th decomposition mode, \(\varepsilon\) is the allowable error, \(\tau\) is the update parameter, and \(\varepsilon\) and \(\tau\) are constants. After satisfying the condition of Equation (7), the \(K\)_IMFs_, \(u_{k}\), are output. Because \(\varepsilon\) is fixed, the residual is approximated as a frequency function of the amplitude \(\varepsilon\), and the original time series can be obtained by accumulating \(u_{k}\) and the residual.  
The corresponding SSA process is shown in Figure 1, where the various steps of the whole optimization process are shown in detail.