## 3 Methods
### Time-Series Decomposition  
Singular spectrum analysis (SSA) is a signal processing technique based on singular value decomposition for analyzing time-series data, which can decompose time-series data into trend, period, and residual terms. Firstly, for the one-dimensional time series, \(x_{t}\) (\(1\leq t\)\(\leq\) N) is embedded into the delayed sequence through a sliding window to form a trajectory matrix, where L is the window length and K = N - L + 1.  
\[X=(x_{ij})_{i,j=1}^{L,K}=\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots&x_{K}\\ x_{2}&x_{3}&x_{4}&\cdots&x_{K+1}\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ x_{L}&x_{L+1}&x_{L+2}&\cdots&x_{N}\\ \end{pmatrix} \tag{5}\]  
The singular value decomposition (SVD) is then performed on the trajectory matrix to decompose the trajectory matrix into the following form:  
\[X=U\Sigma V^{T} \tag{6}\]  
where U \(\in R^{L\times L}\), \(\Sigma\in DIAG^{L\times K}\) and V \(\in R^{K\times K}\), U and V are called You matrices and are unitary orthogonal matrices, and \(\Sigma\) is called singular value and is a diagonal matrix.  
The singular value decomposition decomposes the trajectory matrix into a linear combination for the unitary matrix, the diagonal array, and the unitary matrix; then, \(X\) can be transformed into:  
\[X=\sum_{i=1}^{r}\sigma_{i}U_{i}V_{i}^{T}=\sum_{i=1}^{r}X_{i} \tag{7}\]  
where r is the rank of the matrix, \(\sigma\) is the square root of the eigenvalues of the covariance matrix of the trajectory matrix from smallest to largest, and d = max(i), also known as the singular spectrum of the original sequence.  
Divide the set of subscripts into m disjoint subsets and sum the matrices contained within each group, and the other i\({}_{1}\ldots i_{p}\) are the matrices contained in group I. Then  
\[X_{I}=X_{i_{1}}+\cdots+X_{i_{p}} \tag{8}\]  
\[X=X_{I_{1}}+\cdots+X_{I_{m}} \tag{9}\]  
Finally in the reconstruction step, we transform each matrix \(X_{ij}\), in Equation (5) into a new sequence of length N, i.e., we obtain the decomposed sequence. Define Y as an L*K matrix with elements \(y_{ij}\), \(1\leq I\leq L\), \(1\leq j\leq K\). Let L* = min (L, K), K* = max (L, K), and N = L + K - 1. If L \(<K\), y* \({}_{ij}\) = y\({}_{ij}\); otherwise, y* \({}_{ij}\) = y\({}_{ji}\). Then, the formula for transforming the matrix into a one-dimensional sequence of diagonal averages can be expressed as:  
\[y_{k}=\left\{\begin{array}{ll}\frac{1}{k}\sum\limits_{m=1}^{k}y_{m,k-m+1}^{ *}&1\leq k<L^{*}\\ \frac{1}{L^{*}}\sum\limits_{m=1}^{L^{*}}y_{m,k-m+1}^{*}&L^{*}\leq k\leq K^{*} \\ \frac{1}{N-k+1}\sum\limits_{m=k^{*}+1}^{N-k^{*}+1}y_{m,k-m+1}^{*}&K^{*}<k\leq N \\ \end{array}\right. \tag{10}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Hazard Level** & **Deformation Evaluation** & **Deformation Rate (mm/yr)** \\ \hline Lower risk & Weak development & \(\leq\)10 \\ Moderate risk & Medium development & 10–30 \\ High risk & intensive development & \(\geq\)30 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Indicators for evaluating the risk of surface deformation.  
The final one-dimensional sequence is obtained as:  
\[\left(y_{1},y_{2},y_{3},...,y_{k-1},y_{k},y_{k+1}...y_{L-1},y_{L},y_{L+1},...y_{ K+L-3},y_{K+L-2},y_{K+L-1}\right)\]  
The remaining principal component sequences can be reconstructed similarly. In this way, the signal decomposition of the time series is completed.