## 2 Materials and Methods
### Displacement Back-Analysis Model Based on LSTM  
Displacement back-analysis is a method that can be implemented to find the best valuation of inverted parameters by establishing an objective function model based on the least squares optimization criterion according to the functional relationship among the observed displacement and the forward calculated displacement. The general expression of the objective function model is as follows [24]:  
\[V_{Obj}=\sum_{i=1}^{n}p_{i}v_{i}^{2}=\sum_{i=1}^{n}p_{i}[S_{i}(X)-L_{i}]^{2} \tag{1}\]  
where, if the basic displacement back-analysis data contain \(n\) mutually independent observations \(L=(L_{1}\;L_{2}\;\cdots\;L_{n})^{T}\), then the weight matrix is \(P=diag(p_{1}\;p_{2}\;\cdots\;p_{n})\), \(V_{Obj}\) is the objective function error value, and \(S_{i}(X)\) is the corresponding numerical simulation forward-calculated displacement. The traditional method calculates the objective function error value by multiple forward-analysis iterations, which takes a lot of time and has low computational efficiency. The DBA-ML algorithm is an effective means to solve this problem. The processes of the DBA-BPNN and DBA-SVM algorithms and specific applications for them have been described in detail in the literature [28,39]. However, with the rapid development of artificial intelligence algorithms, deep learning algorithms with multiple hidden layers have been shown to have superior feature learning capabilities compared to shallow machine learning algorithms [32]. In this paper, a deep learning algorithm is employed for displacement back-analysis for the first time, and a DBA-LSTM model is constructed. The LSTM network can avoid the problem of gradient disappearance by changing the internal structure of the cell, adding input gates, forget gates, and output gates into \begin{table}
\begin{tabular}{c c c c} \hline \hline No. & Control & Methods & Instrumentation \\ \hline
1 & Ground displacement & GNSS & Huace H3 GNSS \\ \hline
2 & Rainfall & Rain gauge & GFZ01 digital rain \\  & & & gauge \\ \hline
3 & Environment & Video & Hikvision surveillance camera \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Monitoring content and methods. Figure 4: Location of landslide monitoring stations. its memory cell, which controls whether to forget historical information or to update the state of the cell via the gating unit [28]. The LSTM network model and its cell structure are shown in Figure 5, and the model can be composed of a multilayer network. To facilitate understanding, Figure 5 shows a single-layer LSTM network structure [40]. The one-layer LTSM network generally consists of several cells, and there are three inputs in each cell structure, which are the input \(X_{t}\) of the current moment, the input \(h_{t-1}\), and the memory information \(C_{t-1}\) of the previous moment. Within the cell structure, three gates control the discard and retention of historical and current information: the forget gate \(f_{t}\), the input gate \(i_{t}\), and the output gate \(O_{t}\). The formula is as follows:  
\[\left\{ \begin{aligned} & f_{t}=\sigma\Big{(}W_{f}\cdot[h_{t-1},x_{t}]+b_{f }\Big{)}\\ & i_{t}=\sigma(W_{t}\cdot[h_{t-1},x_{t}]+b_{i})\\ & O_{t}=\sigma(W_{o}\cdot[h_{t-1},x_{t}]+b_{o})\end{aligned} \right. \tag{2}\]  
where \(\sigma\) is the activation function, \(W\) and \(b\) are the weight matrix and offset matrix of the control gates. After obtaining the output information of the three control gates from Equation (2), the short-term memory information \(h_{t}\) and the long-short memory information \(C_{t}\) can be calculated using the following equation:  
\[\left\{ \begin{aligned} &\widetilde{C}_{t}=\tanh(W_{c}\cdot[h_{t-1},x_{t}]+b_{c })\\ & C_{t}=f_{t}\cdot C_{t-1}+i_{t}\cdot\widetilde{C}_{t}\\ & h_{t}=O_{t}\cdot\tanh(C_{t})\end{aligned} \right. \tag{3}\]  
where \(\tanh\) is the activation function that transforms the output to a value between \(-1\) and \(1\). The DBA-LSTM model was constructed as follows:  
Step 1: Prepare the training samples.