## 3 Parameter Identification Methods for Stations through Arch Cover Method
### The Parameter Identification Flowchart  
According to the meaning of the model parameters, the stress and displacement are taken as the objective function to control the target value as well as to reach the minimum objective function.  
\[\begin{array}{l}\min F(x)=\min E(x)+\min D(x)\\ \min E(x_{1},x_{2},\cdots,x_{n})=\frac{1}{m}\sum\limits_{\begin{subarray}{c}i =1\\ i\end{subarray}}^{m}\left[Y_{i}^{0}-Y_{i}\right]^{2}\\ \min D(x_{1},x_{2},\cdots,x_{n})=\frac{1}{T}\sum\limits_{\begin{subarray}{c}i =1\\ i\end{subarray}}^{n}\left[S_{i}^{0}-S_{i}\right]^{2}\\ x_{i}^{a}\leq x_{i}\leq x_{i}^{b}(i=1,2,\cdots,n)\end{array} \tag{10}\]  
where \(\min E(x)\) is a function of the displacement control value; \(\min D(x)\) is the stress control value; \(Y_{i}^{0}\) and \(S_{i}^{0}\) are the measured deformation and stress values of the surrounding rock; \(Y_{i}\) and \(S_{i}\) are implicit functions with the surrounding rock parameters as independent variables, which need to be solved by numerical simulation; \(x_{i}\) is a parameter; \(n\) is the number of parameters; \(x_{i}^{a}\) and \(x_{i}^{b}\) are the upper and lower limits of the parameter \(x_{i}\); and \(m\) is the number of observation values.  
The specific flow of displacement stress joint back analysis of surrounding rock parameters based on GP-DE is as follows:  
1. Orthogonal design of surrounding rock parameters includes: the tensile strength \(T\), Poisson's ratio \(\mu\), cohesion \(c\), and internal friction angle \(\varphi\). Different values of the five surrounding rock parameters of elastic modulus \(E\) are taken within a certain range for an orthogonal design, and an orthogonal scheme is formed.
2. Second item, sample generation: FLAC\({}^{3\text{D}}\) is used to establish a numerical model, and the orthogonal scheme is used to simulate the model. The results of the orthogonal scheme are normalized after they are obtained, and the processed results are test samples, which are regarded as GP learning training samples and prediction test samples, and the influential factors obtained through a sensitivity analysis are used as the variables to be inverted.
3. The following method is used to establish a nonlinear mapping relationship model of surrounding rock parameters: use GP technology to establish the nonlinear mapping relationship, use superparameters to carry out GP learning and prediction on the test samples, and evaluate the fitness of the obtained results obtained. Then, repeat the operation of mutation, crossover, selection, GP prediction, and fitness evaluation on the test samples. When the objective function reaches the preset value, the GP-DE algorithm has been optimized.
4. Optimization of surrounding rock parameters: first of all, take the objective function of Formula (10) as the fitness function of the DE algorithm, generate a random group of parameters in the solution space as the initial population, and repeat the operation of mutation, crossover, selection, GP prediction, and fitness evaluation. The surrounding rock parameters that meet the minimum fitness are the back-analysis parameters. The detailed process is shown in Figure 2.  
Due to the different dimensions of displacement and stress in the results obtained based on the orthogonal design, a direct comparison and calculation will make the final results inaccurate, so they cannot be compared. To make the results accurate and reliable, normalization can be a good choice. After normalization processing, the data of different dimensions can be transformed into dimensionless data for subsequent processing. The normalization formula is as follows:  
\[\begin{split} r_{i}=(k_{i}-k_{\text{min}})/(k_{\text{max}}-k_{ \text{min}})\\ (i=1,2,\cdot\cdot\cdot,25)\end{split} \tag{11}\]  
where: \(k_{i}\) is the calculated value of the \(i\)th output in the sample, \(k_{\text{min}}\) is the minimum value of the output in the orthogonal scheme, \(k_{\text{max}}\) is the maximum value of the output in the orthogonal scheme, and the normalized value of \(r_{i}\) is \(r_{i}\in[0,1]\).