# Prediction of Surface Deformation Induced by Ultra-Shallow-Buried Pilot Tunnel Construction
## 1 Introduction
### Key Parameters  
The back-analysis method is used to infer the following key parameters of the prediction model from existing settlement data: the tangent value of the main influence angle of the surrounding rock \(\tan\beta\), the uniform shrinkage displacement of the section \(\Delta R\) and the second shrinkage displacement \(\delta\). The obtained key parameters can be used for settlement prediction under the same geological conditions. Before back-analysis, a target function between the known settlement data and the key parameters to be determined is first established, transforming the parameter determination problem into a problem of solving the minimum value of the target function. The target function for the parameters to be determined can be defined as  
\[F(p)\ =\ \sum_{i\ =\ 1}^{m}\left[W_{i}^{0}-W_{i}(p)\right]^{2} \tag{16}\]  
\(W_{i}^{0}\) is the measured value of the \(i\)th surface settlement; \(W_{i}(p)\) is the theoretically calculated value of the \(i\)th surface settlement; \(m\) is the total number of measured data; and \(p\) is a parameter to be determined, where \(p=\{\tan\beta,\Delta R,\delta\}\).  
The model in this paper is practical and empirical. The error generated by using the minimum approximation instead of the exact minimum value is acceptable in practical engineering. To obtain the optimal parameter combination, [PERSON]'s optimization algorithm in the principle of optimization is used to find the minimum value of the objective function \(F(p)\). Assuming the initial values of the undetermined parameters are zero, the value of the objective function under the initial conditions is calculated, followed by iterative operations based on the specified output accuracy. First, the extrapolation method is used to determine the search interval for a single golden section, then the step size of the current search is determined. A new coordinate is obtained by moving in the initial direction, and the function variation value corresponding to the current moving direction is recorded. Before the end of the current search, the direction with the maximum variation is identified. At the end of the search, a reflection point is constructed to determine whether it meets the Powell optimization condition. If so, an accelerated search is performed and the direction with the maximum variation is replaced; otherwise, this step is skipped, and only the starting point for the next search is updated. Finally, it is determined whether the target accuracy is met. If so, the data are output to obtain the three key parameters. A corresponding calculation program was developed using Matlab to facilitate the calculation of ground settlement caused by ultra-shallow-buried subsurface pilot tunnel construction.  
The posterior variance test in gray prediction is used to evaluate the back-analysis results. The residual between the theoretical calculation and the measured value at each measuring point is defined as  
\[\epsilon_{i}\ =\ W_{i}-W_{i}^{0} \tag{17}\]  
The average residual of each measuring point is calculated as  
\[\overline{\epsilon}\ =\ \frac{1}{m}\sum_{i\ =\ 1}^{m}\epsilon_{i} \tag{18}\]The average value of the calculated results at each measuring point is calculated as  
\[\overline{W}\ =\ \frac{1}{m}\sum_{i\ =\ 1}^{m}W_{i} \tag{19}\]  
The mean square deviation of the theoretical calculation at each measuring point is calculated as  
\[S_{1}\ =\ \frac{1}{m}\sum_{i\ =\ 1}^{m}\left(W_{i}-\overline{W}\right)^{2} \tag{20}\]  
The mean square deviation of the residual of the theoretical calculation at each measuring point is calculated as  
\[S_{2}\ =\ \frac{1}{m}\sum_{i\ =\ 1}^{m}\left(\epsilon_{i}-\overline{\epsilon} \right)^{2} \tag{21}\]  
The posterior variance ratio is calculated as  
\[C_{\mathrm{P}}\ =\ \frac{S_{2}}{S_{1}} \tag{22}\]  
The evaluation is conducted according to Table 3. Only when the back-analysis results meet the standard of \"qualified\" or above is the parameter combination obtained by back-analysis considered effective.