## 2 Theoretical Basis
### Longitudinal Displacement Profile  
The longitudinal displacement profile (LDP) describes the displacement at the monitoring section versus distance away from tunnel face and provides an analytical solution to establish a distance-convergence relationship. As shown in Figure 3a, the origin of the x-axis is located at the monitoring section and the x-axis represents the relative location from monitoring section to tunnel face; the y-axis represents the displacement of tunnel boundary at the monitoring section. When tunnel face proceeds along the x-axis to the coordinate value of \(\mathbf{x}\), the displacement at the section is denoted as \(\mathbf{u}_{\mathbf{x}}\). Before the arrival of tunnel face, a portion of displacement has occurred. When tunnel face arrives at the monitoring section, initial displacement \(\mathbf{u}_{\mathbf{0}}\), approximately 30% of the ultimate displacement \(\mathbf{u}_{\mathbf{\infty}}\), has been achieved, after which the convergence of surrounding rock will increase dramatically to the peak value and keep stable. In other words, the excavation disturbance effect by the tunnel face will disappear until the tunnel face is far away enough, as shown in Figure 3b. It is commonly believed that the distance between the section and tunnel face is no less than three times the opening diameter. In tunnel design, the appropriate location for the installation of the initial support can be determined by the LDP [8]. To reflect the propagation of the convergence during the tunnel face advancing, the dimensionless displacement, \(\mathbf{\lambda}\), is defined as the ratio of \(\mathbf{u}_{\mathbf{x}}\) to \(\mathbf{u}_{\mathbf{\infty}}\) to normalize the displacement of surrounding rock in Equation (4):  
\[\mathbf{\lambda}=\frac{\mathbf{u}_{\mathbf{x}}}{\mathbf{u}_{\infty}} \tag{4}\]  
Figure 2: Point safety factor based on Mohr–Coulomb criterion. The longitudinal displacement profile can be obtained by the monitoring data. However, the field data usually just contain the excavated part and fail to monitor the unexcavated rock mass. To establish the intact relationship between the dimensionless displacement and distance to tunnel face, numerous authors have proposed alternative expressions. [PERSON] [31] derived the relationship for the normalized LDP from elastic analysis as Equation (5):  
\[\Lambda=0.25+0.75\left[1-\left(\frac{0.75}{0.75+\mathbf{x}/\mathbf{R}}\right)^{2}\right] \tag{5}\]  
where \(\mathbf{R}\) is the tunnel radius; \(\mathbf{x}\) is the monitoring location away from the tunnel face and \(\mathbf{x}>0\) is behind the tunnel face whereas \(\mathbf{x}<0\) is ahead of the face. Based on the measured data, [PERSON] [8] suggested the empirical best-fit formula as Equation (6):  
\[\Lambda=\left[1+\exp\left(\frac{-\mathbf{x}/\mathbf{R}}{1.10}\right)\right]^{-1.7} \tag{6}\]  
The elastic theoretical expression was deduced by [PERSON] et al. [32] as Equation (7):  
\[\Lambda=\left\{ \begin{array}{ll}(1-\Lambda_{0})\left(1-e^{-\frac{\mathbf{x}}{ \mathbf{x}}}\right)+\Lambda_{0}&\quad\mathbf{x}\geq 0\\ \Lambda_{0}\mathbf{e}^{\frac{\mathbf{x}}{\mathbf{x}_{1}}}&\quad\mathbf{x}<0\end{array} \right. \tag{7}\]  
where \(\mathbf{X}\) and \(\mathbf{X_{1}}\) are constants, and, moreover, \(\mathbf{X}_{1}=\frac{\Lambda_{0}}{1-\Lambda_{0}}\mathbf{X}\). Additionally, the expression based on Equation (8) was improved by the numerical analysis as [33]:  
\[\Lambda=\left\{ \begin{array}{ll}(1-\Lambda_{0})\left(1-e^{-\frac{(1-\Lambda_ {0})\mathbf{x}}{\mathbf{X}}}\right)+\Lambda_{0}&\quad\mathbf{x}\geq 0\\ \Lambda_{0}\mathbf{e}^{\frac{\mathbf{x}}{\mathbf{X}_{1}}}&\quad\mathbf{x}<0\end{array} \right.