## 2 Literature Review of Longitudinal Deformation Profile  
The LDP is a graph that expresses the amount of convergence at the tunnel wall as a function of the distance from the tunnel face. Figure 2 presents the concept of a longitudinal cross-section and the LDP for an unsupported tunnel with the excavation radius R. At a distance x from the tunnel face, the convergence displacement is \(u_{r}\). On increasing the distance x from the tunnel face, the convergence displacement converges to \(u_{r}^{m}\). Ahead of the excavation face, negative values of x indicate that the rock mass has begun to deform ahead of the excavation face and measureable displacement is reduced to its minimum value at a distance of about one-half a tunnel diameter ahead of the advancing tunnel face. This implies that longitudinal displacement occurs in front of the tunnel face when tunnel excavation is conducted. Thus, LDP analyses considering the preceding displacement are essential for a complete assessment of the tunnel excavation behavior. [PERSON] and [PERSON] [4] proposed convergence displacement according to the tunnel distance in the following fractional function form by using finite element analysis:  
\[\frac{u_{r}}{u_{r}^{m}}=0.28+0.72\left[1-\left(\frac{0.84}{0.84+x/R_{T}} \right)^{2}\right] \tag{1}\]  
where \(u_{r}=\) radial displacement at a distance \(x\) from the face;  
\(u_{r}^{m}=\) maximum radial displacement;  
\(R_{T}=\) tunnel radius. [PERSON] et al. [10] proposed an empirical formula in the form of an exponential function as follows:  
\[\frac{u_{r}}{u_{r}^{m}}=0.29+0.71\left[1-exp(-1.5(x/R_{T}))^{0.7}\right] \tag{2}\]  
[PERSON] [11] conducted a finite element analysis under elastic conditions and proposed a convergence displacement relationship formula according to tunnel distance, as expressed below. This expression can only be applied to the rear of the tunnel face, as indicated in Figure 3 in the form of a dashed line. \[\frac{u_{r}}{u_{r}^{m}}=0.25+0.75\left[1-\left(\frac{0.75}{0.75+x/R_{T}} \right)^{2}\right] \tag{3}\]  
[PERSON] and [PERSON] [3] presented an improved empirical formula proposed by [PERSON] [8], as expressed in Equation (4). This empirical formula is best fitted with the sigmoid curve function measured by [PERSON] et al. [9] for the Mingtam Power Cavern project (Figure 3). The empirical formula of [PERSON] [8] is applicable to both the front and rear of the tunnel face. \[\frac{u_{r}}{u_{r}^{m}}=\left[1+\exp\left(\frac{-x/R_{T}}{1.10}\right)\right] ^{-1.7} \tag{4}\]  
Figure 2: LDP and cross-section for tunnel excavation. The proposed LDP calculation formula is a function of the excavation distance and does not consider the parameters for the ground conditions. [PERSON] and [PERSON] [12] proposed normalized radial displacements using numerical analyses and statistical regression to account for the effect of [PERSON]'s ratio on circular tunnels under elastic ground conditions. [PERSON] and [PERSON] [12] believed that it was difficult to obtain a good correlation for expressing the normalized LDP in the form of a single sigmoid curve when curve fitting the results of tunnel air displacement obtained via numerical analyses. Moreover, they proposed a formula for determining the LDP by dividing it into two parts, as shown in Figure 4, based on the tunnel face. Figure 4: Statistical LDP models used for the normalized radial displacements occurring around the face [12]. Figure 3: \(u_{r}/u_{r}^{m}\) profiles derived from elastic models [11], measurements in a tunnel [9], and best-fit measurements [1,8].