## 5 Surface Subsidence Model Optimization
### Peak Formula for Two-Line Tunnel  
(1) The study of peak on the Chicago two-lane tunnel shows settling the two-lane tunnel has a symmetrical distribution. In the settlement calculation, the double-track tunnel is replaced by a larger circle, and the radius of the large circle is given by:  
\[R^{\prime}=R+L/2 \tag{2}\]  
where \(L\) is the distance between the two-track tunnel axes.  
This method is a correction formula for symmetrical settlement. However, sometimes the surveying data are not symmetrical at the midline, and the maximum settlement value will deviate from the midline. In addition, this method has some limitations. In fact, for some of the buried depths, \(L\) is small. However, in some tunnels, a \"W\"-shaped settlement curve will appear, which does not satisfy the normal distribution in the peak formula.  
(2) [PERSON] et al. [32] modified the peak formula by considering the maximum land subsidence will be shifted and proposed the following calculation formula:  
\[S(x)=\frac{\pi R^{2}V_{i}}{i_{t}\sqrt{2\pi}}exp[-\frac{(x-a)^{2}}{2i_{t}^{2}}] \tag{3}\]  
where \(i_{t}\) is the width coefficient of the surface settlement tank and \(a\) is the distance from the maximum value of the total settlement to the central axis of the two tunnels (m).  
Equation (3) is then analyzed. Since this project tackles predicting the surface settlement of the double-track tunnel, the maximum settlement may not be on the central axis. In the measurement, the maximum settlement occurs at 4.85 m from the midline, the value of a is 4.85, and the radius of the tunnel is 12 m; therefore, in Equation (3), only two unknown coefficients (i.e., the total width coefficient of the surface settlement tank and the total soil loss rate) are required.  
(3) GaussAmp model fitting.  
It is expressed as:  
\[y=y0+A*exp[-0.5*\left(\frac{x-x_{c}}{\omega}\right)^{2}] \tag{4}\]  
where \(y0\) is the GaussAmp model coefficient used to fit the maximum value of the data (the fixed value is 0), \(A\) is the GaussAmp model coefficient, which is related to the width coefficient of the surface settlement tank and the coefficient of soil loss rate, \(xc\) is the GaussAmp model coefficient, which is characterized by the distance the maximum value of total settlement deviates from the central axis of the two tunnels (equal to 4.85), and \(\omega\) is the GaussAmp model coefficient characterizing the width coefficient of the surface settlement tank.  
The fitting result stood after fitting the settlement values in this project according to Equation (4) is shown in Figure 15. The GaussAmp model is used as the fitting model. According to the surveying data in this project, the values of boundaries \(y0\) and \(x_{c}\) are fixed. The maximum settlement of this project occurs at a position 4.85 m away from the midline, and the value of \(x_{c}\) is 4.85. The adjusted peak formula is used with \(y0\) equal to 0. Fixed parameters and the data are fitted, and after many iterations, the fitting effect is finally achieved according to 35 sets of data. The values of \(A\) and \(\omega\) are \(-7.29076\) and \(20.65042\), respectively. By substituting these values in Equation (3), the stood value of \(V_{i}\) is \(-0.83422\). In summary, the formula for the surface settlement of the double-track tunnel developed in this project is accurate.