## 2 Deformation Monitoring of Inverted Arch Uplifts of Soft Rock Tunnels
### Inverse Calculation of the Deformed Structure Point Cloud Distance  
The normal vector of any point in 3D point cloud data after reconstruction of the inverted arch of a soft rock tunnel is \(\left(x_{i^{\prime}}^{\prime\prime},y_{i^{\prime}}^{\prime\prime},z_{i^{\prime \prime}}^{\prime\prime}\right)\). Then, from the above optimal condition, i.e., Equation (16), parallel to the normal vector of the point, we can obtain:  
\[\hat{a}\left(x_{i^{\prime}}^{\prime\prime}\ +\ \xi_{x_{i^{\prime\prime}}^{\prime \prime}}\right)\ +\ \hat{b}\left(y_{i^{\prime\prime}}^{\prime\prime}\ +\ \xi_{y_{i^{\prime\prime}}^{\prime\prime}}\right)\ +\ \hat{c}\left(z_{i^{\prime \prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)=0 \tag{20}\]  
Combining Formulas (19) and (20) obtains:  
\[\hat{a} =\frac{\hat{c}\left(y_{i^{\prime}}^{\prime\prime}\ +\ \xi_{y_{i^{ \prime\prime}}^{\prime\prime}}\right)\ -\ \hat{b}\left(z_{i^{\prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ }{\hat{b}\left(x_{i^{\prime \prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ -\ \hat{a}\left(x_{i^{\prime \prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ }\hat{c} \tag{21}\] \[\hat{b} =\frac{\hat{c}\left(x_{i^{\prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime \prime}}^{\prime\prime}}\right)\ -\ \hat{a}\left(x_{i^{\prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ }{\hat{a}\left(y_{i^{\prime \prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ -\ \hat{b}\left(x_{i^{\prime \prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ }\hat{c}\]  
Then, according to Formulas (16) and (21), the datum plane formula can be determined as follows:  
\[\frac{\hat{c}\left(y_{i^{\prime}}^{\prime\prime}\ +\ \xi_{y_{i^{\prime}}^{ \prime\prime}}\right)\ -\ \hat{b}\left(z_{i^{\prime\prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ }{\hat{b}\left(x_{i^{\prime \prime}}^{\prime\prime}\ +\ \xi_{x_{i^{\prime\prime}}^{\prime\prime}}\right)\ -\ \hat{a}\left(y_{i^{\prime \prime}}^{\prime\prime}\ +\ \xi_{y_{i^{\prime\prime}}^{\prime\prime}}\right)\ }x^{\prime\prime}\ +\ \frac{\hat{c}\left(x_{i^{ \prime\prime}}^{\prime\prime}\ +\ \xi_{x_{i^{\prime\prime}}^{\prime\prime}}\right)\ -\ \hat{a}\left(z_{i^{ \prime\prime}}^{\prime\prime}\ +\ \xi_{z_{i^{\prime\prime}}^{\prime\prime}}\right)\ }y^{\prime\prime}\ +\ z^{\prime\prime}=0 \tag{22}\]  
Here, \(\left(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime}\right)\) are the best point cloud normal vectors.  
To calculate the average value of the sum of the cosine values of the angle between the normal vector of the inverted arch point cloud and the corresponding datum plane of all soft rock tunnels, \(ave\cos\varepsilon_{i^{\prime}}\), the formula is as follows:  
\[ave\cos\varepsilon_{i^{\prime}}=\frac{\sum\limits_{i^{\prime}=1}^{\eta}\hat{a} |\cos\varepsilon_{i^{\prime}}|}{\eta\rho^{\prime}(t)} \tag{23}\]  
where \(\varepsilon_{i^{\prime}}\) is the included angle between the point cloud normal vector and the corresponding datum plane.  
According to Formula (23), the best datum plane can be found by this formula; for all point cloud normal vectors, the lowest value of \(ave\cos\varepsilon_{i^{\prime}}\) corresponds to the best datum plane. By analogy, after setting a fixed distance from the point cloud to the optimal datum plane, the overall point cloud of the inverted arch of a soft rock tunnel can be cut into multiple slices of point clouds using the optimal datum plane. Similarly, the distance from the optimal datum plane to the point cloud of the deformation structure is inversely deduced and set as a fixed value.