## 3 Performance Evaluation and Discussion
### Geometric Error Evaluation  
We evaluate our tunnel cross sections from three aspects including geometry, topology and LoDs. From the perspective of geometry, we analyze the deviations of radii of the extracted cross sections and the chord length based on a series of the central angles, and calculate the nearest Euclidean distance from the cross-sectional points to their nearest raw point clouds. To evaluate the accuracy of the cross-sectional points, we first analyze the deviations of radii of 1258 cross sections with an interval of 0.1 m and compare them to the designed radius. The mean value and the standard deviation of the radii of these cross sections are 2.7492 m and 1.4 mm, respectively. The statistic results are shown in Figure 9a. Compared with the designed tunnel radius of 2.75 m, the RMSEbetween the extracted radii of tunnel and the designed radius is only 1.6 mm. Considering the influence of the precision of the laser scanner, we select a very flat indoor facade to calibrate the precision of the laser scanner at a distance of 10 m. We fit the flat facade and calculate the mean residual of the fitting error of 1.5 mm between the raw point clouds and the fitted facade. This value is between1.4 mm and 1.6 mm, which implies that the radii of the extracted cross sections are very stable and accurate compared with the design radius of 2.75 m.  
We also analyze the minimum Euclidean distance from each cross-sectional point to its nearest raw point clouds to inform the \"changed\" initial raw point clouds. To this end, we divide the horizontal coordinates of the minimum Euclidean distance shown in Figure 9b into four categories: \(>\)0.100 m, 0.010-0.100 m, 0.007-0.010 m and 0-0.007 m. The cross section points from the first category only take up to 2%, which is mainly because of the missing data at the vicinity of tunnel track. This missing data arises from occlusion of the tunnel track having higher elevation than its associated rail slab. Because of missing data, the generated cross-sectional points are far away from its nearest raw point clouds, causing the deviation over 0.1 m, as is evident in the Figure 10a. In Figure 10b, the cross-sectional points from the second category account for 30%, which mainly arises from the tunnel bottom and the bolts used to fix the tunnel linings and other installations, e.g., bolts, pipes and wires. The third category accounts for 7% of the cross-sectional points created. From the distribution in Figure 10c, we can see that these points are scattered far away from their corresponding scanners denoted by the blue points in Figure 10. In these areas, the raw point clouds have low density; therefore, the local fitted quadric surfaces are not as accurate as the fitted surfaces derived from the high density of points. Furthermore, some installations and other equipment attached to the tunnel linings can also result in the large deviations. The last category takes up to 61% of the extracted cross-sectional points. The detailed distribution of those point clouds is shown in Figure 10d, where it can be seen that most of the cross-sectional points are distributed approximately 10 m away from their associated scanner. From FARO(r) Focus3D X 330 scan data processing and registration software SCENE, we obtain the spacing among point clouds at approximately 10 m is 6.136 mm, which makes it consistent with a maximum of 7 mm and explains why these points are distributed well in the vicinity of the scanner around 10 m.  
Figure 9: Histogram of cross-sectional radii and minimum Euclidean distance from the cross section points to their nearest raw point clouds. (**a**,**b**) represent the statistics of cross-sectional radius and minimum Euclidean distance, respectively. To further evaluate the geometric accuracy of 1258 cross sections, four types of chord lengths associated with central angles of 30\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\) and 180\({}^{\circ}\) are compared and evaluated.