## 4 Analysis of Plane to Points Histogram (P2 PH)
### Analysis Results  
A software, Cloud compare, was used to perform the analysis of the point cloud (CLOUDCOMPARE, 2018 [3]). Since the front view of the failure surface of the point cloud is a rectangle with a vertical to horizontal ratio of about 10:4 on average, the fitting surface is simulated as a rectangle. Compared to the other two samples, the roughness of the gravel has to include both the local roughness and the curvature of the gravel so that the calculation of the fitting surface can affect the result. Therefore, as shown in Figure 7, the analysis results were compared when the sizes of the fitting plane are 25 mm \(\times\) 10 mm, 50 mm \(\times\) 20 mm, and 75 mm \(\times\) 30 mm. It can be seen that when the size of the fitting plane is 25 length and 10 width, the C2M distance is not changed within about 2 mm. If the fitting surface is small, the curvature of some part of the gravel can be simulated, but local roughness is not appropriately simulated. When the rectangular fitting face is 75 mm long and 30 mm wide, it shows that the local roughness and curvature of the gravel are expressed in the histogram within 10 mm. However, since the fitting surface is too large for the sample size, the convex protruding shape irrespective of the roughness of the failure surface affects the result. Therefore, it can be seen that the roughness is calculated even after the C2M distance is 10 mm. When the size of the fitting surface is 50 mm in length and 20 mm in width, it is not only reflected appropriately in the C2M analysis results of local roughness and gravel curvature, but it is also possible to collect a number of analysis samples from the failure surface. As shown in Figure 7, the point clouds of the failure  
Figure 6: Cloud compare analysis using fitting plane. surface were cut into five rectangular shapes in each sample, and the cut surfaces were overlapped with each other in the cases of gravel and grouting samples due to small size of the failure surface. Fitting planes were created from the cut point clouds, and C2M distance can be calculated. Figure 7c shows the location of the fitting plane on the failure surface of the grouting sample and the distribution of points within the fitting plane. In the side view of the distribution of the points within the fitting plane, a uniform distribution pattern is shown without large curvature in the grouting sample. An example of a fitting plane at the failure surface of a sand sample and the distribution of points within the fitting plane are shown in Figure 7d. The points are distributed in a wavy shape relative to the fitting plane by the sand grains and the local curvature on the failure surface (see side view in Figure 7d). In the gravel sample, the sharp curves of the failure surface caused by the protrusion of the gravel and the area from which the gravel is separated from the failure surface are expressed as a distribution of points. Since the fitting plane is 20 mm wide and 50 mm long, the roughness due to gravel is also revealed in the side view. The global distribution of points is not only spread over the fitting surface, but also the points are distributed farther from the fitting plane in the local area than the other samples (see Figure 7e). Figure 8 shows the histogram of the C2M distance for each sample. Figure 8a shows the histogram of the grouting sample. The averaged C2M distance at which the number reached 95% of total points on the five histograms is 3.50 mm. Most points are concentrated within a small C2M distance range of the histogram, which means that the failure surface has a small roughness as discussed in Figure 6d. The averaged C2M distance at which the number of points in the sand sample reached 95% is 3.77 mm, which is slightly larger than that of the grouting sample (see Figure 8b). The C2M distance is increased by the roughness of the sand grain and the local roughness.