## 6 Conclusions  
(1) Plane to points histogram (P2 PH) method was proposed to evaluate the three-dimensional roughness of the failure surface. The point clouds of the failure surfaces were cut by a rectangle of 20 and 50 mm to create a fitting surface representing the cut area. The histogram of the C2M distance calculated between the fitting surface and the point is a criterion for the evaluation of the roughness. For grouting, sand and gravel samples, the roughness at which the point in the histogram reached 95% is 3.50, 3.77 and 5.49 mm, respectively. Points are distributed in a narrow C2M distance range on the flat failure surface, and the points are distributed in a wide C2M distance region on the rough failure surface. In the histogram, the points are predominantly distributed below 1.8 mm for grouting samples, 1.8-3.3 mm for sand samples, and over 3.3 mm for gravel samples. It can show that the roughness of each sample is properly reflected in the histogram, and the proposed method can be a criterion for evaluating the roughness of each pile sample.  
(2) A roughness detection method by the kernel was applied in this study and the elevation of points can be calculated by projecting the point cloud onto the kernel was applied. Since the roughness is affected by the kernel size significantly, the optimized  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No. of Points**} & \multicolumn{3}{c}{**Averaged C2C Distance (mm)**} \\ \cline{2-4}  & **Grouting** & **Sand** & **Gravel** \\ \hline
100,000 & 0.03 & 0.03 & 0.04 \\ \hline
50,000 & 0.17 & 0.19 & 0.22 \\ \hline
30,000 & 0.35 & 0.37 & 0.41 \\ \hline
10,000 & 0.78 & 0.80 & 0.85 \\ \hline
7500 & 0.89 & 0.92 & 0.98 \\ \hline
5000 & 1.05 & 1.08 & 1.16 \\ \hline
2500 & 1.37 & 1.40 & 1.53 \\ \hline
1000 & 1.89 & 1.94 & 2.12 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Averaged C2C distances of subsamples in different density.  
Figure 15: C2C results between subsamples with different density.  
kernel radius reflecting the roughness of the three samples was calculated as 0.05 m. If the kernel radius is increased, the local roughness is diluted by other points and the roughness decreases. If the kernel radius is decreased, it is not able to reflect the global curvature of the failure surface.  
(3) The roughness detection method by the kernel is affected by the density of the point cloud, and hence the roughness analysis to define the effect of the point cloud density was performed. It is proposed that more than 5000 points need to be obtained within the cut point cloud to clearly distinguish the roughness.  
(4) In order to define the density effect, subsamples were extracted from the original point clouds of three pile samples and then the C2C distance between subsamples was calculated. The C2C distance and the C2C distance difference between samples are also rapidly increased with the decrease of density. If the failure surface is rougher such as gravel, it is more affected by density reduction than the other samples.  
The research on 3D roughness analysis performed in this paper is expected to be conducive to studies that require roughness estimation in the future. Since roughness is an important factor in shear behavior, studies using three-dimensional roughness to calculate shear strength can be conducted as a prospective study.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the data protection.  
Assistance with this laboratory study from CSIC (Cambridge Centre for Smart Infrastructure and Construction) at the University of Cambridge is appreciated. We also appreciate support provided by [PERSON] and [PERSON].  
The authors declare no conflict of interest.