## 5 Analysis of Roughness Detection Method by Kernel
### Analysis Results
In the results of Figure 11c,d, the number of points in grouting and sand samples is distributed between about 1700 and 2100 in the roughness range of less than 0.001 m. Although the kernel radius is decreased to 0.01 m, the number of points is not increased in the roughness range of less than 0.001 m. It means that even if the kernel becomes smaller than 0.025 m, the roughness due to the grain or the local curvature of sand is not changed. Therefore, the lower bound of the kernel radius to reflect the roughness of the grouting and the sand samples is about 0.025 m. In this paper, the roughness-dominated section of the three samples was distinguished from the histogram in which the kernel radius that best reflects the roughness was selected as 0.05 m (see Figure 11e). The roughness of the grouting sample is dominant in the roughness range from 0 to 2.90 mm. The effects of local curvature of failure surface and the grain shape in the sand sample is more dominant than the other samples in the roughness range between 2.90 and 4.74 mm. If the roughness is larger than 4.74 mm, it can be seen that the roughness of the gravel sample is overwhelmingly dominant compared to the other two samples. Figure 12 shows the color scale indicating the roughness of each point in three samples and Figure 12a shows the result of a kernel radius of 0.1 m. The kernel size is larger than the gravel sample so that the high roughness value is generated at the edges of the sample, which can be considered as an error. As shown in the results of the top of the gravel sample, the bottom of the grouting sample, and the center of the sand sample, the global curvature of the three samples is partially reflected. When the kernel radius is reduced to 0.05 m, the protrusions of the gravel sample are more clearly distinguished from the failure surface (see Figure 12b). The flat surface of the grouting sample has low roughness and the roughness at an angled section where the angle of the failure surface is sharply changed is high. The effects of grain and the local curvature change are reflected in the sand sample as well. As shown in the results of the gravel sample in Figure 12c, the roughness of the protrusion and the traces of destruction of gravels detected by a kernel radius of 0.025 m is distinguished in detail. However, the grouting sample and the sand sample show similar results because the sharp curvature of the failure surface at the bottom of the grouting sample is not reflected properly due to the small size of kernel radius and the local curvature of the sand sample is not properly reflected. When the kernel radius was reduced to 0.01 m, the roughness  
of all three cases is not reflected in the results and hence the roughness distribution of the three samples cannot be distinguished as with the results in Figure 11(d) (see Figure 11(d)). Figure 11: Roughness histograms for of three samples with different kernel radius. Figure 12: Color scale indicating the degree of roughness of the cloud. The analysis of point cloud is significantly influenced by the density of the points. If each point included accurate coordinate information, accurate analysis results can be obtained with a point cloud containing many points. However, the scanning and analysis time is increased with the increase of the number of points. In this paper, a kernel size of 0.05 m, which reflects the roughness of three samples properly, was used to analyze the calculation of roughness regarding the density change of the point cloud. Subsamples with 100,000, 10,000, 5000, and 1000 points were extracted from 135,000 original point clouds to evaluate the roughness, and compared with the roughness of the original point cloud. Figure 12(a) shows a histogram in which roughness was determined by subsamples with 100,000 points. There is no significant difference with the results of the original point cloud in Figure 10(b). Table 1 shows the roughness at which the number of points reached 95% in each sample.