## 5 Analysis of Roughness Detection Method by Kernel
### Analysis Results  
In order to compare the three samples equivalently, the point clouds of the failure surfaces in each sample were cut to have 135,000 points each so that the histograms of each sample can be compared to each other without normalization. Figure 11a shows the results of the histogram of three samples when the kernel radius is 0.1 m. The histograms of grouting and sand samples have almost similar patterns, and the roughness at which the number of points reached 95% was 7.25 and 6.72 mm, respectively. Contrary to the results in Figure 9, the results of histograms show that the roughness of the grouting sample was slightly rougher than that of the sand sample because the kernel size is too large to reflect the local curvature of failure surface in the sand sample. In the grouting sample, the surface is generally flat, but there is a global curvature where the angle of failure surface is sharply changed at the bottom of the sample. The kernel radius of 0.1 m reflects this global roughness of the grouting sample more than the roughness caused by grains or local curvature of sand. The histogram of the gravel sample showed three inflection points, and the roughness at which the number of points reached 95% is 2.06 mm. Although the roughness caused by the gravel is reflected, the roughness is distributed over a large roughness range in the histogram because the size of the kernel is larger than the sample size. Figure 11b is the result when the kernel radius is 0.05 m, and the distribution of roughness range is narrower than the kernel radius of 0.1 m. The points of the grouting sample are slightly more than that of the sand sample within a small roughness range, and  
Figure 10: Estimation of roughness kernel. the points of sand are more increased with the increase of the roughness. Therefore, the roughness of the failure surface in the sand sample is appropriately reflected as similar as the result obtained in Figure 9. The roughness at which the number of points in the grouting sample and the sand sample reached 95% is 5.73 and 5.94 mm, respectively, showing a relevant result that the roughness of the sand sample is slightly larger. The number of points in the gravel sample linearly decreased with the increase of the roughness. The roughness at which the number of points in the gravel sample reached 95% is 1.21 mm, which is about twice as large as those of the other samples. It was found that the kernel of 0.05 m adequately reflects the roughness of the gravel because it can cover the protrusion and the traces of destruction of several gravels. Therefore, the kernel radius of 0.05 m can be a standard for calculating the three-dimensional point cloud roughness of the three samples, which can reflect the roughness features of the three samples. Figure 11c shows the histogram of three samples when the kernel radius is 0.025 m. The histograms of the grouting and sand samples show almost the same distribution, and the roughness at which the number of points in the sample reached 95% is almost the same at 4.42 and 4.36 mm, respectively. It can be seen that a kernel radius of 0.025 m is not adequate to distinguish the histograms between grouting and sand samples. However, the roughness features of the gravel sample can be appropriately distinguished with the other samples and the roughness at which the number of points reached 95% was 6.69 mm. The kernel radius of 0.01 m is indistinguishable among three histograms because the histograms of the three samples are overlapped in most of the roughness range. The roughness at which the number of points in the grouting, sand and gravel samples reached 95% is 3.16, 3.02 and 3.26 mm, respectively, showing similar values. Therefore, a kernel radius of 0.01 mm is not able to distinguish among histograms of three samples, and the roughness features of gravel cannot be reflected in the histogram.