# Deep Learning (Fast R-CNN)-Based Evaluation of Rail Surface Defects
## 4 Analysis and Discussion  
### Gaussian Probability Density Analysis of Rail Internal Defects  
The Gaussian distribution is a bell-shaped curve. For a clear understanding of the normal distribution, the definitions of \"\(x_{c}\) (mean)\", \"\(w\) (Standard Deviation)\", and \"\(A\) (Area)\" must be known. Here, \"\(x_{c}\)\" is the calculated average of all values, the \"\(w\)\" is the value that represents the range of standard deviation based on the mean, \"\(A\)\" refers to the area of the graph and is required to calculate \(y_{c}\), which is the maximum value on the y-axis (\(x_{c}=y_{c}\)) [24]. In this study, the crack angle (crack propagation rate), according to the rail internal damage depth and crack length, was analyzed using the Gaussian probability density function. The analysis of crack length revealed that in the case of \(x_{c}\), the length in the C direction was 0.902 mm, whereas that in the L direction was 1.504 mm, indicating that the crack length in the L direction was greater than that in the C direction by a factor of 1.67, as shown in Figure 8a,b. Furthermore, in the case of \(w\), the C direction exhibited a deviation of \(\pm\)0.436 mm, whereas the L direction exhibited a deviation of \(\pm\)0.835 mm. The crack length in the L direction was 1.92 times greater than that in the C direction. The analysis of crack depth revealed that, in the case of \(x_{c}\), the depths of the crack were 0.481 and 0.358 mm in the C and L directions, respectively, as shown in Figure 8c,d. The crack depth in the C direction was determined to be greater than that in the L direction by a factor of 1.34. Furthermore, in the case of \(w\), the length in the C direction exhibited a deviation of \(\pm\)0.227 mm, whereas that in the L direction exhibited a deviation of \(\pm\)0.206 mm. It was observed that the crack depth in the C direction was greater than that in the L direction by a factor of 1.10. The analysis of the crack angle (crack propagation rate) revealed that, in the case of \(x_{c}\), the crack angle in the C direction was 25.97\({}^{\circ}\), while that in the L direction was 6.76\({}^{\circ}\) mm. It was determined that the fracture angle in the C direction was larger than in the L direction by a factor of 3.84. Furthermore, in the case of \(w\), the crack angle in the C direction exhibited a \(\pm\)17.43\({}^{\circ}\) deviation, while that in the L direction exhibited a \(\pm\)0.206 mm deviation. The crack angle (crack propagation rate) in the C direction was higher than that in the L direction by a factor of 1.10. The characteristics of rail internal defects manifested as damages of various lengths within the range of a significant standard deviation. As shown in Table 1, An analysis revealed that the average value (\(x_{c}\)) of the damage depth in the C direction was higher than the damage in the L direction. Furthermore, the crack length in the L direction exhibited a relatively large average and standard deviation compared with the crack length in the C direction. It was observed that the large distribution of standard deviation, which is relative to the mean value, indicates the occurrence of significant rail internal defects that have progressed beyond the average level. In terms of the surface damage in the L direction,  
Figure 7: SEM test results: (**a**) C direction (All); (**b**) L direction (All); (**c**) C direction(3D); (**d**) L direction (3D); (**e**) C direction(2D); (**f**) L direction (2D). the defects were observed during curved track conditions. Therefore, it was analyzed that rail surface maintenance should be performed periodically to address the damage in the L direction, which occurs more frequently than the damage on straight tracks. The characteristics of damage length, depth, and crack growth rate were determined according to this probability distribution. Furthermore, depending on the direction of rail surface damage that occurs in urban railways, the size of the defect leading to the interior varies and appropriate maintenance is necessary to manage this.