# Deep Learning (Fast R-CNN)-Based Evaluation of Rail Surface Defects
## 4 Analysis and Discussion
### Gaussian Probability Density Function Analysis According to Crack Depth  
This study analyzed the correlation between crack propagation rate and crack depth based on the Gaussian probability density function. The correlation between crack depth and crack propagation rate, according to the direction of damage, is illustrated in Figures 9 and 10, as well as in Tables 2 and 3. The analysis of crack angle (crack propagation rate) according to the depth of rail sur-face cracks revealed that the crack propagation rate and standard deviation increased and then gradually decreased as the crack depth increased, as shown in Figure 9. It was further observed that the fracture propagation rate exhibited the highest standard deviation for crack depths, ranging from 0.50 mm to 0.75 mm. Furthermore, it was inferred that, at crack depths larger than 0.75 mm, there was a slight attenuation in both the standard deviation and range of the crack propagation rate. Therefore, the range 0.50-0.75 mm was found to be the critical level in terms of fracture propagation rate. As shown in Table 2, it was observed that the relative difference of \(x_{c}\) in the range 0.25-0.50 mm increased by approximately 17% compared with that in the range 0-0.25 mm, and the relative difference of the standard deviation increased by approximately 49.5%. The relative difference of \(x_{c}\) increased by approximately 42.74% in the range 0.50-0.75 mm compared with 0.25-0.50 mm, and the relative difference of the standard deviation increased by approximately 41.1%. The relative difference of \(x_{c}\) decreased by approximately 17.53%  
Figure 10: Correlation analysis result between rail surface crack depth and crack propagation rate (L direction). Figure 9: Correlation analysis result between rail surface crack depth and crack propagation rate (C direction). in the range of 0.75 mm and above, whereas the relative difference of the standard deviation decreased by approximately 38.4%. The analysis of crack angle (crack propagation rate) according to the depth of rail sur-face cracks revealed that the average and range of crack propagation rate increase with the fracture depth. The range of fracture propagation rate was the highest in cases where the crack depth was equal to or greater than 0.75 mm. In contrast, it was observed that the standard deviation range of crack propagation rate for intervals 0-0.75 mm or higher decreased. As shown in Table 3, the relative difference of \(x_{c}\) in the range 0.25-0.50 mm increased by approximately 6.85% compared with that in 0-0.25 mm. Additionally, the relative difference of the standard deviation decreased by approximately 11.2%. The relative difference of \(x_{c}\) increased by approximately 3.28% in the range 0.50-0.75 mm compared with that in 0.25-0.50 mm, whereas the relative difference of the standard deviation decreased by approximately 4.5%. The relative difference of \(x_{c}\) increased by 30% in the range of 0.75 mm and above, whereas the relative difference of the standard deviation decreased by approximately 24.9%. The RCF damage observed in this study can be broadly classified into two types: C-direction damage and L-direction damage. Additionally, differences in defect characteristics depending on the damage direction can be identified using Gaussian probability density analysis. Therefore, as a result of the rail surface defect analysis using the Gaussian probability density function performed in this study, C-direction damage characteristics with a relatively high crack growth rate were analyzed. Through indoor testing, we determined that damage to the rail surface must be removed before reaching an internal crack depth of approximately 0.5 mm or more. The rail internal damage characteristics showed damage of various lengths within the standard deviation size.