## 4 Results and Discussion
### Correlation Analyses through Frequency-Based Technique  
The PSD of vertical rail deflection (_y-PSD_) and longitudinal levels are compared together in Figure 10. As shown in this figure, the vertical rail deflection and the longitudinal levels of the track are closely correlated, with their trends aligning in the frequency domain. The Pearson correlation matrix of the vertical rail displacements and longitudinal levels (for both left and right rails), which were extracted by PSD analyses, are presented in Table 2. Based on the correlation results presented in this table, the correlation between the vertical rail deflection and longitudinal level is 0.668 for the left rail and 0.669 for the right rail. These values indicate a relatively meaningful relationship between the PSD of the data for both rails. These obtained results are synchronized with the outcomes of [PERSON] et al. [49]. Their concluded results show that the Pearson correlation between vertical rail deflections and longitudinal levels are 0.655 and 0.749 for the left and right rails, respectively. Additionally, there is a considerable correlation in the power spectrum density, particularly within the 1-4 rad/m wave number range corresponding to wavelengths of 1.5 and 3 m. This meaningful correlation within specific wave number ranges highlights significant patterns in how vertical rail deflection interacts with longitudinal track alignment. These findings are in agreement with the study of [PERSON] et al. [29; 30] which investigated the contributions of longitudinal track unevenness and track stiffness variations. In this study, track unevenness and stiffness were measured in the Furet railway track of Sweden, and their contribution was evaluated using the PSD function. Their obtained results showed that there is very good agreement between track unevenness and stiffness for wave numbers up to 5 rad/m. Another remark found from the correlation analysis is that the left and right rails are not completely aligned together. Rail misalignments can indicate vertical and horizontal track irregularities but can also be a consequence of a designed cart of railway tracks. So, considering the alignment or misalignment between rails is crucial for several reasons: safety, comfort and passenger experience, infrastructure deterioration, maintenance efficiency, etc. In Table 2, the Pearson correlation between the left and right rail deflections extracted is about 0.90. This shows that the left and right have not completely aligned together, and this could be due to the twist and superelevation of track or rail irregularities. In straight parts of the track without curves, these misalignments can be described as consequences of track settlement and/or short-wave rail surface irregularities. Figure 9: The calculated vertical ballast stiffness along the reference railway line. The relationship between vertical ballast stiffness and the longitudinal level was also analyzed by calculating the PSD of the data, as illustrated in Figure 11. This figure seeks to uncover the correlation between vertical ballast stiffness and the longitudinal levels of the track. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **LLL-PSD** & **LLR-PSD** & **y-PSD** \\ \hline LLL-PSD & 1 & 0.902 & 0.668 \\ \hline LLR-PSD & 0.902 & 1 & 0.669 \\ \hline y-PSD & 0.668 & 0.669 & 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Pearson correlation matrix of vertical rail deflections (y) and longitudinal levels (LLL, LLR) extracted by PSD analyses. Figure 10: The PSD of vertical rail deflections (y) and longitudinal levels: **(a)** LLL, **(b)** LLR. In Table 3, the Pearson correlation matrix of the vertical ballast stiffness (\(k_{b}\)) and track longitudinal levels (for both left and right rails) is extracted using frequency analyses. Based on the correlation results shown in this table, the correlation between vertical ballast stiffness and the longitudinal level is 0.672 for the left rail and 0.674 for the right rail.