## 2 Methodology of Data Acquisition
### Data Acquisition of Vertical Ballast Stiffness
The horizontal distance of the laser beams (\(d\)) is equal to the summation of \(l_{1}\) and \(l_{2}\) (\(d=l_{1}+l_{2}\)), so the vertical height of the measuring device can be derived from the below equation:  
\[d=h(cot\theta_{1}+cot\theta_{2})-(L_{1}+L_{2}) \tag{2}\]  
By determining the distance of laser beams (\(d\)) through image processing, the height of the measuring device (\(h\)) can be calculated according to Equation (2). If the measuring system is rigidly fixed to the railcar frame and wheelset, it can be assumed that the distance between the measuring device and wheel/rail contact point (\(H\)) remains constant as the railcar moves at low speed. Therefore, the relative rail deflection (\(y_{r}\)) can be determined as follows:  
\[y_{r}=H-h \tag{3}\]  
In summary, measuring the laser beam distance (\(d\)) provides the relative displacement between the railcar and railway track (\(h\)), which can then be used to calculate the relative rail displacement by assuming rigidity in the wheel/side frame system. Therefore, the recording of the described camera/laser system can be related to the relative track deflection, enabling the calculation of the track modulus using the Winkler model. In soft track conditions, the rail rises at the wheel/rail contact point, causing the laser tracks to converge. Conversely, in hard track conditions, the distance between the lasers increases. By identifying the vertical deflection of the rail, train axle load, the stiffness of the fastening system, the elastic modulus of the subgrade, the length of the sleeper, and the ballast layer depth, the stiffness of the ballast layer can be calculated using the following equations: [52; 53; 54]  
\[\frac{1}{k_{t}}=\frac{1}{k_{p}}+\frac{1}{k_{b}}+\frac{1}{k_{s}} \tag{4}\]  
\[k_{s}=l_{s}(l_{e}+2h_{b}lg\alpha)E_{f} \tag{5}\]  
\[E_{f}=c_{s}^{2}\rho \tag{6}\]  
where, in Equation (4), \(k_{t}\) is track stiffness, \(k_{p}\) is the stiffness of the railpad of the fastening system, \(k_{b}\) is ballast stiffness, and \(k_{s}\) is subgrade stiffness. In Equations (5) and (6), \(k_{s}\) is subgrade stiffness, \(l_{s}\) is the sleeper length, and \(l_{e}\) is the effective length of the sleeper, typically considered to be 0.6 times the sleeper length; \(h_{b}\) is the height of the ballast, \(E_{f}\) is the subgrade elastic modulus, \(c_{s}\) is the shear wave velocity, \(\alpha\) is the angle of stress distribution, and \(\rho\) is the density of the ballast material. The laser-based rail track inspection system, as described, offers precise measurements by capturing rail deflection and deformation. However, like any technology, it has limitations and challenges, and the following delves into some of these aspects:  
Figure 4: Geometric representation of track deflection measuring using laser/camera system. 1. Sensor Positioning and Stability: Maintaining consistent sensor positioning is critical. Vibrations, track irregularities, and mechanical wear can affect sensor alignment. If sensors deviate from their optimal position, measurement accuracy may suffer. 2. Surface Reflectivity: Rail surfaces vary in reflectivity due to factors like corrosion, dirt, and surface texture. Laser sensors may struggle with highly reflective or uneven surfaces, affecting data quality. 3. Environmental Conditions: Adverse weather (rain, snow, fog) can impact laser performance. Poor visibility affects accurate measurements, especially during increment weather. 4. Speed Dependency: The system's effectiveness depends on train speed (low speed is preferred). At high speeds, rapid data acquisition yields less and inequality measurements. 5. Geometric Defects: Some defects (e.g., cracks, squats) have longitudinal characteristics and so affect obtained results. Traditional 2D imaging struggles to detect such defects accurately. 6. Wheel/Side Frame Assumption: The system assumes the rigidity of the wheel/side frame system. Deviations from this assumption (e.g., wheel wear) may impact measurements. 7.