## 2 Methodology of Data Acquisition
### Data Acquisition of Vertical Ballast Stiffness  
In this study, an onboard technique was utilized to acquire the rail displacements and thus track stiffness. As illustrated in Figure 3, the data collection process involves attaching a solid beam to the bogie of the wagon, with a laser and camera set positioned at the ends of the beam's arms [22; 23; 51]. This rail track measurement system comprises three interconnected units: the sensor unit, the processing unit, and the central unit. These \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Description** & **Value** & **Unit** \\ \hline P & Axle load & 22.5 & Tons \\ \hline K\({}_{\text{P}}\) & Stiffness of fastening system & 400 & kN/mm \\ \hline ls & Sleeper length & 2.6 & m \\ \hline l\({}_{\text{e}}\) & Effective length of sleeper & 1.1 & m \\ \hline h\({}_{\text{b}}\) & Ballast and sub-ballast depth & 0.5 & m \\ \hline \(\alpha\) & Ballast friction angle & 35 & degree \\ \hline E\({}_{\text{f}}\) & Subgrade elastic modulus & 42.7 & MPa \\ \hline \end{tabular}
\end{table} 
Table 1: The characteristics of the reference railway track. Figure 2: A view of the geometric track recording car (EM120). components are securely attached to the railcar body via a girder beam. The sensor unit collects critical data related to rail behavior, including a camera for visual information, lasers for precise measurements, and an incremental rotary encoder. The central unit, which includes a microcontroller, battery, and GPS receiver, manages data flow and communication. Cabling connects the central unit to the sensor and processing units. A 2D laser type is used because it can scan a line across the rail, emitting light at a 60-degree angle relative to the horizontal. The camera captures images of the rail surface and stores them in the processing unit. Camera specifications include high resolution (e.g., Full HD or 4K) and a wide-angle lens (e.g., 140 degrees) to cover the track area. The camera regulation with 720p full HD image captures clear video footage. The processing unit, housed in a laptop, calculates rail displacement using image processing techniques. Known laser light angles and distances allow for the accurate determination of the rail surface height relative to the sensor. This integrated system enables the continuous monitoring of rail conditions, contributing to safety and maintenance efforts. The described setup allows for precise measurements by capturing the deflection and deformation of the track, thereby providing accurate assessments of its stiffness. The installed sensors record the distance between the two lasers, with variations occurring as the sensors' distance from the rail surface fluctuates. These changes allow for the establishment of geometric relationships between the recorded parameters. Integrating these relationships makes it possible to calculate the distance between the lasers and the rail surface. This method ensures the precise tracking of the rail's surface profile, enabling accurate measurements essential for determining track stiffness and overall rail integrity. A schematic of geometric relationships that lead to computing track deflection is shown in Figure 4. As depicted in this figure, the height of the measuring device (\(h\)) from the rail surface can be geometrically extracted using the following equations [23; 51]:  
\[h= \begin{cases}tan\theta_{1}(l_{1}+L_{1})\\ tan\theta_{2}(l_{2}+L_{2})\end{cases}  \tag{1}\]  
Figure 3: Data acquisition of track stiffness: (**a**) recording car; (**b**) laser/camera system installed to bogie frame. where, in the above relationships, \(L_{1}\) and \(L_{2}\) represent the horizontal distances of the lasers from the camera, \(\theta_{1}\) and \(\theta_{2}\) are the angles between lasers and the horizon, \(l_{1}\) and \(l_{2}\) denote the horizontal distances between the center of the camera, the points where the lasers intersect with the rail, and \(h\) is the vertical distance between the camera/lasers and the rail surface.