## 2 Materials and Methods
### Data Preparation and Processing
#### 2.4.3 Curve Fitting Methodology
In this study, since LIDAR measurements were conducted in open fields of view, the atmospheric attenuation coefficient used in Equation (1) at a wavelength of 90 nm was set to \(\gamma\) = 0.01 m\({}^{-1}\) as determined by the Kim-Kruse model [29]. This coefficient represents the propagation and absorption effects of laser light in the atmosphere under open-air conditions. Additionally, in devices such as the VLS-128, the spectral transmittance of the detection optics is generally close to 1 at a wavelength of 903 nm. This allows the LIDAR-emitted laser light to be detected by the sensor with maximum efficiency, thereby enhancing the optical performance of the system. Therefore, the \(\mu_{0}\) value was taken as 1 in this study. Furthermore, an analysis of the relevant LIDAR datasheet was conducted to determine the receiver aperture area. It was found that the VLS-128 features a 12.4 cm\({}^{2}\) receiver aperture area for collecting laser reflections [30]. The receiver aperture area is a crucial parameter that directly influences the amount of laser power collected by the detector, thereby contributing to the overall sensitivity of the system. Accordingly, this parameter was incorporated into the analysis to ensure accurate modeling of LIDAR measurements. The LIDAR system used in the experimental process operates under specific experimental conditions and technical limitations, as stated in the study conducted by [PERSON] et al. Specifically, the laser power was set to 0.39 mW in compliance with the Class 1 laser safety standards of the respective LIDAR model. In the experimental setup, one side of the laser beam was completely blocked by a black panel, ensuring no reflections from this area. As a result, only 96 channels were considered active instead of the full 128 channels of the LIDAR. This adjustment means that reflections from the blocked region were excluded from the intensity data calculations. In this analysis, the curve fitting process applied to the LIDAR data considered the sensitivity of vehicles to factors such as their distance from and angle to the sensor. Specifically, the vehicle angle relative to the sensor was assumed to be between 0\({}^{\circ}\) and 5\({}^{\circ}\) during this study. This assumption was made because vehicles detected by LIDAR under normal traffic conditions typically fall within a narrow angular range. Furthermore, this study focused on LIDAR sensors' ability to analyze vehicles passing in close proximity, particularly at distances of 2.5 m and 5 m. However, discrepancies between theoretical and experimental results were observed due to laser intensity attenuation, sensor optical gain settings, and environmental factors. Consequently, separate curve fitting processes were conducted for 2.5 m and 5 m distances. To enhance the representation of data at different distances, two distinct transformation equations were derived using a third-degree polynomial (poly3) function, as described in Equation (4). These equations were constructed by randomly selecting different reflectivity coefficients and angle values for each distance. This approach enabled the modeling of different scenarios and improved the accuracy of the fitting process. \[\text{Poly(x)}=\text{p}_{1}\text{x}^{3}+\text{p}_{2}\text{x}^{2}+\text{p}_{3} \text{x}+\text{p}_{4} \tag{4}\]First, the curve fitting process was conducted for a distance of 2.5 m. Figure 9 illustrates the curve fitting plots for 2.5 m and 5 m, depicting the relationship between Equation (1) and the intensity values predicted by the GPR model. In the figure, the red curve represents the third-degree polynomial (poly3) function used for curve fitting, while the blue dots correspond to the theoretical and experimental intensity values.