# A Methodology to Model the Rain and Fog Effect on the Performance of Automotive LiDAR Sensors
## 5 Results
### Validation of the Rain Effect Modeling on the Time Domain Level  
The primary reason for verifying the LiDAR model on the time domain is to ensure that the link budget, detector, and circuit modules work as intended. Furthermore, comparing the time domain signals (TDS) establishes the association between measured and modeled noise and the amplitude levels because it is difficult to compare the simulated and measured noise at the point cloud level. It is, therefore, convenient to quantify the LiDAR signal attenuation \(\sigma_{ext}\) and the decrease in the SNR due to the rain and fog droplets on the time domain level. A 3%- and a 10%-reflective Lambertian plate were placed in front of the sensor at 20 m in the rain rate \(r_{rate}\) of 16 mm/h, 32 mm/h, 66 mm/h, and 98 mm/h. An exemplary scenario for a 3% Lambertian plate is shown in Figure 6.  
Figure 6: (a) Real setup to validate the time domain and point cloud data. (b) Static simulation scene to validate the time domain and point cloud data. The 3%-reflective target with an area of 1.3 m \(\times\) 1.3 m was placed in front of the sensor at different distances. The actual and the simulated sensor and target coordinates are the same. The ground truth distance \(d_{GT}\) is calculated from the sensor’s origin to the target’s center. Both the simulated and the real measured TDS obtained from the 10%-reflective plate are shown in Figure 7, both with and without rain at 16 mm/h. There is a good match between the target peaks and the amplitude level of the backscattered raindrops, but still there is a slight difference in the amplitude level of the backscattered raindrops because real-world rainfall is a random process, and it is impossible to replicate the exact behavior of real-world rain in the simulation, yet the simulation and the real measurements agree well with each other. Figure 8 shows the LiDAR signal attenuation \(\sigma_{ext,rain}\) at different rain rates \(r_{rates}\). The results show that the LiDAR signal attenuation significantly depends on the rain distribution. Therefore, the simulation and real measurements will match well if the simulated rain distribution is close to the real distribution. Figure 9 shows the SNR of the simulated and real measured signals in different rain rates \(r_{rate}\). The SNR can be calculated as:  
\[SNR=20\,log_{10}\Big{(}\frac{\mu}{\sigma}\Big{)}, \tag{36}\]  
where \(\mu\) is the mean or expected signal value and \(\sigma\) is the standard deviation of noise [44]. The result shows that at the lower rain rates \(r_{rate}\), the SNR of the simulated and measured signals match well, but just as the rain rate \(r_{rate}\) increases, the SNR mismatch also increases, especially for the simulation results with the Marshall-Palmer rain distribution. Therefore, the simulation and real measurements for SNR will also match well if the simulated rain distribution is close to the real distribution. The backscattering from the raindrops is a random process, and it is not always possible to generate the same backscattering amplitude from the virtual raindrops as from real-world raindrops, and this is also a reason for the mismatch between the real and simulated LiDAR signals' SNR. To quantify the difference between the simulated and real measured signals' attenuation \(\sigma_{ext,rain}\) and SNR, we use the mean absolute percentage error (MAPE) metric:  
\[MAPE=\frac{1}{n}\sum_{i=1}^{n}\ |\ \frac{y_{i}-x_{i}}{y_{i}}\ |, \tag{37}\]  
where \(y_{i}\) is the measured value, the simulated value is denoted by \(x_{i}\), and \(n\) shows the total number of data points [45]. The MAPE of the signal attenuation \(\sigma_{ext}\) for the NIED rain distribution is 16.2% and 31.6% for the Marshall-Palmer rain distribution model. Moreover, the MAPE of the SNR for the NIED rain distribution is 4.3% and 9.1% for the Marshall  
Figure 7: (**a**) LiDAR FMU and real measured TDS comparison obtained from the surface of a 10%-reflective Lambertian plate at 20 m without rain. The target peaks and noise levels match well.