# A Methodology to Model the Rain and Fog Effect on the Performance of Automotive LiDAR Sensors
## 5 Results
### Validation of the Fog Effect Modeling on the Time Domain Level  
To verify the modeling of the fog effect, we have placed the 3%-reflective Lambertian plate at 15.3 m in front of the sensor, as shown in Figure 11. Both the simulated and the real measured signals that reflect from the surface of the 3%-reflective Lambertian plate, with and without fog, are shown in Figure 12.  
The results show that the simulated and the real measured target peaks and noise levels match well without and with fog. The peak location of backscattering from the fog droplets in the simulation and the real measurement varies because it is a random process, and replicating the exact real-world fog behavior in the simulation is challenging. For instance, the fog drop size changes during the fog life cycle [47; 48], affecting the LiDAR sensor performance. Measuring and replicating the change in the fog droplet size during the fog life cycle in the virtual environment is quite challenging. Moreover, in a virtual environment, it is easy to maintain constant visibility to a certain distance. It was a challenge, however, to retain constant visibility to a certain distance, both in the real world and in laboratory-controlled conditions, due to the limitations of the measurement instruments and the fog generation setup. Figures 13 and 14 show the simulated and the real LiDAR signal attenuation \(\sigma_{ext, fog}\) and SNR with different visibility distances \(V\). The result shows that the LiDAR signal attenuation \(\sigma_{ext, fog}\) caused by fog droplets increases while the visibility distances \(V\) decreases; for instance, the LiDAR signal attenuation increases to 9.2 dB with a visibility distance of 50 m. The result also shows that the dense fog increases the mismatch of the simulated and the real measured signals for the attenuation and SNR. As mentioned above, the possible reason behind these deviations is the non-constant distribution of fog in the measurement area and the random backscattering process from fog droplets.  
Figure 11: (**a**) The real setup for the fog measurement. (**b**) The static simulation scene for the validation of the fog effect. The 3%-reflective Lambertian plate was placed at a 15.3 m distance. The real and virtual LiDAR sensor and targets coordinates are the same.  
Figure 10: The exemplary visualization of simulated and real point clouds obtained in 32 mm/h rain rate \(r_{rate}\).  
Figure 14: The SNR of the simulated and real measured signals with different visibility distances \(V\). The simulation and real measurement results match well at the higher visibility distances, but the mismatch between the simulated and real measured SNR increases as the visibility distance decreases.  
Figure 12: (a) LiDAR FMU and real measured TDS obtained from the surface of a 3%-reflective plate placed at 15.3 m without fog. The target peaks and noise levels match well. (b) LiDAR FMU and real measured TDS obtained from the surface of a 3%-reflective plate at 15.3 m with a fog visibility \(V\) of 140 m. It should be noted that the relative distance is calculated from the internal reflection to the target peaks.  
Figure 13: The real and virtual LiDAR signals attenuation \(\sigma_{ext, fog}\) due to the different visibility distances \(V\). The simulation and real measurement results match well at the higher visibility distances. However, as the visibility distance decreases due to fog, the simulated and real measured signal attenuation mismatch increases.  
We use the MAPE metric as given in Equation (37) to quantify the difference between the simulated and the real measured results for fog. The MAPE for the signal attenuation due to fog \(\sigma_{ext, fog}\) is 13.9% and 15.7% for SNR.