# A Methodology to Model the Rain and Fog Effect on the Performance of Automotive LiDAR Sensors
## 6 Conclusions  
In this work, we introduce a novel approach to model the rain and fog effect in the virtual LiDAR sensor with high fidelity. The presented approach allows for the rain and fog effect simulation in the LiDAR model on the time domain and point cloud level. Furthermore, validating the LiDAR sensor model on the time domain enabled us to benchmark the LiDAR signal attenuation \(\sigma_{ext}\) and SNR. The results show that the virtual and real LiDAR sensor signal attenuation \(\sigma_{ext,rain}\) and SNR match well if both simulated and real rain distributions are the same; for instance, the MAPE of the signal attenuation for the NIED rain distribution is 16.2%, while the MAPE for the Marshall-Palmer rain distribution is 31.6%. In addition, the MAPE of the SNR for the NIED rain distribution is 4.3% and 9.1% for the Marshall-Palmer rain distribution model. Therefore, the simulation and real measurement results will show a good correlation if the simulated and real rain distribution are the same.  
The results also show an increase in the LiDAR signal attenuation \(\sigma_{ext, fog}\) caused by fog droplets, while the visibility distance \(V\) decreases; for instance, the LiDAR signal attenuation increases to 9.2 dB with a visibility distance of 50 m. Furthermore, the simulation and the real measurements show good agreement for the signal attenuation and SNR due to fog droplets. The MAPE for the signal attenuation is about 13.9% and 15.7% for SNR due to fog droplets.  
Figure 17: The distance error \(d_{error}\) of the LiDAR sensor for the real and virtual 3%-reflective Lambertian plate with different visibility distances \(V\). The distance error \(d_{error}\) increases with the decrease in visibility distances. The simulation and real measurements show a good correlation.  
To validate the modeling of the effect of rain and fog at the point cloud level, we have introduced three KPIs: DR, FDR, and distance error \(d_{error}\). The results show that with the increasing rain rate \(r_{rate}\), DR decreases, while both the real and virtual LiDAR sensors could detect the 3%-reflective target at 20 m with a rain rate of 98 mm/h. On the other hand, both FDR and distance error increase with an increase in the rain rate. Moreover, the simulation and the real measurements show a good correlation; for instance, the MAPE for DR is 2.1%, and the MAPE for FDR is 14.7%.  
DR significantly decreases in fog with a decrease in the visibility distance because the density of fog droplets is \(10^{3}\) times higher than the density of raindrops in 1 m\({}^{3}\) of air. As a consequence, LiDAR rays cannot easily penetrate through fog. FDR also increases with the decrease in visibility distances \(V\), but the distance error is less than 2 cm. As a fog droplet is several magnitudes smaller than a raindrop, LiDAR rays become less misaligned by the scattering from fog droplets as opposed to the scattering from raindrops. However, the simulation and the real measurements correlate well for these KPIs. For example, the MAPE for DR is 7.5%, while MAPE for FDR is 48.2%. It should be noted that the backscattering from the fog droplets is a random process, and it is, therefore, impossible to model the exact behavior of real-world fog in the simulation. Furthermore, we have not measured nor modeled the real fog distribution due to limited resources. This could be another reason for the mismatch between the simulation and the real measurements. For the LiDAR sensor, it is easier to detect the low-reflective target easily in heavy rain, but it is very challenging for the LiDAR sensor to detect the target reliably once the visibility distance drops below 100 m.