# 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 Point Cloud Level  
We have introduced three KPIs to validate the rain effect modeling on the point cloud level: the LiDAR detecting rate (DR), the false detection rate (FDR), and the distance error \(d_{error}\). 1. The DR is defined as the ratio between the number of returns obtained from both real and simulated objects of interest (OOI) in rainy (\(\#\,returns\,OOI_{rain,\,sim/real}\)) and dry (\(\#\,returns\,OOI_{try,\,sim/real}\)) conditions. It can be written as: \[DR=\frac{\#\,returns\,OOI_{rain,\,sim/real}}{\#\,returns\,OOI_{dry,\,sim/ real}}.\] (38) It should be noted that the number of points obtained from OOI in rainy and dry conditions are the mean over all measurements of the same scenario. Figure 8: Simulated and real LiDAR signals attenuation \(\sigma_{ext,rain}\) due to different rain rates \(r_{rate}\). The simulation results with Marshall–Palmer, NIED rain distribution, and real measurements match very well at lower rain rates. However, as the rain rate increases, the simulation and real measurement signal attenuation mismatch also increases, especially for the simulation results with the Marshall–Palmer rain distribution. Figure 9: The SNR of the simulated and real measured signals in different rain rates \(r_{rate}\). The simulation and real measurement results match very well at lower rain rates, but the mismatch between the simulated and real SNR increases as the rain rate increases, especially for the simulation results with the Marshall–Palmer rain distribution. 2. The FDR of the LiDAR sensor in rainy conditions can be written as: \[FDR=\frac{\#\,returns_{rain,\,sim/real}-\#\,returns\,OOI_{rain,\,sim/real}}{\#\,returns\,OOI_{dry,\,sim/real}},\] (39) where \(\#\,returns_{rain,\,sim/real}\) is the total number of reflections from the sensor minimum detection range to the simulated and real OOI, and it does not contain any reflection from the OOI's surroundings. \(\#\,returns\,OOI_{rain,\,sim/real}\) and \(\#\,returns\,OOI_{dry,\,sim/real}\) depict the LiDAR returns from the surface of the simulated and real OOI in rainy and dry conditions. It should be noted that the number of LiDAR reflections obtained under rainy and dry conditions are the mean over all measurements of the same scenario. 3. The distance error \(d_{error}\) of the point cloud received from OOI in rainy and dry conditions, both simulated and real, can be written as: \[d_{error}=d_{GT}-d_{mean,\,dry/rain,\,sim/real},\] (40) where the ground truth distance is denoted by \(d_{GT}\) and \(d_{mean,\,dry/rain,\,sim/real}\) is the mean distance of reflections received from the surface of the simulated and the real OOI in rainy and dry conditions. The ground truth distance \(d_{GT}\) is calculated from the sensor's origin to the target's center, and it can be written as: \[d_{GT}=\sqrt{(x_{t}-xs)^{2}+(y_{t}-y_{s})^{2}+(z_{t}-z_{s})^{2}},\] (41) where the target's \(x\), \(y\), and \(z\) coordinates are denoted by subscript \(t\) and the sensors by \(s\)[46]. The OSI ground truth interface _osi3::GroundTruth_ is used to retrieve the sensor origin and target center position in 3D coordinates. Table 3 gives the DR of the real and virtual LiDAR sensor in different rain conditions for a 3%-reflective Lambertian plate. The results show that up to 20 m, the LiDAR sensor can reliably detect a very low-reflective target with a rain rate of 98 mm/h. It should be noted that these results were obtained in a rain facility area where the rain was homogenously distributed. For the benchmarking of the DR, we have filtered the point clouds from the edges of the Lambertian plate, meaning that the effective area of the plate becomes 1.1 m \(\times\) 1.1 m. The simulation and the real measurements show a good correlation with each other. The MAPE for the DR is 2.1%.