# A Methodology to Model the Rain and Fog Effect on the Performance of Automotive LiDAR Sensors
## 4 Modeling of the Rain and Fog Effect in the Virtual LiDAR Sensor
### Fog Module  
There are two main approaches to model the effects of fog on the LiDAR signals: modeling via empirical formulas or modeling based on statistical distributions, as given in Section 4.2. All of these approaches have advantages and disadvantages. For instance, the LiDAR signal attenuation due to fog droplets can be modeled using empirical formulas introduced in [34; 35; 36], but it is not possible to model the backscattering from fog droplets. These approaches are easy to implement and computationally less expensive, while the statistical distributions that can be used to model the LiDAR signal attenuation and backscattering from the fog droplets are computationally expensive. In this work, we have therefore calculated the extinction coefficient \(\sigma_{ext, fog}\) as a function of visibility distance \(V\) using the empirical formula recommended by the International Commission on Illumination (CIE) given in [9], which can be written as:  
\[\sigma_{ext, fog}=\frac{3}{V}. \tag{22}\]  
Figure 5: The geometry of a LiDAR ray from near-field to range \(r_{1}\) considering beam divergence.  
The backscattering coefficient \(\beta_{back,\mathit{fog}}\) is calculated using the Mie scattering theory and the Deirmendjian gamma distribution, and it can be written as:  
\[N(D)=\frac{\gamma\rho b^{\frac{\theta+1}{\gamma}}}{\Gamma\Big{(}\frac{\alpha+1}{ \gamma}\Big{)}}\left(\frac{D}{2}\right)^{\alpha}e^{\left(-b\left(\frac{D}{2} \right)^{\gamma}\right)}, \tag{23}\]  
where \(\Gamma(\bullet)\) is the gamma function and \(\alpha\), \(\gamma\), \(b\), \(\rho\), and \(D\) are the parameters of \(N(D)\),  
\[b=\frac{\alpha}{\gamma(D_{\mathrm{C}}/2)^{\gamma}}, \tag{24}\]  
where \(D_{c}\) denotes the mode diameter of maximum frequency droplets [9; 12; 37]. The distribution parameters for typical environmental conditions are given in Table 2.  
The concentration of fog particles in the atmosphere is significantly higher than the number of rain particles, and their size \(D\) ranges from 0.1 \(\upmu\)m to 20 \(\upmu\)m [9; 31]. The Monte Carlo Simulation approach adopted in Section 4.2 would be computationally highly expensive if used to generate virtual fog. Unlike the rain, we have therefore used an analytical method based on the Deirmendjian gamma distribution to generate homogenously distributed fog particles. We have used Equation (8) to calculate the backscattered efficiency \(Q_{back}\). First, the virtual fog field is divided into range cells \(R_{cells}\) based on the LiDAR range resolution \(\Delta R\) to determine the backscattered coefficient \(\beta_{back,\mathit{fog}}\). In the next step, we will consider all of the fog droplets within the beam volume of LiDAR as given in Equation (21) at any range cell \(R_{cell}\) to determine the backscattered coefficient \(\beta_{back,\mathit{fog}}\) from fog droplets. This can be written as:  
\[\beta_{back,\mathit{fog}}=R_{cell}\frac{\tau}{4}\int_{D=0}^{\infty}D^{2}N(D)Q_ {back}\,dD, \tag{25}\]  
where \(N(D)\) is the fog distribution, \(D\) denotes the drop diameter, and \(Q_{back}\) is the backscattered efficiency [38].