## 2 Monte Carlo Point Probability Model and the Modeling of Light Sources
### Monte Carlo Point Probability Method  
For the simulation of photon atmospheric transport, we used the Rayleigh phase function (Equation (5)) and generalized Henyey-Greenstein function (Equation (6)) to model the scattering caused by air molecules and aerosols, respectively, according to References [34; 35]. \[P^{mic}(\mu)=\frac{1-g^{2}}{4\pi}\Bigg{[}\frac{1}{\left(1+g^{2}-2g\mu\right)^ {1.5}}+f\frac{3\mu^{2}-1}{2(1+g^{2})^{1.5}}\Bigg{]}, \tag{7}\]  
\[P^{nay}(\mu)=\frac{3\Big{[}1+3\gamma+(1-\gamma)\mu^{2}\Big{]}}{16\pi(1+2\gamma) }, \tag{8}\]  
where \(\mu=\cos\theta_{s}\), \(\gamma=0.17\), \(g=0.72\), and \(f=0.5\). The total scattering phase function can be written as a weighted sum of the Rayleigh scattering (Equation (8)) and the Mie scattering phase function (Equation (7)). \[P(\mu)=\frac{k_{sr}}{k_{s}}p^{nay}(\mu)+\frac{k_{sm}}{k_{s}}p^{mie}(\mu), \tag{9}\]  
where \(k_{sr}\) and \(k_{sm}\) are the Rayleigh and Mie scattering coefficients, respectively. The total scattering coefficient \(k_{s}=k_{sr}+k_{sm}\). As for the ultraviolet light at wavelength 266 nm, \(k_{sm}=0.25\) km\({}^{-1}\), \(k_{sr}=0.24\) km\({}^{-1}\) and the absorption coefficient can be set as \(k_{a}=0.74\) km\({}^{-1}\)[36]. The algorithm of the MC point probability method includes four steps: photon emission, propagation, scattering, and detection. **Step 1. Emissions:** For the scattering, the propagation direction vectors at the \(n\)th scattering position can be expressed as \(\left(s_{m},s_{yn},s_{zm}\right)\), and the acceptance probability is \(P_{reci}\). The acquisition probability of all photons is assumed to be 1 at the beginning. Factors such as atmospheric absorption and the limit of the receiver's FOV (field of view) can both make it much less than 1. The photon is initially emitted from point T and the coordinate is T\((x_{0},y_{0},z_{0})\). The direction cosines of a photon can be described by two angles \(\theta_{\text{si}}\) and \(\phi_{\text{si}}\) in a 3D Cartesian coordinate system. The initial direction cosines can be derived by \(\theta_{s0}\) and \(\phi_{s0}\). **Step 2. Propagation:** The transmission distance of a photon needs to be determined during the simulation. According to [PERSON]'s Law [27], the random propagation distance of photons between each scattering point can be set as  
\[d=-\frac{\ln\xi_{d}}{k_{s}}, \tag{10}\]  
Figure 3: Plot of the different light sources’ normalized light intensity \(I(\theta)/I_{0}\) vs. off-axis angle \(\theta\). where \(\xi_{d}\) is a random number that satisfies uniform distribution in the interval [0,1]. Thus, the coordinates of the \(n\)th scattering point can be obtained using (Equation (11)). \[\left\{ \begin{array}{l}x_{n}=x_{0}+\sum\limits_{i=0}^{n-1}s_{xi}d_{i}\\ y_{n}=y_{0}+\sum\limits_{i=0}^{n-1}s_{yi}d_{i}\\ z_{n}=z_{0}+\sum\limits_{i=0}^{n-1}s_{zi}d_{i}\end{array} \right., \tag{11}\]  
**Step 3. Scattering:** According to Reference [37], the direction cosines \(\left(s_{x,i+1},s_{y,i+1},s_{z,i+1}\right)\) at the \(i\)th scattering point after scattering can be derived by \(\left(s_{x,i},s_{y,i},s_{z,i}\right)\) and the spatial scattering angle, which is derived by \(\left(\theta_{si},\phi_{si}\right)\). \[ \begin{array}{l}s_{x,i+1}=\frac{\sin\theta_{si}}{\sqrt{1-s_{zi}^{2}}}\left( s_{xi}s_{zi}\cos\phi_{si}-s_{yi}\sin\phi_{si}\right)+s_{xi}\cos\theta_{si},\\ s_{y,i+1}=\frac{\sin\theta_{si}}{\sqrt{1-s_{zi}^{2}}}\left(s_{yi}s_{zi}\cos \phi_{si}+s_{xi}\sin\phi_{si}\right)+s_{yi}\cos\theta_{si},\\ s_{z,i+1}=-\sin\theta_{si}\cos\phi_{si}\sqrt{1-s_{zi}^{2}}+s_{zi}\cos\theta_{ si}.\end{array}  \tag{12}\]  
**Step 4. Photon detection:** The probability of a photon being successfully received after passing through \(S_{n}\) is \(P_{2n}=\exp(-k_{e}|S_{n}-R|).A_{r}\) refers to the effective area of the detector. We set \(A_{r}=\pi(d/2)^{2}\) and \(d=1.5 cm\) according to Reference [38]. \(|S_{n}-R|^{2}\) is the distance between the \(n\)th scattering position \(S_{n}\) and the receiver \(R\).