## 2 System Model
### MC Simulation Method  
We use the MC simulation method to simulate the propagation trajectory of photons through complex seawater environments, and evaluate the transmission characteristics by counting the state information of photons at the receiving terminal. The flow chart of the MC simulation method is shown in Figure 1.  
The LED light source used in our UOWC system model can be defined by the generalized Lambertian light sources. The distribution of radiant intensity \(\Psi(\theta_{0})\) can be expressed as [23]  
\[\psi(\theta_{0})=\frac{1+m}{2\pi}\cos^{m}(\theta_{0}) \tag{2}\]  
\[m=-\ln 2/\ln[\cos(\theta_{1/2}/2)] \tag{3}\]  
where \(\theta_{0}\) is the zenith angle, \(m\) is the radiation mode coefficient of the light source, and \(\theta_{1/2}\) is the semi-angle at half-power of the light source. \begin{table}
\begin{tabular}{c c c c c} \hline
**Typical Seawater** & \(a\) **(m\({}^{-1}\))** & \(b\) **(m\({}^{-1}\))** & \(c\) **(m\({}^{-1}\))** & **Albedo** \\ \hline clean seawater & 0.114 & 0.037 & 0.151 & 0.25 \\ coastal seawater & 0.179 & 0.219 & 0.398 & 0.55 \\ harbor seawater & 0.366 & 1.824 & 2.190 & 0.83 \\ \hline \end{tabular}
\end{table} 
Table 1: Attenuation coefficient of different seawater environments. The step size \(s\) of a photon is the distance that the photon moves in the seawater before scattering and absorption. It can be expressed as [24]  
\[s=-\frac{\ln(\eta)}{c} \tag{4}\]  
where \(\eta\) is a uniform random number between 0 and 1. The emitted photons are affected by the seawater channel during underwater transmission, where the absorption effect reduces the weight of the photon, and the scattering effect changes the propagation direction. The weight \(W_{pos}\) of the photon after the scattering can be expressed as  
\[W_{pos}=W_{pre}\cdot r \tag{5}\]  
where \(W_{pre}\) is the weight before photon scattering. The albedo \(r\) describes the weight loss of photons after the scattering and is given by \(r=b/c\). After the photon weight is determined, we need to further obtain the scattering angle of the photon to evaluate its movement direction. And the probability distribution of photon propagation directions usually follows the volume scattering function (VSF). It is difficult to measure the VSF, and the most widely cited scattering measurement is the Petzold average phase function. Moreover, several types of analytical phase functions, such as the Henyey-Greenstein (HG), two-term Henyey-Greenstein (TTHG), and Fournier-Forand (FF), have been proposed. The variation of different scattering phase functions is shown in Figure 2.  
Figure 1: MC simulation method flow chart. The HG phase function is a convenient and simple way to achieve an easy computation of the RTE. However, the phase function has a large difference compared with the Petzold average when the scattering angle approaches 0\({}^{\circ}\) and 180\({}^{\circ}\). The TTHG function also underestimates Petzold's measurement of scattering angle < 1\({}^{\circ}\). The FF function is used to fit the measurement results. The root mean square value of the fitting results and the experimental results is 15.3, which is closer to the real seawater scattering phase function. Therefore, the FF function can be used as the seawater scattering phase function to generate the zenith angle and azimuth angle of photons. The zenith angle using the FF scattering phase function \(\beta_{FF}(\theta)\) can be obtained by [9]  
\[2\pi\int_{0}^{\theta}\beta_{FF}(\theta^{\prime})\sin\theta^{\prime}d\theta^{ \prime}=\varepsilon \tag{6}\]  
\[\beta_{FF}(\theta)=\beta_{0}(\theta)+\beta_{0}(\pi)(3\cos^{2}\theta-1)/4 \tag{7}\]  
\[\beta_{0}(\theta)=\frac{1}{4\pi}\frac{1}{\left(1-\delta\right)^{2}\delta^{ \circ}}\times\left\{\left[v(1-\delta)-(1-\delta^{v})\right]+\frac{4}{u^{2}}[ \delta(1-\delta^{v})-v(1-\delta)\right\} \tag{8}\]  
\[\delta=\frac{u^{2}}{3(n-1)^{2}} \tag{9}\]  
\[v=\frac{(3-u)}{2} \tag{10}\]  
\[s=-\frac{\ln(\eta)}{c} \tag{4}\]0  
where \(u\) is the exponential term and \(n\) is the refractive index.