## 3 Underwater-Air Channel Model
### Sea Surface Channel  
In Section 2, two sea surface models, statistical and spatial, are introduced. To compute the interaction of laser rays with the sea surface, the wave front normal vector \(\vec{n}\) of the surface must first be determined. In the statistical model, the vector is derived from the PDF of sea surface slopes and the assumption of isotropy in the direction. In contrast, regarding the spatial model, we employ a spatial discretization approach, in which each rectangular grid cell is subdivided into two triangular facets--one in the upper half and one in the lower half. Given the coordinates of the three vertices for each triangle, the corresponding normal vector is computed using the cross product. Tracing ray paths in three-dimensional space over a sea surface is akin to determining the intersection of a line with a plane [25]. The laser rays are transmitted from seawater to air, where the refractive index of water is \(n_{1}=1.34\) and that of air is \(n_{2}=1\). The incidence vector \(\vec{I}\) and the wave front normal vector \(\vec{n}\) are unit vectors, meaning that \(\left|\vec{I}\right|=\left|\vec{n}\right|=1\). By applying [PERSON]'s law directly with the vectors, the transmission vector \(\vec{T}\) for the laser ray can be derived as follows [49]:  
\[\vec{T}=n_{1}\ \vec{I}-\mathrm{c}\ \vec{n} \tag{15}\]  
where \(\mathrm{c}\) can be considered constant and is calculated from  
\[\mathrm{c}=n_{1}\ \vec{I}\ \cdot\ \vec{n}-\sqrt{n_{1}\left(\vec{I}\ \cdot\ \vec{n}\right)^{2}-n_{1}^{2}+1}. \tag{16}\]  
Before Equation (15), it is necessary to verify that the ray impinges on the exact wave facet and that the vector direction always transmits upward (defined as the \(z\)-axis positive) [17]. When the angle of incidence is greater than the critical angle, \(\sim\)48.3\({}^{\circ}\), no beam is transmitted, and total reflection occurs. The spatial sea surface with temporal characteristics is further simulated, and the wave spectrum is the Elfouhaily spectrum. The time interval is 1 s, totaling 50 sea surface topographies per wind speed. Figure 5 displays spatial surfaces for a time series. The waves propagate toward the \(x\)-axis positively as time progresses, and their statistical \begin{table}
\begin{tabular}{c c c} \hline \hline
**Layer [km]** & **Aerosol Type** & \(\tau\) \\ \hline
0–2 & maritime & 0.26 \\
2–12 & continental & 0.05 \\
12–50 & sulfate & 0.005 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Vertical distribution and optical thickness of aerosol stratifications [from Ref. [48]]. properties remain consistent. The average MSS of the spatial model is shown in Table 4, and the results are closer to those of the Hu model than to those of the Cox-Munk model, especially at 1 m s\({}^{-1}\). Therefore, in this study, the statistical model utilizes a combination of Equations (1) and (4). The parameters for the cross-media uplink laser transmission channel in the simulation are presented in Table 5. The laser operates at 532 nm, and the telescope parameters are assumed to be optimized at the receiver plane. To increase the accuracy of the Monte Carlo simulations, each simulation is conducted with more than \(10^{9}\) photon packets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{\(\sigma^{2}[\times 10^{-3}]\) at Wind Speed [m s\({}^{-1}\)]} \\ \cline{2-5}  & **1** & **5** & **10** & **15** \\ \hline [PERSON] \& [PERSON] [16] & 8.12 & 28.6 & 54.2 & 79.8 \\ [PERSON] et al. [29] & 14.6 & 32.6 & 54.2 & 78.3 \\ Elfouhaily Spectrum & origin & 6.89 & 31.5 & 53.9 & 78.1 \\ Refined & 16.6 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of sea surface mean square slopes across different models. Figure 5: Spatial surface topographies with temporal variations for wind speeds of (top to bottom) 1, 5, 10 and 15 m s\({}^{-1}\) and moments of (left to right) 1, 10, 25, and 50 s. Each spatial sea surface scale contains at least one wave wavelength.