## 3 Simulation Results  
### Model Validation  
To validate our model, the results of unpolarized light (i.e., \(S_{0}=[1,0,0,0]^{\mathsf{T}}\)) were compared with those of [PERSON] [30]. [PERSON] generalized a matrix method of solving the transfer of solar radiation in an atmosphere-ocean system with a wind-roughed ocean surface. He used an isotropic Gaussian distribution to assume the density function of the wave slope as in Equation (1), with \(\sigma^{2}=0.00534\ W\) in his model.  
The unidirectional reflectivity was calculated using Equation (24a) where \(\mu_{1}=0,\ \mu_{2}=1,\ \varphi_{1}=0,\ \text{and}\ \varphi_{2}=2\pi\) (i.e., the upper hemispherical integration), and Equation (24b) where \(\mu_{1}=-1,\ \mu_{2}=0,\ \varphi_{1}=0,\ \text{and}\ \varphi_{2}=2\pi\) (i.e., the lower hemispherical integration).  
Figure 5a compares the reflectivity in our model, shown as solid lines, with [PERSON]'s outputs, which are highlighted with asterisks on dashed lines. The solid lines match the dashed lines well, which indicates an excellent agreement both in the air-incident case (light across the surface from air to sea) and in the water-incident case (light across the surface from sea to air). Figure 5b presents the absolute percentage deviation (APD) between the two models. A large difference happened when the wind speed was 0 m/s, which was due to the little difference between \(\sigma^{2}\) and \(W\) in the two models. In [PERSON]'s model, \(\sigma^{2}=0\) (i.e., a flat sea surface), when \(W\) was 0. In our model, \(\sigma^{2}\) (\(W=0\)) was 0.003. In reality, the sea surface is not flat, although there is no wind blowing, (i.e., \(\sigma^{2}_{W=0}\
eq 0\)). Our model is consistent with a previous study [31], indicating that our model is more precise. The maximum value of APD occurs near the critical angle for total internal reflection, but in our model, a little upwelling energy still leaked through the surface into the air due to the roughness. As the wind speed increased, the APD between the two models narrowed rapidly. Figure 5c,d present the regression plots of outputs between the two models for air-incident and water-incident cases, respectively. It shows that the adjusted \(R\)-squared between them was nearly one. As shown in Table 1, the root mean squared error (RMSE) between them was very small except at the zero speed.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **W (m/s)** & \(\mathbf{R^{2}}\) & **RMSE** & **Max\({}_{\text{AD}}\)** & **Max\({}_{\text{APD}}\)** & **MAD** & **MAPD** \\ \hline \multirow{8}{*}{Air-incident} & 0 & 0.9938 & 0.0491 & 0.2513 & 25.4646\% & 0.0143 & 2.3642\% \\  & 0.83 & 0.9998 & 0.0138 & 0.0599 & 8.4826\% & 0.0054 & 1.5342\% \\  & 3.33 & 1.0000 & 0.0044 & 0.0178 & 3.4032\% & 0.0020 & 0.8736\% \\  & 7.49 & 1.0000 & 0.0013 & 0.0049 & 1.2275\% & 0.0006 & 0.3722\% \\  & 15.29 & 1.0000 & 0.0002 & 0.0006 & 0.1988\% & 0.0001 & 0.0677\% \\ \hline \multirow{8}{*}{Water-incident} & 0 & 0.9952 & 0.0489 & 0.2926 & 59.9937\% & 0.0188 & 5.0894\% \\  & 0.83 & 0.9995 & 0.0155 & 0.0543 & 25.1187\% & 0.0085 & 2.9464\% \\ \cline{1-1}  & 3.33 & 0.9999 & 0.0059 & 0.0154 & 9.1151\% & 0.0041 & 1.5886\% \\ \cline{1-1}  & 7.49 & 1.0000 & 0.0025 & 0.0062 & 3.0189\% & 0.0017 & 0.6922\% \\ \cline{1-1}  & 15.29 & 1.0000 & 0.0005 & 0.0010 & 0.3833\% & 0.0003 & 0.1180\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Statistics of the differences between our results and [PERSON]’s results. AD—absolute difference; APD—absolute percentage deviation; MAD—mean absolute difference; MAPD—mean absolute percentage deviation;Table 1 shows the statistics of the differences. AD means the absolute value of the difference. MAD and MAPD are the mean values of AD and APD, respectively. The table shows that the MADs between the two models were no more than 0.01, and the MAPDs were no more than 3% when the wind was larger than 0 m/s. Both statistics indicate that our model is accurate enough for simulating light propagation across a sea surface.