## 2 Sea Surface Model
### Spatial Model
#### 2.2.3 Sea Surface Simulations and Statistical Properties  
The above results suggest that all the candidate wave spectra maintain stable performance across variable wind speeds and wavenumbers, allowing for robust calculation in the simulation of spatial surfaces. After specific environmental parameters, primarily wind field data, are prescribed, combining a wave spectrum with FFT techniques facilitates the efficient simulation of virtual realistic sea surfaces (publicly available from Refs. [24; 43]). The simulated sea surface parameters are presented in Table 2. The wave length L is set to be at least twice the characteristic wavelength to ensure the simulation captures the full wavenumber spectrum. And the grid number N is chosen based on computational efficiency and empirical considerations. Notably, to compensate for the lack of sea surface information due to the discrete grid size used during the simulation process [44], [PERSON] suggested a spectral enhancement sampling method (Ref. [24], Equation (9)), which can effectively ensure that the simulated sea surface finite difference sampling results are consistent with the theoretical calculations. The simulation can capture the dynamics of waves more fully at temporal and spatial scales. Discrete, 2D random elevation grids are generated, as illustrated in Figure 3, which shows examples of simulated spatial wind-blown surfaces for a wind speed of 10 m s\({}^{-1}\). The topographies of the sea surface obtained from different wave spectra are visually quite comparable, with wave heights distributed within \(\pm\)2.5 m. Regarding the fine surface characteristics, the simulated sea surface displays greater roughness with the composite spectrum (Figure 3d-f) than with the typical gravity spectrum (Figure 3a \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{4}{c}{**U [m s\({}^{-1}\)]**} \\ \cline{2-5}  & **1** & **5** & **10** & **15** \\ \hline L [m] & 6 & 50 & 300 & 480 \\ N & 240 & 2000 & 3000 & 2400 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the simulated sea surface. When spatial models are applied, the key statistical properties of sea surfaces are elevation and slope [8]. The elevation provides insights into the physical features of topographies and the spatial distribution of slopes. In contrast, the slope characterizes the optical properties of surfaces and considerably influences the interaction of the beam with surfaces. Under these circumstances, finite differences must be applied to calculate the elevation and total MSS for each spatial surface. Therefore, Equations (7) and (8) can be reformulated accordingly as  
\[ \begin{array}{ll}\left\langle h^{2}\right\rangle&=\sum z^{2}\\ &=\frac{1}{N_{x}N_{y}}\sum\limits_{i,j}^{N_{x},N_{y}}z\left(x_{i,j}\,\ y_{i,j} \right)^{2}\end{array}  \tag{11}\]  
and  
\[ \begin{array}{ll}\sigma^{2}=\sum[(\frac{\Delta z}{\Delta x})^{2}+(\frac{ \Delta z}{\Delta y})^{2}]\\ \\ =\frac{1}{(N_{x}-1)N_{y}}\sum\limits_{i,j}^{N_{x},N_{y}}\left[\frac{z\left(x_{ i+1,j}\,\ y_{i,j}\right)-z\left(x_{i,j}\,\ y_{i,j}\right)}{x_{i+1,j}-x_{i,j}}\right]^{2}\\ \\ +\frac{1}{N_{x}\big{(}N_{y}-1\big{)}}\sum\limits_{i,j}^{N_{x},N_{y}}\left[ \frac{z\left(x_{i,j}\,\ y_{i,j+1}\right)-z\left(x_{i,j}\,\ y_{i,j}\right)}{y_{i,j+1}-y_{i,j}} \right]^{2}\end{array}  \tag{12}\]  
where \(z\left(x_{i,j}\,\ y_{i,j}\right)\) is the sea surface elevation at spatial coordinates \(\left(x_{i,j}\,\ y_{i,j}\right)\), and \(N_{x}\), \(N_{y}\) are the numbers of grids in the \(x\) and \(y\) directions, respectively. The significant wave heights and total MSS are calculated for each state via Equations (11) and (12). Figure 4 shows the statistical distributions of \(H_{s}\) and \(\sigma^{2}\) across various types of sea surfaces for wind speeds ranging from 1 m s\({}^{-1}\) to 15 m s\({}^{-1}\). Figure 4a depicts the increase in \(H_{s}\) across different spatial surfaces as the wind speed increases.