## 2 Sea Surface Model
### Spatial Model
#### 2.2.1 Sea Spectra  
Several previous studies have simulated the impact of sea surfaces on water-air laser transmission links, primarily concentrating on typical gravity wave spectra [2; 14; 25]. However, existing studies have indicated that wave height is predominantly influenced by gravity wave dynamics, whereas the slope of the sea surface is affected by short gravity waves and capillary waves [24; 26; 32]. When wave spectra are used to model the spatial characteristics of the sea surface, ensuring that the spectra exhibit robust performance across all wavenumbers is essential. To conduct a systematic evaluation of diverse wave spectra for the application of laser transmission channel simulation, three representative composite spectra containing high-frequency wavenumbers were also selected, in addition to the typical gravity wave spectra. Table 1 presents detailed descriptions of the typical candidate spectra utilized in this comparative analysis. [PERSON] [33] pioneered the empirical relationship between wave height and period assisted by apparent observations. [PERSON] first formulated a wave spectrum that correlated wind speed variations to the functional form of the long gravity wave spectrum. [PERSON] and [PERSON] [34] and [PERSON] et al. [35], drawing on the similarity hypothesis from [PERSON], meticulously processed and analyzed spectral data from the North Atlantic and North Sea westward from Syll, respectively. Their efforts resulted in one-dimensional spectra that are highly effective and robust over time in characterizing gravity waves in fully developed sea waves [26; 36; 39]. Remote sensing technologies, through advanced observation equipment such as radar, altimeters, and LIDAR, offer new insights into the complex structures of waves. These technologies aid in detailing high-frequency wave patterns and improve the understanding \begin{table}
\begin{tabular}{c c c} \hline \hline
**Spectrum** & **Input Parameter** & **Description** \\ \hline Neumann [33] & wind speed & Growth of wind-generated ocean waves \\ [PERSON] [34] & wind speed & Steady wind \& fully developed sea state model \\ JONSWAP [35] & parameters of fetch & Enhanced peakness \& rapid energy growth under wind \\ Apel [36] & wind speed \& wave age & Model for Radar applications \\ Elfouhaily [37] & wind speed \& wave age & Unifies low and high wavenumbers for optical data \\ Hwang [38] & wind speed, wave age & Model from tropical cyclones \\  & \& influence parameters & Model from tropical cyclones \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Several typical candidate wave spectra. of sea surface roughness [40]. The Apel [36], Elfouhaily [37], and Hwang [38] spectra are comprehensive wave spectrum models across full wavenumbers, utilizing extensive field and water tank data and citing radar backscatter, optical observations, and hurricane observation experiments, respectively. The simulation of the sea surface employs the directional spectrum function:  
\[\Psi(k,\theta)=\frac{1}{k}S(k)D(k,\theta) \tag{5}\]  
where \(S(k)\) is the omnidirectional spectrum and \(D(k,\theta)\) is the spreading function [36]:  
\[D(k,\theta)=C_{s}\text{cos}^{2s_{p}}(\theta/2) \tag{6}\]  
Here, \(s_{p}\) is the wave propagation parameter, which is associated with frequency \(k\), wind speed \(U\), and wave age \(\Omega_{c}\), and \(C_{s}\) is the normalization parameter. On the basis of the above equations, the variance of the zero-mean wave height and the total MSS can be calculated as follows [24]:  
\[ \begin{array}{ll}\left\langle{{h^{2}}}\right\rangle&=\int_{0}^{\infty}\int_ {0}^{2\pi}\Psi(k,\theta)k\ dkd\theta\\ &=\int_{0}^{\infty}\int_{0}^{2\pi}S(k)D(k,\varphi)\ dkd\theta\\ &=\int_{0}^{\infty}S(k)\ dk\end{array}  \tag{7}\]  
and  
\[ \begin{array}{ll}\sigma^{2}&=\int_{0}^{\infty}\int_{0}^{2\pi}k^{2}\Big{(} \text{sin}^{2}\varphi+\text{cos}^{2}\varphi\Big{)}\Psi(k,\varphi)k\ dkd\varphi \\ &=\int_{0}^{\infty}\int_{0}^{2\pi}k^{2}\Psi(k,\varphi)k\ dkd\varphi\\ &=\int_{0}^{\infty}k^{2}S(k)\ dk\end{array}