## 2 Sea Surface Model
### Spatial Model
#### 2.2.2 Refinement of the Elfouhaily Spectrum  
To explicate the spectral truncation depicted in Figure 1a,e, the process of modeling the Elfouhaily spectrum is reviewed. [PERSON] et al. [37] introduced the generalized Phillips-Kitaigorodskii equilibrium range parameter \(\alpha_{m}\) to describe the short-wave curvature spectrum. A two-regime logarithmic law was fitted on the basis of field experimental data as (Ref. [37], Equation (44))  
\[\alpha_{m}=\begin{cases}0.01[1+ln(\frac{Ut}{c_{m}})]\|\|&\text{if}Ut\leqslant c _{m}\\ 0.01[1+3 ln(\frac{Ut}{c_{m}})]\|\|&\text{if}Ut>c_{m}\end{cases} \tag{9}\]  
where \(U_{s}\) is the friction velocity, which is approximately equal to 0.0378 \(U\); \(c_{m}\) can be considered a constant for 0.23 m s\({}^{-1}\); and \(\alpha_{m}\) is also called the saturation level of the secondary peak. It is important to note that due to the fitting data used by the Elfouhaily spectrum at low wind speeds, the measurements may be close to the noise level of the instrument, which may affect the fitting accuracy and lead to bias or failure at low wind speeds [37; 41]. Thereafter, a threshold friction wind speed of approximately 0.085 m s\({}^{-1}\), equivalent to a wind speed of approximately 2.25 m s\({}^{-1}\), can be calculated for the Elfouhaily spectrum. If the wind speed is less than the threshold condition, the calculation of the spectral energy from the gravity waves to capillary ranges would take on negative values, resulting in the failure of the spectral bands (as shown by the red line in Figure 2). Therefore, it is necessary to combine a variety of observation instruments for comprehensive comparison to improve the reliability and consistency of data.  
Figure 1: Comparison of elevation spectrum **(a-d)** and slope spectrum **(e-h)** across six different wave spectra for wind speed of (left to right) 1, 5, 10 and 15 m s\({}^{-1}\), respectively. Each spectrum is represented by a unique color. **(a,e)** reveal truncations due to negative values in the Elfouhaily spectrum (red solid). The subplot examples in **(d,h)** feature detailed zoom-in on the wavenumber range from 0.001 to 0.1 rad m\({}^{-1}\), enhancing the visibility of spectral nuances.  
Figure 2: The Elfouhaily elevation spectrum **(a,b)** and slope spectrum **(c,d)** for wind speed of (left to right) 1 and 2 m s\({}^{-1}\), respectively. The red line represents the values derived from the original spectrum, where the missing part is negative, and the blue line shows the refined results.  
Similarly, [PERSON] [36] also modeled a saturation exponential function from radar data, which is particularly effective for fitting low-wind-speed conditions, as (Ref. [36], Equation (15))  
\[\alpha_{m}=\exp\biggl{\{}\biggl{[}-4.95+3.45\biggl{(}1-e^{\frac{-U}{\Omega_{\eta} }}\biggr{)}\biggr{]}\text{ln}10\biggr{\}} \tag{10}\]  
where \(U_{\eta}\) is the saturation speed, which is 4.7 m s\({}^{-1}\), and a secondary peak is assumed at 750 rad m\({}^{-1}\).  
Equation (10) can fit the saturation parameter accurately, whereas Equation (9) aligns closely with the experimental data at medium to high wind speeds [36; 37; 41; 42]. Thus, in this study, the above two equations are combined to compensate for the instability of the Elfouhaily spectrum under low-wind-speed conditions.  
In this instance, the refined Elfouhaily spectrum applies Equation (10) for wind speeds below 2.25 m s\({}^{-1}\) and Equation (9) for all other conditions. The blue lines in Figure 2 illustrate examples of the results. This implementation enhances and broadens the detail regarding wind saturation effects and provides a reference for the spectral integrity of the Elfouhaily spectrum at low wind speeds.