## 2 Methodology
### The Impact of Oceanic Atmospheric Stability on the Scattering Coefficient  
According to the theory of electromagnetic wave scattering, the scattering coefficient generated via Bragg resonance can be expressed under the first-order approximation as follows:  
\[\sigma_{pp}^{0}(\theta_{i})=16\pi k_{e}^{4}\cos^{4}\theta_{i}|g_{pp}|^{2}S(k _{B},\varphi_{B}) \tag{1}\]In this equation, \(k_{e}=2\pi/\lambda_{e}\) represents the wave number of the incident electromagnetic wave, \(k_{B}=2k_{e}\sin\theta_{i}\) denotes the wave number of the gravity-capillary wave that induces Bragg resonance with the incident electromagnetic wave, \(\theta_{i}\) is the incident angle, and \(\varphi_{B}\) is the azimuth angle between the radar beam projection on the sea surface and the wind direction. \(pp\in\{HH,VV\}\), and \(g_{pp}\) is a factor related to the polarization state, which is expressed as follows in Equation (2) [27]:  
\[g_{pp}= \begin{cases}\frac{\varepsilon_{r}-1}{[\cos\theta_{i}+\sqrt{\varepsilon _{r}-\sin^{2}\theta_{i}}]}^{2}&\text{HH-pol}\\ \frac{(\varepsilon_{r}-1)[\varepsilon_{r}(1+\sin^{2}\theta_{i})-\sin^{2} \theta_{i}]}{[\varepsilon_{r}\cos\theta_{i}+\sqrt{\varepsilon_{r}-\sin^{2} \theta_{i}}]}&\text{VV-pol}\end{cases}  \tag{2}\]  
In Equation (2), \(\varepsilon_{r}\) denotes the relative permittivity of seawater, and the value of \(\varepsilon_{r}\) can be well calculated by the Debye expression [28]:  
\[\varepsilon_{r}(S,T,\omega)=\varepsilon_{\infty}(S,T)+\frac{\varepsilon_{1}( S,T)-\varepsilon_{\infty}}{1-i\omega\tau(S,T)}-\frac{i\sigma(S,T)}{\omega \varepsilon_{0}} \tag{3}\]  
Here, \(S\) represents seawater salinity; \(T\) denotes \(SST\); and the other parameters as \(\varepsilon_{1}\), \(\varepsilon_{\infty}\), \(\omega\), and \(\tau\) can be found in reference [28]. In Equation (1), \(S(k_{B},\varphi_{B})\) represents the sea roughness spectrum, which is associated with wind speed. In reference [26], it is mentioned that when considering the influence of the wind field, the roughness spectrum for the capillary and short gravity waves proposed by [PERSON] has limitations under low-wind-speed conditions [29; 30]. [PERSON] [31; 32] established an ocean surface roughness spectral model (referred to as H-spectrum) by analyzing the field measurements of the sea roughness spectrum, which is adequate for describing the capillary and short gravity waves. Therefore, in this work, we employ the E-spectrum to represent the long gravity waves and use the H-spectrum to describe the capillary and short gravity waves, ultimately forming a full sea roughness spectrum (referred to as the E-H spectrum). The combination of the E-H spectrum can be represented with the following equations:  
\[S(k,\varphi)_{E_{-H}}=f(k,\varphi)\left\{ \begin{array}{ll}M_{L}(k)&k<k_{Lc} \\ M_{M}(k)&k_{Lc}\leq k<k_{Hc}\\ M_{H}(k)&k\geq k_{Hc}\end{array} \right. \tag{4}\]  
\[M_{L}(k)=\frac{\alpha_{g}v_{g}F_{g}}{2k^{4}v_{ph}}\kappa^{\exp{[-\frac{(\sqrt{ \kappa/k_{B}}-1)^{2}}{2\rho^{2}}]}}\exp(-\frac{5k_{p}^{2}}{4k^{2}}) \tag{5}\]  
\[M_{H}(k)=\frac{A(k)}{k^{4}}\bigg{(}\frac{u_{f}}{c}\bigg{)}^{a(k)} \tag{6}\]  
\[M_{M}(k)=\frac{(k_{Hc}-k)}{(k_{Hc}-k_{Lc})}M_{L}(k)+\frac{(k-k_{Lc})}{(k_{Hc}- k_{Lc})}M_{H}(k) \tag{7}\]  
The directional spreading function is defined as follows:  
\[f(k,\Phi)=\frac{1}{2\pi}[1+\Delta_{E}\cos(2\varphi)] \tag{8}\]  
In Equations (4)-(8), the values of the parameters can be found in reference [26]. \(A(k)\) and \(a(k)\) are empirical expressions [32]. And the middle branch \(M(k)\) is used to maintain the continuity of the E-H spectrum. In the case of microwave scattering, \(k_{B}>k_{HC}\).