## 2 Methodology
### The Impact of Oceanic Atmospheric Stability on the Scattering Coefficient
The first-order response of the scattering coefficient to the _SATD_ can be derived from Equations (1)-(4) as follows:  
\[\frac{\mathrm{d}\sigma_{pp}^{0}(\theta_{i})}{\mathrm{d}\Delta T}=16\pi k_{e}^{4} \cos^{4}\theta_{i}\big{|}g_{pp}\big{|}^{2}\left\{\frac{\partial M_{H}}{\partial \Delta T}f(k,\Phi)+M_{H}\frac{\partial f(k,\Phi)}{\partial\Delta T}\right\} \tag{9}\]  
Here, \(\frac{\partial M_{H}}{\partial\Delta T}\) can be represented as follows:  
\[\frac{\partial M_{H}}{\partial\Delta T}=\frac{A(k)a(k)}{k^{4}c}(\frac{u_{f}}{ c})^{a(k)-1}\frac{du_{f}}{d\Delta T} \tag{10}\]  
In Equation (10), \(u_{f}\) represents the friction wind speed, and \(c\) denotes the speed of light. Figure 1a shows the correlation between the scattering coefficient and _SATD_ under various wind speeds. The scattering coefficient increases with an increasing _SATD_. Under a stable atmospheric condition (i.e., \(\Delta T<0\)), the scattering coefficient increases rapidly with an increasing _SATD_; in contrast, when the atmosphere is unstable (i.e., \(\Delta T>0\)), the scattering coefficient exhibits a slower increase with the _SATD_. Figure 1b presents the first-order derivative of the scattering coefficient for different wind speeds. It is found that the first-order derivative of the scattering coefficient increases with the _SATD_ under a stable atmospheric condition and decreases with the _SATD_ under an unstable atmospheric condition. The values of the first-order derivative under different wind speeds remain consistently greater than 0. These results indicate that the scattering coefficient increases with the increase in atmospheric instability. Therefore, there is a positive correlation between the scattering coefficient and _SATD_, as well as between the _SCA_ and _SATDA_. The theoretical formulas in Equations (1) and (2) indicate that the scattering coefficient is also influenced by the wind field and permittivity of seawater. Given that the relationship between the scattering coefficient and the wind field has been extensively studied, we only focus on the effects of the permittivity of seawater, _SST_, and salinity on the scattering coefficient. Figure 2a illustrates the scattering coefficient as a function of wind speed under different seawater permittivity values. Here, the radar frequency is 13.8 GHz. The results show that the scattering coefficient is basically unchanged under different seawater  
Figure 1: (**a**) Effects of _SATD_ on scattering coefficient at different wind speeds. (**b**) First–order derivative of scattering coefficient versus _SATD_ at different wind speeds. permittivity values, indicating that the effect of the permittivity of seawater on the scattering coefficient can be ignored. As shown in the Debye formula (Equation (3)), the permittivity of seawater is influenced by both \(SST\) and seawater salinity. Figure 2b illustrates the variation in the scattering coefficient if the effect of \(SST\) on the permittivity is considered. Figure 2c displays the changes in the scattering coefficient with different salinities at a constant sea surface temperature. The results reveal that the influences of \(SST\) and salinity on the permittivity of seawater have a minor effect on the scattering coefficient. Furthermore, a comparison with Figure 1a indicates that the \(SATD\) has a more obvious influence on the scattering coefficient, confirming that air-sea coupling is a key factor affecting the scattering coefficient.