## 3 Data and Methods
### Sea Surface Emissivity Models  
1. Small perturbance/small scale approximation model (SPM/SSA):  
In the SPM/SSA model, the Stokes vector of the emission from the sea surface is expressed as an integral over the surface directional spectrum [21] as:  
\[\left[ \begin{array}{c}T_{Bh}^{\rm ss}\\ T_{Bv}^{\rm ss}\end{array} \right]=SST\left(\left[ \begin{array}{c}1-\left|R_{ hh}^{(0)}\right|^{2}\\ 1-\left|R_{vv}^{(0)}\right|^{2}\end{array} \right]-\int_{0}^{\infty}kdk\int_{0} ^{2\pi}W(k,\phi)\left[ \begin{array}{c}g_{h}\\ g_{v}\end{array} \right]d\phi\right), \tag{3}\]  
where \(R_{hh}^{(0)}\) and \(R_{vv}^{(0)}\) are the horizontally and vertically polarized Fresnel reflection coefficients, respectively; \(W(k,\phi)\) is the surface directional spectrum; \(g_{h}\) and \(g_{v}\) are the weighting functions in the formulation for correcting the reflectivity of a flat sea surface induced by small-scale waves. The weighting functions in Equation (3) are based on some theoretical understanding [18] and may require modifications based on measurements. For this, the incoherent Bragg scatter and second order correction terms were combined; the developed coordinates in [18] were modified by [21] and induced the modification of scattering coefficients. In this study, the surface directional spectrum used in the SPM/SSA model was developed by [PERSON] and [PERSON] [41] (hereafter DV spectrum). For well-developed ocean waves, the sea surface wave spectrum can be obtained as:  
\[W(k,\phi)=\frac{1}{2\pi k}S(k)\Phi(k,\phi), \tag{4}\]  
where \(S(k)\) is the omnidirectional part of the surface wave spectrum and \(\Phi(k,\phi)\) is the angular portion of the spectrum. Following [PERSON] and [PERSON], the omnidirectional part of the surface spectrum takes the form:  
\[ \begin{cases}S(k)=a_{0}k^{-3}\left(1.25 klU_{*}^{2}/g_{*}\right)^{0.225\log_{10 }\left(k/2\right)}&k>2\ \rm{rad}\cdot\rm{m}^{-1}\\ S(k)=b_{0}k^{-3}\exp\left[-0.74\left(g/klU_{19.5}^{2}\right)^{2}\right]&k<2\ \rm{rad}\cdot\rm{m}^{-1}\end{cases} , \tag{5}\]  
where \(g^{*}=9.81+7.25\times 10^{-5}k^{2}\); \(k\) is the wavenumber and \(U_{*}\) is the friction velocity, which is defined in Equation (2); \(U_{19.5}\) is the equivalent neutral wind speed given at an elevation of 19.5 m; a0 rangesfrom 0.004 to 0.008 and \(b_{0}\) is selected in such a way that \(S(k)\) is continuous at \(k=2\) rad\(\cdot\)m\({}^{-1}\) for a given a\({}_{0}\). The angular part is represented by:  
\[\Phi(k,\phi)=1+c\Big{(}1-e^{-sk^{2}}\Big{)}\cos 2\phi, \tag{6}\]  
where \(s\) is a constant set as \(1.5\times 10^{-4}\), and \(c\) is given by:  
\[c=\Big{(}\frac{1-R}{1+R}\Big{)}\frac{2}{(1-D)}, \tag{7}\]  
with  
\[R=\frac{0.003+0.00192 Ul_{12.5}}{0.00316 Ul_{12.5}}. \tag{8}\]  
\[D=\frac{\int_{0}^{\infty}k^{2}S(k)e^{-sk^{2}}dk}{\int_{0}^{\infty}k^{2}S(k)dk}. \tag{9}\]  
Following [23], some misprints of coefficients in the DV spectrum described in [41] is corrected. (2) Two-scale model:  
The two-scale model assumes that small-scale waves are rippled and superimposed on to the large-scale waves. In this model, a cutoff wavenumber \(k_{c}\) to determine the large-scale waves and small-scale waves is given as:  
\[W(k,\phi)= \begin{cases}W_{s}(k,\phi)&k>k_{c}\\ W_{l}(k,\phi)&k<k_{c}\end{cases} . \tag{10}\]  
As described in the SPM/SSA model, the DV spectrum was used in this study. For small-scale waves, the Stokes vector of the emission from a local surface patch is exactly the same as Equation (3), and only the cutoff wavenumber is applied in the second terms of Equation (3). For large-scale waves, the integration is used to obtain the Stokes vector as:  
\[\left[ \begin{array}{c}T_{Bh}\\ T_{Bv}\end{array} \right]=\int_{-\infty}^{\infty}\!\!dS_{y}\!\int_{-\infty}^{ \cot\theta}\!\!dS_{z}\mathbf{R}\left[ \begin{array}{c}T_{Bh}^{\mathrm{ss}}\\ T_{Bv}^{\mathrm{ss}}\end{array}