## 2 Theory
### A Model of Microwave Radar Backscatter from the Sea Surface  
Total normalized radar cross-section (NRCS) in the Ka-band from a wavy water surface can be written as a combination of polarized (Bragg) and non-polarized (non-Bragg) components, BC and NP, respectively (see, e.g., [31; 32]):  
\[\sigma^{pp}=\sigma_{BC}^{pp}+\sigma_{NP} \tag{13}\]  
The term, \(\sigma_{BC}^{pp}\) is determined by the spectrum \(\widetilde{F}\) (\(k_{br}\)) of mm-scale wind ripples, including both free waves and parasitic capillary ripples, at the Bragg wavenumber [33]\(k_{br}=2k_{e}\sin\theta\) (\(k_{e}\) and \(\theta\) are the wavenumber and the angle of incidence of microwaves), and by the scattering coefficient \(g_{pp}^{2}(\theta)\), depending on the polarization (p) of the emitted/received signal, the incidence angle, and dielectric properties of water (see, [33]), and can be written as  
\[\sigma_{BC}^{pp}=16\pi k_{e}^{4}g_{pp}^{2}\left(\theta\right)\widetilde{F}\) (\(k_{br}\))  
The second, non-polarized component in Equation (13), \(\sigma_{NP}\), is determined by scattering from quasi-specular facets on the wind wave profile and is obviously equal for vertical and horizontal emitted/received polarizations, VV and HH, respectively. The appearance of these facets is related to high steepness/curvature parts of the nonlinear structures, such as bulge/toes and partly to parasitic capillary ripples on the profile of steep cm-dm-scale wind waves (see, e.g., [18; 19; 20; 26]).  
A relationship between NP radar return and characteristics of the nonlinear structures has not been established thus far. However, some results of our wave tank experiments, which will be published elsewhere, indicate that the NP Ka-band radar return can be described by a power function of the steepness of basic cm-dm-scale GCW. Accordingly, we suppose that  
\[\sigma_{NP}\sim\left[k_{g}^{2}\widetilde{F}(k_{g})\right]^{q} \tag{15}\]  
where \(q\approx 5\).  
We define the radar slick contrasts as a ratio of total radar return, BC and NP in non-slick (_nsl_) and slick (_sl_) areas, according to:  
\[K^{pp}=\frac{\sigma^{pp}\big{|}_{nsl}}{\sigma^{pp}\big{|}_{sl}},\ K^{pp}_{BC}= \frac{\sigma^{pp}_{BC}\big{|}_{nsl}}{\sigma^{pp}_{BC}\big{|}_{sl}}=\frac{F_{0 }(k)\big{|}_{nsl}}{F_{0}(k)\big{|}_{sl}},\ K_{NP}=\frac{\sigma_{NP}\big{|}_{nsl }}{\sigma_{NP}\big{|}_{sl}}, \tag{16}\]It can be shown clearly that the slick contrast for the total radar return is related to the partial, NP and BC contrasts according to the formula:  
\[K^{pp}=\chi_{BC}(\Gamma_{0})\frac{\sigma_{BC}^{pp}\big{|}_{nsl}}{\sigma_{BC}^{ pp}\big{|}_{sl}}+\chi_{NP}(\Gamma_{0})\frac{\sigma_{NP}\big{|}_{nsl}}{\sigma_{NP} \big{|}_{sl}}, \tag{17}\]  
where \(\chi_{BC}(\Gamma_{0})=\frac{\sigma_{BC}^{pp}|_{sL}}{\sigma\sigma\mathcal{P}|_{ sL}}\) and \(\chi_{NP}(\Gamma_{0})=\frac{\sigma_{NP}|_{sL}}{\sigma\sigma\mathcal{P}|_{sL}}\) are the relative Bragg and non-polarized radar backscatter components in total radar return at a given mean surfactant concentration \(\Gamma_{0}\).  
Variations of radar return due to internal waves \(\Delta\sigma^{pp}\) can be characterized by the radar modulation transfer function (RMTF) \(m^{radar\_pp}\), according to its definition, namely  
\[\Delta\sigma^{pp}/\sigma^{pp}=m^{radar\_pp}\frac{U_{0}}{C}e^{jk(x-Ct)}, \tag{18}\]  
RMTF can be described as follows:  
\[m^{radar\_pp}=\chi_{BC}(\Gamma_{0})\cdot m^{hydro}(k)+\chi_{NP}(\Gamma_{0}) \cdot qm^{hydro}(k_{g}) \tag{19}\]