## 2 Data and Methods
### Wave Spectra and Spreading Functions  
The wave spectrum covering a wide range of wavenumbers, e.g., from millimeter wavelengths to wavelengths on the order of hundreds of meters, is required for the backscattering calculation. Over the past decades, many full-wave-number spectra have been proposed, of which the D (Donelan-Banner-Plant) spectrum [35], the A ([PERSON]) spectrum [36], the E ([PERSON] et al.) spectrum [37], and the most recent version of the [PERSON] ([PERSON] et al.) spectrum, H18 [38], are used in this paper. The D spectrum is the synthesis of the results of measurements with wave gauge arrays, stereo photography, and microwave backscatter [39; 40; 41]. The A spectrum is derived from the in situ measurements in the gravity wave region and the wave tank results in gravity-capillary wave region. It addresses the consistency with the measured NRCS. The E spectrum emphasizes the consistency with the measured mean square slopes in [42], and is developed based on the unification of optical, in situ, and wave tank data. The H18 spectrum is a combination of the H15 spectrum [43] and the G spectrum [38]. The details of the four spectral models can been seen in [35; 36; 37; 38; 44]. Figure 1 illustrates the curvature spectra \(B(K)\) of D, A, E, and H18 for wind speeds varying from 2 m/s to 24 m/s, and the step is 2 m/s. We can see that the intermediate- and short-scale waves roughly from 1 rad/m to 1000 rad/m make the dominant contribution to the surface roughness. As wind speed increases, the wind speed sensitivity of spectral density decreases gradually. Furthermore, the spectral density exhibits very little wind speed sensitivity over the whole wind speed range for decimeter- to meter-scale waves. This may be partly due to the fact that waves of these scales are the main wave-breaking regions that generate short-scale waves [6]. In addition, there is a spectral peak in the capillary wave region, followed by a rapid drop-off. This peak is at 190 rad/m (3.3 cm wavelength) for the D spectrum, 700 rad/m (0.9 cm wavelength) for the A spectrum, 370 rad/m (1.7 cm wavelength) for the E spectrum, and 530 rad/m (1.19 cm wavelength) for the H18 spectrum. More importantly, the magnitude of the spectral density varies from one model to another, especially in the gravity-capillary wave region. For example, at a 16 m/s wind speed, the curvature spectral density is about 0.0132 rad\({}^{2}\) at 143 rad/m (the  
C-band resonance Bragg wave number at 40\({}^{\circ}\) incidence angle) for the D spectrum, about 0.0225 rad\({}^{2}\) for the A spectrum, about 0.0159 rad\({}^{2}\) for the E spectrum, and about 0.0135 rad\({}^{2}\) for the H18 spectrum. The prediction of the directional anisotropy of ocean backscatter requires the knowledge of the directional distribution of ocean waves. It is known that this distribution can be described by the directional spreading function \(f(K,\varphi)\). Three widely used spreading functions are the Cosine [37], Sech [35], and Gaussian [27] spreading functions. 1. []
2. []  
Cosine spreading function:  
\[f(K,\varphi)=\frac{1}{2\pi}[1+\Delta(K)\cos(2\varphi)] \tag{3}\]  
where \(K\) is ocean wave number magnitude and \(\varphi\) is its angle with respect to the direction toward which the wind blows. \(\Delta(K)\) can be expressed as:  
\[\Delta(K)=\tanh\Bigl{\{}a_{0}+a_{p}\bigl{(}c/c_{p}\bigr{)}^{2.5}+a_{m}(c_{m}/c )^{2.5}\Bigr{\}} \tag{4}\]  
where \(c\) is the wave phase velocity:  
\[c=\sqrt{\frac{g}{K}+TK} \tag{5}\]  
where \(g\) is the acceleration due to gravity, \(T=7.25\times 10^{-5}\). \(c_{p}\) is the phase speed at the dominant wave, \(a_{0}=\ln(2)/4\), and \(a_{p}=4\). \(a_{m}\) is:  
\[a_{m}=0.13\frac{u_{*}}{c_{m}} \tag{6}\]  
where \(u_{*}\) is the friction velocity and \(c_{m}\) is the minimum phase speed that is equal to 0.32 m/s. Figure 1: Curvature spectra \(B(K)\) in log\(-\)log scales: (**a**) the D spectrum, (**b**) the A spectrum, (**c**) the E spectrum, and (**d**) the H18 spectrum. Wind speed increases upward from 2 m/s to 24 m/s in 2 m/s steps. 2.