## 3 Modeling
Figure 3e,f depict two integration factor dependences on the incidence angles, in which the SWH is at 3 m. From Figure 3e, the \(IF_{range}\) is basically constant at first and then increases with incidence angles larger than 6\({}^{\circ}\). From Figure 3f, the \(IF_{azimuth}\) shows a similar trend with \(IF_{range}\). This means that the relationship between the SWH and each integration  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]7 Two InIRA data cases: (**a**) NRCS image for swell and (**b**) image spectra extracted from (**a**); (**c**) NRCS image for crossing wind waves and (**d**) image spectra extracted from (**c**). factor varies with the incidence angle. The incidence angle should also be considered in building the SWH model. From the above analysis, the SWH increases with both \(IF_{range}\) and \(IF_{azimuth}\), and the relationships between them vary with the incidence angle. In particular, the SWH values in the range and azimuth directions are more dependent on \(IF_{range}\) and \(IF_{azimuth}\), respectively. In the framework of the linear ocean wave theory, any wave can be divided into two components in the range and azimuth directions. We considered combining the \(IF_{range}\) (describing the wave component in the range direction) and \(IF_{azimuth}\) (describing the wave component in the azimuth direction) factors to build an empirical SWH model. We inferred that there is a coupling effect between the range modulation (tilt and range bunching) and azimuth modulation (velocity bunching) based on previous SAR studies on the wave transfer relation. Thus, we used a quadratic orthogonal polynomial to model the coupling effect. To reduce the modeling difficulty, the model was segmented using the incidence angle at steps of 0.5\({}^{\circ}\). The final model function was described by  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]8  
where \(C_{ij}\) are model coefficients. The model coefficients in Equation (18) were estimated by fitting randomly selected 50% collocations consisting of InIRA and WW3 data. Note that the InIRA data collocated with traditional altimeter data were not used for fitting, as they were reserved for an independent validation later. The estimated model coefficients are listed in \[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]9  
\[T^{rb}(k)=\frac{\mathrm{i}k_{x}}{tan\theta} \tag{3}\]0 InIRA data dependence analysis: (**a**) SWH dependence on the range integration factor for data close to the range direction; (**b**) SWH dependence on the azimuth integration factor for data close to the azimuth direction; (**c**) same as (**a**) but for data close to the azimuth direction; (**d**) same as (**b**) but for data close to the range direction; (**e**) range integration factor dependence on the incidence angle for data close to the range direction; (**f**) the azimuth integration factor dependence on the incidence angle for data close to the azimuth direction. The green points represent the data close to the range direction, while the blue points are for the azimuth direction.