## 3 Modeling
Meanwhile, the InIRA velocity bunching modulation function \(T^{vb}\) can be expressed as [9]  
\[T^{vb}(k)=-\beta k_{y}\omega\bigg{(}cos\theta-i\ sin\theta\ \frac{k_{x}}{k} \bigg{)} \tag{4}\]  
where:  
\[w^{2}=gk \tag{5}\]  
\[\beta=\frac{R}{V} \tag{6}\]  
where \(k_{y}\) is the azimuth component of \(k\), \(g\) is the acceleration of gravity, \(R\) is the slant range, and \(V\) is the satellite velocity. For the InIRA, \(\theta<8^{\circ}\); therefore, \(sin\theta\frac{k_{x}}{k}\ll cos\theta\). In this case, Equation (4) can be approximated as  
\[T^{vb}(k)=-\beta k_{y}\sqrt{g}kcos\theta \tag{7}\]  
Equations (1) and (3) show that the tilt and range bunching modulations are only related to the range component of waves, while Equation (7) shows that the velocity bunching modulation is only related to the azimuth component. When there are only waves toward the range direction, the transfer relation between the InIRA image spectra \(P(k)\) and wave spectra \(F(k)\) can be described in terms of tilt and range bunching modulation by  
\[P(k)=\mathrm{e}^{-k_{y}^{2}\xi^{2}}(\alpha^{2}(\theta)+\frac{1}{tan^{2}\theta })k_{x}^{2}F(k) \tag{8}\]  
Similarly, when there are only waves toward the azimuth direction, the transfer relation can be described in terms of velocity bunching modulation by  
\[P(k)=\mathrm{e}^{-k_{y}^{2}\xi^{2}}\beta^{2}k_{y}^{2}gkcos^{2}\theta F(k) \tag{9}\]  
From above analysis, InIRA has a weak cutoff effect owing to the low satellite altitude, however, we still added the cutoff factor (\(\mathrm{e}^{-k_{y}^{2}\xi^{2}}\)) for improving retrieval accuracy according to quasi-specular SAR ocean mapping relation [9; 10; 11]. The \(\xi^{2}\) is the mean square azimuthal displacement of a scattering element, which can be estimated using cutoff \(\lambda_{c}\) by [40]  
\[\xi^{2}=\bigg{(}\frac{\lambda_{c}}{\pi}\bigg{)}^{2} \tag{10}\]Here, the \(\lambda_{c}\) can be estimated by fitting the azimuthal autocorrelation function using a Gaussian function (Equation (7) of [40]). As we know, the SWH can be estimated by integrating \(F(k)\) as  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]0  
Using Equations (8), (9) and (11), the measured SWH for waves toward the range direction (\(SWH_{tilt+rb}\)) and azimuth direction (\(SWH_{vb}\)) can be estimated, respectively, by  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]1  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]2  
We defined two image spectra-related integration factors. The first is the range integration factor (\(IF_{range}\)) and the second is the azimuth integration factor (\(IF_{azimuth}\)):  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]3  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]4  
Substituting Equations (14) and (15) into Equations (12) and (13), the two SWH expressions can be rewritten as  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]5  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]6  
From the above two Equations, the SWH can be represented as a one-order function of \(IF_{range}\) (for waves toward the range direction) or \(IF_{azimuth}\) (for waves toward the azimuth direction) when the incidence angle is constant. In the actual modeling process, we do not expect a perfect first-order relationship between them. As long as there is a linear dependence between them, which can be used for modeling by fitting. Below, we attempt to confirm the linearity of the dependence by analyzing the InIRA data. Figure 2 depicts two cases of InIRA data in terms of NRCS image and image spectra.