## 3 Modeling  
Based on previous studies on SARs [12; 13; 14; 15], the ocean SWH can be estimated by establishing an empirical relationship to the SAR observed parameters. Motivated by these methods, this study employed an empirical model to estimate the SWH. The modulation functions for the InIRA data were first analyzed to determine the model input parameters. From the radar ocean wave modulation theory, the InIRA data were modulated by four wave effects including tilt, hydrodynamic, range bunching, and velocity bunching modulations [35; 36; 37; 38]. These four modulations, respectively, arise from the local incidence angle change induced by the long wave slope, hydrodynamic interaction between short and long waves, effective backscattering area change due to the long wave slope, and displacement of the backscattering facet by the long wave orbital velocity. The first two are due to the cross-section change, while the last two are from the area change and orbit motion of water particles, respectively. The first two modulation functions applied for the InIRA are different from those of the conventional SAR at moderate incidence angles and need to be deduced by the quasi-specular scattering theory at low incidence angles. For the last two, the InIRA modulation function are the same as that for the SAR, as they are independent of the sea surface scattering mechanism. The tilt, range bunching and velocity bunching modulation functions for the InIRA data were derived. The first is deduced from a quasi-specular scattering mechanism in studies on OWSs [4; 5; 6; 7]. For the last two, they are the same as that applied on the SAR. However, the hydrodynamic modulation function suitable for low incident InIRA data has not been proposed. The hydrodynamic modulation function at moderate incidence angles is proposed based on the Bragg scattering mechanism in the framework of a complex weak interaction theory [36]. Following this concept, the hydrodynamic modulation at low incidence angles should consider quasi-specular scattering in similar frameworks. In this case, the deduction for hydrodynamic modulation is very complex. On the other hand, the hydrodynamic modulation is usually considered relatively weak in long wave retrieval studies based on quasi-specular mechanism. This modulation is thus neglected and seen as an error source of wave retrievals. For examples, in long-wave retrieval for CFOSAT SWIM (with incidence angles from \(0^{\circ}\) to \(10^{\circ}\)) [7] and Sentinel-2 Multi-Spectral Instrument (sun glitter imagery) [39], both of them only consider the tilt modulation. From the above analysis, we realized that the hydrodynamic modulation for InIRA is probably weak and complex. To reduce the analysis difficulty, we neglected the hydrodynamic modulation and only used the tilt, range bunching and velocity bunching modulations to explore the SWH empirical model. The possible implications for SWH retrieval due to the neglectwere arranged in Discussion. The following describes the specific analysis of the three modulation functions. From previous studies, the InIRA tilt modulation function \(T^{tilt}\) can be expressed as [4; 5; 6; 7]  
\[T^{tilt}(k)=\alpha(\theta)k_{x} \tag{1}\]  
where:  
\[\alpha(\theta)=\cot\theta-4 tan\theta-\frac{1}{cos^{2}\theta}\frac{\partial lnP( tan\theta,0)}{\partial tan\theta} \tag{2}\]  
The range bunching modulation function \(T^{rb}\) can be expressed as [37]  
\[T^{rb}(k)=\frac{\mathrm{i}k_{x}}{tan\theta} \tag{3}\]  
Here, \(k_{x}\) is the range component of wave number \(k\), \(\alpha\) is a scaling factor, \(\theta\) is the incidence angle, and \(P\) is the probability density function of wave slopes.