## 3 Modeling
Based on the spectra distribution, we believe that the first case (Figure 2a,b) is for a swell from the concentrated energy distribution, and the second case (Figure 2c,d) is for crossing wind waves from the dispersed energy distribution. Here, the swell case shows clearer wave-like stripes than the crossing wind wave case due to the mixing of the two wave systems. Both cases exhibit clear image spectra. This indicates that the InIRA can image ocean waves similar to the SAR. From the image spectra presented in Figure 2b,d, there is a weak cutoff phenomenon in the azimuth similar to that of the SAR, but the spectra patterns are still relatively complete (the spectra edges are not cutoff abruptly). In the SAR ocean wave imaging theory, the cutoff is attributed to the azimuthal displacement of the apparent position of a backscattering element. It determines the minimum detectable wavelength for the InIRA or SAR data. The cutoff wavelength in Figure 2c is only approximately 80 m, whereas the conventional SAR cutoff is approximately 150-200 m in the azimuth direction [7]. This shows that the InIRA cutoff is weaker than that of the SAR. As the InIRA has a lower satellite altitude (between 380 and 400 km) than the conventional SAR (with an altitude of approximately 755 km for the Gaofen-3 SAR), which brings a small cutoff factor as described in Equation (56) of [9]. To a certain extent, it helps to reduce the cutoff effects in Equations (8) and (9). Based on the estimated image spectra, the two integration factors (\(IF_{range}\) and \(IF_{azimuth}\)) were calculated using Equations (14) and (15). Note that only the image spectra with wavelengths from 30 m to 500 m were preserved in the calculation, while the rest were removed as interference information. Then, some dependence analyses were performed for confirming the relation between SWH and the two integration factors. First, Figure 3a,b illustrate the dependence of SWH on \(IF_{range}\) and \(IF_{azimuth}\), respectively. To ensure that the analysis of \(IF_{range}\) was performed in the cases dominated by tilt and range bunching modulation, only the data close to the range direction (with relative wave directions between 0\({}^{\circ}\) and 15\({}^{\circ}\)) were used. Similarly, the analysis of \(IF_{azimuth}\) was performed using data close to the azimuth direction (with relative wave directions between 75\({}^{\circ}\) and 90\({}^{\circ}\)), making the velocity bunching the dominant modulation. The relative wave direction was defined as the angle between the wave direction (from WW3 data) and range direction (from InIRA auxiliary data). By using 180\({}^{\circ}\) symmetry feature of radar returns, the relative wave direction was converted and concentrated into the range of 0\({}^{\circ}\)-90\({}^{\circ}\). The concentration of data helps plotting clear trends, especially when the data are limited. From the two figures, the SWH increases with both \(IF_{range}\) and \(IF_{azimuth}\) with an approximate one-order function. This is basically consistent with the theoretical dependence in Equations (16) and (17). Comparing the two, the dependence on \(IF_{range}\) is clearer. These dependences reveal the potential of using the \(IF_{range}\) or \(IF_{azimuth}\) factor to build an SWH model. We further conducted a similar analysis for Figure 3c,d; however, the data used were exchanged with each other. In this case, Figure 3c shows a bifurcated trend, whereas the trend in Figure 3d is more dispersed than that in Figure 3b. Although there are still some dependences, they are significantly weaker than those in Figure 3a,b. This means that the \(IF_{range}\) factor is more suitable for describing waves in the range direction, whereas the \(IF_{bb}\) factor is more appropriate for those in the azimuth direction. This also indicates that the single factor defined in this study cannot describe waves in different directions simultaneously. The two factors should be considered together for estimating the SWH.