# Ocean Wave Inversion Based on a Ku/Ka Dual-Band Airborne Interferometric Imaging Radar Altimeter
## 2 Method  
### DInIRA Ocean Waves Inversion Methodology  
The DInIRA exploited two sets of antennas (viz. four antennas) emitting different frequency bands for observation, and the pulse signals were emitted toward the ground-range direction at 6-18\({}^{\circ}\) (Ku) and 4-17\({}^{\circ}\) (Ka) incidence angles, respectively. As shown in Figure 1, the DInIRA measurement principle is similar to that in InSAR sensors.  
Taking the Ku band (in black) as an example, where \(A_{1}\) is the primary antenna and \(A_{2}\) is the auxiliary antenna, \(B\) denotes the baseline length, \(\alpha\) refers to the baseline roll angle, and \(r_{1}\) and \(r_{2}\) are the distances from the primary and auxiliary antennas to point \(P\), respectively. The angle of incidence relative to the target point \(P\) on the sea surface is \(\theta_{1}\), the \(alt\) is the altitude of \(A_{1}\) above the WGS-84 reference ellipsoid, and the height of the target point \(P\) is \(h\). The different phases are further measured by the \(A_{1}\) and \(A_{2}\) antennas due to the existence of the baseline. Therefore, the relationship between the slant range and the interference phase difference \(\varphi\) can be expressed as [27]:  
\[\varphi=\frac{2\pi(r_{2}-r_{1})}{\lambda}=\frac{2\pi B\sin(\theta-\alpha)}{\lambda} \tag{1}\]  
where \(\lambda\) represents the return echo wavelength and the interference phase \(\varphi\) can be obtained by the interferometric processing. Finally, the elevation of the target point \(P\) can be calculated according to Equation (2) by combining the information of the primary antenna parameters (including antenna height, baseline roll angle, and baseline length) [27]:  
\[h=alt-r_{1}\cos\left(\frac{\pi}{2}+\alpha-\arccos\left(-\frac{\lambda\varphi} {2\pi B}\right)\right) \tag{2}\]  
Figure 1: Principle of height measurement (DInIRA).  
Then, the SSH is inversed using the abovementioned methods, and the 2D OWS can be subsequently calculated based on the 2D Fourier transform (_FFT_2) [6]:  
\[S\big{(}k_{x},k_{y}\big{)}=\frac{\left|FFT2(\Delta h)\right|^{2}}{mn\Big{(} \frac{4\pi^{2}}{res_{y}}\Big{)}} \tag{3}\]  
where \(k_{x}\) and \(k_{y}\) denote the wavenumber vectors in the ground-range (cross-track) direction and azimuth direction, respectively; \(m\) and \(n\) denote the number of pixels in two directions; and \(res_{x}\) and \(res_{y}\) refer to the pixel resolution in different directions. Finally, the ocean wave parameters (OWPs) are further extracted via the OWS [28]:  
\[\text{SWH}=4.005\times\sqrt{\iint S\big{(}k_{x},k_{y}\big{)}dk_{x}dk_{y}} \tag{4}\]  
\[\text{MWW}=2\pi/k_{p} \tag{5}\]  
where \(k_{p}\) shows the wave number corresponding to the maximum spectral value in the OWS, and the WPD denotes the angle relative to the forward direction of the \(k_{x}\) axis once \(k=k_{p}\).  
In addition, the main wave period (MWP) can be obtained based on the dispersion formula [29]:  
\[\text{MWP}=\frac{2\pi}{\omega_{p}} \tag{6}\]  
where \(\omega_{p}=\sqrt{gk_{p}\text{tanh}\big{(}hk_{p}\big{)}}\), \(g\) denotes the gravity constant, and \(h\) is the depth of water.