# Performance Analysis of Ocean Surface Topography
## 2 Altimetry Simulation of OST
### Altimetry Principle of OST  
Before the simulation of altimetric precision, we briefly introduce the OST altimetry principle. As shown in Figure 3, the baseline of InIRA is perpendicular to its orbital track, and \(\alpha\) is the baseline tilt angle. Antenna A1 sends pulses to the ocean surface, and both of the antennas receive the reflected pulses; \(r_{1}\) and \(r_{2}\) are the distances between the antennas and point P, respectively. This one-transmit-two-receive working mechanism makes the effective baseline half of its physical length [10]. The received pulses of A1 and A2 are processed into two SAR complex images separately. Because of the return path difference \(\Delta r\), which is caused by the existence of baseline, the corresponding phases of point P in the complex images are different. Therefore, the relationship between phase difference \(\phi\) and return path difference \(\Delta r\) can be expressed by:  
\[\phi=\frac{2\pi(r_{2}-r_{1})}{\lambda}=\frac{2\pi\Delta r}{\lambda}=\frac{2 \pi B\sin(\theta-\alpha)}{\lambda} \tag{2}\]  
where \(\lambda\) is the wavelength, \(B\) is the physical baseline length, and \(\Delta r\) is the return path difference, which can be approximated by the parallel component of the baseline along the look direction.  
After acquiring the phase difference \(\phi\), with precise knowledge of \(r_{1}\), the height of point P above the reference plane (could be the local mean sea level) can be deduced by:  
\[h=H-r_{1}\cos\theta \tag{3}\]  
where \(H\) is the altitude of Tiangong II above the reference plane, \(r_{1}\) is measured by timing the signal round trip delay, and \(\theta\) is the incidence which is related to the phase difference \(\phi\).  
Figure 2: (**a**) Swath width and (**b**) ground-range resolution of InIRA within the incidence range.  
According to the geometrical relationship in Figure 3, two equations are given:  
\[\cos\beta=\frac{r_{1}^{2}+B^{2}-{r_{2}}^{2}}{2r_{1}B}\approx-\frac{\Delta r}{B} \tag{4}\]  
\[\beta=\frac{\pi}{2}+\alpha-\theta \tag{5}\]  
where \(\beta\) is the angle between the baseline and view direction. Combining Equations (2)-(5), the relationship between height \(h\) and phase difference \(\phi\) is given by:  
\[h=H-r_{1}\cos(\frac{\pi}{2}+\alpha-\arccos(-\frac{\lambda\phi}{2\pi B})) \tag{6}\]  
In fact, Equation (1) is deduced from the differential of Equation (6).  
The interferometric phase \(\phi_{12}\) which is derived from the phase difference between two SAR complex images, cannot be directly translated into ocean surface height. According to Equation (2), the return path difference \(\Delta r\) can be expressed by the number of wavelengths, normally \(\Delta r\) is longer than one wavelength. Whereas the phases recorded by complex images are confined within \([-\pi,\pi]\), this means the phases are wrapped. The relationship between unwrapped phase \(\phi\) and interferometric phase \(\phi_{12}\) is given in Equation (7). Phase wrapping leads to the missing of integer number \(n\), so phase calibration must be done before acquiring the absolute ocean surface height.  
\[\phi=2\pi(\text{n}+\frac{\phi_{12}}{\lambda}) \tag{7}\]