# Performance Analysis of Ocean Surface Topography
## 2 Altimetry Simulation of OST
### Altimetric Precision Simulation  
In Equation (1), \(\Delta h\) can also be regarded as the relative height change above a reference sea level, \(\Delta\phi\) is the corresponding phase change. Since phases are confined within \([-\pi,\pi]\), a too-large relative height change will cause phase ambiguity, also referred to as phase wrapping [10]. The altimetric  
Figure 3: Ocean surface topography (OST) measurement principle (slant-range profile). ambiguity is defined as the relative height change corresponding to a phase ambiguity of 2\(\pi\). When \(\Delta\phi~{}=~{}2\pi\) in Equation (1), then the altimetric ambiguity is given by:  
\[h_{2\pi}=\frac{\lambda r_{1}\sin\theta}{B\cos(\theta-\alpha)} \tag{8}\]  
Obviously, a longer baseline, smaller incidence and shorter wavelength will decrease the altimetric ambiguity or, in other words, cause the phase to be wrapped more easily by the same relative height change. Phase unwrapping is difficult, hence the altimetric ambiguity of InIRA should be large enough to avoid this problem. Figure 4a shows that the altimetric ambiguity has changed over eight times from -64 m in the near range to -518 m in the far range. This indicates that a relative height change larger than ~32 m in the near range will wrap the phase (since the corresponding phase is larger than \(\pi\)). However, phase wrapping is quite unlikely to happen, since ocean surface height variation is usually only within a few meters. While a too-large altimetric ambiguity also brings a challenge to InIRA, for a 10-cm relative height change, the corresponding phase change is only 0.01 rad in the near range, then drops down to 0.001 rad in the far range, as shown in Figure 4b. This indicates that the phase error in the near range should be better than 0.01 rad to detect the relative height change effectively, and the phase error in the far range should be even 10 times better than in the near range. The stringent requirement on phase error demands that the system thermal noise be constrained at a certain level, and the data processing must be extremely accurate. According to the analysis mentioned above, phase error will eventually determine the altimetric precision of InIRA. Phase errors are mainly from systematic errors and random errors. Systematic errors are constant or slow drifting, such as differential phase drift error or baseline roll angle error, most of which can be eliminated through calibration. In this paper we focus on the relative altimetric precision of ocean surface height, thus only random errors are considered during the simulation. Random phase error is introduced in the simulation by multiple decorrelation factors. For any InSAR, thermal noise influences phase error the most [21], the thermal correlation factor \(\gamma_{N}\) is given by:  
\[\gamma_{N}=\frac{SNR}{SNR+1} \tag{9}\]  
where \(SNR\) is the signal-to-noise ratio. Benefitting from low orbit and near-nadir incidence, the SNR of InIRA received data is very high within most of its swath, as show in Figure 5a (the SNR  
Figure 4: (a) Altimetric ambiguity of InIRA corresponds to a phase ambiguity of 2\(\pi\); (b) Altimetric sensitivity of InIRA corresponds to a 10-cm ocean surface height change. curve is derived from the acquired data of InIRA), and high SNR guarantees high coherence for the interferometric data. Apart from thermal decorrelation, the presence of ocean waves will introduce a volume-scattering layer on the angle subtended by a range resolution grid, thereby introducing the volumetric decorrelation [22], which is given approximately by:  
\[\gamma_{v}\approx e^{-2(\frac{2\pi\gamma_{h}B}{\gamma_{1}4\Delta\varpi})^{2}} \tag{10}\]  
where \(\sigma_{h}\) is the ocean surface height standard deviation, related to the significant wave height (SWH) as \(\text{SWH}~{}=~{}4\sigma_{h}\). The simulated volumetric decorrelation coefficient is shown in Figure 5b. It obviously has a much bigger influence in the near range because of the steeper incidence, and it decreases significantly as the SWH increases.