# Ocean Surface Topography Altimetry by Large Baseline Cross-Interferometry from Satellite Formation
## 2 Measurement Principle  
### Altimetry Principle  
The altimetry principle of IRA is basically triangulation, inherited from InSAR. As shown in Figure 1, two antennas measure the ranges to the target at slightly different geometry due to the existence of baseline, \(B\), the range difference, \(\Delta r\), related to the target height is recorded by the interferometric phase [6]. Based on triangulation, the height of the target is expressed as:  
\[h=H-r_{1}\cos\theta \tag{1}\]  
where \(H\) is the altitude of antenna phase center on Sat1, \(r_{1}\) is the range between the target and the phase center, and \(\theta\) is the radar look angle from Sat1 to the target. Altitude, \(H\), and range, \(r_{1}\), can be precisely measured by other means, only leaving the look angle, \(\theta\), which varies with the target height unknown. However, the interferometric phase is directly related to the look angle as:  
\[\theta=\alpha+\arcsin\frac{16\pi^{2}B^{2}+8\pi r_{1}\phi\lambda-(\phi\lambda) ^{2}}{32\pi^{2}Br_{1}} \tag{2}\]  
where \(\phi\) is the so-called interferometric phase, i.e., the phase difference between two complex images of IRA, \(B\) is the baseline length, \(\alpha\) is the baseline inclination angle, and \(\lambda\) is the radar wavelength determined by center carrier frequency. Based on Equations (1) and (2), the target height can be expressed as [23]:  
\[ \begin{split}& h=H-r_{1}\cos\Bigl{(}\alpha+\arcsin\frac{16\pi^{2}B ^{2}+8\pi r_{1}\Delta\phi\lambda-(\Delta\phi\lambda)^{2}}{32\pi^{2}Br_{1}} \Bigr{)}\\ &\approx H-r_{1}\cos\Bigl{\{}\alpha+\arcsin\frac{\Delta\phi \lambda}{4\pi B}\Bigr{\}}\end{split}  \tag{3}\]  
the approximation on the right side of the equation is true when \(B<<r_{1}\). The baseline length of IRA is no larger than a few kilometers, which meets the approximation condition. The approximated equation can be used to derive altimetric error transfer functions for simplification, however, only the non-approximation equation should be used to calculate the target height, since any approximation may introduce major altimetric errors. Figure 1: Schematic diagram of triangulation altimetry of Imaging Radar Altimeter (IRA). The altimetric accuracy is determined by all kinds of errors, the most significant ones are baseline length error, baseline inclination error, and interferometric phase error. Based on Equation (3), the altimetric error transfer functions are derived from partial differentiation:  
\[\sigma_{h}^{B}\approx\frac{H\tan\theta}{B\cot(\theta-\alpha)}\cdot\sigma_{B}= \frac{X}{B\cot(\theta-\alpha)}\cdot\sigma_{B} \tag{4}\]  
\[\sigma_{h}^{\alpha}\approx H\tan\theta\cdot\sigma_{\alpha}=X\cdot\sigma_{\alpha} \tag{5}\]  
\[\sigma_{h}^{\phi}\approx\frac{H\tan\theta\lambda}{4\pi B\cos(\theta-\alpha)} \cdot\sigma_{\phi}=\frac{X\lambda}{4\pi B\cos(\theta-\alpha)}\cdot\sigma_{\phi} \tag{6}\]  
where \(X\) is the target position along the range direction. Equation (4) indicates that the altimetric error introduced by baseline length error, \(\sigma_{B}\), is affected by the baseline length, baseline inclination, altitude, and radar look angle. The selection of altitude and baseline inclination has limited degrees of freedom; therefore, small look angle and large baseline should be set to reduce the altimetric error. Equation (5) indicates that small look angle decreases the altimetric error induced by baseline inclination error, \(\sigma_{\alpha}\), and the altimetric error increases linearly with target position along the range direction.