# Phase Imbalance Analysis of GF-3 Along-Track InSAR Data for Ocean Current Measurement
## 2 Ocean Current Measurement Method Based on GF-3 ATI Data
### Error Analysis of the Phase Imbalances between Channels  
The phase imbalances between channels for ATI-SAR are caused by many factors, such as the radar electronic system, the phase mismatch of the antenna pattern, satellite attitude errors, antenna electronic miss-pointing, antenna phase center position errors, sampling time delay error, and target elevation [13]. Therefore, the interferometric phase directly obtained from the two SAR images cannot truly reflect the Doppler information of the radar irradiation area. Firstly, the radar electronic system error will introduce constant phase errors. Secondly, due to the attitude error and antenna electronic miss-pointing, the squint angle will appear in SAR imaging, which directly leads to the inconsistent distance between the center of dual receiving channels and the target. Thirdly, the antenna phase center position of each receiving channel may change due to installation errors, thermal deformation, etc. Therefore, the antenna phase center position errors and satellite attitude errors will cause an additional cross-track baseline, which will produce phase imbalances that vary with the slant range and target elevation. Finally, sampling time errors between the two channels caused by inconsistent synchronous circuits and analog-to-digital devices will also lead to a dual-channel phase imbalance. In the range-Doppler domain, phase imbalance can be expressed as [23]:  
\[\angle E\{S_{1}^{*}(f_{\tau},\eta)\cdot S_{2}(f_{\tau},\eta)\}=2\pi f_{dc} \Delta\eta_{21}+2\pi f_{\tau}\Delta\tau_{21}+\Delta\phi_{21}, \tag{14}\]  
where \(\angle\) stands for calculation angle operation, \(f_{dc}\) is the Doppler centroid frequency, which can be measured by real data, \(f_{\tau}\) is the range frequency, \(\Delta\tau_{21}\) is the sampling time delay, and \(\Delta\phi_{21}\) is the phase imbalance caused by the baseline error and constant phase imbalance between channels. In short, after compensating the sampling time delay error, the space-varying phase imbalance may be caused by squint imaging, baseline error, and the phase mismatch of the antenna pattern. Although GF-3 ATI-SAR is a vertical side-looking imaging system, due to satellite attitude error, antenna electronic miss-pointing, and other reasons, it will be actually squint imaging. The geometric model with baseline error under the condition of squint imaging is shown in Figure 2, where \(\psi\) is the squint angle. Figure 2: The geometric model with baseline error under the condition of squint imaging. In the case of squint imaging with an antenna baseline error, which is \([\Delta x_{m},\Delta y_{m},\Delta z_{m}]\), the distance from the actual antenna receiving center to the corresponding point \(\Gamma\) can be expressed as:  
\[r_{1}(\eta)=\sqrt{(x_{0}-\Delta x_{1})^{2}+(V_{s}\eta-d/2+\Delta y_{1}-Y_{0})^{ 2}+(H-h+\Delta z_{1})^{2}}, \tag{15}\]  
\[r_{2}(\eta)=\sqrt{(x_{0}-\Delta x_{2})^{2}+(V_{s}\eta+d/2+\Delta y_{2}-Y_{0})^{ 2}+(H-h+\Delta z_{2})^{2}}. \tag{16}\]  
The phase imbalance between channels in the case of squint imaging caused by baseline error can be expressed as [13; 24]:  
\[\Delta\phi(\eta)= \frac{2\pi}{\lambda}(r_{2}(\eta)-r_{1}(\eta)) \tag{17}\] \[\approx\frac{2\pi}{\lambda}(\frac{V_{s}\cos\psi}{R_{0}}\eta( \Delta y_{2}-\Delta y_{1}))-(d+\Delta y_{2}-\Delta y_{1})\cdot\sin\psi\] \[+(\Delta z_{2}-\Delta z_{1})\cdot\frac{(H-h)\cos\psi}{R_{0}}-( \Delta x_{2}-\Delta x_{1})\cdot\frac{x_{0}\cos\psi}{R_{0}}).\]  
Let \(\Delta x=\Delta x_{2}-\Delta x_{1}\), \(\Delta y=\Delta y_{2}-\Delta y_{1}\), and \(\Delta z=\Delta z_{2}-\Delta z_{1}\). When \(\eta=0\), that is at the time of the Doppler centroid, the phase imbalance can be expressed as:  
\[\Delta\phi\approx\frac{2\pi}{\lambda}(\Delta z\cos\theta\cos\psi-\Delta x\sin \theta\cos\psi)+2\pi f_{dc}\Delta\eta_{21}^{\prime}, \tag{18}\]  
where \(\theta\) is the look angle, \(f_{dc}=-\frac{2V_{s}}{\lambda}\sin\psi\) and \(\Delta\eta_{21}^{\prime}=(d+\Delta y_{2}-\Delta y_{1})/2V_{s}\).