# 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
### Calibration Method of Phase Imbalance between Channels  
For phase imbalance calibration of azimuth multi-channel (AMC) SAR, researchers have done much research and put forward many methods such as the orthogonal subspace method [25], the signal subspace comparison method [26], etc. However, the space-varying of phase imbalance error is not considered. In the SAR instrument with joint XTI-ATI mode, the known elevation and velocity of land or ships are needed for phase imbalance calibration. In [27], the phase imbalance introduced by surface height was removed by the high precision DEM information, and the phase trend in range was removed using a second-order polynomial fit. In [28], the phase imbalance was calibrated by using the ships as the reference of the known velocity. However, since the satellite attitude and velocity will change with the azimuth time and the antenna phase center position error and antenna phase pattern are unknown, the cross-track baseline error is uncertain. Therefore, the phase imbalance cannot be calculated by a theoretical formula. According to [29; 30], there is a method to calibrate the trajectories of SAR systems using the multisquint phase. In this paper, a method based on SAR echo data without the actual satellite state was used to remove the space-varying of phase imbalance. Through proper block processing in the azimuth and range, the echo signal delay and phase imbalance between two channels are solved by using the least squares method in the Doppler frequency domain, where the land scene is needed as a reference. For stationary targets, the dual channel echo signal satisfies the following formula in the Doppler frequency domain [14]:  
\[\angle E\{S_{1}^{*}(\tau,f_{a})\cdot S_{2}(\tau,f_{a})\}=2\pi\Delta\eta_{21}^{ \prime}f_{a}+\Delta\phi_{21}, \tag{23}\]  
where \(f_{a}\) is the Doppler frequency and \(\Delta\phi_{21}\) represents the phase imbalance between channels caused by various errors. \(\angle E\{S_{1}^{*}(\tau,f_{a})\cdot S_{2}(\tau,f_{a})\}\) is a linear function of the Doppler frequency \(f_{a}\), where the coefficient of the first term represents the time delay of the received echo of the two channels and the constant term represents the phase imbalance of the two channels. Therefore, image registration and moving target velocity measurement \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pitch Error** & **Yaw Error** & \(\Delta x_{max}\) & \(\Delta z_{max}\) \\ \hline \(-0.03^{\circ}\) & \(0.03^{\circ}\) & \(-3.93\) mm & \(-3.93\) mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The max attitude control errors of GF-3. Figure 4: The change of phase imbalance in range caused by maximum attitude control errors of GF-3. can be carried out directly by using the calculated \(\Delta\eta_{21}^{\prime}\), without knowing the exact along-track interferometric baseline and satellite velocity. The steps of phase imbalance error calibration are shown in Figure 5. The process flow is as follows:  
(1) According to the appropriate block size and step size, the dual channel images are divided into blocks along the azimuth and the range. The premise of this is to ignore the space-time variation of the phase imbalance between channels in each block. (2) The azimuth FFT is applied to each block image of the dual channel, and the corresponding conjugate multiplication is performed in the Doppler frequency domain and averaged along the range. Finally, the phase change along the Doppler frequency is obtained. (3) The curve near zero frequency is intercepted and solved by the least squares method to obtain the time delay and phase imbalance between each small piece of dual channel data, which is to avoid the influence of high-frequency noise. (4) Image registration and phase imbalance calibration are performed by using the time delay and phase imbalance of dual channel data calculated in the previous step. (5) Finally, further moving target detection and velocity measurement are carried out.