# First Retrieval of Sea Surface Currents Using L-Band SAR in Satellite Formation
## 4 Data Processing
### Cross-Track Baseline Component Phase Removal
To accurately calculate \(\theta_{L}\), it is essential to determine the precise position vector of each pixel in the SAR image within WGS-84; the coordinate system can first be established according to (16). \[\left\{ \begin{array}{l}\frac{(X_{p}+Y_{p})^{2}}{(\theta\cdot\beta)^{2}}+ \frac{Z_{p}^{2}}{(b\cdot\beta)^{2}}=1\\ R=|\mathbf{P}-\mathbf{S}|\\ f_{\text{Dop}}=2\mathbf{V}\cdot(\mathbf{P}-\mathbf{S})/\lambda\\ \beta=1+h_{\text{sea}}\,/\,R_{e}\end{array} \right. \tag{16}\]  
where \(\mathbf{P}=\big{(}x_{p},y_{p},z_{p}\big{)}\) is the position vector of the ground point in the WGS-84 coordinate system, \(a\) and \(b\) represent the semi-major and semi-minor axes of the Earth, respectively, \(R\) is the distance vector from the platform to the sea surface target, and \(S\) is the platform position vector at each azimuth moment. \(H_{sea}\) indicates the local elevation of the sea surface, and \(R_{e}\) refers to the Earth's localized radius. \(\mathbf{P}=\big{(}x_{p},y_{p},z_{p}\big{)}\) in (16) can be solved using [PERSON]'s iteration method. Furthermore, \(\theta_{L}\) can be precisely calculated in accordance with (17) and (18). \[\cos\beta_{e}=\frac{R_{e}^{2}+(R_{e}+h)^{2}-R^{2}}{2R_{e}(R_{e}+h_{\text{sea}} )}. \tag{17}\]  
\[\sin\theta_{L}=\frac{R_{e}\cdot\sin\beta_{e}}{R}. \tag{18}\]  
where \(\beta_{e}\) represents the local geocentric angle. The meanings of the other parameters are the same as before. \[\varphi=\frac{2\pi B_{XT}\sin(\theta_{L}-\alpha)}{\lambda}+\varphi_{0}= \varphi_{\text{fringe}}\,+2\pi\text{n} \tag{19}\]  
\[(B_{XT},\hat{\alpha},\varphi_{0})=\arg\min\Bigl{[}\varphi_{\text{fringe}}\,- \varphi(B_{XT},\alpha,\varphi_{0})\Bigr{]} \tag{20}\]  
After determining the value of \(\theta_{L}\) using the aforementioned steps, this value of \(\theta_{L}\) is substituted into (14) to obtain the interference phase after initially removing the cross-track interferometric phase. Figure 9a shows the phase diagram after the initial removal using the orbit information provided by LuTan-1. Compared to Figure 7b, the frequency of the interferometric fringes is significantly reduced. However, there are still significant cross-track interferometric stripes. Two potential sources of this error were identified. The first stems from discrepancies in the altitude, including the satellite altitude and sea surface height. It is worth emphasizing that the height error alone is insufficient to account for the observed consistently shifting stripes. The problem may instead be attributed to baseline errors, encompassing variationsin both baseline length and inclination angle. Therefore, it is recommended that refinement techniques be employed to enhance the elimination of the cross-track interferometric phase. Based on the geometric relationships, the expression for the cross-track baseline phase component is given by (19), where \(\varphi_{\text{fringe}}\) represents the interferometric phase fringes in the actual data. It can be regarded as the phase fringes after applying the orbit-based method, as shown in Figure 9a. The interferometric fringes in the figure contain the calibration parameter information for the LuTan-1 system. First, the phase data in Figure 9a undergo multi-looking to reduce phase noise. Since (19) describes the unwrapped phase, the phase in Figure 9a must be unwrapped. Based on the characteristics of Figure 9a, the unwrapping is performed in both the azimuth and range directions, yielding the unwrapped phase. Subsequently, the correlation coefficients from Figure 7a are used as weights for a weighted least-squares fit between the unwrapped phase and the theoretical phase described by (19). By adjusting the cross-track baseline length, tilt angle, and initial phase, the best-fit parameters are obtained when the root-mean-square error (RMSE) between the actual and fitted phases is minimized, thus satisfying the condition in (20). This process is repeated for each azimuth gate.