# First Retrieval of Sea Surface Currents Using L-Band SAR in Satellite Formation
## 4 Data Processing
### Ocean Current Inversion  
The phase of the ATI-SAR is directly proportional to the Doppler velocity of the ocean surface. After acquiring the along-track interference phase, the horizontal Doppler velocity can be derived using (4) and (5). Owing to the intricate characteristics of the ocean environment and the complicated SAR imaging mechanism for sea surfaces, the Doppler velocity obtained from the ATI phase serves as a comprehensive estimate of different velocities, as shown in (21). \[u_{\mathrm{d}}\,=u_{c}+u_{p}. \tag{21}\]  
where \(u_{c}\) denotes the sea surface current, encompassing the tidal currents and ocean currents. \(u_{p}\) denotes the pseudo current. In most cases, \(u_{c}\) is the physical quantity of concern, and the remaining components (\(u_{p}\)) can be regarded as interference terms. \(u_{p}\) is the sea surface velocity related to wind and waves. Specifically, it contains surface wind-generated currents, the orbital velocity of large-scale waves, and the net Bragg wave phase velocity, where wind-generated currents are the dominant disturbance terms. To obtain \(u_{c}\), it is crucial to reduce the effect of the wind field on the Doppler velocities. While the C-band Doppler frequency (CDOP) empirical model may effectively address this issue for C-band SAR, such as ENVISAT, Gaofen-3, and Sentinel-1 [50], no equivalent empirical model is currently available for L-band spaceborne InSAR systems such as LuTan-1. The modeling system for SAR sea surface simulation (M4S) [26] was used to remove the wind-generated currents to obtain a precise sea surface current field. The M4S model is an advanced tool used for SAR imaging simulations of ocean surfaces under various scenarios. It is highly efficient in simulating the SAR interference phase of a given sea surface current and wind field. Figure 10 is a schematic flowchart illustrating the correction inversion approach for generating the current field in ATInSAR based on the M4S model [51]. The initial step involves configuring crucial parameters for the current field correction procedure, such as the maximum number of iterations and predefined threshold values. Subsequently, the Doppler phase computed in Section 4.2, referred to as the initial-guess phase, along with wind field data, radar parameters, and platform parameters, were provided as inputs to the M4S model to calculate the initial-guess current field. Following processing by two modules, M4 Sw and M4 Sr, which, respectively, handle the computation of the wave spectrum and the calculation of the current field, a comparative analysis was conducted between the simulated and initial-guess current fields. The process of determining the \"optimal current field\" followed specific criteria. The process terminated when the RMSE of two adjacent iterations was lower than the set threshold and gradually tended to a specific value. Otherwise, the iteration continued unless the maximum number of iterations was exceeded. The current field underwent correction for the subsequent iteration. The corrected current field can be expressed as follows:  
\[u_{ij}^{n+1}=u_{ij}^{n}+\lambda\Delta u \tag{22}\]  
where \(n\) denotes the \(n\) th iteration, \(u_{ij}\) denotes the velocity of the \((i,j)\)th pixel, \(\lambda\) is the scaling factor, ranging from 0 to 1, and \(\Delta u\) signifies the speed difference between two successive iterations, where \(\lambda=0.238\) m, and the RMSE threshold is 0.015 rad. Figure 11 displays a graph of the RMSE change as the number of iterations increased. It is evident that the RMSE gradually decreased from 0.045 rad and approached 0.011 rad. Figure 11b shows a graph of the percentage of points that satisfy the convergence conditions as the number of iterations increases. The figure shows that the satisfaction rate varies from below 30% to over 95%, indicating that the retrieved current converged. Figure 12a shows the final alternative output current field obtained using the hybrid InSAR images of LuTan-1. Figure 11: (**a**) Variation in RMSE with number of iterations.