# Measuring Ocean Surface Current in the Kuroshio Region Using Gaofen-3 SAR Data
## 2 Measurement of the Ocean Surface Current
### Measurement Method with Single Antenna SAR Data
#### 2.2.3 Vector Estimation of Ocean Current  
In this section, we assume that the Doppler frequency caused by the Bragg wave phase velocity is the same at the same spatial position in each sub-aperture image. In addition, in order to improve the efficiency of the algorithm, we deal with the influence of orbital velocity through space averaging, and consider that the residual errors in the same Doppler estimation block are the same. Set the Doppler frequency shift caused by the influence of Bragg wave phase velocity and residual orbit velocity to \(f_{B}\), and combine Equations (8) and (9), then we can get:  
\[f_{i,j}^{k} =\triangle f_{a}^{k}+f_{c}^{k}(i,j)+f_{B}(i,j) \tag{12}\] \[=-\frac{2}{\lambda}v_{p}\triangle\varphi\sin\varphi_{k}\sin\theta\] \[+\frac{2}{\lambda}\left(u_{x}(i,j)+\triangle\varphi u_{y}(i,j) \right)\cos\varphi_{k}\sin\theta\] \[+\frac{2}{\lambda}\left(u_{y}(i,j)-\triangle\varphi u_{x}(i,j) \right)\sin\varphi_{k}\sin\theta+f_{B}(i,j),\]  
where \(f_{i,j}^{k}\) represents the Doppler results of the data block at the central coordinate \((i,j)\) of  
Figure 4: (**a**) Doppler power spectrum with sub-apertures bandwidth. (**b**) Weighting function used for Doppler centroid estimation.  
the \(k\)-th sub-aperture image. Let \(\mathbf{Y}_{i,j}\) represent the vector composed of Doppler centroids of all \(L\) sub-aperture resolution units at spatial coordinates \((i,j)\), expressed as  
\[\mathbf{Y}_{i,j}=[f_{i,j}^{1},f_{i,j}^{2}\cdots f_{i,j}^{k}\cdots f_{i,j}^{L}]^{ \mathsf{T}}. \tag{13}\]  
Then, the overdetermined linear equations are established and expressed in matrix form as:  
\[\mathbf{Y}_{i,j}=\mathbf{W}\mathbf{X}_{i,j}, \tag{14}\]  
where the coefficient matrix \(\mathbf{W}\) and the unknown vector \(\mathbf{X}_{i,j}\) containing the current information are respectively expressed as  
\[\mathbf{W}=\frac{2\sin\theta}{\lambda}\begin{bmatrix}-v_{p}\sin\varphi_{1}& \cos\varphi_{1}&\sin\varphi_{1}&\lambda/2\sin\theta\\ -v_{p}\sin\varphi_{2}&\cos\varphi_{2}&\sin\varphi_{2}&\lambda/2\sin\theta\\ \vdots&\vdots&\vdots&\vdots\\ -v_{p}\sin\varphi_{L}&\cos\varphi_{L}&\sin\varphi_{L}&\lambda/2\sin\theta \end{bmatrix} \tag{15}\]  
\[\mathbf{X}_{i,j}=[\bigtriangleup\varphi,u_{x}(i,j)+\bigtriangleup\varphi u_ {y}(i,j),u_{y}(i,j)-\bigtriangleup\varphi u_{x}(i,j),f_{y}(i,j)]^{\mathsf{T}}. \tag{16}\]  
Then, after the least squares method is used to solve \(\mathbf{X}_{i,j}\), the azimuth velocity component \(u_{x}\) and range velocity component \(u_{y}\) corresponding to the data block at coordinates \((i,j)\) can be expressed as  
\[u_{x}=\frac{X_{i,j}(2)-X_{i,j}(1)X_{i,j}(3)}{1+X_{i,j}(1)^{2}} \tag{17}\]  
\[u_{y}=\frac{X_{i,j}(1)X_{i,j}(2)+X_{i,j}(3)}{1+X_{i,j}(1)^{2}}. \tag{18}\]  
Therefore, the current velocity \(u_{c}\) and direction \(\phi_{c}\) can be obtained:  
\[u_{c}=\sqrt{u_{x}^{2}+u_{y}^{2}}=\sqrt{\frac{X_{i,j}^{2}(2)+X_{i,j}^{2}(3)}{1+ X_{i,j}^{2}(1)}}. \tag{19}\]  
\[\phi_{c}=\arctan\biggl{(}\frac{u_{x}}{u_{y}}\biggr{)}=\arctan\biggl{(}\frac {X_{i,j}(2)-X_{i,j}(1)X_{i,j}(3)}{X_{i,j}(1)X_{i,j}(2)+X_{i,j}(3)}\biggr{)}. \tag{20}\]  
The current field corresponding to the whole SAR image can be measured by solving the current vector of each spatial position in turn.