# Measuring Ocean Surface Current in the Kuroshio Region Using Gaofen-3 SAR Data
## 2 Measurement of the Ocean Surface Current  
### Ocean Doppler Information of Single Antenna SAR Data  
Due to the relative motion between the SAR platform and the ocean surface, the azimuth Doppler information of the echo signal would change, suggesting that the Doppler frequency can be used to retrieve the ocean surface motion. In fact, considering the complexity of ocean motion and the particularity of SAR for ocean imaging, the estimated Doppler centroid shift in SAR ocean data usually includes the following items [18]:  
\[f_{dc}=f_{dc}^{phys}+f_{dc}^{geo}+f_{dc}^{ele}+\triangle f_{dc}. \tag{1}\]  
In Equation (1), \(f_{dc}^{phys}\) is the geophysical term, \(f_{dc}^{geo}\) is the geometric attitude term, \(f_{dc}^{ele}\) indicates electronic misdirection of antenna, \(\triangle f_{dc}\) represents the residual error in data processing. As one of our concerns, \(f_{dc}^{phys}\) is proportional to the ocean surface Doppler velocity in the line-of-sight direction, which contains not only the information of ocean currents, but also the contributions of wind and wave motion. In Bragg theory, wind and wave motion primarily involve Bragg wave phase velocity and large-scale wave orbit velocity. Therefore, \(f_{dc}^{phys}\) can be expressed as  
\[f_{dc}^{phys}=f_{c}+f_{b}+f_{o}, \tag{2}\]where \(f_{c}\), \(f_{b}\) and \(f_{o}\) denote the Doppler centroid frequency variation caused by ocean currents, Bragg wave phase velocity, and large-scale wave orbit velocity, respectively. How to estimate \(f_{dc}^{\text{phys}}\) from \(f_{dc}\) has been studied in detail in reference [8], and the relevant data processing process will not be discussed. The focus of this paper is to separate \(f_{c}\) from \(f_{dc}^{\text{phys}}\) and further extract the ocean current information. According to the geometric relationship between ocean current and SAR platform shown in Figure 1, \(f_{c}\) can be expressed as:  
\[f_{c}=\frac{2}{\lambda}(u_{x}\cos\varphi\sin\theta+u_{y}\sin\varphi\sin\theta), \tag{3}\]  
where \(u_{x}\) and \(u_{y}\) represent the azimuth and range velocity components of the current, respectively. \(\varphi\) is the angle between the beam center and the flight direction in the horizontal plane, which is called the azimuth angle. \(\theta\) is the incidence angle, and \(\lambda\) is the electromagnetic wavelength. We can also obtain the Doppler frequency caused by platform movement as:  
\[f_{p}=\frac{2v_{p}}{\lambda}\cos\varphi\sin\theta, \tag{4}\]  
where \(v_{p}\) is the radar speed. In Bragg scattering theory, Bragg resonance is the main SAR imaging mechanism for the ocean surface. Furthermore, the phase velocity of the Bragg wave is the main factor affecting the deviation of current measurements, which is expressed as [19; 20]:  
\[c=\sqrt{g/k_{b}+Tk_{b}}, \tag{5}\]  
where \(g\) is the acceleration of gravity, and \(T\) is the ratio of water surface tension to seawater density. \(k_{b}=2\pi/\lambda_{b}\) is the wavenumber of Bragg waves, which can be expressed by electromagnetic wavenumber \(k_{e}\) and incidence angle \(\theta_{e}\) as  
\[k_{b}=2k_{e}\sin\theta_{e}. \tag{6}\]  
Therefore, the Doppler centroid variation caused by the Bragg wave phase velocity is  
\[f_{b}=\frac{2}{\lambda}\sqrt{g/k_{b}+Tk_{b}}. \tag{7}\]  
In the composite surface model, as one form of Bragg theory, there are not only small-scale Bragg waves but also large-scale modulating waves on the ocean surface. In this  
Figure 1: Schematic diagram of the geometric relationship between current and radar. situation, the large-scale waves will be displayed in SAR images by modulating the energy distribution of Bragg waves, and the Doppler centroid will also shift with the variation of large-scale wave orbital velocity, that is, there is \(f_{o}\), which also results in the deviation of current measurements. Generally, the orbital velocity is affected by the weighted average of the local backscatter within the SAR resolution cell and during the SAR integration time [4; 5].