# 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
#### 2.2.2 Doppler Centroid Estimation  
Accurate Doppler centroid estimation of real data is the basis of subsequent processing through the above analysis. According to Reference [29], the methods used for Doppler centroid estimation include the energy balance method, the power spectrum correlation method, the correlation Doppler centroid estimation method and the optimal estimation method. They use different weighting functions \(B(f)\). The results of the optimal estimation method are more accurate, very close to the Cramer-Rao bound, so this paper uses this method to estimate Doppler centroid. The weighting function of optimal estimation method is shown in Equation (11), where \(E(f)\) denotes the Doppler power spectrum. According to [29], Figure 4 shows \(E(f)\) and \(B(f)\), where \(F_{a}\) is pulse repeat frequency (PRF). In Figure 4a, we show the Doppler bandwidth of the sub-aperture image, which corresponds to \(S\) in Figure 3. In data processing, we use a band-pass filter to extract the sub-aperture frequency band from the azimuth spectrum. The specific process is referred to in Reference [30]. The frequency point at the center of each sub-aperture bandwidth corresponds to a sub-aperture azimuth angle, and the sub-aperture azimuth angles at both ends are the extreme values, so we can get  
\[B(f)=\frac{E^{\prime}(f)}{E^{2}(f)}. \tag{11}\]  
As shown in Figure 4, the Doppler power spectrum \(E(f)\) is an even function, while the weighting function \(B(f)\) is an odd function. The convolution of the two functions has only one zero-crossing point, which is then calculated by mathematical methods such as the least squares. The zero-crossing point is the Doppler centroid frequency. We convolute  
Figure 3: Schematic diagram of sub-aperture processing.  
the Doppler spectrum of each sub-aperture with the weighted function in Figure 4b to find the zero-crossing point, so as to estimate the Doppler centroid.  
In order to accurately estimate the Doppler centroid in actual data, enough sampling points need to be taken in the azimuth and range direction for calculation. Therefore, it is necessary to block sub-aperture image data for Doppler estimation and eliminate the influence of non-geophysical terms. Finally, an average Doppler frequency for several kilometers' range of ocean surface is acquired, which is expressed as \(f_{ij,j}\) (\(i=1,2,\cdots,M\), \(j=1,2,\cdots,N\), \(M\) and \(N\) represent the number of blocks in azimuth and range direction of each sub-aperture datapoint, respectively). Generally, there is a time interval between sub-aperture images, and the variation of the resulting ocean surface is usually at the pixel level. However, the Doppler estimation of sub-apertures requires thousands of pixels to obtain the mean motion state of the ocean surface over several kilometers, so it is considered that the changes of Doppler frequency between different sub-apertures are mainly caused by different beam directions, rather than the ocean surface evolution.  
It should be noted that the Doppler centroid estimation process in this paper acts on the correctly imaged single look complex data for the purpose of current inversion. In some of the literature, the Doppler estimation process acts on SAR raw data to improve imaging quality, such as in [31,32]. Compared with imaging processing, this paper pays more attention to the application process of imaging data. In addition, the Doppler estimation algorithm in References [31,32] is applied to airborne SAR data with large squint and high contrast. In this paper, the squint angle of the Gaofen-3 satellite is close to zero, and the SAR data scattering is uniform and the contrast is low.