# First Retrieval of Sea Surface Currents Using L-Band SAR in Satellite Formation
## 4 Data Processing
### Image Registration  
The interference phase's quality is the primary determinant of velocity precision. Obtaining a high-quality interference phase requires high-precision registration of complicated pictures. For interferometric data processing, complicated image registration is therefore essential. As per the various metric functions employed, the registration techniques may be broadly classified into three categories: the correlation function method, the average fluctuation method, and the maximum spectrum method [32]. The correlation function approach is now the most popular among them because of its strong stability and excellent effectiveness [33]. For interferometric data processing, the correlation function approach is used as the registration procedure in the two global interference mapping projects, TanDEM-X and SRTM [34]. The correlation function can be classified as either real or complex correlation [35]. The complex correlation function's registration accuracy is  
Figure 6: Flowchart of surface current extraction processor for distributed SAR interferometry. superior to the actual correlation function in regions with strong coherence and comparatively consistent scattering properties, such the sea surface, according to studies [34]. A complex correlation function was used as the metric function in this investigation. The complex correlation coefficient can be expressed using Equation (10). \[\gamma(u,v)=\frac{\left|\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}u_{1}(m,n)u_{2}(m+u,n+v) ^{*}\right|}{\sqrt{\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}|u_{1}(m,n)|^{2}}\sqrt{\sum_ {m=0}^{M-1}\sum_{n=0}^{N-1}|u_{2}(m+u,n+v)|^{2}}} \tag{10}\]  
Here, \(\mu_{1}\) and \(\mu_{2}\) denote the two SAR Single Look Complex (SLC) images, while \(M\) and \(N\) symbolize the size of the window used for computation. \((u,v)\) denotes the sliding position of the correlation calculations, and \((m,n)\) denotes the pixel position in the SAR image. The symbol \(|~{}~{}|\) indicates the absolute value of the conjugate operation. The Fourier transformation of the cross-correlation function of the two images equals the product of the Fourier transformation of one image and the conjugate of the Fourier transformation of the other. Therefore, the correlation functions in (10) can be implemented using Fast Fourier Transform (FFT) to enhance computational efficiency, as shown in Equation (11). \[\gamma=\text{norm}(|~{}IFFT(FFT(u_{1}(m,n)) \tag{11}\] \[FFT(u_{2}(m,n))^{*})~{}|)\]  
Here, \(\text{norm}(~{}~{})\) represents the normalization operation. The term \(FFT\) stands for Fast Fourier Transform, and \(IFFT\) represents inverse Fast Fourier Transform. For SAR interferometric images, the registration precision should generally be better than 1/8 of the pixel. Sub-pixel-level displacement can be estimated through interpolation. The final step involves adjusting the secondary image based on the calculated offsets, as described in (12). Here, \(S\) and \(s\) represent the secondary images before and after registration, respectively, \(\delta_{A}\) signifies the offset of the block in the azimuth direction, and \(\delta_{R}\) refers to the offset of the block in the range direction. Additionally, size \(A\) and size \(R\) represent the block lengths in the azimuth and range directions, respectively. \[S=\text{IFFT}\bigg{(}FFT(s)\cdot\exp\bigg{(}-2\pi i\bigg{(}\frac{\delta_{A}}{ \text{size}\,A}+\frac{\delta_{R}}{\text{size}\,R}\bigg{)}\bigg{)}\bigg{)}. \tag{12}\]  
Figure 7a shows the correlation coefficients between the primary image and the secondary image of LuTan-1 after registration. The figure illustrates a correlation coefficient ranging from 0.4 to 0.95. The central portion of the image has a larger correlation coefficient because of its comparatively elevated signal-to-noise ratio. These results suggest a strong correlation between primary and secondary images after registration.