## 3 Ambiguity Suppression Method
### Ambiguous Area Imaging
According to the SAR imaging algorithm, \(R_{ref}\) and \(V\) are calculated in the standard imaging process apparently for main area signals [20]. Considering the long slant range of the spaceborne SAR, \(V\) of the ambiguous area varies little to the main area. Thus, we just need an appropriate \(R_{ref}\) for the ambiguous area. According to the range ambiguity theory, it can be expressed as:  
\[R_{ref}=R_{main}+c/2\cdot n/prf \tag{15}\]  
where \(R_{main}\) represents the calculated reference slant range from standard imaging process. We can get the ambiguous area image result of certain \(n\) through the following matching filtering process:  
\[s_{im}(\tau,t,n)=F^{-1}\{[S_{rd\_odd}(\tau,f_{t},n)+S_{rd\_even}(\tau,f_{t},n) ]*H_{azimuth}(f_{t})\} \tag{16}\]  
where \(S_{rd\_even}(\tau f_{t},n)\) represents the range compression results of even \(n\) in Doppler frequency domain. \(\bullet\) Finding the main ambiguous area  
For better ambiguity suppression, we need to find the right \(n\) and \(R_{ref}\) for main ambiguous area which shows strongest intensity in the focused image. Considering the mismatched \(R_{ref}\) leads to defocusing targets, we just need to test the quality of images acquired by different \(R_{ref}(n)\) and select the best focusing one. Generally, the focusing quality of a SAR image can be measured by image entropy [21]:  
\[ \begin{array}{l}E_{im}{=}-\sum\limits_{x=1}^{M}\sum\limits_{y=1}^{N}\frac{I (x,y)}{S}\log\frac{I(x,y)}{S}\\ S{=}\sum\limits_{x=1}^{M}\sum\limits_{y=1}^{N}I(x,y)\end{array}  \tag{17}\]  
where \(I(x,y)\) represents the gray scale value of the SAR image, \(M\) and \(N\) represent the total number of sample points in azimuth and range direction. The smaller the image entropy, the better the focused image. What needs to be specified is that larger \(n\) can bring about smaller power of the corresponding ambiguous area as a result of farther distance from the antenna center. Thus, in this paper we choose \(n=1\) and \(n=-1\) to find the location of the main ambiguous area. * Main ambiguous area location  
The right \(n\) can not only bring right focused main ambiguous area image, but also the center longitude and latitude of the main ambiguous area. Supposing \(\mathbf{R}(R_{x}R_{y}R_{z})\) represents the center position coordinates of the main ambiguous area in the earth coordinate system, \(\alpha\) and \(\beta\) represent the longitude (positive the East while negative the West) and latitude (positive the North while negative the South) respectively, we can get the following expression:  
\[ \begin{array}{l}R_{x}=R_{T}\cdot\cos\alpha\cdot\cos\beta\\ R_{y}=R_{T}\cdot\cos\alpha\cdot\sin\beta\\ R_{z}=R_{T}\cdot\sin\alpha\end{array}  \tag{18}\]  
where \(R_{T}\) represents the Earth radius at the location of the imaging target. According to the geometric relationship of the satellite and Earth, we can get the following equations [22]:  
\[ \begin{array}{l}|\mathbf{{R_{s}}-R}|={R^{\prime}}_{cm}\\ \frac{\mathbf{{V_{s}}}\times(\mathbf{{R_{s}}-R})}{R^{\prime}}=\frac{\lambda f _{dc}}{2}\\ \frac{\left(R_{x}^{2}+R_{y}^{2}\right)}{R_{equ}}+\frac{R_{z}^{2}}{R_{pol}}=1 \end{array}  \tag{19}\]  
where \(\mathbf{{R_{s}}}\) and \(\mathbf{{V_{s}}}\) represent the position and velocity vector of satellite at the center time,\(f_{dc}\) the Doppler center frequency, \(R_{equ}\) and \(R_{pol}\) the radius of the equator and pole of the Earth. \({R^{\prime}}_{cm}\) represents the center slant range of main ambiguous area that can be obtained by:  
\[cent_{R}^{\prime}=R_{ccn}+n/prf\cdot c/2 \tag{20}\]  
where \(R_{ccn}\) represents the center slant range of main imaging area. By solving the Equations (18)-(20), we can get the value of \(\alpha\) and \(\beta\). The optical image can be obtained by the information of longitude and latitude, which can be compared with the acquired SAR image to verify the validity of the imaging process.