# Modeling and Analysis of RFI Impacts on Imaging between Geosynchronous SAR and Low Earth Orbit SAR
## 4 Discussion on Probability of Specular Scattering Case  
According to the analysis in the previous section, when the specular scattering case occurs, the image SINR will fall below 5 dB and the image quality will be poor, leading to the loss of detailed information. In fact, specular scattering requires a very special geometric configuration, and its probability between a single GEO SAR and a single LEO SAR is generally small. However, the satellite constellation will be an important form of spaceborne SAR in the future. The growing number of LEO SAR constellations is expected to increase the specular scattering RFI occurrence between GEO SARs and LEO SARs. Therefore, considering the LEO SAR constellations, we evaluated the probability of specular scattering RFI and presented the probability for different orbital elements. Taking in account observations of China, the probability of the specular scattering case was analyzed. The probability of GEO SAR receiving specular scattering RFI can be written as  
\[p_{\text{GEO}}=\frac{t_{\text{specular}\_in\_China}}{T_{\text{GEO,illuminate, China}}} \tag{29}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Case** & **Relationship between \(\theta_{1}\theta_{2}\),\(\varphi\) in Figure 1** \\ \hline \
\begin{tabular}{c} Specular scattering \\ Non-specular scattering \\ \end{tabular} &
\begin{tabular}{c} \(\varphi=0,\theta_{1}=\theta_{2}\) \\ out-of-plane case : \(\varphi\
eq 0,\theta_{1}=\theta_{2}\) \\ in-plane case : \(\varphi=0,\theta_{1}\
eq\theta_{2}\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Specular scattering case and non-specular scattering case. where \(t_{\text{specular\_in\_China}}\) is the total time for the specular scattering case in China, and \(T_{\text{GEO\_illuminate\_China}}\) is the total time for GEO SAR illuminating in China. Similarly, the probability of LEO SAR receiving specular scattering RFI can be written as  
\[p_{LEO}=\frac{t_{\text{specular\_in\_China}}}{T_{\text{LEO\_illuminate\_China}}} \tag{30}\]  
where \(T_{\text{LEO\_illuminate\_China}}\) is the total time for LEO SAR illuminating in China. Since specular scattering needs the same incident angle, we suppose that the beam shape of both the GEO SAR and the LEO SAR are circular. Furthermore, we think that GEO SARs have the capability of right look function and LEO SARs have the capability of both the right look function and the left look function. Generally, the number of GEO SARs is much smaller than that of LEO SARs. Therefore, we considered a case involving both a GEO SAR and a LEO SAR constellation. Figure 10 illustrates the geometry of the GEO SAR and the LEO SAR constellation. The orbit elements of the GEO SAR are given in Table 8. The LEO SAR constellation is a Walker Patten constellation for the sun-synchronous orbit. Then, we obtained the position of GEO SARs, LEO SARs, and their boresight using the STK. Finally, according to Table 8, (29) and (30), we evaluated the probability. To illustrate how the altitude and the satellite number of the LEO SAR constellation affect the probability of specular scatter, we estimated \(p_{GEO}\) and \(p_{LEO}\) in different orbit altitudes (varying from 500 km to 1000 km), different pattens (20/1/1 and 10/1/1), and different true anomaly of the seed satellite. The constellation parameters are detailed in Table 8 and the results are shown in Figures 11 and 12. Every box line draws on the probability of different true anomaly in the same orbit altitude and the same Walker pat  
Figure 10: The sketch of simultaneous observation of China by a GEO SAR and a LEO SAR constellation (Walker Patten 20/1/1); (**b**) is the zoom of the green box in (**a**).