# High-Resolution and Wide-Swath 3D Imaging for Urban Areas Based on Distributed Spaceborne SAR
## Subsatellite  
Figure 5: The range ambiguity of the SAR system.  
Figure 6: The geometric diagram of range ambiguity.  
In Figure 6, the solid line represents the useful signal, and the dotted line represents the ambiguous signal.  
\[R_{tm}+R_{rm}=R_{t}+R_{r}+mC/F_{PR} \tag{6}\]  
where \(R_{tm}\) and \(R_{rm}\) denote the distances from the \(m\)th ambiguity point to the main star point and the auxiliary star point, respectively; \(R_{t}\) and \(R_{r}\) represent the distance from the target point to the main star point and the auxiliary star point, respectively; and \(F_{PR}\) denotes PRF.  
According to its definition, the range ambiguity is expressed as [33]:  
\[R_{ASR}=(\sum_{M=-\infty}^{\infty}\frac{G_{\rm T}(\theta_{1m})G_{\rm R}( \theta_{2m})\sigma_{0}(\phi_{m})A(\phi_{m})}{R_{tm}R_{rm}^{2}})/(\frac{G_{\rm T }(\theta_{1})G_{\rm R}(\theta_{2})\sigma_{0}A(\phi)}{R_{t}R_{r}^{2}}) \tag{7}\]  
where \(\theta_{1}\) and \(\theta_{2}\) represent the bottom view of the main satellite and the sub-satellite, respectively; \(\phi\) denotes the beam incidence angle; \(G_{\rm T}(\theta)\) and \(G_{\rm R}(\theta)\) represent the antenna range gain of the main satellite and the sub-satellite, respectively; \(\sigma_{0}\) is the backscattering coefficient of ground scattering points; and \(A\) denotes antenna area.  
Then, the coordinates of ambiguity points are found. Firstly, a coordinate system is established by taking the midpoint of the connecting line between the main satellite and the sub-satellite as the coordinate origin and the connecting line between the main satellite and the sub-satellite as the x-axis. Pointing from the main satellite to the sub-satellite, the y-axis in the plane of the main satellite, the sub-satellite, and the earth center, pointing to the earth center, and the z-axis is determined by the right-hand spiral law.  
\[(x-x_{2})^{2}+(y-y_{2})^{2}+(z-z_{2})^{2}=R^{2} \tag{8}\]  
\[\frac{x^{2}}{A^{2}}+\frac{y^{2}+z^{2}}{B^{2}}=1 \tag{9}\]  
\[\left|\begin{matrix}x-x_{1}&y-y_{1}&z-z_{1}\\ x_{2}-x_{1}&y_{2}-y_{1}&z_{2}-z_{1}\\ x_{3}-x_{1}&y_{3}-y_{1}&z_{3}-z_{1}\end{matrix}\right|=0 \tag{10}\]  
Formula (8) shows the equation of the earth in the coordinate system, where \((x_{2},y_{2},z_{2})\) represents the earth's center and \(R\) denotes the earth's radius. Formula (9) shows the ellipsoid equation with the main satellite and sub-satellite as the focus. \(A=(R_{tm}+R_{rm})/2\), \(B=\sqrt{(A^{2}-D^{2})}/2\), where \(D\) is half the distance between the main satellite and sub-satellite. Formula (10) shows the plane equation, where the main satellite, the earth center, the beam center point, and the ambiguity point are coplanar, where \((x_{1},y_{1},z_{1}),(x_{2},y_{2},z_{2})\), \((x_{3},y_{3},z_{3})\) are the coordinates of the main satellite, the earth center, and the beam center point, respectively. The coordinates of the ambiguity point can be obtained by combining the above equation. Then, \(R_{tm}\) and \(R_{rm}\) can be obtained to acquire the range ambiguity.  
After calculating the ambiguity, the weight of the area with low ambiguity is increased, and image registration is performed on this area.