## 4 Spatial Variance Analysis and Correction for GTLR-BiSAR
### Range-Doppler Geolocation Functions for GTLR-BiSAR System  
As shown in Figure 3, the bistatic signal propagation path starts from the GEO transmitter, it propagates from GEO transmitter to the target (the red arrow solid line points to the red star). Through back-scattering, the signal is received by the LEO receiver (the blue arrow solid line points to LEO satellite). The coordinate frame O-\(XYZ\), dashed lines \(L_{1}\)\(L_{2}\) and the virtual rotation center are consistent with the previous definitions in Figure 1. We use \(\vec{S}_{G}\) and \(\vec{V}_{G}\) to represent the position and velocity vector of GEO transmitter. Correspondingly, \(\vec{S}_{L}\) and \(\vec{V}_{L}\) stand for the position and velocity vector of LEO receiver.  
The position vector of the star target is assumed as \(\vec{S}_{T}\). The descriptions of bistatic RD geolocation functions are not dependent on the operation mode, so we only give the geometry of GTLR-BiSAR system operating in the SS mode in Figure 3.  
According to Figure 3 above, we obtain \(\vec{S}_{T}\) by simultaneously solving the three geolocation equations, including the ellipsoidal equation, the Doppler equation, and the round-trip slant range equation [28; 29]:  
\[\begin{cases}\frac{\vec{S}_{T+S_{T}}^{2}\vec{S}_{T}^{2}}{\vec{S}_{T}^{2}}+ \frac{\vec{S}_{T}^{2}}{\vec{S}_{T}^{2}}=1\\ \frac{(\vec{S}_{T}-\vec{S}_{C})\cdot\vec{V}_{C}}{\lambda\|\vec{S}_{T}-S_{C}\|} +\frac{(\vec{S}_{c}-\vec{S}_{T})\cdot\vec{V}_{L}}{\lambda\|\vec{S}_{L}-\vec{S} _{T}\|}=f_{dc}\\ \|\vec{S}_{T}-\vec{S}_{C}\|+\|\vec{S}_{L}-\vec{S}_{T}\|=R_{G}+R_{L}\end{cases} \tag{26}\]  
where \(\vec{S}_{T}=\left(S_{Tx},S_{Ty},S_{Tz}\right)\), \(R_{G}\) and \(R_{L}\) stand for the distances from GEO transmitter and LEO receiver to the target, respectively. \(\lambda\) is the wavelength of transmitted signal. \(f_{dc}\) is the Doppler center of the GTLR-BiSAR echo data, which is mainly determined by LEO receiver since GEO transmitter is geostationary. \(R_{e}\) is the equatorial radius of Earth, and \(R_{q}\) is the polar radius of Earth.It is obvious that the GTLR-BiSAR RD geolocation equations are nonlinear, Newton iterative method is adopted to obtain the target position (See Appendix B for details).  
According to Equation (26) and the Newton iterative method expatiated in Appendix B, we can obtain the target geolocations in the illuminated scene. We calculate the latitude and longitude coordinates of 25 point targets and mark them in Google Earth map with pushpins shown as Figure 4. The pushpins with the same color indicate the targets with the same minimal range \(Min(R_{L})\) to the moving LEO receiver. Referring to the previous Equation (21), which shows the RD spectrum expression of a certain target \(P\), we know that \(R_{Tp}\) and \(R_{r0p}\) (\(Min(R_{L})\)) varying with the geolocation distribution of targets in the illuminated scene result in the spatial variance of the RD spectrum. The Doppler FM rate mainly depends on \(R_{r0p}\) and the RCMs are relevant with both \(R_{Tp}\) and \(R_{r0p}\).  
For the echo of GTLR-BiSAR in the RD domain, the targets confined in the same bistatic range gate differ from each other in \(R_{Tp}\) and \(R_{r0p}\). The spatial variance of \(R_{Tp}\) and \(R_{r0p}\) causes the variance of Doppler FM rate and RCMs of the targets along the azimuth direction. As a result, the traditional RD and CS algorithms become invalid. Moreover, the existing azimuth perturbation function generated by nonlinear chirp scaling (NLCS) or local fit method [11; 23] cannot be applied to equalize the targets' azimuth FM rates directly  
Figure 3: Geometry of GTLR-BiSAR range-Doppler geolocation in the SS mode (not scaled).  
because the signal is aliasing in time azimuth domain after the deramping operation in Section 3. Therefore, we propose a novel method to solve the GTLR-BiSAR azimuth and range variance problem in Sections 4.2 and 4.3.