## 2 GTLR-BiSAR Imaging Geometry and Signal Properties
### GTLR-BiSAR Signal Model and Properties  
In this subsection, we will deduce the signal expressions in different domains, including time domain, two-dimensional (2D) frequency domain (FD), and range-Doppler (RD) domain. Assume that a certain target point \(P\) is located in the illuminated scene, the minimal range from the target \(P\) to the LEO receiver is \(R_{r_{0}p}\), and the range from the target \(P\) to the GEO transmitter is \(R_{T_{p}}\). The effective radar velocity (see Appendix A for details) of the target \(P\) is \(V_{p}\), and the azimuth time is \(t_{a}\). It should be noted that all the velocity quantities in the rest of this paper refer to equivalent velocity. Then, the bistatic time-varying range from the target \(P\) to the GEO transmitter and LEO receiver is expressed as follows:  
\[R_{bi}(t_{a};P)=R_{T_{p}}+\sqrt{R_{r_{0}p}^{2}+{V_{p}}^{2}t_{a}^{2}} \tag{1}\]  
Furthermore, the signal model of \(P\) can be written as:  
\[S_{echo}(\tau,t_{a};P)=W(\frac{t_{a}}{T_{a}})W(\frac{\tau}{T_{\tau}})exp[-j2 \pi f_{c}\frac{R_{bi}(t_{a};P)}{c}]exp[j\pi K_{r}(\tau-\frac{R_{bi}(t_{a};P)}{ c})^{2}] \tag{2}\]  
where \(K_{r}\) is the FM rate of the chirp signal. \(f_{c}\) is the carrier frequency. \(c\) is the velocity of light. \(\tau\) is defined as the range time. \(W(\frac{t_{a}}{t_{a}})\) and \(W(\frac{\tau}{T_{\tau}})\) are the azimuth and range time window. \(T_{a}\) is the azimuth synthetic aperture time and \(T_{\tau}\) is the pulse duration. Ignoring unimportant amplitude window, we pay more attention to the phase terms in the following derivation. Firstly, we perform the Fourier transform of Equation (2) to obtain the range spectrum by adopting the principle of the stationary phase (POSP) [27]:  
\[S_{echo}(f_{\tau},t_{a};P)=\int_{-T_{\tau}/2}^{T_{\tau}/2}S_{echo}(\tau,t_{a };P)exp(-j2\pi f_{\tau}\tau)\,d\tau \tag{3}\]  
where \(f_{\tau}\) is the range frequency. Assuming that the integral phase of Equation (3) is \(\theta\), then  
\[\theta(\tau,t_{a},f_{\tau};P)=-2\pi f_{c}\frac{R_{T_{p}}+\sqrt{R_{T_{0}p}^{2} t_{a}^{2}}}{c}+\pi K_{r}(\tau-\frac{R_{T_{p}}+\sqrt{R_{r_{0}p}^{2}t_{a}^{2}}}{c})^{2} -2\pi f_{\tau}\tau \tag{4}\]  
Figure 1: Imaging geometry of the GTLR-BiSAR system (not scaled). (**a**) Geometry of the GTLR-BiSAR system working in SS mode. (**b**) Geometry of the GTLR-BiSAR system working in TOPS mode. By calculating the partial derivative of \(\theta\) to \(\tau\) and finding the zero point, we obtain the relationship between \(\tau\) and \(f_{\tau}\). \[\frac{\partial\theta(\tau,t_{a},f_{\tau};P)}{\partial\tau}=2\pi K_{\tau}(\tau- \frac{R_{T_{p}}+\sqrt{R_{\tau_{0}p}^{2}+{V_{p}}^{2}t_{a}^{2}}}{c})-2\pi f_{ \tau}=0 \tag{5}\]  
\[\tau=\frac{f_{\tau}}{K_{\tau}}+\frac{R_{T_{p}}+\sqrt{R_{\tau_{0}p}^{2}+{V_{p}} ^{2}t_{a}^{2}}}{c} \tag{6}\]  
Let us bring \(\tau\) (Equation (6)) into \(\theta\) (Equation (4)), the range spectrum expression is obtained as follows:  
\[S_{echo}(f_{\tau},t_{a};P)=exp[-j2\pi(f_{\tau}+f_{c})\frac{R_{T_{p}}+\sqrt{R_{ \tau_{0}p}^{2}t_{a}^{2}}}{c}-j\frac{\pi}{K_{\tau}}f_{\tau}^{2}] \tag{7}\]  
Secondly, we perform the Fourier transform of Equation (7) in the azimuth time to obtain the 2D frequency domain expression:  
\[S_{echo}(f_{\tau},f_{a};P)=\int_{-T_{a}/2}^{T_{a}/2}S_{echo}(f_{\tau},t_{a};P )exp(-j2\pi f_{a}t_{a})\,dt_{a} \tag{8}\]  
where \(f_{a}\) is the azimuth Doppler frequency. The integral phase \(\theta\) of Equation (8) becomes:  
\[\theta(f_{\tau},f_{a},t_{a};P)=-2\pi(f_{\tau}+f_{c})\frac{R_{T_{p}}+\sqrt{R_{ \tau_{0}p}^{2}t_{a}^{2}}}{c}-\frac{\pi}{K_{\tau}}f_{\tau}^{2}-2\pi f_{a}t_{a} \tag{9}\]  
Similar to the former range operation, we obtain the relationship between azimuth time \(t_{a}\) and Doppler frequency \(f_{a}\).