## 2 Equivalent Range Model for GEO-LEO BiSAR
### Range Model Analysis for GEO-LEO BiSAR without 'Stop-and-Go' Assumption  
In this section, we derive the accurate range model for GEO-LEO BiSAR, which is indispensable for the later imaging method formulation. Two major problems will be encountered to accurately model the range history of GEO-LEO BiSAR in sliding-spotlight mode. First, the synthetic aperture time in sliding-spotlight mode is long and the error introduced by the curved trajectory of the LEO receiver cannot be ignored. Second, due to the large observation distance of the GEO transmitter, the commonly adopted'stop-and-go' assumption would introduce significant error and should be accounted for. In the past decade, several range models have been proposed to describe the relative motion between LEO-SAR and the target. The conventional hyperbolic range equation [53; 54] approximates the motion trajectory \(\widehat{AB}\) of the LEO-SAR as a uniform linear motion, which leads to significant range error in high-resolution modes. In order to improve the precision of the conventional hyperbolic range equation, uniformly accelerated linear motion with equivalent radar acceleration is introduced to approximate the circular arc \(\widehat{AB}\) in Figure 1, which is the approach adopted by MESRM [50], EARM [51] and squinted-EARM [46]. However, the traditional range models for LEO-SAR did not exploit the essential characteristics of its circular orbital motion, which fail to accurately describe the actual motion trajectory of the LEO satellite with long integration time. In this paper, a novel equivalent squinted range model based on circular orbit trajectory (CoT-ESRM) is proposed to accurately represent the range history of the LEO receiver for GEO-LEO BiSAR in sliding-spotlight mode. In Figure 1, assume that the arc \(\widehat{AB}\) is part of an equivalent circular orbit of LEO-SAR (for elliptical orbit, \(\widehat{AB}\) can be approximated by a circular arc within a short period). Let \(R_{s}\) and \(\omega_{s}\) denote the radius and angular velocity of the equivalent circular orbit in RTLC, respectively. Then, the length of the chord \(AB\), i.e., the straight-line segment from \(A\) to \(B\), can be calculated as  
\[L_{chord}=2\sin\frac{\omega_{s}t}{2}R_{s} \tag{2}\]  
where \(\omega_{s}t\) represents the central angle of the arc \(\widehat{AB}\) with respect to the center of the equivalent circular orbit of LEO receiver. Therefore, the instantaneous slant range from the receiver position \(B\) to an arbitrary target \(P\) can be formulated by exploiting the triangle \(\triangle ABP\) as  
\[R_{R}(t)=\sqrt{R_{R0}^{2}+L_{chord}^{2}-2R_{R0}L_{chord}\sin\theta_{s}}. \tag{3}\]  
\(L_{chord}\) essentially contains higher-order components of azimuth time \(t\) and can better approximate the actual motion trajectory of LEO-SAR. In contrast, \(L_{chord}\) is replaced by \(v_{0}t\) in the case of conventional hyperbolic range equation, and a quadratic polynomial \(v_{0}t+A_{0}t^{2}/2\) for MESRM [50], EARM [51] and squinted-EARM [46], where \(v_{0}\) and \(A_{0}\) are the equivalent radar velocity and acceleration, respectively. It should be noted that  
Figure 1: Imaging geometry of GEO-LEO BiSAR in RTLC. and \(\omega_{s}\) can be calculated by letting the expanding coefficients of (3) equal those of the fourth-order polynomial expansion of range history. As for the GEO transmitter, a fourth-order polynomial range model [55] is adopted to accurately represent its relative motion  
\[R_{T}(t)=R_{T0}+k_{T1}t+k_{T2}t^{2}+k_{T3}t^{3}+k_{T4}t^{4} \tag{4}\]  
where \(k_{Ti},i=1,2,\cdots,4\) are the expanding coefficients of the GEO-SAR range model. Next, we will analyze the error of'stop-and-go' assumption in GEO-LEO BiSAR and therefore propose an accurate range model for GEO-LEO BiSAR without'stop-and-go' assumption. The propagation paths of GEO-LEO BiSAR are illustrated in Figure 2. Assume that the signal is transmitted at time instant \(t+\tau\).