# An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems
## 2 Signal Modelling  
### Range Equation Modelling  
The MEO SAR geometrical model containing one moving target is illustrated in Figure 1, where OXYZ refers to the Earth-centered rotating (ECR) coordinate system. The origin of the ECR coordinate system coincides with the earth's core, the OX axis points from the core of Earth to the prime meridian in the equatorial plane, the OZ axis points to the north pole, and the OY axis is determined by the right-hand rule. The target's longitude and latitude at \(t_{a}=0\) are assumed to be \(\delta_{\mathit{lng}}\) and \(\delta_{\mathit{lat}}\), where \(t_{a}\) is the azimuth slow time. Besides, \(v_{\mathit{lng}}\), \(a_{\mathit{lng}}\), \(v_{\mathit{lat}}\) and \(a_{\mathit{lat}}\) respectively represent the velocity along the longitude, acceleration along the longitude, velocity along the latitude, and acceleration along the latitude of the moving target. In addition, the radar's beam centerline is assumed to cross the target at \(t_{a}=0\). \[ \begin{array}{l}x_{s}(t_{a})=R_{s}(t_{a})\cos(\omega+f(t_{a}))\cos(\Omega_{G}(t_{ a})-\Omega)+R_{s}(t_{a})\sin(\omega+f(t_{a}))\sin(\Omega_{G}(t_{a})-\Omega)\cos \theta_{i}\\ y_{s}(t_{a})=-R_{s}(t_{a})\cos(\omega+f(t_{a}))\sin(\Omega_{G}(t_{a})-\Omega)+R_ {s}(t_{a})\sin(\omega+f(t_{a}))\cos(\Omega_{G}(t_{a})-\Omega)\cos\theta_{i}\\ z_{s}(t_{a})=R_{s}(t_{a})\sin(\omega+f(t_{a}))\sin\theta_{i},\\ \end{array}  \tag{1}\]  
with  
\[R_{s}(t_{a})=\frac{a(1-e^{2})}{1+e\cdot\cos(f(t_{a}))}, \tag{2}\]  
where \(\omega\) means the argument of perigee, \(f\) (\(t_{a}\)) represents the satellite's true anomaly, \(a\) indicates the orbit's semi-major axis, and \(e\) represents the eccentricity, \(\Omega\) is the right ascension of ascending node, \(\theta_{i}\) defines the orbit's inclination, and \(\Omega_{G}(t_{a})=\Omega_{G0}+\omega_{e}t_{a}\) presents the Greenwich Hour Angle (GHA), where \(\omega_{e}\) refers to Earth's rotational angular velocity, and \(\Omega_{G0}\) indicates the GHA at \(t_{a}=0\). The detailed derivation of (1) can be found in [20]. Based on Figure 1, the target's coordinates in the ECR coordinate system are given by [20]  
\[ \begin{array}{l}x_{t}(t_{a})=R_{e}\cos\delta_{lat}\cos\delta_{lng}-\left(v_{ lat}t_{a}+0.5a_{lat}t_{a}^{2}\right)\sin\delta_{lng}-\left(v_{lng}t_{a}+0.5a_{ lng}t_{a}^{2}\right)\sin\delta_{lat}\cos\delta_{lng}\\ y_{t}(t_{a})=R_{e}\cos\delta_{lat}\sin\delta_{lng}+\left(v_{lat}t_{a}+0.5a_{ lat}t_{a}^{2}\right)\cos\delta_{lng}-\left(v_{lng}t_{a}+0.5a_{lng}t_{a}^{2} \right)\sin\delta_{lat}\sin\delta_{lng}\\ z_{t}(t_{a})=R_{e}\sin\delta_{lat}+\left(v_{lng}t_{a}+0.5a_{lng}t_{a}^{2} \right)\cos\delta_{lat},\\ \end{array}  \tag{3}\]  
where \(R_{e}\) refers to Earth's radius. Assuming that the MEO multichannel SAR has \(N\) channels in azimuth direction, and the distance between two adjacent channels is \(d\), as shown in Figure 2. The dotted line in the figure represents the motion track of the satellite, and the azimuth axis of the antenna is parallel to the satellite's velocity vector \(v_{\mathbf{s}}(t_{a})\) because of the zero-Doppler steering. Therefore, the position vector of the effective phase center of the \(n\)th (\(n=1\), \(2\),..., \(N\)) channel can be represented by \(r_{\mathbf{s},n}(t_{a})=r_{\mathbf{s}}(t_{a})-(n-1)d\cdot v_{\mathbf{s}}(t_{a })/v_{\mathbf{s}}(t_{a})\), and the range from the target to the \(n\)th channel can be expressed as  
\[ \begin{array}{l}R_{n}(t_{a})=|\mathbf{r}_{\mathbf{f}}(t_{a})-r_{\mathbf{s}, n}(t_{a})|\\ =|\mathbf{r}_{\mathbf{f}}(t_{a})-r_{\mathbf{s}}(t_{a})+(n-1)d\frac{v_{\mathbf{ s}}(t_{a})}{v_{\mathbf{s}}(t_{a})}|,\\ \end{array}  \tag{4}\]  
Figure 1: MEO SAR geometry. where \(\mathbf{r_{t}}(t_{a})=[x_{t}(t_{a})\ y_{t}(t_{a})\ z_{t}(t_{a})]\) and \(\mathbf{r_{s}}(t_{a})=[x_{s}(t_{a})\ y_{s}(t_{a})\ z_{s}(t_{a})]\) are the position vectors of the moving target and the satellite, respectively, and \(\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\mid\!\!\) indicates modulo operation to the vector.