# Research on Synthetic Aperture Radar Processing for the Spaceborne Sliding Spotlight Mode
## 2 Model of Spaceborne Sliding Spotlight SAR
### The Acquisition of Parameters in Spaceborne SAR  
The geometric pattern of spaceborne SAR is shown in Figure 2. Suppose the radius of the Earth is \(R_{e}\), the height of the satellite orbit from the ground is \(h\), and if the center \(\phi_{0}\) is known, the other main parameters can be calculated according to the following formulas [3]:  
Corresponding center angle:  
\[\phi_{0}=\text{asin}[(R_{e}+h)/R_{e}\cdot\text{sin}\,\varphi_{0}] \tag{1}\]  
Center slanting distance:  
\[R_{s0}=(R_{e}+h)\cos\varphi_{0}-\sqrt{R_{e}^{2}\cos^{2}\varphi_{0}-(h^{2}+2 hR_ {e})\sin^{2}\varphi_{0}} \tag{2}\]  
The orbit of the satellite is approximately an elliptical eccentricity, and in general, the orbit can be treated as a circle. If the orbit is a circumference, the relationship between the orbital period \(P\) and the orbit radius \(R_{s}\) is as follows:  
\[P^{2}=\frac{4\pi^{2}{R_{s}}^{3}}{\mu_{e}} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Description** & **Value** \\ \hline \(A/\text{Km}\) & Semimajor axis & 7000 \\ \(e\) & Eccentricity & 0.001 \\ \(i/^{\circ}\) & Inclination & 97.0 \\ \(\Omega/^{\circ}\) & Ascending node & \(-\)0.0999845 \\ \(\omega/^{\circ}\) & Argument of perigee & \(-\)0.0121851 \\ \hline \end{tabular}
\end{table}
Table 1: Orbit simulation parameters.  
Figure 1: Basic model of sliding spotlight SAR.  
Figure 2: Geometric model of spaceborne sliding spotlight SAR.  
\(\mu_{e}=3.986\times 10^{14}\) m\({}^{3}\)/s\({}^{2}\) is the Earth gravitational constant. Correspondingly, the angular velocity of the satellite is:  
\[\omega_{s}=\frac{2\pi}{P}=\sqrt{\frac{\mu_{e}}{{R_{s}}^{3}}} \tag{4}\]  
The satellite inertia speed is:  
\[V_{s}=R_{s}\omega_{s}=\sqrt{\frac{\mu_{e}}{R_{s}}} \tag{5}\]  
\(R_{s}=R_{e}+h\), \(R_{e}\) is the radius of the Earth, and \(h\) is the height of satellite orbit. Then the velocity of the satellite can be obtained.  
The slanting distance between satellite and targets is the most important parameter in SAR processing. This distance varies with azimuth and time. Assuming that the flight path is a partial line, the Earth is locally flat and not rotating. The distance between the satellite and the target point is \(R(\eta)\), which is given by the following hyperbolic equation:  
\[R(\eta)=\sqrt{{R_{0}}^{2}+\left({V_{r}\eta}\right)^{2}} \tag{6}\]  
In this hypothetical case, hyperbolic equation is also suitable for spaceborne situations, but \(V\) is not the physical speed, but the virtual speed chosen to make the real distance equation conform to the hyperbolic model.  
As shown in Figure 3, the distance between the target and the satellite at time \(\eta\) is:  
\[R(\eta)=\sqrt{\left({R_{e}\sin\beta_{e}}\right)^{2}+\left({H\sin\omega_{s}\eta }\right)^{2}+\left({H\cos\omega_{s}\eta R_{e}-R_{e}\cos\beta_{e}}\right)^{2}} \tag{7}\]  
In the case of small angle, the distance in the upper form can be expressed as:  
\[\begin{array}{ll}R(\eta)&\approx\sqrt{H^{2}+R_{e}^{2}-2 HR_{e}\cos\beta_{e}+ \left({H\omega_{s}}\right)\left({R_{e}\omega_{s}\cos\beta_{e}}\right)\eta^{2}} \\ &=\sqrt{R_{0}^{2}+\left({H\omega_{s}}\right)\left({R_{e}\omega_{s}\cos\beta_{ e}}\right)\eta^{2}}\end{array} \tag{8}\]  
\[R_{0}=\sqrt{H^{2}+R_{e}^{2}-2 HR_{e}\cos\beta_{e}} \tag{9}\]  
According to the hypothesis of the local circular orbit, \(H\omega_{s}\) is the orbital velocity of the satellite \(V_{s}\), while \(R_{e}\omega_{s}\cos\beta_{e}\) is the velocity of the beam coverage rate, that is the value of the ground speed \(V_{g}\) based on the assumption that the earth is locally spherical near the point \(C\), so \(V_{g}\) is parallel to \(V_{s}\).  
The final result is as follows:  
\[R(\eta)=\sqrt{{R_{0}}^{2}+V_{s}V_{g}\eta^{2}} \tag{10}\]  
The equivalent velocity can be obtained by:  
Figure 3: The geometric relationship, indicating the speed of satellite and beam coverage area.  
\[V_{r}\approx\sqrt{V_{g}V_{s}} \tag{11}\]