# Interferometric DEM-Assisted High Precision Imaging Method for ArcSAR
## 3 Analyzing the Error Caused by ArcSAR Imaging on the Reference Plane
### The Error Model of ArcSAR Imaging on the Reference Plane  
As shown in Figure 2, the antenna rotates counterclockwise from S\({}_{1}\) to S\({}_{2}\). When the antenna rotates to S, it is closest to the target P. The _xoy_ plane is defined as the reference plane (the _xoy_ plane is defined as the reference plane in subsequent parts of this paper). We use the backward projection algorithm (BP algorithm) to achieve the imaging of the target P on the reference plane. The P\({}_{0}\) is the imaging result on the reference plane. It has the same range and Doppler from the antenna with target P. However, referring to the geometric model given in Figure 2, target P and P\({}_{0}\) has different range migration because the height of the reference plane is inconsistent with the actual height of the target P.  
The inverse Fourier transform is performed on \(S(\theta,f)\) at the range direction to obtain its time domain form, which can be expressed as:  
\[S_{\rm c}(\theta,t_{\rm r})=\sin\zeta\big{(}B_{\rm r}\big{(}t_{\rm r}{-}2R_{ \rm P}/c\big{)}\big{)}\exp\!\left(-{\rm j}\frac{4\pi f_{\rm c}}{c}R_{\rm P} \right) \tag{8}\]  
Figure 1: The geometric model of ArcSAR system. Figure 2: Target P and its imaging result on the reference plane. The exponential term of \(S_{\rm c}(\theta\),\(t_{\rm r}\)) represents the range migration, which must be compensated during the process of focusing. Therefore, the matched filter is used to compensate for the range migration of \(S_{\rm c}(\theta\),\(t_{\rm r}\)). The ideal matched filter is expressed as follows:  
\[H(\theta,t_{\rm r})=\exp\left\{\mathrm{j}\frac{4\pi f_{\rm c}}{c}\cdot R_{\rm p }\right\} \tag{9}\]  
However, the actual matched filter used to compensate for the range migration of target P is different from the ideal matched filter because target P and P\({}_{0}\) have different range migrations. The actual matched filter contains the phase error, which means the range migration of target P cannot be compensated correctly. The actual matched filter can be expressed as:  
\[H(\theta,t_{\rm r})=\exp\left\{\mathrm{j}\left(\frac{4\pi f_{\rm c}}{c}\cdot R _{\rm p}+\Delta p\right)\right\} \tag{10}\]  
where \(\Delta p\) represents the phase error:  
\[\Delta p=\frac{4\pi f_{\rm c}}{c}\cdot(R_{\rm p0}-R_{\rm p}) \tag{11}\]  
where \(R_{\rm p0}\) represents the distance between P\({}_{0}\) and the APC, which can be expressed as:  
\[R_{\rm p0}=\sqrt{\left(R_{\rm OA}-r\right)^{2}+r^{2}+h^{2}-2r\sqrt{\left(R_{ \rm OA}-r\right)^{2}+h^{2}\cos\theta}} \tag{12}\]  
\[R_{\rm OA}=\sqrt{R_{0}^{2}-h^{2}} \tag{13}\]  
where \(R_{\rm OA}\) is the ground range from the target to the rotation center. Therefore, the imaging result of the target P on the reference plane is  
\[G_{\rm p0}=\int_{\theta}S_{\rm c}(\theta,t_{\rm r})\cdot H(\theta,t_{\rm r}) \mathrm{d}\theta=\int_{\theta}B_{\rm r}\sin\zeta\left(B_{\rm r}\left(t_{\rm r} -2R_{\rm p}/c\right)\right)\exp(\mathrm{j}\Delta p)\mathrm{d}\theta \tag{14}\]  
The matched filter in Equation (10) cannot correctly compensate for the range migration. Thus, the imaging quality of the target P on the reference plane will decrease. Then, we derive the condition for the defocusing of the ArcSAR imaging result on the reference plane. According to Equation (11), the occurrence of \(\Delta p\) is due to the slant range error caused by \(R_{\rm p}\) and \(R_{\rm p0}\). We define the slant range error as the \(\Delta R\), which can be expressed as:  
\[\Delta R=R_{\rm p0}-R_{\rm p} \tag{15}\]  
Equations (6) and (12) indicate that the \(\Delta R\) is the function of \(\theta\). The relationship curve of \(\theta\) and \(\Delta R\) is shown in Figure 1:0 S\({}_{1}\), S and S\({}_{2}\) are marked in the curve. As can be seen from the curve, when the antenna is at point S, the value of slant range error is zero. As the antenna deviates from position S, the value of slant range error gradually increases. When the antenna is at S\({}_{1}\) and S\({}_{2}\), the value of slant range error is the largest. We define the maximum slant range error as \(\Delta R_{\rm max}\).