# Interferometric DEM-Assisted High Precision Imaging Method for ArcSAR
## 3 Analyzing the Error Caused by ArcSAR Imaging on the Reference Plane
### The Relationship between Height Difference Threshold and System Parameters  
In this part, we calculate the height difference threshold for determining whether the ArcSAR imaging result on the reference plane will severely defocus, and analyze the relationship between the height difference threshold and the system parameters. We define the height of the reference plane as \(h_{\text{ref}}\), which the value is 0 m. The geometric relationship is shown in Figure 2. The necessary parameters are given in Table 1.  
\(R_{\text{sp}}\) is the minimum slant range between the antenna and the target during the rotation. The relationship between height of the target and \(\Delta R_{\max}\) is as follows:  
\[R_{\text{p0 max}}-R_{\text{pmax}}=\Delta R_{\max} \tag{17}\]  
\[R_{\text{p0 max}}=\sqrt{\left(r\sin\frac{\theta}{2}\right)^{2}+\left(R_{\text{ sp}}+r-r\cos\frac{\theta}{2}\right)^{2}} \tag{18}\]  
\[R_{\text{pmax}}=\sqrt{\left(r\sin\frac{\theta}{2}\right)^{2}+\left(\sqrt{R_{ \text{sp}}-h^{2}}+r-r\cos\frac{\theta}{2}\right)^{2}+h^{2}} \tag{19}\]  
where \(R_{\text{pmax}}\) is the maximum slant range between the antenna and the target P during system rotation, \(R_{\text{p0 max}}\) represents the maximum slant range between the antenna and P\({}_{0}\) during system rotation.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**r (m)**} & \(\theta_{\text{bw}}\) **(rad)** & \(\theta\) **(rad)** & \(\lambda\) **(mm)** & \(R_{\text{sp}}\) **(m)** & \(Br\) **(MHz)** \\ \hline \(1\) & \(\pi\)/3 & \(\pi\)/3 & 17.50 & 300 & 150 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The ArcSAR system parameters.  
Figure 3: The \(\theta\)–\(\Delta R\) relationship curve.  
According to Equations (17)-(19) and the parameters given in Table 1, we can get the \(h\)-\(\Delta R_{\max}\) relationship curve (Figure 4).  
With reference to Figure 4, it can be seen that the \(\Delta R_{\max}\) increases with \(h\). Applying the parameters of Table 1 to Equations (17)-(19), we calculate that the height difference threshold is 38.26 m. When \(h\) exceeds 38.26 m, the ArcSAR image of target P on the reference plane will severely defocus. We give the simulated imaging results of the target P on the reference plane in Figure 5. When \(h\) is 0 m, the imaging result of target P on the reference plane has no defocusing because the height of the reference plane is consistent with the actual height of the target at this time. When \(h\) is 80 m, the height of the target P exceeds height difference threshold, thus the imaging result of target P on the reference plane appears severe defocusing.  
The height difference threshold is determined by the system parameters given in Table 1. Therefore, the change of system parameters has an impact on height difference threshold: under the premise of changing only single system parameter, the increase of \(\theta_{\rm bw}\), \(\theta\) and \(R_{\rm sp}\) will cause the height difference threshold to rise, while the increase of the \(f_{\rm c}\), \(r\) will cause the height difference threshold to decrease.