# Spatial Baseline Optimization for Spaceborne Multistatic SAR Tomography Systems
## 4 Spatial Baseline Optimization Method for SMS-TomoSAR systems
### Maximum Perturbation Estimation Method  
To further explore the effect of baseline perturbation on the imaging performance, the experiment defined the ratio of the vertical baseline deviation of the actual track and the predetermined track to double baseline interval under uniform sampling as the baseline perturbation \(e\), and assumed that \(e\) was subject to Gaussian distribution, and satisfied  
\[e=\frac{\Delta b_{\perp}}{2B_{\text{span}}/(MN-1)}\sim N\Big{(}0,\left(e_{ \text{max}}/2\right)^{2}\Big{)}, \tag{7}\]  
where \(\Delta b_{\perp}\) is the vertical baseline deviation and \(e_{\text{max}}\) is the maximum baseline perturbation, in which the probability of the vertical baseline perturbation falling in \((-e_{\text{max}},\,e_{\text{max}})\) is 99.45%. The actual simulated baseline can be expressed as:  
\[\left\{b_{\perp 1}+e_{1},\;\ldots,\;b_{\perp p}+e_{p},\;\ldots,\;b_{\perp MN-1}+ e_{MN-1}\right\},\;p=1,\;\ldots,\;MN-1, \tag{8}\]  
where \(\left\{e_{1},\;\ldots,\;e_{p},\;\ldots,\;e_{MN-1}\right\}\), \(p=1,\;\ldots,\;MN-1\) is a random sample of the baseline perturbation \(e\).  
The experiment simulated \(T\) groups of actual simulated baselines for every maximum vertical baseline perturbation, performed height-dimensional imaging performance analysis, and then used the root mean square error (RMSE) to reflect the impact of the maximum vertical baseline deviation on Peak side lobe ratio (PSLR) and integrated side lobe ratio (ISLR). The RMSE expression on PSLR and ISLR for maximum baseline perturbation \(e\) is given as follows:  
\[\left\{\begin{array}{l}RMSE_{PSLR\_e}=\sqrt{\frac{\frac{T}{\gamma}\left(PSLR _{i}-\overline{PSLR}\right)^{2}}{T}}\\ \\ RMSE_{ISLR\_e}=\sqrt{\frac{\frac{T}{\gamma}\left(ISLR_{i}-\overline{ISLR}\right) ^{2}}{T}}\\ \end{array}\right. \tag{9}\]  
Figure 3: The baseline distribution with changing \(q\) (take \(MN=21\) as an example).  
where \(RMSE_{PSLR\_e}\) and \(RMSE_{ISLR\_e}\) are the RMSE values of PSLR and ISLR for \(e\); \(PSLR_{i}\) and \(ISLR_{i}\) are the PSLR and ISLR values of the \(i\)th group of experiments; and \(\overline{PSLR}\) and \(\overline{ISLR}\) are the mean values of \(T\) groups of experiments.  
The experimental simulation analysis and summary showed that as the baseline perturbation increased, the RMSE value of PSLR and ISLR exhibited a growing trend, that is, the system became more unstable to image in the height direction.  
We then define \(\sigma_{e}\) as the performance control factor (PCF),  
\[\left\{\begin{array}{l}\sigma_{e\_PSLR}=\frac{RMSE_{PSLR\_\max}-RMSE_{PSLR \_\min}}{RMSE_{PSLR\_\min}}\\ \\ \sigma_{e\_ISLR}=\frac{RMSE_{ISLR\_\max}-RMSE_{ISLR\_\min}}{RMSE_{ISLR\_ \min}}\end{array}\right. \tag{10}\]  
where \(\sigma_{e\_PSLR}\) and \(\sigma_{e\_ISLR}\) are the PSLR and ISLR performance control factors; \(RMSE_{PSLR\_\min}\) and \(RMSE_{PSLR\_\max}\) respectively represent the optimal value and the performance control threshold of PSLR; and similarly, \(RMSE_{ISLR\_\min}\) and \(RMSE_{ISLR\_\_\max}\) respectively represent the optimal value and performance control threshold of ISLR.  
We then use  
\[e_{\max}=\min(PSLR\_e_{\max},\;ISLR\_e_{\max}) \tag{11}\]  
to obtain the maximum allowable baseline perturbation. The experimental simulation showed that the SMS-TomoSAR system had better height-dimensional imaging performance when the baseline met the maximum baseline perturbation.