## 2 System Model and Atmospheric Error Analysis
### Analysis of Ionospheric Effects on GNSS-BSA Imaging  
The Earth's surface is not an ideal vacuum environment. As shown in Figure 4, the troposphere is from the Earth's surface to about 20 km, and the ionosphere is from about 60 km to 1000 km. When signals are transmitted between the antenna and the target area, the troposphere and ionosphere will introduce additional amplitude and phase errors, affecting the focusing of SAR imaging. For the studied GNSS-BSA system, the distance between the target area and the receiver is relatively close, and the influence of atmospheric effects on the receiving path can be negligible (the red dotted line in Figure 4). The transmitting path will be affected by the ionosphere and troposphere, which will be analyzed in detail below. The troposphere is a non-dispersive medium, which has a consistent impact on different frequencies of the signal and does not affect the range compression effect. However, differences in the geometric relationship at different azimuth time and the time variability of the troposphere will introduce different delays. For the L-band signal, the range delay is usually on the order of meters [27]. In this paper, the GPS L5 signal was used, and its range resolution was about 15 m. The range delay caused by the troposphere is less than half a range resolution unit. Therefore, this work mainly studies the impact of ionospheric effects on GNSS-BSA imaging. The density of the atmosphere above the Earth decreases with height. The density of the low-altitude atmosphere is relatively high, and molecules collide frequently. The ionized electrons and ions can recombine quickly and are macroscopically neutral. The density of the upper atmosphere is thin, the ionization is faster, and the recombination is slower [26]. The formation of the ionosphere is due to the large number of electrons and ions  
Figure 3: Schematic diagram of two-dimensional SAR matrix formation. produced by the ionization of the upper atmosphere by radiation such as solar ultraviolet and X-rays. The group delay introduced by the ionosphere is determined by the total electron content (_TEC_) and the electromagnetic wave frequency. _TEC_ includes a uniform part and a fluctuating part. The uniform part is equivalent to the background ionospheric or the large-scale electron density irregularity, which causes the range dispersion effect. The fluctuations are partly caused by small and medium-scale irregularities, which mainly lead to azimuthal phase fluctuations, called the scintillation effect. After complex analysis [26], and in the GNSS-BSAR system, only the one-way additional phase introduced by ionospheric dispersion needs to be considered. The group delay, range delay, and introduced quadratic phase error (_QPE_) can be derived as:  
\[\left\{ \begin{array}{c}\Delta t=\frac{\text{K-TEC}}{c_{c}^{2}}\\ \Delta R=\frac{\text{K-TEC}}{f_{c}^{2}}\\ QPE=\frac{\pi\kappa R_{c}^{2}}{2c_{f}^{2}}TEC\end{array} \right. \tag{6}\]  
where \(K=40.28\). _TEC_ refers to the total electron content, which represents the integral of electron density along the electromagnetic wave propagation path, the unit is TECU, 1 TECU = \(10^{16}\) m\({}^{-2}\), \(f_{c}\) is the electromagnetic wave frequency. \(B_{w}\) represents the signal bandwidth. Equation (6) shows that the group delay and range delay error are proportional to the _TEC_ in the ionosphere and inversely proportional to the \(f_{c}^{2}\) of the transmitted signal. Figure 5 takes the GPS L5 signal as an example to simulate the range delay and _QPE_ introduced by ionospheric dispersion under different _TEC_. When the _TEC_ is less than 50 TECU (the peak value of _TEC_ throughout the day in the experimental area is less than 50 TECU), the range delay is less than one resolution unit. The _QPE_ is less than 2 degrees. Therefore, it can be considered that ionospheric dispersion has no impact on the studied GNSS-BSAR imaging. The ionospheric scintillation effect is caused by irregularities.