## 2 System Model and Atmospheric Error Analysis
### Analysis of Ionospheric Effects on GNSS-BSA Imaging
The general method to describe the scintillation effect is to use the phase screen model, which treats the ionosphere as a thin layer (phase screen). As the satellite moves, the signal passing through the ionosphere sweeps across the ionospheric phase screen producing different additional phases. Due to the complexity of irregularities, the additional phase changes are random  
Figure 4: Atmospheric distribution in the GNSS-BSAR system. phases, and only statistical methods can be used to describe their characteristics. The power spectral density can be expressed as [28]:  
\[PSD_{\varphi}(\kappa)=T^{\prime}\bigg{(}\sqrt{\kappa_{o}^{2}+\kappa^{2}}\bigg{)}^ {-p} \tag{7}\]  
where \(\kappa_{o}=2\pi/L_{o}\), \(L_{o}\) represents the turbulence outer scale of irregularities. \(\kappa\) represents the wave number, \(p\) is the power spectrum index, and \(T^{\prime}\) represents a constant proportional to the ionospheric disturbance, expressed as:  
\[T^{\prime}=A\cdot C_{k}L \tag{8}\]  
where \(A\) represents a quantity that is related to parameters such as wavelength, radar viewing angle, and power spectral index, with the specific expression and value available from [28]. \(C_{k}L\) represents the intensity of ionospheric disturbance at the 1 km scale. Due to the phase error is random, only its statistical property can be analyzed. Using the power spectral density shown in Equation (7), the root mean square (RMS) of the phase error can be expressed as [28]:  
\[\phi_{RMS}=\sqrt{2\int_{\kappa_{C}}^{\infty}PSD_{\varphi}(\kappa)d\kappa} \tag{9}\]  
where \(\kappa_{C}=2\pi/L_{C}\). \(L_{C}\) represents the synthetic aperture length on the ionosphere. As the synthetic aperture time becomes longer, the azimuth resolution become finer, and the phase error introduced becomes larger. The scintillation effect will produce high-order phase errors in the azimuthal dimension. Figure 6 simulates the scintillation phase error of the GPS L5 signal under the ionospheric disturbance intensities \(C_{k}L\) are \(10^{32}\), \(10^{33}\), and \(10^{34}\), respectively, when it passes through the ionosphere. Moreover, \(10^{32}\), \(10^{33}\), and \(10^{34}\) represent weak scintillation, medium scintillation, and strong scintillation respectively. And the synthetic aperture time is 300 s. Figure 6 shows that even when the ionospheric scintillation is weak, the phase error caused by it can reach about 45 degrees. For GNSS-BSAR system, this phase error cannot be ignored and must be compensated to achieve precise azimuth focusing of SAR image. The above analysis can be summarized as follows:  
Figure 5: The variation of range delay error and \(QPE\) with the \(TEC\) of the ionosphere. (**a**) The variation curve of range delay with \(TEC\); (**b**) the variation curve of \(QPE\) with \(TEC\). 1. The atmosphere on the Earth's surface is mainly divided into the troposphere and the ionosphere. In the studied GNSS-BSAR system, combining the signal and system characteristics, only the ionospheric effect is considered. 2. The ionospheric dispersion effect is caused by the background ionosphere, which causes delays and pulse width changes in the range dimension of the echo, resulting in range delay error. Due to the limitation of GNSS signal resolution, the impact of this error can be negligible. 3. The ionospheric scintillation effect will introduce random phase errors in the azimuth direction, thereby affecting the azimuth focusing and causing an overall increase in the side lobe level. The greater the intensity of the scintillation is, the worse the SAR image quality is. In next Section 3, the proposed ionospheric scintillation error extraction and compensation method based on the reference channel will be introduced in detail.