## 6 Conclusions  
This paper explored GNSS-BSAK imaging, focusing on the extraction and compensation of phase errors due to atmospheric effects and receiver's time-frequency errors under long synthetic aperture time. Based on the dual-channel receiving characteristic of the system, a modified BP imaging algorithm using two-step azimuth phase compensation was proposed. Initially, the reference phase was extracted from the received direct wave of the reference channel, extracting both low-order phase errors from receiver local oscillator drift and high-order phase errors due to ionospheric scintillation. Subsequently, the target echo's azimuth phase in the imaging channel was divided into reference phase and difference phase. The former can be corrected using the Doppler phase extracted from the reference channel, while the latter can be solved based on the positional information. This algorithm can eliminate the azimuth phase error in the imaging channel and effectively improve the focusing accuracy of SAR images. Experiments using GPS L5 signals as illuminators were conducted at different times in the morning, midday, and evening, performing coherent accumulation imaging about 300 s on a 1.5 km by 0.8 km ground scene. The comparative processing results with PGA show that the proposed method was not limited by scene contrast, prominent scatters, high-order error estimation capability, or iterative convergence. It can directly compensate for azimuth phase errors, demonstrating more precise azimuth focusing capability, which verifies the effectiveness and advantages of the proposed method in the studied dual-channel GNSS-BSAK system. Moreover, ionospheric scintillation errors extracted from these experiments indicate diurnal variations in ionospheric scintillation intensity, suggesting potential research applications.  
The proposed method can also be adapted to other passive SAR imaging systems, offering certain applicability. Future work will aim to enhance the algorithm's processing efficiency. In addition, the intensity of ionospheric scintillation is related to factors such as solar activity, geomagnetic activity, geographic location, and seasonal variations. Therefore, more experiments and analysis will be designed, which may provide helpful references for enriching the ionospheric research.  
Conceptualization, T.T. and J.C.; data curation, Z.R. and P.Z.; formal analysis, T.T. and H.Y.; investigation, Z.R.; methodology, T.T. and P.W.; software, T.T.; supervision, P.W. and J.C.; validation, H.Y. and H.Z.; writing--original draft, T.T.; writing--review and editing, J.C. and H.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (NNSFC), grant number 62101014, the Beijing Natural Science Foundation, grant number 4222006 and the National Natural Science Foundation of China (NNSFC), grant number U2241202.  
The data presented in this study are available on request from the corresponding author.  
Thanks to the editors and reviewers for their careful review, constructive suggestion, and reminding, which helped improve the quality of the paper.  
The authors declare no conflicts of interest.