## 3 The Ionospheric Scintillation Compensation-Based Modified BP Imaging Method
### The Proposed Modified BP Imaging Algorithm  
#### 3.2.1 Ionospheric Scintillation Error Extraction Based on Reference Channel Processing  
In the GNSS-BSAR system, the direct wave signal received by the reference channel has the advantages of high SNR and simple equivalent range model. The dual-channel receiving system uses the same local oscillator, which provides coherence. In the studied scenario, the target scene area was close to the receiver, and it could be considered that the two channels had the same Doppler phase error. Here, the reference channel signal was first analyzed, and the overall Doppler phase error was extracted for imaging channel phase compensation. In addition, the ionospheric scintillation error is separated from the overall errors to provide possible reference information for ionospheric research. In actual data processing, the reference channel directly receives the navigation satellite signals through the right-hand circularly polarized antenna. The signal processing flow is shown in Figure 8. First, it was necessary to capture the navigation signal and determine the currently visible satellites. Then, the accurate initial code phase and carrier frequency were obtained through tracking, which was used for bistatic synchronization and Doppler phase (or carrier) extraction. Finally, through navigation data decode and position computation, the position coordinate of the navigation satellite with the receiver as the coordinate origin was obtained, which was used for subsequent SAR imaging. The principle of navigation signal acquisition and tracking is very mature [29] and will not be described in detail in this paper. Taking the intermediate frequency receiver as an example (the experimental data in this work are also collected through IF receiver), the tracked carrier frequency contains the Doppler frequency/phase information for every millisecond of the radar observation duration. The tracked carrier frequency is very important for azimuth focus of the signals in imaging channel. The tracked carrier frequency is recorded as \(Tf_{d}(t)\), \(t\) represents the azimuth time (also called slow time), and the interval is 1 ms. A carrier frequency update value can be tracked and obtained after each equivalent pulse. It is worth noting that this paper uses an \(IF\) receiver as an example, and the tracked carrier frequency has not undergone \(IF\) demodulation. Because \(Tf_{d}(t)\) is discrete, the measured Doppler phase of the reference channel can be obtained by accumulation as follows:  
\[\phi_{ref}(t)=\sum\limits_{i=1}^{N(t)}\left(Tf_{d}(t_{i})-IF_{c}\right)\times T _{p} \tag{12}\]  
where \(IF_{c}\) represents the tracked carrier frequency at the radar observation center time. \(T_{p}\) represents the equivalent pulse repetition period, which is 1 ms. \(i\) represents the azimuth time corresponding to the \(i\)-th pulse. In the above equation, \(N(t)\) changes with the independent variable \(t\), corresponding to the number of pulses when the independent variable takes a certain value. In actual measurements, the reference phase \(\phi_{ref}\) is not only the Doppler phase generated by the movement of the navigation satellite relative to the receiver, but also includes the frequency error of the receiver's local oscillator and the influence of ionospheric scintillation effects. Therefore, the reference phase can also be expressed by the following form:  
\[\left\{ \begin{matrix}\phi_{ref}(t)=-2\pi\frac{R(x_{r},y_{r},t)}{\lambda}-2\pi f _{dc}t+\phi_{e}\\ \phi_{e}=\phi_{le}+\phi_{he}\end{matrix} \right. \tag{13}\]  
where \(f_{dc}\) represents the satellite Doppler central frequency. \((x_{r},y_{r})\) represents the coordinates of the receiver. \(\phi_{le}\) represents the low-order phase error from zero to third order, which is caused by the fixed frequency offset of the receiver intermediate frequency and the drift of the local oscillator of the receiver. \(\phi_{he}\) represents the residual high-order phase, which is the phase error caused by atmospheric effects and thermal noise.