## 3 The Ionospheric Scintillation Compensation-Based Modified BP Imaging Method
### The Proposed Modified BP Imaging Algorithm
#### 3.2.2 The Two-Step Azimuth Phase Compensation-Based Modified BP Imaging Algorithm  
Utilizing the characteristics of dual-channel receiving of the GNSS-BSAR system, this paper proposes a two-step azimuth phase compensation algorithm to solve the problem of errors compensation, such as ionospheric scintillation phase error under ultra-long synthetic aperture time. This method decomposes the azimuth phase of the target echo into the reference phase extracted from the reference channel and the difference phase of the target point relative to the direct wave signal of the reference channel. For the ground-based fixed receiver platform, the imaging area is close to the receiver, and the reference channel and imaging channel use the same local oscillator. And Figure 4 also shows that the ionospheric effect has almost the same impact on \(R_{d}\) and \(R_{t}\) (blue dashed line and red solid line). Therefore, the phase errors of the signals of the two channels can be considered consistent. The azimuth doppler phase extracted from the reference channel signal not only includes the Doppler phase caused by the motion of the satellite transmitter relative to the receiver, but also includes the error caused by the local oscillator drift of the receiver, the receiver thermal noise error, and the ionospheric scintillation error.  
Taking the receiver as the coordinate origin, the ENU coordinate was established as Figure 4. For the target point \(k\) located at \((x_{k},y_{k})\), its azimuth phase can be expressed as:  
\[\phi_{k}(t)=-2\pi\frac{R(x_{k},y_{k};t)}{\lambda}-2\pi f_{dc}t+\phi_{e} \tag{15}\]  
where \(\phi_{e}\) represents the sum of various azimuth phase errors after removing the satellite motion's Doppler phase. Based on the analysis in Section 3.2.1, the phase error \(\phi_{e}\) was approximately consistent in the reference channel and imaging channel.  
According to the analysis in Section 3.2.1, the azimuth reference phase of the reference channel signal can be expressed as the Equation (13). Because the ENU coordinate system uses the receiver as the origin, \((x_{r},y_{r})=(0,0)\) in the Equation (13). According to the previous analysis, \(\phi_{e}\) cannot be calculated accurately, but it can be estimated and eliminated using the Doppler phase tracked by the reference channel.  
By deforming the expression of the azimuth phase in Equation (15) of the target point \(k\), we can obtain:  
\[\phi_{k}(t) =-2\pi\frac{R(x_{k},y_{k};t)}{\lambda}-2\pi f_{dc}t+\phi_{e} \tag{16}\] \[=-2\pi\frac{R(x_{r},y_{r};t)}{\lambda}-2\pi f_{dc}t+\phi_{e}+2 \pi\frac{R(x_{r},y_{r};t)-R(x_{k},y_{k};t)}{\lambda}\] \[=\phi_{ref}(t)+\phi_{diff}(x_{k},y_{k};t)\]  
where we define \(\phi_{diff}(x_{k},y_{k};t)=2\pi[R(x_{r},y_{r};t)-R(x_{k},y_{k};t)]/\lambda\) as the difference phase. In this way, the azimuth phase of the target point \(k\) is decomposed into two parts: the reference phase and the difference phase, which facilitates phase compensation in actual data processing.  
The flow chart of the two-step phase compensation algorithm is shown in Figure 9. First, the Doppler phase extraction technology was used to extract the carrier frequency observations tracked by the reference channel and obtain the reference phase \(\phi_{ref}(t)\) through integration. Then, the satellite position information obtained from ephemeris calculation and the receiver position information obtained from positioning were used to calculate the difference Doppler phase \(\phi_{diff}(x_{k},y_{k};t)\) between target point \(k\) and the receiver. Finally, the two types of phases were added to give the actual azimuth phase \(\phi_{k}(t)\) required for compensation. Use the resulting \(\phi_{k}(t)\) to replace the target Doppler phase in step 3 in Section 3.1. Then, continue to complete steps 4 and 5 to obtain a SAR image with high-precision focus in azimuth.