## 3 The Ionospheric Scintillation Compensation-Based Modified BP Imaging Method
### Basic Process and Analysis of BP Imaging Algorithm  
The BP algorithm is a classic imaging algorithm in time domain. It calculates the position and phase of echo at each grid point on the ground mesh falling on the signal mesh. The echo of that grid point on the ground mesh is coherently integrated. The echoes from other grid points are incoherently superimposed, resulting in a very low intensity. Therefore, the integrated intensity can reflect the electromagnetic scattering intensity at that grid point on the ground mesh. For the entire ground scene area, the above process is operated point by point, and finally, a well-focused SAR image can be obtained. The advantage of the BP algorithm is that the principle is clear. There are no restrictions on the motion mode of the radar platform, and no approximation is made during the derivation process. Therefore, the BP algorithm can be competent for various modes of SAR imaging processing. It is an imaging processing algorithm with high precision and wide applicability. In the GNSS-BSAR system, the echo after pulse compression in the imaging channel can be expressed as:  
\[S_{r3}(t,\tau)=\text{rect}\bigg{[}\frac{\tau-R(t)/c}{T_{p}}\bigg{]}P\bigg{[} \tau-\frac{R(t)}{c}\bigg{]}\exp\bigg{[}-j2\pi\frac{R(t)}{\lambda}-j2\pi f_{dc}t \bigg{]} \tag{10}\]  
where \(P(\cdot)\) represents the autocorrelation function of the ranging code. The meanings of other variables are the same as in Equation (5). Figure 7 shows the process of data processing using classic BPA, which is described in detail below. 1. The imaging area is divided into a two-dimensional ground mesh, where \(x\) and \(y\) are the two-dimensional coordinates corresponding to the grid points respectively. And the ground grid spacing \(dx\) and \(dy\) should be smaller than the imaging resolution in the corresponding directions. 2. For a certain pixel point \((x_{k},y_{k})\) on the ground mesh, the range migration trajectory \(\tau=\tau(x_{k},y_{k};t)\) of the point is calculated according to its coordinate value and the positions of the GNSS transmitter and receiver. And then the energy of the target echo along the trajectory can be extracted. Finally, the azimuth-dimensional signal \(S_{r3}[t,\tau(x_{k},y_{k};t)]\) can be recombined. 3. Based on the coordinate value \((x_{k},y_{k})\) and the positions of the transmitter and receiver, Doppler phase history of the grid point can also be calculated. The azimuth phase compensation is operated on the one-dimensional signal \(S_{r3}[t,\tau(x_{k},y_{k};t)]\), that is, multiply it by the complex conjugate of the Doppler phase \(\exp[-j2\pi R(x_{k},y_{k};t)/\lambda-j2\pi f_{dc}t]\) of the target echo at that point, eliminate its azimuth phase modulation. 4. Accumulate the one-dimensional signal \(S_{r3}[t,\tau(x_{k},y_{k};t)]\) along the azimuth direction to obtain the coherent integrated energy of the target echo at that ground grid point and assign the integration result to \(S_{g}(x_{k},y_{k})\). 5. Traverse all points on the ground mesh and repeat the above steps 2-4 to obtain the final SAR image \(S_{g}(x,y)\) projected onto the ground. Figure 7: Principle diagram of the classic BPA. The whole process can be summarized as follows:  
\[S_{g}(x_{k},y_{k})=\int_{0}^{NT_{p}}S_{r3}[t,\tau(x_{k},y_{k};t)]\times\exp\left[ j2\pi\frac{R(x_{k},y_{k};t)}{\lambda}+j2\pi f_{dc}t\right]dt \tag{11}\]  
After BPA processing, the focused SAR image backprojected onto the ground scene mesh can be obtained. In the step 3 of the above processing flow, the Doppler phase history of the echo needs to be calculated and compensated based on the relative geometric relationship between the target, receiver, and GNSS satellite. In theory, there is no problem with this operation. However, for GNSS-BSAR imaging, the synthetic aperture time is usually long, especially for ground-based receiving configurations. To obtain fine azimuth resolution, it is often necessary to accumulate hundreds of seconds.