## 4 Spatial Variance Analysis and Correction for GTLR-BiSAR
### Azimuth Variance Analysis and Correction Strategy  
As shown in Figure 4, the yellow pushpins along the azimuth direction represent the targets cluster 3, 8, 13, 18, 23. On one hand, the cluster is characterized with the same minimal range to the moving LEO receiver, denoted by \(R_{r0\_ref}\). On the other hand, we suppose the distances from these targets to GEO transmitter as \(R_{T3}\), \(R_{T8}\), \(R_{T13}\), \(R_{T18}\) and \(R_{T23}\). It is obvious that these distances are different from each other. We suppose \(R_{T13}\) as the reference GEO distance \(R_{T\_ref}\), if we can adjust the GEO distance differences and rearrange them with the same \(R_{T\_ref}\). Then the signal azimuth variance will be removed and the traditional RD or CS method can be applied to process the azimuth variance corrected GTLR-BiSAR echo data. The azimuth variance correction schematic is shown in Figure 5, and we only give three target clusters for distinct illustration. Figure 5 shows the range-Doppler signal of 15 point targets corresponding to Figure 4, the supporting area of the targets cluster with the same \(R_{r0}\) spread to different bistatic range gates \(R_{bi}\). The vertical direction stands for the Doppler frequency \(f_{a}\), and \(f_{a\_ref}\) is the azimuth reference frequency. The discussion in this paper is assumed as the zero-Doppler, so \(f_{a\_ref}=0\). The horizontal direction represents the bistatic range gate, \(R_{bi\_near}\), \(R_{bi\_ref}\) and \(R_{bi\_far}\) are the near, reference and far bistatic range gate, \(R_{bi\_ref}=R_{T\_ref}+R_{r0\_ref}\). The specific correction steps are shown in Figure 6.  
Figure 4: The target geolocations obtained from GTLR-BiSAR RD equation. Figure 5: The azimuth variance correction schematic. Firstly, we select the range-Doppler signal of targets within similar Doppler frequency range, these targets are illuminated with similar azimuth time, shown in Figure 7. The time-frequency diagram (Figure 7b) of 25 targets is obtained according to the orbit tracks and beam steering data of GEO transmitter and LEO receiver. Take the targets 1-5 as an example to illustrate the correction strategy, we should adjust the GEO distance differences according to the GEO distance of the reference row targets 11-15. Secondly, we multiply a linear phase function \(h_{i}(f_{r})\) in the RD domain with the selected RD data (the red and blue dashed boxes in Figure 6). After inverse Fast Fourier transform (IFFT) operation in range direction, the RD data migrates to the desired positions (the right part of Figure 6). The migrations of targets 1-5 are \(R_{T_{i}}-R_{T_{(i+10)}}(i\in[1,5]\), \(i\in\mathbb{Z}^{+})\), recorded as \(\Delta R_{T_{i}}\). According to GEO satellite orbit tracks and target geolocations, the migrations are calculated and they change slightly along the range direction. Therefore, we choose \(\Delta R_{T_{3}}\) to calculate \(\Delta t_{1}\) in Figure 6 to complete the signal migration. It should be noted that the phase compensation is needed after the signal migration. Since the phase is sensitive, each \(\Delta R_{T_{i}}\) is utilized to calculate the compensation phase, and the compensation function is shown as follows:  
Figure 6: The specific correction steps for the azimuth variance. Figure 7: Doppler frequency changes with azimuth time of the targets. (**a**) The illuminated time of targets. (**b**) The relationship between Doppler frequency and azimuth time of targets. The pink cross, green triangle, red cross, blue square and light green diamond represent the time-frequency diagrams of targets 1–5, 6–10, 11–15, 16–20, and 21–25, respectively. The same color and shape mean the similar Doppler frequency range. \[h_{com}=exp(j2\pi\frac{R_{T_{i}}-R_{T_{(i+10)}}}{\lambda}) \tag{27}\]  
After the migration and compensation operations, the azimuth variances of targets 1-5 are removed. For the other targets, the similar operations are applied to complete the azimuth variance removals.