## 3 Results and Discussion
### Formation Configuration Sensitivity Study  
A sensitivity study was undertaken to investigate the impact of the variation of the LEO satellite radial offset on the vertical _Ne_ measurements. Based on the 8 March 2019 almanac, three formation flying configurations were simulated respectively with different radial separations during formation keeping: (i) 100 m, (ii) 1000 m, and (iii) 3000 m. The _Ne_ retrieval using Galileo PRN 1 and the E1 and E5b TEC are compared between the three configurations. By simulating configuration (i) with a radial offset of 100 m in the ESF scenario, very noisy _Ne_ measurements are obtained as shown in Figure 8i. The measurement errors (in green) were computed by differencing the SR modeled _Ne_ (in red) and raw measured _Ne_ (in blue). The average absolute error is 8.8472 \(\times\) 10\({}^{5}\) cm\({}^{-3}\), and standard deviation of errors is 1.1143 \(\times\) 10\({}^{6}\) cm\({}^{-3}\). The raw _Ne_ measurements here are too noisy to distinguish the three plasma bubbles (located approximately at 340-380 km, 395-435 km, and 480-520 km, respectively). The main reason causing such scattering features is the relatively short altitude separation (\(\sim\)100 m) between the LEO satellites. The VTEC measured at each altitude contains a certain amount of noise and such TEC noise levels are comparable to the real TEC difference between 100 m. Therefore, 100-meter radial separation is not large enough to resolve the E1 and E5b TEC noise from the OEM628 receiver. By simulating configuration (ii) with a radial offset of 1000 m in the ESF scenario, _Ne_ measurements with a limited amount of noise are retrieved as shown in Figure 8 ii. This is the same formation configuration implemented in Sections 3.2 and 3.3. The average absolute error is 1.0981 \(\times\) 10\({}^{5}\) cm\({}^{-3}\), and standard deviation of error is 1.3423 \(\times\) 10\({}^{5}\) cm\({}^{-3}\). The raw _Ne_ measurements look much better  
Figure 7: Wavenumber spectrum comparison between GPS-only and multi-constellation global navigation satellite system (GNSS). than the result of configuration (a), however, applying a low pass filter can further smooth the results and help visualize the plasma bubbles as demonstrated in the earlier two subsections. By simulating configuration (iii) with a radial offset of 3000 m in the ESF scenario, _Ne_ measurements are retrieved without the data scattering as shown in Figure 8 iii. The average absolute error is \(4.8102\times 10^{4}\) cm\({}^{-3}\), and standard deviation of error is \(6.4383\times 10^{4}\) cm\({}^{-3}\). The raw _Ne_ measurements are \"clean\" enough to distinguish the three plasma bubbles without further applying filtering. A wavenumber spectrum analysis was performed to compare the resolution results between the three different configurations. As shown in Figure 9, the wavenumber spectra of _Ne_ retrieval and SR model are plotted together to determine the multi-scale resolution using each formation configuration in this scenario. Clearly, the 100-m case exhibit the worst performance across the whole spatial range. The 1-km and 3-km cases show comparable agreements with the spectrum of SR _Ne_ model. The agreement for the 1-km case outperforms the 3-km case for smaller scales (\(k>1\)), vice versa. The two GNSS receivers vertically separated by 3000 m in LEO allows a near optimal _Ne_ retrieval for the larger scale plasma bubbles using Galileo TEC, however, this will decrease the resolution of GPS _Ne_ retrieval because the GPS's TEC noise level is lower than Galileo's. Therefore, the 1000 m radial separation was chosen due to a (measurement noise) trade-off between the GPS and Galileo given the OEM628 receiver. The results indicate there is an optimal range of vertical separation (between the satellites in formation) to resolve the spatial scale of a specific ionospheric structure. In future  
Figure 8: Vertical _Ne_ retrieval from Galileo PRN 1 with different radial offsets: (i) 100-m; (ii) 1-km; (iii) 3-km.