## 2 Methods
### Case 1: High-Latitude Region
#### 3.1.2 FR and VTEC Estimations  
For the collected SAR data with raw format (L0 level), they have to be processed to the single look complex (SLC) format (L1 level). During this procedure, the system-dependent terms of \(A\), \(e^{i\Phi}\), \(R\), and  
Figure 2: Experimental region and Global Positioning System (GPS)-measured ionospheric variations at Synthetic aperture radar (SAR) acquisition time. (**a**) Experimental area. The black rectangle denotes the spatial coverage of SAR images and red pentagram shows the location of a dual-frequency GPS station, (**b**,**c**) GPS-measured the time rate of changes of total electronic content(ROT) values on 1 April and 17 May 2007, respectively, (**d**,**e**) AE variations on April and May 2007, respectively. \(T\) in (1) are calibrated [50]. Then, measured scattering matrices \(M\) are formed from the full-polarimetric SLC data. The linear polarization scattering matrix M is transformed into circular polarized scattering matrix \(Z\) based on (4) and complex matrices of \(Z_{LR}\times Z_{RL}^{*}\) are created. For suppressing the noises and preserving the same ground resolution in both of range and azimuth directions, multi-looking operation of complex image with two looks in the range direction and 14 looks in the azimuth direction is applied to the \(Z_{LR}\times Z_{RL}^{*}\). To reduce phase noise, the Goldstein adaptive filter with a window size of 32 is applied to the multi-looked data [51]. Finally, FR angles from two images are produced by (7), as shown in Figure 3.  
Figure 3a,b represents the spatial distributions of FR on 1 April 2007 and 17 May 2007, respectively. Figure 3a shows that prominent inhomogeneity of FR distribution appears in the whole image, particularly for the upper part of image, where a maximum value of \(16.9^{\circ}\) is observed. The mean  
Figure 3: **Faraday rotation (FR)** angles. (**a,b**) FR angle maps from 1 April and 17 May 2007, respectively, (**c**) the FR angle values in the profiles A-A\({}^{\prime}\) of (**a**) and (**b**). value and standard deviation of FR angles are respectively 6.8\({}^{\circ}\) and 2.9\({}^{\circ}\) in Figure 3a. Compared to Figure 3a, Figure 3b shows spatially homogenous FR angles. The standard deviation of FR angles in Figure 3b is 0.18\({}^{\circ}\). Figure 3c depicts the FR differences between profiles A-A' of Figure 3a,b. It can be observed from Figure 3c that a gradient is recorded for FR angles on 1 April 2007. Because the system-dependent terms and noise have been calibrated and mitigated, the estimated FR angles in Figure 3a,b are considered to be primarily introduced by the ionosphere. For Figure 3a, it can be assumed that there are the ionospheric irregularities, which results in the FR angle inhomogeneity and SAR image distortion. However, for Figure 3b, there is the weak ionospheric irregularities. On the other hand, it is also observed from Figure 3a,b that there is a certain FR estimation error, particularly for Figure 3b. According to FR estimation theory, this error may be caused by the calibration parameters, such as channel imbalance \(f_{1}\) and \(f_{2}\), crosstalk terms \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\) and \(\delta_{4}\), additional noise \(N\), as well as backscatter characteristics of the imaged surface. Once FR angles are determined, the vertical total electronic content (VTEC) distribution can be generated from (9). For this process, the geomagnetic field parameters are required. The eleventh generation International Geomagnetic Reference Field (IGRF), which is a standard mathematical description of the Earth's main magnetic field [52], was used for this case. Using SAR acquisition time and area, the geomagnetic field parameters are extracted from the IGRF and projected to the SAR coordinate system. Then, the geomagnetic field factor \(\overline{B\cdot cos\theta\cdot sec\phi}\) at a constant height of 400 km was determined using geomagnetic field and SAR imaging geometry.