## 2 Materials and Methods
### Atmospheric Delay Model
#### 2.3.1 Ionospheric Delay  
The ionosphere comprises the upper layer of the earth's atmosphere, extending from around 60 km to nearly 2000 km above the surface, and terminates at the magnetic apex. Commonly, the study of ionospheric properties employs a single-layer model [55]. This model posits that the ionospheric electrons are distributed within an infinitesimally thin spherical shell at a designated equivalent height above the earth's surface. As shown in Figure 3, \(O\) is the center of the earth, \(R_{e}\) is the radius of the earth, \(H\) is the height of the thin layer, \(p\) is the puncture point, \(\alpha\) is the zenith distance of the satellite at the receiver, and \(\alpha^{\prime}\) is the zenith distance of the satellite at the puncture point.  
In practice, the VTEC at the ionospheric puncture point is usually converted to STEC in TECU, where \(1\) TECU \(=10^{16}\) electronics/m\({}^{3}\).  
According to the geometric relationship in Figure 3, the ionospheric mapping function can be obtained as follows:  
\[m(\alpha)=\frac{1}{\cos\alpha^{\prime}}=\frac{1}{\sqrt{1-\left(\frac{R_{e}\sin \alpha}{R_{e}+R_{i}}\right)^{2}}} \tag{11}\]  
where \(R_{e}\) is the earth's radius, and \(h_{i}\) is the ionospheric puncture point height. Therefore, the ionospheric delay model is as follows:  
\[\Delta R_{ion}=\frac{K\cdot VTEC_{SAR}}{f^{2}\cdot\sqrt{1-\left(\frac{R_{e}\sin \alpha}{R_{e}+h_{i}}\right)^{2}}} \tag{12}\]  
where \(\Delta R_{ion}\) is the ionospheric delay; \(K\) is the coefficient, and \(K=40.3\) m\({}^{3}\)/s\({}^{2}\); \(VTEC_{SAR}\) is the vertical total electron content at the SAR puncture point, and \(f\) is signal frequency.  
Figure 3: Ionospheric single-layer model.  
Ionospheric VTEC can be obtained from CODE GIM data. According to the bilinear interpolation principle, the spatial interpolation of GIM grid data is first carried out:  
\[\begin{array}{l}I_{(lat,lon)}=I_{1,1}\times(1-\frac{lat-lat_{1}}{\Delta lat})(1- \frac{lon-lon_{1}}{\Delta lon})+I_{1,2}\times(1-\frac{lat-lat_{1}}{\Delta lat})( \frac{lon-lon_{1}}{\Delta lon})+\\ I_{2,1}\times(\frac{lat-lat_{1}}{\Delta lat})(1-\frac{lon-lon_{1}}{\Delta lon})+I _{2,2}\times(\frac{lat-lat_{1}}{\Delta lat})(\frac{lon-lon_{1}}{\Delta lon})\\ \end{array} \tag{13}\]  
where \(I_{(lat,lon)}\) is the VTEC at the piercing point; \(I_{1,1},I_{1,2},I_{2,1},I_{2,2}\) represent the VTEC at the lower-left, upper-left, lower-right, and upper-right corners of the piercing point; \(lat_{1},lon_{1}\) are the latitude and longitude of the lower-left corner; \(lat,lon\) are the latitude and longitude of the piercing point; and \(\Delta lat\) and \(\Delta lon\) are the longitude and latitude intervals of the global ionosphere map. Next, perform time interpolation.  
\[I_{(lat,lon,t)}=I_{i}\times\frac{T_{i+1}-t}{T_{i+1}-T_{i}}+I_{i+1}\times\frac{ t-T_{i}}{T_{i+1}-T_{i}} \tag{14}\]  
where \(I_{(lat,lon,t)}\) refers to the VTEC associated with the piercing point at the imaging moment; \(I_{i},I_{i+1}\) denote the VTEC at the piercing point corresponding to the epochs before and after the imaging instance; \(T_{i},T_{i+1}\) represent the epochs before and after the imaging moment; and \(t\) is the imaging moment.  
Additionally, it is essential to consider the total electron content of the entire ionosphere, as derived from the current VTEC obtained from the GIM for GNSS. Since the SAR satellite orbit is in low earth orbit and only includes a portion of the ionosphere, the SAR VTEC is theoretically smaller than the GNSS VTEC. The conversion rate can be determined by their ratio.  
\[ratio=\frac{VTEC_{SAR}}{VTEC_{GNSS}}=\frac{\int_{h_{0}}^{h_{1}}N_{e}dh}{\int_{ h_{bottom}}^{h_{top}}N_{e}dh} \tag{15}\]  
where \(h_{bottom},h_{top}\) are the total ionospheric boundaries; \(h_{0},h_{1}\) represent the specific elevation ranges; and \(N_{e}\) is the electron density.