## 2 Materials and Methods  
The BDS is an integral component of the GNSS, including GEO satellites, inclined geosynchronous orbit (IGSO) satellites, and medium Earth orbit (MEO) satellites [37; 38; 39]. Presently, there are five operational GEO satellites, each maintaining a fixed position relative to the Earth and orbiting at an approximate altitude of 35,787 km [40]. The 5 GEO satellites are located at longitudes of 58.75\({}^{\circ}\)E, 80\({}^{\circ}\)E, 110.5\({}^{\circ}\)E, 140\({}^{\circ}\)E, and 160\({}^{\circ}\)E along the equator, respectively [38], ensuring comprehensive coverage across mid- and low-latitude regions. The data utilized in this study are obtained from the CMONOC, with a temporal resolution of 30 s. The slant TEC (STEC) data along the signal propagation path is acquired through an inversion calculation process employing carrier phase observations and satellite orbit parameters, both of which are furnished by dual-frequency receivers [41]. The principle of STEC inversion is that when satellite signals pass through the ionosphere, the phase will advance. The specific steps for the inversion calculation are as follows. The ionosphere may be regarded as a dispersive medium, and the propagation of electromagnetic wave signals in the ionosphere is usually described by magnetic ion theory. Therefore, it can be studied by the Appleton-Hartree (A-H) formula, which reveals the theoretical relationship between the phase refractive index and electron density:  
\[\mu_{p}=n=\left[1+\left(-\frac{f_{0}^{2}}{f^{2}}\right)\right]^{\frac{1}{2}}=1 +\frac{1}{2}\left(-\frac{f_{0}^{2}}{f^{2}}\right)=1-\frac{\frac{N_{e}e^{2}}{4 \pi^{2}c_{0}m}}{2f^{2}}\approx 1-40.28\frac{N_{e}}{f^{2}}, \tag{1}\]  
where \(N_{e}\) denotes the electron density, \(f_{0}\) is the plasma frequency, and \(f\) is the electromagnetic wave frequency. TEC is defined as the integrated count of electrons within a unit propagation cross-section [42]. Hence, the total electron content along the signal path between the satellite and receiver can be mathematically represented by Equation (2). \[STEC=\int N_{e}(s)ds \tag{2}\]  
The phase velocity in the carrier phase measurements \(v_{p}\) and the phase refraction index \(\mu_{p}\) relationship exists as in Equation (3), and the signal feature propagates at the phase velocity. Therefore, the distance between the receiver and the transmitter \(S\) can be quantified by integrating the phase velocity. Considering the linear relationship between the time and the ratio of phase and frequency, the relationship between electron density, electron density, and phase can be expressed as follows:  
\[v_{p}=\frac{c}{\mu_{p}}=c(1+40.28\frac{N_{e}}{f^{2}}), \tag{3}\]  
\[S=\int_{\Delta t}v_{p}dt=c\Delta t+\frac{40.28}{f^{2}}\int_{\Delta t}N_{e}dt=c \frac{\Phi}{f}+\frac{40.28}{f^{2}}\int_{s}N_{e}ds. \tag{4}\]If the hardware delays of the satellite and receiver are ignored, the dual-frequency carrier phase observation equation, taking into account the integer ambiguity, can be expressed as  
\[\Phi_{1}={\lambda_{1}}^{-1}[S+c(\delta t_{u}-\delta t_{s})-I_{1}+T]-N_{1}+ \varepsilon_{\varphi}, \tag{5}\]  
\[\Phi_{2}={\lambda_{2}}^{-1}[S+c(\delta t_{u}-\delta t_{s})-I_{2}+T]-N_{2}+ \varepsilon_{\varphi}, \tag{6}\]  
where \(\Phi\) denotes the carrier phase observation; \(\lambda\) denotes the carrier wavelength; \(S\) denotes the real distance from the satellite to the receiver; \(c(\delta t_{u}-\delta t_{s})\) denotes the receiver clock difference as well as the satellite clock difference; \(I\) is the ionospheric delay; \(T\) is the tropospheric delay; \(N\) is the integer ambiguity; and \(\varepsilon_{\varphi}\) is the measured random noise term; where \(I\) represents the phase advance corresponding to the frequency caused by the ionosphere.