## 2 System Model and Problem Formulation
### Communication Model  
The channel quality between LEO satellites and TTGs varies rapidly due to their relative movement, making it a crucial factor in determining handover policies. Given the significant distance between them, which leads to relatively slow spatial signal variations, we primarily focus on large\(-\)scale fading (LSF). According to document [19] proposed by the International Telecommunication Union Radio Communications Sector (ITU\(-\)R), the LSF in satellite\(-\)to\(-\)ground communication mainly comprises four parts, i.e., free\(-\)space path loss, rain attenuation, ionospheric attenuation, and shadowing, expressed as  
\[\mathrm{LSF}_{k,m}^{t}\left[\mathrm{dB}\right]=\mathrm{PL}_{k,m}^{t,fre}+ \mathrm{PL}_{k,m}^{t,rai}+\mathrm{PL}_{k,m}^{t,loom}+\gamma_{k,m}^{t,sha}, \tag{6}\]  
where \(\mathrm{PL}_{k,m}^{t,fre}\), \(\mathrm{PL}_{k,m}^{t,rai}\), \(\mathrm{PL}_{k,m}^{t,loom}\), and \(\gamma^{sha}\) represent the free\(-\)space path loss, rain attenuation, ionospheric attenuation, and shadowing from satellite \(m\) to TTG \(k\) at time slot \(t\), respectively. The free\(-\)space path loss is calculated by  
\[\mathrm{PL}_{k,m}^{t,fre}\left[\mathrm{dB}\right]=32.45+20\log_{10}(f_{c})+20 \log_{10}(d_{k,m}^{t}), \tag{7}\]  
where \(f_{c}\) denotes the carrier frequency. The rain attenuation, accounting for the rainfall rate, elevation angle, frequency, and polarization, is computed based on the ITU\(-\)R P.837 [20] using  
\[\mathrm{PL}_{k,m}^{t,rai}\left[\mathrm{dB}\right]=\gamma_{R}^{t}\cdot d_{k,m} ^{t,eff}, \tag{8}\]  
where \(\gamma_{R}^{t}\) represents the rain absorption coefficient, calculated by  
\[\gamma_{R}^{t}=k\cdot R_{g}^{t}, \tag{9}\]where \(k\) and \(\alpha\) are coefficients dependent on the frequency and polarization, respectively, and \(R_{a}^{t}\) represents the polarization\(-\)adjusted rainfall rate. Additionally, \(d_{k,m}^{t,eff}\) represents the effective path length of the signal through the rain region, expressed as  
\[d_{k,m}^{t,eff}=\frac{d_{R}^{t}}{\cos\theta_{k,m}^{t}}, \tag{10}\]  
where \(d_{R}^{t}\) denotes the projected length of the rain region along the signal propagation direction, calculated by  
\[d_{R}^{t}=\frac{h_{0}}{\sin\theta_{k,m}^{t}}, \tag{11}\]  
where \(h_{0}\) represents the height of the \(0\,^{\circ}\)C isotherm. Ionospheric attenuation primarily arises from two components: absorption loss and scintillation loss, expressed as  
\[\mathrm{PL}_{k,m}^{t,ion}\ [\mathrm{dB}]=\mathrm{PL}_{k,m}^{t,abs}+\mathrm{PL}_{k,m}^{ t,sci}, \tag{12}\]  
where \(\mathrm{PL}_{k,m}^{t,abs}\) and \(\mathrm{PL}_{k,m}^{t,sci}\) represent the ionospheric absorption loss and ionospheric scintillation loss, respectively, from satellite \(m\) to TTG \(k\) at time slot \(t\). The absorption loss primarily results from the interaction between the signal and electrons in the ionospheric \(D\) and \(E\) layers, expressed as  
\[\mathrm{PL}_{k,m}^{t,abs}=\int_{h_{start}}^{h_{end}}\kappa(h)\cdot N_{e}(h)\, dh,\ m\in\mathcal{M}_{k}^{t}, \tag{13}\]  
where \(h_{start}\) and \(h_{end}\) represent the starting and ending heights of the ionospheric D and E layers, respectively, while \(N_{e}(h)\) and \(\kappa(h)\) denote the height\(-\)dependent electron density and ionospheric absorption coefficient, respectively. Ionospheric scintillation loss arises from the irregular structure of the electron density in the ionospheric \(F\) layer. According to ITU\(-\)RP.618 [19], its probability density function can be expressed as  
\[p(\mathrm{PL}_{k,m}^{t,sci})=\frac{1}{\sqrt{2\pi}\cdot\sigma_{sci}}\cdot\exp \left(-\frac{(\mathrm{PL}_{k,m}^{t,sci}-\mu_{sci})^{2}}{2\cdot\sigma_{sci}^{2} }\right),\ m\in\mathcal{M}_{k}^{t}, \tag{14}\]  
where \(\mu_{sci}\) and \(\sigma_{sci}\) represent the mean and standard deviation of the scintillation attenuation, respectively. As for the shadowing, we model it as a log\(-\)normal distribution with a mean of 0 and a variance of \(a_{k,m}^{t,shu}\), which is related to the satellite elevation angle \(\theta_{k,m}^{t}\) and can be found in TR38.811 [21].