## 3 System Model
### Channel Model  
In the IHS network scenarios, there are three types of channels. These are terminal-to-satellite (T-S) direct channels, terminal-to-HAP (T-H) relay channels and HAP-to-satellite (H-S) relay channels, which are denoted as \(h_{1}\), \(h_{2}\) and \(h_{3}\) in Figure 1. Channel \(h_{1}\) is Rician small-scale fading with serious free-space loss, a long-delay and high elevation in mid- and low-latitude regions, due to the remote communication distance between the terminal and satellite. Channel \(h_{2}\) is Rician small-scale fading with mild free-space loss, a low-delay and various elevation angles, due to the relatively close distance between the terminal and the HAP. Compared to \(h_{1}\), obstruction by buildings and other tall objects is serious for \(h_{2}\), due to its various elevation angles. Finally, channel \(h_{3}\) is almost an ideal channel with additive white Gaussian noise (AWGN), due to the quasi-vacuum between the HAP and satellite. We assume no multipath signals, resulting in a slow flat fading channel, where the fading of the channel can be considered to be constant over the packet duration, but changing over time. Since the user is motionless or moving at a walking speed in the disaster area, this is practical in our proposed model. The corresponding gains of all three channels vary over time based on the spatial separation between the nodes. Then, the three channels can be uniformly represented as:  
\[C_{i}=f\left(n_{i},d_{i},\mathsf{\kappa}_{i},h_{i}\right),i=1,2,3 \tag{1}\]  
where \(n_{i}\) is the path-loss exponent, \(d_{i}\) denotes the send-receive (S-R) distance, \(\mathsf{\kappa}_{i}\) denotes the Rician factor and \(h_{i}\) is the small-scale slow flat fading gain. For channel \(h_{3}\), the Rician factor is ideally \(\mathsf{\kappa}_{3}=\infty\). The values of \(n_{i}\) depend on the specified propagation environment. For example, in free space, the value of \(n_{i}\) is equal to two, and when obstructions are present, \(n_{i}\) will have a layer value. By using \(d_{i}\), the average path loss can be calculated as \(L_{m}(d_{i})=L_{f}(d_{0})(d_{i}/d_{0})^{n_{i}}\), where \(d_{0}\leq d_{i}\) is the reference distance. \(L_{f}(d_{0})=(4\pi fd_{0}/c)^{The mean signal-to-noise ratio (SNR) of the received signal is given by \(\gamma=\alpha E_{b}/N_{0}\), where \(E_{b}=P_{s}G_{s}G_{r}/(R_{b}L_{m}(d_{i})BO_{o})\) denotes the energy per bit. \(G_{s}\), \(G_{r}\) and \(\alpha=\frac{1}{t}\int_{0}^{t}h^{2}(t)dt\) are the transmitter antenna gain, receiver antenna gain and mean fading channel power gain, respectively. \(R_{b}\) (bit/s) denotes the data rate. \(BO_{o}\) denotes the back-off of the output power. If differential phase-shift keying (DPSK) modulation is used, the corresponding bit error rate (BER) for the T-S direct channel and T-H relay channel is given in [34,35] by:  
\[\rho_{e}=\int_{0}^{\infty}P_{e}(X)p(X)dX=\frac{(1+\kappa)}{2(1+\kappa+\gamma)} \exp\left(\frac{-\kappa\gamma}{1+\kappa+\gamma}\right) \tag{3}\]  
where \(X=\beta^{2}E_{b}/N_{0}\) denotes the temporal SNR at the receiving end, \(P_{e}(X)=\frac{1}{2}\exp(-X)\) is the BER for DPSK in the AWGN channel, \(p(X)=\frac{1+\kappa}{\gamma}\exp(-\frac{X(1+\kappa)+\kappa\gamma}{\gamma})K_{ 0}(\sqrt{\frac{4(1+\kappa)\kappa X}{\gamma}})\) is the Rician probability density function (PDF) in terms of \(X\), \(K_{0}(\cdot)\) denotes the zero order modified Bessel function of the first kind and \(\beta\) denotes the temporal gain of the fading channel.