## 3 System Model
### Aerial-UD Linkage Model  
For aerial-UD communication linkage, we leveraged the model of [24], where the total path loss in dB between aerial \(j\) and UD \(k\) in cluster \(i\) can be expressed as follows:  
\[\lambda_{j_{k}}(d_{j_{k}})=\mathbb{P}^{LoS}\lambda_{j_{k}}^{LoS}(d_{j_{k}})+ \mathbb{P}^{NLoS}\lambda_{j_{k}}^{NLoS}(d_{j_{k}}) \tag{1}\]  
where  
\[\lambda_{j_{k}}^{LoS}\Big{(}d_{j_{k}}\Big{)} = 20\log{(\frac{4\pi f_{V}d_{j_{k}}}{c})}+\rho^{LoS}\xi\kappa \lambda_{j_{k}}^{NLoS}(d_{j_{k}}) = 20\log{(\frac{4\pi f_{V}d_{j_{k}}}{c})}+\rho^{NLoS} \tag{2}\]  
Figure 1: LEO-Sat and aerials integration in post-disaster scenario.  
Herein, \(\lambda_{jk_{i}}(d_{jk_{i}})\), is the total path loss. \(\lambda_{jk_{i}}^{LoS}\Big{(}d_{jk_{i}}\Big{)}\) and \(\lambda_{jk_{i}}^{NLoS}\Big{(}d_{jk_{i}}\Big{)}\) are the line of sight (LoS), and non-LoS (NLoS) components between aerial \(j\) and UD \(k\) in cluster \(i\) influenced by their related distance \(d_{jk_{i}}\), respectively. Furthermore, \(f_{V}\) and \(c\) are the operating frequency of the aerial-UD link and the speed of light, respectively, where aerial is assumed to use the sub 6 GHz wireless communication technology as given in [24]. \(\rho^{LoS}\) and \(\rho^{NLoS}\) indicate the LoS and NLoS losses in dB, respectively. \(\mathbb{P}^{LoS}\) and \(\mathbb{P}^{NLoS}\), \(\mathbb{P}^{NLoS}=1-\mathbb{P}^{LoS}\), define the LoS and NLoS linkage probabilities, mathematically modeled as [24].  
\[\mathbb{P}^{LoS}=\Big{[}1+a\text{exp}\Big{(}-b\Big{(}\varphi_{jk_{i}}-a\Big{)} \Big{)}\Big{]}^{-1} \tag{3}\]  
where \(a\) and \(b\) are the environmental constant parameters based on the ratio of built-up land area to the total land area, the mean number of buildings per unit area, and the buildings' height distribution. Details of these parameters and how they affect the values of \(a\) and \(b\) can be found in [25]. \(\varphi_{jk_{i}}\) is the elevation angle between aerial \(j\) and UD \(k\) in cluster \(i\), which is equal to \(\varphi_{jk_{i}}=\tan^{-1}\left(\frac{h_{j}}{d_{jk_{i}}}\right)\), where \(h_{j}\) is the aerial \(j\) height and \(d_{Hjk_{i}}\) is the horizontal distance between aerial \(j\) and UD \(k\) within cluster \(i\). An assumption is made that uplink transmissions from UDs in cluster \(i\) to aerial \(j\), where the total data rate between them at time \(t\) can be formulated as:  
\[\Psi_{j_{i}}(t)=B_{V}\sum_{k=1}^{K_{i}}\text{log}_{2}\left(1+\frac{P_{rjk_{i}} (t)}{\sum_{u=1,\ u\
eq k}^{K_{i}}P_{rju_{i}}(t)+\sigma^{2}}\right) \tag{4}\]  
where \(B_{V}\) is the total available bandwidth for aerial \(j\). \(P_{rjk_{i}}(t)\) is the received power (RX) at aerial \(j\) from UD \(k\) in cluster \(i\) at time \(t\) determined by its TX power \(P_{tk_{i}j}\) and \(\lambda_{jk_{i}}(d_{jk_{i}}(t))\), where \(d_{jk_{i}}(t)\) is the separation distance between them at time \(t\). \(\sum_{u=1,\ u\
eq k}^{K_{i}}P_{rju}(t)\) is the sum of interfering RX powers at aerial \(j\) from other UDs in its covered cluster \(i\), while \(\sigma^{2}\) is the additive white Gaussian noise (AWGN) power. For further details about these link models and their related parameters, readers are advised to check [24].