# Distributed Coordination of Space-Ground Multiresources for Remote Sensing Missions
## 2 System Model
### Network Model
#### 2.2.2 Ordinary Arcs  
The ordinary arcs in TEG are divided into three categories, namely, link arcs, storage arcs, and observation arcs. The set of ordinary arcs is denoted by \(A_{OD}=A_{L}\cup A_{S}\cup A_{O}\), where \(A_{L}\), \(A_{S}\), and \(A_{O}\) are the set of link arcs, storage arcs, and observation arcs, respectively. The link arcs represent the transmission ability of each time slot in the network. The link arc set is denoted by \(A_{L}=A_{RL}\cup A_{GL}\), where \(A_{RL}\) represents the set of link arcs corresponding to the links between RSSs and DRSs, and \(A_{GL}\) denotes the set of link arcs corresponding to the links between the RSSs and ground stations of the same RSS system. We use \(s^{k}_{n,i,j}\) to denote the link arc from vertex \(s^{k}_{n,i}\) to vertex \(r^{k}_{j}\) in TEG, which represents the link from RSS \(s_{n,i}\) to DRS \(r_{j}\) in the \(k\)-th slot. Similarly, \(sg^{k}_{n,i,j}\) denote the link arc from \(s^{k}_{n,i}\) to \(g^{k}_{n,j}\). Then, the link arc set \(A_{GL}\) and \(A_{RL}\) can be expressed as  
\[\begin{split} A_{RL}&=\Big{\{}sr^{k}_{n,i,j}|lck^{ \prime}(r_{j})\in R^{k}_{C}(s_{n,i}),1\leq k\leq K,1\leq i\leq|S_{n}|,1\leq j \leq|R|,1\leq n\leq N\Big{\}},\\ A_{GL}&=\Big{\{}sg^{k}_{n,i,j}|lc(g_{n,i})\in R^{k}_ {C}(s_{n,i}),1\leq k\leq K,1\leq i\leq|S_{n}|,1\leq j\leq|G_{n}|,1\leq n\leq N \Big{\}},\end{split} \tag{1}\]  
where \(lc^{k}(r_{j})\) represents the position of \(r_{j}\) in the \(k\)-th slot, and \(R^{k}_{C}(s_{n,i})\) denotes the communication range of \(s_{n,i}\) in the \(k\)-th slot. Similarly, \(lc(g_{n,i})\) is the position of \(g_{n,i}\).  
The storage arcs represent the storage ability of the nodes in SGCN. Let \(ss^{k}_{n,i}\) denote the storage arc from vertex \(s^{k}_{n,i}\) to vertex \(s^{k+1}_{n,i}\) in TEG, which represents the storage capability of RSS \(s_{n,i}\) in the \(k\)-th slot. Similarly, \(rr^{k}_{i}\) denotes the storage arc from \(r^{k}_{i}\) to \(r^{k+1}_{i}\), and \(gg^{k}_{n,i}\) denotes the storage arc from \(g^{k}_{n,i}\) to \(g^{k+1}_{n,i}\). The set of storage arcs \(A_{S}\) is expressed as  
\[\begin{split} A_{S}&=\Big{\{}ss^{k}_{n,i}|1\leq k \leq K-1,1\leq i\leq|S_{n}|,1\leq n\leq N\Big{\}}\\ &\cup\{rr^{k}_{i}|1\leq k\leq K-1,1\leq i\leq|R|\Big{\}}\\ &\cup\{gg^{k}_{n,i}|1\leq k\leq K-1,1\leq i\leq|G_{n}|,1\leq n \leq N\Big{\}}.\end{split} \tag{2}\]  
The observation arcs model the opportunities for RSSs to observe the observation objects. We use \(os^{k}_{n,i,j}\) to denote the observation arc from vertex \(o^{k}_{n,i}\) to vertex \(s^{k}_{n,i}\) in TEG, which represents the opportunity for RSS \(s_{n,j}\) to observe the observation object \(o_{n,i}\) in the \(k\)-th slot. The set of observation arcs \(A_{O}\) is expressed as  
\[A_{O}=\Big{\{}os^{k}_{n,i,j}\Big{|}|lc^{k}(o_{n,i})\in R^{k}_{O}(s_{n,j}),1 \leq k\leq K,1\leq i\leq|OM_{n}|,1\leq j\leq|S_{n}|,1\leq n\leq N\Big{\}}, \tag{3}\]  
where \(lc^{k}(o_{n,i})\) represents the location of observation object \(o_{n,i}\), and \(R^{k}_{O}(s_{n,j})\) denotes the visible range of \(s_{n,j}\) in the \(k\)-th slot. The capacity of a link/storage/observation arc represents the maximum amount of data that the corresponding communication/storage/observation resource can transmit/storage/observe in a slot, which is denoted by \(C(a)\), \(\forall a\in A\). For more details of arc capacity, please refer to reference [18; 19].