# AUV-Aided Optical--Acoustic Hybrid Data Collection Based on Deep Reinforcement Learning
## 4 Multi-Modal Data Collection Analysis
### Definition of AoI  
The AoI is an important metric to portray the freshness of collected data and is defined as the time elapsed between the data collected by the AUV from the CHs until its delivery to the sink node [41]. We use \(\delta_{i,t}\) to denote the AoI collected from \(c_{i}\) in the navigation trajectory at time \(t\). When \(t<T_{i}\), the information of CH \(c_{i}\) is not sampled since it is not visited, and thus, \(\delta_{i,t}=0\) holds. Otherwise, \(\delta_{i,t}=t-T_{i}\) holds. Then, the AoI of \(c_{i}\) at the start of time slot \(t\) is given by the following relation.  
\[\delta_{i,t}=\left\{\begin{array}{ll}0,&\text{if}\;\;t<T_{i}\\ t-T_{i},&\text{otherwise}\end{array}\right.. \tag{10}\]  
The primary factors affecting AoI during data collection include data transmission delay and AUV sailing time. We use \(\mathcal{T}_{i}^{ac}\) and \(\mathcal{T}_{i}^{gp}\) to denote the data transmission time using UAC and UOC, respectively. The time to transmit \(B_{i}\) bits by the UAC can be written as  
\[\mathcal{T}_{i}^{ac}=\frac{B_{i}}{R_{ac}}+\frac{d_{i}}{V_{ac}}, \tag{11}\]  
where \(R_{ac}\) and \(V_{ac}\) indicate the data rate and transmission velocity of the underwater acoustic modem, respectively. Similarly, the data transmission time of the UOC at data rate \(R_{op}\) and transmission velocity \(V_{op}\) can be obtained as follows.  
\[\mathcal{T}_{i}^{op}=\frac{B_{i}}{R_{op}}+\frac{d_{i}}{V_{op}}. \tag{12}\]  
To collect the monitored data, the AUV travels from the sink \(p_{0}\), collects data from each of the \(N\) CHs according to a pre-determined trajectory, and then returns to the sink node after completing the task. Assume that the travel trajectory of the AUV \(P=\{p_{0},p_{i},\ldots,p_{j},p_{0}\}\), and thus the travel time of the AUV can be expressed as  
\[\mathcal{T}_{travel}=\frac{D(P)}{V_{AUV}}, \tag{13}\]  
where \(D(P)\) and \(V_{AUV}\) denote the total distance and velocity traveled by the AUV, respectively.  
According to (11)-(13), from the moment \(T_{i}\) when AUV arrives at CH \(c_{i}\) to the moment \(T_{i+1}\) when it finishes collecting data and moves to the next data collection point, the AoI of CH \(c_{i}\) can be expressed as \(\mathcal{T}_{i}^{m}+\mu_{i,i+1}^{travel}\). The optical communication has a much smaller transmission delay compared to the acoustic waves. However, the acoustic communication enables long-range transmission that significantly reduces the travel time of the AUV. The time delay caused by data transmission is mainly determined by the data size \(B_{i}\), and so the data collection time and traveling time need to be considered jointly to reduce the decline of data freshness. Then, at moment \(t=T_{i+1}\), the AoI collected from CH \(c_{i}\) refers to  
\[\delta_{i,T_{i+1}}=\left\{\begin{array}{ll}\mathcal{T}_{i}^{ac}+\mu_{i,i+1} ^{travel},&\text{if}\;b_{i}=1\;\text{and}\;x_{i,t}^{ac}=1\\ \mathcal{T}_{i}^{op}+\mu_{i,i+1}^{travel},&\text{if}\;b_{i}=1\;\text{and}\;x_{ i,t}^{gp}=1\end{array}\right., \tag{14}\]  
where \(b_{i}=1,\;i=\{1,2,\ldots,N\}\) indicates that the data of CH \(c_{i}\) has been collected; otherwise, \(b_{i}=0\) holds. When the AUV arrives at the sink node, the AoI of \(c_{i}\) is  
\[A_{i}=\sum_{k=i}^{N}\eta_{i}\delta_{k,T_{k+1}}, \tag{15}\]  
where \(\eta_{i}\) denotes the importance weight of the data collected by CH \(c_{i}\), and \(\sum_{i=1}^{N}\eta_{i}=1\). The higher its value, the greater the data importance.