# An AUV-Aided Cross-Layer Mobile Data Gathering Protocol for Underwater Sensor Networks +
## 4 Description of Our Proposed Cross-Layer Mobile Data-Gathering Protocol
### Distributed Clustering Phase  
In order to determine the cluster heads; CHs; and their members, CMs, we use the same procedure that used in [6; 29; 30]. In this model, we extend our previous work by integrating the routing protocol in [6] with an efficient MAC protocol [29; 30] which can enhance the mobile data-gathering efficiency. In this cross-layer design, we take advantage of the AUV discovery phase to assign the time frames to each cluster head, and subsequently assign time slots to each member of the cluster by CH. In our cross-layer mobile data-gathering protocol, a cluster head can be determined based on the priority value of a sensor node, which depends on a sensor degrees (one-hop neighboring degrees), distance (from a sensor to the sink), and energy. The priority value, \(P\), of each sensor at time step \(t\) can be calculated by using the following equation,  
\[P(t,i)=\alpha(N_{Deg}(i))+\beta(N_{Dis}(i))+\gamma(N_{Ex}(t,i)), \tag{1}\]  
where \(t\) is the current time step and \(i\) is a unique identifier (ID) of a sensor node. \(\alpha\), \(\beta\), and \(\gamma\) are the degree weight, distance weight, and expected energy weight of a sensor node, respectively, which all of them should be equal to 1 (\(\alpha+\beta+\gamma=1\)). \(N_{Deg}(i)\) is a normalized degrees of each node which can be obtained by (\(\frac{Degree(i)}{MaxDegree(\Delta)}\)) and \(N_{Dis}(i)\) is a normalized distance to the sink which can be given by (\(1-(\frac{Distance(i)}{MaxDistance(\Delta)})\)). The normalized expected energy, \(N_{Ex}(\lambda,i)\), can be calculated by using the following equation,  
\[N_{Ex}(t,i)=(1-log(\frac{9\times E_{i}(t+1)}{InitialEnergy}+1)), \tag{2}\]  
Figure 1: Timeline of Cross-Layer Mobile Data gathering (CLMD) protocol. where \(E_{i}(t+1)\) denotes the expected consumed energy at time step \(t+1\). Expected Consumed Energy Calculation  
The sensor set can be represented as \(S=\{s_{1},s_{2},s_{3},...,s_{n}\}\) while \(s_{i}\) is a node in the network. We have the following definition. \[\forall s_{i}\in S,\qquad\qquad\delta=\lambda_{i}+\Delta_{i}\] (3) where (4) \[\delta=\{t_{0},t_{1},t_{2},...,t_{n}\}\] represents the time steps (5) \[\Delta_{i}=\text{ Set of time steps that }s_{i}\text{ was the member} \tag{6}\] \[\lambda_{i}=\text{ Set of time steps that }s_{i}\text{ was CH} \tag{7}\]  
For instance, if \(\Delta_{i}=\{t_{2},t_{4}\}\) and \(\lambda_{i}=\{t_{1},t_{3}\}\), it means that \(s_{i}\) has been a CH at \(t_{1},t_{3}\) and has been a member at \(t_{2},t_{4}\). The expected consumed energy of each node at \(t+1\) can be calculated as  
\[E_{i}(t+1)=E_{\lambda_{i}}(t+1)+E_{\Delta_{i}}(t+1)+E_{\lambda_{i}\zeta_{CH,AUV }}(t+1) \tag{8}\]  
where \(E_{\lambda_{i}}(t+1)\) is the expected consumed energy as a CH at \(t+1\) and so far, \(E_{\Delta_{i}}(t+1)\) is the expected consumed energy as a member so far, and \(E_{\lambda_{i}\zeta_{CH,AUV}}(t+1)\) is the expected consumed energy in the CH and AUV communication at \(t+1\) and so far. We have the following equations,  
\[E_{\Delta_{i}}(t+1)=|\Delta|\times OP\times\phi\times E_{tran} \tag{9}\] \[E_{\lambda_{i}}(t+1)=(\sum_{N\in\lambda_{i}}D_{i}(N)+D_{i}(t+1)) \times\times OP\times\phi\times E_{rec}\] (10) \[N_{Ex}(t,i)=(1-log(\frac{9\times E_{i}(t+1)}{InitialEnergy}+1)), \tag{2}\]0  
where \(OP\) denotes the operational period/phase, \(\phi\) indicates the traffic (data generated) rate, and \(E_{rec}\) and \(E_{tran}\) represent the receiving energy and transmission energy, respectively. \(D_{i}(N)\) denotes the degree of node \(i\) at time step \(N\). By substituting Equations (9)-(11) in Equation (8), it can be rewritten as the following. \[ \begin{split} E_{i}(t+1)=\Bigg{(}\bigg{(}\Big{(}\sum_{N\in \lambda_{i}}D_{i}(N)+D_{i}(t+1)\Big{)}\times OP\times\phi\bigg{)}\times\\ (E_{rec}+E_{tran})\Bigg{)}+\Bigg{(}|\Delta|\times OP\times\phi \times E_{tran}\Bigg{)}\end{split}  \tag{12}\]  
It can also be reduced to the following equation,  
\[