## 3 System Model
### Network Model: Basic Definitions  
We consider a UWSN in which the nodes are responsible for gathering the information of interest from the underwater environment and transmitting these sensed data to the GNs, which in turn forward the received data to the AUV. The AUV, which gathers data from all of the GNs, transmits these to the nearby sink. We model the UWSN as a graph \(G=(V,E)\), where \(V\) represents sensor nodes and \(E\) represents links (edges) between them; \(\forall(i,j)\in E\). The network area is divided into two parts; direct communication area \((A_{DC})\) and multi-hop communication area \((A_{MC})\). Nodes lying in the first sub-area are selected as GNs based on their residual energy. Later on, these GNs are rotated to balance the network's energy consumption. Sensor nodes, other than GNs, are called member nodes (MNs). The MNs are associated with GNs on the basis of localized information for the shortest path. Subject to maximization of data gathering while improving the network's energy efficiency, the details are as follows. In our scenario, \(n\) number of sensor nodes are randomly deployed over the seabed. Nodes close to the trajectory of AUV are selected as GNs on the basis of the RSSI value, and after that, their role is rotated on the basis of the residual energy of individual nodes. The GNs are rotated at the cost of signalling messages (overhead) exchanged between the AUV and the nodes to be selected as GNs. However, this overhead has no significant impact on the energy efficiency and throughput degradation as compared to the achievements made due to the GNs' rotation. In order to efficiently communicate within the assigned time slots (to prevent a single point of failure), the AUV needs to be synchronized with the nodes (to be selected as GNs). Within the expected time slot, the AUV listens for the packet (data or control). Upon reception, the AUV compares its current reception time with the expected reception time and calculates a drift value. If \(D_{a}\) is the acceptable delay, a drift value is calculated. If the drift value is greater than \(D_{a}\), a piggyback mechanism is used for future synchronization. The nodes away from the AUV choose different GNs as their local destinations. AUV moves along the elliptical path to collect data, as shown in Figure 1. We assume that the AUV has unlimited energy and computing resources. Data are relayed to the GNs through the SPT algorithm. Efficient data gathering in UWSNs is a big challenge due to the dynamic and harsh conditions of the ocean, like high propagation delay, packet loss due to multiple transmissions and receptions, high attenuation and multi-path fading. The sub-optimal elliptical path of the AUV may overcome these challenges to some extent. Note: The GNs are rotated on the basis of residual energy, and the SPT is used to forward the sensed data of the MNs to their respective GNs. Since the SPT takes into consideration the communication distance, it is a feasible choice because the distance is directly related to the energy consumption of the MNs. In other words, our proposed work takes into consideration both residual energy and communication distance. More specifically, residual energy is considered at the time of the GNs rotation, and communication distance is considered at the time of the MNs association with their respective GNs. According to the Monterey-Miami parabolic equation (MMPE) model, distance is directly proportional to path loss. In other words, received signal strength increases as the communication distance between the sender and receiver decreases. Distant communication between nodes leads to a high bit error rate (BER). More specifically, received data with a low signal strength need retransmission(s) to achieve an acceptable level. However, in doing so, surplus energy is consumed. Thus, to avoid this surplus energy consumption, we have considered communication distance at the time of the MNs association with their respective GNs. In this way, the energy of the network is conserved.