## 3 Proposed Methodology
### Implementation Procedure
#### 3.3.3 Performance Evaluation Metrics of the Model  
Several key performance indicators are used to evaluate the effectiveness of the model.  
Firstly, the path length measurement is used to assess the total distance traveled by the AUV when it starts its journey from the surface station up to visiting sensor nodes and then returns to the surface station (the ship). The surface station then forwards the gathered information to the central department for further analysis via an antenna. The path length evaluation formula is expressed as follows:  
\[D=\sum_{i=1}^{\text{SN}}\sqrt{\left(x_{i}-x_{i+1}\right)^{2}+\left(y_{i}-y_{i+ 1}\right)^{2}+\left(z_{i}-z_{i+1}\right)^{2}} \tag{15}\]  
On the other hand, the Vol is such an important metric to capture the full worth of data delivered at the surface station. The purpose of this is that the sensor nodes record information depending on the event that took place in their close proximity. Intuitively, we propose that each sensor has an event importance value as a heuristic, where with the sensor node having a high value, it means that very important information had been recorded, thus increasing the priority to visit this node to maximize the Vol. Thus, the process of information collection has sensor node-specific data, including node coordinates, time to transmit a packet, and node event importance. In addition, the time required for the AUV to reach the boundaries of the sensor nodes also affects the information collectionprocess. Collecting the information that has the highest importance would maximize the VoI, and it can be determined as follows [24]:  
\[\mathcal{V}\mathrm{oI}=\beta\sum_{i=1}^{n}EI_{i}+(1-\beta)\sum_{i=1}^{n}EI_{i}e^ {-\sum_{q=q+1}^{n}\frac{t_{q}+q+1}{q}} \tag{16}\]  
where \(\beta\) is the trade-off factor that contributes to balancing both the severity of the event and how quickly it decays over time. \(\sigma\) is a scaling factor indicating the level of sensitivity with timeliness. Moreover, \(t\) represents the time required for the AUV to sail from the \(qth\) node to the next node along with the time to transmit the packets and can be calculated as follows [24]:  
\[\begin{array}{c}t_{tra,\,q}=\frac{h}{V}+\frac{D_{q}}{R}\\ t_{sail,q\to q+1}=\frac{D_{q,q+1}^{n}}{\sqrt{S}}\\ t_{q\to q+1}=t_{tra,\,q}+t_{sail,q\to q+1}\end{array} \tag{17}\]  
where \(t_{tra,\,q}\) represent the time required to send information from the \(qth\) sensor node to the AUV. \(\frac{h}{V}\) denotes the propagation delay, and \(\frac{D_{q}}{R}\) represent the data transmission time. \(D_{q,q+1}\) is the desired distance for the AUV to sail from node boundary \(q\) to node boundary \(q+1\). \(VS\) denotes the static velocity of the AUV in meters/second (m/s).  
The final measure is the overall energy consumption by the AUV during its journey. The total energy consumption for data collection depends on data transmission, data reception, and the energy consumed by the AUV during its journey, and this is calculated as follows:  
\[EC=(P_{tra}+P_{rec})\times t_{tra}+P_{mov}\times t_{mov} \tag{18}\]  
where \(EC\) denotes the total energy consumption. \(P_{tra}\), \(P_{rec}\), and \(P_{mov}\) represent the power consumption for sending messages and receiving messages, and the power consumption during the movement of the AUV, respectively. \(t_{mov}\) denotes the time required for the AUV to complete its journey from the start node and return to the same node [25].