## 2 Proposed Coverage Enhancement Modeling
### Enhanced Coverage and Protocol Modeling  
The distribution of underwater nodes can get too dense or too sparse when affected by the random distribution of sensor nodes and water flow force, leading to poor coverage and more coverage holes in the sensor networks. Therefore, underwater nodes need to be deployed to improve coverage rate and reduce coverage holes in as many grid points as possible [34]. With the expansion of the application of artificial intelligence algorithms to UWSNs, the underwater nodes are considered as particles and the improved PSO is used for UWSN deployment. Each particle is guided to the optimal path and its particle best (_pbest_) and global best (_gbest_) positions are obtained in the solution set to discover a better position. To search for more accurate particle positions, the entire population is divided into several groups and the particle with the best fitness in several groups is called _lbest_. At the \(k^{\text{th}}\) iteration, the particle velocity is determined by the optimal position of the \(i^{\text{th}}\) particle search \(\mathbf{P}_{i}^{k}\), the locally best position within several groups \(\mathbf{P}_{i}^{k  
\(E_{i}\) is large, the population average performance is higher than the particle performance. Therefore, the adaptive position update strategy can be expressed as  
\[\mathbf{x}_{i}^{k+1}= \begin{cases}\omega^{k}\mathbf{x}_{i}^{k}+\left(1-\omega^{k}\right) \mathbf{v}_{i}^{k+1}+rand\cdot\mathbf{P}_{g}^{k}\cdot\omega^{k}&E_{i}>rand\\ \mathbf{x}_{i}^{k}+\mathbf{v}_{i}^{k+1}&\text{otherwise}\end{cases}  \tag{14}\]  
where the ratio \(E_{i}\) is written as \(\frac{\exp\left(\text{fit}(\mathbf{x}_{i}^{k+1})\right)}{\exp\left(\frac{\sum_{i=1 }^{n}\text{fit}(\mathbf{x}_{i}^{k+1})}{n}\right)}\); fit\((\cdot)\) is the particle fitness function; and \(n\) is the number of particle populations. However, underwater node coverage based on artificial intelligence search is prone to falling into local optimal. Virtual forces are used to mutate the optimal position and the underwater node position is updated by comparing the optimal values before and after the mutation. Underwater nodes are abstracted as particles in the potential field, and corresponding repulsive and attractive forces are generated by comparing the geometric distance of the nodes with the threshold. At the same time, there is a repulsive force between the underwater node and the boundary region. Under the combined action of three forces, underwater nodes distributed in dense or sparse areas are driven to move in the direction of a higher coverage rate. Based on Coulomb law, the virtual force \(\overset{\rightarrow}{\mathbf{F}}_{ij}\) between the \(i^{\text{th}}\) and \(j^{\text{th}}\) nodes is expressed as  
\[\overset{\rightarrow}{\mathbf{F}}_{ij}= \begin{cases}\left(\varepsilon_{a} \left(d_{ij}-d_{th}\right),\alpha_{ij}\right)\text{, if }\ d_{th}<d_{ij}\leq R_{c}\\ 0\text{, if }\ d_{ij}>R_{c}\text{ or }d_{th}=d_{ij}\\ \left(\varepsilon_{r}\left(\frac{1}{d_{th}^{2}}-\frac{1}{d_{ij}^{2}}\right), \alpha_{ij}+\pi\right)\text{, if }\ d_{ij}<d_{th}\end{cases}  \tag{15}\]  
where \(\varepsilon_{r}\) is the repulsive force coefficient; \(\varepsilon_{a}\) is the attractive force coefficient; \(d_{ij}\) is the geometric distance between nodes; \(d_{th}\) is the distance threshold; \(R_{c}\) is the communication radius of the underwater node; and \(\alpha_{ij}\) is the azimuth angle between nodes. The distribution density of underwater nodes causes the distance between nodes to be too far or too close, resulting in corresponding repulsive or attractive forces between nodes. It is easy to reduce the node coverage efficiency when there are many underwater nodes clustered in the boundary area.