## 2 Proposed Coverage Enhancement Modeling
### Enhanced Coverage and Protocol Modeling
The repulsive force \(\overset{\rightarrow}{\mathbf{F}}_{ib}\) between underwater nodes and the boundary can be calculated as  
\[\overset{\rightarrow}{\mathbf{F}}_{ib}=\begin{cases}\left(\frac{\varepsilon _{b}}{\left(d_{th}-d_{ib}\right)^{2}},\ \alpha_{ib}+\pi\right)\text{, \ if }d_{ib}<d_{thb}\\ 0\text{, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
In summary, the total virtual force \(\overset{\rightarrow}{\mathbf{F}}\) on underwater node \(S_{i}\) is expressed as  
\[\overset{\rightarrow}{\mathbf{F}}=\sum_{j=1,j\
eq i}^{N}\overset{\rightarrow}{\mathbf{F }}_{ij}+\overset{\rightarrow}{\mathbf{F}}_{ib}+\overset{\rightarrow}{\mathbf{F}}_{S_{i}G} \tag{19}\]  
where \(j\) represents neighboring underwater nodes. Virtual forces act on randomly distributed underwater nodes, causing them to move towards sparse areas from dense areas, thereby maximizing the coverage rate. The underwater node moves from its initial position to the final position under the action of virtual forces, which can be expressed as  
\[x_{new}=\left\{ \begin{aligned} x_{old},&\text{if} \left|\overset{\rightarrow}{\mathbf{F}}_{x}\right|\leq\overset{\rightarrow}{\mathbf{F }}_{th}\\ x_{old}+\frac{\overset{\rightarrow}{\mathbf{F}}_{x}^{i}}{\mathbf{F}_{i}} \times\eta\times e^{-\frac{1}{\mathbf{F}}_{i}},\text{if}\left|\overset{ \rightarrow}{\mathbf{F}}_{x}\right|>\overset{\rightarrow}{\mathbf{F}}_{th}\end{aligned} \right. \tag{20}\]  
\[y_{new}=\left\{ \begin{aligned} y_{old},&\text{if} \left|\overset{\rightarrow}{\mathbf{F}}_{x}\right|\leq\overset{\rightarrow}{ \mathbf{F}}_{th}\\ y_{old}+\frac{\overset{\rightarrow}{\mathbf{F}}_{x}^{i}}{\mathbf{F}_{i}} \times\eta\times e^{-\frac{1}{\mathbf{F}}_{i}},\text{if}\left|\overset{ \rightarrow}{\mathbf{F}}_{x}\right|>\overset{\rightarrow}{\mathbf{F}}_{th}\end{aligned} \right. \tag{21}\]  
\[z_{new}=\left\{ \begin{aligned} z_{old},&\text{if} \left|\overset{\rightarrow}{\mathbf{F}}_{x}\right|\leq\overset{\rightarrow}{ \mathbf{F}}_{th}\\ z_{old}+\frac{\overset{\rightarrow}{\mathbf{F}}_{x}^{i}}{\mathbf{F}_{i}} \times\eta\times e^{-\frac{1}{\mathbf{F}}_{i}},\text{if}\left|\overset{ \rightarrow}{\mathbf{F}}_{x}\right|>\overset{\rightarrow}{\mathbf{F}}_{th}\end{aligned} \right. \tag{22}\]  
where \(x_{i}^{k+1}=(x_{old},y_{old},z_{old})\) are the optimal coordinates of underwater nodes generated by improved particle swarm optimization; \(x_{i}^{{}^{\prime}}=(x_{new},y_{new},z_{new})\) are the final coordinates of underwater nodes after using virtual force; \(\overset{\rightarrow}{\mathbf{F}}_{x^{i}}\overset{\rightarrow}{\mathbf{F}}_{y}\), and \(\overset{\rightarrow}{\mathbf{F}}_{z}^{i}\) are the virtual force components on the nodes \(\mathbf{i}\); \(\mathbf{\eta}\) is the moving step coefficient. The optimization tends to fall into the local optimal in the late iteration, thus global interference is introduced to change the particle motion mode [38]. The optimal particle position is defined as [39]  
\[x  
\(ST=RT+HT\). When a candidate relay node comes across the information package, it delays forwarding it until it compares its holding period \(HT\), that is, the forwarding delay of the packet. Packets are scheduled for forward forwarding based on their retained times at each node. Shorter holding times result in greater forwarding priority for nodes, thus packets are forwarded first. The \(HT\) calculation is associated with the depth difference, \(\Delta d\), between nodes [29]:  
\[HT=\frac{2\tau}{\delta}(R-\Delta d) \tag{25}\]  
where \(\Delta d\) is the depth difference between the candidate node and the previous hop node; \(R\) is the maximum communication range of the node; \(\tau=\frac{R}{v}\), \(v\) represents the acoustic velocity in water; and \(\delta\) is the global constant. When \(\delta\) is small, node \(HT\) is longer, fewer nodes participate in packet forwarding, and less energy is consumed. In the traditional DBR routing protocol, all nodes keep working and are ready to receive data packets at any time; this wastes a lot of node energy.