# Underwater Sensor Network Redeployment Algorithm Based on Wolf Search
## 3 System Model
### Node Perception Model  
Network coverage rate reflects the coverage degree of the underwater sensor network covering the monitoring area or targets. The probabilistic sensing model was utilized in this study to calculate the network coverage rate. The sensing range of the node is a spherical region with the node as the center and radius r. Among them, \(p_{j}\)\(\left(a_{j},b_{j},c_{j}\right)\) expresses the position of the \(j\)-th monitored target, and \(d(s_{i},p_{j})\) express the distance between nodes and the monitoring target. The formula is defined as follows:  
\[d(s_{i},p_{j})=\sqrt{\left(x_{i}-a_{j}\right)^{2}+\left(y_{i}-b_{j}\right)^{2 }+\left(z_{i}-c_{j}\right)^{2}} \tag{1}\]  
\(f(s_{i},p_{j})\) expresses the probability that the \(i\)-th node ensures coverage for \(j\)-th event, and is formulated as follows:  
\[f(s_{i},p_{j})=\left\{\begin{array}{rl}&0,\,r+Re\,\leq\,d\left(s_{i},p_{j} \right)\\ &e^{-\alpha\lambda^{\beta}},\,\,\,r-Re\,\,<d\left(s_{i},p_{j}\right)\,\,<\,r \,+\,Re\\ &1,\,r-Re\,\geq\,d\left(s_{i},p_{j}\right)\end{array}\right. \tag{2}\]where \(Re\) is an uncertain factor in the measurement range of underwater nodes and satisfies \(0<Re<r\). \(\alpha\) and \(\beta\) are related to the parameters of the measurement equipment. \(\lambda\) is the input parameter defined as follows:  
\[\lambda=\ d\left(s_{i},p_{j}\right)-\left(Rs-Re\right) \tag{3}\]  
To calculate the coverage rate of UWSNs, the probability of monitoring should be greater than 0. The ratio of the value to the total target monitoring points (\(n\)) is the network coverage rate (\(C\)). The formulation is as follows:  
\[C=\left(\sum\limits_{1}^{n}f\left(p_{j},s_{i}\right)\
eq 0\right)/n \tag{4}\]  
where \(f\left(p_{j},s_{i}\right)\
eq 0\) denotes the \(i\)-th node covering the \(j\)-th event. It means that the \(i\)-th node is covered. The \(\sum_{1}^{n}f\left(p_{j},s_{i}\right)\
eq 0\) denotes the covered nodes' total number. The \(n\) denotes the nodes' total number. Thus, we define that the coverage rate \(C\) is the ratio of the covered nodes' total number and the nodes' total number.  
To calculate node coverage, it must consider the node density and obstacle density. As shown in Figure 1, the scope of node communication is a sphere, and it is equally divided into \(k\) districts. When the \(k\) is large, the density calculation is more accurate. The irregular polyhedron represents obstacles in Figure 1. The node communication radius is \(R\). The perceived radius is \(r\). Regions 1 to 4 distribute in the above hemisphere from front the back and from left to right, and the regions 5 to 8 distribute in the below hemisphere in the same way. The node density of S1 is as follows:  
\[D=N_{A}/N_{t}, \tag{5}\]  
where \(D\) represents the node density, \(N_{A}\) is the number of nodes in the \(S1\) region, and \(N_{t}\) represents the minimum number of nodes to cover the entire partition. \(N_{t}\) depends on the sensing and communication radii of the node. It can be calculated according to the formula as follows:  
\[N_{t}=C\frac{K_{m}}{V_{ball}}=3\times(1-\frac{3V_{z}}{4\pi r^{3}})\cdot\frac{K _{m}}{4\pi r^{3}}, \tag{6}\]  
where \(C\) represents the effective coverage rate, \(K_{m}\) represents the volume of a single partition of the node, and \(V_{z}\) represents the volume of the overlapping area. In the same manner, the density of the obstacle is defined as follows:  
\[D_{o}=\frac{O_{m}}{K_{m}}, \tag{7}\]  
where \(D_{o}\) is the obstacle density and \(O_{m}\) is the volume of the obstacles detected by the node.  
Figure 1: Division map of an underwater node communication area.  
First of all, the node perception model has been described. It can better reflect the monitoring process of node coverage targets. Then, the density models of obstacles and nodes have been shown. The RAWS algorithm described below will involve the problem of the relationship between nodes and obstacles. Thus, using the three-dimensional model displays intuitively better.