## 2 Proposed Coverage Enhancement Modeling
### Coverage Description and Node Motion Model  
To make the problem easier to solve, the monitoring area is defined as \(L\times W\times H\), with several \(M\) cubic grids. The center point of each cubic grid corresponds to a grid dot, and all grid dots constitute the UWSNs. Defining \(G_{w}\) as the \(w^{\text{th}}\) grid point, the sets of all grid dots is represented by \(\mathbf{G}=\{G_{1},G_{2},\cdots G_{w},\cdots G_{M}\}\), where \(G_{w}=(x_{w},y_{w},z_{w})\). There are \(N\) underwater nodes deployed in the UWSNs. Defining \(S_{i}\) as the \(i^{\text{th}}\) underwater node, the sets of all underwater nodes are expressed as \(\mathbf{S}=\{S_{1},S_{2},\cdots S_{i},\cdots S_{N}\}\), where \(S_{i}=(x_{i},y_{i},z_{i})\). The sphere-sensing area of each underwater node has the same sensing radius whose sphere center is the node coordinates. The Euclidean distance \(d(S_{i},G_{w})\) between the underwater node \(S_{i}\) and the grid dot \(G_{w}\) is expressed as  
\[d(S_{i},G_{w})=\left\|S_{i}-G_{w}\right\| \tag{1}\]  
where \(\left\|\cdot\right\|\) is a binary norm indicating the distance between points. Based on the Boolean perception model, the probability \(P_{G_{w}}(S_{i})\) that the grid point \(G_{w}\) is perceived by the underwater node \(S_{i}\) is expressed as  
\[P_{G_{w}}(S_{i})= \begin{cases}1&d(S_{i},G_{w})\leq R_{s}\\ 0&d(S_{i},G_{w})>R_{s}\end{cases}  \tag{2}\]  
Figure 1: The overall architecture of the proposed coverage algorithm. where \(R_{s}\) is the sensing radius of the underwater node. If \(P_{G_{w}}(S_{i})\) is equal to 1, the grid point \(G_{w}\) is covered by the underwater node \(S_{i}\); otherwise, the grid point \(G_{w}\) is a coverage hole. Hence, the joint sensing probability \(P_{G_{w}}(\mathbf{S})\) by all underwater nodes \(\mathbf{S}\) is expressed as  
\[P_{G_{w}}(\mathbf{S})=1-\prod_{i=1}^{N}(1-P_{G_{w}}(S_{i})) \tag{3}\]  
In UWSNs, the underwater nodes are affected by environmental factors such as underwater currents, tides, and wind, and their positions change, to some extent [31]. Therefore, it is crucial to analyze the movement of underwater nodes. The oceans in Physical Oceanography and Pure Kinematics are viewed as homogeneous fluids of rotation that manifest themselves as layers in terms of density. Horizontal flows occur in nearly unperturbed fashion at every point due to vertical flows; thus, such motions display a finite damp behavior. When the node moves in UWSNs with node drift, the coverage of the node also changes. Therefore, it is necessary to introduce a water flow model that conforms to the actual situation. As deep-sea environments are often highly complex, the meandering current mobility model (MCM) [32] is used to establish the motion model for underwater nodes under ocean currents, tides, and other marine environmental conditions. All plane motions that cannot be compressed can be represented by the fluid equation \(\varphi(x,y,t)\). This model was also used in this study to predict the velocity of point \(i\) and the coordinate positions \((x_{i},y_{i},z_{i})\) in the horizontal direction at time \(t\). Nodes are initially distributed unevenly in the surveillance zone, and their positions change in the mobility model given in the following equation [33]:  
\[\varphi(x,y,t)=-\text{tanh}\left[\frac{y-B(t)\sin(k(x-ct))}{\sqrt{1+k^{2}B(t)^{ 2}\cos^{2}(k(x-ct))}}\right] \tag{4}\]  
\[B(t)=A+\varepsilon\cos(\omega t) \tag{5}\]  
where \(A\) is the average meander width; \(k\) is the number of bends per unit length; \(c\) is the displacement rate of ocean current in direction \(Y\); \(B(t)\) is the width of the control curve; \(\varepsilon\) is used to control the amplitude of the entire ocean current field; and \(\omega\) is the frequency of the ocean current in the flow field. In general, attention is only paid to the movement of nodes in the \(X\) and \(Y\) directions, and the variability in the vertical direction is ignored.