## 4 Proposed Routing Protocol
### Routing Establishment  
In underwater sensor networks, the routing setup is crucial for reliable data delivery. The process begins with the surface buoy node broadcasting a beacon at maximum power. The beacon includes key metrics such as depth, residual energy, neighbor density, average neighbor energy, and link stability. When a node receives a beacon, it checks whether the sender meets the neighbor's criteria. If valid, it updates its neighbor table, estimates the optimal transmission power, calculates the forwarding probability, and updates the routing table. To propagate the information further, the node rebroadcasts its own beacon--also at maximum power--embedding its updated status unless the forwarding window has expired.  
As shown in Figure 2a, this top-down propagation continues until all nodes, including the source node, have updated their neighbor and routing tables, thereby completing the routing setup. Even nodes that are not selected as relays update their tables using the received beacon data, thereby improving the overall network awareness and robustness. To reduce the overhead, each beacon is forwarded only once within a limited time window. The detailed process is shown in Figure 2b.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**SID** & **RE** & **DEP** & **ND** & **ARE** & **LSF** & **PL** \\ \hline SID\({}_{1}\) & RE\({}_{1}\) & DEP\({}_{1}\) & ND\({}_{1}\) & ARE\({}_{1}\) & LSF\({}_{1}\) & PL\({}_{1}\) \\ SID\({}_{2}\) & RE\({}_{2}\) & DEP\({}_{2}\) & ND\({}_{2}\) & ARE\({}_{2}\) & LSF\({}_{2}\) & PL\({}_{2}\) \\ SID\({}_{3}\) & RE\({}_{3}\) & DEP\({}_{3}\) & ND\({}_{3}\) & ARE\({}_{3}\) & LSF\({}_{3}\) & PL\({}_{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Neighbor table structure.  
Figure 2: (**a**) Illustration of the routing establishment process showing the beacon broadcasting, neighbor discovery, and next-hop selection mechanism; (**b**) Flowchart detailing the step-by-step procedure of routing table update and forwarding node selection.  
As shown in Figure 2a, let node A receive a beacon from node B and measure the received signal power \(P_{r}\) in watts. For calculation purposes, the received signal level in decibels is defined as \(R_{e}\):  
\[R_{e}=10\log_{10}P_{r} \tag{5}\]  
Assuming the transmit power of node B is \(P_{t}\), the path loss \(L_{ab}\) between A and B (in dB) is given by:  
\[L_{ab}=10\log_{10}\frac{P_{t}}{P_{r}} \tag{6}\]  
This expression represents the attenuation of the signal during transmission. To ensure node B can successfully receive a signal, the minimum required receive power threshold is \(P_{th}\). Then, the minimum required transmit power \(P_{abl}\) from node A must satisfy:  
\[P_{abl}=P_{th}\times 10^{\frac{L_{ab}}{L_{a}}/10} \tag{7}\]  
Since practical communication devices typically support only a limited set of discrete transmission power levels, node A cannot transmit signals using a continuously adjustable power value \(P_{abl}\). To ensure that the receiving node B can successfully receive the data packet while minimizing energy consumption, node A should select the smallest transmission power level from the supported power level set that is not less than \(P_{abl}\) denoted as \(P_{l}\). The selection of the transmission power level must satisfy the following constraint:  
\[\left\{\begin{array}{l}P_{l}\in power\_levels\\ P_{l}\geq P_{abl}\\ P_{l}=\min\{P_{l}\}\end{array}\right. \tag{8}\]  
where power levels represent the set of transmission power levels supported by the device.  
\[E_{ab}=\frac{P_{l}\times L}{R_{a}} \tag{9}\]  
where \(L\) is the packet length (in bits), \(R_{a}\) is the transmission rate (in bits per second).