## 3 Improved LEACH Protocol
### Clustering Based on Underwater Energy Propagation Model  
#### 3.2.1 Optimization of the Cluster Number  
In UASNs, the energy consumption for network operations can be categorized into transmission energy consumption, received energy consumption, and standby energy consumption. Received energy consumption is defined as the energy required by a node to receive and fuse data packets. It is determined by the packet size. Standby energy consumption is defined as the energy consumption of a node in the standby state, which is related to the standby power. These parameters are not controlled by the LEACH protocol, so optimizing received energy and standby energy is beyond the scope of this paper. Transmission energy consumption is defined as the transmission energy required for a packet to be correctly received by the destination node. It is related by the packet size and the acoustic propagation loss, where the latter can be related to the routing algorithm. The acoustic propagation loss in the water can be calculated by the Marsh-Schulkin model [20] as follows:  
\[TL=\left\{ \begin{array}{ll}20\lg R+\alpha R+60-K_{L},&R<H\\ 15\lg R+\alpha R+\alpha_{r}\left(\frac{R}{H}-1\right)+5\lg H+60-K_{L},&H<R<8H \\ 10\lg R+\alpha R+\alpha_{r}\left(\frac{R}{H}-1\right)+10\lg H+60-K_{L},&R>8H \end{array} \right. \tag{1}\]  
where \(R\) denotes the distance from transmitter to receiver, \(\alpha\) denotes the seawater absorption coefficient, \(H\) denotes the seawater depth, \(\alpha_{r}\) denotes the shallow sea attenuation coefficient, and \(K_{L}\) denotes the near-field propagation anomaly. In (1), \(\alpha\), \(H\), \(\alpha_{r}\), and \(K_{L}\) are not controllable in the LEACH protocol, so \(R\) is the key parameter for optimizing the transmission energy consumption. Note that \(dTL/dR\) is always non-negative, so \(TL\) increases with \(R\) in all the three cases of the Marsh-Schulkin model, i.e., minimizing the transmission energy consumption is equivalent to minimizing \(R\). In other words, to minimize the energy consumption of network operations, the total transmission distance of all packets throughout the network should be as small as possible. Based on this observation, the optimal number of clusters \(K_{opt}\) that minimizes the total transmission distance can be given by:  
\[K_{opt}=\underset{K}{argmin}R_{S} \tag{2}\]  
where \(R_{S}\) denotes the total transmission distance of all packets, and \(K\) denotes the number of clusters. Because the packets sent  
where \(N\) denotes the number of underwater sensor nodes and \(\overline{r}\) denotes the mean of the distances between the sensor nodes and cluster head nodes in the network as follows:  
\[\overline{r}=\frac{1}{N-K}\sum_{i=1}^{K}\sum_{m}r_{i}^{m} \tag{5}\]  
where \(\overline{r^{BS}}\) denotes the mean of the distances between cluster head nodes and base stations in the network as follows:  
\[\overline{r^{BS}}=\frac{1}{K}\sum_{i=1}^{K}r_{i}^{BS} \tag{6}\]  
Assuming that \(N\) underwater sensor nodes are uniformly distributed in a square with the length of \(l\), when \(N\) is relatively large, (5) can be approximated using integral calculus to represent the expected value of \(\overline{r}\), since the discrete summation converges to a continuous distribution by the law of large numbers. So, (5) can be approximated as follows:  
\[\overline{r}\approx E(r_{i}^{m})=\int_{0}^{2\pi}\int_{0}^{\sqrt{\frac{2}{K\pi} }}r\rho_{1}(r,\theta)\cdot rdrd\theta=\frac{2\pi}{3}\cdot\frac{l^{3}}{(K\pi)^ {\frac{3}{2}}}\cdot\frac{N}{l^{2}}=\frac{2 NI}{3\sqrt{\pi}}\cdot\frac{1}{K^{ \frac{3}{2}}} \tag{7}\]  
where \(\rho_{1}(r,\theta)\) denotes the probability density function of the sensor node positions, and \(\rho_{1}(r,\theta)=\frac{N}{l^{2}}\) when this distribution is uniform.