## 2 System Model
### Energy Model  
Underwater sensor nodes consume energy for sensing, packet transmission, packet reception, data processing, network maintenance, staying awake, etc. Among them, data transmission induces the main energy consumption, which is related to the packet size and transmission distance [41]. It includes the power consumed by wireless optical transmitting equipment and the power consumed by the transmitter power amplifier. For the receiver, only the power consumed in the device is considered to be related to the received packet. When the transmission packet size is invariant, the value of the power is constant.  
Given the line of sight (LOS) transmission, the estimation of received optical power is determined by the empirical path loss model [40], which gives an effective model regarding the product of transmitter power, telescope gain, and loss as follows:  
\[\mathrm{P}_{\mathrm{r}}=\mathrm{P}_{\mathrm{r}}\mathrm{\eta}_{\mathrm{r}} \mathrm{\eta}_{\mathrm{r}}\mathrm{e}^{\frac{-\mathrm{c}(\lambda)\mathrm{R}}{ \mathrm{cos}\theta}}\frac{\mathrm{A}_{\mathrm{r}}\mathrm{cos}\theta}{2\pi \mathrm{R}^{2}(1-\mathrm{cos}(\theta_{0}))} \tag{8}\]  
where R represents the vertical distance between the transmitting plane and the receiving plane (approximately equal to the transmission range), \(\mathrm{\eta}_{\mathrm{r}}\) and \(\mathrm{\eta}_{\mathrm{r}}\) represent the optical efficiencies of the corresponding transmitter and receiver, respectively, with \(\mathrm{A}_{\mathrm{r}}\), \(\mathrm{\theta}_{0}\), and \(\mathrm{\theta}\) being the aperture region at the receiver, the divergence angle of the transmitter beam, and the angle perpendicular to the receiver plane and the transmitter-receiver trajectory, respectively. Without considering the beam divergence, only the effective area of the receiving antenna and the optical effect of the optical system are taken into account, and the transmission power \(\mathrm{P}_{\mathrm{t}}\) can be expressed as [40]  
\[\mathrm{P}_{\mathrm{t}}=\frac{\mathrm{P}_{\mathrm{r}}\mathrm{e}^{\mathrm{K}( \lambda)\mathrm{R}}\mathrm{\pi R}^{2}}{\mathrm{\delta\eta}_{\mathrm{r}}\mathrm{ n}_{\mathrm{r}}}, \tag{9}\]  
The transmission energy consumed by the transmitter can be calculated as follows:  
\[\mathrm{E}=\mathrm{P}\,*\,\mathrm{t}, \tag{10}\]  
where \(\mathrm{\delta}\) denotes the effective area of the receiving antenna, t denotes the transmission time, and K(\(\lambda\)) represents the attenuation coefficient of seawater. The energy consumed by node #i during its delivery of 1-bit data to node #j can be derived as follows [40]:  
\[\mathrm{E}_{\mathrm{t}}^{\mathrm{i}\mathrm{j}} =\mathrm{E}_{\mathrm{elec}}+\frac{\mathrm{P}_{\mathrm{r}}\mathrm{ e}^{\mathrm{K}(\lambda)\mathrm{d}_{\mathrm{ij}}}\mathrm{\pi d}_{\mathrm{ij}}^{2}}{ \mathrm{\delta\eta}_{\mathrm{r}}\mathrm{n}_{\mathrm{r}}\mathrm{R}_{\mathrm{b}}} \tag{11}\]where E\({}_{\rm{elec}}\) indicates the energy consumed in processing 1-bit information, d\({}_{\rm{ij}}\) indicates the distance from node #i to #j, and R\({}_{\rm{b}}\) indicates the data-rate in UWOC. The energy consumption for the node to receive and process 1-bit information can be derived as follows [40]:  
\[\rm{E_{r}}{=}\ E_{\rm{elec}}+\frac{P_{r}}{R_{b}} \tag{12}\]