## 3 Preliminaries  
Underwater optical signals experience a wavelength (\(\lambda\))-dependent attenuation \(c(\lambda)\), which is a linear sum of absorption \(a(\lambda)\) and scattering \(b(\lambda)\) coefficients as shown in Equation (1) based on the composition of water. The magnitudes of light scattering and absorption may be used to categorize water into Jerlov types I, IA, IB, II, and III, which are oceanic waters, and types 1C, 3C, 5C, and 7C, which are coastal waters of increasing turbidity. Their spectral distributions for attenuation values are given in Figure 2. One limitation of the Jerlov classification is the assumption that the water body is homogeneous, which may not hold in a realistic scenario of underwater optical communication if the optical link crosses the physical boundaries (pycro-, halo-, and thermoclines) that structure coastal and open-ocean systems [24]. The nominal absorption, scattering, and attenuation coefficient values for clear, coastal, and turbid waters are given in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Manufacturer** & **Model** & \begin{tabular}{c} **Transmitter** \\ **Type** \\ \end{tabular} & **Color** & \begin{tabular}{c} **Range** \\ **(m)** \\ \end{tabular} & \begin{tabular}{c} **Data Rate** \\ **(max)** \\ \end{tabular} & \
\begin{tabular}{c} **Receiver Port** \\ **Shape** \\ \end{tabular} &
\begin{tabular}{c} **Power Consumption** \\ **(W)** \\ \end{tabular} \\ \hline Aqumodem [21] & OP2/OP2L & LED & Cyan & 1 & 80 kbps & Flat & Not available \\ \hline \multirow{3}{*}{Sonardyne [22]} & 8361 Bluecomm 200 & LED & Blue & 150 & 10 Mbps & Dome & Receiver: 10 W \\  & 8361 Bluecomm 200 UV & LED & UV & 75 & 10 Mbps & Dome & Transmitter: 30 W \\ \hline \multirow{3}{*}{Hydromea [23]} & Luma 100 & LED & Blue & 2 & 115 kbps & Flat, potted & Receiver: 0.5 W \\  & Luma 250 LP & LED & Blue & 7 & 250 kbps & Flat, potted & Receiver: 2-5 W \\ \cline{1-1}  & Luma 500 ER & LED & Blue & 50 & 500 kbps & Flat, potted & Receiver: 1 W \\ \cline{1-1}  & Luma X & LED & Blue & 50 & 10 Mbps & Flat & Transmitter: 2-17 W \\ \cline{1-1}  & Luma X UV & UV LED & UV & 50 & 10 Mbps & Flat & Receiver: 2 W \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of commercially available UOWC receivers and transmitters.  
The attenuation of the initial light intensity \(I_{0}\) over a distance \(d\) follows the Beer-Lambert law as shown in Equation (2). The power transmitted \(P_{t}\) is evaluated as Equation (3), where \(P_{e}\) is the electrical power and \(\eta_{e}\) is the optical power conversion efficiency. The power \(P_{r}\) received by a photodiode of surface area \(A_{r}\), and inclination angle \(\beta\), is given by Equation (4), where \(A_{s}\) is the geometric surface over which it is distributed and \(I_{0}=\frac{I_{0}}{A_{s}}\). \(A_{s}\) is commonly estimated as Equation (5) for P2P-LOS and DLOS links where the emitter beam divergence angle \(\theta\) (the angle at which intensity is 50% of its maximum) is much smaller than \(\frac{\pi}{20}\) or is greater than \(\frac{\pi}{20}\), respectively [26]. Substituting (5) in (4) derives a model for \(P_{r}\) as Equation (6).  
\[c(\lambda)=a(\lambda)+b(\lambda) \tag{1}\]  
\[I_{d}=I_{0}e^{-c(\lambda)} \tag{2}\]  
\[P_{t}=\eta_{e}P_{e} \tag{3}\]  
\[P_{r}=\left(\frac{P_{t}}{A_{s}}\right)^{-c(\lambda)d}A_{r}\text{cos}(\beta) \tag{4}\]  
\[A_{s}=\left\{\begin{array}{ll}\pi(d\tan\theta)^{2}&\left(\theta\ll\frac{\pi }{20}\right)\\ 2\pi d^{2}(1-\cos\theta)&\left(\theta>\frac{\pi}{20}\right)\end{array}\right. \tag{5}\]  
\[P_{r}=\left\{\begin{array}{ll}\left(\frac{\eta_{e}P_{e}}{\pi d^{2}\tan^{2} \theta}\right)e^{-c(\lambda)d}A_{r}\text{cos}\,\beta&\left(\theta\ll\frac{\pi }{20}\right)\\ \left(\frac{\eta_{e}P_{e}}{2\pi d^{2}(1-\cos\theta)}\right)e^{-c(\lambda)d}A_ {r}\text{cos}\,\beta&\left(\theta>\frac{\pi}{20}\right)\end{array}\right. \tag{6}\]