## 4 Devised Model for Received Power  
The geometric surface \(A_{s}\) for a DLOS link in the literature is commonly identified with that of a sphere [20] or a spherical cap [26], where the former assumes that the intensity distribution is uniform, and the latter tends to overestimate the received power by assuming a uniform distribution within the beam divergence angle but negating any power dissipation beyond the half angle \(\theta\). It is also common to generalize LED emission to Lambertian radiation patterns [27]. However, unlike in indoor free-space optical wireless communication systems where the fixed nodes are on the roof, and the mobile nodes often move on a parallel plane below (the floor), mobile UOWC consists of the UUVs moving in the third axis as well. Since LED-based emitters are preferred for DLOS links, \begin{table}
\begin{tabular}{c c c c} \hline
**Water Type** & \(a\) **(m\({}^{-1}\))** & \(b\) **(m\({}^{-1}\))** & \(c\) **(m\({}^{-1}\))** \\ \hline Clear ocean & 0.114 & 0.037 & 0.151 \\ Coastal ocean & 0.179 & 0.220 & 0.339 \\ Turbid harbor & 0.366 & 1.829 & 2.195 \\ \hline \end{tabular}
\end{table} 
Table 2: Nominal values of absorption (a), scattering (b), and attenuation (c) coefficients for different water types [3]. Figure 2: Spectral attenuation variation of Jerlov water types [25]. idealized approximations like those mentioned above may not be accurate for determining the received power within the established link range. This is because even a slight change in the polar angle at the transmitter can result in a significant region projected at the receiver end. Moreover, manufacturers typically only supply information regarding the overall transmitting optical power and the polar distribution curve of the relative intensity normalized to the LED's center axis. This makes it difficult to determine and adopt the actual polar intensities of non-uniform or irregular patterns into the DLOS UOWC channel models. In addition to this, \(\beta\) in Equation (4) is true only when the reference view of the axes of the emitter and receiver are coaxial. Therefore, herein we have adapted Equation (4) to approximate the received power more accurately in a mobile link scenario. For simplicity, we assume that the efficiency of emission is a uniform property across all angles at 100%, i.e., \(\eta_{e}=1\). The hemispheric area illuminated by the LED as the light source is divided into \(N\) annuli concentric about the LED's center axis as shown in Figure 3. The intensity \(I_{n}\) of the \(n_{th}\) annulus (\(n=1...N\)) is determined at distance \(d\) from the point of emission that subtends an angle \(\alpha_{n}\) with the center axis and has a surface area \(A_{n}\), where \(\tau\) is the corresponding relative intensity extracted from the datasheet. The total transmit power \(P_{T}\) is given by the sum of powers at each of the \(N\) annuli, corresponding to the number of samples from the intensity distribution. With receiver (Rx) and transmitter (Tx) being non-coaxial, Equations (7)-(13) illustrate the derivation of the adapted total transmit power, while Equation (14) gives the power received by the receiver PD at each annulus:  
\[I_{n}=\tau_{\alpha_{n}}I_{0} \tag{7}\]  
\[A_{\alpha_{1}}=2\pi d^{2}(1-\cos\alpha_{1}) \tag{8}\]  
\[A_{\alpha_{2}}=2\pi d^{2}(1-\cos\alpha_{2}) \tag{9}\]  
\[A_{2}=A_{\alpha_{2}}-A_{\alpha_{1}}\to A_{n+1}=A_{\alpha_{n+1}}-A_{ \alpha_{n}} \tag{10}\]  
\[A_{n}=2\pi d^{2}(\cos\alpha_{n-1}-\cos\alpha_{n})\ ;\ \left(\alpha_{0}=0^{ \circ}\right) \tag{11}\]  
\[P_{n}=I_{n}A_{n}=\tau_{\alpha_{n}}I_{0}A_{n} \tag{12}\]  
\[P_{T}=\sum_{n=1}^{N}P_{n}=2\pi d^{2}I_{0}\sum_{n=1}^{N}\tau_{\alpha_{n}}(\cos \alpha_{n-1}-\cos\alpha_{n}) \tag{13}\]  
Figure 2:0  
where \(\alpha\) is the angle subtended at Tx by the ray and center axis of Tx, and \(\beta\) is the inclination of the Rx to the horizontal axis parallel to and facing the Tx including any rotation; their linear sum gives the overall AOI, \(\varphi\), as shown in Equation (15).