## 2 System and Channel Model
It can be seen from Equation (1) that the instantaneous received signal-to-noise ratio (SNR) of the Radio Frequency link is as follows:  
\[\gamma_{1}=\frac{P_{1}{h_{1}}^{2}}{L_{\text{{U}}IR}\sigma_{1}^{2}}=\frac{{h_{ 1}}^{2}\overline{\gamma}_{1}}{L_{\text{{U}}IR}}, \tag{2}\]  
in the formula, \(\overline{\gamma}_{1}=P_{1}/\sigma_{1}^{2}\) corresponds to the average SNR of the RF link. Given that the channel fading of the RF link follows a Generalized K distribution, the Probability Density Function (PDF) of the instantaneous SNR is \(\gamma_{1}\) (The exact derivation of (3) can be found in Appendix A):  
\[f_{\gamma_{1}}(\gamma_{1})=\frac{1}{\gamma_{1}\Gamma(m)\Gamma(k)}G_{0,2}^{2,0 }\Bigg{[}\frac{mk\cdot L_{\text{{U}}tk}\cdot\gamma_{1}}{\overline{\gamma}_{1 }}\Bigg{|} \begin{array}{c}-\\ k,m\end{array} \Bigg{]}, \tag{3}\]  
In the above formula, \(G_{p,q}^{m,H}(\cdot)\) is the Meijer-G function [14] (Equation 9.301). Derivation of the Cumulative Distribution Function of the Instantaneous Received SNR for the S-R Composite Fading Link  
Substitute the above Formula (3) into the definition formula of the Cumulative Density Function (CDF) of the instantaneous received SNR \(\gamma_{1}\), and we have the following:  
\[F_{\gamma_{1}}(\gamma_{1})=\int_{0}^{\gamma_{1}}f_{\gamma_{1}}(x)dx=\frac{1} {\Gamma(m)\Gamma(k)}\int_{0}^{\gamma_{1}}\frac{1}{x}G_{0,2}^{2,0}\Bigg{[}\frac {mkL_{\text{{U}}tk}\cdot x}{\overline{\gamma}_{1}}\Bigg{|} \begin{array}{c}- \\ k,m\end{array} \Bigg{]}dx. \tag{4}\]  
By using Formula (2.24.2) in Reference [15] and making relevant parameter comparisons, Formula (4) can be transformed into the following:  
\[ \begin{array}{ll}F_{\gamma_{1}}(\gamma_{1})&=\frac{1}{\Gamma(m)\Gamma(k)} \int_{0}^{\gamma_{1}}\frac{1}{x}G_{0,2}^{2,0}\Bigg{[}\frac{mkL_{\text{{U}}tk} \cdot x}{\overline{\gamma}_{1}}\Bigg{|}\begin{array}{c}-\\ k,m\end{array}\Bigg{]}dx\\ &=\frac{1}{\Gamma(m)\Gamma(k)}G_{1,3}^{2,1}\Bigg{[}\frac{mkL_{\text{{U}}tk} \cdot\gamma_{1}}{\overline{\gamma}_{1}}\Bigg{|}\begin{array}{c}1\\ k,m,0\end{array}\Bigg{]}\end{array} . \tag{5}\]  
Derivation of the Probability Density Function of the Instantaneous Received SNR of the R-D Composite Fading Link  
Assume that the R-D link adopts the DF (Decode-and-Forward) relay protocol, and the expression of the received signal is as follows:  
\[y_{\text{{R}}D}=\eta\sqrt{P_{2}}h_{2}\cdot\xi+n_{2}, \tag{6}\]in the above formula, \(\eta\) is the photoelectric conversion coefficient of the UWOC link, \(\hat{s}\) represents the decoded signal at the relay node R, and \(P_{2}\) is the transmission power of the reconstructed signal; \(n_{2}\) is a Gaussian random variable with a mean of 0 and a variance of \(\sigma_{2}^{2}\). The underwater channel gain \(h_{2}=h_{f}h_{p}h_{l2}\) is a mixed fading that includes the underwater path loss \(h_{l2}\), the zero line-of-sight pointing error \(h_{p}\), and the GGD (Generalized Gamma Distribution) ocean turbulence fading \(h_{f}\). Now, first derive the PDF (Probability Density Function) of the mixed fading coefficient \(h_{2}\). The PDF of the GGD is known as follows [16]:  
\[f_{h_{f}}\Big{(}h_{f}\Big{)}=\frac{c\cdot h_{f}^{ac-1}}{b^{ac}}\frac{\exp\! \left(-\Big{(}\frac{h_{f}}{b}\Big{)}^{c}\right)}{\Gamma(a)}, \tag{7}\]  
In the formula, \(b\) is the scale parameter; \(a\) and \(c\) are the shape parameters. In addition, the zero line-of-sight pointing error \(r\) with a vertical distance of \(z\) from the transmitting point and a radial error of \(h_{p}\) can be approximately expressed as follows [17]:  
\[h_{p}(r;z)\approx A_{0}\exp(-\frac{2r^{2}}{w_{z_{eq}}^{2}}), \tag{8}\]  
In the equation, \(A_{0}=\left[erf(v)\right]^{2}\) represents the power ratio of the laser beam collected by the receiver at the center of the detector \(r=0\), where \(v=(\sqrt{\pi}r_{a})/(\sqrt{2}w_{z})\) represents the ratio of \(r_{a}\), the receiver aperture to the beam width \(w_{z}\), and \(erf(\cdot)\) is the error function. \(w_{z_{eq}}^{2}=w_{z}^{2}\sqrt{\pi}erf(v)/2v\exp(-v^{2})\) represents the equivalent beam width.