## 3 Performance Analysis of UOWC Links under Air Bubble-Induced Fading
### Outage Probability Analysis  
An outage probability analysis for a SISO UOWC system under the UOWC channel model in the presence of air bubbles proposed in [22] is presented here to assess the completeness of the UOWC system performance. The outage probability is defined as the instantaneous SNR falls below a predefined threshold, \(\gamma_{th}\) as [29]  
\[\text{OP}=P(\gamma_{T}\leq\gamma_{th}), \tag{14}\]  
where \(\gamma_{T}=4\cdot\gamma\cdot(h_{b}\cdot L)^{2}\). Therefore, Equation (14) can be expressed as  
\[\text{OP}=P(4\cdot\gamma\cdot(h_{b}\cdot L)^{2}\leq\gamma_{th})=\int_{0}^{ \sqrt{\gamma_{th}/4L^{2}\gamma}}f_{h_{b}}(h)dh=F_{h_{b}}\left(\sqrt{\frac{1}{4 L^{2}\gamma}}\right), \tag{15}\]  
where \(\overline{\gamma}=\frac{\gamma}{\gamma_{th}}\) is the normalized SNR, and \(F_{h_{b}}(\cdot)\) is the cumulative distribution function (CDF). The definition of the normalized SNR allows for the extrapolation of outage probability results to any predefined threshold value. The corresponding CDF when considering small air bubbles is obtained as follows ([28], Equation (2)).  
\[F_{h_{\text{m}}\text{m}}(h;a,d,p)=\frac{\
u(\frac{d}{p},(h/a)^{p})}{\Gamma \left(\frac{d}{p}\right)}, \tag{16}\]  
where \(\
u(\cdot,\cdot)\) is the lower incomplete Gamma function [33]. Hence, by substituting Equation (16) into Equation (2.3) of [33], \(F_{h_{\text{m}}\text{m}}(\cdot)\) can be easily derived, as follows:  
\[F_{h_{\text{m}}\text{m}\text{m}}(h;a,d,p)=1-\frac{\Gamma(\frac{d}{p},(h/a)^{p} )}{\Gamma\left(\frac{d}{p}\right)}, \tag{17}\]  
where \(\Gamma(\cdot,\cdot)\) is the upper incomplete Gamma function [33]. Therefore, the corresponding outage probability when small air bubbles are considered can be easily derived by substituting Equation (17) into (15), as follows:  
\[\text{OP}_{\text{m}\text{m}\text{m}\text{m}}=1-\frac{\Gamma\left(\frac{d}{p},\frac{1}{(aL)^{p}}\overline{\gamma}^{-\frac{p}{2}}\right)}{\Gamma\left( \frac{d}{p}\right)}. \tag{18}\]  
In the same way that the average BER, the outage probability at high SNR also tends to \(\text{OP}\doteq(O_{c}\gamma)^{-O_{d}}\), where \(O_{c}\) is the coding gain, and \(Od\) is the diversity order. Hence, by replacing Equation (10) into Equation (15), we obtain the asymptotic behavior of the outage probability at high SNR for small air bubbles as follows:  
\[\text{OP}_{\text{m}\text{m}\text{m}}\doteq\frac{p}{(aL)^{d}\,\Gamma\left( \frac{d}{p}\right)}\overline{\gamma}^{-\frac{d}{2}}. \tag{19}\]  
Based on Equation (3), the CDF of large air bubble-induced fading can be expressed as  
\[F_{h_{\text{h}_{\text{m}}}}(h;a_{1},d_{1},p_{1},a_{2},d_{2},p_{2},w)=w\cdot F_{ h_{\text{m}}\text{m}}(h;a_{1},d_{1},p_{1})+(1-w)F_{h_{\text{m}}\text{m}}(h;a_{2},d_{2},p_ {2}). \tag{20}\]Hence, the outage probability in the presence of large air bubbles is derived by substituting Equation (17) into Equation (20) as follows:  
\[\text{OP}_{\text{large}}=w\left(1-\frac{\Gamma\Big{(}\frac{d_{1}}{p_{1}},\frac{1 }{(a_{1}L)^{\text{PI}}}\gamma^{-\frac{p_{1}}{2}}\Big{)}}{\Gamma\Big{(}\frac{d_{ 1}}{p_{1}}\Big{)}}\right)+(1-w)\left(1-\frac{\Gamma\Big{(}\frac{d_{2}}{p_{2}}, \frac{1}{(a_{2}L)^{\text{PI}}}\gamma^{-\frac{p_{2}}{2}}\Big{)}}{\Gamma\Big{(} \frac{d_{2}}{p_{2}}\Big{)}}\right). \tag{21}\]  
Finally, by applying Equation (11) into Equation (15), we obtain the asymptotic of the outage probability at high SNR for large air bubbles as follows:  
\[OP_{\text{large}}\doteq w\frac{\Big{(}\frac{1}{a_{1}L}\Big{)}^{d_{1}}}{ \Gamma\Big{(}\frac{d_{1}+p_{1}}{p_{1}}\Big{)}}\gamma^{-\frac{d_{1}}{2}}+(1-w )\frac{\Big{(}\frac{1}{a_{2}L}\Big{)}^{d_{2}}}{\Gamma\Big{(}\frac{d_{2}+p_{2} }{p_{2}}\Big{)}}\gamma^{-\frac{d_{2}}{2}}. \tag{22}\]