## 2 Light Propagation in Water
### Turbulence
As the spatial coherence radius decreases, a greater number of these larger turbulent cells become ineffective, preventing the scintillation index from increasing further and in some cases even reducing it [21; 22]. Scintillation refers to the fluctuations in the irradiance received when a beam propagates through a turbulent medium [23]. The variance in these fluctuations, known as the scintillation index, serves as a measure of turbulence strength. The scintillation index is defined as follows:  
\[\sigma_{I}^{2}=\frac{\langle I^{2}\rangle-\langle I\rangle^{2}}{\langle I \rangle^{2}}, \tag{4}\]  
where \(\langle x\rangle\) is the long-time average over \(x\). The scintillation index in weak turbulence is defined based on the Kolmogorov power-law spectrum by  
\[\sigma_{I,sp}^{2}=0.4\,\sigma_{R}^{2}, \tag{5}\]  
where \(\sigma_{R}^{2}=1.23\,C_{n}^{2}\,k^{7/6}L^{11/6}\) is called the Rytov variance for a plane wave. \(k=2\pi/\lambda\) is the wave number, \(L\) is the link distance, and \(C_{n}^{2}\) is the refractive index structure constant, which is also a measure of the strength of turbulence. Note that \(C_{n}^{2}\) is not constant. It varies with seasonal, daily, and hourly fluctuations, as well as by geographical location and altitude [24]. The refractive index structure constant is given by [16]:  
Figure 1: Scintillation index vs. distance for varying ratios of temperature-to-salinity fluctuations \(w\). The parameters for numerical simulations are taken from Table 1.  
\[C_{n}^{2}=16\pi^{2}k^{-\frac{7}{6}}L^{-\frac{11}{6}}\,\text{Re}\bigg{\{}\int_{0}^{ L}\int_{0}^{\infty}\kappa\Big{[}E(z,\kappa)E(z,-\kappa)+|E(z,\kappa)|^{2}\Big{]} \Phi(\kappa)\,d\kappa\,dz\bigg{\}}. \tag{6}\]  
Assuming a point source size,  
\[E(z,\kappa)=jk\,\exp\bigg{(}0.5j\,\frac{z(z-L)}{kL}\,\kappa^{2}\bigg{)}. \tag{7}\]  
Compared to atmospheric turbulence, where strong turbulence typically arises after a few kilometers, strong turbulence in the ocean can occur after just a few meters [17]. The Equation (5) is not suitable for strong turbulence. The extended Rytov method provides a theoretical approach for estimating the scintillation index in both weak and strong turbulence scenarios, making it more suitable for oceanic conditions compared to the conventional Rytov method, which is traditionally limited to weak turbulence solutions. The scintillation index \(\sigma_{I}^{2}\) in the extended Rytov method is determined by the parameters \(\sigma_{SS}^{2}\) and \(\sigma_{LS}^{2}\), representing independent complex phase disturbances related to small-scale (SS) and large-scale (LS) fluctuations, respectively [20; 25]:  
\[\sigma_{I,sp}^{2}=\exp(\sigma_{SS}^{2}+\sigma_{LS}^{2})-1,\ \ \sigma_{SS}^{2}=\frac{0.196\, \sigma_{R}^{2}}{(1+0.224\,\sigma_{R}^{12/5})^{7/6}},\ \ \sigma_{LS}^{2}=\frac{0.204\, \sigma_{R}^{2}}{(1+0.276\,\sigma_{R}^{12/5})^{5/6}}, \tag{8}\]  
A stochastic model based on a probability density function (PDF) is required to represent turbulence-induced irradiance fluctuations and to study their fading statistics. Over the years, many PDF models of irradiance have been proposed with varying degrees of success. The log-normal distribution is suitable for weak irradiance fluctuations, which is defined as follows:  
\[p_{I}(I)=\frac{1}{I\sqrt{2\pi\,\sigma_{I,sp}^{2}}}\exp\Bigg{\{}-\frac{[\ln{(I) }+\sigma_{I,sp}^{2}/2]^{2}}{2\,\sigma_{I,sp}^{2}}\Bigg{\}}, \tag{9}\]  
where \(I\) is the normalized irradiance.