## 3 Scintillation Index for Spherical Wave with Aperture Average Effect  
In the anisotropic ocean turbulence, the scintillation index is usually used to measure the performance level of the UWOC systems under intensity fluctuations. Under weak fluctuation conditions, the scintillation index is approximately equal to the log-irradiance variance, i.e., \(m^{2}=\sigma_{\ln I}^{2}\)[37]. Previous studies [15; 41] show that the scintillation index \(\sigma_{\ln I}^{2}\) is strongly related to the log-amplitude variance \(\sigma_{\chi}^{2}\). The log-irradiance variance can be expressed as [42], where \(\langle I\rangle\) is the statistical mean of the optical intensity. The log-amplitude variance is used to describe the strength of amplitude fluctuation and is defined as [43], where \(A\) is the amplitude of an optical wave and \(\langle A\rangle\) is the statistical average of the amplitudes. The relationship between the logarithmic intensity fluctuation variance and the logarithmic amplitude fluctuation variance can be derived as \(\sigma_{\ln I}^{2}=4\sigma_{\chi}^{2}\). Therefore, the scintillation index can be obtained  
\[m^{2}=4\sigma_{\chi}^{2}. \tag{4}\]By using the Rytov method, the variance of the log-amplitude for a spherical wave propagating in the anisotropic turbulent medium is derived as [44]  
\[\begin{split}\sigma_{\chi}^{2}=&\pi\,\text{Re}\bigg{\{} \int_{0}^{L}dz\int_{-\infty}^{+\infty}dx_{x}\int_{-\infty}^{+\infty}dx_{y}\; \big{[}P\big{(}z,\kappa_{x},\kappa_{y}\big{)}P\big{(}z,-\kappa_{x},-\kappa_{y} \big{)}\\ &+\big{|}P\big{(}z,\kappa_{x},\kappa_{y}\big{)}\big{|}^{2}\big{]} \cdot\Phi_{n}\big{(}\mu_{x},\mu_{y},\kappa_{x},\kappa_{y}\big{)}\bigg{\}}\end{split}, \tag{5}\]  
It should be noted that only the radial component is considered and used for the point-like scintillation case, that is, the normalized variance amount of scintillation at a specific position (normally on-axis) of the transverse receiver plane. In addition, the symbol Re in Equation (5) denotes the real part, \(z\) is the distance along the propagation axis, and \(L\) is the wave propagation distance. In addition, we have  
\[P\big{(}z,\kappa_{x},\kappa_{y}\big{)}=jk\exp\left[-0.5(kL)^{-1}jz(L-z)\Big{(} \kappa_{x}^{2}+\kappa_{y}^{2}\Big{)}\right], \tag{6}\]  
which is the propagation expression in Equation (5), \(j=\sqrt{-1}\), \(k=2\pi/\lambda\) and is the number of the wave and \(\lambda\) is the wavelength.  
In addition, according to the previous study [33], the logarithmic amplitude variance of the spherical wave over a circular aperture, which is over a certain area of the receiver aperture and sometimes called \"power scintillation\" can be obtained by  
\[\begin{split}&\sigma_{XD}^{2}=\pi\,\text{Re}\bigg{\{}\int_{0}^{L}dz \bigg{\{}\int_{-\infty}^{+\infty}dx_{x}\int_{-\infty}^{+\infty}dx_{y}\;\big{[} P\big{(}z,\kappa_{x},\kappa_{y}\big{)}\cdot P\big{(}z,-\kappa_{x},-\kappa_{y} \big{)} \tag{7}\]  
where \(D\) denotes the diameter of the receiver aperture. It is noted that anisotropy also affects the beam vector, which is denoted by \(\kappa\) in the aperture averaging effect. In addition, the aperture averaging effect can be expressed by the integral of the radial parameters \(\kappa_{x}\) and \(\kappa_{y}\).  
Finally, the intensity fluctuations of a spherical optical beam in an anisotropic oceanic medium are quantified by the scintillation index by inserting Equation (7) into Equation (4). Note that we mainly focus on the effect of oceanic turbulence on the scintillation index, thus the closed-form expression of the scintillation index is ignored in this study.