## 2 Comprehensive Multiparameter Model for UWOC Channels
In real underwater environments, the refractive index of water will slightly increase as the pressure increases or the temperature decreases within the normal range [24], and the dissipation rate of mean-squared temperature \(\chi_{T}\) in the PSD function in Equation (4) is related to the temperature gradient of water [23]. In this way, the phase values on the screen can be obtained to model the effect of underwater turbulence. In addition to spatial randomness of static turbulence, the model is extended to dynamic scenarios based on the [PERSON] frozen-turbulence hypothesis [25], which treats temporal variations of the turbulence at a location as its advection by the turbulence flow. Here, we also introduce the long-size phase screens and the parameter of water flow velocity to simulate the temporal properties of the channel. Regarding the diffraction part, the absorption and scattering effects can be concisely described as the exponential light attenuation as [26]:  
\[I=I_{0}e^{-(a+b)z} \tag{5}\]  
where \(I_{0}\) is the initial light intensity, z is the transmission distance, \(a\) and \(b\) are the absorption coefficient and the scattering coefficient of seawater, respectively. The coefficients \(a\) and \(b\) will vary with different water types, climatic conditions, and optical wavelengths [1]. The light transmission path in each propagation segment (typically on the scale of meter) is much longer than the light position movement on the transverse plane (typically on the scale of millimeter). Therefore, it is reasonable to neglect the distance difference among each light path and unify the transmission distance of the whole light beam as \(\Delta z\). Sincethe light intensity satisfies \(I\propto\left|U\right|^{2}\), the optical field after passing through a \(\Delta z\) distance of seawater should be:  
\[U=U_{vac}\sqrt{e^{-(a+b)\Delta z}} \tag{6}\]  
where \(U_{vac}\) represents the optical field after passing through the same distance of vacuum. Moreover, we develop the known transfer function for vacuum \(\hat{H}_{vac}\)[27] to the transfer function for seawater transmission as:  
\[\hat{H}=\sqrt{e^{-(a+b)\Delta z}}\hat{H}_{vac}=\sqrt{e^{-(a+b)\Delta z}}e^{b \Delta z}e^{-\frac{\left|\Delta z\right|(b^{2}+ky^{2})}{4\pi}} \tag{7}\]  
thereby integrating underwater absorption and scattering effects into the diffraction parts of the model. Based on the above channel model, the comprehensive analysis of all the relevant parameters' effects on the UWOC channel can be carried out, including the absorption and scattering coefficients \((a,b)\), the turbulence inherent properties \((\epsilon,\,\chi_{\Upsilon},\,\omega)\), the macroscopic velocity of water flow \((\upsilon)\), and the total transmission distance \((z)\). In view of the complexity of the transmission process and the multitude of parameters, a qualitative evaluation of the static received spots is first conducted at the lens position. The results can also visually prove the availability of our model. Then, the quantitative analysis of the dynamic underwater channel is completed by studying the probability density distribution function (PDF) of the received light intensity in succession. Furthermore, the UWOC performance is evaluated through the PDF data. The BER of the on-off keying modulation system can be calculated by the Gauss error function as:  
\[\text{BER}=erfc(\zeta)=\frac{2}{\sqrt{\pi}}\int_{\zeta}^{\infty}e^{-\frac{p^{2 }}{\pi}}dt \tag{8}\]  
where the parameter \(\zeta\) is the square root of the signal-to-noise ratio (SNR) as:  
\[\zeta=\sqrt{\text{SNR}}=\sqrt{\frac{\left\langle I\right\rangle}{{\sigma_{ \text{s}}}^{2}+{\sigma_{\text{N}}}^{2}}} \tag{9}\]  
where \(\left\langle I\right\rangle\) represents the average received light intensity and \({\sigma_{\text{s}}}^{2}\) is its variance. Both can be obtained from the PDFs through numerical simulations. \({\sigma_{\text{N}}}^{2}\) is the additive white Gaussian noise (AWGN) introduced by the communication system.