## 3 Propagation Model
### Temporal Response of the Channel  
We design a unified system model that accommodates channel loss due to multiple scattering and measures its respective performances in terms of magnitude response for different system configurations and water types. Depending upon the degree of spatial dispersion and system variables, the scattered photons incur distinct path lengths. These path length differences between photons give \begin{table}
\begin{tabular}{c c c} \hline \hline
**Water Type** & **Attenuation Coefficient \(\mathbf{c}\)** & **Water Albedo \(\mathbf{w_{o}}\)** \\ \hline Clear & 0.15 m\({}^{-1}\) & 0.25 \\ Coastal & 0.4 m\({}^{-1}\) & 0.55 \\ Harbor & 2.19 m\({}^{-1}\) & 0.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Petzold’s measured water albedo and attenuation coefficients for the three water types. rise to temporal dispersion. In addition to the spreading of light, it also causes loss of received power, resulting in reduced signal levels. We first estimate the channel using a combination of water parameters including water albedo \(w_{o}\), absorption, scattering coefficients, and system parameters, i.e., receiver FoV, aperture, and link distance. Our simulations track each photon path for a homogeneous medium to generate photon time-of-flight (ToF). ToF is a measure of the speed of photon propagation under the water. It is directly proportional to the product of photon distance traveled \(d_{pro}\) and refractive index of water \(n_{w}\); however, it is inversely proportional to the speed of light in vacuum \(c\), as given in Equation (16):  
\[t_{ToF}=\frac{d_{pro}n_{w}}{c}. \tag{16}\]  
The ToF equation gives information about channel's temporal response and its frequency. In order to determine the discrete time channel IR in terms of the received power, we first form a histogram of the total received power and then normalize it by the weight of the histogram, i.e., the total number of received photons. In principle, impulse response gives the dispersal of power over time at the receiver. For an equivalent magnitude response, we simply take the z-transform of the IR and arrive at Figures 3-5. Figure 4: Comparison of the magnitude response of coastal water under various aperture sizes and FoVs for a 3 m LoS link. Figure 3: Comparison of the magnitude response of ocean water under various aperture sizes and FoVs for a 3 m LoS link. Discrete Histograms and Their Respective Bin Widths  
The histogram is a conventional method of estimating the densities of ordinal data sets. It also offers a true estimate of the probability density function. For the least mean squared error, the appropriate choice of bin size is of utmost importance. If the bin size is too small, the histogram will be too uneven; conversely, if the bin size is too large, the histogram will be too flat, equivalent to a large variance. The appropriate choice of bin size not only balances the variance, but it also minimizes the mean squared error. For a fairly large data set such as the received number of photons, it is crucial to obtain the optimal number of bins to place all the photons' received powers in the appropriate bins based on the time-of-flight information. Therefore, for the ideal number of bins and respective bin sizes \(t_{bw}\) we use [PERSON]'s rule [26]. The optimal bin size requires an estimate of the true elemental density \(f\); however, [PERSON] [26] derived an equation for the optimal choice of histogram bin size that either uses the knowledge of the density function \(f\) or determines the bin width based on normally distributed data. \[h_{n} =\left\{\frac{6}{{f^{\prime}\left(x\right)^{2}dx}}\right\}^{ \frac{1}{3}}n^{-\frac{1}{3}}, \tag{17}\] \[h_{n} =3.49\sigma n^{-\frac{1}{3}}, \tag{18}\]  
where \(\sigma\) is the estimated standard deviation, and \(n\) denotes the sample size.