## 2 Seawater Channel Modelling Elements
### Scattering Phase Function Formats
#### 2.7.2 Henyey-Greenstein Scattering Phase Function  
The study in [76] introduced the HG-SPF to characterise the scattered light via interstellar dust, given by Equation (25) in Table 4. Here \(g\) is the HG asymmetry parameter, which relies on the medium properties. It is equal to the average cosine of the scattering angle across all paths of scattered light [48]. The mean of the cosine of the scattering angle cos ranges from \(-1\) to \(1\). There is entire backscattering at \(g=-1\), isotropic scattering at \(g=0\), and complete forward scattering at \(g=1\). As in [66], we produced Figure 5 to illustrate the behaviour of HG-SPF over g set {0.0, 0.5, 0.9}. We can observe that when \(g=0.924\) the HG becomes a suitable approximation for underwater light transmission and the closest match for Petzold's data. The HG-SPF is widely utilised through ocean optics simulations, as well as in MCS and other channel modelling studies [66]. The HG-SPF is defined in terms of the \(g\) as follows [76]:  
\[\widetilde{\beta}(\theta_{s},g,\ \lambda)=\frac{1}{2\pi}\Bigg{\{}\frac{\left(1-g^{2 }\right)}{2\left(\ 1+g^{2}-2\ g\text{cos}\,\theta_{s}\ \right)^{3/2}}\Bigg{\}} \tag{25}\]  
where \(\theta_{s}\) is the scattering angle and \(-1\ \leq\ g\ \leq\ 1\); when \(g=0\) Equation (25) describes the isotropic scattering, and when \(g\ \sim\ 1\), Equation (25) represents a forward scattering, with a simple manipulation of Equation (25) and satisfying Equation (22), we can write the PDF of \(\theta_{s}\) as follows [73]:  
\[p(\theta_{s})=\left\{\frac{\left(1-g^{2}\right)sin\theta_{s}}{2\left(\ 1+g^{2}-2\ g\text{cos}\,\theta_{s}\ \right)^{\frac{3}{2}}}\right\}\ \ \ \ \ \text{when}\ g\
eq 0 \tag{26}\]  
Then, the CDF of \(cos\theta_{s}\) over the range \({}_{s}\) of \(1\) is given as follows:  
\[P(cos\theta_{s})=\frac{1-g^{2}}{2g}\Bigg{\{}\frac{1}{1-g}-\frac{1}{\left(1+g^{ 2}-2\ g\text{cos}\,\theta_{s}\right)^{\frac{1}{2}}}\Bigg{\}}=q \tag{27}\]  
where \(q\) is an RV and has the range \(0\ \leq\ q\ \leq\ 1\). Solving for \(\cos\theta_{s}\), we obtain the following:  
\[cos\theta_{s}=\Bigg{\{}\frac{1+2g-2 gq-2q+2 gq^{2}+g^{2}-2g^{3}q-2g^{2}q+2g^{3}q ^{2}}{(-1-g+2 gq)^{2}}\Bigg{\}} \tag{28}\]For \(g\
eq 0\),  
\[cos\theta_{s}=\frac{1}{2g}\Bigg{\{}1+g^{2}-\left(\frac{1-g^{2}}{1-g+2g\,q}\right) ^{2}\Bigg{\}}\] (29) For \(g=0\),  
\[cos\theta_{s}=1-2g \tag{30}\]  
The HG-SPF is widely utilised through ocean optics simulations, as well as in MCS and other channel modelling studies [66].  
In Figure 5, the main conclusion from the \(\cos\theta_{s}\) family of curves in Figure 5 is that the main bulk of the scattering occurs within a small range of \(\theta_{s}\) values \(0~{}\leq~{}\theta_{s}~{}\leq~{}3\). From the family of curves of the Photon Travelled Step, we notice that the photon travels in large steps over the \(0~{}\leq~{}\theta_{s}~{}\leq~{}3\) value range. The curves in Figure 5 can be used as unit tests for the RTE numerical solution use cases.  
Figure 6 shows the difference between PAP-SPF and HG-SPF for various g values and indicates that the three single-scattering types (forward, isotropic, backward) share practically the same value of \(\overset{\sim}{\beta}(\theta_{s},g,~{}\lambda)\) for the radian range \(1.0~{}\leq~{}\theta_{s}~{}\leq~{}2.5\).