## 2 Seawater Channel Modelling Elements
### Scattering Functions
#### 2.6.2 Scattering Phase Function  
From the above equations, we can define the spectral volume SPF as follows:  
\[\overset{\sim}{\beta}(\theta,\ \Phi,\lambda)=\frac{\beta(\theta,\ \lambda)}{b( \lambda)}\ \ \ \ \ \ \ \text{in }(sr)^{-1} \tag{16}\]  
where \(\Phi\) is the scattered azimuthal angle. From Equations (13) and (16), we can deduce the normalisation condition for the SPF, with the assumption that the integral of \(\overset{\sim}{\beta}(\theta,\Phi)\) overall directions is unity for a single-mode wavelength \(\lambda\) optical beam:  
\[\overset{2\pi}{\underset{0}{\int}}\overset{\pi}{\underset{0}{\int}}\overset{ \pi}{\beta}(\theta,\Phi)\ sin\theta\ d\theta\ d\Phi=1 \tag{17}\]  
The SPF \(\overset{\sim}{\beta}(\theta,\Phi)\) can be defined as the PDF that gives the probability of a scattered photon at \(\theta\) and \(\Phi\) away from the incident direction. The angles \(\theta\) and \(\Phi\) are two independent random variables, which implies that their PDFs \(p(\theta)\) and \(p(\Phi)\) are independent of one another; hence, \(p(\theta,\Phi)=p(\theta)\ p(\Phi)\). It is safe to assume that the \(\Phi\) with respect to the incident direction is uniformly distributed over its value range \([0,2\pi]\), and consequently \(\int_{0}^{2\pi}p(\Phi)d\Phi=1\); i.e., the PDF \(p(\Phi)=\frac{1}{2\pi}\). In the UOWC water channel, it is practically acceptable to conclude the following:  
\[2\pi\int_{0}^{\pi}\overset{\sim}{\beta}(\theta)\ sin\theta\ d\theta=1\ \ \text{ normalisationconditionfortheSPF} \tag{18}\]  
We should keep in mind that \(\overset{\sim}{\beta}(\theta)\) is equivalent to \(\overset{\sim}{\beta}(\theta,\lambda)\). On the left-hand side of Equation (18), we consider  
\[p(\theta)=2\pi\overset{\sim}{\beta}(\theta)\ sin\theta \tag{19}\]  
where \(p(\theta)\) is the PDF of \(\overset{\sim}{\beta}(\theta)\ sin\theta\); then, the CDF over the value range \(0\ \leq\ \theta\ \leq\ \theta_{s}\) takes the following format [73]:  
\[P(\theta_{s})=2\pi\int_{0}^{\theta_{s}}\ p(\theta)\ \ d\theta=q \tag{20}\]  
where \(q\) is an RV with a value range \(0\ \leq\ q\ \leq\ 1\). Alternatively, SPF can be expressed in terms of the \(\mu_{s}=\cos\theta_{s}\) where \(d\mu_{s}=-\sin\theta_{s}d\theta_{s}\).  
\[-2\pi\int\limits_{1}^{-1}\overset{\sim}{\beta}(\mu_{s})\ d\mu_{s}=2\pi\int \limits_{-1}^{1}\overset{\sim}{\beta}(\mu_{s})\ d\mu_{s} \tag{21}\]  
This leads to  
\[P(\mu_{s})=\int_{\mu_{s}}^{1}\ p(\mu)d\mu\text{=2}\pi\int_{\mu_{s}}^{1} \overset{\sim}{\beta}(\mu)d\mu=q \tag{22}\]The average overall scattering directions of the cosine of the scattering angle \(\theta\) can be expressed as follows:  
\[g=2\pi\int_{0}^{\pi}\widehat{\beta}(\theta,\lambda)\ sin\theta\ cos\theta\ d\theta\ \ \text{average cosine/asymmetry parameter of the SPF} \tag{23}\]  
In Equation (23), \(g\) is a convenient measure of the \"shape\" of the SPF. If \(\widehat{\beta}(\theta,\lambda)\) is very large for small \(\theta\), then \(g\) is near one. If \(\widehat{\beta}(\theta,\lambda)\) is symmetric about \(\theta=90^{\circ}\), then \(g=0\). This average cosine \(g\) is often termed as the asymmetry parameter of the SPF. For the isotropic scattering scenario, \(\widehat{\beta}(\theta,\lambda)\) is constant, as shown in Figure 5.