## 3 Propagation Model
### Model Requirements
#### 3.1.2 Light Transport Underwater  
The a priori known variables given to the model are the underwater coefficients, i.e., water albedo \(w_{o}\), attenuation coefficient \(c\), and volume scattering phase function \(\gamma\). The albedo is the ratio of scattering coefficient \(b\) in \(\mathbf{m}^{-1}\) to the attenuation coefficient \(c\), also known as the extinction coefficient.  
It defines the total loss of energy as a sum of absorption and scattering coefficient \(c(\lambda)=a(\lambda)+b(\lambda)\). Moreover, the attenuation coefficient is also wavelength dependent. Different attenuation coefficient values determine the amount of attenuation the optical signal will face underwater. Table 2 lists Petzold's measured water albedo and attenuation coefficients for the three water types. The simulation model is based on a Cartesian coordinate system with photons traveling along the \(z\)-axis. The initial location of the photon is determined by the \(x,y,z\) cartesian coordinates; whereas, the direction of the photon is described by the direction cosines \(\mu_{x},\mu_{y},\mu_{z}\). Instead of calculating the probability of absorption at every event (i.e., scattering or boundary interaction), we suppose that absorption would have occurred; hence, the weight of the photon is updated using the relation  
\[w_{n+1}=w_{n}*\left(1-\frac{a}{c}\right) \tag{6}\]  
where \(a\) is absorption coefficient in m\({}^{-1}\) and \(c\) is attenuation coefficient. To simulate scattering, we utilize scattering phase functions measured by [PERSON] [23], given as  
\[\gamma(\lambda,\psi)=\frac{\phi_{\mathrm{s}}(\psi,\lambda)}{\phi_{\mathrm{i}}( \lambda)l\Delta\Omega}. \tag{7}\]  
where \(\phi_{\mathrm{s}}(\psi,\lambda)\) is the power scattered into the solid angle \(\Omega\) after leaving the sample volume of length \(l\) at an angle \(\psi\) with respect to the incident power \(\phi_{\mathrm{i}}(\lambda)\). The total volume scattering function is given by  
\[b=2\pi\int_{0}^{\pi}\gamma(\lambda,\psi)\sin\theta d\theta. \tag{8}\]  
We implemented the anisotropic scattering (see Figure 2) model in which scattering angles are determined using [PERSON]'s phase functions [23]. They help to study temporal spreading in detail and give a more realistic channel response, as compared to a Beer-Lambert model which is a rather simple but unrealistic approach. Thus, when we closely study scattering, we come across different characteristics of water that indicate significant aspects of channel losses, as discussed in Section 4.  
Figure 2: Illustration of photon travel, where \(\theta\) is the scattering angle determined using [23], and \(\phi\) is the azimuth angle \([0,2\pi]\).  
Photon position and direction update whenever it hits the water surface (or any particle) and when it reaches the receiver and has not yet been marked as received or terminated. The successive position of the photon is determined using the pseudo-random path length \({}^{\prime}\). The path length is the distance a photon travels without being scattered or absorbed and is calculated at the beginning of the simulation. The variables \(x^{\prime},y^{\prime},z^{\prime}\) are the updated positions, \(x_{0},y_{0},z_{0}\) are the old positions (before scattering event) and \(\mu_{x},\mu_{y},\mu_{z}\) are initial directions.  
\[x^{\prime} = x_{0}+\mu_{x}l, \tag{9}\] \[y^{\prime} = y_{0}+\mu_{y}l,\] (10) \[z^{\prime} = z_{0}+\mu_{z}l. \tag{11}\]  
Given the scattering angles \(\theta\) and \(\phi\) and the initial directions \(\mu_{x},\mu_{y},\mu_{z}\), we also update the directions. If the photon is close to the \(z\)-axis, i.e., \(|\mu_{z}|>0.999\), then the direction cosines are updated according to [24]  
\[\mu_{x}^{\prime} = \sin\theta\cos\phi, \tag{12}\] \[\mu_{y}^{\prime} = \sin\theta\cos\phi,\] (13) \[\mu_{z}^{\prime} = sign(\mu_{z})\cos\theta. \tag{14}\]