# Energy Harvesting Hybrid Acoustic-Optical Underwater Wireless Sensor Networks Localization
## 2 System Model and Proposed Technique
### Acoustic Underwater Ranging  
Underwater acoustic channels suffer from two kinds of major losses: attenuation loss and spreading loss [24]. Attenuation loss is a result of scattering, diffraction, absorption, and leakage from ducts while spreading loss is a combination of cylindrical and spherical losses [25]. In general, the sea water transmission loss between any two nodes \(i\) and \(j\) is modeled as  
\[\eta_{t}=\eta_{s}+\eta_{c}+10^{-3}\alpha d_{ij}, \tag{1}\]  
where \(\eta_{s}\) is spherical spreading loss, \(\eta_{c}\) is the cylindrical spreading loss, \(\alpha\) is the absorption coefficient, and \(d_{ij}\) is the Euclidean distance between nodes \(i\) and \(j\). Using the Thorp absorption model [26], \(\alpha\) only depends on the frequency \(f\), i.e.,  
\[\alpha=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^{2}}. \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Variable** & **Symbol** & **Variable** \\ \hline \(m\) & Number of anchor nodes & \(\sigma_{ij}\) & Noise variance \\ \(n\) & Number of sensor nodes & \(\theta_{0}\) & Divergence angle \\ \(\eta_{s}\) & Spherical spreading loss & \(T\) & Time duration \\ \(\eta_{c}\) & Cylindrical spreading loss & \(I\) & Actual two-dimensional location of a node \\ \(\alpha\) & Absorption coefficient & \(\mathbf{\Lambda}\) & Matrix of Estimated distances \\ \(d_{ij}\) & Euclidean distance & \(\eta_{ij}\) & Weighting coefficients \\ \(\tilde{d}_{ij}\) & Estimated distance & \(\Phi\) & Importance of an observation \\ \(\lambda\) & Wavelength & \(\gamma\) & Controlling parameter \\ \(e(\lambda)\) & Extinction coefficient & \(\mathbf{L}\) & Actual locations of all the nodes \\ \(r_{j}\) & Number of photons & \(\mathbf{L}\) & Estimated locations of all the nodes \\ \(\kappa_{ij}\) & Propagation loss & \(\kappa\) & Scaling factor \\ \(P_{r_{i}}\) & Received power at node \(j\) & \(\Omega\) & Rotation factor \\ \(P_{t_{i}}\) & Transmitted power by node \(i\) & \(\mathbf{\Psi}\) & Translation factor \\ \(\eta_{i_{i}}\) & \(\eta_{j}\) & Optical efficiencies & \(E_{t}\) & Energy consumption \\ \(\theta\) & Trajectory angle & \(\mathbf{\Gamma_{d}}\) & Noise co-variance matrix \\ \(r\) & Transmission range & \(\delta^{2}\) & Mean square error \\ \(\tilde{d}_{a_{ij}}\) & Estimated acoustic distance & \(\tilde{d}_{a_{ij}}\) & Estimated optical distance \\ \hline \hline \end{tabular}
\end{table}
Table 2: List of symbols.  
Spherical spreading loss fits the measured data appropriately. Therefore, ignoring the cylindrical loss and substituting \(\alpha\) in (1) for spherical losses \(\eta_{s}=20\log(d_{ij})\), we get  
\[\eta_{t}=20\log(d_{ij})+10^{-3}\left(\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4 100+f^{2}}\right)d_{ij}. \tag{3}\]  
Figure 1 shows the transmission loss between any two nodes \(i\) and \(j\) with respect to distance. The acoustic ranging based distance \(\tilde{d}_{a_{ij}}\) is estimated from (3) using the real part of Lambert \(W_{0}\) function [27]  
\[\tilde{d}_{a_{ij}}=\frac{20000W_{0}(1.15\exp^{-4}\alpha\exp^{0.11\eta_{t}})}{2.3\alpha}. \tag{4}\]