# Path Diversity Improved Opportunistic Routing for Underwater Sensor Networks
## 3 Path Diversity Improved Opportunistic Routing  
### System Model  
We model a typical 3D underwater sensor network composed of some autonomous underwater sensor nodes and sink nodes, where sensor nodes transmit sensed information to the sink nodes on the surface through the wireless acoustic communication channel. Assume that the nodes are enumerated as \(\mathbb{N}=\{1,2,...,N\}\), the depth of node \(i\) is denoted as \(d_{i}\), and the Euclidean distance between node \(i\) and node \(j\) is represented by \(d_{ij}^{e}\). The neighbor set of node \(i\) is denoted as \(V_{i}\) and the candidate set of node \(i\) is denoted as \(\mathcal{C}_{i}\). The node communication range is \(R\). \(\tau_{ij}=d_{ij}^{e}/c\) is propagation delay between node \(i\) and node \(j\), while \(c\) is the acoustic speed in water.  
We consider a lossy channel in which path loss and bit error depend on the traversed distance and signal frequency [24]. The path loss over a distance \(r\) for a signal of frequency \(f\) due to large scale fading is given as:  
\[A(r,f)=n\cdot 10\text{log}r+\alpha(f)r\times 10^{-3} \tag{1}\]  
The first term in the above summation represents the spreading loss, and the second term represents the absorption loss. The spreading factor \(n\) describes the geometry of propagation, and its commonly used values are \(n=2\) for spherical spreading, \(n=1\) for cylindrical spreading, and \(n=1.5\) for the so-called practical spreading. The absorption coefficient \(\alpha(f)\) can be expressed empirically, using the Thorp's formula, which gives \(\alpha(f)\) in dB/km for \(f\) in kHz.  
\[\alpha(f)=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^{2}}+3.0\times 10^{-4} f^{2}+3.3\times 10^{-3} \tag{2}\]  
The average signal-to-noise ratio (SNR) over distance \(d_{si}^{e}\) is given by:  
\[\gamma(s,i)=\frac{PA(d_{si}^{e},f)}{N(f)\Delta f} \tag{3}\]  
where \(P\) is the transmitter power, \(N(f)\) is the ambient noise and \(\Delta f\) represents the receiver noise bandwidth.  
The periodic beacon mechanism is used to pick up one-hop neighbor information, including depth, address and the pairwise distance. All nodes can get their depths using a pressure sensor [13]. The periodic beacon mechanism adopts a simple time-based schedule. Nodes broadcast the beacon message to its neighbors during the specific slot in turn. The slot length is related to the maximum propagation delay of node communication range. When neighbors receive the beacon message, they can compute the distance information by ToA (Time of Arrival) technique [25], and pick up depth and address information embedded in the beacon. The periodic beacon mechanism is scheduled with a period \(T\) to update the neighbor information. Two-hop neighbor information is exchanged following the beacon process with the same time-based schedule as the beacon period. Each node broadcasts its neighbor information in turn, and the others collect the two-hop neighbor information. Besides, the time-based schedule requires time synchronization for transmission scheduling; in recent years, several works [26; 27; 28] have provided time synchronization mechanisms for UWSNs. Hence, the time synchronization issue is out of the scope of this paper.