# A Network Coding Based Hybrid ARQ Protocol for Underwater Acoustic Sensor Networks
## 2 Network and Channel Model
### Underwater Channel Model  
We present a brief introduction to the underwater channel and propose an empirical method to estimate the packet error rate between any two nodes according to the underwater channel. The acoustic transmission in underwater channel is affected by three main factors: path loss, ambient noise and fading [18]. For a given distance \(d\) and frequency \(f\), the path loss for sound propagation \(TL\left(d,f\right)\) is described by [PERSON] [19] and can be expressed as:  
\[TL\left(d,f\right)=A_{0}d^{k}a\left(f\right)^{d} \tag{2}\]  
where \(A_{0}\) is a normalizing parameter. \(k\) is the spread factor and \(k=1.5\) for practical spreading. \(a\left(f\right)\) denotes the sound wave absorption loss and is modeled by the [PERSON] formula [20] as:  
\[10\log a\left(f\right)=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^{2}}+ \frac{2.75f^{2}}{10^{4}}+0.003\ \ \left[dB/km\right] \tag{3}\]  
The ambient noise in underwater environment is affected by several factors. According to the famous [PERSON] noise model [21], there are four main noise sources in the ocean: turbulence \(N_{t}\), shipping \(N_{s}\), waves \(N_{w}\) and thermal noise \(N_{th}\). The total power spectral density of noise level \(NL(f)\) for a given frequency \(f\) is the accumulation of these factors as follows:  
\[NL\left(f\right)=N_{t}\left(f\right)+N_{s}\left(f\right)+N_{w}\left(f\right)+ N_{th}\left(f\right) \tag{4}\]  
where  
\[\begin{array}{l}10\log N_{t}\left(f\right)=17-30\log f,\\ 10\log N_{s}\left(f\right)=40+20\left(s-0.5\right)+26\log f-60\log\left(f+0.03 \right),\\ 10\log N_{w}\left(f\right)=50+7.5w^{1/2}+20\log f-40\log\left(f+0.4\right),\\ 10\log N_{th}\left(f\right)=-15+20\log f,\end{array} \tag{5}\]  
where \(s\) is the shipping activity factor that takes values between 0 and 1 for low and high activity, respectively. \(w\) is the wind velocity in m/s. Then, we can obtain the SNR (signal-to-noise ratio) per bit \(\gamma_{b}\) from passive sonar equation described in [19]:  
\[\gamma_{b}=SL-TL-NL+DI \tag{6}\]  
Figure 1: An illustration of network model for underwater acoustic sensor networks.  
where \(SL\) is the source level and \(DI\) denotes the directivity factor. For omnidirectional hydrophones, \(DI=0\). \(TL\) and \(NL\) come from Equations (2) and (4), respectively. If we use binary phase shift keying and suppose Rayleigh fading for the underwater multi-path channel as many researchers do [9; 18; 22; 23], the average bit error rate (BER) \(P_{b}\left(i,j\right)\) between node \(i\) and node \(j\) can be obtained [24] as follows:  
\[P_{b}\left(i,j\right)=\frac{1}{2}\left(1-\sqrt{\frac{\hat{\gamma}_{s}\left(i,j \right)}{1+\hat{\gamma}_{s}\left(i,j\right)}}\right) \tag{7}\]  
where  
\[\hat{\gamma}_{s}=10^{\hat{\gamma}_{s}/10} \tag{8}\]  
Thus, for a packet with \(L\) bits, the packet error rate \(P_{pac}\left(i,j\right)\) between node \(i\) and node \(j\) can be obtained as follows:  
\[P_{pac}\left(i,j\right)=1-\left(1-P_{b}\left(i,j\right)\right)^{L} \tag{9}\]  
Then, for a given source level \(SL\) and distance \(d_{i,j}\), we can obtain its packet loss rate from Equations (6), (7) and (9).