# Double-Scale Adaptive Transmission in Time-Varying Channel for Underwater Acoustic Sensor Networks
## 3 Double-Scale Adaptive Transmission Mechanism for UASNs
### Underwater Acoustic Channel Model  
Throughout this article, a shallow-water acoustic propagation environment is considered. According to the principle of underwater acoustic, the signal-to-noise ratio of an underwater acoustic signal at the receiver can be calculated using the passive sonar equation as follows [45]:  
\[SNR\ =\ SL\ -PL\ -\ NL+DI\geq DT,\ (in\ \text{dB}\ \text{re}\ \text{\mu Pa}), \tag{2}\]  
where \(SL\) is the source level, \(PL\) is the transmission loss, \(NL\) is the noise level, \(DI\) is the directive index, and \(DT\) is the detection threshold of the sonar for specific modulation and coding mode. All these quantities are in dB re \(\text{\mu Pa}\), where the reference value of \(1\ \text{\mu Pa}\) equals to \(0.67\times 10^{-18}\ \text{Watts/m}^{2}\)[45]. For the convenience of expression, the notation dB is used to signify dB re \(\text{\mu Pa}\) in the rest of this paper. The path loss for a signal over distance \(d\) is given by Reference [46] as  
Figure 2: Multi-hop clustered underwater acoustic sensor network.  
\[PL=f_{s}\cdot 10\text{lg}d+\alpha\cdot d\times 10^{-3}, \tag{3}\]  
where \(f_{s}\) is the spreading factor, which can be taken as 1.5 for a practical scenario, and \(\alpha\) is the absorption coefficient, which can be modeled by [PERSON]'s formula [45] as follows:  
\[\alpha=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^{2}}+\frac{2.75f^{2}}{1 0^{4}}+0.003, \tag{4}\]  
where \(\alpha\) is given in dB/km, and \(f\) is in kHz. The power spectrum density of the ambient noise in the ocean can be calculated by  
\[NL=N_{turb}(f)+N_{ship}(f)+N_{wave}(f)+N_{ther}(f), \tag{5}\]  
where \(N_{turb}(f)\), \(N_{ship}(f)\), \(N_{wave}(f)\), and \(N_{ther}(f)\) in dB per Hz represent the turbulence noise, the shipping noise, the waves noise, and the thermal noise, respectively. These noises can be modeled by the following empirical formulas [46]:  
\[10 log(N_{turb}(f))=17-30 logf, \tag{6}\]  
\[10 log\Big{(}N_{ship}(f)\Big{)}=40+20(s-0.5)+26 logf-60 log(f+0.03), \tag{7}\]  
\[10 log(N_{wave}(f))=50+7.5\sqrt{w}+20 logf-40 log(f+0.4), \tag{8}\]  
\[10 log(N_{ther}(f))=-15+20 logf, \tag{9}\]  
where \(s\) and \(w\) denote the shipping activity factor and wind speed, respectively. The channel state \(h_{t}\) is defined as the received signal-to-noise ratio corresponding to a source level \(SL_{0}\) with a transmission power of one unit. When the source level is \(SL_{1}\) and the signal-to-noise ratio at the receiver is \(SNR_{1}\), channel state is recorded as  
\[h_{t}=SNR_{1}-SL_{1}+SL_{0}. \tag{10}\]  
According to the above formula, channel state can vary with the dynamic environmental factors. The underwater channels could be affected by various environmental factors, such as water temperature, wind speed, tidal, ocean swell, and so on, and these natural phenomena occur at different time scales, such as seasonal, diurnal, minutes, and seconds. These environmental factors result in complex fluctuation of the underwater acoustic channel on various time scales. Specially, the well-known \"afternoon effect\" caused by the diurnal and seasonal change in water surface temperature can result in the diurnal and seasonal change in the signal and noise strength [9]. As a result, the fluctuation characteristics of channel state can be analyzed for channel state prediction, and an optimized configuration scheme can be derived to improve the energy efficiency of data transmission.