## 4 Channel Models
### Relationship between the SNR and Successful Delivery Ratio  
After computing the SNR (by using the models of the relationship between the transmitter power and SNR), we further investigate the models of the relationship between the SNR and successful delivery ratio. The goal of the second part (of our channel models) is to calculate the successful delivery ratio, that is, the reliability of links over IoUT.  
To achieve the above goal, we need to choose the suitable models for underwater environments. First, for modulation scheme, we choose the BPSK modulation because BPSK is widely used in underwater acoustic sensor networks [49]. Second, for signal propagation, we choose the Rayleigh fading channel because Rayleigh fading is an appropriate model for the multipath effect in both the shallow and deep water [47]. According to the models we choose, the bit error rate (BER) (the bit error rate (BER) means the number of bit errors per unit time) of BPSK in a Rayleigh fading channel can be derived [50], as follows  
\[BER(\gamma)=\frac{1}{2}\left(1-\sqrt{\frac{10\gamma^{/10}}{1+10\gamma^{/10}}} \right), \tag{14}\]  
where \(BER(\gamma)\) is the estimated number of bit errors (the number of bit errors means the number of received bits (of a data stream over a communication channel) that have been altered due to noise, interference, distortion, or bit synchronization errors) per unit time over a communication channel when the SNR is equal to \(\gamma\).  
Based on the error rate of a single bit given by Equation (14), we can then compute the successful delivery ratio of a single bit, as follows  
\[\begin{array}{ll}P^{1}_{successful}(\gamma)&=1-BER(\gamma)\\ &=1-\frac{1}{2}\Bigl{(}1-\sqrt{\frac{10\gamma^{/10}}{1+10\gamma^{/10}}}\Bigr{)} \\ &=\frac{1}{2}+\frac{1}{2}\sqrt{\frac{10\gamma^{/10}}{1+10\gamma^{/10}}},\end{array} \tag{15}\]  
where \(P^{1}_{successful}(\gamma)\) is defined as the probability that a bit is successfully delivered when the SNR is equal to \(\gamma\). Following the definition, let \(P^{m}_{successful}(\gamma)\) be the probability that a packet (a packet is a formatted unit of data supported by computer communications links) is successfully delivered when the SNR is equal to \(\gamma\) and the size of the packet is equal to \(m\) bits. Finally, given the SNR \(\gamma\), the successful delivery ratio of a packet with size \(m\) bits can be calculated by the following equation:  
\[\begin{array}{ll}P^{m}_{successful}(\gamma)&=\left[1-BER(\gamma)\right]^{m} \\ &=\left[\frac{1}{2}+\frac{1}{2}\sqrt{\frac{10\gamma^{/10}}{1+10\gamma^{/10}}} \right)^{m}.\end{array} \tag{16}\]  
Based on all of the channel models proposed in this section, given the transmitter power \(P\), transmission range \(r\), transmission distance \(d\), and frequency \(f\), we can first compute the SNR \(\gamma\) for IoUT by using the first part of the channel models, i.e., Equation (13). Given the SNR \(\gamma\), we can then calculate the bit error rate \(BER(\gamma)\) for IoUT by using the second part of the channel models, i.e., Equation (14). Finally, we compute the successful delivery ratio of any pair of sensor nodes using Equation (16). The delivery ratio represents the probability that a data packet can be successfully delivered from a sender to a receiver, and stands for the reliability of an underwater wireless link. In summary, given the transmitter power, the channel models can provide researchers with a quick and accurate way of estimating the SNR, BER, and successful delivery ratio, that is, the reliability for UWSNs/IoUT.