# Retransmission Avoidance for Reliable Data Delivery in Underwater WSNs
## 3 FLMPC: Proposed Protocols
### Propagation Model  
In this section, we present the propagation model of acoustic signals and their absorption in underwater environment. Energy Model  
The attenuation of acoustic channel over distance \(d\), for the signal of frequency \(f\) can be expressed as [20]:  
\[A(d,f)=d^{k}\alpha(f)^{d} \tag{1}\]  
Figure 2: Network Architecture of forward layered multipath power control (FLMPC)-One and FLMPC-Two. where \(k\) shows the spreading factor of the signal and \(\alpha(f)\) is the absorption coefficient. \(k\) defines the geometry of propagation (i.e., \(k=1\) is cylindrical spreading (signal does not propagate in all directions, but gets trapped within the boundaries and wave fronts move in cylindrical shape without any attenuation in the power of the signal), \(k=1.5\) is practical spreading (spreading loss in which transmission loss and attenuation are considered while performing experiments in order to achieve more accurate and precise results.) and \(k=2\) is spherical spreading (signal propagates in all directions, where wave fronts grow as it move away from the source and intensity of the signal remains the same)). [PERSON]'s propagation model is used to describe \(\alpha(f)\) given by [7]:  
\[\alpha(f)=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^{2}}+2.75\times 10^{ -4}f^{2}+0.003 \tag{2}\]  
where \(\alpha(f)\) is measured in dB/Km and \(f\) is given in KHz. Figure 3 shows the relation of \(\alpha(f)\) and frequency which is obtained from [PERSON]'s model. As the frequency increases, absorption of acoustic signals also increases. The total noise power spectral density (\(NP\)) of all noises at frequency \(f\) is given by [5]:  
\[NP(f)=\sum_{i=1}^{ns}NP_{i}(f) \tag{3}\]  
where \(ns\) shows the total number of noise sources like shipping, turbulence, etc. and \(NP\) is measured in dB. The more noise signals propagate away from the source, the more they attenuate. Their attenuation over distance \(d\) can be calculated from:  
\[NP^{d}(f)=\sum_{i=1}^{ns}\frac{NP_{i}}{d^{k}\alpha(f)^{d}} \tag{4}\]  
The signal to noise ratio (\(SNR\)) of the received signal can be written as:  
\[SNR=\frac{P/A(d,f)}{NP^{d}(f)}=\frac{P}{d^{k}\alpha(f)^{d}\sum_{i=1}^{ns} \frac{NP_{i}}{d^{k}\alpha(f)^{d}}} \tag{5}\]  
where \(P\) is the power of acoustic signal and \(d\) is the distance among \(n-1\) and \(n\)th nodes. \(SNR\) follows additive white gaussian noise channel (\(AWGN\)). As \(SNR\) depends on channel modulation scheme through which we can calculate \(BER\). The \(BER\) for \((\pi/4)\)\(QPSK\) modulation scheme is given by:  
\[p^{ber}(SNR)=\frac{1}{\sqrt{2\pi}}\int_{SNR}^{\infty}e^{-t^{2}/2}dt \tag{6}\]  
\(BER\) depends on channel coding scheme and it can be converted to (\(PER\)) which is given by:  
\[P^{per}=1-[1-p^{ber}(SNR)]^{Pl} \tag{7}\]  
where \(Pl\) is the data packet size. Energy consumption of a sensor node while sending a data packet can be calculated from the values of transmission power (\(Tp\)), \(Pl\), data rate (\(DR\)) of acoustic channel and \(d\) is the distance between \(n-1\) and \(n\)th node. The mathematical expression is given as follows [5]:  
\[E_{tx}=\frac{Tp\times Pl}{DR}\times d \tag{8}\]  
While, reception energy (\(E_{rx}\)) is calculated according to Equation (9). \[E_{rx}=\frac{P_{r}\times Pl}{DR}\times d \tag{9}\]where \(P_{r}\) is the reception power. Total leftover energy of the network nodes after forwarding all the data packets is computed as follows:  
Figure 2:0  
where \(N\) shows the total number of data packets, \(m\) is the total hop count for one data packet from the source node to the destination, \(n_{i}\) is the \(i\)th node of the network. When the hop count increases by one, it updates \(n_{i}\). We can find the total remaining energy of the network for receiving data packets from:  
Figure 2:1  
Total consumed energy \(E_{total}\) of the network is computed by Equation (12). Figure 2:2  
where \(E_{i}\) is the total initial energy of the network.