# An Improved Forwarding of Diverse Events with Mobile Sinks in Underwater Wireless Sensor Networks
## 3 Underwater Channel Model  
Underwater wireless channel is severely affected by issues like multi path and fading. Signal to Noise Ratio (\(SNR\)) in UWSN is computed using passive sonar equation [42] given below:  
\[SNR=SL-TL-NL+DI \tag{1}\]  
where \(SL\) is source level of the target or noise generated by the target, \(TL\) is transmission loss in aquatic environment, \(NL\) is noise loss and \(DI\) is directive index. \(DI\) is the capability of receiving sensor to direct its antenna to avoid unwanted noise. SNR value must be greater than detection threshold (\(DT\)). All the above quantities are measured in \(dBre\mu Pa\). Transmission loss is given in following:  
\[TL=10\log(d)+\alpha\times d\times 10^{-3} \tag{2}\]  
Absorption loss \(\alpha(f)\) is calculated by the following equation.  
\[\alpha_{dB}(f)=\begin{cases}\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f}+ 2.75\times 10^{-4}f^{2}+0.003&\text{if}\quad f\geqslant 0.4\\ 0.002+0.11(\frac{f}{(1+f)})+0.011f&\text{if}\quad f<0.4\end{cases} \tag{3}\]  
Thorp attenuation formula [43; 44] is used to calculate the totals attenuation \(A(l,f)\) given in the following formula.  
\[A(l,f)=l\times\alpha(f)+k\times 10\log(l) \tag{4}\]where \(l\) is the depth difference \(k\) is spreading coefficient with value 1.5 for practical spreading. The first term in Equation (4) represents the absorption loss while the second term is about the scattering loss. Passive sonar equation is used to find out Source Level (\(SL\)) which is used to find intensity of transmitted signal \(I_{T}\) using the following formula.  
\[I_{T}=10^{5L/10}\times 0.67\times 10^{-18} \tag{5}\]  
Transmission power of source is \(P_{T}(d)\) calculated by the following equation.  
\[P_{T}(d)=2\pi\times 1m\times H\times I_{T} \tag{6}\]  
Energy consumed in transmitting the \(k\) bits of packet is given as:  
\[E_{Tx}(k,d)=2P_{T}(d)\times T_{Tx} \tag{7}\]  
where \(T_{Tx}\) is the transmission time in seconds. Speed of the acoustic signal is calculated by the following equation.  
\[v=1499.05+45.7t-5.21t^{2}+0.23t^{3}+(1.333-0.126t+0.009t^{2})(S-35)+16.3z+0.18 z^{2} \tag{8}\]  
Average end to end delay consists of two type of delays, calculated and explain in the following equations.  
* Transmission delay: Transmission delay to a packet of length \(l\) caused by a node \(n\) is given by following formula. \[D_{t}^{n}=\frac{l}{r}\] (9) where \(r\) is data rate with value 250,000 bps.
* Propagation delay: Propagation delay of an acoustic signal moving with velocity \(v\) between a sender \(i\) and receiver \(j\) separated by distance \(s^{i,j}\) is given by: \[D_{p}^{i,j}=\frac{s^{i,j}}{v}\] (10) The speed of acoustic signal \(v=1500\) ms\({}^{-1}\) which is calculated using Equation (8). Sum of both type of delay constitutes to one hop delay. \[D_{n}^{i,j}=D_{t}^{n}+D_{p}^{i,j}\] (11)  
The sum of all hops delay equals to the total delay in a packet transmission from a certain node \(n\) to sink. It is also termed as end to end delay.  
\[D_{e2e}^{n}=\sum_{h=1}^{h_{max}}D_{h} \tag{12}\]  
The mean of all end to end delays gives average end to end delay.  
\[\overline{D_{e2e}}=\frac{\sum_{n=1}^{n_{max}}D_{e2e}^{n}}{n_{max}} \tag{13}\]