# An Energy Balanced and Lifetime Extended Routing Protocol for Underwater Sensor Networks
## 3 Models and Assumptions
### Channel and Energy Consumption Model  
The energy consumption is affected by the underwater channel. Here we present the description of path loss and ambient noise of underwater channel and derive the energy consumption model for UWSNs. [PERSON] gave an empirical formula in 1967 [27] to describe the underwater path loss and attenuation as in (1).  
\[A\left(d,f\right)=A_{0}d^{k}a(f)^{d} \tag{1}\]  
\(A\left(d,f\right)\) is the path loss for a given distance \(d\) and frequency \(f\). \(A_{0}\) is a unit-normalizing constant. \(k\) is the spreading factor and \(k=2\) is for spherical spreading in deep sea communications. \(a(f)\) is the absorption coefficient. The path loss can be expressed as TL in dB [28] by (2).  
\[TL=10\log A(d,f)/A_{0}=k\cdot 10\log d+d\cdot 10\log a(f) \tag{2}\]  
Then \(10\log a(f)\) is given by [PERSON] [29] in dB/km for \(f\) in kHz as in (3).  
\[10\log a(f)= 0.11\frac{f^{2}}{1+f^{2}}+44\frac{f^{2}}{4100+f^{2}}\] \[+2.75\times 10^{-4}f^{2}+0.003,f>200\mathrm{Hz} \tag{3}\]  
The underwater ambient noise is modeled by [PERSON] [6] as in (4).  
\[NL\left(f\right)=N_{t}\left(f\right)+N_{s}\left(f\right)+N_{w}\left(f\right)+ N_{th}\left(f\right) \tag{4}\]  
\(N_{t}(f)\), \(N_{s}(f)\), \(N_{w}(f)\), and \(N_{th}(f)\) represent the effect of turbulence, shipping, waves, and thermal noise at a given frequency \(f\), respectively. The total ambient noise \(NL(f)\) is the sum of the four parts. These four parts can be obtained from [6,28]. Then the signal-to-noise ratio (SNR) can be derived according to the passive sonar equation [27] as in (5).  
\[\gamma_{b}=SL-TL-NL+DI\geq DT \tag{5}\]  
\(\gamma_{b}\) is the SNR at the receiver and \(DI\) is the directive coefficient. For omnidirectional hydrophones, \(DI=0\). \(DT\) is the minimal SNR required for signal acquisition at the receiver. \(SL\) is the output acoustic source level in dB re \(\upmu\)Pa at the sender and the reference value of 1 \(\upmu\)Pa equals to \(0.67\times 10^{-18}\)Watts/m\({}^{2}\)[25]. Derived from (5), we can obtain the minimal \(SL\) at the sender as in (6).  
\[SL=DT+TL+NL-DI=10\log\frac{I_{T}}{0.67\times 10^{-18}} \tag{6}\]  
Here \(I_{T}\) is the transmitted signal intensity at 1 m in Watts/m\({}^{2}\) from the source. For spherical spreading, the transmit power \(P_{tx}\) at the sender can be derived from (7).  
\[P_{tx}=4\pi\times(1m)^{2}\times I_{T} \tag{7}\]  
From (6) and (7), the transmit power can be expressed as in (8).  
\[P_{tx}=4\pi\times 0.67\times 10^{-18}\times 10^{(DT+TL+NL-DI)/10} \tag{8}\]  
Suppose that there are \(N_{tx}\) nodes transmitting data, \(N_{rx}\) nodes receiving data, and \(N_{idle}\) nodes staying idle in the network within a short time \(t\). The total energy consumption of the network in \(t\) can be expressed as in (9).  
\[E_{t} =E_{tx}+E_{rx}+E_{idle}\] \[=t\left(\sum_{i=1}^{N_{tx}}P_{tx}(i)+\sum_{j=1}^{N_{tx}}P_{rx}+\sum_{k =1}^{N_{idle}}P_{idle}\right) \tag{9}\]  
Here \(E_{tx}\), \(E_{rx}\), and \(E_{idle}\) denote the energy consumption in transmitting, receiving, and idle state, respectively. \(P_{tx}(i)\) is the transmit power for node \(i\). \(P_{rx}\) and \(P_{idle}\) are the receiving and idle power in Watts for one node. We assume that all sensor nodes are homogeneous, so \(P_{rx}\) and \(P_{idle}\) are the same for each node. Our aim is to minimize the total energy consumption before some nodes in the network deplete their energy as in (10).  
\[\left\{\begin{array}{l}\min\left(\int_{0}^{T_{d}}E_{t}dt\right)\\ \exists i\in[1,N]\;\;s.t.\;E_{res}(i,T_{d})=0\end{array}\right. \tag{10}\]  
\(N\) is the total number of sensor nodes in the network. \(T_{d}\) is the time when a first node is depleted of its battery power. \(E_{res}(i,T_{d})\) is the residual energy of node \(i\) at time \(T_{d}\). Note that the sink node can usually collect energy from solar power, so the sink node is considered to have infinite energy and the energy consumption at sink node is not considered in protocol design.