## 4 Energy Balanced Strategies for Sparsely Deployed UWA-SNs
### Overview of EBH  
We divide the node's initial energy into \(m\) units and each unit denotes as one gradation of energy. Initially, each node works in \(MODE0\) and the linear sensor network is a multi-hop network. The nodes closer to the sink will dissipate more energy. Let's again consider two adjacent nodes: \(S_{i}\), \(S_{i-1}\) in Figure 3. \(S_{i-1}\) first consumes one gradation of energy and drops into the next energy gradation. Then it sends a control message which includes its energy information to upstream node \(S_{i}\). When the upstream node \(S_{i}\) receives the control message, it compares its own grade number with the grade number in the control message. If the grade number is higher than the downstream node's, it changes working mode to \(MODE1\). Just as Figure 4 shows, \(S_{i}\) will transmit data directly to the sink, which mitigates its downstream node's loads. Otherwise, it relays the control message to upstream node until the message finds a node which has higher energy. In the same way, when \(S_{i}\) has consumed one gradation of energy, it does the same thing as \(S_{i-1}\) and changes back to \(MODE0\). In other words, it links to \(S_{i-1}\) again just as Figure 3 shows. By alternatively changing node's transmit mode between multi-hop and direct transmission based on residual energy, the nodes in the linear network dissipate energy evenly, so the lifetime of the network prolongs. Figure 4: Unlinked sensors. Figure 3: Linked sensor network. Optimum gradation number of linear network  
In EBH, when node's residual energy gradation changes, the node changes the working mode. However, how many energy grades are required to maximize the network lifetime? On one hand, if the number of grades \(m\) is too small, EBH tends to be a multi-hop transmission scheme which leads to uneven energy consumption. On the other hand, if the number of grades \(m\) is too large, the transmission mode of nodes will alternate frequently between \(MODE0\) and \(MODE1\), which incurs a large amount of control messages and consumes tremendous energy. Therefore, finding the appropriate energy gradations of node is very important. Next we calculate the appropriate grades \(m\). We assume the initial energy of each sensor is \(E>0\), and the nodes implement optimal sleep schedule protocol, in which the nodes are in sleep mode and only wake up during active transmission and reception [17]. The total energy consumption in each round includes transmitting, receiving and sensing. The total energy grades denote as \(m\), then one gradation of energy is \(E/m\). We notice that the farthest node from the sink consumes more energy when transporting data directly to the sink. Just as Figure 2 shows the node \(S_{n}\) is the farthest node from the sink. Let \(E_{d}\) be the energy dissipation that the farthest node spent on transmitting one packet directly to the sink and \(E_{TX}(k,nr)\) be the energy dissipation of sending a \(k\) bits packet over distance \(nr\). Then we have  
\[E_{d}=E_{TX}(k,nr)=P_{T}(nr)\times T_{TX} \tag{13}\]  
where \(E_{d}\) is in \(Joule\), \(P_{T}(nr)\) is the transmit power which can reach distance \(nr\), \(T_{TX}\) is the duration of transmit time in second. If \(E_{d}\) is less than one gradation of energy, then we have:  
\[E_{d}<E/m. \tag{14}\]  
We use Equation (14) as a condition to calculate \(m\). The reason is that if the farthest node from the sink consumes less than one unit energy, then the other nodes also consume less than one unit energy when transmitting one packet directly to the sink. Therefore, any two adjacent nodes at most differ only one unit of energy. Next, we give our proof in Lemma 1. We first give some definitions to deduce Lemma 1. As shown in Figure 3, the distance between \(S_{i-1}\) and the sink is \(D\), then \(D=(i-1)r\). Thus the distance between \(S_{i}\) and the sink is \(D+r\) and \(D+r=ir\). Let \(RG(s_{i})\) be the residual energy gradations of node \(S_{i}\).