# Cross-Layer-Aided Opportunistic Routing for Sparse Underwater Wireless Sensor Networks
## 5 Protocol Analysis
### Effectiveness of Negotiation  
The effectiveness of negotiation is related to multi-hop interferences. However, the interference range \(R_{i}\) is large underwater. Additionally, the noticeable ambient noise increases the demand of high transmission power. Considering these factors, we derive the relational expression of effectiveness of negotiation \(E_{ngt}\) as below. Considering the network scenario where the nodes are sparsely deployed in shallow underwater and the transmission range \(R_{tx}\) is much larger than the depth of the sea region, we use the two-dimension network model to analyze \(E_{ngt}\). To facilitate analysis, we use \(d\) to represent the distance between sender \(S\) and forwarder \(R\). As shown in Figure 3, we classify all the situations into three categories with three relationship between \(R_{tx}\) and \(R_{i}\)[56; 57; 58; 59]. As shown in Figure 3, interference region in Scenario 1 contains three zones. In zone I, the interference nodes can overhear both RTF and CTF packets. In zone II, the interference nodes only can overhear RTF or CTF. In zone III, any control packets cannot be overheard correctly. We divide the interference region of Scenario 2 into three zones as well. In Scenario 3, the interference region contains two zones. Then, \(E_{ngt}\) can be expressed as:  
\[E_{ngt}=\frac{S_{I\cap(ngt)}}{S_{I}} \tag{26}\]  
where \(S_{I}\) represents the area of the region with interferers, \(S_{ngt}\) denotes the area of the region controlled by negotiation control packets, and \(S_{I\cap(ngt)}\) is the intersection of the area of these two regions. We introduce \(R_{i}=\gamma d\), where \(\gamma\) is spatial reuse factor. Moreover, we use \(R_{tx}=d/\lambda\) to represent \(R_{tx}\). Referring to the process of [59], we derive the expression of \(E_{ngt}\) as follows:  
\[E_{ngt}=\left\{ \begin{array}{l}\frac{2\pi-2\theta_{0}+\lambda\sin(\theta_{0} )}{\gamma_{1}\lambda^{2}\tau^{2}},\frac{1}{(\gamma-1)}\leq\lambda\leq 1\\ \frac{4\pi-4\theta_{0}+2\lambda\sin(\theta_{0})+2\theta_{1}\lambda^{2}\tau^{ 2}-\sin(2\theta_{1})\lambda^{2}\tau^{2}-2\theta_{2}+\sin(2\theta_{2})}{2\pi \lambda^{2}\tau^{2}},\frac{1}{\gamma}<\lambda<\frac{1}{(\gamma-1)}\\ 1,\ 0<\lambda\leq\frac{1}{\gamma}\end{array} \right. \tag{27}\]  
where angles \(\theta_{0}\), \(\theta_{1}\), and \(\theta_{2}\) are shown in Figure 4 and are represented as follows. \[\left\{ \begin{array}{l}\theta_{0}=\arccos(\frac{d}{2R_{tx}})\\ \theta_{1}=\arccos(\frac{d^{2}+R_{tx}^{2}-R_{tx}^{2}}{2 dR_{tx}})\\ \theta_{2}=\arccos(\frac{R_{tx}^{2}-d^{2}-R_{tx}^{2}}{2 dR_{tx}})\end{array} \right. \tag{28}\]  
The spatial reuse factor \(\gamma\) is related to the frequency and distance. We firstly use the formula of receiver detection threshold \(T\) given in [56] to derive the spatial reuse factor \(\gamma(f,d)\). \[T=\frac{P_{s}/A(f,d)}{P_{i}/A(f,d_{i})+\sigma_{N}(f)} \tag{29}\]  
The \(\gamma(f,d)\) can be calculated as below. \[\gamma^{-k}(\alpha(f)^{d})^{1-\gamma}=\frac{P_{s}}{P_{i}T}-\frac{A_{p}\sigma_ {N}(f)}{P_{i}d^{-k}\alpha(f)^{-d}} \tag{30}\]where \(P_{s}\) is the transmit power of the node \(S\), and \(P_{l}=P_{s}\) is the transmit power of node \(R\), since we assume that each node has the same transmit power. As a result of no closed solution for \(\gamma(f,d)\), we solve this equation numerically. On the basis of BER = \(10^{-4}\), the detection threshold with modulation of BPSK in the Gaussian channel is corresponding to 8.4 dB. Considering the scale factor \(\sigma=4\) dB that represents the receiver inefficiencies [37], thus, we set \(T=12.4\) dB. The central frequency is set to 25.6 kHz. We set \(k=\)1.75 to give a better asymptotic of the Bellhop model used in the simulation. In addition, we assume that the distribution of \(A_{p}\) is \(10\times\lg(A_{p})\sim N(10,\delta_{p}{}^{2})\). Figure 4(a) shows the relationship between the spatial reuse factor \(\gamma\) and normalized node separation \(d/R_{tx}\) for several SL. \(\gamma\) decreases as SL increases and reaches a certain value with high values of SL.