## 4 Multi-Hop Data Forwarding Strategy Based on Annulus-Sectors
### Energy Consumption of Nodes in an Annulus-Sector  
#### 4.2.1 Energy Consumption of the Non-Cluster Heads  
For the \(j\)-th annulus-sector of the \(i\)-th annulus, it is easy to know that the distance from the center of its RCCH to the base station (BS) is \((2i-1)R/2k\), as shown in Figure 4. \(x\) and \(d_{s}\) are defined as the Euclidean distances from any one node in this annulus-sector to BS or to the center of the RCCH, respectively. It is assumed that during a round of data collection, the amount of data collected by one  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** & **Unit** \\ \hline \(r_{i}\) & Radius of the RCCH in the \(i\)-th Annulus & m \\ \(W\) & Weight of the CCH & - \\ \(l\) & The Amount of Data Collected by one Node during a Round of Data Gathering Time & bi \\ \(N_{i}\) & Average Number of Nodes in Each Annulus-sector of the \(i\)-th Annulus & - \\ \(E_{C\alpha i}\) & Energy Consumption of One CH in the \(i\)-th Annulus during a Round of Data Collection & - \\ \(E_{\alpha\alpha i}\) & Energy Consumption of the Sending and Receiving Circuit & nJ \(>\) b\({}^{-1}\) \\ \(\varepsilon_{\alpha\alpha}\) & Energy Consumption of the Amplifier in Free-Space Model & pJ \(\times\) (b/m\({}^{2}\))\({}^{-1}\) \\ \(\varepsilon_{\alpha\alpha\beta}\) & Energy Consumption of the Amplifier in Multi-Path Fading Transmission & pJ \(\times\) (b/m\({}^{4}\))\({}^{-1}\) \\ \(d_{0}\) & The Threshold Distance between the Sender and the Receiver & m \\ \(E_{\beta}\) & The Total Energy Consumption of the Non Cluster Headers in One Annulus-Sector & J \\ \(\varepsilon_{0}\) & The Initial Energy of One Node & J \\ \(\varepsilon_{r}\) & The Residual Energy of One Node & J \\ \(d_{\text{scattler}}\) & The Euclidean Distance between the CCH and the Center of the RCCH & m \\ \(\alpha\) & Adjustable Parameter & - \\ \(\beta\) & Adjustable Parameter & - \\ \(\delta\) & Adjustable Coefficient & - \\ \(x\) & The Euclidean Distance from any One Node in this Annulus-sector to BS & m \\ \(d_{s}\) & The Euclidean Distance from any One Node in this Annulus-sector to the Center of the RCCH & m \\ \(\theta\) & The Angle Deviated from the Center of the RCCH to any One Node & - \\ \(\varepsilon_{t}\) & Energy Consumption on Sending One Bit of Data & J \\ \(\varepsilon_{r}\) & Energy Consumption on Receiving One Bit of Data & J \\ \(C\) & Buffer Size of Node & bit \\ \(c_{i}\) & The CH in the \(i\)-th Annulus & - \\ \hline \hline \end{tabular}
\end{table}
Table 2: Definition of data collection parameters.  
node is \(l\) bit. Therefore, according to [37], the total energy consumption of the non-cluster heads in one annulus-sector (denoted as \(E_{ij}\)) during a round of data gathering time is  
\[E_{ij}=2l\rho\int_{(i-1)R/k}^{iR/k}\int_{0}^{\pi/n}\Big{(}E_{elec}+\varepsilon_{ fs}{d_{s}}^{2}\Big{)}xdxd\theta \tag{4}\]  
So, in Figure 4, according to the law of cosines, it is known that:  
\[{d_{s}}^{2}=x^{2}+(2i-1)^{2}(R/2k)^{2}-2x(2i-1)(R/2k)\cos\theta \tag{5}\]  
By combining Equations (4) and (5), we get:  
\[E_{ij}=4l\rho\varepsilon_{fs}\Big{(}\frac{R}{2k}\Big{)}^{4}\Big{(}\frac{2\pi}{ n}\Big{(}t^{4}-(i-1)^{4}\Big{)}+\frac{\pi}{n}(2i-1)^{3}-\frac{8}{3}\Big{(}t^{3}-(i-1)^{ 3}\Big{)}(2i-1)\sin\frac{\pi}{n}\Big{)}+4l(2i-1)\Big{(}\frac{R}{2k}\Big{)}^{2 }\rho\big{(}\frac{\pi}{n}\big{)}E_{elec}. \tag{6}\]