# Data Aggregation Based on Overlapping Rate of Sensing Area in Wireless Sensor Networks
## 3 Network Model and Aggregation Rules
### Network Model
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & \multicolumn{1}{c}{**Description**} \\ \hline \(R_{S}\) & Sensing radius of a sensor node. \\ \(R_{C}\) & Communication radius of a sensor node. \\ \(E_{T}\) & Consumed energy of sending per unit data. \\ \(E_{R}\) & Consumed energy of receiving per unit data. \\ \(d\) & The size of data collected by a sensor node. \\ \(OR_{i,j}\) & Overlapping rate of sensing area of two nodes \(v_{i}\) and \(v_{j}\). \\ \(OR_{T}\) & Threshold of overlapping rate of sensing area. \\ \(GA_{i}\) & Gathering area with \(v_{i}\) as the gathering node. \\ \(GN\) & Gathering node, which aggregates the data collected in a gathering area. \\ \(CG_{i}\) & Candidate gathering node set of \(v_{i}\). \\ \(CP_{i}\) & Candidate parent node set of \(v_{i}\), including all the upper-level nodes that could \\ \(CP_{i}\) & communicate with \(v_{i}\) directly. \\ \(N_{i}\) & Neighbor node set of \(v_{i}\), which consists of the nodes at the same level that could \\  & communicate with \(v_{i}\) directly. \\ \(L_{i}\) & Level of \(v_{i}\), and \(L_{i}\in N^{+}\). \\ \(d_{i}^{T}\) & The total amount of data transferred from \(v_{i}\). \\ \(d_{i}^{R}\) & The total amount of data received by \(v_{i}\). \\ \(E_{GN_{i}}\) & The energy consumed for delivering aggregated data from the gathering node \(v_{i}\). \\ \(E_{GA_{i}}\) & The total energy cost of the nodes in the gathering area \(GA_{i}\). \\ \(E_{i}^{L}\) & The total energy cost of transmitting data of node \(v_{k}\) to the sink via another node \(v_{i}\). \\ \(FN_{n}\) & Free nodes at the level \(n\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols. Figure 1: Transmission hierarchy diagram. Definition 2: Overlapping rate of sensing area, \(OR_{i,j}\): The ratio of the overlapping sensing area of two nodes \(v_{i}\) and \(v_{j}\) to a node's entire sensing range (\(\pi{R_{S}}^{2}\)). Furthermore, considering that the two nodes with a larger overlapped sensing area probably have a larger similarity of the collected data, we assume that the amount of duplicated data at two sensors is proportional to the overlapping rate of sensing area. Our aggregation only removes redundant data, and thus the entropy of the sensing data in the whole network is not lost. In other words, the aggregation works like a lossless compression approach whose compression ratio is the overlapping rate of sensing area. In order to construct gathering areas, a threshold of overlapping rate of sensing area, denoted by \(OR_{T}\), is utilized. The assignment of \(OR_{T}\) affects the aggregation efficiency. The smaller \(OR_{T}\) is, the larger the gathering area is, but the smaller the amount of duplicated data between nodes is. Further analysis of \(OR_{T}\) is in Section 5.3.2. As shown in Figure 2, take the overlapping sensing area of \(v_{2}\) and \(v_{5}\) in Figure 1 as an example. Two dashed circles represent the sensing ranges of two nodes, respectively, and the hatched area is the overlapping sensing area, denoted by \(S_{C}\). Therefore the overlapping rate of sensing area of \(v_{2}\) and \(v_{5}\) is computed by \(OR_{2,5}=\frac{S_{C}}{\pi{R_{S}}^{2}}\). In order to calculate \(S_{C}\), we denote the distance between \(v_{i}\) and \(v_{j}\) by \(ds_{i,j}\). According to geometric theory, the overlapping sensing area is \(S_{C}=2{R_{S}}^{2}\arccos\frac{ds_{i,j}}{2{R_{S}}}-\frac{ds_{i,j}}{2}\sqrt{4{R_ {S}}^{2}-{ds_{i,j}}^{2}}\). To remove the complicated calculation of inverse trigonometric functions, using curve fitting mechanism [25], \(S_{C}\) could be computed as \(S_{C}=0.34{ds_{i,j}}^{2}-2.26{R_{S}}{ds_{i,j}}+\pi{R_{S}}^{2}\). Therefore, \(OR_{i,j}=\frac{0.34{ds_{i,j}}^{2}}{\pi{R_{S}}^{2}}-\frac{2.26{ds_{i,j}}}{\pi{R _{S}}}+1\). Definition 3: Gathering area, \(GA\): A gathering area is composed of a gathering node and several member nodes, and the overlapping rate of sensing area between the gathering node and each member node equals or is larger than \(OR_{T}\).