## 2 WSNs Data Acquisition Model
### Our HRM_CS Model  
Inspired by the above theory and method, we put forward our HRM_CS model, which jointly applies the hierarchical routing method and compressive sensing to effectively compress and recover the original signal by exploiting the correlation of sensor readings. Next we describe the HRM_CS model in the formula form. In the HRM_CS model, based on the description of Equation (2), we design a hierarchical routing method to get the value of \(\,Y\,\)\(\in\,R^{M}\,\). According to LEACH in [4], we divide the \(\,N\,\) nodes into \(\,M\,\) clusters and randomly rotate cluster-heads, where the \(\,M\,\) cluster-head nodes, denoted by \(\,\{Y_{1},Y_{2},Y_{3},...,Y_{M}\,\}\,\). However, in order to further reduce energy dissipation and enhance system lifetime, differing from _LEACH_, we only select a part of nodes to transmit data to the cluster head. Assume that \(\,\{C_{1},C_{2},C_{3},...,C_{M}\,\}\,\) corresponds to the number of nodes in each of the \(\,M\,\) clusters and \(\,\sum_{e=C_{i}}^{C_{M}}c=N\,\). Note that the original signal is \(\,X\,\)\(\in\,R^{N}\,\) and the compressive signal from the \(\,M\,\) clusters is \(\,Y\,\)\(\in\,R^{M}\,\).  
In order to balance the energy cost among all the nodes, when we create the measurement matrix \(\,\Phi\,\), in which each row corresponds to the whole \(\,N\,\)nodes in the WSN. We introduce random coefficients \(\,W_{k,c}\,\)set to 0 or 1, where \(\,k\,\) denotes the \(\,k\)th \(\,\)cluster \(\,(Y_{k}\,)\) and \(\,c\,\) denotes the \(\,c\)th \(\,\)node \(\,(x_{k,c}\,)\) in this cluster. Each sensor contributes its readings to the sink with a non zero coefficient or zero otherwise. And now \(\mathbf{X}\) can be expressed as \(\mathbf{X}=\left\{x_{{}_{1,1}},x_{{}_{1,2}},...,x_{{}_{1,C_{1}}},...,x_{{}_{M,1}},x_{{ }_{M,2}},...,x_{{}_{M,C_{M}}}\right\},\) where \(x_{{}_{k,c}}\) is the sensor value and \(1\leq k\leq M,C_{1}\leq c\leq C_{{}_{M}}\). In this way, a randomly chosen subset of nodes participates in the sensing process so that the lifetime of the whole network is prolonged as long as the energy left in these nodes is higher than the threshold. At first, the sink node picks a random subset of \(M\) sensors for sampling and broadcasts the selected set of nodes at each round. Then, the selected nodes sample the physical value \(x_{{}_{k,c}}\) from the sensor. Furthermore, HRM_CS model of jointly applying the hierarchical routing method and compressive sensing process can be formally expressed as Equation (5):  
\[\begin{cases}Y_{x}=\sum\limits_{c=c_{i}}^{c_{i}}w_{{}_{k,c}}x_{{}_{k,c}}+z_{{} _{k,c}}\\ Y=\sum\limits_{k=1}^{N}Y_{k}+Z_{k}\\ w_{{}_{k,c}}\sim\textit{Gaussian}\ N(0,\!1)\\ z_{{}_{k,c}},Z_{{}_{k}}\sim\textit{Gaussian}\ N(0,\beta^{-1})\end{cases} \tag{5}\]  
where \(z_{{}_{k,c}}\) and \(Z_{{}_{k}}\) denote white Gaussian noise from sensor node \(x_{{}_{k,c}}\) and cluster \(Y_{{}_{k}}\), respectively, and \(\beta^{-1}\) denotes the covariance of the noise. Here \(Y_{k}\) and \(Y\) can act as the superposition of the signal on both the cluster-head node and the sink node. It is clear that the coefficients of \(\mathbf{\Phi}\) are not fixed at all rounds and the measurement matrix would be changed at each round. Because we compress the data while we transmit the data, we reduce the number of transmissions to the sink, with a corresponding reduction in the energy consumed by the WSN. We provide insight into the data collection based on the hierarchical routing method, which differs from traditional schemes discussed in [3]. Moreover, using the CS technique ensures that the HRM_CS is able to support the online recovery of the original signal \(\mathbf{X}\in R^{{}_{N}}\) from its compressed signal \(\mathbf{Y}\in R^{{}^{M}}\) with higher accuracy at the sink. It is worth noting that our HRM_CS is not simply about combining the hierarchical routing method and compressive sensing because it is not an efficient and stable way in a real WSNs environment. Next we will analyze the HRM_CS further and give the optimized variant model of HRM_CS.