## 2 Network Model and Optimal Cluster Number Calculation
### Energy Consumption Model  
This paper quotes the energy consumption model proposed in [12]. According to the actual transmission distance from the CHs to the BS, the free space model and the multipath fading channel model both need to be comprehensively analyzed, which is different from [12] considering only the multipath fading channel model. Therefore, the expression of the total energy consumption of the model will undergo some changes.  
\(E_{T}(eJ)\) indicates the energy consumed by the wireless transmitter to transmit a set of \(e\) bits of information. The expression is as follows:  
\[E_{T}(e,d)=\left\{\begin{array}{ll}e\times\left(E_{elce}+\varepsilon_{fs}d^{2 }\right),&d<d_{0}\\ e\times\left(E_{elce}+\varepsilon_{mp}d^{4}\right),&d\geq d_{0}\end{array}\right. \tag{1}\]  
\(E_{R}(e)\) indicates the energy required to receive the information of the \(e\) bit. The expression is as follows:  
\[E_{R}(e)=e\times E_{elce} \tag{2}\]  
In Equations (1) and (2), \(E_{elce}\) is the energy consumed per bit by the transmitter or receiving circuit and \(d\) is the distance between the transmitter and the receiver. In Equation (2), when \(d<d_{0}\), we use the free space model and \(\varepsilon_{fs}\) acts as the energy factor per bit. Otherwise, the multipath fading channel model is used, and \(\varepsilon_{mp}\) acts as the energy factor per bit. In addition, \(d_{0}\) is used as the distance threshold. As long as it is input as an independent variable into the free space model and the multipath fading channel model to establish an equation, the following expression can be obtained:  
\[d_{0}=\sqrt{\frac{\varepsilon_{fs}}{\varepsilon_{mp}}} \tag{3}\]  
In each round of data transmission, the cluster member nodes are responsible for sensing information from the environment, then transmitting it to the CH of the corresponding cluster. Therefore, the calculation formula for the energy consumed by transmitting \(e\) bit information is defined as follows:  
\[E_{non-CH}=e\cdot E_{elce}+e\cdot\varepsilon_{fs}d_{toCH}^{2} \tag{4}\]  
In Equation (4), \(d_{toCH}\) represents the distance from the cluster member node to the CH.  
The CH receives information from the cluster member nodes in the cluster, then fuses the information with that which it senses from the environment, eventually transmits the merged information to the BS. In this paper, we assume that in each round of data transmission, the information size obtained after processing by the CH is \(e\) bit. The energy consumed in the process is calculated as follows:  
\[E_{CH}=(\frac{n}{k}-1)\cdot e\cdot E_{dec}+\frac{n}{k}\cdot e\cdot E_{DA}+e \cdot E_{elce}+\left\{\begin{array}{ll}e\varepsilon_{fs}d_{toBS}^{2},&d_{toBS}<d_{0} \\ e\varepsilon_{mp}d_{toBS}^{4},&d_{toBS}\geq d_{0}\end{array}\right. \tag{5}\]  
In the above formula, the energy consumed consists of three parts: receiving energy consumption, processing energy consumption, and transmitting energy consumption. In Equation (5), \(n\) is the number of nodes surviving in the monitored area, \(k\) is the number of clusters to be divided, \(E_{DA}\) is the energy consumed by the CH to process each bit of data (including received data and sensed data), and \(d_{toBS}\) is the distance between the CH and the BS.