## 2 WSN Propagation Models  
The wireless channel refers to the transmitter antenna, the receiver antenna, and the propagation path between them. Radio channels use electromagnetic waves to transmit signals, and the main forms of radio wave propagation are space waves, including direct wave, refracted wave, scattered wave, and their synthesis wave [20]. A propagation model is a set of mathematical expressions or diagrams which can describe the radio characteristics and estimate the signal strength drop. It can be estimated either by theoretical modeling or based on direct experimental measurements. The free-space model [21] is the most widely used model. The model is based on the Friis transmission formula, which indicates that, in an ideal environment, the power is spread uniformly over the surface of a sphere surrounding the antenna. The free space path loss (FSPL) can be defined as  
\[PL_{F}\ =\ \frac{P_{t}}{P_{r}}G_{t}G_{r}\ =\ (\frac{4\pi d}{\lambda})^{2} \tag{1}\]  
where \(P_{t}\) and \(P_{r}\) are the transmitted and received power, respectively; \(d\) is the distance between the transmitter and the receiver; \(\lambda\) is the wavelength; and \(G_{t}\) and \(G_{r}\) are the transmitter and receiver antenna gains, respectively. The free-space model is a theoretical model, where the path loss \(PL_{F}\) is only determined by the distance and frequency. The fundamental assumption behind the free-space model is that there are no obstacles between or beside the transmitter and receiver. Therefore, the free-space model is considered an extremely optimistic model. The two-ray (plane earth) model considers both the direct path and the ground reflected waves [22]. Assuming the antenna heights are small compared with the total path length, the plane earth loss can be expressed in decibels as  
\[PL_{PE}\ =\ 40\log_{10}\left(d\right)-20\log_{10}\left(H_{r}\right)-20\log_{10 }\left(H_{t}\right) \tag{2}\]  
where \(H_{t}\) and \(H_{r}\) are defined as the height of the transmitting and receiving antenna, respectively. In comparison to the free-space model, the two-ray model is more practical because it takes the reflection path and the ground characteristics into account. However, it still makes a few assumptions and simplifications that are somewhat unrealistic. For example, the model only holds for long distances and for cases where the amplitude and phase of the reflected wave is very close to \(-1\). Factors that affect the radio propagation are very complex. Environmental clutter, obstacles on the propagation path, and all reflection and scattering from other objects can lead to random variations in path loss. As a result, path loss at different locations with the same distance to the transmitter can differ considerably. It is difficult to use a generalized formula or a single model to accurately describe the path loss in various environments. Accordingly, using statistical models becomes an appropriate alternative to express these non-deterministic characteristics of path loss. The log-normal shadowing model is the most commonly used model to characterize these non-deterministic effects. The one-slope log-normal model can be expressed as  
\[PL(d)\ =\ PL(d_{0})+10n\log_{10}\left(\frac{d}{d_{0}}\right)+X_{\sigma} \tag{3}\]  
where \(PL(d)\) is the average path loss at the distance \(d\) (expressed in dB) and \(PL(d_{0})\) is the path loss at a reference distance \(d_{0}\) in the vicinity of the transmitter, which can be determined by measurements at \(d_{0}\) or the free-space path loss at \(d_{0}\). In this study, \(d_{0}\) was set to \(1\) m in consideration of the near-groundWSN scenario. \(n\) denotes the path loss exponent. \(X_{\sigma}\) is a zero mean log-normally distributed random variable with \(\sigma\) standard deviation in dB, expressing the uncertainty caused by shadow fading. In contrast to the one-slope model, the two-slope log-normal model divides the distance into near field and far field by a breakpoint. The two-slope model can be expressed as  
\[P\!L(d)\,=\,\,\,\left\{