## 3 Communication at Ground Level: Radio Channel Modeling
### Received Power  
When an electromagnetic signal propagates, its strength decreases not only because of the distance between the transmitter and the receiver, but also because of other physical phenomena, such as diffraction, reflection or scattering [30]. The signal attenuation, or path-loss, is related to the environment characteristics. Its value depends on the distance between the transmitter and the receiver. It also depends on the position, the shape and the dielectric properties of objects on the signal path. One of the most common radio propagation models, widely used in WSN, is the log-normal path-loss model [17; 30] which is given by Equation (1):  
\[PL(d)=PL(d_{0})+10\alpha log_{10}\left(\frac{d}{d_{0}}\right)+X_{\sigma} \tag{1}\]  
In Equation (1), \(PL(d)\) represents the path-lost at a distance \(d\) between the transmitter and the receiver. On the right side of Equation (1), the first two terms (\(PL(d_{0})+10\alpha log_{10}\left(\frac{d}{d_{0}}\right)\)) represent the signal path loss produced by the distance \(d\) between the transmitter and the receiver (ideal path-loss), while \(X_{\sigma}\) is a Gaussian random variable with mean 0 and variance \(\sigma\) (standard deviation due to multipath effect), which represents the path loss due to shadowing effects. The parameters \(\alpha\) (also known as the path loss exponent) and \(\sigma\) depend on the propagation environment and they are generally obtained from field tests in different scenarios. However, from our knowledge, suitable values have never been determined for these parameters in a ground-level deployment scenario. Instead, for propagation that approximately follows a free-space or two-ray model, \(\alpha\) can be set to 2 or 4, respectively [30]. Furthermore, in [30; 31], the authors proposed an approach to calculate the value of \(\alpha\) in a given environment, given measured data. This approach is used to estimate the values of \(\alpha\) and \(\sigma\) in Section 6.  
The parameter \(PL(d_{0})\) is the path loss at a reference distance (distance at which the communication area can be assumed to be a perfect disc) \(d_{0}\); its value can be obtained empirically or analytically using the free-space propagation model at distance \(d_{0}\). Using an analytical approach, \(PL(d_{0})=20 log_{10}(\frac{4\pi d_{0}}{\lambda})\), where \(\lambda\) is the wave-length of the transmitted signal. Finally, the received power \(P_{r}(d)\) at a distance \(d\) from the transmitter is given by Equation (2):  
\[P_{r}(d)=P_{t}-PL(d) \tag{2}\]  
Figure 1 shows theoretical values of the received power \(P_{r}\) (see Equation (2)) as a function of distance, using a transmission power of 0 dBm and a reference distance \(d_{0}\) of 1 m. \(PL(d_{0})\) is evaluated using the free space propagation model [30] and is equal to 40.33 dB. The values of \(\alpha\) and \(\sigma\) (3.56 and 6.06, respectively) are computed from our empirical data, as discussed in Section 6. To demonstrate the impact of the multi-path fading effect, this figure also depicts the received power in an ideal scenario, with no multi-path fading (i.e., \(X_{\sigma}=0\) in Equation (1)). In this simple ideal channel, the path-loss is a logarithmic function of the distance between the transmitter and the receiver.