# Comparison of Radio Frequency Path Loss Models in Soil for Wireless Underground Sensor Networks
## 1 Introduction  
A wireless Underground Sensor Network (WUSN) consists of groups of nodes which are responsible for data collection, transmission and reception in an underground environment [1; 2; 3; 4]. Recent developments in electronics and wireless communications have enabled the development of low cost and low power WUSNs [5; 6]. WUSNs can be used in a variety of applications including precision agriculture, environmental monitoring, pipe monitoring and mine monitoring [4]. However, WUSNs have two major limitations; namely, Radio Frequency (RF) transmission in soil and lack of access to an energy supply. RF transmission is the most challenging aspect of WUSNs due to the high attenuation of the electromagnetic (EM) signal in soil [1]. Signals are often required to travel through dense soils with a high density or water content, and generally through a non-isotropic and non-homogenous soil mixture with different boundaries; both of which can attenuate the signal sharply. The variability of these conditions and complexity of the behaviour of signal attenuation with respect to these parameters makes accurate estimation of EM signal attenuation in soil challenging. The signal attenuation is caused by reflection, refraction, absorption and scattering of the electromagnetic wave, which affects the electromagnetic signal travel through different media [2]. The complex dielectric permittivity is one of the main parameters of a medium that expresses thisinteraction with electromagnetic radiation, and for soil it is dependent on the constituent components of the soil including volumetric water content. There are several models which predict dielectric properties of the soil from soil mixtures, all of which aim to estimate the effect of these parameters on signal attenuation [7, 8]. Hence, estimation of signal attenuation in soil also depends on the estimation of the complex dielectric permittivity; which can also be determined directly from Vector Network Analyser (VNA) measurements [9, 10] or extracted from Time Domain Reflectometry (TDR) measurements [11] without the need to know soil composition. Electromagnetic waves are also attenuated as they travel in free space, this attenuation is usually described as the Free Space Path Loss (_FSPL_). A model exists which is used for the calculation of the attenuation of the signal between two points (transmitter and receiver of signal) based on a clear line of sight (no obstacles at either of nodes [12, 13, 14]. The Friis transmission equation for electromagnetic signals in open space is given by Equation (1) [12, 13, 14]:  
\[\frac{P_{t}(d)}{P_{r}(d)}=\frac{(4\pi d)^{2}L}{G_{t}G_{r}\lambda^{2}} \tag{1}\]  
where \(P_{r}(d)\) is the received transmission power as a function of distance, \(P_{t}(d)\) is the transmission power, \(G_{t},G_{r}\) are the transmitter and receiver gain, \(d\) is the distance between transmitter and receiver, \(\lambda\) is the wavelength of the signal and \(L\) is a propagation independent system loss factor. This transmitted power relationship can also be represented in logarithmic form using the transmission link budget, which describes the general relation between the transmitted and received power \(P_{r}\) considering the loss in free space, Equation (2) [1, 12, 11, 13], where the symbols have the same meaning as in Equation (1) but now \(L_{0}\) is the path loss in free space:  
\[P_{r}=P_{t}+G_{r}+G_{t}-L_{0} \tag{2}\]  
Considering propagation dependent factors, then from Equation (1), the ratio of the transmitter power and receiver power is given by the ratio of the distance over the signal wavelength and the FSPL (\(L_{0}\)) can therefore be derived in decibel form as [5]:  
\[L_{0}=20\ log_{10}\ d\ (m)+20\ log_{10}\ f\ (Hz)-20\log(\frac{4\pi}{C}) \tag{3}\]  
where \(L_{0}\) is the total free space path loss in dB, \(d\) is the distance in meters, between the sensors, \(f\) is frequency in Hz and \(c\) is the speed of light in meters per second.