# Comparison of Radio Frequency Path Loss Models in Soil for Wireless Underground Sensor Networks
## 1 Introduction
From Equation (3) it is clear that the FSPL depends on the distance between two nodes and the frequency of the signal. To model the attenuation of the signal in a medium such as soil, the FSPL can be modified to include the attenuation due to the medium (\(L_{m}\)), such that Equation (2) becomes:  
\[P_{r}=P_{t}+G_{r}+G_{t}-(L_{0}+L_{m}) \tag{4}\]  
This is referred to as the modified-Friis model. The total path loss (\(P_{L}\)) of the signal is the summation of the medium path loss and the free space path loss, and has been shown to be given by Equation (5) [5]:  
\[P_{L}=6.45+20\log(\mathrm{d}\ (\mathrm{m}))+20\log(\beta)+8.69\alpha\mathrm{d} (\mathrm{m}) \tag{5}\]  
where \(\alpha\) is the signal attenuation coefficient and \(\beta\) is the phase shifting coefficient, and which are given by Equations (6) and (7), respectively;  
\[\alpha=\omega\sqrt{\frac{\mu e^{\prime}}{2}\Bigg{(}\sqrt{1+\left(\frac{e^{ \prime\prime}}{\varepsilon^{\prime}}\right)^{2}}-1\Bigg{)}} \tag{6}\]\[\beta=\omega\sqrt{\frac{\mu\varepsilon^{\prime}}{2}\Bigg{(}\sqrt{1+\left(\frac{ \varepsilon^{\prime\prime}}{\varepsilon^{\prime}}\right)^{2}+1}\Bigg{)}} \tag{7}\]  
where \(\omega=2\pi f\), \(\mu\) is the magnetic permeability of the soil, \(\varepsilon^{\prime}\) is the real part of the complex dielectric permittivity and \(\varepsilon^{\prime\prime}\) is the imaginary part of the dielectric permittivity. As shown in Equations (6) and (7), the values of \(\alpha\) and \(\beta\) are directly related to the dielectric properties of the soil. Research published by [PERSON] [1] used the Peplinski mixing model to calculate the real and imaginary parts of the complex dielectric permittivity of the soil based on the composition and volumetric water content of the soil [1; 15]. By substituting Equation (5) into Equation (4) [16], the modified-Friis model for total power received through a medium by the receiver is given by Equation (8):  
\[P_{r}=P_{t}+G_{r}+G_{t}-[6.45+20\log(\mathrm{d}\ (\mathrm{m}))+20\log(\beta)+8.69 \alpha\mathrm{d}(\mathrm{m})] \tag{8}\]  
The CRIM-Fresnel model, initially proposed by [PERSON] [17], is another model for estimating the attenuation of a signal in soil and uses complex refractive index. The CRIM-Fresnel model also takes the attenuation of the signal due to reflection into account. The total attenuation of the signal is introduced as the sum of soil attenuation as well as the signal reflection [17]. The total attenuation of the EM signal (\(A_{tot}\)) based on this model is given by Equation (9) [2]:  
\[A_{tot}=\alpha_{CRIM}\Bigg{(}\frac{dB}{m}\Bigg{)}\times d\ (m)+R_{c} \tag{9}\]  
where \(\alpha_{CRIM}\) is the soil attenuation coefficient and \(\mathrm{d}\) is the distance, \(R_{c}\) is the attenuation due to the reflection of the signal. The attenuation coefficient, \(\alpha_{CRIM}\), is calculated by Equation (10) and the attenuation due to the reflection, \(R_{c}\), is given by Equation (11):  
\[\alpha_{CRIM}=8.68\frac{60\pi(2\pi f\varepsilon_{0}\varepsilon^{\prime\prime} +\alpha)}{\sqrt{\frac{\varepsilon^{\prime}}{2}\Bigg{(}1+\sqrt{1+\left(\frac{ \varepsilon^{\prime\prime}+\frac{\varepsilon^{\prime}}{2\pi f\varepsilon_{0}} }{\varepsilon^{\prime}}\right)^{2}}\Bigg{)}}} \tag{10}\]  
\[P_{r}=P_{t}+G_{r}+G_{t}-L_{0} \tag{2}\]0  
where R is the reflection coefficient and is given by Equation (12) as:  
\[P_{r}=P_{t}+G_{r}+G_{t}-L_{0} \tag{2}\]1  
The permittivity of the soil in the CRIM-Fresnel model is calculated based on the Complex Refractive Index Model (CRIM). This model calculates the soil dielectric permittivity by using the permittivity of solid (\(\varepsilon_{s}\)), water (\(\varepsilon_{w}\)) and air (\(\varepsilon_{a}\)) at a certain frequency, based on the fact that the only loss source is water [17]. The CRIM-Fresnel model is claimed by [17] to be a better estimation for the signal propagation in soil compared to the modified-Friis model due to the fact that the proposed mixing model to gain soil properties used in the modified-Friis was not developed based on a large data set.