# A New Approach to Estimating the Path Loss in Underground Wireless Sensor Networks
## 2 RF Signals in Soil
### Propagation Models  
Accurately estimating the signal attenuation in soil can benefit a variety of fields where WUSNs are required. Therefore, multiple empirical and semi-empirical models exist in the literature, which aim to predict the attenuation of electromagnetic signals in soil. The two main models in the literature are the CRIM-Fresnel model and the Modified-Friis model [23; 24; 25]. Both of these models are based on the \"link budget\" formula, which was proposed by [PERSON] and [PERSON] [3] to act as a framework for EM wave propagations models in soil. This formula adds a correction factor to the well-known Friis transmission equations in free space [26] to account for the extra losses in soil. The link budget formula is given by Equation (2). \[P_{r}=P_{t}+G_{r}+G_{t}-L_{0}-L_{m} \tag{2}\]  
where \(L_{m}\) is the path loss in soil (medium) due to material absorption (dB), \(L_{0}\) is the path loss in free space (dB), \(G_{t}\) and \(G_{r}\) are the transmitter and receiver antenna gains (dB) and \(P_{t}\) and \(P_{r}\) are the transmitter and receiver powers (dB). Based on the Modified-Friis model, the total path loss caused by the medium, \(L_{m}\), (i.e., soil) consists of losses caused by the change in the wavelength, and the attenuation losses caused by thematerial absorption [24]. Therefore, based on this model, the total losses of an EM signal in soil, \(L_{p}\), including losses in free space, is given by Equation (3). \[L_{p}=6.4+20\log(d)+20\log(\beta)+8.68\alpha d \tag{3}\]  
where \(d\) is the distance between the transmitter and receiver and \(\alpha\) and \(\beta\) are given by Equations (4) and (5). \[\propto =\omega\sqrt{\frac{\mu\epsilon^{\prime}}{2}\left[\sqrt{1+\left( \frac{\epsilon\,^{\prime\prime}}{\epsilon^{\prime}}\right)^{2}-1}\right]} \tag{4}\] \[\beta=\omega\sqrt{\frac{\mu\epsilon^{\prime}}{2}\left[\sqrt{1+ \left(\frac{\epsilon\,^{\prime\prime}}{\epsilon^{\prime}}\right)^{2}+1}\right]} \tag{5}\]  
where \(\omega\) is the angular frequency (\(\omega=2\pi f\)) and \(\mu\) is the relative magnetic permeability; and where the real and imaginary parts of the complex permittivity are estimated using the Peplinski dielectric mixing formula [17; 18]. The CRIM-Fresnel model proposed by [PERSON] et al. [23] is a semi empirical model based on the Complex Refractive Index Model (CRIM). This model is also based on the link budget formula presented in Equation (2). However, unlike the Modified-Friis model, the CRIM-Fresnel model also considers losses due to reflection in the calculations of total attenuation, A\({}_{tot}\) in soil [15; 21; 23]. Another difference between these two models is that the CRIM-Fresnel model uses the CRIM mixing model for estimating the complex permittivity of soil. The total attenuation loss, A\({}_{tot}\), based on the CRIM-Fresnel model, is given by Equation (6). \[A_{tot}=\alpha_{c}d+R_{c} \tag{6}\]  
where \(\alpha_{c}\) is the attenuation due to material absorption [23; 27] and \(R_{c}\) is the attenuation due to reflection [23]. [PERSON] et al. [23] claim that their proposed model has a better performance at estimating the attenuation in soil compared to the Modified-Friis due to the lack of a large supporting database for the Peplinski mixing model used by the Modified-Friis. However, the CRIM-Fresnel model itself was developed based on a very limited soil types and conditions. The soil used in the research by [PERSON] et al. [23] was stated to be \"Sieved Sand\", without providing further information on the type and particle size of the sand that was used; furthermore, it has not been validated by field trials [23]. A common drawback of both of these models is that their accuracy is largely dependent on the accuracy of the mixing model used. In addition, both of the mixing models used in the CRIM-Fresnel and Modified-Friis models require a sample of the soil to be analysed in the laboratory to accurately measure the parameters required for the mixing models.