## 2 Materials and Methods
### Underground Probe Design
#### 2.4.3 Dielectric Permittivity Estimation  
The following flowchart describes how to extract the dielectric permittivity (Figure 19):  
Based on Figure 14, the complex admittance expressions for the three branches can be deduced as follows:  
\[Y_{1}=\frac{1}{R} \tag{9}\]  
\[Y_{2}=jC_{\text{tot}}\omega \tag{10}\]  
\[Y_{3}=\frac{1}{j\omega L} \tag{11}\]  
The total complex admittance of the probe is given by their summation:  
\[Y=Y_{1}+Y_{2}+Y_{3} \tag{12}\]  
\[Y=G+j\omega C_{\text{tot}}+\frac{1}{j\omega L} \tag{13}\]  
\[Y=G+j\bigg{(}\omega C_{\text{tot}}-\frac{1}{\omega L}\bigg{)} \tag{14}\]  
where \(C_{\text{tot}}=C_{s}+C_{o}\).  
The conductance \(G\) and susceptance \(H\), therefore, have the following expressions:  
Figure 19: Measured vegetal soil electromagnetic properties.  
\[G=\frac{1}{R} \tag{15}\]  
\[H=j\bigg{(}\omega C_{\text{tot}}-\frac{1}{\omega L}\bigg{)} \tag{16}\]  
To determine \(\varepsilon_{r}\) at resonant frequency,  
\[\omega=\omega_{r} \tag{17}\]  
\[Y(\omega_{r})=G+\bigg{(}\omega_{r}(C_{s}+C_{v})-\frac{1}{\omega_{r}L}\bigg{)}^{2} \tag{18}\]  
From this, we can deduce  
\[\bigg{(}\omega_{r}(C_{s}+C_{v})-\frac{1}{\omega_{r}L}\bigg{)}^{2}=0 \tag{19}\]  
\[Y(\omega_{r})=G \tag{20}\]  
From the conductance, we can deduce  
\[\sigma=\omega\varepsilon_{0}\varepsilon_{r}^{\prime\prime} \tag{21}\]  
We can characterize the electrical properties of the soil using the complex relative permittivity, denoted as \(\varepsilon_{r}\), which can be expressed as  
\[\varepsilon_{r}=\varepsilon_{r}^{\prime}-je_{r}^{\prime\prime}. \tag{22}\]  
In the above equation, the terms are defined as follows:  
* \(\varepsilon_{r}^{\prime}\): energy storage of the dielectric permittivity;
* \(\varepsilon_{r}^{\prime\prime}\): dielectric losses.  
\[\varepsilon_{r}^{\prime}=\sqrt{\frac{1}{\omega C_{0}}\bigg{(}Y^{2}-G^{2}+ \frac{1}{\omega L}\bigg{)}-\frac{C_{v}}{C_{0}}} \tag{23}\]  
\[\Gamma=\frac{Y_{0}-Y}{Y_{0}+Y}\quad Y=\frac{1}{Z_{0}}\bigg{(}\frac{1-\Gamma}{ 1+\Gamma}\bigg{)} \tag{24}\]  
Following the methodology outlined below, we developed an algorithm in MATLAB to calculate the estimated conductance from resonance and the dielectric permittivity based on the measured gamma as a function of the applied voltage on the diode. By evaluating symmetric points around the resonance, as depicted in the graph below, we confirmed that the estimation of the dielectric permittivity was accurate and closely aligned with the actual value.  
From Figure 20, at the resonance frequency of 433 MHz and for a dielectric constant of 10, the reflection coefficient reached \(-15\) dB at a diode voltage of approximately 4 volts. The tables below present the results of the dielectric permittivity obtained using the two methodologies described.  
The previous figure shows (Figure 21) the results of the estimation of the dielectric permittivity; with the two methodologies, we, despite this table, have a reasonable estimation of the dielectric permittivity between 6 and 15. Hence, we explain the bad estimation of the dielectric permittivity from 2 to 5 as being due to the low value of the reflection coefficient, which did not allow information to be extracted properly.