## 2 Materials and Methods
### Underground Probe Design
#### 2.4.1 Coil Inductance Selection  
The selection of proper coil inductance is crucial for optimizing gamma variation in a parallel resonant probe across various soil permittivities. A key parameter is achieving self-resonance at 433 MHz, which requires a balance between capacitance and soil permittivity. Our simulation and testing indicated that dielectric constants ranging from 2 to 15 require this resonance to achieve the maximum reflection coefficient. The figure illustrates the ideal inductance needed to obtain this reflection coefficient. The relationship between the self-resonance frequency and the inductance and capacitance values is given by the following equations:  
\[\omega_{r}=2\pi f_{r} \tag{6}\]  
\[f_{r}=\frac{1}{2\pi\sqrt{L_{r}c}}. \tag{7}\]  
In the above equations, the terms are defined as follows:  
* \(\omega_{r}\): angular frequency;
* \(f_{r}\): resonance frequency;
* \(L_{r}\): coil inductance; \begin{table}
\begin{tabular}{c c} \hline \hline
**FDR** & **TDR** \\ \hline
**Definition** & **Definition** \\ \hline The Frequency Domain Reflectometry approach is the foundation of the FDR sensor, a soil sensor. By transmitting a high-frequency signal into the soil and analyzing the properties of the returned signal, it calculates the soil’s moisture content. FDR sensors analyze the frequency response and translate it into a moisture content reading, using sophisticated circuitry and algorithms. & Based on the Time Domain Reflectometry technique, the TDR sensor is a soil sensor. By delivering a brief pulse signal into the soil and analyzing the properties of the reflected signal, it ascertains the soil’s moisture content. Based on how quickly the pulse signal passes through the soil, the TDR sensor determines the moisture level. & Based on the Time Domain Reflectometry technique, the TDR sensor is a soil sensor. By delivering a brief pulse signal into the soil and analyzing the properties of the reflected signal, it ascertains the soil’s moisture content. Based on how quickly the pulse signal passes through the soil, the TDR sensor determines the moisture level. & **Advantages** \\ \hline
1. The FDR sensor operates reliably in a variety of salinity and soil types. & 1. The TDR sensor is very stable and accurate. \\
2. With comparatively minimal impact on salinity and temperature, it can offer excellent accuracy and stability. & 2. Its impact on soil conductivity and temperature is negligible. \\
3. Generally speaking, FDR sensors last a long time. & 3. TDR sensors monitor moisture quickly and precisely. \\ \hline
**Limitations** & **Limitations** \\ \hline
1. FDR sensors are more costly and sophisticated. & 1. Installing and using TDR sensors requires specialized knowledge and is costly. \\
2. Specialized knowledge and abilities are needed for installation and operation. & 2. In high-salinity soils, TDR sensor accuracy may be hampered. \\
3. Because of the resolution of the frequency response, FDR sensors are comparatively sensitive to variations in the conductivity of the soil material. & 3. Because TDR sensors require accurate time measurement and data analysis, they can be more susceptible to interference. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between TDR and FDR methods. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Measurement** & **Advantages** & **Disadvantages** \\ \hline Coaxial probe & High accuracy for high-loss materials. & Air gaps cause errors. \\  & High accuracy for high-loss materials. & Repetitive calibrations. \\ Transmission & Suitable for high frequency. & Cannot use below a few GHz, due to practical sample length limitation. \\ line & Support for both solids and liquids. & Sample preparation is difficult (fills fixture cross-section). \\ Parallel plate & Measurements are easier. & Support for low frequency (best results). \\ capacitance & Suitable for high-loss materials. & Electrode polarization effect. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between the measurement methods.