## 2 Materials and Methods
### Measurement of Dielectric Properties  
In this study, we used a vector network analyzer (VNA) equipped with an open-ended coaxial probe to measure the dielectric properties of the soil samples used in our experiment. The VNA (Keysight E5071C ENA network analyzer, Keysight Technologies Inc., San Jose, CA, USA) had a wide frequency range of 300 kHz to 20 GHz, and we used a dielectric slim form probe (Keysight N1501A, Keysight Technologies Inc., San Jose, CA, USA) as  
Figure 2: Schematic diagram of RF heating system: (**a**) Soil samples used for radio frequency heating system. (**b**) Radio frequency heating system.  
the open-ended coaxial probe. This probe is non-destructive to materials and can be used to measure the dielectric properties of solid, liquid, and powder materials, as advised in [26]. The VNA was connected to a computer desktop with the Keysight Materials Measurement Suit software (Keysight Technologies Inc., Santa Rosa, CA, USA) for data acquisition and dielectric property calculations as suggested in [27]. The measurement of dielectric properties involves the determination of both the dielectric constant and dielectric loss factor, as described in the following equation:  
\[\varepsilon=\varepsilon^{\prime}-je^{\prime\prime} \tag{3}\]  
where \(\varepsilon^{\prime}\) is the dielectric constant and \(\varepsilon^{\prime\prime}\) is the dielectric loss factor.  
The process of calibrating the network analyzer involved using air, a shot probe, and deionized water, as recommended in [28]. The calibrations and measurements were conducted in a closed laboratory setting with a temperature of 25 \({}^{\circ}\)C. The soil samples were measured on site within a frequency range of 20 MHz to 10 GHz, utilizing a VNA configured with a total of 1001 sampling points. This frequency range is relevant for industrial, scientific, and medical applications [24; 25]. The soil samples were placed in a container that was wide enough to allow the probe to be inserted and the soil sample was thick enough to prevent any reflection coefficient interference from the glass dish and ground floor on the measurements and errors in the measurement of the dielectric properties. A dielectric slim form probe was used to measure the dielectric properties of the soil samples in situ. The dielectric properties values were measured with three replications, as suggested in [29], and the mean values were used for data analysis. The setup for measuring dielectric properties is depicted in Figure 3.  
The complex relative permittivity, or the measure of a material's ability to allow an electric field to pass through it, often varies with frequency when subjected to an applied electric field. The frequency-dependent behavior of the complex permittivity can be described using the Debye equation, which is expressed as follows:  
\[\varepsilon=\varepsilon_{\infty}+\frac{\varepsilon_{s}-\varepsilon_{\infty}}{ 1+j\omega\tau} \tag{4}\]  
Figure 3: Experimental setup of dielectric properties measurement.  
where \(\varepsilon_{s}\) is the static dielectric constant, \(\varepsilon_{\infty}\) is the dielectric constant at high frequency, \(\omega\) is the angular frequency and \(\tau\) is the relaxation time. Equation (4) can be separated into the dielectric constant \(\varepsilon^{\prime}\), and dielectric loss factor \(\varepsilon^{\prime}\), as follows:  
\[\varepsilon^{\prime}=\varepsilon_{\infty}+\frac{\varepsilon_{s}-\varepsilon_{ \infty}}{1+\left(\omega\tau\right)^{2}} \tag{5}\]  
\[\varepsilon^{\prime\prime}=\frac{\left(\varepsilon_{s}-\varepsilon_{\infty} \right)\omega\tau}{1+\left(\omega\tau\right)^{2}} \tag{6}\]  
Here, the permittivity values in this paper are obtained directly from our experimental equipment, which simplifies the process of recording results.