## 2 Theoretical Background  
### Complex Dielectric Permittivity  
The dielectric permittivity (\(\varepsilon\)) is a physical quantity that describes the ability of a material to oppose an external electric field through the polarization of its molecules. It is often measured in soil, for agronomic or geotechnical purposes, since it is closely related to the water content. A dielectric material, placed in an electric field, undergoes a polarization phenomenon in which the dipole moment of its molecules aligns according to the external electric field. In this way, a new electric field is generated in the dielectric material, which is opposed to the external one: the greater the permittivity, the greater the opposition to the external electric field and the lower the net field strength [20]. When an electromagnetic wave propagates through a material, for example the soil, an electric field and a magnetic field are generated, oscillating perpendicularly to each other and toward the direction of wave propagation. Its signal is continuously modulated by the soil properties and by the presence of water, changing in amplitude, frequency, and speed. The nature of the material plays a crucial role in defining its behavior when an electromagnetic wave is propagating inside it. For example, while the water particles can rotate and orientate according to the oscillating electric field, those of the soil remain mostly stationary. The dielectric permittivity \(\varepsilon\) governs the correlation between the electric field \(E\) and the reorganization of electric charges in the material, measured by the electric displacement field \(D\), according to the law:  
\[D\ =\ \varepsilon E \tag{1}\]  
If the electric field is static, and in case of homogeneous and isotropic medium, the dielectric permittivity is constant. Otherwise, in case of an alternate electric field, it depends on the frequency. In case of an anisotropic and non-homogeneous medium, it can change with the spatial coordinates and with the direction. The dimensionless relative permittivity (or dielectric constant) is defined as:  
\[\varepsilon_{r}\ =\ \frac{\varepsilon}{\varepsilon_{vacuum}} \tag{2}\]  
The typical values of the relative permittivity are 78.5 for water (at the reference temperature of 25 \({}^{\circ}\)C), 1 for air, 2.1 for diesel oil, and about 3-4 for soil grains. Dielectric permittivity is a complex quantity, and it can be written as the sum of a real component and an imaginary component:  
\[\varepsilon_{r}\ =\ \varepsilon_{r}^{\prime}\ +\ i(\varepsilon_{r}^{\prime \prime}\ +\ \frac{\sigma_{dc}}{2\pi f\varepsilon_{v}}) \tag{3}\]  
The real part (\(\varepsilon_{r}^{\prime}\)) represents the ability of a medium to store energy in the form of the polarization of molecules. It can be defined considering a parallel plate capacitor: more charge is stored when a dielectric material is put between the plates than if a vacuum is between them. The ratio between the capacitance \(C\) (with the material) and \(C_{0}\) (with vacuum) is the real part \(\varepsilon_{r}\)[21]:  
\[\frac{C}{C_{0}}\ =\ \varepsilon_{r}^{\prime} \tag{4}\]  
The imaginary component of the complex dielectric permittivity is a measure of how much dissipative the material is to an external electric field. The ratio between the imaginary component and the real one is called dissipation factor and its inverse is called quality factor (\(Q\)). It is equal to the tangent of \(\delta\) (\(\delta\) is the angle between \(\varepsilon_{r}^{\prime}\) and \(\varepsilon_{r}\) in the complex plane):  
\[\frac{\varepsilon_{r}^{\prime\prime}\ +\ \frac{\sigma_{dc}}{2\pi f\varepsilon_{v}} }{\varepsilon_{r}^{\prime}}\ =\ tan(\delta) \tag{5}\]The first term (\(\varepsilon_{r}\)\") represents the dielectric losses of the signal due to the vibration or rotation of the molecules. Dielectric losses are frequency-dependent and have a maximum at the so-called relaxation frequency, which for water is 17 GHz at 20 \({}^{\circ}\)C [22]: to estimate them, the Debye equation can be used.