## 2 Material and Methods
### GPR Forward Modeling  
The electromagnetic model used to describe the GPR wave propagation in the antenna-air-soil system is considered as a multilayered medium, details of which can be found in [PERSON] _et al._[35]. The model is applied to an off-ground horn antenna connected to the SFCW radar. In the model, the ratio between the backscattered electromagnetic field \(b(\omega)\) and the incident electromagnetic field \(a(\omega)\) is expressed in the frequency-dependent complex ratio \(S_{11}(\omega)\), where \(\omega\) is the angular frequency (\(\omega=2\pi f\)). In the far-field assumption, the parameters of the antenna are assumed independent of the scatterer, _i.e._, the plane wave field approximation is distributed over the antenna aperture. The complex, frequency-dependent global transmission and reflection coefficients can be used to model the GPR system precisely via [PERSON]'s equations [35]:  
\[S_{11}(\omega)=\frac{b(\omega)}{a(\omega)}=H_{i}(\omega)+\frac{H_{t}(\omega)H_ {r}(\omega)G_{xx}^{\uparrow}(\omega)}{1-H_{f}(\omega)G_{xx}^{\uparrow}( \omega)} \tag{1}\]  
where \(H_{f}(\omega)\), \(H_{t}(\omega)\), \(H_{r}(\omega)\) and \(H_{i}(\omega)\) are the characteristics of the antenna for the feedback loss, transmitting, receiving and the complex return loss transfer functions of the antenna, respectively. The \(G_{xx}^{\uparrow}(\omega)\) term is the Green function impulse reflection of the air-soil system modeled as a multilayered medium. \(H_{i}(\omega)\) represents the variation of impedance between the antenna aperture, feed point of the antenna and multiple wave reflections occurring within the antenna, which are independent of the backscattered electromagnetic field \(G_{xx}^{\uparrow}(\omega)\). The \(H_{t}(\omega)\) and \(H_{r}(\omega)\) transfer functions correspond to the phase delay and antenna gain between the source-receiver virtual point and the measurement point. To reduce the number of transfer functions, Equation (1) can be defined as \(H(\omega)=\)\(H_{t}(\omega)H_{r}(\omega)\). The characteristics of the antenna transfer functions (\(H_{f}(\omega)\), \(H_{i}(\omega)\) and \(H(\omega)\)) can be estimated by solving a system of equations as Equation (1) for these three unknown functions, by performing \(S_{11}\) measurements above a perfect electrical conductor (e.g., a copper sheet), for which the Green functions \(G_{xx}^{\uparrow}\) can be simulated [36, 37, 38]. The antenna effects can be removed from the raw radar measurements, and once the antenna transfer functions are known together with Green's function, the response from the air-soil medium can be derived using Equation (1). The earth medium properties govern the electromagnetic wave propagation and include the electric conductivity \(\sigma\) (S\(\cdot\)m\({}^{-1}\)), electric permittivity \(\varepsilon\) (F\(\cdot\)m\({}^{-1}\)) and magnetic permeability \(\upmu\) (H\(\cdot\)m\({}^{-1}\)). In this study, \(\upmu\) is assumed to be equal to the permeability of free space, \(\upmu_{0}=4\pi\times 10^{-7}\) H\(\cdot\)m\({}^{-1}\), which is valid for non-magnetic soil materials, such as natural soil. The subsurface is modeled as a 3D multilayered medium consisting of \(N\) horizontal layers separated by \(N-1\) interfaces. The characteristics of the \(n\)-th layer is considered homogeneous and expressed as \(\varepsilon_{n}\)\(\sigma_{n}\) with a thickness \(h_{n}\). The electromagnetic waves propagating in a multilayered media are defined by [PERSON]'s equations. The antenna is monostatic and modeled as a point source and receiver, and hence, the radiation pattern is emulated by that of a point dipole.