## 4 Results and Discussion
With respect to the simple topography of the objective function dealing with two unknown parameters (\(\mathbf{p}=[\varepsilon_{r},h]\)), the initial guess for the relative dielectric permittivity can be chosen arbitrarily. Here, it was fixed to eight. To illustrate the uniqueness of the inverse solution, tests were performed with randomly-selected initial guesses of \(\varepsilon_{r}\) and showed that the solution found by the algorithm was independent of the initial guess. Additionally, the GPR estimated \(\varepsilon_{r}\) matches closely to the \(\varepsilon_{r}\) recorded by the sensor installed at a 3-cm depth and followed a similar trend, with the effect of the freeze-thaw cycles well reproduced. [PERSON] _et al._[45] reported that the time domain inversion of surface reflection is sensitive to the soil moisture dynamics in the top 4 cm of soil. Furthermore, [PERSON] _et al._[46] tested the full-waveform inversion of off-ground GPR measurements to estimate the depth profile of soil moisture in the top 60 cm. Figure 5: (**a**) Radar signal in the time domain for a single measurement and (**b**) peak-to-peak (\(PtP\)) maximum amplitude of the time-lapse GPR data, with the effect of the five freezing and thawing events clearly observed as marked by arrows. As shown by the experiment, the freezing front and the freeze-thaw cycles could be detected by the off-ground monostatic GPR system, without disturbing the natural snow cover or the frozen soil. Additionally, time-lapse measurements are feasible even if the (sub-)surface may be exposed to natural environmental conditions, such as precipitation in the form of snow, wind, _etc_. Furthermore, antenna effects can be removed from the raw radar signal due to the accuracy of the radar electromagnetic model. In comparison, commercially available GPR systems used to detect freezing zones and permafrost dynamics in the subsurface are mostly ground coupled. For time-lapse GPR measurements, the structure of the top snow cover or frozen soil, as well as environmental conditions (e.g., precipitation in terms of snow or radiation) can be significantly disturbed and, hence, affect the fidelity of measurements. Additionally, the tested off-ground GPR system was stable with respect to ambient temperature and recorded precise measurements even at fairly low ambient temperature (\(<-10\,^{\circ}\mathrm{C}\)). The tested time domain inversion approach is based on surface reflection and, therefore, mainly sensitive to the top few centimeters. As such, this approach can be very useful to calibrate different remote sensing products and to deliver precise information that can be used to improve remote sensing retrieval algorithms. In the setup presented here, soil surface roughness as well as vegetation effects, were excluded by placing the GPR system over a smooth bare soil field. In general, a differentiation between rough and smooth surfaces can be made with respect to the wavelength of the signal based on the Rayleigh criterion (\(h_{c}=\lambda/8\)), where \(\lambda\) is the wavelength and \(h_{c}\) is the critical height of the surface protuberances. If the surface is smooth and homogeneous (in terms of electrical properties), most of the reflected energy will be in the specular direction (coherent component), while if the surface is rough, diffuse reflections  
Figure 6: Time series of the relative permittivity (\(\varepsilon_{r}\)) of the soil recorded at shallow depths (2, 3, 4, 5, 7, 8 cm) and estimated from time-lapse GPR data. Five freezing and thawing cycles can be clearly observed over a period of nine days, as indicated by arrows. or scattering (incoherent component) can occur, leading to less energy being recorded in the specular direction. To account for rougher surfaces, [PERSON] _et al._[47] introduced a roughness sub-model that can be incorporated into the full-waveform inversion of the off-ground GPR model. Similar surface roughness models can be included for spatial off-ground GPR measurements operating on a large scale.