## 4 Results and Discussion
It is important to note that the reflection from the copper sheet has been delayed by the noise if we compare the time of reflection in the raw and processed GPR signal. The slight ripples in the time domain signal may be caused by the inverse Fourier transform, as the signal frequencies between 200 and 800 MHz were measured. Figure 5a,b shows the time series of [PERSON]'s function in the time domain and the maximum peak-to-peak (\(PtP\)) amplitude of the processed signal, respectively. GPR measurements were carried out with a constant time step of 15 min for nine days, resulting in nearly 860 measurements. In general, GPR propagates the electromagnetic energy from the monostatic horn antenna and records the reflected energy from the subsurface with different electrical properties. In Figure 5a, [PERSON]'s function is obtained by filtering out noise effects via the antenna transfer functions in Equation 1. When the soil is in the process of freezing, the presence of ice (instead of liquid water) in some pores reduces the dielectric permittivity of the soil and results in small reflections in the GPR signal. At the time of thawing, the soil dielectric permittivity increases due to more water in the soil system, and the amplitude of [PERSON]'s function is relatively large due to the strong reflection at the wet soil surface. The amplitude of [PERSON]'s function varies across the nine-day period due to the different reflections received during the soil freeze and thawing process. To provide a better visual interpretation of this, the amplitude of maximum \(PtP\) is plotted in Figure 5b. All five of the freeze-thaw cycles, as indicated by arrows, can be clearly observed in the \(PtP\) plot of Figure 5b and follow the trend of freeze-thaw cycles observed in temperature measurements detectable in Figure 3b. Figure 4: Time domain representation (b-scan) of radar measurement performed with the horn antenna at different heights above a perfect electrical conductor (copper sheet). (**a**) Measured raw GPR signal \(S_{11}(t)\) and (**b**) the filtered signal \(g_{xx}^{\uparrow}(t)\) using the antenna model. Reflections from snow water films or wet snow were not observed in the GPR signal. In the far-field assumption, the layer thickness of surface water films or wet snow should be more than one tenth the minimal wavelength (\(\upl_{min}\)) to obtain a reflection in the GPR radargram. Considering the \(\varepsilon_{r}\) for snow water films to be 80 and for wet snow as 50, the vertical resolution of the GPR signal in the frequency range of 200-800 MHz can be 0.53 cm and 0.41 cm for the surface water films and wet snow, respectively. Reflections in the GPR radargram can be observed in the case of surface water films or wet snow exceeding this calculated layer thicknesses. The electrical conductivity of the snow cover was recorded to be below 0.001 mS/cm, which cannot attenuate the GPR signal in the low frequency range. Figure 6 shows the relative dielectric permittivity (\(\varepsilon_{r}\)) recorded by the _in situ_ sensors and estimated from time-lapse GPR data. The dynamics of the freeze-thaw cycle can be observed in the \(\varepsilon_{r}\) measured by the _in situ_ sensors installed at a 2-4 cm depth over the nine-day measurement period, as indicated by arrows. Below a 4 cm depth, the \(\varepsilon_{r}\) variation is less pronounced and indicates that the freezing front did not reach this depth, which is in correspondence with the recorded soil temperature measurements at these deeper depths, which never drop below \(0\,^{\circ}\mathrm{C}\) (Figure 3b). As can be seen, the GPR signal is highly sensitive to the variation of \(\varepsilon_{r}\) in the subsurface and the contrast between the lower \(\varepsilon_{r}\) of the frozen compared to higher \(\varepsilon_{r}\) of the unfrozen soil. As already mentioned, the surface reflection in the measured time-lapse GPR data was used to estimate \(\varepsilon_{r}\) of the soil by inversion.