# Detection of Frozen Soil Using Sentinel-1 SAR Data
## 4 Results
### Signal Modeling on Frozen Soils  
The dielectric constants calculated in Section 4.1 were used in the IEM_B to calculate the backscattering coefficient of the soil surface. Figure 4 shows the difference \(\Delta\sigma^{\circ}\) between the backscattering coefficient (\(\sigma^{\circ}\)) of unfrozen soil that corresponds to a soil temperature higher than \(0^{\circ}\)C (in this study the reference soil temperature of unfrozen soil was considered equal to \(0^{\circ}\)C) and the backscattering coefficient (\(\sigma^{\circ}\)) of frozen soil with a temperature between \(-20\) and \(0^{\circ}\)C. The maximum difference \(\Delta\sigma^{\circ}\) between unfrozen soil and frozen soil is between 4.6 and 5.6 dB depending on the soil  
Figure 3: Dielectric constant of a frozen soil–water mixture \(\varepsilon_{mf}\) according to the soil temperature for three soil types (Table 1): (**a**) real part \(\varepsilon^{\prime}_{mf}\); and (**b**) imaginary part \(\varepsilon^{\prime}_{mf}\). \(m_{v}=30\) vol%, \(\rho_{w}=1\) g/cm\({}^{3}\), \(\rho_{i}=0.9175\) g/cm\({}^{3}\), and A, B and \(\rho_{b}\) are given in Table 1.  
Figure 2: Air (at 5 cm) and soil temperatures (ground surface and at 5 cm depth). Temperature was recorded using HOBO U12 sensors. type (temperature = \(-20\)\({}^{\circ}\)C). For soil with the most clay (silty clay soil type), the difference \(\Delta\sigma^{\circ}\) is lower than that for soils with less clay (sandy loam followed by silt loam). For temperatures between \(-5\) and \(-20\)\({}^{\circ}\)C, the increase of \(\Delta\sigma^{\circ}\) is smaller than 1 dB regardless of soil type. This result in consistent with Figure 3 where the dielectric constant of a frozen soil water mixture increases only slightly with the soil temperature Ts for Ts-values between \(-20\)\({}^{\circ}\)C and \(-5\)\({}^{\circ}\)C. Figure 4 also shows that, even when the temperature drops to \(-2\)\({}^{\circ}\)C, \(\Delta\sigma^{\circ}\) ranges between 2.6 and 4.7 dB. This decrease in the radar signal due to freezing (increase in \(\Delta\sigma^{\circ}\)) is more important for sandy soils than for clay soils because sandy soils freeze more quickly than clay soils because of their lesser capacity to store water. To understand the behavior of the radar signal (\(\sigma^{\circ}\)) with the temperature for the three soil types, the unfrozen volumetric moisture content (\(m_{vu}\)) and volumetric ice content (\(m_{vi}\)) were ploted according to the temperature. Figure 5 shows that when there is more clay, the \(m_{vu}\) of frozen soil is higher than those of the other soils, and, therefore, \(\sigma^{\circ}\) is higher. Therefore, \(\sigma^{\circ}\) for frozen soil with more clay is higher than \(\sigma^{\circ}\) for frozen soil with less clay. As for all unfrozen soils (whatever the soil type), with more or less clay, \(\sigma^{\circ}\) is the same since the total volumetric moisture content is the same (in this paper, \(m_{v}\) = 30 vol%) and \(\Delta\sigma^{\circ}\) (\(\sigma^{\circ}\) of frozen soil \(-\)\(\sigma^{\circ}\) of unfrozen soil) for clay soils is lower than \(\Delta\sigma^{\circ}\) for soils that are less clay. Figure 4: Difference \(\Delta\sigma^{\circ}\) between the modeled backscattering coefficient of unfrozen soil with reference temperature Ts equal to 0 \({}^{\circ}\)C \"\(\sigma^{\circ}\) (Ts = 0\({}^{\circ}\))\" and the modeled backscattering coefficient of frozen soil with soil temperature between \(-20\)\({}^{\circ}\)C and 0 \({}^{\circ}\)C \"\(\sigma^{\circ}\) (Ts)\". For soil temperatures above 0 \({}^{\circ}\)C, \(\Delta\sigma^{\circ}\) (\(\sigma^{\circ}\) (Ts = 0) \(-\)\(\sigma^{\circ}\) (Ts > 0\({}^{\circ}\))) is equal to 0 dB. Radar frequency f = 5.405 GHz, \(m_{v}\) = 30 vol%, \(Hrms\) = 1 cm, incidence angle \(\theta\) = 40\({}^{\circ}\), and polarization = VV. Figure 5: Unfrozen volumetric moisture content (\(m_{vu}\)) and corresponding volumetric ice content (\(m_{vi}\) ) of soil during freezing according to the soil temperature for three soil types (Table 1).