# Detection of Frozen Soil Using Sentinel-1 SAR Data
## 2 Radar Signal Modeling
### Semi-Empirical Dielectric Mixing Model for Frozen Soil-Water  
[PERSON] et al. [10] developed a semi-empirical soil-water dielectric mixing model for the 1.4-18 GHz frequency range of the following form:  
\[\varepsilon_{m}=\varepsilon_{m}^{\prime}-je_{m}^{\prime\prime} \tag{1}\]  
where \(\varepsilon_{m}\) is the relative complex dielectric constant of the soil-water mixture, \(\varepsilon_{m}^{\prime}\) is the real part and \(\varepsilon_{m}^{\prime\prime}\) is the imaginary part. \(\varepsilon_{m}^{\prime}\) and \(\varepsilon_{m}^{\prime\prime}\) are given by:  
\[\varepsilon_{m}^{\prime}=\left[1+\frac{\rho_{b}}{\rho_{s}}(\varepsilon_{s}^{a }-1)+m_{p}^{g^{\prime}}\varepsilon_{fw}^{\prime a}-m_{v}\right]^{1/\alpha} \tag{2}\]  
\[\varepsilon_{m}^{\prime\prime}=\left[m_{v}^{g^{\prime\prime}}\varepsilon_{fw }^{\prime\prime}\right]^{1/\alpha} \tag{3}\]where \(\rho_{b}\) is the bulk density (g/cm\({}^{3}\)), \(\rho_{s}\) is the specific density of the solid soil particles (g/cm\({}^{3}\)), \(\varepsilon_{s}\) is the dielectric constant of soil solid, \(m_{v}\) is the volumetric moisture content (vol%), \(\varepsilon^{\prime}_{fw}\) is the real part of the dielectric constant of free water, \(\varepsilon^{\prime}_{fw}\) is the imaginary part of the dielectric constant of free water, \(\alpha\) is an empirically determined constant (=0.65 [10]), and \(\beta^{\prime}\) and \(\beta^{\prime}\) are soil-texture-dependent coefficients. The dielectric constant of soil solid, \(\varepsilon_{s}\), given in Equation (2), is expressed by [PERSON] et al. [10]:  
\[\varepsilon_{s}=\left(1.01+0.44\rho_{s}\right)^{2}-0.062 \tag{4}\]  
For a given soil with \(\rho_{s}\) = 2.66 g/cm\({}^{3}\), \(\varepsilon_{s}\) = 4.7. To calculate \(\beta^{\prime}\) and \(\beta^{\prime\prime}\) of Equations (2) and (3), the following expressions are used [10,11]:  
\[\beta^{\prime}=\left(127.48-0.519S-0.152C\right)/100 \tag{5}\]  
\[\beta^{\prime\prime}=\left(133.797-0.603S-0.166C\right)/100 \tag{6}\]  
where \(S\) and \(C\) are the percentages of sand and clay, respectively. The quantities \(\varepsilon^{\prime}_{fw}\) and \(\varepsilon^{\prime}_{fw}\) presented in Equations (2) and (3) are defined in [PERSON] et al. [10] and [PERSON] et al. [11] as follows:  
\[\varepsilon^{\prime}_{fw}=\varepsilon_{w00}+\frac{\varepsilon_{w0}- \varepsilon_{w00}}{1+\left(2\pi f\tau_{w}\right)} \tag{7}\]  
\[\varepsilon^{\prime\prime}_{fw}=\frac{2\pi f\tau_{w}(\varepsilon_{w0}- \varepsilon_{w00})}{1+\left(2\pi f\tau_{w}\right)}+\frac{\sigma_{eff}}{2\pi \varepsilon_{0}f}\frac{(\rho_{s}-\rho_{b})}{\rho_{s}m_{v}} \tag{8}\]  
where \(\varepsilon_{w00}\) = 4.9 is the high frequency limit of \(\varepsilon^{\prime}_{fw}\), \(\varepsilon_{w0}\) is the static dielectric constant of water,\(f\) is the frequency (Hz), \(\tau_{w}\) is the relaxation time for water, \(\varepsilon_{0}\) = 8.854 \(\times\) 10\({}^{-12}\) F/m is the permittivity of free space, and \(\sigma_{eff}\) is the effective conductivity. In Equations (7) and (8), the expressions for the static dielectric constant of water, \(\varepsilon_{w0}\), and the relaxation time for water, \(\tau_{w}\), are given as a function of the water temperature (\(T_{w}\) in \({}^{\circ}\)C) [12,13]:  
\[\varepsilon_{w0}=\)88.045-0.4147 \(T_{w}+6.295.10^{-4}\ T_{w}{}^{2}+1.075\times 10^{-5}\ T_{w}{}^{3} \tag{9}\]  
\[2\pi\tau_{w}=\left(1.1109.10^{-10}-3.824.10^{-12}\ T_{w}+6.938.10^{-14}\ T_{w}{}^{2}-5.096.10^{-16}\ T_{w}{}^{3}\right) \tag{10}\]  
For \(T_{w}\) = 20 \({}^{\circ}\)C, \(\varepsilon_{w0}\) = 80.1 and \(2\pi\tau_{w}\) = 0.58 \(\times\) 10\({}^{-10}\) s.  
The effective conductivity, \(\sigma_{eff}\), defined in Equation (8), is given by [PERSON] et al. [11]:  
\[\varepsilon_{m}^{\prime}=\left[1+\frac{\rho_{b}}{\rho_{s}}(\varepsilon_{s}^{a }-1)+m_{p}^{g^{\prime}}\varepsilon_{fw}^{\prime a}-m_{v}\right]^{1/\alpha} \tag{2}\]0  
An extension of this semi-empirical dielectric mixing model was proposed by [PERSON] et al. [14] to include freezing conditions.