# Near Real-Time Freeze Detection over Agricultural Plots Using Sentinel-1 Data
## 3 Materials and Methods
### Threshold Analysis over \(\sigma^{0}\)  
In order to detect surface states over a given plot p at date '\(d_{i}\)', we need to calculate a threshold over \(\Delta\sigma^{0}\left(\mathrm{d_{i}},\mathrm{p}\right)\equiv\left(\mathrm{ ref\_o^{0}-\sigma^{0}\left(\mathrm{d_{i}},\mathrm{p}\right)}\right)\) which allows the plot surface to be flagged as being unfrozen, mild-to-moderately frozen, or severely frozen. Since the behavior of the backscatter varies according to the characteristics of the soil and vegetation, a threshold was determined for each land cover class. In this study, crop types are classified into three main land cover classes: cereal grains (LC1), meadows (LC2), and orchards and vineyards (LC3). The thresholds were determined using an approach based on the use of an additional S1 SAR time-series from 1 September 2017, till 31 May 2018. First, the approach consists of calculating the difference \(\Delta\sigma^{0}\left(\mathrm{d_{i}},\mathrm{p},\mathrm{t}\right)\) for each plot \(p\) of our three land cover classes, at date \(d_{i}\) when the temperatures t are below 0 \({}^{\circ}\)C (temperatures with freezing potential). The temperatures t corresponding to the S1 acquisition dates are acquired from the nearest weather stations. Thus, for each of our three land cover classes (LC) we have a set (S\({}_{\mathrm{LC}}\)) of \(\Delta\sigma^{0}\) containing N plots and M dates (when t \(<\)0 \({}^{\circ}\)C). Then, from each set S\({}_{\mathrm{LC}}\) we calculate the distribution of \(\Delta\sigma^{0}\) at each S1 polarization (VH and VV) separately for two temperature ranges, t \(\in\) [-3 \({}^{\circ}\)C, 0 \({}^{\circ}\)C] and t \(<\) -3 \({}^{\circ}\)C. The obtained distributions of each LC at each polarization will be fitted with a normal distribution of mean \(\mu\left(\mathrm{LC_{k}},\mathrm{pol_{p}}\right)\) and standard deviation \(\sigma\left(\mathrm{LC_{k}},\mathrm{pol_{p}}\right)\), where k is the land cover class (between 1, and 3), and p the polarization, with \(\mathrm{pol_{1}}\) designating VH polarization and \(\mathrm{pol_{2}}\) the VV polarization. In [PERSON] et al. [20], it was shown by simulation that the difference between the backscattering coefficient of unfrozen soil and the backscattering coefficient of frozen soil with soil temperature (t) lower than 0 \({}^{\circ}\)C abruptly increases for t between 0 and \(-\)3 \({}^{\circ}\)C. The radar signal decreased from roughly 3.3 dB for silty clay soil to \(\sim\)5 dB for sandy loam soil when the temperature decreased from 0 to \(-\)3 \({}^{\circ}\)C. For t between \(-\)3 and \(-\)20 \({}^{\circ}\)C, the radar signal slows its decline to less than 1 dB. Therefore, in this study, a temperature of \(-\)3 \({}^{\circ}\)C is thus considered as the temperatureat which it is highly probable to have severe freezing. The mean \(\mu_{1}(\mathrm{LC_{k}},\mathrm{pol_{p}})\) obtained from the distribution calculated from the range of temperatures lower than \(-3\)\({}^{\circ}\)C, determines the threshold over which the surface is considered as being severely frozen. Thus, if \(\Delta\sigma^{0}\geq\mu_{1}(\mathrm{LC_{k}},\mathrm{pol_{p}})\), the surface is assumed to be severely frozen (50% of the area of the normal function). The part where the normal function that corresponds to \(\Delta\sigma^{0}\)\(<\)\(\mu_{1}(\mathrm{LC_{k}},\mathrm{pol_{p}})\) designates surfaces that are either unfrozen or having a mild-to-moderate freezing episode (e.g., plots sheltered from the cold). The mean value \(\mu_{2}(\mathrm{LC_{k}},\mathrm{pol_{p}})\), obtained from the distribution using the temperature ranges [\(-3\)\({}^{\circ}\)C, \(0\)\({}^{\circ}\)C], determines the threshold below which the surface is supposed to be unfrozen (i.e., \(\Delta\sigma^{0}\)\(<\)\(\mu_{2}(\mathrm{LC_{k}},\mathrm{pol_{p}})\)). Finally, the zone that corresponds to \(\mu_{2}(\mathrm{LC_{k}},\mathrm{pol_{p}})\)\(\leq\)\(\Delta\sigma^{0}\)\(<\)\(\mu_{1}(\mathrm{LC_{k}},\mathrm{pol_{p}})\) is considered as having a mild-to-moderate freezing episode.