# First Vegetation Optical Depth Mapping from Sentinel-1 C-band SAR Data over Crop Fields
## 3 Methodology  
In this study, VOD was computed for two crop growth cycles that occurred between September 2017 and September 2019. VOD computation was based on the use of C-band Sentinel-1 time series and the Water Cloud Model (WCM) ([PERSON] et al., 2019). In the WCM, the total backscattered signal (\(\sigma^{0}_{\rm tot}\)) is the sum of vegetation contribution (\(\sigma^{0}_{\rm veeg}\)) and soil contribution (\(\sigma^{0}_{\rm soil}\)) attenuated by the vegetation canopy (Equations (1) and (2)). In addition to sensor parameters (SAR wavelength, incidence angle and polarization), soil contribution is a function of soil moisture and surface roughness. WCM does not  
Figure 2: Precipitation (mm) and mean temperature (\({}^{\circ}\)C) registered from a weather station in the study zone. consider high order scattering mechanisms like double scattering from ground-stem interactions. The total backscatter, in the power domain, is expressed as follows:  
\[\sigma_{\text{tot}}^{0}=\sigma_{\text{veg}}^{0}+T^{2}\sigma_{\text{soil}}^{0} \tag{1}\]  
\[T^{2}=\text{e}^{-2\frac{\text{VOD}}{\text{VOD}}} \tag{2}\]  
where, \(\theta\) is the SAR incidence angle (in degrees), and \(T^{2}\) is the two-way attenuation. Using two SAR images acquired at two different dates (t\({}_{1}\) and t\({}_{2}\)) and by supposing that the vegetation characteristics are constant between these two dates (i.e., \(\Delta\sigma_{\text{veg,t2-t1}}^{0}=\sigma_{\text{veg,t2}}^{0}-\sigma_{\text{ veg,t1}}^{0}=0\)), Equations (1) and (2) can be solved to estimate VOD as:  
\[\text{VOD}=-\frac{\cos\theta}{2}\ln\left(\frac{\Delta\sigma_{\text{tot},\text {t2-t1}}^{0}}{\Delta\sigma_{\text{soil},\text{t2-t1}}^{0}}\right) \tag{3}\]  
For a given plot, VOD can be analytically computed from Equation (3) if \(\sigma_{\text{tot}}^{0}\) and \(\sigma_{\text{soil}}^{0}\) are known from the SAR images on dates t\({}_{1}\) and t\({}_{2}\). However, to compute \(\sigma_{\text{soil}}^{0}\) from radar backscattering models (e.g., [23; 24; 25]), estimates of soil moisture and surface roughness values are needed. These values are often unavailable or available only for a few plots within the study area. It is worth mentioning that VOD estimation from WCM (Equation (3)) may be inaccurate in the case of important double-scattering mechanisms as this mechanism is not considered in Equation (1) as mentioned above. To estimate VOD over vegetated plots, \(\sigma_{\text{tot}}^{0}\) is computed for a given plot as the average of the SAR pixel values within that plot. Only plots with NDVI values higher than 0.3 are considered for VOD computation in order to ensure that VOD will be computed only over plots covered by vegetation. To compute soil contribution (\(\sigma_{\text{soil}}^{0}\)), we assume that the underlying soil of a given vegetated plot has the same properties (soil moisture and roughness) as the soil of the bare agricultural plots situated in a grid of 5 km \(\times\) 5 km centered at the considered plot. Bare agricultural soils within SAR images can be spotted using NDVI images calculated from S2 acquisitions. It is assumed that bare soils correspond to NDVI values lower than 0.3 [26; 27]. Thus, \(\sigma_{\text{soil}}^{0}\) for a given plot at a given date corresponds to \(\sigma_{\text{soil}}^{0}\) averaged over all S1 bare pixels located in the 5 km \(\times\) 5 km grid at the same date. This assumption is valid only for non-irrigated plots where the increase and decrease in soil moisture is mainly related to rainfall and temperature conditions which are assumed to be homogeneous in a grid of 5 km \(\times\) 5 km. For this reason, VOD was computed only for non-irrigated crops (winter crops) in the present study. VOD values were computed separately from SAR data time series in ascending and descending modes. For each acquisition mode, one VOD value is computed for each set of four consecutive SAR images.