# Assessing the Added Value of Sentinel-1 PolSAR Data for Crop Classification
## 2 Materials
### Satellite Data
#### 2.3.2 Polarimetric Data Representation
The main polarimetric parameters derived by Cloude-Pottier decomposition are Entropy \(H\), Anisotropy \(A\) and mean alpha angle \(\overline{\pi}\) and they are determined by the following equations:  
\[H=\sum_{i=1}^{2}p_{i}\log_{2}p_{i} \tag{6}\]\[A=\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}} \tag{7}\]  
\[\overline{\alpha}=\sum_{i=1}^{2}p_{i}\alpha_{i}=p_{1}\alpha_{1}+p_{2}\alpha_{2} \tag{8}\]  
where pseudo-probabilities \(p_{i}\) represent the relative importance of \(C_{2}\) eigenvalues \(\lambda_{i}\) (Equation (9)), allowing for a probabilistic interpretation of the scattering phenomenon. \[p_{i}=\frac{\lambda_{i}}{\sum_{i=1}^{2}\lambda_{i}},\quad i=1,2 \tag{9}\]  
According to this view, mean delta angle \(\overline{\delta}\) is mathematically expressed in a similar way to mean alpha angle \(\overline{\alpha}\), as expressed in Equation (10). Furthermore, Shannon Entropy (SE) can be obtained by the covariance matrix (Equation (11)), expressing the degree of randomness of the scattering process. \[\overline{\delta}=\sum_{i=1}^{2}p_{i}\delta_{i} \tag{10}\]  
\[SE=\log{(\pi^{2}e^{2}\mid C^{2}\mid)}=SE_{I}+SE_{P} \tag{11}\]  
This polarimetric parameter can also be expressed as a sum of two measures \(SE_{i}\) and \(SE_{P}\), quantifying the degree of randomness related with changes in the back-scattered signal intensity or phase, respectively [63] (Equations (12) and (13)). \[SE_{i}=2\log{(\frac{\pi eTr[C_{2}]}{2})} \tag{12}\]  
\[SE_{p}=\log{(4\frac{det[C_{2}]}{Tr[C_{2}]^{2}})} \tag{13}\]  
where \(Tr\) and det represent the trace and determinant of the matrix, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Parameter Notation** \\ \hline mean scattering alpha angle & alpha \\ first scattering alpha angle & alpha1 \\ second scattering alpha angle & alpha2 \\ Anisotropy & _anisotropy_ \\ H-A combination 1 \([H\cdot A]\) & combination\_HA \\ H-A combination 2 \([H\cdot(1-A)]\) & combination\_H1 mA \\ H-A combination 3 \([(1-H)\cdot A]\) & combination\_1 mHA \\ H-A combination 4 \([(1-H)\cdot(1-A)]\) & combination\_1 mH1 mA \\ mean scattering delta angle & delta \\ first scattering delta angle & delta2 \\ second scattering delta angle & entropy \\ Shannon entropy & entropy\_shannon \\ Shannon entropy intensity & entropy\_shannon\_I \\ Shannon entropy intensity normalized & entropy\_shannon\_P \\ Shannon entropy polarization & entropy\_shannon\_P\_norm \\ first eigenvalue & 11 \\ second eigenvalue & 12 \\ mean eigenvalue & lambda \\ probability 1 & p1 \\ probability 2 & p2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of H/A/\(\overline{\pi}\) polarimetric parameters. Sentinel-1 multi-temporal images underwent a series of pre-processing steps, in order to obtain time-series data for each one of the 23 polarimetric parameters. In more detail, we followed the pre-processing workflow recommended by [PERSON] et al. (2019) [100], using ESA's Sentinel Application Platform (SNAP) v.7.0.0 and PolSARpro v.6.0 tools [101], which is schematically presented in Figure 3. The first step of this pre-processing chain was the splitting step. As our study site was situated within two successive sub-swatths of the Sentinel-1 images, these sub-swatths were separated and processed individually, using Sentinel-1 TOPS Split operator. Splitting operation was followed by performing orbit correction at each sub-swath, using an Apply-Orbit operator. This pre-processing step allows for accurate image geocoding, making use of the precise orbit files contained in each product's metadata. After applying orbit files, each sub-swath was radiometrically calibrated and saved in complex-valued format, required for the computation of C\({}_{2}\) matrix. SAR calibration allows for directly relating image's pixel values to the backscattering signals of the scene [102].