# Assessing the Added Value of Sentinel-1 PolSAR Data for Crop Classification
## 2 Materials
### Satellite Data
#### 2.3.2 Polarimetric Data Representation  
Fully polarimetric SAR data, captured in different polarization channels (HH, HV, VV, VH), are often combined into a 2 \(\times\) 2 complex scattering matrix for each image cell/pixel (Equation (1)), which is considered an adequate descriptor of point/pure single scatterers [59]. \[[S]= \begin{bmatrix}S_{HH}&S_{HV}\\ \hat{S}_{VH}&S_{VV}\end{bmatrix}  \tag{1}\]  
The scattering matrix for dual-polarized SAR systems (e.g., Sentinel-1) has lower dimensionality and contains half of the polarimetric information involved in quad-polarised data [98]. Specifically, for the Sentinel-1 case, backscatter measurements collected in VV and VH polarization channels, is often incorporated into a scattering vector \(\vec{k}=[S_{VV},S_{VH}]^{T}\). By spatially averaging this scattering vector, 2 \(\times\) 2 multi-looked covariance matrix \(\mathcal{C}_{2}\) is generated at pixel level, as expressed in Equation (2):  
\[\langle\mathcal{C}_{2}\rangle=\langle\vec{k}\cdot\vec{k}^{*T}\rangle= \begin{bmatrix} C_{11}&C_{12}\\ C_{21}&C_{22}\end{bmatrix} = \begin{bmatrix}\langle\mid S_{VV}\mid^{2}\rangle& \langle S_{VV}\hat{S}_{VH}^{*}\rangle\\ \langle S_{VH}S_{VV}^{*}\rangle&\langle\mid S_{VH}\mid^{2}\rangle\end{bmatrix}  \tag{2}\]  
where \(\langle...\rangle\) denotes multi-looking processing, * complex conjugation, and \(T\) transposition. The diagonal elements of the covariance matrix (\(C_{11}\), \(C_{22}\)) represent the backscattering coefficients in VV, VH polarization channels [11], respectively, while \(C_{12}\), \(C_{21}\) elements express the complex correlation of back-scattered information between VV, VH polarization channels [67]. Deriving the covariance matrix from the vectorized scattering matrix is essential for the accurate characterization of distributed targets, such as vegetation [99]. For the purposes of this study, multi-temporal data for 23 polarimetric parameters (Table 1) were derived from the dual-pol Sentinel-1 PolSAR data by employing the H/A/\(\overline{\pi}\) polarimetric decomposition technique in order to obtain information about the scattering mechanisms of each crop type. By applying this method, covariance matrix data of each SAR image was subjected to an eigen-decomposition process. Each one of the two eigenvectors derived from a 2 \(\times\) 2 covariance matrix is related to one of the two scattering mechanisms occurring at the particular image cell, while the corresponding eigenvalues quantify the relative importance of each scattering mechanism to the total measured backscattered signal [33]. Specifically, through the process of eigen-decomposition, covariance matrix is expressed as a weighted sum of two matrices, \([C_{2}]_{1}\) and \([C_{2}]_{2}\), representing the two different scattering mechanisms occurring at pixel level:  
\[\langle[C_{2}]\rangle=[U] \begin{bmatrix}\lambda_{1}&0\\ 0&\lambda_{2}\end{bmatrix} [U]^{T*}=\sum_{i=1}^{2}\lambda_{i}\vec{u}_{i}\vec{u }_{i}^{T*}=\lambda_{1}\vec{u}_{1}\vec{u}_{1}^{T*}+\lambda_{2}\vec{u}_{2}\vec {u}_{2}^{T*}=\lambda_{1}[C_{2}]_{1}+\lambda_{2}[C_{2}]_{2} \tag{3}\]  
\[[U]= \begin{bmatrix}U_{11}&U_{12}\\ U_{21}&U_{22}\end{bmatrix} = \begin{bmatrix}\vec{u}_{1}\vec{u}_{2}\end{bmatrix}  \tag{4}\]  
where \([U]\) is the orthogonal unitary matrix, \(\vec{u}_{i}\) (\(i=1,2\)) is the eigenvector i and \(\lambda_{i}\) (\(i=1,2\)) is the corresponding eigenvalue \(i\). Additionally, each \(C_{2}\) eigenvector \(\vec{u}_{i}\) can be parameterized, using two angular variables, \(\alpha_{i}\) and \(\delta_{i}\), denoting the scattering mechanism and the phase angle, respectively [61] (Equation (5)). \[\vec{u}_{i}=[\cos{a_{i}},\sin{a_{i}}e^{i\delta_{i}}]^{T},\quad i=1,2 \tag{5}\]  
Using the calculated eigenvectors and eigenvalue we are able to compute polarimetric parameters.