## Materials and Methods
### PolSAR Observables  
Different from single and dual polarization SAR systems that can only obtain one or two linear channel images over the target site, a fully polarimetric SAR system can acquire four images by alternately transmitting two radar signals with an orthogonal polarization basis (e.g., H or V), and receiving backscattered signals simultaneously (H and V). The acquired data can be represented, for instance in an H-V polarization basis, by a \(2~{}\times~{}2\) complex Sinclair scattering matrix \(S\) as [28; 29]  
\[S=\left[ \begin{array}{cc}S_{HH}&S_{HV}\\ S_{VH}&S_{VV}\end{array} \right] \tag{1}\]  
where \(S_{HH}\) and \(S_{VV}\) represent the complex backscattering coefficients as the transmitted polarization and received polarization are the same (i.e., the copolarized channels), while \(S_{HV}\) and \(S_{VH}\) denote the complex backscattering coefficients in the cross-polarized channels. Under the assumption of reciprocal scattering, \(S_{HV}\) and \(S_{VH}\) are considered equal [28; 29]. Since the scattering matrix is commonly affected by speckle noise, and distributed targets are common in natural media, incoherent polarimetric analysis by using second-order statistics, such as the covariance matrix or the coherency matrix, are given more attention [28; 29]. The corresponding covariance matrix \(C\) and the coherency matrix \(T\) are represented as [28; 29]:  
\[C=\left[ \begin{array}{cc}\left\langle\left|S_{HH}\right|^{2} \right\rangle&\sqrt{2}\langle S_{HH}S_{HV}^{*}\rangle&\left\langle S_{HH}S_{ VV}^{*}\right\rangle\\ \sqrt{2}\langle S_{HV}S_{HH}^{*}\rangle&2\left\langle\left|S_{HV}\right|^{2} \right\rangle&\sqrt{2}\langle S_{HV}S_{VV}^{*}\rangle\\ \langle S_{VV}S_{HH}^{*}\rangle&\sqrt{2}\langle S_{VV}S_{HV}^{*}\rangle& \left\langle\left|S_{VV}\right|^{2}\right\rangle\end{array} \right] \tag{2}\]  
\[T=\frac{1}{2}\left[ \begin{array}{cc}\left\langle\left|S_{HH}+S_{VV}\right|^ {2}\right\rangle&\left\langle\left(S_{HH}+S_{VV}\right)\left(S_{HH}-S_{VV} \right)^{*}\right\rangle&2\langle\left(S_{HH}+S_{VV}\right)S_{HV}^{*}\right\rangle \\ \left\langle\left(S_{HH}-S_{VV}\right)\left([PERSON]_{HH}+[PERSON]_{VV}\right)^{*}\right\rangle &\left\langle\left|[PERSON]_{HH}-[PERSON]_{VV}\right|^{2}\right\rangle&2\langle\left([PERSON]_{ HH}-[PERSON]_{VV}\right)S_{HV}^{*}\rangle\\ 2\langle S_{HV}\left([PERSON]_{HH}+[PERSON]_{VV}\right)^{*}\rangle&2\langle S_{HV}\left([PERSON]_{ HH}-[PERSON]_{VV}\right)^{*}\rangle&4\langle\left|[PERSON]_{HV}\right|^{2}\rangle\end{array} \right] \tag{3}\]  
In past studies, a large number of polarimetric observables were extracted from these polarimetric matrices (i.e., C or T in Equations (2) and (3)) for crop monitoring applications [24; 42; 43]. For instance, a global sensitivity analysis of 20 PolSAR observables for crop biophysical variable estimation were investigated in [43]. However, the number and types of selected PolSAR observables for investigating temporal evolutions and crop classification were usually limited. Besides those 20 polarimetric observables in sensitivity studies, according to wide usage of PolSAR observables in crop monitoring studies, a total of 27 polarimetric observables were selected for temporal evolution inspection and considered as input features for crop classification in this study, as listed in Table 2. For the convenience of subsequent analysis, the 27 polarimetric observ \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Land Cover**} & \multicolumn{2}{c}{**Training Samples**} & \multicolumn{2}{c}{**Testing Samples**} \\ \cline{2-5}  & **Number of** & **Number of** & **Number of** & **Number of** & **Number of** \\  & **Pixels** & **Fields** & **Pixels** & **Fields** \\ \hline Corn & 6258 & 4 & 20,246 & 16 \\ Soybean & 6505 & 4 & 15,995 & 12 \\ Forage & 3700 & 5 & 3615 & 7 \\ Winter wheat & 6018 & 3 & 17,723 & 16 \\ Watermelon & 310 & 1 & 309 & 1 \\ Tobacco & 416 & 1 & 301 & 1 \\ Forest & 5148 & 4 & 7292 & 6 \\ Built-up & 1267 & 1 & 1117 & 1 \\ Soil & 2331 & 1 & 1592 & 1 \\ \hline \hline \end{tabular}
\end{table}