## 2 Methods
### Estimation of the FR Angle from Full-Polarimetric SAR Data  
For a linear polarization SAR system, the measured scattering matrix M can be written as [37]:  
\[M=Ae^{i\Phi}R^{T}R_{F}SR_{F}T+N \tag{1}\] \[R^{T}=\left[\begin{array}{cc}1&\delta_{1}\\ \delta_{2}&f_{1}\end{array}\right]\;T=\left[\begin{array}{cc}1&\delta_{3}\\ \delta_{4}&f_{2}\end{array}\right]\;N=\left[\begin{array}{cc}N_{hh}&N_{eh}\\ N_{hv}&N_{vv}\end{array}\right]\]  
where \(R_{F}\) is a one-way FR matrix; \(\Omega\) is the FR angle; \(S\) is the true scattering matrix; \(R\) and \(T\) are the receiving and transmitting distortion matrices, respectively;\(f_{1}\) and \(f_{2}\) are the channel imbalance (the complex ratio of signals Vertical(V)/Horizontal(H) on receive and transmit, respectively); \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), \(\delta_{4}\) are the complex antenna crosstalk terms resulting from the incomplete isolation of H and V polarizations on transmit and receive; \(A\) is the overall gain of radar system, which is a function of radar range and elevation angle; \(e^{i\Phi}\) corresponds to the round-trip phase delay and system-dependent phase effects on signal; and \(N\) is an additive noise term due to earth radiation, thermal fluctuations in the receiver and digitization noise [37]. A calibration technique applied to SAR data can estimate and correct the system-dependent terms of \(A\), \(e^{i\Phi}\), \(R\) and \(T\) in (1) [13]. Furthermore, filtering methods, such as boxcar, band-pass and low-pass filters, can be used to suppress the additive noise \(N\). After these calibrations, the measured scattering matrix M can be reformed as given by:  
\[M=R_{F}SR_{F}\]  
Figure 1: The flowchart of the proposed method. Synthetic aperture radar (SAR), SAR interferometry (InSAR), Faraday rotation (FR), vertical total electronic content (VTEC).  
After expanding (2) and invoking true backscatter reciprocity (\(S_{lm}=S_{vth}\)), the components of matrix M can be defined as:  
\[\begin{array}{l}M_{hh}=S_{hlt}cos^{2}\Omega-S_{vv}sin^{2}\Omega,\\ M_{vth}=S_{lhv}+(S_{hh}+S_{vv})sinTcos\Omega,\\ M_{hv}=S_{hv}-(S_{hh}+S_{vv})sinTcos\Omega,\\ M_{vv}=S_{vv}cos^{2}\Omega-S_{hlt}sin^{2}\Omega.\end{array} \tag{3}\]  
For cross-polarization, a nonzero FR angle means that the measured scattering matrix will not be invertible (\(M_{vth}\
eq M_{hv}\)). Suppose that FR angle is the only error source and full-polarization SAR data is available, then the FR angle can be easily estimated from (3). A robust algorithm has been proposed to estimate the FR angle through circular polarization scattering matrix Z [9], as given by:  
\[Z=\left[\begin{array}{cc}Z_{RR}&Z_{RL}\\ Z_{LR}&Z_{LL}\end{array}\right]=\left[\begin{array}{cc}1&j\\ j&1\end{array}\right]\times\left[\begin{array}{cc}M_{hh}&M_{hv}\\ M_{vh}&M_{vv}\end{array}\right]\times\left[\begin{array}{cc}1&j\\ j&1\end{array}\right] \tag{4}\]  
\[Z_{LR}=M_{vth}-M_{hv}+j\times(M_{hh}+M_{vv}) \tag{5}\]  
\[Z_{RL}=M_{hv}-M_{vth}+j\times(M_{hh}+M_{vv}) \tag{6}\]  
From (3) and (4), FR angle \(\Omega\) can be calculated by [9]:  
\[\Omega=-\frac{1}{4}\times arg(Z_{RL}\times Z_{LR}^{*}),\text{ for }-\frac{\pi}{4}< \Omega<\frac{\pi}{4}. \tag{7}\]  
where arg denotes the argument of a complex number and * represents the complex conjugate. From (7), it can be seen that estimation of the FR angle becomes a phase estimation problem, which is a fairly well-understood problem in radar polarization [46]. Further, a reduction in noise is required in order to increase the quality of FR angle estimation. Some filtering methods can be used for the reduction.