# C-SAR/02 Satellite Polarimetric Calibration and Validation Based on Active Radar Calibrators
## 3 Experimental Procedures and Methods
### Freeman's Polarization Calibration Method  
In 1990, [PERSON] et al. [30] proposed an algorithm for polarization calibration using ARC, which was applied to the airborne polarimetric SAR system NASA/IPL DC-8 Aircraft SAR. This algorithm utilizes standard polarized ARCs with three known scattering matrices for polarization calibration. Additionally, it employs artificially deployed dihedral and trihedral CRs in the scene for accuracy assessment. The polarization calibration model is as follows:  
\[ \begin{split}& M=Ae^{i\varphi}R^{\dagger}ST+N\\ &=Ae^{i\varphi}\left(\begin{array}{cc}R_{hh}&R_{hv}\\ R_{vth}&R_{vv}\end{array}\right)\left(\begin{array}{cc}S_{hh}&S_{vth}\\ S_{hv}&S_{vv}\end{array}\right)\left(\begin{array}{cc}T_{hh}&T_{vth}\\ T_{hv}&T_{vv}\end{array}\right)+\left(\begin{array}{cc}N_{hh}&N_{vth}\\ N_{hv}&N_{vv}\end{array}\right)\\ &=\widetilde{A}e^{i\varphi}\left(\begin{array}{cc}1&\delta_{2}\\ \delta_{1}&f_{1}\end{array}\right)\left(\begin{array}{cc}S_{hh}&S_{vth}\\ S_{hv}&S_{vv}\end{array}\right)\left(\begin{array}{cc}1&\delta_{3}\\ \delta_{4}&f_{2}\end{array}\right)+\left(\begin{array}{cc}N_{hh}&N_{vth}\\ N_{hv}&N_{hh}\end{array}\right)\end{split}  \tag{1}\]  
where \(A\) represents the gains and losses of the radar system (expressed in magnitude); \(\varphi\) denotes the phase shift caused by round-trip delay between the target and radar as well as losses within the system; \(S\) is the target scattering matrix; \(R\) and \(T\) are the radar  
Figure 5: Polarization calibration experimental flowchart. receiving and transmitting matrices, respectively; t denotes the transpose operation; the elements in matrix \(N\) represent the noise in each channel; \(\delta_{1}\) represents the crosstalk signal ratio of the V-polarization channel introduced by the received H-polarization channel signal; \(\delta_{2}\) represents the crosstalk signal ratio of the H-polarization channel introduced by the received V-polarization channel signal; \(\delta_{3}\) represents the crosstalk signal ratio of the V-polarization channel introduced by the emitted H-polarization channel signal; \(\delta_{4}\) represents the crosstalk signal ratio of the H-polarization channel introduced by the emitted V-polarization channel signal; \(f_{1}\) represents the ratio of the amplitude and phase imbalance of V-polarization at the receiving end with respect to H-polarization; and \(f_{2}\) represents the ratio of the amplitude and phase imbalance of H-polarization at the receiving end with respect to V-polarization [25]. Representing the above matrix using the Kronecker delta matrix format gives the following form:  
\[M= \begin{pmatrix}M_{hh}\\ M_{hv}\\ M_{vv}\end{pmatrix} =\widetilde{A}e^{i\varphi} \begin{pmatrix}1&\delta_{2}&\delta _{4}&\delta_{2}\delta_{4}\\ \delta_{1}&f_{1}&\delta_{4}\delta_{1}&\delta_{4}f_{1}\\ \delta_{3}&\delta_{2}\delta_{3}&f_{2}&\delta_{2}f_{2}\\ \delta_{1}\delta_{3}&\delta_{3}f_{1}&\delta_{1}f_{2}&f_{2}f_{1}\\ \end{pmatrix}\begin{pmatrix}S_{hh}\\ S_{hv}\\ S_{vv}\end{pmatrix} +N \tag{2}\]  
The algorithm uses three standard polarized ARCs with known Sinclair matrices to calibrate the image, which is followed by an accuracy evaluation using dihedral or trihedral corner calibrators allocated in the scene. The three actively polarized ARCs are preset as \(S_{a}\), \(S_{b}\), \(S_{c}\) and described by the corresponding Sinclair matrices as  
\[S_{a}= \begin{bmatrix}0&0\\ 1&0\end{bmatrix} ,\;S_{b}= \begin{bmatrix}0&1\\ 0&0\end{bmatrix} ,\;S_{c}= \begin{bmatrix}-1&-1\\ 1&1\end{bmatrix}  \tag{3}\]  
The observed values of the polarization scattering matrix obtained from the three ARCs are expressed as follows:  
\[X=A_{x}e^{i\varphi_{x}}R^{l}S_{x}T,\;Y=A_{y}e^{i\varphi_{y}}R^{l}S_{y}T,\;Z=A_ {z}e^{i\varphi_{z}}R^{l}S_{z}T \tag{4}\]  
The implicit assumption in this formulation is that the distortion matrices \(R\) and \(T\) do not change during the operation of the three devices.