# C-SAR/02 Satellite Polarimetric Calibration and Validation Based on Active Radar Calibrators
## 3 Experimental Procedures and Methods
### SAR Image Processing Procedure
#### 3.3.3 Solving Distortion Matrices \(R\) and \(T\)  
Denote \(R\) by \(R_{\upsilon\upsilon}\) and \(T\) by \(T_{hh}\):  
\[R=R_{\upsilon\upsilon}\begin{bmatrix}Q_{1}&Q_{1}\cdot Y_{\upsilon\upsilon}/Y _{\upsilon h}\\ X_{hh}/X_{h\upsilon}&1\end{bmatrix}=R_{\upsilon\upsilon}\widetilde{R}_{1} \tag{10}\]  
where  
\[Q_{1}=\frac{Z_{hh}/Z_{h\upsilon}-X_{hh}/X_{h\upsilon}}{Z_{hh}/Z_{h\upsilon} \cdot(Y_{\upsilon\upsilon}/Y_{\upsilon h})-1}\;or\;\frac{Z_{\upsilon h}/Z_{ \upsilon\upsilon}-X_{hh}/X_{h\upsilon}}{(Z_{\upsilon h}/Z_{\upsilon\upsilon} )\cdot(Y_{\upsilon\upsilon}/Y_{\upsilon h})-1} \tag{11}\]  
\[T=T_{hh}\begin{bmatrix}1&X_{\upsilon\upsilon}/X_{h\upsilon}\\ Q_{2}\cdot Y_{hh}/Y_{\upsilon h}&Q_{2}\end{bmatrix}=T_{hh}\widetilde{T}_{1} \tag{12}\]  
Figure 7: Schematic diagram of the phase angle.  
where  
\[Q_{2}=\frac{Z_{vv}/Z_{htv}-X_{vv}/X_{hv}}{1-Z_{vv}/Z_{hv}\cdot(Y_{hth}/Y_{vh})}\;or \;\frac{Z_{vh}/Z_{hh}-X_{vv}/X_{hv}}{1-(Z_{vh}/Z_{hh})\cdot(Y_{ht}/Y_{vh})} \tag{13}\]  
From the above equation, the six unknown (complex) variables of the radar system can be determined, namely,  
\[\delta_{1}=\frac{Y_{vv}}{Y_{vh}},\;\delta_{2}=f_{1}\frac{X_{hh}}{X_{hv}},\; \delta_{3}=\frac{X_{vv}}{X_{hv}},\;\delta_{4}=f_{2}\frac{Y_{hh}}{Y_{vh}},\;f_{1 }=\frac{1}{Q_{1}},\;f_{2}=Q_{2} \tag{14}\]  
Denote \(R\) by \(R_{hh}\) and \(T\) by \(T_{vv}\):  
\[R=R_{hh}\begin{bmatrix}1&Y_{vv}/Y_{vh}\\ Q_{3}\cdot X_{hh}/X_{hv}&Q_{3}\end{bmatrix}=R_{hh}\tilde{R}_{2} \tag{15}\]  
where  
\[Q_{3}=\frac{Z_{hv}/Z_{hh}-Y_{vv}/Y_{vh}}{Z_{htv}/Z_{hh}\cdot X_{hh}/X_{hv}-1} \;or\;\frac{Z_{vv}/Z_{vh}-Y_{vv}/Y_{vh}}{(Z_{vv}/Z_{vh})\cdot X_{hh}/X_{hv}-1} \tag{16}\]  
\[T=T_{vv}\begin{bmatrix}Q_{4}&Q_{4}\cdot X_{vv}/X_{hv}\\ Y_{vh}/Y_{vh}&1\end{bmatrix}=T_{vv}\tilde{T}_{2} \tag{17}\]  
where  
\[Q_{4}=\frac{Z_{hh}/Z_{vh}-Y_{hh}/Y_{vh}}{1-(Z_{hh}/Z_{vh})\cdot X_{vv}/X_{hv} }\;or\;\frac{Z_{hv}/Z_{vv}-Y_{hh}/Y_{vh}}{1-Z_{hv}/Z_{vv}\cdot X_{vv}/X_{hv}} \tag{18}\]  
where  
\[\delta_{1}=\frac{Y_{vv}}{Y_{vh}},\;\delta_{2}=f_{1}\frac{X_{hh}}{X_{hv}},\; \delta_{3}=\frac{X_{vv}}{X_{hv}},\;\delta_{4}=f_{2}\frac{Y_{hh}}{Y_{vh}},\;f _{1}=Q_{3},\;f_{2}=\frac{1}{Q_{4}} \tag{19}\]  
The distortion matrices in Equations (10), (12), (14), and (16) are in normalized form, with one of the elements serving as the normalization factor. The parameters were all computed as ratios between the elements of the quad-polarized matrices of the same calibrator, so that the complex factors before the matrix multiplication of Equation (4) were eliminated. In the absence of noise, these solutions should be identical. However, for noise, each \(Q\) was obtained using different elements from the distortion matrix \(Z\), providing a second estimate for each distortion matrix, balancing the impact of different computational factors on the results by taking the average. In addition, the factor in the denominator is a large weighting factor in the quad-polarization measurements and therefore has minimal sensitivity to noise, and so the effect of noise on the computed results was ignored [50].