Performance of an Effective SAR Polarimetric Calibration Method Using Polarimetric Active Radar Calibrators: Numerical Simulations and LT-1 Experiments

Abstract

This paper presents a new approach to polarimetric calibration, extending classical PARC-based methods by exploring new matrix combinations that broaden the applicability of the existing techniques. By investigating alternative matrix configurations, the proposed method not only enhances the flexibility of conventional calibration approaches but also identifies matrix combinations that offer superior performance advantages. The influence of the SNR and scattering matrix error of PARC on the proposed method is evaluated by numerical simulations. The results demonstrate that the proposed method is highly accurate for PARCs with an SNR greater than 34 dB and with single-channel scattering matrix deviations less than −40 dB and four-channel scattering matrix deviations less than 0.5 dB. The effectiveness and precision of the method were validated through calibration experiments conducted on the L-band polarimetric synthetic-aperture radar aboard the LT-1 satellite. The experimental results demonstrate that the amplitude and phase estimation errors of channel unbalance are less than 0.6 dB and 4.5°, respectively, and that the crosstalk estimation error is less than −33 dB. Furthermore, the effectiveness of the method is validated through trihedral corner reflector correlation experiments and the synthesis of pseudo-color images via Pauli decomposition. The theoretical polarization characteristics of the reference target exhibited a high degree of agreement with the calibrated polarization characteristics.

1. Introduction

The polarimetric synthetic-aperture radar (PolSAR) system is capable of transmitting and receiving horizontally and vertically polarized pulse signals, thereby enabling the acquisition of polarization information pertaining to ground objects [1]. In comparison to conventional SAR systems, PolSAR is capable of reflecting the scattering characteristics of ground objects in a more comprehensive manner, with a wide range of potential applications in environmental monitoring, terrain exploration, and target recognition [2]. Nevertheless, the performance discrepancy between polarization channels and the mutual leakage of energy result in polarization distortion [3]. Polarization distortion results in an imbalance in the polarization scattering matrix, rendering it unable to accurately represent the physical characteristics and backscattering characteristics of ground objects [4]. Consequently, polarization data are affected, potentially limiting their processing and application [5]. Consequently, the PolSAR data must initially be accurately polarization-calibrated in order to eliminate the influence of polarization distortion, so as to accurately extract the ground object information contained in the radar data and invert the relevant parameters of the ground object [6,7].

A variety of polarimetric calibration methods have been proposed. Currently, the predominant research on polarimetric calibration encompasses point-target polarimetric calibration schemes [7,8,9,10,11] and distributed-target calibration schemes [12,13,14,15,16,17,18].

In the domain of point-target calibration techniques, Whitt et al. [7] developed a method utilizing three calibration targets without the distortion matrix assumption and solved it using the eigenvalue method. Freeman et al. [8] developed a technique that involves the use of three polarimetric active radar calibrators in an imaging scenario to derive the target scattering matrix. Fujita et al. [9] used selective angular reflectors to calibrate the polarization of spaceborne and airborne SAR systems. Sarabandi et al. [10] improved this technique by using a conducting sphere and a depolarizing calibration target to determine the distortion parameters. Yueh et al. [11] provided a summary of the conditions for calibrating the scattering matrix of the target. Due to the limitation of the spatial location of the point target, the polarimetric calibration parameters obtained by the point target calibration algorithm are generally only valid for the azimuth position of the point target. In order to ensure the stability of the whole range direction calibration, a certain number of devices should be deployed, which increases the cost of the experiment. Nevertheless, the high accuracy of point target-based methods renders the aforementioned algorithms still widely employed in PolSAR calibration.

In the field of distributed-target calibration, J.J.van Zyl [12] proposed a unified algorithm that uses at least one trihedral CR and one distributed target area, assuming that the azimuthal symmetry of the target is preserved. Quegan [13] compensates for crosstalk by using distributed targets without considering the reciprocity of the system. Shimada [14] applied the incoherent decomposition model to the uncalibrated distributed target covariance measurement data and then estimated the polarization distortion matrix. Furthermore, he employed distributed targets to rectify the system gain and channel imbalance and calibrated and validated the polarization pattern of a phased-array L-band synthetic-aperture radar utilizing symmetrically scattered target tilt angles in a subsequent study [15]. Klein [16] proposed a polarimetric SAR data calibration method based on backscatter reciprocity and the lack of correlation between co- and cross-polarized acquisition of distributed targets. Kimura [17] proposed a new calibration method using the building-induced polarization direction and applied it to PolSAR polarimetric calibration by exploiting the reflection symmetry on the target. Ainsworth et al. [18] proposed the concept of azimuth preservation and the method of using polarization azimuth in built-up areas. This is a method for a posteriori correction of PolSAR data using scattering reciprocity as a minimum constraint for the correlated PolSAR system. Zhou et al. [19] proposed a distributed target-based SAR radiometric calibration method, which selects stable distributed targets through time-series stability analysis. Different stable pixel extraction methods are employed, and the Oh model is used to correct scattering differences caused by variations in incidence angles, thereby significantly improving the accuracy of the backscattering coefficients of the calibration targets. The algorithms based on distributed targets require a relatively large, uniform, and stable distribution of targets that satisfy scattering reciprocity and reflection symmetry. The Amazon rainforest and other distributed targets meet the requisite conditions, thereby reducing the cost of distributed-target calibration algorithms. However, the accuracy of these algorithms is not as high as those of calibration algorithms designed for point targets [20,21], because distributed targets typically exhibit channel imbalance and phase ambiguity after calibration.

PARCs are characterized by an exceptionally high SNR ratio, which is one of their primary advantages. In addition, they exhibit excellent polarization isolation and enable the generation of various polarization scattering matrices through antenna rotation or digital signal processing. These features establish PARCs as indispensable instruments for the on-orbit calibration and performance evaluation of SAR systems. Our research team has been dedicated to polarimetric calibration based on PARCs for nearly a decade [22,23,24,25]. Most of these projects are based on Freeman’s classic method. In practical engineering applications, the PARC devices we have designed are required to operate with the three scattering matrix combinations mentioned in Freeman’s method. However, through the development of these engineering projects, we have also discovered that other matrix combinations can effectively perform polarimetric calibration. By investigating these new matrix combinations, we can expand and enrich the existing PARC-based polarimetric calibration methods, thereby better serving the polarimetric calibration needs of SAR systems. Based on these research backgrounds, a polarimetric calibration method based on PARCs is proposed in this paper. This method necessitates the use of one equal-amplitude four-channel PARC and two single-channel PARCs for polarimetric calibration. The method is straightforward and does not make any presumptions. The remainder of this paper is organized as follows. Section 2 provides an overview of the fundamentals of polarimetric calibration, the calibration model, and the solution of the calibration matrix that will be employed to calibrate polarimetric SAR data. In Section 3, the calibration method is validated through numerical simulations. The calibration performance and error analysis are discussed, and the effect of the four-channel PARC scattering matrix on the performance of the method is discussed. Section 4 presents the polarimetric calibration experiments and results based on the LT-1 system. It compares the polarization characteristics of the reference target before and after calibration and verifies the performance of the method through experiments related to the trihedral corner reflector and the generation of pseudo-color images based on the Pauli decomposition. The final section, Section 5, serves to conclude the paper.

2. Calibration Models and Methods

In a practical non-ideal PolSAR system, there are inherent polarimetric crosstalk errors and channel imbalance between the transmitter and receiver. In Figure 1, $E_h^t$ and $E_v^t$ represent the horizontal and vertical polarimetric components of the incident field, respectively. Similarly, $E_h^r$ and $E_v^r$ represent the horizontal and vertical polarimetric components of the scattered field. $T_{hh}$, $T_{vv}$, $R_{hh}$, and $R_{vv}$ represent channel imbalance, while $T_{hv}$, $T_{vh}$, $R_{hv}$, and $R_{vh}$ represent crosstalk. During transmission and reception, wave propagation causes the target scattering matrix S to be distorted into the measured scattering matrix M. The objective of polarimetric calibration is to recover the target scattering matrix S from the measured scattering matrix M as completely as possible.

2.1. Polarimetric Signal Model

In Figure 1, the relationship between the measured polarization matrix M and the theoretical polarization matrix S in the polarized SAR system can be expressed by

$$\begin{array}{c}\hfill \mathbf{M}=A{e}^{j\phi }\mathbf{RST}+\mathbf{N}\end{array}$$

(1)

where M and S represent the measured and theoretical 2 × 2 scattering matrices, respectively. T and R represent the transmitting and receiving distortion matrices, respectively, and N accounts for random noise. A represents the amplitude gain of the SAR system, and $\phi$ represents the sum of the propagation delay phase shift between the target and the SAR and all phase losses in the system. By expanding the matrix, (1) can be expanded into the following form:

$$\left[\begin{array}{cc}{M}_{hh}& M_{hv}\\ M_{vh}& M_{vv}\end{array}\right]=A{e}^{j\phi }\left[\begin{array}{cc}{R}_{hh}& R_{hv}\\ R_{vh}& R_{vv}\end{array}\right]\left[\begin{array}{cc}{S}_{hh}& S_{hv}\\ S_{vh}& S_{vv}\end{array}\right]\left[\begin{array}{cc}{T}_{hh}& T_{hv}\\ T_{vh}& T_{vv}\end{array}\right]+\left[\begin{array}{cc}{N}_{hh}& N_{hv}\\ N_{vh}& N_{vv}\end{array}\right]$$

(2)

The normalization of (2) based on $R_{vv}$ and $T_{vv}$ can be further rewritten as

$$\begin{array}{cc}\hfill & \begin{array}{c}\left[\begin{array}{cc}{M}_{hh}& M_{hv}\\ M_{vh}& M_{vv}\end{array}\right]=A{e}^{j\phi }{R}_{vv}{T}_{vv}\left[\begin{array}{cc}{\displaystyle \frac{R_{hh}}{R_{vv}}}& {\displaystyle \frac{R_{hv}}{R_{vv}}}\\ {\displaystyle \frac{R_{vh}}{R_{vv}}}& 1\end{array}\right]\hfill \\ \\ \left[\begin{array}{cc}{S}_{hh}& S_{hv}\\ S_{vh}& S_{vv}\end{array}\right]\left[\begin{array}{cc}{\displaystyle \frac{T_{hh}}{T_{vv}}}& {\displaystyle \frac{T_{hv}}{T_{vv}}}\\ {\displaystyle \frac{T_{vh}}{T_{vv}}}& 1\end{array}\right]+\left[\begin{array}{cc}{N}_{hh}& N_{hv}\\ N_{vh}& N_{vv}\end{array}\right]\hfill \\ \\ =A^{\prime }{e}^{j\phi }\left[\begin{array}{cc}{f}_1& {\delta }_1\\ {\delta }_2& 1\end{array}\right]\times \left[\begin{array}{cc}{S}_{hh}& S_{hv}\\ S_{vh}& S_{vv}\end{array}\right]\hfill \\ \\ \times \left[\begin{array}{cc}{f}_2& {\delta }_3\\ {\delta }_4& 1\end{array}\right]+\left[\begin{array}{cc}{N}_{hh}& N_{hv}\\ N_{vh}& N_{vv}\end{array}\right]\hfill \end{array}\hfill \end{array}$$

(3)

where ${\delta }_1=R_{hv}/R_{vv}$ is the normalized crosstalk from the H-polarized receive channel into the V-polarized receive channel, ${\delta }_2=R_{vh}/R_{vv}$ is the normalized crosstalk from the V-polarized receive channel into the H-polarized receive channel, ${\delta }_3=T_{hv}/T_{vv}$ is the normalized crosstalk from the H-polarized transmit channel into the V-polarized transmit channel, ${\delta }_4=T_{vh}/T_{vv}$ is the normalized crosstalk from the V-polarized transmit channel into the H-polarized transmit channel, $f_1=R_{hh}/R_{vv}$ is the imbalance of the V-polarized receive channel with respect to the H-polarized receive channel, $f_2=T_{hh}/T_{vv}$ is the normalized crosstalk of the V-polarized transmit channel with respect to the H-polarized transmit channel, and $A^{\prime }$ is the equivalent absolute gain. Since $A^{\prime }$ can be determined in the course of radioactive calibration, the distortion matrices are all normalized using the cells on the diagonal as normalization factors, and the calculation of its parameters are all expressed as the ratio of four polarization values of the same calibrator, so that the complexity factor before the matrix product can be canceled out. So the task of full polarimetric SAR calibration translates into solving for the six distortion parameters ${\delta }_1$, ${\delta }_2$, ${\delta }_3$, ${\delta }_4$, $f_1$, and $f_2$.

2.2. Method

In 1990, Freeman [8] developed a polarimetric calibration algorithm based on PARCs in accordance with the mathematical model for polarized SAR calibration expressed in (1). Freeman postulated that the effect of noise could be disregarded, given that the active calibrator target exhibited a markedly high SNR. This assertion will be validated through subsequent simulations and experiments.

Freeman’s calibration uses active calibrators with three different polarimetric states, defined as

$${\mathbf{S}}_x=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right], {\mathbf{S}}_y=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right], {\mathbf{S}}_z=\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right]$$

(4)

In the case of ignoring the effect of noise, bring ${\mathbf{S}}_x$, ${\mathbf{S}}_y$, and ${\mathbf{S}}_z$ into the polarization calibration model (1), giving, respectively,

$$\begin{array}{c}\mathbf{X}=A_x{e}^{j{\phi }_x}\mathbf{R}{\mathbf{S}}_x\mathbf{T}\hfill \\ \mathbf{Y}=A_y{e}^{j{\phi }_y}\mathbf{R}{\mathbf{S}}_y\mathbf{T}\hfill \\ \mathbf{Z}=A_z{e}^{j{\phi }_z}\mathbf{R}{\mathbf{S}}_z\mathbf{T}\hfill \end{array}$$

(5)

Solving the above system of equations yields solutions for the six system polarization distortion parameters as follows:

$$\begin{array}{c}{f}_1={\displaystyle \frac{X_{hv}{Y}_{hh}{Z}_{vh}-X_{hv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}, f_2={\displaystyle \frac{X_{hh}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vv}{Z}_{hh}-X_{hv}{Y}_{vh}{Z}_{hv}}},\hfill \\ \\ {\delta }_1={\displaystyle \frac{Y_{hh}}{Y_{vh}}}, {\delta }_2={\displaystyle \frac{X_{vv}{Y}_{hh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}, {\delta }_3={\displaystyle \frac{X_{hv}{Y}_{vv}{Z}_{hh}-X_{hh}{Y}_{vv}{Z}_{hv}}{X_{hv}{Y}_{vv}{Z}_{hh}-X_{hv}{Y}_{vh}{Z}_{hv}}}, {\delta }_4={\displaystyle \frac{X_{hh}}{X_{hv}}}\hfill \end{array}$$

(6)

where $X_{hh}$, $X_{hv}$, $X_{vh}$, $X_{vv}$ are the components of the observation polarization matrix $\mathbf{X}$ corresponding to different polarization transmission and reception combinations, respectively. $Y_{hh}$, $Y_{hv}$, $Y_{vh}$, $Y_{vv}$ are the corresponding components of the observation polarization matrix $\mathbf{Y}$, and $Z_{hh}$, $Z_{hv}$, $Z_{vh}$, $Z_{vv}$ are the relevant components of the observation polarization matrix $\mathbf{Z}$.

Freeman’s algorithm uses three PARCs with specific scattering matrices, while other kinds of combinations of scattering matrices are equally good for the polarimetric calibration task. If we fix the scattering matrices of ${\mathbf{S}}_x$ and ${\mathbf{S}}_y$ to be the same as Freeman’s algorithm, then there are three kinds of ${\mathbf{S}}_z$ scattering matrices to choose from, as shown in (8). We combine the four kinds of ${\mathbf{S}}_z$ matrix, together with the ${\mathbf{S}}_x$ and ${\mathbf{S}}_y$ matrix, respectively, to obtain four different matrix combinations to complete the polarimetric calibration work. We refer to these four matrix combinations as F. Comb and Comb. 1–3. By substituting these matrix combinations into the linear equation shown in (5) and solving it, the analytical solutions of the three sets of distortion parameters shown in

Table 1. Analytical solutions for the distortion parameters corresponding to four matrix combinations.

	F. Comb	Comb. 1	Comb. 2	Comb. 3
$f_1$	${\displaystyle \frac{X_{hv}{Y}_{hh}{Z}_{vh}-X_{hv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$	${\displaystyle \frac{X_{hv}{Y}_{hh}{Z}_{vh}-X_{hv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$	${\displaystyle \frac{X_{hv}{Y}_{vh}{Z}_{hh}-X_{hv}{Y}_{hh}{Z}_{vh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$	${\displaystyle \frac{X_{hv}{Y}_{vh}{Z}_{hh}-X_{hv}{Y}_{hh}{Z}_{vh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$
$f_2$	${\displaystyle \frac{X_{hv}{Y}_{vh}{Z}_{hh}-X_{hh}{Y}_{vh}{Z}_{hv}}{X_{hv}{Y}_{vv}{Z}_{hh}-X_{hv}{Y}_{vh}{Z}_{hv}}}$	${\displaystyle \frac{X_{hh}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vv}{Z}_{hh}}}$	${\displaystyle \frac{X_{hh}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vv}{Z}_{hh}}}$	${\displaystyle \frac{X_{hv}{Y}_{vh}{Z}_{hh}-X_{hh}{Y}_{vh}{Z}_{hv}}{X_{hv}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vv}{Z}_{hh}}}$
${\delta }_1$	${\displaystyle \frac{Y_{hh}}{Y_{vh}}}$	${\displaystyle \frac{Y_{hh}}{Y_{vh}}}$	${\displaystyle \frac{Y_{hh}}{Y_{vh}}}$	${\displaystyle \frac{Y_{hh}}{Y_{vh}}}$
${\delta }_2$	${\displaystyle \frac{X_{vv}{Y}_{hh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$	${\displaystyle \frac{X_{vv}{Y}_{hh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$	${\displaystyle \frac{X_{vv}{Y}_{vh}{Z}_{hh}-X_{vv}{Y}_{hh}{Z}_{vh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$	${\displaystyle \frac{X_{vv}{Y}_{vh}{Z}_{hh}-X_{vv}{Y}_{hh}{Z}_{vh}}{X_{hv}{Y}_{vh}{Z}_{vh}-X_{vv}{Y}_{vh}{Z}_{hh}}}$
${\delta }_3$	${\displaystyle \frac{X_{hv}{Y}_{vv}{Z}_{hh}-X_{hh}{Y}_{vv}{Z}_{hv}}{X_{hv}{Y}_{vv}{Z}_{hh}-X_{hv}{Y}_{vh}{Z}_{hv}}}$	${\displaystyle \frac{X_{hh}{Y}_{vv}{Z}_{hv}-X_{hv}{Y}_{vv}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vv}{Z}_{hh}}}$	${\displaystyle \frac{X_{hh}{Y}_{vv}{Z}_{hh}-X_{hv}{Y}_{vv}{Z}_{hh}}{X_{hv}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vv}{Z}_{hh}}}$	${\displaystyle \frac{X_{hv}{Y}_{vv}{Z}_{hh}-X_{hh}{Y}_{vv}{Z}_{hv}}{X_{hv}{Y}_{vh}{Z}_{hv}-X_{hv}{Y}_{vv}{Z}_{hh}}}$
${\delta }_4$	${\displaystyle \frac{X_{hh}}{X_{hv}}}$	${\displaystyle \frac{X_{hh}}{X_{hv}}}$	${\displaystyle \frac{X_{hh}}{X_{hv}}}$	${\displaystyle \frac{X_{hh}}{X_{hv}}}$

can be obtained.

$$\begin{array}{c}{\mathbf{S}}_x=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right], {\mathbf{S}}_y=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]\hfill \\ \\ {\mathbf{S}}_z^1=\left[\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right], {\mathbf{S}}_z^2=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right], {\mathbf{S}}_z^3=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right], {\mathbf{S}}_z^4=\left[\begin{array}{cc}1& -1\\ 1& -1\end{array}\right]\hfill \end{array}$$

(7)

In this paper, three additional combinations of PARC scattering matrices are proposed, which are developed based on the well-known Freeman algorithm. All three combinations and the Freeman algorithm are capable of completing the polarimetric calibration task with zero error under ideal, noise-free conditions. In comparison to the approach proposed by Freeman, Comb. 1, as part of the improved method, results in a reduction in the design requirements for PARC. Specifically, in the ${\mathbf{S}}_z$ matrix of Comb. 1, all four polarization channels have a phase of 0, while the ${\mathbf{S}}_z$ matrices of the other three combination matrices all have two polarimetric channels with a phase of 180°. Nevertheless, a 180° phase necessitates that the PARC either phase-shift or invert the received polarimetric signal. This not only increases the design requirements for the data processing module of the PARC, but also introduces the potential for phase errors. The use of Comb. 1 circumvents the aforementioned issues.

Regarding matrix combinations 3 and 4, they share similar performance prerequisites for the PARC as Freeman’s method. However, the value of these new combinations lies in their potential for verification through comparison. By comparing their calibration results with those of the Freeman algorithm and other established methods, we can gain a more comprehensive understanding of their performance. This comparative analysis allows us to identify any potential advantages or areas for improvement, thereby contributing to the enhancement of the overall accuracy of polarimetric calibration.

Figure 2 shows the flowchart of the new calibration method proposed in this paper. Firstly, it is necessary to select the matrix combination to be used for polarimetric calibration in order to complete the matrix setting of PARC. The subsequent step is to obtain the polarization measurement matrix, which is derived from two sources: simulation experiments and LT-1. The calibration experiment is then conducted using the enhanced algorithm proposed in this paper, resulting in the theoretical measurement matrix. Finally, the efficacy of the algorithm is validated through the use of corner reflector and Pauli decomposition to generate pseudo-color images.

3. Validation by Numerical Simulation

In this section, we evaluate the noise immunity performance of the new method and define the parameter $\xi$ to evaluate the sensitivity of the method to errors in the theoretical scattering matrix of PARC.

3.1. Analysis of Anti-Noise Performance

All the four matrix combinations proposed in Section 2 can provide unbiased estimation of distortion parameters without noise. In order to test the anti-noise performance of the new method, we performed 100 Monte Carlo simulations. The distortion parameters ${\delta }_1$, ${\delta }_2$, ${\delta }_3$, ${\delta }_4$, $f_1$, and $f_2$ mentioned above are all complex numbers and consist of two parts: amplitude and phase. In the current airborne and spaceborne SAR designs, the amplitudes of the distortion parameters ${\delta }_1$, ${\delta }_2$, ${\delta }_3$, and ${\delta }_4$ are usually below −25 dB or less, so we are generally concerned only with their amplitude rather than their phase in the calibration experiments. The amplitudes of the distortion parameters $f_1$ and $f_2$ can reach −2 dB or even higher, so their phases cannot be ignored [26]. For this reason, we set the values of the distortion parameters as shown in

Table 2. Parameter settings for experiments on the anti-noise performance of the calibration method.

Parameter Name	Numerical Value
Amplitude of crosstalk (|${\delta }_1$|, |${\delta }_2$|, |${\delta }_3$|, |${\delta }_4$|)	−30 dB
Amplitude of channel imbalance (|$f_1$|, |$f_2$|)	−2 dB
Phase of channel imbalance (Arg($f_1$), Arg($f_2$))	15°
Signal-to-noise ratio (SNR)	15∼40 dB

for the numerical simulation experiments. The experimental results are shown in Figure 3.

Figure 3(i–viii) show that there is no significant difference in the anti-noise performance of the method composed of the four matrix combinations. With an increase in SNR, the calculation error of the polarization distortion parameter decreases continuously. In Figure 3(i,iv), the estimation error of crosstalk ${\delta }_1$ and ${\delta }_4$ is less than 1 dB when the SNR is greater than 34 dB. In Figure 3(ii,iii), the estimation error of crosstalk ${\delta }_2$ and ${\delta }_3$ is less than 1 dB when the SNR is greater than 30 dB. Figure 3(v,vi) show that $f_1$ and $f_2$’s amplitudes are less than 0.5 dB for an SNR greater than 20 dB. In Figure 3(vii,viii), $f_1$ and $f_2$’s phases are less than 1° when the SNR is greater than 25 dB. When the SNR falls below 30 dB, the estimation errors of ${\delta }_1$ and ${\delta }_4$ will exceed 5 dB. Similarly, when the SNR drops below 25 dB, the estimation errors of ${\delta }_2$ and ${\delta }_3$ will exceed 5 dB. Furthermore, as the SNR continues to decrease below 15 dB, significant deviations in the estimation of the channel imbalance parameters will occur. In conclusion, the methodology proposed in this paper demonstrates a high level of accuracy when the SNR exceeds 34 dB, with the performance of the current PARC device meeting the necessary requirements for parameter estimation [18]. Given that the new matrix combination exhibits noise resistance comparable to the classical Freeman method, the proposed approach can be implemented without requiring improvements to the SNR specifications of the PARC. Coupled with the previously discussed advantages of reduced PARC design complexity and error, the new method holds significant potential for a wide range of engineering applications. It is important to note, however, that the accuracy of polarimetric calibration methods for point targets using PARC is highly dependent on the SNR of the calibration device. In the proposed method, the crosstalk parameters (${\delta }_1$, ${\delta }_2$, ${\delta }_3$, ${\delta }_4$) are estimated with almost the same order of magnitude accuracy as the SNR. Therefore, improving the SNR of PARC is the key to improving the accuracy of polarimetric calibration. With the improvement in the transceiver channel isolation of today’s polarimetric SAR system, the SNR requirement of PARC is also increasing. Otherwise, the estimation of crosstalk will have a large deviation due to excessive noise.

3.2. The Impact of PARC’s Theoretical Scattering Matrix Bias

During the polarimetric calibration process, the deviation of the PARC theoretical scattering matrix also affects the calibration results. For example, the deviation of the PARC theoretical scattering matrix can be represented by the parameters ${\xi }_1$ and ${\xi }_2$ in the figure. The parameter ${\xi }_1$ corresponds to the scattering matrix deviation for the single-channel PARC, while the parameter ${\xi }_2$ corresponds to the scattering matrix deviation for the four-channel PARC. The smaller the magnitude of these parameters, the closer the theoretical scattering matrix of PARC is to the set value. In order to determine the requirements of the proposed method for the parameters ${\xi }_1$ and ${\xi }_2$, 100 Monte Carlo simulations were performed. For PARCs in current engineering practice, the amplitude of ${\xi }_1$ can be as low as −45 dB, while that of ${\xi }_2$ can be as low as 0.4 dB [27]. Typically, ${\xi }_1$ has a random phase and ${\xi }_2$ has a phase in the range of −10° to 10°. In summary, the parameters of the simulation experiment were set as shown in

Table 3. Parameter settings for experiments on the effects of parameters ${\xi }_1$ and ${\xi }_2$.

Parameter Name	Numerical Value
Amplitude of crosstalk (|${\delta }_1$|, |${\delta }_2$|, |${\delta }_3$|, |${\delta }_4$|)	−30 dB
Amplitude of channel imbalance (|$f_1$|, |$f_2$|)	−2 dB
Phase of channel imbalance (Arg($f_1$), Arg($f_2$))	15°
Signal-to-noise ratio (SNR)	35 dB
Amplitude of the matrix deviation (|${\xi }_1$|)	−50∼−10 dB
Phase of the matrix deviation (Arg(${\xi }_1$))	Any value within the range of −180° to 180°.
Amplitude of the matrix deviation (|${\xi }_2$|)	0 dB, 0.5 dB, 1 dB, 3 dB
Phase of the matrix deviation (Arg(${\xi }_2$))	Any value within the range of −10° to 10°.

. According to the experimental results, parameter a has basically the same influence on the algorithms corresponding to the four matrix combinations, so only the experimental result of Comb. 1 in Figure 4 is shown.

As illustrated in Table 1, the value of ${\xi }_2$ has no impact on the calculation of ${\delta }_1$ and ${\delta }_4$. When the amplitude of parameter ${\xi }_1$ is less than −40 dB, the estimation error of ${\delta }_1$ and ${\delta }_4$ remains below 1 dB. When the amplitude of ${\xi }_1$ is less than −38 dB and the amplitude of ${\xi }_2$ is less than 0.5 dB, the estimation error of ${\delta }_2$ and ${\delta }_3$ stays below 1 dB. The channel imbalance amplitude is found to be sensitive to ${\xi }_2$ and insensitive to ${\xi }_1$. When the amplitude of ${\xi }_1$ is less than −20 dB and the amplitude of ${\xi }_2$ is less than 1 dB, the estimation errors of the amplitudes of $f_1$ and $f_2$ are both within 1 dB. The channel imbalance phase is insensitive to both ${\xi }_1$ and ${\xi }_2$. When the amplitude of ${\xi }_1$ is less than −15 dB and the amplitude of ${\xi }_2$ is less than 3 dB, the phase estimation errors of $f_1$ and $f_2$ remain within 1°. In conclusion, when the amplitude of ${\xi }_1$ is less than −40 dB, and when the amplitude of ${\xi }_2$ is less than 0.5 dB, their impact on the method proposed herein is within acceptable limits. As previously stated, the amplitude of the current PARC employed in actual engineering practice may be as low as −45 dB for ${\xi }_1$ and as low as 0.4 dB for ${\xi }_2$, which is sufficient to fulfill the requirements of the method proposed in this paper [27].

$$\begin{array}{c}{\mathbf{S}}_x=\left[\begin{array}{cc}0& {\xi }_1\\ 0& 0\end{array}\right], {\mathbf{S}}_y=\left[\begin{array}{cc}0& 0\\ {\xi }_1& 0\end{array}\right]\hfill \\ \\ {\mathbf{S}}_z^1=\left[\begin{array}{cc}-{\xi }_2& -1\\ 1& 1\end{array}\right], {\mathbf{S}}_z^2=\left[\begin{array}{cc}{\xi }_2& 1\\ 1& 1\end{array}\right], {\mathbf{S}}_z^3=\left[\begin{array}{cc}{\xi }_2& -1\\ -1& 1\end{array}\right], {\mathbf{S}}_z^4=\left[\begin{array}{cc}{\xi }_2& -1\\ 1& -1\end{array}\right]\hfill \end{array}$$

(8)

4. Calibration Experiments Using LT-1 Data

In this section, we conduct calibration experiments using the polarimetric data of the LT-1 satellite to evaluate the practical performance of the proposed method.

4.1. Experiment Setup

The LT-1 group satellite, comprising satellites A and B, operates in a sun-synchronous orbit at an altitude of 607 km. The satellite is equipped with an advanced L-band multi-polarimetric multichannel SAR payload, which enables it to operate in multiple imaging modes, with a maximum resolution of 3 meters and a maximum observation width of 400 km. It is currently the largest in-orbit synthetic-aperture radar satellite in China.

In order to calibrate the L-band fully polarized multichannel SAR on board the LT-1 satellite, we conducted calibration experiments at the experimental site in Inner Mongolia, China. Figure 5(i) shows an optical satellite image of the experimental site, with the blue box indicating the deployment range of the experimental equipment. For this experiment we used four PARCs developed by the Aerospace Information Research Institute, Chinese Academy of Sciences, as shown in Figure 5(ii). These PARCs can provide different scattering matrices with good point-target characteristics and polarization isolation indices [22].

The scattering matrices of these PARCs are shown in

Table 4. PARCs set for the LT-1 calibration experiment.

The Serial Number of the PARC	Polarization Scattering Matrix
01	$\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$
02	$\left[\begin{array}{cc}1& -1\\ 1& -1\end{array}\right]$
03	$\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]$
04	$\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$

, and they are arranged in sequence from left to right in the azimuth direction. According to the previous content of this paper, these four PARCs can be divided into two groups. The first set is 01, 03, and 04, and the second set is 02, 03, and 04, which we will refer to as Set-1 and Set-2, respectively, in the following. Set-1 and Set-2 correspond to Comb. 2 and Comb. 3 in the proposed method, and we will verify the effectiveness of the new method through this calibration experiment.

We set up the PARC equipment in the field to ensure that there are no targets with high RCS around to affect the calibration results. At the same time, these PARCs are more than 2 km away from each other to ensure that the imaging results of each PARC are independent.

In addition, a number of L-band trihedral corner reflectors were installed at the experimental site, as shown in Figure 5(iii). The polarization scattering matrix of these reflectors is a second-order standard matrix These reflectors permit the evaluation of the accuracy of the calibration results after the PARC calibration experiment is completed.

4.2. SAR Images and Calibration Results

The fully polarimetric SAR images of the experimental site taken by LT-1 in this experiment are shown in Figure 6. One set contains four images of polarization (HH, HV, VH, VV) states. The PARC we set can be observed in these images, and their observation matrix is consistent with what we expected. Before we perform the calibration calculation, we confirm the SNR of these pictures. We use PARC-01 as a representative device and calculated its SNR with respect to the surrounding background. In the four polarimetric images, the SNRs of PARC-01 is 43.52 dB, 61.28 dB, 62.10 dB, and 43.89 dB. These values are much better than the SNR requirement value of the proposed method determined in our simulation experiments. Therefore, from the perspective of the SNR ratio, the image acquired by LT-1 in this experiment can obtain a better calibration effect through the image presented in this paper. we extract the PARC observation matrix from the original data of the image and carry out the calibration experiment. The calibration results are shown in

Table 5. Calibration results.

Parameter Name	Set-1	Set-2
Amplitude of ${\delta }_1$	−25.96 dB	−25.96 dB
Amplitude of ${\delta }_2$	−26.53 dB	−26.43 dB
Amplitude of ${\delta }_3$	−26.25 dB	−26.02 dB
Amplitude of ${\delta }_4$	−25.81 dB	−25.81 dB
Amplitude of $f_1$	−0.31 dB	−0.20 dB
Phase of $f_1$	−0.57°	−1.39°
Amplitude of $f_2$	−0.48 dB	−0.23 dB
Phase of $f_2$	10.33°	10.71°

.

4.3. Calibration Result Verification by a Trihedral Corner Reflector

The results of the polarimetric calibration can be corrected by a trihedral corner reflector. Ignoring the noise and the absolute amplitude and phase of the system, Equation (9) can be rewritten as follows:

$$\mathbf{S}={\mathbf{R}}^{-1}\mathbf{M}{\mathbf{T}}^{-1}$$

(9)

Put the R and T matrices determined by calibration and the observation matrix M of the trihedral corner reflector extracted from the picture into Equation (9), calculate the matrix S, and compare the difference between S and the second-order identity matrix to evaluate the accuracy of calibration. We write the calibration results written in Table 5 in the form of matrices R and T, as shown in

Table 6. The calibration result written in the form of the received distortion matrix R and the transmitted distortion matrix T.

	Set-1	Set-2
R	$\left[\begin{array}{cc}0.97-0.0031i& -0.050+0.0069i\\ 0.047+0.0054i& 1\end{array}\right]$	$\left[\begin{array}{cc}0.98-0.0075i& -0.050-0.0070i\\ 0.047+0.0052i& 1\end{array}\right]$
T	$\left[\begin{array}{cc}0.95-0.054i& -0.048-0.0081i\\ 0.051-0.0032i& 1\end{array}\right]$	$\left[\begin{array}{cc}0.97+0.058i& -0.049-0.0084i\\ 0.051-0.0032i& 1\end{array}\right]$

. The results of matrix S after calculation and normalization are shown in

Table 7. Results of theoretical scattering matrix S calculation for a trihedral corner reflector.

Set-1	Set-2
$\left[\begin{array}{cc}1.06\angle 0.{41}^{\circ }& 0.01\angle 78.7^{\circ }\\ 0.02\angle 27.3^{\circ }& 1\angle 0^{\circ }\end{array}\right]$	$\left[\begin{array}{cc}1.02\angle 0.{42}^{\circ }& 0.01\angle -77.6^{\circ }\\ 0.02\angle -26.6^{\circ }& 1\angle 0^{\circ }\end{array}\right]$

. In the field of SAR polarimetric calibration, we usually use the degree of difference between the matrix S and the second-order identity matrix to measure the accuracy of the calibration work. The conversion of these differences into log values is shown in

Table 8. Errors in calibration results determined by trihedral corner reflectors.

Error of Distortion Parameter	Before Calibration	Set-1	Set-2
Magnitude of channel imbalance	−0.68 dB	0.56 dB	0.21 dB
Phase of channel imbalance	7.27°	4.04°	4.18°
Crosstalk on the HV channel	−20.28 dB	−39.39 dB	−39.47 dB
Crosstalk on the VH channel	−18.62 dB	−33.22 dB	−33.46 dB

.

According to the calibration results, The PARCs of Set-1 and Set-2 have successfully completed the calibration task. The calibration error of amplitude imbalance is below 0.6 dB, the calibration error of phase imbalance is within 5°, and the calibration error of crosstalk is below −33 dB. The above calibration errors are within the reasonable requirements of the project [28]. This verifies the engineering effectiveness and implementability of the PARC-based polarimetric calibration method proposed in this paper.

The polarimetric signatures of the calibrator can be employed to validate the effectiveness of the calibration process, thereby enabling the visualization of the scattering properties of the target. A comparison was conducted between the polarimetric characteristics of the trihedral corner reflector before calibration, after calibration, and in the ideal state. As illustrated in Figure 7, the polarization signatures before calibration exhibited notable distortion, whereas the polarimetric signatures after calibration were found to be more consistent with the theoretical values. These results thus confirm the reliability of the calibration method.

4.4. Effect of Calibration on Pauli Decomposition

The purpose of the Pauli decomposition [29] is to decompose the measured scattering matrix into four components corresponding to different scattering mechanisms and associated with each basis matrix, which is expressed in the form of

$$\begin{array}{c}\hfill \mathbf{S}=\left[\begin{array}{cc}{S}_{hh}& S_{hv}\\ S_{vh}& S_{vv}\end{array}\right]={\displaystyle \frac{a}{\sqrt{2}}}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+{\displaystyle \frac{b}{\sqrt{2}}}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]+{\displaystyle \frac{c}{\sqrt{2}}}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]+{\displaystyle \frac{d}{\sqrt{2}}}\left[\begin{array}{cc}0& -j\\ j& 0\end{array}\right]\end{array}$$

(10)

where a, b, c, and d in (10) correspond to odd scattering, even scattering, $\pi$/4 even scattering, and spiral scattering, respectively, and are expressed as

$$\begin{array}{cc}\hfill & a={\displaystyle \frac{S_{hh}+S_{vv}}{\sqrt{2}}}, b={\displaystyle \frac{S_{hh}-S_{vv}}{\sqrt{2}}},\hfill \\ \hfill & c={\displaystyle \frac{S_{hv}+S_{vh}}{\sqrt{2}}}, d=j{\displaystyle \frac{S_{hv}-S_{vh}}{\sqrt{2}}}\hfill \end{array}$$

(11)

Considering the reciprocity principle, that is, $S_{hv}$ = $S_{vh}$, an RGB image can be represented by three components: ${\left|a\right|}^2$, ${\left|b\right|}^2$, and ${\left|c\right|}^2$. Here, ${\left|a\right|}^2$ corresponds to the red color, ${\left|b\right|}^2$ corresponds to the blue color, and ${\left|c\right|}^2$ corresponds to the green color. These components are used to represent different physical scattering mechanisms [30,31].

Figure 8(i,ii) show the comparison of Pauli decomposition of LT-1 data before and after calibration, where the blue rectangular box indicates PARCs as well as the deployment area of the trihedral corner reflectors. It can be observed that the features before and after calibration are predominantly red, which represents the single scattering in Pauli decomposition. The scenes captured in this experiment are predominantly bare deserts, which are indeed characterized by single scattering as the primary scattering feature. This is consistent with the representation of these pictures.

In order to further elucidate the impact of calibration on the pseudo-color image, we selected a representative feature, as illustrated in Figure 9, and a trihedral corner reflector, as depicted in Figure 10, for analysis.

Figure 9 depicts a substantial building. The scattering from the building, following Pauli decomposition, exhibits an odd scattering component of 36.7%, an even scattering component of 60.5%, and a $\pi$/4 even scattering component of 2.8% [30,32]. Consequently, the building should appear as light purple in the pseudo-color image synthesized after Pauli decomposition. It is evident that the calibrated buildings exhibit a greater degree of blue components and are more closely aligned with the light purple hue.

We compare the images of the deployed trihedral corner reflectors before and after calibration, as shown in Figure 10. Ideally, the theoretical scattering matrix of the trihedral corner reflector is the identity matrix, so its odd scattering (represented by the value of a) is the largest, and the remaining scattering characteristics are zero. Therefore, the device color in the image should be red, which is encoded in RGB as (255, 0, 0). In Figure 10, before calibration, the trihedral corner reflector is pink in the image, indicating a certain degree of crosstalk and channel imbalance. After calibration, the trihedral corner reflector is corrected to red color and an accurate recovery is obtained. Pseudo-color images of both typical features and trihedral corner reflectors verify the correctness of the calibration method in this paper.

5. Conclusions

The objective of this paper was to evaluate a new polarimetric calibration method based on PARCs approach. The proposed method represents an improvement upon Freeman’s method, as it introduces three additional matrix combinations. The evaluation employed both numerically simulated data and fully polarimetric real-world data from LT-1. The results of the numerical simulations indicate that the SNR requirement of the proposed method is greater than 34 dB under the parameter settings described in this paper. Furthermore, the threshold for deviation from the scattering matrix is −40 dB for a single-channel PARC and 0.5 dB for a four-channel PARC. It should be noted that all current PARC devices are capable of meeting these specifications. The proposed method was successfully employed in calibration experiments based on LT-1 fully polarimetric images. The amplitude imbalance parameter estimation error is less than 0.6 dB, the phase imbalance parameter estimation error is less than 5°, and the crosstalk parameter estimation error is less than −33 dB, which is better than the engineering standard. Furthermore, correlation experiments based on trihedral corner reflectors and pseudo-color images generated by Pauli decomposition demonstrated the effectiveness of the proposed method.