Analysis and Correction of Antenna Pattern Errors for In-Orbit Fully Polarimetric Aperture Synthesis Radiometer

Abstract

The fully polarimetric aperture synthesis radiometer (FPASR) is capable of acquiring the fully polarimetric brightness temperature (BT), which has become increasingly significant in remote sensing. Antenna pattern errors can introduce significant errors to the reconstructed image of the FPASR. Analyzing and correcting the antenna pattern errors is crucial for obtaining high-quality BT images. In this paper, the antenna pattern errors are analyzed and classified into additive and multiplicative errors. A two-step correction method is proposed to reduce the influence of antenna pattern errors on the reconstructed BT. An end-to-end simulator for FPASR has been developed to assess both the antenna pattern errors and the effectiveness of the correction method. The simulation results show that the two-step correction method can reduce the brightness temperature error caused by the antenna pattern errors by over 70%. The successful image of the flight experiment validates the correction method as well.

1. Introduction

The aperture synthesis radiometer uses an antenna array to observe the target scene [1]. The complex cross-correlations of signals, referred to as visibilities, are collected by every pair of antennas in the array. The reconstructed BT of the scene can be obtained from the visibility function. The fully polarimetric aperture synthesis radiometer (FPASR) combines aperture synthesis technology with fully polarimetric information acquisition. FPASR is capable of measuring four Stokes parameters, thereby enhancing the polarization detection capabilities beyond those offered by dual polarimetric radiometers [2,3,4]. FPASR plays an important role in Earth remote sensing. It improves the retrieval ability of important geophysical parameters such as sea surface wind speed, wind direction, sea surface salinity, and soil moisture [5,6,7]. The integration of full polarization and aperture synthesis technology not only enhances the capabilities of current remote-sensing systems but also opens up new fields for multidimensional data fusion and interdisciplinary research.

The antenna plays a vital role in FPASR, exerting a substantial influence on the system’s performance. The maximum aperture of the antenna array dictates the spatial resolution, while the minimum spacing establishes an alias-free field of view. The sensitivity is closely related to the minimum antenna spacing and array arrangement. The antenna pattern serves as a critical auxiliary data set for the BT reconstruction of FPASR. In an ideal case (equal antenna and neglecting the fringe-washing function), the BT can be obtained by solving the visibility function through inverse Fourier transform. In the real case, due to the non-negligible differences in antenna patterns, it is necessary to measure all antenna patterns in the array accurately. Before the radiometers are put into operation, each antenna element within the system undergoes measurement in a grounded anechoic chamber. The limitations of antenna machining accuracy, the insufficiency of the test system’s precision, and the changes in the in-orbit environment may lead to inaccurate antenna pattern measurements and antenna pattern differences. These antenna pattern errors may introduce distortions in the reconstructed images. Furthermore, non-negligible backlobe antenna patterns and the cross-polar antenna patterns will bring additional reconstruction errors.

Some studies have found that the antenna pattern has an important impact on the aperture synthesis radiometer. After taking into account the coupling effect between antennas, a new formulation of the visibility function equation (called the Corbella equation) was derived in [8,9]. A. Camps et al. analyzed the impact of antenna errors on radiometric accuracy and emphasized that antenna patterns are critical and need to be calibrated [10]. Díez-García, R., et al. investigated the relationship between the reconstruction error and antenna pattern differences. They took the SMOS antenna as the reference and analyzed the influence of the pattern differences and the element spacing on the reconstruction error [11]. At present, some methods have been proposed to mitigate antenna pattern error. The flat target transformation (FTT) method proposed in reference [12] can be used to correct the antenna pattern error. This method reduces the antenna pattern error by measuring a “flat” target such as the cold sky near the galactic poles. Refs. [13,14] proposed to observe a reference stable ocean scene instead of the cold sky in [12], thereby further reducing the image errors caused by antenna pattern error. However, it is not easy to find an ideal reference target for observation, and the quality of its correction effect depends on the flat target being observed. In addition, these methods only correct part of the antenna pattern errors.

In this paper, antenna pattern errors are analyzed for FPASR and a two-step correction method is proposed to correct the antenna pattern errors. This method does not require observing specific targets and has wider applicability. In the following, Section 2 analyzes the antenna pattern errors, including the antenna pattern differences, antenna pattern measurement errors, backlobe contributions, and cross-polar contributions. In Section 3, these errors are classified as multiplicative errors and additive errors, and a two-step correction method is proposed to correct the antenna pattern errors step by step. The first step is to correct the antenna pattern additive errors using the artificial brightness temperature distribution. The second step is to reduce the antenna pattern multiplicative errors by using the average antenna brightness temperature. In Section 4, the simulator was used to verify the effectiveness of the correction method quantitatively. Additionally, flight experiment results also show that the image quality is improved after the two-step correction method is adopted. Section 5 analyses the results. Finally, the conclusions are summarized in Section 6.

2. Analysis of Antenna Pattern Errors

The relationship between the system input and output can be established for FPASR, where the BT distribution serves as the system input, while the visibility function represents the system output. After canceling the contribution of the Corbella term addressed in [8,9], and ignoring the antenna backlobe and cross-polar contributions [15], the relationship between the visibility function and the BT can be articulated as follows [16]:

$$\begin{array}{l}{V}_{kj}^{pq}(u,v)=\\ {\displaystyle \underset{{\xi }^2+{\eta }^2<1}{\iint }{T}_B^{pq}(\xi ,\eta )\frac{F_{nk}^p(\xi ,\eta )F_{nj}^{q*}(\xi ,\eta )}{\sqrt{{\Omega }_k{\Omega }_j}\sqrt{1-{\xi }^2-{\eta }^2}}\tilde{r}(-\frac{u\xi +v\eta }{f_c})e^{-j2\pi (u\xi +v\eta )}d\xi d\eta }\end{array}$$

(1)

where $p,q$ = {X, Y} are the polarization configuration of antennas. $T_B^{pq}(\xi ,\eta )$ is the BT of the scene. $u=(x_j-x_k)/{\lambda }_0,v=(y_j-y_k)/{\lambda }_0$ are the normalized spatial frequencies corresponding to the pair of antennas k and j. $(x_j,y_j)$ and $(x_k,y_k)$ are two antenna positions, respectively. $V_{kj}^{pq}(u,v)$ is the visibility function. $F_{nk}^p(\xi ,\eta )$ denotes the normalized antenna co-polar voltage patterns at p polarization, while $F_{nj}^q(\xi ,\eta )$ represents the normalized antenna co-polar voltage patterns at q polarization. $F_{nj}^{q*}(\xi ,\eta )$ is the conjugate of $F_{nj}^q(\xi ,\eta )$. ${\Omega }_k$ and ${\Omega }_j$ are the corresponding solid angles of the antenna. $\xi =\sin \theta \cos \phi$, $\eta =\sin \theta \sin \phi$ are the director cosines with respect to the (X, Y) axes. $\tilde{r}(-{\displaystyle \frac{u\xi +v\eta }{f_c}})$ is the fringe-washing function, which is determined by the frequency responses of two receivers and accounts for spatial decorrelation effects [17,18].

To reconstruct the scene BT from the measured visibility function, the so-called G-matrix method can be used [19,20]. The G-matrix represents the impulse response matrix of the system, with its elements being functions of the antenna pattern. The elements of the G-matrix can be written as

$$G_{kj}^{pq}(\xi ,\eta )={\displaystyle \frac{F_{nk}^p(\xi ,\eta )F_{nj}^{q*}(\xi ,\eta )}{\sqrt{{\Omega }_k{\Omega }_j}\sqrt{1-{\xi }^2-{\eta }^2}}}\tilde{r}(-{\displaystyle \frac{u\xi +v\eta }{f_c}})e^{-j2\pi (u\xi +v\eta )}\Delta \xi \Delta \eta$$

(2)

where $\Delta \xi \Delta \eta$ is the elementary area in $(\xi ,\eta )$.

The reconstructed BT $\mathrm{inv}T$ can be obtained by solving the G-matrix and the measured visibility function

$$V=GT\Rightarrow invT={(G_S)}^{-1}V$$

(3)

where $V$ represents the visibility vector; $G$ represents the G-matrix; $T$ represents the BT vector. $invT$ is the reconstructed BT vector. Note that the left $G$ in the above equation has the same expression as the right $G_S$, but their respective grid ranges are different. The grid points $(\xi ,\eta )$ corresponding to $G$ fall inside the unit circle, while the grid points $(\xi ,\eta )$ corresponding to $G_S$ are reciprocal to the ones in (u, v), which means that $G_S$ is a subset of $G$ [21].

The expression of the G-matrix indicates that the input data required for its construction include the following: (1) the antenna pattern of all elements in the array; (2) the antenna position of all elements in the array; (3) the fringe-washing function. The antenna position is the reference position of the antenna phase pattern. In general, it is assumed that the antenna positions are equal to the nominal values. For wide-beam antennas of synthetic aperture radiometers, this position is typically close to the phase center of the antenna [21]. The fringe-washing function can be measured by injecting the correlated noise signals with different delay times. However, this paper uses a simpler formulation and does not consider the fringe-washing function. Antenna patterns are measured in an anechoic chamber. To mitigate the influence of the frequency dependence of the antenna voltage patterns, the weighted average of the antenna voltage pattern is adopted [22].

It is evident from Equations (1) and (2) that both the visibility function and the G-matrix are functions of the antenna pattern. Therefore, antenna patterns play an important role in BT reconstruction. In practical application, the antenna pattern will bring a variety of errors to the reconstructed BT, thereby affecting the radiometric accuracy of the system. The subsequent analysis will focus on these antenna pattern errors.

2.1. Antenna Pattern Differences

Under ideal conditions, the antenna patterns of all the elements in Equation (1) are assumed to be identical. The relationship between the visibility function and the reconstructed BT satisfies the Fourier transform $V(u,v)=F\{T(\xi ,\eta )\}$. The G-matrix is analogous to an inverse Fourier transform. When backlobe and cross-polarization effects are disregarded, the reconstructed BT can be expressed as an inverse Fourier transform IF of the visibility function:

$$invT=IF(V)=G^{-1}V$$

(4)

The elements of the G-matrix $G_{kj}^0(\xi ,\eta )$ are expressed as follows:

$$G_{kj}^0(\xi ,\eta )={\displaystyle \frac{{\left|F_{n0}(\xi ,\eta )\right|}^2}{{\Omega }_0\sqrt{1-{\xi }^2-{\eta }^2}}}{e}^{-j2\pi (u\xi +v\eta )}\Delta \xi \Delta \eta$$

(5)

where $F_{n0}(\xi ,\eta )$ represents the identical ideal antenna pattern applicable to all elements in the array, and ${\Omega }_0$ is the corresponding solid angle of the antenna.

For an FPASR system, the limitations in machining accuracy and the variations in the electromagnetic environments surrounding each antenna result in disparities in electrical performance among different antennas. It is reflected in the antenna pattern differences of each element. In addition, the processing and assembly of the feed network and antenna often introduce amplitude and phase errors into their respective patterns. For the antenna array, these errors are not the same for each element, which is also the reason for the antenna pattern differences. At this point, the visibility function equation deviates from the simple Fourier transform. The elements in G-matrix are expressed as Equation (2), where $F_{nk}(\xi ,\eta )\ne F_{nj}(\xi ,\eta )$ and ${\Omega }_k\ne {\Omega }_j$.

The antenna pattern differences can be characterized as an additive error to the average antenna pattern, which is different for each antenna within the array. Then, the antenna pattern can be expressed as follows:

$$\begin{array}{l}\left|F_{nk}(\xi ,\eta )\right|=\left|\overline{F_n}(\xi ,\eta )\right|+\left|D_k(\xi ,\eta )\right|,{\phi }_k(\xi ,\eta )=\overline{\phi }(\xi ,\eta )+{\phi }_{Dk}(\xi ,\eta )\\ \left|F_{nj}(\xi ,\eta )\right|=\left|\overline{F_n}(\xi ,\eta )\right|+\left|D_j(\xi ,\eta )\right|,{\phi }_j(\xi ,\eta )=\overline{\phi }(\xi ,\eta )+{\phi }_{Dj}(\xi ,\eta )\end{array}$$

(6)

where $\left|F_{nk,nj}(\xi ,\eta )\right|$ represents the amplitudes of the antenna pattern. ${\phi }_{k,j}(\xi ,\eta )$ represents the phases of the antenna pattern. $\overline{F_n}(\xi ,\eta )$ is the average antenna pattern, $\overline{F_n}(\xi ,\eta )={\displaystyle \frac{F_{n1}(\xi ,\eta )+F_{n2}(\xi ,\eta )+\dots +F_{nM}(\xi ,\eta )}{M}}$. M indicates the number of antenna elements in the array. $\left|\overline{F_n}(\xi ,\eta )\right|$ represents the amplitude of the average pattern. $\overline{\phi }(\xi ,\eta )$ represents the phase of the average pattern. $D_{k,j}(\xi ,\eta )$ represents the deviation from the array average pattern. $\left|D_{k,j}(\xi ,\eta )\right|$ represents the amplitude of the deviation, and ${\phi }_{Dk,Dj}(\xi ,\eta )$ represents the phase of the deviation.

The impact of antenna pattern differences on BT reconstruction is highly complex, and there is currently no universally accepted metric for quantifying the extent of these differences. In ref. [11], the SMOS pattern is used as a reference to quantify the level of antenna pattern differences. However, this method lacks general applicability. This paper adopts the average antenna pattern as a reference standard. Within the range of the antenna main beam ($\xi \in \left[{\xi }_1, \dots , {\xi }_m\right]$, $\eta \in \left[{\eta }_1, \dots , {\eta }_n\right]$), the amplitude difference $A{M}^k$ and phase difference $P{H}^k$ of the antenna k are defined as follows:

$$A{M}^k=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{\xi ={\xi }_1}^{{\xi }_m}{\displaystyle \sum _{\eta ={\eta }_1}^{{\eta }_n}{(\frac{\left|F_{nk}(\xi ,\eta )\right|-\left|\overline{F_n}(\xi ,\eta )\right|}{\left|\overline{F_n}(\xi ,\eta )\right|})}^2}}}{\mathrm{m}\ast n}}}$$

(7)

$$P{H}^k=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{\xi ={\xi }_1}^{{\xi }_{\mathrm{m}}}{\displaystyle \sum _{\eta ={\eta }_1}^{{\eta }_n}{({\phi }_k(\xi ,\eta )-\overline{\phi }(\xi ,\eta ))}^2}}}{m\ast n}}}$$

(8)

where m and n represent the number of measurement points in direction $\xi$ and $\eta$ of the antenna pattern within the range of the main beam.

The amplitude difference and phase difference of the array are defined as the average of the differences of all elements in the array:

$$C_{am}={\displaystyle \frac{A{M}^1+A{M}^2+\dots +A{M}^M}{M}}$$

(9)

$$C_{ph}={\displaystyle \frac{P{H}^1+P{H}^2+\dots +P{H}^M}{M}}$$

(10)

$C_{am}$ and $C_{ph}$ can be used to quantitatively evaluate the degree of antenna pattern differences for the aperture synthesis radiometer.

2.2. Antenna Pattern Measurement Errors

When antenna patterns are measured in a grounded anechoic chamber, measurement errors arise due to the limitations of the testing system’s accuracy. For the FPASR operating in orbit, achieving the requisite high precision necessitates a larger array size. The existing anechoic chamber struggles to meet the requirements for the antenna pattern measurement of large-size arrays. The array needs to be segmented and grouped for measurement [23]. However, this segmentation alters the boundary conditions of the antennas compared to those of the complete array, leading to errors in the measured antenna patterns. After the system is in orbit, the antenna pattern will change with the in-orbit environment. Therefore, inconsistencies likely exist between the actual antenna patterns in orbit and the ground measurement results. The antenna pattern with measurement errors is expressed as follows:

$$F_n^{er}(\xi ,\eta )=\left|F_n(\xi ,\eta )\right|(1+{\tilde{F}}_n(\xi ,\eta ))e^{j({\phi }_F(\xi ,\eta )+{\tilde{\phi }}_F(\xi ,\eta ))}$$

(11)

where $F_n^{er}$ represents the antenna pattern with measurement errors. $\left|F_n(\xi ,\eta )\right|$ and ${\phi }_F(\xi ,\eta )$ are the actual amplitude and phase of the in-orbit antenna pattern. ${\tilde{F}}_n(\xi ,\eta )$ is the amplitude error of the pattern caused by ground measurement and in-orbit change. ${\tilde{\phi }}_F(\xi ,\eta )$ is the phase error of the pattern. $(1+{\tilde{F}}_n(\xi ,\eta ))e^{j{\tilde{\phi }}_F(\xi ,\eta )}$ is the difference between the actual antenna patterns in orbit and those characterized on the ground. The antenna pattern is an important component of the G-matrix. Thus, any measurement error in the antenna pattern will propagate as an error within the G-matrix, subsequently affecting the reconstructed BT image.

2.3. Backlobe Antenna Pattern

When FPASR is observing the Earth in orbit, the main lobes of the antenna points toward the Earth. The antennas capture not only the BT of the Earth directly in front of the array but also that of the cold sky entering through their backlobe. Due to the fact that the gain of the backlobe antenna pattern is lower than that of the main lobe, and the BT of the cold sky is significantly lower than that of the Earth, the contribution of the backlobe is very small in normal mode. However, since the cold sky provides a well-known and stable reference, FPASR can achieve in-orbit external calibration by observing the cold sky. In cold sky mode, the attitude of FPASR is different from the nominal mode. The main lobes of the antennas point towards the cold sky, while Earth is located within its backlobe [12,24]. At this time, the BT contribution of the backlobe cannot be ignored [25]. When the backlobe antenna pattern is considered, the visibility function expression is as follows:

$$\begin{array}{l}{V}_{kj}^{pq}(u,v)={\displaystyle \underset{{\xi }^2+{\eta }^2\le 1}{\iint }\frac{T{F}_{front}(\xi ,\eta )+T{F}_{back}(\xi ,\eta )}{\sqrt{{\Omega }_i{\Omega }_k}\sqrt{1-{\xi }^2-{\eta }^2}}{e}^{-j2\pi (u\xi +v\eta )}d\xi d\eta }\\ T{F}_{front}(\xi ,\eta )=T_B^{pq}(\xi ,\eta )F_{nk}^p(\xi ,\eta )F_{nj}^{q*}(\xi ,\eta )\\ T{F}_{back}(\xi ,\eta )=T_{B\_back}^{pq}(\xi ,\eta )F_{back\_k}^p(\xi ,\eta )F_{back\_j}^{q*}(\xi ,\eta )\end{array}$$

(12)

where $T_{B\_back}^{pq}(\xi ,\eta )$ is the BT entering the backlobe. $F_{back\_k}^p(\xi ,\eta )$ represents the backlobe antenna pattern at p polarization. $F_{nj}^{q*}(\xi ,\eta )$ is the conjugate of the backlobe antenna pattern at q polarization. The contribution of the backlobe is equivalent to adding an additive error to the visibility function.

2.4. Cross-Polar Antenna Pattern

Equation (1) represents the visibility function when only the co-polar patterns are considered. If nonzero cross-polar antenna patterns are taken into account, the signal received by the antenna element k in the array is as follows:

$$\begin{array}{l}{M}_k^X(t)=\left[\begin{array}{cc}{b}_k^x(t)& b_k^y(t)\end{array}\right]\left[\begin{array}{c}{C}_{Xk}\\ X_{Xk}\end{array}\right]\\ M_k^Y(t)=\left[\begin{array}{cc}{b}_k^x(t)& b_k^y(t)\end{array}\right]\left[\begin{array}{c}{X}_{Yk}\\ C_{Yk}\end{array}\right]\end{array}$$

(13)

where $C_{X,Y}$ is the co-polar pattern and $X_{X,Y}$ is the cross-polar pattern. $b^x(t)$ and $b^y(t)$ represent two orthogonal signals. For FPASR, the correlations between the output of antenna k and j are as follows:

$$\begin{array}{l}{V}_{kj}^{XX}=<M_k^X(t)M_j^X(t)>\\ V_{kj}^{XY}=<M_k^X(t)M_j^Y(t)>\\ V_{kj}^{YX}=<M_k^Y(t)M_j^X(t)>\\ V_{kj}^{YY}=<M_k^Y(t)M_j^Y(t)>\end{array}$$

(14)

$V_{kj}^{XX}$ and $V_{kj}^{YY}$ are the correlations of the antennas k and j operating in the same polarization. $V_{kj}^{XY}$ and $V_{kj}^{YX}$ are the correlations operating in different polarizations. The horizontal and vertical BT distribution of the scene and the complex polarimetric BT distribution of the scene constitute the fully polarimetric visibility function.

The fully polarimetric discretized visibility function equation can be written in terms of the fully polarimetric G-matrix as

$$\left[\begin{array}{c}{V}^{XX}\\ V^{YY}\\ V^{XY}\\ V^{YX}\end{array}\right]=\left[\begin{array}{cccc}{G}_{XX}^{CC}& G_{XX}^{XX}& G_{XX}^{CX}& G_{XX}^{XC}\\ G_{YY}^{XX}& G_{YY}^{CC}& G_{YY}^{XC}& G_{YY}^{CX}\\ G_{XY}^{CX}& G_{XY}^{XC}& G_{XY}^{CC}& G_{XY}^{XX}\\ G_{YX}^{XC}& G_{YX}^{CX}& G_{YX}^{XX}& G_{YX}^{CC}\end{array}\right]\left[\begin{array}{c}{T}_B^{XX}\\ T_B^{YY}\\ T_B^{XY}\\ T_B^{YX}\end{array}\right]$$

(15)

The elements of the fully polarimetric G-matrix in the above equation are as follows:

$$\begin{array}{l}{G}_{\mathrm{pq}}^{RS}(\xi ,\eta )={\displaystyle \frac{R_k^p(\xi ,\eta )S_j^{q*}(\xi ,\eta )}{\sqrt{{\Omega }_k{\Omega }_j}\sqrt{1-{\xi }^2-{\eta }^2}}}\tilde{r}(-{\displaystyle \frac{u\xi +v\eta }{f_c}})e^{-j2\pi (u\xi +v\eta )}\Delta \xi \Delta \eta \\ {\xi }^2+{\eta }^2\le 1\end{array}$$

(16)

where $R_k^p(\xi ,\eta )$ represents the pattern of the co-polar (C) or cross-polar (X) of the antenna element k at p polarizations. $S_j^q(\xi ,\eta )$ represents the pattern of the co-polar (C) or cross-polar (X) of the antenna element j at q polarizations. $S_j^{q*}(\xi ,\eta )$ is the conjugate of $S_j^q(\xi ,\eta )$. It is essential to emphasize that the G-matrix required in Equation (16) must be computed over the complete unit circle. In cases where the cross-polar pattern is zero, the fully polarimetric G-matrix in Equation (15) has zero values except for the diagonal. However, since the cross-polar antenna pattern of a real system cannot be zero, all terms in the fully polarimetric G-matrix are nonzero. Furthermore, given that the co-polar pattern significantly exceeds the cross-polar pattern, it follows that the diagonal values of the fully polarimetric G-matrix are the largest. And $G_{pq}^{CC}>G_{pq}^{CX}\approx G_{pq}^{XC}>G_{pq}^{XX}$. For $V^{XY}$ and $V^{YX}$, $G_{pq}^{CX},G_{pq}^{XC}$ are multiplied by $T_B^{XX}$, $T_B^{YY}$. Therefore, the cross-polarization has a non-negligible contribution to both $V^{XY}$ and $V^{YX}$. If only the co-polar pattern is considered in the BT reconstruction, the contribution of the cross-polar pattern is equivalent to adding an additive error to the visibility function. The visibility function error caused by the cross-polar pattern is as follows:

$$\begin{array}{l}\Delta V^{XX}=G_{XX}^{XX}{T}_B^{YY}+G_{XX}^{CX}{T}_B^{XY}+G_{XX}^{XC}{T}_B^{YX}\\ \Delta V^{YY}=G_{YY}^{XX}{T}_B^{XX}+G_{YY}^{XC}{T}_B^{XY}+G_{YY}^{CX}{T}_B^{YX}\\ \Delta V^{XY}=G_{XY}^{CX}{T}_B^{XX}+G_{XY}^{XC}{T}_B^{YY}+G_{XY}^{XX}{T}_B^{YX}\\ \Delta V^{YX}=G_{YX}^{XC}{T}_B^{XX}+G_{YX}^{CX}{T}_B^{YY}+G_{YX}^{XX}{T}_B^{XY}\end{array}$$

(17)

3. Correction Method for Antenna Pattern Errors

Based on the analysis of antenna pattern errors presented in the previous section, the impact of these errors on the visibility function can be categorized into two types: multiplicative errors and additive errors.

Antenna pattern differences and antenna pattern measurement errors can be uniformly expressed in the form of amplitude error and phase error:

$$F_n^{er}(\xi ,\eta )=F_n^{id}(\xi ,\eta )(1+\Delta F_n(\xi ,\eta ))e^{j\Delta {\phi }_F(\xi ,\eta )}$$

(18)

where $F_n^{er}(\xi ,\eta )$ is the measured antenna pattern. $F_n^{id}(\xi ,\eta )$ is an ideal antenna pattern for all elements in the array. $\Delta F_n(\xi ,\eta )$ and $\Delta {\phi }_F(\xi ,\eta )$ is the amplitude error and phase error. These errors arise from the process of antenna fabrication, assembly, and measurement. As a result, the antenna patterns of each element are inconsistent, and the measured antenna pattern on the ground is different from the actual antenna pattern in orbit. Antenna pattern differences and antenna pattern measurement errors are multiplicative errors and scale with the BT of the target. As indicated by Equation (1), $T_B(\xi ,\eta )$ is the amplification factor.

The errors introduced by the backlobe and cross-polar antenna pattern can be uniformly expressed as additive errors to the visibility function:

$$V_{kj}^{er}=V_{kj}^{id}+\Delta V_{kj}^F$$

(19)

$\Delta V_{kj}^F$ is the contribution of the backlobe pattern and the cross-polar pattern to the visibility function.

The in-orbit calibration of FPASR mainly involves injecting correlated and uncorrelated noise signals into the receiver channels. It relies on the response to a known signal to correct for various errors, mainly errors in the receiver channels. However, it is important to note that this correction mechanism does not include the antenna; consequently, the errors related to the antenna pattern still cannot be solved by noisy signal injection. Multiplicative errors and additive errors are difficult to correct by a single correction method, so a two-step correction method is required to correct the antenna pattern errors. Initially, corrections are applied to backlobe contributions and cross-polar contributions utilizing an artificial BT distribution. Subsequently, differential processing of the visibility function is conducted to mitigate the amplification factor attributed to antenna pattern errors. The flowchart illustrating this correction process for antenna pattern errors is presented in Figure 1.

3.1. Correction of Antenna Pattern Additive Errors

The first step is to correct the visibility function additive errors caused by the backlobe antenna pattern and the cross-polar antenna pattern. This additive error can be alleviated by subtracting a model visibility function from the measured visibility function.

The model visibility function is composed of the contribution of the backlobe antenna pattern and the cross-polar antenna pattern. It is founded on an artificial BT distribution and the antenna pattern measured in the grounded anechoic chamber. Notably, the closer the artificial BT distribution to the observation scene, the better the correction effect is.

The contribution of the backlobe can be calculated from the artificial BT distribution associated with the backlobe and its corresponding backlobe antenna pattern:

$$\left[\begin{array}{c}\Delta V_{back}^{XX}\\ \Delta V_{back}^{YY}\\ \Delta V_{back}^{XY}\\ \Delta V_{back}^{XY}\end{array}\right]=\left[\begin{array}{cccc}{H}_{XX}^{CC}& H_{XX}^{XX}& H_{XX}^{CX}& H_{XX}^{XC}\\ H_{YY}^{XX}& H_{YY}^{CC}& H_{YY}^{XC}& H_{YY}^{CX}\\ H_{XY}^{CX}& H_{XY}^{XC}& H_{XY}^{CC}& H_{XY}^{XX}\\ H_{YX}^{XC}& H_{YX}^{CX}& H_{YX}^{XX}& H_{YX}^{CC}\end{array}\right]\left[\begin{array}{c}{T}_{M\_back}^{XX}\\ T_{M\_back}^{YY}\\ T_{M\_back}^{XY}\\ T_{M\_back}^{YX}\end{array}\right]$$

(20)

where $T_{M\_back}^{pq}$ represents the artificial BT distribution that corresponds to the scene observed through the backlobe antenna pattern. The computation of the H-matrix elements closely resembles that of the G-matrix, with the distinction that the H-matrix is derived using the backlobe pattern. Its expression is as follows:

$$\begin{array}{l}{H}_{\mathrm{pq}}^{RS}(\xi ,\eta )={\displaystyle \frac{R_{back\_k}^p(\xi ,\eta )S_{back\_j}^{q*}(\xi ,\eta )}{\sqrt{{\Omega }_k{\Omega }_j}\sqrt{1-{\xi }^2-{\eta }^2}}}\tilde{r}(-{\displaystyle \frac{u\xi +v\eta }{f_c}})e^{-j2\pi (u\xi +v\eta )}\Delta \xi \Delta \eta \\ {\xi }^2+{\eta }^2\le 1\end{array}$$

(21)

where $R_{back\_k}^p(\xi ,\eta )$ represents the backlobe pattern of the co-polar (C) or cross-polar (X) of the antenna element k at p polarizations.

The contribution of the cross-polar antenna pattern can be calculated as follows:

$$\left[\begin{array}{c}\Delta V_{cross}^{XX}\\ \Delta V_{cross}^{YY}\\ \Delta V_{cross}^{XY}\\ \Delta V_{cross}^{XY}\end{array}\right]=\left[\begin{array}{cccc}0& G_{XX}^{XX}& G_{XX}^{CX}& G_{XX}^{XC}\\ G_{YY}^{XX}& 0& G_{YY}^{XC}& G_{YY}^{CX}\\ G_{XY}^{CX}& G_{XY}^{XC}& 0& G_{XY}^{XX}\\ G_{YX}^{XC}& G_{YX}^{CX}& G_{YX}^{XX}& 0\end{array}\right]\left[\begin{array}{c}{T}_M^{XX}\\ T_M^{YY}\\ T_M^{XY}\\ T_M^{YX}\end{array}\right]$$

(22)

The G-matrix required in Equation (22) is calculated in the same way as in Equation (16). The grid points $(\xi ,\eta )$ fall inside the complete unit circle, unlike the one used in the image reconstruction process, which is only computed in the fundamental period. $T_M^{pq}$ represents the artificial BT distribution.

The observation scenes of the backlobe and the cross-polarization are divided into two regions: sky and Earth. The BT of the sky adopts a default value of 2.7 K. It is convenient to divide the artificial BT of the Earth into two components: (1) land and (2) ocean. The BT of the land pixel is set to a constant value, while that for ocean pixels employs a simplified Fresnel model to represent polarimetric temperatures. In nominal mode, the artificial BT distribution corresponding to the backlobe $T_{M\_back}^{pq}$ is the sky, while that of the cross-polarization $T_M^{pq}$ is mainly the Earth. In cold sky mode, $T_{M\_back}^{pq}$ is mainly the BT of the Earth, and $T_M^{pq}$ is the BT of the sky.

The measured visibility function subtracted the contribution of the backlobe and cross-polarization:

$$\left[\begin{array}{c}{V}_{C1}^{XX}\\ V_{C1}^{YY}\\ V_{C1}^{XY}\\ V_{C1}^{YX}\end{array}\right]=\left[\begin{array}{c}{\widehat{V}}^{XX}\\ {\widehat{V}}^{YY}\\ {\widehat{V}}^{XY}\\ {\widehat{V}}^{YX}\end{array}\right]-\left[\begin{array}{c}\Delta V_{back}^{XX}\\ \Delta V_{back}^{YY}\\ \Delta V_{back}^{XY}\\ \Delta V_{back}^{XY}\end{array}\right]-\left[\begin{array}{c}\Delta V_{cross}^{XX}\\ \Delta V_{cross}^{YY}\\ \Delta V_{cross}^{XY}\\ \Delta V_{cross}^{XY}\end{array}\right]$$

(23)

where the hat (^) stands for the measured visibility function and $V_{C1}$ stands for the visibility function corrected in the first step, that is, the visibility function after correcting the contributions of the backlobe and cross-polarization.

3.2. Correction of Antenna Pattern Multiplicative Errors

The second step is to correct the visibility function multiplicative errors caused by antenna pattern differences and antenna pattern measurement errors. Since these antenna pattern errors are inseparable multiplicative errors, they cannot be corrected merely by accurately measuring the error of each individual term. To mitigate the impact of these multiplicative errors, it is essential to reduce the amplification factor associated with the antenna pattern. The antenna brightness temperature of each element can be obtained from the measured data. The average antenna brightness temperature ${\overline{T}}_A$ is calculated by taking the mean of the BTs across all elements. The antenna pattern measured on the ground and the average antenna brightness temperature can be used to calculate the visibility function corresponding to the average antenna brightness temperature:

$$V_{T_A}^{XX}=G_{XX}^{CC}{\overline{T}}_A^{XX},V_{T_A}^{YY}=G_{YY}^{CC}{\overline{T}}_A^{YY}$$

(24)

Since the first step has corrected the contribution of cross-polarization, the above equation would contain only the product terms of the two co-polar antenna patterns.

By subtracting the visibility function corresponding to the average antenna brightness temperature, the differential visibility function can be obtained:

$$V_{C2}^{XX}=V_{C1}^{XX}-V_{T_A}^{XX},V_{C2}^{YY}=V_{C1}^{YY}-V_{T_A}^{YY}$$

(25)

The differential visibility function of $V_{C1}^{XY}$ and $V_{C1}^{YX}$ is not considered. By reconstructing the BT from $V_{C2}^{XX}$ and $V_{C2}^{YY}$, the reconstructed BT $T_B^{XX}-{\overline{T}}_A^{XX}$ and $T_B^{YY}-{\overline{T}}_A^{YY}$ can be obtained. Therefore, the amplification factor can be reduced to $T_B-{\overline{T}}_A$. The antenna errors would be amplified only by the difference between the BT and the average antenna brightness temperature. Consequently, the effect of antenna pattern differences and antenna pattern measurement errors is reduced. Finally, the average antenna brightness temperature is added back to the reconstructed BT in the end.

After correction by the above two steps, the BT is reconstructed. Given that Equation (3) is underdetermined, a regularization method has been proposed in [26] to achieve a stable solution. In full polarization mode, the reconstructed BT obtained from the corrected visibility can be expressed as follows:

$$invT=U^{*}Z{(J_{pq}^{CC})}^{-1}({\widehat{V}}^{pq}-\Delta V_{back}^{pq}-\Delta V_{cross}^{pq}-{\delta }_{pq}{V}_{T_A}^{pq})+{\delta }_{pq}{T}_A^{pq}$$

(26)

$J_{pq}^{CC}=G_{pq}^{CC}{U}^{*}Z$, U stands for discrete Fourier transform operator, and Z is the zero-padding operator. Since the cross-polarization has been corrected, only the co-polarization is considered for the G-matrix $G_{pq}^{CC}$. ${\delta }_{pq}$ is the Kronecker delta. ${\delta }_{pq}=1$ if $p=q$ and ${\delta }_{pq}=0$ if $p\ne q$.

4. Results

4.1. Fully Polarimetric Simulation

In order to analyze the antenna pattern error, an end-to-end simulator for FPASR has been developed. The antenna array is a Y-shaped array of 55 elements, with 18 elements in one arm. The position of each antenna element in the array is illustrated in Figure 2. The ideal normalized co-polar and cross-polar antenna patterns are depicted in Figure 3. The antenna patterns for both polarizations are similar. A Fresnel model of the ocean has been employed as the observation scene for simulation. Figure 4 shows the four expected polarimetric BTs: $\{T^{XX},T^{YY},T^3,T^4)$. $T^3$ and $T^4$ are the third and fourth Stokes parameters, defined as $T^3=2\mathrm{Re}(T^{XY})$, $T^4=2\mathrm{Im}(T^{XY})$. The BT error $\Delta T$ is computed within a circle ${\xi }^2+{\eta }^2<0.3$. The calculation of $\Delta T$ is as follows:

$$\Delta T=\sqrt{{\displaystyle \frac{1}{Num}}{\displaystyle \sum _{k=1}^{N\mathrm{um}}{\left[invT(x_k)-T(x_k)\right]}^2}}$$

(27)

where $T$ indicates the original BT. $invT$ is the reconstructed BT obtained by visibility function. $Num$ is the number of pixels in the circle ${\xi }^2+{\eta }^2<0.3$.

Table 1. Antenna pattern errors for simulation.

Antenna Pattern Errors	Error
Amplitude differences of the antenna patterns	5%
Phase differences of antenna patterns	5°
Amplitude error of antenna pattern measurement	1%
Phase error of antenna pattern measurement	1°

presents the antenna pattern errors taken into account in the simulation. Backlobe antenna patterns and cross-polar antenna patterns are also taken into account. The BT of the cold sky observed by the backlobe is set to 2.7 K. The simulation results are shown in Figure 5, Figure 6 and Figure 7. It is clear that the image quality deteriorates with the introduction of antenna pattern errors, as shown in Figure 5. To mitigate the impact of these errors on BT, the two-step correction method described in the previous section is adopted. The corrected results are shown in Figure 6, where the reconstructed BT images after correction exhibit a closer resemblance to the original BT image. For comparison, the FTT method was also simulated. In the simulation of the FTT method, the simulator simulates the FPASR’s observation of cold sky and uses the observation data of cold sky to correct the antenna pattern error. The results are shown in Figure 7. The BT error results are summarized in

Table 2. The BT error $\Delta T$ in four polarimetric BTs.

$\mathbf{\Delta }\mathit{T}$	X Polarization	Y Polarization	T3 Polarization	T4 Polarization
Before calibration (K)	12.32 K	14.54 K	4.84 K	4.51 K
After calibration with the two-step correction method (K)	2.64 K	2.58 K	3.18 K	3.14 K
After calibration with the FTT method (K)	3.56 K	3.31 K	4.84 K	4.51 K

. The qualitative results confirm that the two-step correction method is effective in correcting the antenna pattern error and has a better correction effect than the FTT method.

4.2. Flight Experiment

Experiments were carried out on an airborne demonstrator. The instrument is an 11-element fully polarimetric aperture synthesis radiometer operating at 1.415 GHz and arranged in a Y-shape array with three elements per arm. Before the flight experiment, antenna patterns for all elements’ X and Y polarization were measured in an anechoic chamber, as illustrated in Figure 8a. These antenna patterns are utilized for image reconstruction and the correction of antenna pattern errors. The average antenna pattern is obtained by averaging the measured antenna pattern of all elements. The co-polar (CO) and cross-polar (CR) average antenna patterns at X polarization and Y polarization are illustrated in Figure 9. After calculation, the amplitude difference of the array is 6.8% and the phase difference is 4.4°.

Before the flight experiment, indoor experiments were carried out in an anechoic chamber. The results of the indoor experiments were used to conduct a quantitative assessment of the effectiveness of the correction method. The absorptive material background wall of the anechoic chamber is used as the observed scene for imaging. The absorbing material’s physical temperature is measured using a temperature sensor, and it is approximately 293 K. The BT of the absorbing material is approximately equal to the physical temperature. Therefore, the original BT of the observed scene can be set as 293 K. By using Equation (27) and combining the original BT with the reconstructed BT, the BT error can be calculated. The reconstructed images are shown in Figure 10 and Figure 11. The absorptive material background wall has a uniform and stable temperature, so the reconstructed BT in the alias-free field of view is theoretically a constant value. The experimental results obtained by the two-step correction method are more consistent with expectations. Before the correction, the reconstructed BT in the alias-free field of view exhibited significant errors, with substantial variations in brightness temperature. After correction, the brightness temperature of the image became more uniform and consistent. From the results of BT errors before and after correction in

Table 3. The BT error $\Delta T$ of the indoor experiment.

$\mathbf{\Delta }\mathit{T}$	X Polarization	Y Polarization
Before calibration (K)	11.56 K	10.27 K
After calibration with the two-step correction method (K)	3.01 K	2.64 K

, it can be seen that after the two-step correction method is applied, the error has been reduced from over 10 K to approximately 3 K.

Following the initial validation through indoor experiments, the instrument was subsequently transported to the coastal site for the execution of flight experiments. During the flight experiment, the instrument is mounted on the aircraft, with the antenna array located in the underpart of the aircraft’s fuselage, as shown in Figure 8b. The signal of the target scene enters the receiving channel through the antenna array. The visibility function is obtained after the complex correlation operation of signals. The reconstructed BT image can be obtained by the measured visibility function. The experiment selected a coastal-land junction area as the observation scene.

Figure 12 presents an optical image of a coastline scene, with the black line indicating the flight path of the aircraft. Its corresponding reconstructed BT image is shown in Figure 13. There is a significant contrast in BT between land and water. However, the BT image exhibits distortion. The boundary of the blue low-temperature zone in the experimental results delineates the coastline. As illustrated in Figure 11a,b, the X polarization and Y polarization reconstructed images present significantly different coastal features, and these features are all different from those depicted in the optical images. The two-step correction method described above was used to correct the measured visibility function without considering the contribution of the backlobe. After the antenna pattern correction method is applied, the corrected image results are shown in Figure 14. The image quality of the corrected image is improved and the geometry of the coastline is well preserved. In Figure 14, the distribution areas and shapes of the ocean and land are highly consistent with those of the optical image. Through imaging analysis of this area, the validity of the correction method was qualitatively verified.

5. Discussion

The quantitative simulation results are presented in Table 2. The brightness temperature of the ocean scene in X polarization and Y polarization is much higher than in T3 and T4 polarization. When the antenna pattern error is considered, the brightness temperature error of the reconstructed brightness temperature images of X polarization and Y polarization is larger than that of T3 and T4. Before the correction, the BT errors of $T^{XX}$ and $T^{YY}$ both exceeded 12 K, and the BT errors of T3 and T4 both exceeded 4 K. After correction by the two-step correction method, the BT errors of the four polarizations are all reduced. Especially, the correction effects of $T^{XX}$ and $T^{YY}$ are significant, with the error reduced to within 3 K. It must be noted that the second step of the two-step correction method is only used for X polarization and Y polarization. The results of T3 and T4 have only undergone the first step of correction. That is, T3 and T4 only corrected the antenna pattern additive errors, but did not correct the antenna pattern multiplicative errors. Therefore, the results in Figure 6a,b show a better correction effect.

The FTT method focuses on correcting the antenna pattern multiplicative errors of the X polarization and Y polarization, without correcting T3 and T4. Consequently, the errors of T3 and T4 remain unaltered after the application of the FTT method. The results in Table 2 show that the BT error of the two-step correction method is smaller than that of the FTT method. The main differences between the two correction methods are as follows: (1) Compared with the FTT method, the proposed two-step correction method corrects not only the antenna pattern multiplicative errors but also the antenna pattern additive errors. (2) The FTT method requires a “flat” target for observation, and the flatness of the target will affect the correction effect. The FTT method is applicable to in-orbit aperture synthesis radiometers, which correct antenna pattern errors by observing the cold sky. The two-step correction method does not require observation of a specific target, and it is suitable for both in-orbit FPASR and ground-based FPASR.

The two-step correction method proposed in this paper mainly focuses on correcting the brightness temperature in X-polarization and Y-polarization components, especially the second step of the correction method. For the third and fourth Stokes parameters, this method corrects only the antenna pattern additive error. The correction of the antenna pattern multiplicative errors for the third and fourth Stokes parameters requires further investigation.

6. Conclusions

For FPASR, the antenna pattern significantly influences the imaging quality. This paper analyzes various aspects including the antenna pattern differences, antenna pattern measurement errors, backlobe antenna pattern, and cross-polar antenna pattern. According to the action form of the pattern errors on the visibility function, they can be divided into additive pattern errors (contribution of the backlobe and cross-polar antenna pattern) and multiplicative pattern errors (antenna pattern differences and antenna pattern measurement error). A two-step correction method is proposed to systematically correct these antenna pattern errors: first, artificial brightness temperature is employed to mitigate additive errors; subsequently, antenna brightness temperature is utilized to alleviate multiplicative errors.

The effectiveness of this two-step method has been validated through both simulations and experimental results. Firstly, the Fresnel model of the ocean is used for fully polarimetric simulations, and the quantitative analysis of the correction method was conducted by comparing it with the original brightness temperature. When antenna pattern errors are introduced into the simulation, the BT errors for both X polarization and Y polarization exceed 12 K. After the two-step correction method, the BT errors are reduced to within 3 K. Then, to further verify the effectiveness of the correction method, flight experiments were carried out. An 11-element fully polarimetric aperture synthesis radiometer was carried on an aircraft for the flight experiment. The coastline was imaged in the flight experiment. By comparing the reconstructed image with the optical image, it can be seen that the corrected BT image has a higher imaging quality and is more consistent with the optical image.