# Calibrating VLBI Polarization Data Using GPCAL. I. Frequency-Dependent
Calibration

Jongho Park Department of Astronomy and Space Science, Kyung Hee University,
1732, Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 17104, Republic of
Korea  Korea Astronomy and Space Science Institute, Daedeok-daero 776,
Yuseong-gu, Daejeon 34055, Republic of Korea  Institute of Astronomy and
Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan  Keiichi
Asada Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box
23-141, Taipei 10617, Taiwan  Do-Young Byun Korea Astronomy and Space Science
Institute, Daedeok-daero 776, Yuseong-gu, Daejeon 34055, Republic of Korea
University of Science and Technology, Gajeong-ro 217, Yuseong-gu, Daejeon
34113, Republic of Korea

(Received –; Revised –; Accepted –)

###### Abstract

In this series of papers, we present new methods of frequency- and time-
dependent instrumental polarization calibration for Very Long Baseline
Interferometry (VLBI). In most existing calibration tools and pipelines, it
has been assumed that instrumental polarization is constant over frequency
within the instrument bandwidth and over time. The assumption is not always
true and may prevent an accurate calibration, which can result in degradation
of the quality of linear polarization images. In this paper, we present a
method of frequency-dependent instrumental polarization calibration that is
implemented in GPCAL, a recently developed polarization calibration pipeline.
The method is tested using simulated data sets generated from real Very Long
Baseline Array (VLBA) data. We present the results of appyling the method to
real VLBA data sets observed at 15 and 43 GHz. We were able to eliminate
significant variability in cross-hand visibilities over frequency that is
caused by frequency-dependent instrumental polarization. As a result of the
calibration, linear polarization images were slightly to modestly improved as
compared to those obtained without frequency-dependent instrumental
polarization calibration. We discuss the reason for the minor impact of
frequency-dependent instrumental polarization calibration on existing VLBA
data sets and prospects for applying the method to future VLBI data sets,
which are expected to provide very large bandwidths.

high angular resolution — techniques: interferometric — techniques:
polarimetric — methods: data analysis

††journal: ApJ††facilities: VLBA (NRAO)††software: AIPS (Greisen, 2003),
Difmap (Shepherd, 1997), GPCAL (Park et al., 2021b), ParselTongue (Kettenis et
al., 2006), Scipy (Virtanen et al., 2020)

##  1 Introduction

The Very Long Baseline Interferometry (VLBI) technique is a method for
obtaining very high angular resolution by using telescopes that are widely
separated. The telescopes in a VLBI array can split the sky signal into two
orthogonal polarization channels, which are typically right and left
circularly polarized (RCP and LCP, respectively), and each can record
independently. The observed polarized visibilities, which are the time average
of the complex correlations between the electric fields for each polarization
incident at each telescope in the array, are directly related to the
distributions of the four Stokes parameters in the sky by means of the van
Cittert-Zernike theorem (van Cittert, 1934; Zernike, 1938; Thompson et al.,
2017).

The relationship between the measured polarimetric visibilities and the
source’s intrinsic polarization can be distorted due to imperfections in the
instrument’s response. This distortion is primarily caused by antenna
polarization that is not perfectly circular, which results in “instrumental
polarization” in the visibility data. Instrumental polarization is commonly
described in terms of an antenna’s response to a wave of polarization
orthogonal to the nominal antenna response (e.g., Thompson et al., 2017, their
Chapter 4.7), which is often referred to as “polarization leakages” or
“D-terms”.

LPCAL, a task implemented in the Astronomical Image Processing System (AIPS,
Greisen, 2003) based on the linearized leakage model (Leppanen et al., 1995),
has been a standard method for calibrating VLBI data for nearly three decades.
This method has been successful in a variety of studies using a variety of
VLBI arrays (e.g., Gómez et al., 2016; Casadio et al., 2017; Jorstad et al.,
2017; Lister et al., 2018; Park et al., 2018, 2019; Event Horizon Telescope
Collaboration et al., 2021). However, LPCAL has some limitations that prevent
accurate calibration. It assumes that the total intensity and linear
polarization structures of a calibrator are similar, and also it cannot fit
the instrumental polarization model to data from multiple calibrators at the
same time. The former is referred to as the “similarity approximation”111As
far as we are aware, the term ”similarity approximation” was first introduced
in Leppanen et al. (1995). The original idea was proposed by Cotton (1993),
who also cautioned that the assumption of a direct proportionality between
total intensity and linear polarization emission may not hold. (Cotton, 1993;
Leppanen et al., 1995), and it is generally violated for VLBI data, especially
at high frequencies at which nearly all calibrators are resolved.

The Generalized Polarization CALibration pipeline (GPCAL; Park et al. 2021b)
has been developed to overcome these limitations222Other calibration/imaging
pipelines that have recently been developed are polsolve (Martí-Vidal et al.,
2021), similar to GPCAL but based on CASA (Janssen et al., 2019; THE CASA TEAM
et al., 2022; van Bemmel et al., 2022), the eht-imaging software library using
the regularized maximum likelihood technique (Chael et al., 2016, 2018, 2022),
D-term Modeling Code (DMC, Pesce 2021), THEMIS (Broderick et al., 2020), and
Comrade (Tiede, 2022) using Markov chain Monte Carlo schemes. See Park &
Algaba (2022) for more details. (Park et al., 2021b). The GPCAL pipeline is
written in ParselTongue (Kettenis et al., 2006) and is based on AIPS and the
Caltech Difmap package (Shepherd, 1997). It enables the use of more accurate
linear polarization models of calibrators for D-term estimation without being
limited by the similarity approximation. It can also fit the instrumental
polarization model to data from multiple calibrator sources simultaneously in
order to increase the accuracy of the fitting.

GPCAL was applied to the polarization analysis of the first M87 Event Horizon
Telescope (EHT) results (Event Horizon Telescope Collaboration et al., 2021)
as well as other recent studies (e.g., Takamura et al., 2023). We have also
demonstrated that GPCAL is capable of detecting polarizations from sources
that are very weakly polarized (Park et al., 2021a). The linear polarization
structure of the subparsec core of the M87 jet observed with the Very Long
Baseline Array (VLBA) at 43 GHz is compact and has a very low fractional
polarization ($\sim 0.2$-$0.6\%$). It was not possible to well constrain this
structure in previous studies with LPCAL using the same data set due to a
breakdown in the similarity approximation, resulting in less accurate
calibration of the instrumental polarization (Walker et al., 2018; Kravchenko
et al., 2020).

In GPCAL, one assumption was made that can limit the accuracy of instrumental
polarization calibration: it assumes that polarimetric leakages remain
constant across the frequency bandwidth of a receiver and over time. The
assumption of constant leakages over frequency may not apply to many VLBI
arrays that have offered a wide bandwidth in recent years. In general, the
magnitudes of leakages tend to increase as the operating frequency deviates
further from the nominal frequency within the band. This is due to the design
of the instruments, including polarizers, which are optimized to minimize
leakages at the nominal frequency (e.g., Hull & Plambeck, 2015). Based on
recent wide-bandwidth VLBA data, some previous studies have suggested that
frequency-dependent D-terms can limit the dynamic range of linear polarization
images (Kravchenko et al., 2020). Also, the assumption of constant leakages
over time is incorrect since polarimetric leakages are not constant across
telescope beams; they are direction-dependent (Smirnov, 2011). The effect is
more severe for large telescopes and at high frequencies, where antenna
pointing is usually inaccurate.

In this series of papers, we present GPCAL implementations of novel methods of
frequency and time-dependent instrumental polarization calibration of VLBI
data. In this paper (Paper I hereafter), we present a method for calibrating
frequency-dependent instrumental polarization. An accompanying publication
(Park et al. 2023; Paper II) describes a method of calibrating instrumental
polarization that varies with time. These methods are applied to real VLBI
data sets and demonstrate that, depending on the circumstances, they can
significantly improve calibration accuracy.

This paper is organized as follows. In Section 2, we explain the radio
interferometer measurement equation, which GPCAL is based on to model the
instrumental polarization. In Section 3, we introduce the calibration
procedure for frequency-dependent leakage calibration implemented in GPCAL. In
Section 4, we validate the method using synthetic data. In Section 5, we
present results of applying the method to real observations from the VLBA at
15 and 43 GHz. We conclude in Section 6.

##  2 The Radio Interferometer Measurement Equation

The observed polarized visibilities on a baseline between stations $m$ and $n$
are given by the correlation products

$\displaystyle r^{RR}_{mn}$ $\displaystyle\equiv$ $\displaystyle\langle
E_{R,m}E^{*}_{R,n}\rangle$ $\displaystyle r^{LL}_{mn}$ $\displaystyle\equiv$
$\displaystyle\langle E_{L,m}E^{*}_{L,n}\rangle$ $\displaystyle r^{RL}_{mn}$
$\displaystyle\equiv$ $\displaystyle\langle E_{R,m}E^{*}_{L,n}\rangle$
$\displaystyle r^{LR}_{mn}$ $\displaystyle\equiv$ $\displaystyle\langle
E_{L,m}E^{*}_{R,n}\rangle,$ (1)

where $E$ is an electric field, $R$ indicates RCP, $L$ indicates LCP, angular
brackets denote a time average, and an asterisk denotes complex conjugate. One
can represent the visibilities using a coherency matrix formalism:

${\bm{V}}_{mn}=\begin{pmatrix}r^{RR}_{mn}&r^{RL}_{mn}\\\
r^{LR}_{mn}&r^{LL}_{mn}\end{pmatrix}.$ (2)

The observed ${\bm{V}}_{mn}$ are corrupted by antenna gains and polarimetric
leakages. The relationship between the true $\bar{\bm{V}}_{mn}$ and the
observed ${\bm{V}}_{mn}$ can be described via a sequence of linear
transformations, which is the so-called radio interferometer measurement
equation (RIME; Hamaker et al. 1996, see also Sault et al. 1996; Hamaker &
Bregman 1996; Hamaker 2000; Smirnov 2011),

${\bm{V}}_{mn}={\bm{J}}_{m}\bar{\bm{V}}_{mn}{\bm{J}}^{H}_{n},$ (3)

where H is the Hermitian operator and ${\bm{J}}$ is the so-called Jones matrix
(Jones, 1941):

$\displaystyle{\bm{J}}_{m}$ $\displaystyle=$
$\displaystyle{\bm{G}}_{m}{\bm{D}}_{m}{\bm{P}}_{m}$ (4) $\displaystyle=$
$\displaystyle\begin{pmatrix}G^{R}_{m}&0\\\
0&G^{L}_{m}\end{pmatrix}\begin{pmatrix}1&D^{R}_{m}\\\
D^{L}_{m}&1\end{pmatrix}\begin{pmatrix}e^{-j\phi_{m}}&0\\\
0&e^{+j\phi_{m}}\end{pmatrix}.$

$G$ is the complex antenna gain, $D$ is the leakage factor (“D-term”), and
$\phi$ is the antenna field rotation angle. Subscripts denote antenna numbers,
and superscripts denote polarization. The field rotation angle takes the form

$\phi=f_{\rm el}\theta_{\rm el}+f_{\rm par}\psi_{\rm par}+\phi_{\rm off},$ (5)

where $\theta_{\rm el}$ is the source’s elevation, $\psi_{\rm par}$ the
parallactic angle, and $\phi_{\rm off}$ is a constant offset for the rotation
of antenna feed with respect to the azimuth axis. Cassegrain mounts have
$f_{\rm par}=1$ and $f_{\rm el}=0$ and thus the field rotation angle is
equivalent to the parallactic angle, except for the constant offset. Nasmyth
mounts have $f_{\rm par}=1$ and $f_{\rm el}=+1$ for Nasmyth-Right type and
$f_{\rm par}=1$ and $f_{\rm el}=-1$ for Nasmyth-Left type.

For circular feeds, the $\bar{\bm{V}}$ are related to the Fourier transforms
of the Stokes parameters ($\tilde{I}$, $\tilde{Q}$, $\tilde{U}$, and
$\tilde{V}$) via

$\bar{\bm{V}}_{mn}\equiv\begin{pmatrix}\mathscr{RR}&\mathscr{RL}\\\
\mathscr{LR}&\mathscr{LL}\end{pmatrix}=\begin{pmatrix}\tilde{I}_{mn}+\tilde{V}_{mn}&\tilde{Q}_{mn}+j\tilde{U}_{mn}\\\
\tilde{Q}_{mn}-j\tilde{U}_{mn}&\tilde{I}_{mn}-\tilde{V}_{mn}\end{pmatrix}.$
(6)

Thus, the source’s brightness distribution of each Stokes parameter can be
derived once ${\bm{G}}$ and ${\bm{D}}$ matrices are estimated and removed from
the data. ${\bm{P}}$ consists of purely geometric terms and can be determined
straightforwardly.

Nearly all calibrators are resolved with VLBI, and their full polarization
structures change rapidly (typically over several weeks or months; for
example, Jorstad et al. 2017; Lister et al. 2018; Weaver et al. 2022). This
means that $\bar{\bm{V}}$ are unknown and cannot be easily modeled using a
simple assumption such as $\tilde{I}$, $\tilde{Q}$, $\tilde{U}$, and
$\tilde{V}$ are proportional to each other333This assumption holds when
calibrators are point sources, which is typical for connected
interferometers.. As a result, a standard calibration strategy for VLBI data
has been to estimate ${\bm{G}}$ first using the parallel-hand visibilities and
then ${\bm{D}}$ using the cross-hand visibilities after removing ${\bm{G}}$.
It is because parallel-hand visibilities are sensitive to source’s total
intensity emission and antenna gains, but insensitive to linear polarization
emission and D-terms, whereas cross-hand visibilities are the opposite. GPCAL
follows this strategy as well (Park et al., 2021b). Some recently developed
calibration/imaging pipelines, such as polsolve (Martí-Vidal et al., 2021),
THEMIS (Broderick et al., 2020), DMC (Pesce, 2021), and Comrade (Tiede, 2022),
utilize the full RIME for fitting.

The antenna field-rotation angles are usually corrected at an upstream
calibration stage (before performing global fringe fitting). Thus, Equation 3
becomes:

${\bm{V}}_{mn}={\bm{P}}^{-1}_{m}{\bm{G}}_{m}{\bm{D}}_{m}{\bm{P}}_{m}\bar{\bm{V}}_{mn}{\bm{P}}^{H}_{n}{\bm{D}}^{H}_{n}{\bm{G}}^{H}_{n}({\bm{P}}^{H}_{n})^{-1},$
(7)

which can be rewritten element-wise as (e.g., Roberts et al., 1994):

$\displaystyle r^{RR}_{mn}$ $\displaystyle=$ $\displaystyle
G^{R}_{m}G^{R*}_{n}[\mathscr{RR}+D^{R}_{m}e^{2j\phi_{m}}\mathscr{LR}+$
$\displaystyle
D^{R*}_{n}e^{-2j\phi_{n}}\mathscr{RL}+D^{R}_{m}D^{R*}_{n}e^{2j(\phi_{m}-\phi_{n})}\mathscr{LL}]$
$\displaystyle r^{LL}_{mn}$ $\displaystyle=$ $\displaystyle
G^{L}_{m}G^{L*}_{n}[\mathscr{LL}+D^{L}_{m}e^{-2j\phi_{m}}\mathscr{RL}+$
$\displaystyle
D^{L*}_{n}e^{2j\phi_{n}}\mathscr{LR}+D^{L}_{m}D^{L*}_{n}e^{-2j(\phi_{m}-\phi_{n})}\mathscr{RR}]$
$\displaystyle r^{RL}_{mn}$ $\displaystyle=$ $\displaystyle
G^{R}_{m}G^{L*}_{n}[\mathscr{RL}+D^{R}_{m}e^{2j\phi_{m}}\mathscr{LL}+$
$\displaystyle
D^{L*}_{n}e^{2j\phi_{n}}\mathscr{RR}+D^{R}_{m}D^{L*}_{n}e^{2j(\phi_{m}+\phi_{n})}\mathscr{LR}]$
$\displaystyle r^{LR}_{mn}$ $\displaystyle=$ $\displaystyle
G^{L}_{m}G^{R*}_{n}[\mathscr{LR}+D^{L}_{m}e^{-2j\phi_{m}}\mathscr{RR}+$
$\displaystyle
D^{R*}_{n}e^{-2j\phi_{n}}\mathscr{LL}+D^{L}_{m}D^{R*}_{n}e^{-2j(\phi_{m}+\phi_{n})}\mathscr{RL}].$

The last two equations and their approximations are used in GPCAL (Park et
al., 2021b) and LPCAL (Leppanen et al., 1995), respectively. In the original
GPCAL pipeline, the D-terms are assumed to remain constant over frequency
within the bandwidth and over time. We present new methods for calibrating
frequency- (the present paper) and time-dependent polarimetric leakages (Paper
II) that have been implemented in GPCAL.

##  3 Calibration Procedure

We present the method of correcting for frequency-dependent polarimetric
leakages in VLBI data. The method assumes that antenna gains are unity, same
as the original GPCAL pipeline, assuming that they are accurately constrained
during the upstream calibration procedure and the imaging and self-calibration
procedure. There is still a single phase offset ($e^{j\phi_{RL,\rm ref}}$)
remaining in the cross-hand visibilities of all baselines, which originates
from the instrumental phase offset between polarizations at the reference
antenna.

The equations of our interest are:

$\displaystyle r^{RL}_{mn}(\nu)$ $\displaystyle=$
$\displaystyle\left[\tilde{Q}_{mn}(\nu)+j\tilde{U}_{mn}(\nu)\right]+D^{R}_{m}(\nu)e^{2j\phi_{m}}r^{LL}_{mn,{\rm
cal}}(\nu)$ $\displaystyle+$ $\displaystyle
D^{L*}_{n}(\nu)e^{2j\phi_{n}}r^{RR}_{mn,{\rm cal}}(\nu)$ $\displaystyle+$
$\displaystyle
D^{R}_{m}(\nu)D^{L*}_{n}(\nu)e^{2j(\phi_{m}+\phi_{n})}\left[\tilde{Q}_{mn}(\nu)-j\tilde{U}_{mn}(\nu)\right]$
$\displaystyle r^{LR}_{mn}(\nu)$ $\displaystyle=$
$\displaystyle\left[\tilde{Q}_{mn}(\nu)-j\tilde{U}_{mn}(\nu)\right]+D^{L}_{m}(\nu)e^{-2j\phi_{m}}r^{RR}_{mn,{\rm
cal}}(\nu)$ $\displaystyle+$ $\displaystyle
D^{R*}_{n}(\nu)e^{-2j\phi_{n}}r^{LL}_{mn,{\rm cal}}(\nu)$ $\displaystyle+$
$\displaystyle
D^{L}_{m}(\nu)D^{R*}_{n}(\nu)e^{-2j(\phi_{m}+\phi_{n})}\left[\tilde{Q}_{mn}(\nu)+j\tilde{U}_{mn}(\nu)\right].$

$\mathscr{RL}$ and $\mathscr{LR}$ are replaced by Equation 6 and
$\mathscr{RR}$ and $\mathscr{LL}$ by the final calibrated parallel-hand
visibilities $r^{RR}_{mn,{\rm cal}}$ and $r^{LL}_{mn,{\rm cal}}$,
respectively. It is assumed that $e^{j\phi_{RL,{\rm ref}}}$ has already been
corrected, which can be done by comparing the integrated electric vector
position angles (EVPAs) of calibrators to known values. This can usually be
accomplished by using sources with stable integrated EVPAs over time or from
contemporaneous observations of single dishes or connected arrays.
$\tilde{Q}_{mn}(\nu)$ and $\tilde{U}_{mn}(\nu)$ are obtained from the Fourier
Transforms of the source’s linear polarization images. In order to obtain
these images, CLEAN is applied to the polarimetric leakage corrected data
acquired by running the GPCAL pipeline on the frequency-averaged data. It is
assumed that the linear polarization images are constant over frequency for
each baseband channel (often referred to as “IF”). We make the assumption that
the fluctuations of polarimetric leakages across different frequencies are
more significant than those of the source’s intrinsic linear polarization
emission. To obtain the values of the former, we consider the latter to be a
constant and solve for it accordingly444In spite of this,
$\tilde{Q}_{mn}(\nu)$ and $\tilde{U}_{mn}(\nu)$ are not exactly constant over
frequency since the visibility at different $\nu$ has a different $(u,v)$,
although the variation due to the $(u,v)$ coverage effect is generally
expected to be small.. Field-rotation angles are independent of the frequency.

The method fits Equation 3 to the data for each channel to determine
$D^{R}(\nu)$ and $D^{L}(\nu)$ using the Scipy curve_fit package. As with the
GPCAL pipeline, it is possible to fit the model to data from multiple
calibrator sources simultaneously to enhance the fitting accuracy.

The method uses a multi-source UVFITS dataset (AIPS data catalog file)
provided by users. To reduce calculation time and memory requirements, it bins
the data in frequency, if specified by users. This approach is usually
recommended because recent VLBI data often contain a large number of channels,
while the D-terms are unlikely to vary significantly across adjacent channels.
Determining the optimal bin size may be data-dependent and therefore, it is
recommended that users determine this parameter themselves. If the phase
offset ($e^{j\phi_{RL,{\rm ref}}}$) still exists in the provided data, the
method can remove the offset using the offset value provided by users555As
explained above, the offset value can be derived by performing the EVPA
calibration of the frequency-averaged data., which may vary depending on the
IF used. The multi-source UVFITS data is usually not self-calibrated. The
method performs amplitude and phase self-calibration with the task CALIB in
AIPS, if requested, using the total intensity CLEAN models (saved in Difmap
fits image files) provided by users.

The method runs the GPCAL pipeline to derive the D-terms for each channel. It
then calibrates the multi-source UVFITS data using the derived D-term spectra
to obtain $\bar{V}$ in Equation 3. This results in the final output being a
multi-source UVFITS dataset saved in an AIPS catalog after correction of the
frequency-dependent D-terms.

The delay offset between polarizations at the reference antenna is corrected
during data pre-processing in AIPS after global fringe fitting. It should be
noted, however, that the frequency-dependent instrumental polarization may
prevent accurate delay offset corrections. Thus, it is recommended to perform
additional delay offset corrections using the output data after the frequency-
dependent D-terms have been removed. It is recommended that users perform this
additional correction after checking the output file, as it is not included in
the calibration procedure. The method is implemented in GPCAL, which is
publicly available at https://github.com/jhparkastro/gpcal.

##  4 Validation using Synthetic Data

We validate our method by employing synthetic data generated from actual VLBA
data. Our objective is to evaluate the effectiveness of our method on
synthetic data that closely resembles real data in terms of properties. To
achieve this, we acquired the ground-truth models of total intensity and
linear polarization for the sources, as well as the noise characteristics and
frequency-dependent leakages from real VLBA 15 GHz data. These real data
components served as the basis for generating the synthetic data. The data we
selected is part of the Monitoring of Jets in Active Galactic Nuclei with VLBA
Experiments666https://www.physics.purdue.edu/MOJAVE/ (MOJAVE; Lister et al.
2018) data sets, which have been monitoring a number of AGN jets with the VLBA
at 15 GHz for decades. The data were collected from the observation of 25
sources on 2020 June 14 (project code: BL273AK). The data were recorded in
both RCP and LCP, with two-bit quantization in four IFs, using the digital
downconverter (DDC) observing system, at a recording rate of 4 Gbps, yielding
a total bandwidth of 128 MHz for each IF and polarization.

We performed a standard data postcorrelation process with AIPS according to
Park et al. (2021c) and hybrid imaging with CLEAN and self-calibration in
Difmap. We ran the GPCAL pipeline on the self-calibrated data. We used several
calibrators with linear polarization structures dominated by cores for the
initial D-term estimation using the similarity approximation. Further
instrumental polarization self-calibration was performed with 10 iterations
using several calibrators with high signal-to-noise ratios in the cross-hand
visibilities. The phase offset between polarizations at the reference antenna
was corrected by comparing the integrated EVPAs of several calibrators using
the D-term corrected data provided by GPCAL with the data already calibrated
in the monitoring program’s public database. We produced Stokes $Q$ and $U$
images of the sources with CLEAN in Difmap.

Figure 1: Comparison of the ground-truth D-term components (real and imaginary
parts) assumed in the synthetic data sets shown on x-axis with the
reconstructed D-term components derived by the method shown on y-axis for each
frequency channel. Each station’s result is presented using distinct colors.
For each antenna, the correlation between the actual and estimated D-term
components is shown in units of percentage. The norm $L_{1}\equiv|D_{\rm
Truth}-D_{\rm Recon}|$ is averaged over real and imaginary components of the
D-terms for RCP and LCP for all antennas, and is denoted in each panel.
Figure 2: Linear polarization images of 3C 273 from the synthetic data set
generated based on the real VLBA 15 GHz data. Colored ticks indicate EVPAs,
with colors corresponding to linearly polarized intensities. We have made
Ricean de-biasing corrections (i.e., $P_{\rm corr}=P_{\rm
obs}\sqrt{1-(\sigma_{P}/P_{\rm obs})}$) to the images (Wardle & Kronberg,
1974), where $P_{\rm corr}$ and $P_{\rm obs}$ denote the corrected and
original linearly polarized intensity, respectively. The white crosses in the
bottom right corner represent the shape of the synthesized beam. The top left
image is the ground-truth image used for synthetic data generation. The top
right image is obtained from the data without being corrupted by polarimetric
leakage (“No D-Terms”). Hence, this image comprises solely of two potential
sources of error, namely thermal noise within the data and inaccuracies caused
by the utilization of the CLEAN algorithm. Consequently, this image could
serve as a benchmark reference for the purpose of scrutinizing the influence
of polarimetric leakages within the data. In the bottom left image, we have
applied our method of frequency-dependent instrumental polarization
calibration to data where frequency-dependent leakage corruption has been
introduced (“channel-by-channel D-term correction”). Using the same data, the
bottom right image was produced by averaging the data over frequency for each
IF and then performing instrumental polarization calibration based on the
averaged data (“D-term correction using frequency-averaged data”). A
comparison of the bottom left and right images can be used to evaluate the
effect of frequency-dependent leakage calibration. With the exception of the
ground-truth image, which is noise-free, each panel exhibits the dynamic range
of the linear polarization images.  Figure 3: Dirty images of the residual
visibilities of synthetic data for 3C 273, which represent the visibilities
obtained after subtracting the CLEAN model visibilities for Stokes $Q$ (upper)
and $U$ (lower). White contours represent the contour of the Stokes I image,
with the contour level set at three times the off-source image rms noise,
delineating the source region. Images on the left, obtained with frequency-
dependent D-term correction, exhibit significantly reduced noise levels
compared to those on the right, which use frequency-averaged data for D-term
correction.

We generated synthetic data for three sources, 3C 273, 3C 279, and OJ 287,
using the synthetic data generation function implemented in GPCAL, as
described by Park et al. (2021a). In brief, the function generates a synthetic
data set based on Equation 7, using the $I$, $Q$, and $U$ CLEAN models as
ground-truth source structures (i.e., their Fourier Transforms are
$\tilde{I}$, $\tilde{Q}$, $\tilde{U}$ in Equation 6), and assuming
$\tilde{V}=0$. The multi-channel real VLBA 15 GHz data are replaced by the
source model visibility corresponding to each $(u,v)$, and we added thermal
noise based on the uncertainties of the real data for each channel. We
introduce frequency-dependent D-terms, which were derived from the real data
using GPCAL (Section 5). For example, the frequency-dependent leakages of the
VLBA BR station shown in the left panel of Figure 4 are introduced to the
synthetic data set of the same station using Equation 7. Since the purpose of
the synthetic data test is to validate our method of correcting for frequency-
dependent leakages, we assumed unity antenna gains, i.e., ${\bm{G}}={\bm{I}}$
(However, refer to the discussion below and Appendix B for the analysis of
synthetic data, incorporating antenna gain distortions.).

From the synthetic data set, we derived frequency-dependent D-terms with
GPCAL. Firstly, the synthetic data set was averaged over frequency for each
IF. The total intensity CLEAN models are reconstructed using Difmap for each
source from this data set. Secondly, we corrected antenna leakages by running
the GPCAL pipeline on the frequency-averaged data, from which we obtained
Stokes $Q$ and $U$ CLEAN images. For this leakage correction, an initial
D-term estimation is performed with OJ 287 based on the similarity
approximation, followed by instrumental polarization self-calibration with 10
iterations using all three sources. Thirdly, frequency-dependent leakages were
derived with the method from the frequency-unaveraged data set and using the
$Q$ and $U$ CLEAN images. For this estimation, all three sources were used
simultaneously. Lastly, we averaged the frequency-dependent leakage corrected
data set over frequency and determined Stokes $Q$ and $U$ images based on that
data set.

In Figure 1, we compare the components of the ground-truth D-terms (real and
imaginary parts) with those of the reconstructed D-terms. Each panel indicates
an average of the $L_{1}\equiv|D_{\rm Truth}-D_{\rm Recon}|$ norm. The method
was able to reproduce ground-truth D-terms with an accuracy of an $\langle
L_{1}\rangle$ norm of $\lesssim 0.1\%$ for all VLBA stations.

Figure 4: D-terms for RCP (blue) and LCP (green) for BL273AK (left, 15 GHz)
and BM462M (right, 43 GHz) data. The D-terms of specific stations, such as BR
for BL273AK and PT for BM462M, are presented as examples. The open circles
indicate the D-term spectra derived from GPCAL’s frequency-dependent leakage
calibration ($D(\nu)$), while the solid lines indicate D-terms obtained from
the frequency-averaged data set ($D(\bar{\nu})$). We note that in the right
figure, the green and blue solid lines representing the amplitudes of the 4th
IF overlap, making them difficult to distinguish from each other. The D-term
spectra exhibit smooth and continuous variation over frequency within and
across IFs. The $D(\bar{\nu})$ is a good approximation of the $D(\nu)$ for
both data sets. It is worth noting a noticeable discrepancy between
$D(\bar{\nu})$ and $D(\nu)$ in the right figure. However, the numerical
difference is relatively small, and it is important to emphasize that
$D(\bar{\nu})$ is not necessarily an average of $D(\nu)$, as discussed in
Appendix A..

Figure 2 showcases the impact of frequency-dependent leakage correction on the
resulting linear polarization images of 3C 273, as compared to the ground-
truth linear polarization image employed in the generation of synthetic data
(top left, labeled as “Ground-Truth”). Aside from the linear polarization
image obtained after correcting for frequency-dependent D-terms (bottom left,
“channel-by-channel D-term correction”), two reference polarization images
were produced. The first is an image that has been reconstructed from a
synthetic data set generated in accordance with the above description, but
without taking into account leakage (top right, “No D-terms”). There are only
two sources of error in this image: thermal noise in the data and errors
introduced by the CLEAN algorithm. As a result, this image could function as a
standard point of reference for evaluating the impact of polarimetric leakages
present in the data. Another image is reconstructed using the synthetic data
set, however, instead of a frequency-dependent leakage correction, the
leakages are corrected using the data set averaged over frequency for each IF
(bottom right, “D-term correction using frequency-averaged data”). It is
through this method that LPCAL and the original GPCAL pipeline would perform
leakage calibration. This image can then be used as a reference to test the
effect of frequency-dependent leakage calibration.

In each panel, we note the dynamic range (DR) of the linear polarization
images, defined as $P_{\rm peak}/\sigma_{P}$, where $P_{\rm peak}$ represents
the peak polarization intensity and
$\sigma_{P}\equiv(\sigma_{Q}+\sigma_{U})/2$ represents the average of the rms
noise for the off-source regions in the Stokes $Q$ and $U$ images (Hovatta et
al., 2012). The DR of the image obtained with frequency-dependent D-term
correction (bottom left) is slightly lower than that of the top right image,
while the DR of the image obtained with leakage correction using the
frequency-averaged data (bottom right) is about half that of the top right
image.

In Figure 3, we present the dirty images of the residual visibilities, which
are the visibilities after subtracting the CLEAN model visibilities for Stokes
$Q$ and $U$. These images showcase the noise levels within the residual
visibilities, which can serve as an indicator of the accuracy of the leakage
calibration. It is evident that the application of frequency-dependent leakage
correction has resulted in a significant reduction in image noise compared to
the image obtained with leakage correction using frequency-averaged data.
Based on this difference, it can be concluded that frequency-dependent leakage
correction with the method improves the quality of linear polarization images
significantly when antenna gains are perfectly calibrated.

##  5 Application to Real Data

###  5.1 Data & Analysis

In this section, we present the results of applying the frequency-dependent
leakage calibration method to two real VLBA data sets. The first is the same
VLBA 15 GHz data set that was presented in Section 4. The second is part of
the VLBA-BU-BLAZAR monitoring
program777https://www.bu.edu/blazars/VLBAproject.html, which has monitored
many $\gamma$-ray bright AGN jets with the VLBA at 43 GHz for decades. We used
the data from the observation of 33 sources on 2018 December 8 (project code:
BM462M). The data were recorded in both RCP and LCP with two-bit quantization
in four IFs, using the digital downconverter (DDC) observing system, at a
recording rate of 2 Gbps, yielding a total bandwidth of 64 MHz for each IF and
polarization.

We first obtained source’s total intensity and linear polarization images from
the frequency-averaged data set as described in Section 4. Based on these
images, frequency-dependent D-terms were derived using the method following
the same procedure described in Section 4. We used multiple bright calibrators
simultaneously to derive the D-terms (OJ 287, 3C 273, and 3C 279 for the 15
GHz data and OJ 287, 3C 273, 3C 279, 3C 84, and 3C 454.3 for the 43 GHz data).
After correcting for frequency-dependent D-terms, we averaged the data set
over frequency and ran the GPCAL pipeline. This is because we derived
frequency-dependent D-terms using the data for each channel, which have
smaller S/Ns than the frequency-averaged data, D-terms with small amplitudes
may remain in the data set.

Figure 5: Cross power spectra (amplitude; $|V|$, phase; $\phi$) of the cross-
hand visibilities (RL; upper, LR; lower) of 3C 279 obtained from the BL273AK
data. The baselines associated with BR station averaged over one scan are
shown. The upper and lower spectra are presented before and after correcting
for frequency-dependent polarimetric leakage, respectively.

Figure 6: Same as Figure 5 but for PT baselines of the BM462M data observed at
43 GHz.

Figure 7: Linear polarization images of 3C 273 from the BL273AK data (upper,
15 GHz) and BM462M data (lower, 43 GHz). Colored ticks indicate EVPAs, with
colors corresponding to linearly polarized intensities. The white crosses in
the bottom right corner of the right panel represent the shape of the
synthesized beam. The images obtained by using the GPCAL pipeline using the
data averaged over frequency within eacn IF are presented in the left panels.
The images obtained by correcting frequency-dependent polarimetric leakages
using the method presented in this paper are shown in the right panels. The
polarization image DR is noted in each panel.

###  5.2 Results

In Figure 4, we present the derived D-term spectra ($D(\nu)$) for BR and PT
stations for the BL273AK and BM462M data sets, respectively. The D-terms
derived from the frequency-averaged data using GPCAL are overplotted
($D(\bar{\nu})$; solid lines). In both data sets, the D-term spectra display
smooth variation over frequency across different IFs, indicating that these
spectra accurately reflect the characteristics of the instruments. As
expected, $D(\bar{\nu})$ follows $D(\nu)$ closely, but only approximates the
rapidly varying spectrum.

Figures 5 and 6 show the cross-hand visibilities of 3C 279 averaged over one
scan (with a scan length of $\sim 1.5$ and $\sim 4.5$ minutes for BL273AK and
BM462M data, respectively) for both data sets before and after correction for
frequency-dependent polarimetric leakages. We present data on the baselines
associated with specific stations (BR station for BL273AK and PT station for
BM462M) as an example. Cross-hand visibility amplitudes of many BR baselines
in the BL273AK data exhibit a similar pattern, which is a rapid increase in
amplitude with frequency for IF 4888It is not surprising to observe the rapid
increase for IF 4, which has a frequency coverage of 15.496 – 15.624 GHz. The
nominal frequency range of the VLBA 2 cm band is 12.0 – 15.4 GHz according to
the VLBA Observational Status Summary
(https://science.nrao.edu/facilities/vlba/docs/manuals/oss/bands-perf). The IF
is near the band edge, where the polarizer characteristic can become poor.. It
is noteworthy that this behavior is very similar to the trend of the derived
D-term spectra for BR station (the left panel of Figure 4). This indicates
that the frequency-dependent polarimetric leakage of this station is
responsible for the pattern in the cross-hand visibilities. The behavior
disappears when the D-terms are removed from the data (lower panel). Similar
results were obtained for the BM462M data as well (Figure 6). As an example,
after the correction, the zigzag pattern in the RL baselines associated with
PT station has disappeared. In conclusion, these results demonstrate that the
frequency-dependent calibration method implemented in GPCAL is effective.

###  5.3 Evaluation

In this subsection, we evaluate the effects of frequency-dependent
instrumental polarization calibration on linear polarization images. In Figure
7, we present the linear polarization image of the bright quasar 3C 273, which
has a complex linear polarization structure, after correcting for frequency-
dependent D-terms. It is compared with the image obtained from frequency-
averaged data. It was found that the linear polarization images obtained with
and without frequency-dependent leakage correction were almost identical. The
results showed that the polarization image dynamic range slightly improved
with the correction. The results for other sources were similar.

According to our results, frequency-dependent instrumental polarization
calibration can slightly improve the quality of resulting linear polarization
images in some cases, however, the improvement is generally not very
significant. Our hypothesis is that this is because the primary contribution
of polarimetric leakages to the cross-hand visibilities is linear, and the
distortion caused by averaging the data over frequency before correcting for
frequency-dependent leakages is generally small (see Appendix A for more
details). Also, other systematic errors in the data, such as residual antenna
gain errors, may dominate the errors caused by averaging data over frequency
before correction for frequency-dependent leakage. In Appendix B, we test this
claim by using synthetic data sets that include both antenna gain and
frequency-dependent leakage corruptions. In fact, we found that the quality of
linear polarization images does not improve much with frequency-dependent
instrumental polarization calibration, which is similar to what we found in
real data sets.

Generally, if D-term amplitudes are large and significantly vary over
frequency within an IF, the distortion in cross-hand visibility caused by
averaging data over frequency before correcting for frequency-dependent
leakages (see Equation 3) will increase and degrade the quality of linear
polarization images. Therefore, our method of frequency-dependent instrumental
polarization calibration implemented in GPCAL will be particularly useful for
calibrating such data. There has been a substantial increase in the bandwidth
of modern radio interferometers in recent years. The calibration of frequency-
dependent instrumental polarization becomes more critical for future VLBI
data, where the D-terms are expected to vary significantly over a wide
bandwidth. Connected interferometers, such as the Very Large Array and the
Atacama Large Millimeter/submillimeter Array, have already observed this
phenomenon (e.g. Nagai et al., 2016; Bower et al., 2018; Jagannathan et al.,
2017, 2018).

##  6 Conclusions

Compared to the past, modern VLBI arrays offer a much wider bandwidth. The
large bandwidth of the data allows for a significant reduction in thermal
noise. However, systematic errors in the data do not decrease as bandwidth
increases, and this can pose a significant limitation in terms of image
quality. In the case of linear polarization images, this effect is more
severe, since conventional calibration methods are not able to easily remove
the systematic errors. Some of the systematic errors are polarimetric leakages
varying over frequency within the bandwidth and over time. Leakages usually
become larger at frequencies far from the nominal frequency in the band, as
the instruments are designed to have small leakages at the nominal frequency.
Also, polarization leakage is direction-dependent effect (see, e.g., lecture 3
in Taylor et al. 1999; Thum et al. 2008; Smirnov 2011). Accordingly, D-terms
can vary depending on the antenna pointing accuracy, which varies from scan to
scan. Most existing calibration tools, however, assume that leakages are
constant over time and frequency.

We present in this series of papers new calibration methods for frequency-
(this paper) and time-dependent (Paper II) polarimetric leakages in VLBI data,
which have been implemented in GPCAL. The method presented in this paper
determines leakages for each channel based on the Stokes $Q$ and $U$
calibrator models, which are derived from the frequency-averaged data set. As
in the original GPCAL pipeline, data from multiple calibrator sources can be
used simultaneously for fitting to improve accuracy.

We verified the method using synthetic data generated based on real VLBA 15
GHz data. The method was able to reconstruct the ground-truth D-terms with an
accuracy of $\langle L_{1}\rangle\lesssim 0.1\%$. The linear polarization
images are significantly improved with the frequency-dependent leakage
calibration using the method in the case of no antenna gain corruption assumed
in the synthetic data.

We applied the method to two sets of real data obtained with the VLBA at 15
and 43 GHz. The D-term spectra were derived using several bright and
moderately to highly polarized calibrators. There is continuous and smooth
variation in the spectra across all bandwidths, which indicates that the
spectra are an accurate representation of the characteristics of the VLBA’s
instruments. The D-terms obtained from the data averaged over frequency within
each IF are a good approximation of the D-term spectra. After correction of
frequency-dependent D-terms, substantial variations in the cross-hand
visibility spectra over frequency disappear. This result indicates that the
variation is caused by frequency-dependent polarimetric leakages, rather than
by intrinsic signals of the source. This result also indicates that the method
is effective in removing the frequency-dependent leakages from real VLBI data.

We evaluated the effects of frequency-dependent instrumental polarization
calibration on linear polarization images. There were no significant
differences between the images obtained after correcting for frequency-
dependent D-terms and those obtained from frequency-averaged data, although
the dynamic range of the linear polarization images with frequency-dependent
leakage corrections were slightly higher.

Based on these results, the calibration of D-terms based on frequency-averaged
data appears to be a reasonable approximation for the particular VLBA data
sets analyzed in this paper. The reason for this is that the primary
contribution of D-terms to the cross-hand visibilities is linear. When the
cross-hand visibilities are averaged over frequency before correcting
frequency-dependent D-terms, there are second-order terms that can cause
distortion. The distortion is expected to be small if the D-term amplitudes of
frequency channels are small and do not vary significantly over frequency
within each IF. Furthermore, other systematic errors in the data, such as
residual antenna gain errors, may dominate the errors caused by averaging data
over frequency before correcting for frequency-dependent leakage. This appears
to be the case for the VLBA data sets presented in this paper. Nevertheless,
we observe some improvement in the quality of polarization images as a result
of calibration of frequency-dependent D-terms, indicating that users should
consider calibration of frequency-dependent D-terms to increase calibration
precision. Additionally, it will contribute significantly to the calibration
of future VLBI data, for which significant variations in D-terms are expected
over frequency within large bandwidths.

We express our gratitude to the anonymous referee for conducting a
comprehensive review of our manuscript, which greatly enhanced the quality of
the paper. J.P. acknowledges financial support through the EACOA Fellowship
awarded by the East Asia Core Observatories Association, which consists of the
Academia Sinica Institute of Astronomy and Astrophysics, the National
Astronomical Observatory of Japan, Center for Astronomical Mega-Science,
Chinese Academy of Sciences, and the Korea Astronomy and Space Science
Institute. This work is supported by the Ministry of Science and Technology of
Taiwan grant MOST 109-2112-M-001-025 and 108-2112-M-001-051 (K.A). The VLBA is
an instrument of the National Radio Astronomy Observatory. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated by Associated Universities, Inc.

##  Appendix A Distortion in cross-hand visibilities caused by frequency-
dependent polarimetric leakages

The original GPCAL pipeline uses the cross-hand visibilities averaged over
frequency within each IF. This is equivalent to take average of the left and
right hand sides of Equation 3:

$\displaystyle\langle r^{RL}_{mn}(\nu)\rangle_{\nu}$ $\displaystyle=$
$\displaystyle\langle\tilde{Q}_{mn}(\nu)+j\tilde{U}_{mn}(\nu)\rangle_{\nu}+\langle
D^{R}_{m}(\nu)e^{2j\phi_{m}}r^{LL}_{mn,{\rm cal}}(\nu)\rangle_{\nu}$
$\displaystyle+$ $\displaystyle\langle
D^{L*}_{n}(\nu)e^{2j\phi_{n}}r^{RR}_{mn,{\rm cal}}(\nu)\rangle_{\nu}$
$\displaystyle+$ $\displaystyle\langle
D^{R}_{m}(\nu)D^{L*}_{n}(\nu)e^{2j(\phi_{m}+\phi_{n})}(\tilde{Q}_{mn}(\nu)-j\tilde{U}_{mn}(\nu))\rangle_{\nu},$

where $\langle\cdots\rangle_{\nu}$ indicates averaging over frequency. The
source’s emission is not expected to change significantly within the frequency
range. As a result, $\tilde{Q}_{mn}$, $\tilde{U}_{mn}$, $r^{LL}_{mn,{\rm
cal}}$, and $r^{RR}_{mn,{\rm cal}}$ can be approximated as a constant if we
ignore the small variations in these quantities with frequency caused by the
different $(u,v)$ coverage. Thus, Equation A can be rewritten as follows:

$\displaystyle\langle r^{RL}_{mn}(\nu)\rangle_{\nu}$ $\displaystyle=$
$\displaystyle{\rm const}+\langle D^{R}_{m}(\nu)\rangle_{\nu}\times{\rm
const}+\langle D^{L*}_{n}(\nu)\rangle_{\nu}\times{\rm const}$ (A2)
$\displaystyle+$ $\displaystyle\langle
D^{R}_{m}(\nu)D^{L*}_{n}(\nu)\rangle_{\nu}\times{\rm const}.$

In the absence of the second-order term, the D-terms derived using frequency-
averaged visibilities must equal the average of the spectra of the true
D-terms. In other words, if the second-order term is not present, correcting
for D-terms by using frequency-averaged data is equivalent to correcting for
D-terms for each frequency channel and then performing frequency averaging.
Due to the second-order terms, however, the two cases differ as $\langle
D^{R}_{m}(\nu)D^{L*}_{n}(\nu)\rangle_{\nu}\neq\langle
D^{R}_{m}(\nu)\rangle_{\nu}\langle D^{L*}_{n}(\nu)\rangle_{\nu}$. There is an
exception to this relationship when the D-terms are constant over frequency,
which we do not take into account in this paper. Thus, if the D-term
amplitudes are small, the effect of calibration of frequency-dependent
instrumental polarization will be small.

##  Appendix B Test using synthetic data with antenna gain corruptions

A synthetic data set with antenna gain matrices assumed to be identity
matrices was used in Section 4 to verify GPCAL’s frequency-dependent leakage
calibration method. Any realistic VLBI data, however, will contain antenna
gain corruptions. In this Appendix, we test the method with a synthetic data
set containing gain corruptions.

We generated a synthetic data set in accordance with Section 4, but we
included the corruption of antenna gain in the data set. Antenna gains were
assumed to be constant over frequency and to vary from scan to scan, but to
remain constant within a scan. The gain amplitudes for each station are
randomly drawn from a Gaussian distribution with a standard deviation of 0.05.
An overall shift in the gain amplitudes from unity has been introduced by a
uniform random value between $-0.1$ and $0.1$ for each station. Following the
synthetic data generation conducted for the EHT analysis (e.g., Event Horizon
Telescope Collaboration et al., 2019), a random station phase is adopted for
each scan to simulate the absence of absolute phase.

We derived frequency-dependent D-terms based on the synthetic data sets, as
described in Section 4. Due to the presence of gain corruptions in this data
set, we performed iterative CLEAN and self-calibration on Stokes I data prior
to frequency-dependent leakage calibration. The reconstructed D-term
components (real and imaginary parts) are compared to the ground truth
components in Figure 8. For all VLBA stations, we were able to reproduce the
ground-truth D-terms with an accuracy of the average $L_{1}$ norm that was
slightly higher than the accuracy obtained in the synthetic data test without
gain corruption (Section 4). In Figure 9, we present the linear polarization
images of 3C 273 obtained from the data without including any leakage (“No
D-terms”; top right), as well as the images obtained from the leakage
corrupted data with (bottom left) and without (bottom right) frequency-
dependent leakage correction.

The DR of the image obtained from the data without including D-terms is
already much lower than the DR obtained from the data without inclusion of
gain corruptions (the top right panel in Figure 2). This value is similar to
that of the DR values for images with and without frequency-dependent leakage
corrections from gain corrupted synthetic data. We present the dirty images of
the residual visibilities for Stokes $Q$ and $U$ in Figure 10. The noise
levels did not exhibit significant improvement with the frequency-dependent
leakage correction for this synthetic data, which contrasts with the results
observed for the synthetic data without gain corruptions (Section 4). It
appears that the gains could not be perfectly calibrated, which has a
significant impact on the quality of the polarization images. Relative to the
degradation caused by residual gain errors, the quality improvement due to
frequency-dependent leakage correction is marginal.

The present result is not surprising since the major contribution of D-terms
to cross-hand visibilities is linear, as outlined in Appendix A. The
distortion in cross-hand visibilities caused by averaging data over frequency
prior to correcting the frequency-dependent D-terms introduced by the second-
order terms in Equation 3 is expected to be small if the amplitudes of the
D-terms are small. It is possible that this distortion may be dominated by
other systematic errors, such as residual gain errors. It is expected that
realistic VLBI data will contain gain errors and this is probably why we did
not see much improvement in the image DR for the real VLBA data sets in
Section 5.

Figure 8: Same as Figure 1 but for the synthetic data including both
frequency-dependent polarimetric leakage and antenna gain corruptions.  Figure
9: Same as Figure 2 but for the synthetic data including both frequency-
dependent polarimetric leakage and antenna gain corruptions.  Figure 10: Same
as Figure 3 but for the synthetic data including both frequency-dependent
polarimetric leakage and antenna gain corruptions.

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