11institutetext: Aalto University Metsähovi Radio Observatory, Metsähovintie
114, FI-02540 Kylmälä, Finland
11email<EMAIL_ADDRESS>22institutetext: Aalto University Department of
Electronics and Nanoengineering, PL15500, FI-00076 Aalto, Finland
33institutetext: Technische Universität Berlin, Institut für Geodäsie und
Geoinformationstechnik, Fakultät VI, Sekr. KAI 2-2, Kaiserin-Augusta-Allee
104-106, DE-10553 Berlin, Germany 44institutetext: Max-Planck-Institut für
Radioastronomie, Auf dem Hügel 69, DE-53121 Bonn, Germany 55institutetext:
Finnish Geospatial Research Institute, Geodeetinrinne 2, FI-02430 Masala,
Finland 66institutetext: Deutsches GeoForschungsZentrum (GFZ), Potsdam,
Telegrafenberg, DE-14473 Potsdam, Germany

# Impacts of the image alignment over frequency for VLBI Global Observing
System

Ming H. Xu 112233 Tuomas Savolainen 112244 James M. Anderson 3366 Niko
Kareinen 55 Nataliya Zubko 55 Susanne Lunz 66 Harald Schuh 6633

(Received ***; accepted ***)

###### Abstract

Aims. The VLBI Global Observing System, which is the next generation of
geodetic VLBI and is called VGOS, observes simultaneously at four frequency
bands in the range 3.0–10.7 GHz (expected to be extended to 14 GHz). Because
source structure changes with frequency, we aim to study the source position
estimates from the observations of this new VLBI system.

Methods. Based on an ideal point source model, simulations are made to
determine the relation between the source positions as determined by VGOS
observations and the locations of the radio emission at the four bands.

Results. We obtained the source positions as determined by VGOS observations
as a function of the source positions at the four frequency bands for both
group and phase delays. The results reveal that if the location of the radio
emission at one band is offset with respect to that at the other bands, the
position estimate can be shifted to the opposite direction and even by more
than three times that offset.

Conclusions. The VGOS source positions will be very variable with time and
very imprecise in the sense of relating to the locations of the radio emission
at the four bands, if the effects of source structure are not modeled. The
image alignment over frequency is essential in order to model the effects of
source structure in VGOS observations, which is the only way to mitigate these
strong frequency-dependent impacts on VGOS source positions.

###### Key Words.:

reference systems / astrometry / galaxies: active / galaxies: jets / radio
continuum: galaxies

##  1 Introduction

In fundamental astronomy, two cardinal improvements in the precision of
astrometric measurements of positions of celestial objects have been achieved
based on very long baseline interferometry (VLBI) at radio wavelengths and the
European Space Agency mission Gaia111https://sci.esa.int/web/gaia (Gaia
Collaboration et al., 2016) at optical wavelengths. A detailed comparison
between the third realization of the International Celestial Reference Frame
(ICRF) (ICRF3; Charlot et al., 2020) and the Gaia Early Data Release 3 (EDR3;
Gaia Collaboration et al., 2020) shows that the median difference of radio and
optical source positions for more than 2000 common sources is about 0.5
milliarcsecond (mas) and for the sources with optical $G$ magnitudes ¡ 18.0
mag the median difference is on the order of 0.3 mas (Xu et al., 2021b). On
the other hand, for the sources that have radio to optical position
differences larger than 3$\sigma$, which is $\sim$24% of the common sources,
it has become more and more convincing that the position differences are
parallel to the directions of the radio jets (Kovalev et al., 2017; Petrov &
Kovalev, 2017; Plavin et al., 2019a; Xu et al., 2021b). Our recent work showed
that these sources are more likely to have extended structure at radio
wavelengths (Xu et al., 2021b). VLBI and Gaia already have the potential of
detecting astrophysical properties about the radio and optical emission at mas
scales for these objects, which are mostly active galactic nuclei (AGNs)
(Plavin et al., 2019a; Petrov et al., 2019).

In the last several decades, celestial reference frame (CRF) sources were
observed predominantly at S/X bands by geodetic VLBI to derive their positions
for building the ICRFs (Ma et al., 1998; Fey et al., 2015; Charlot et al.,
2020). The geodetic VLBI observations are coordinated by the International
VLBI Service for Geodesy and Astrometry
(IVS222https://ivscc.gsfc.nasa.gov/index.html; Schuh & Behrend, 2012;
Nothnagel et al., 2017) since 2000. In order to achieve 1 mm station position
accuracy and 0.1 mm/yr velocity stability on global scales, the IVS is
developing the next-generation, broadband geodetic VLBI system, known as the
VLBI Global Observing System (VGOS; Niell et al., 2007; Petrachenko et al.,
2009). Even though the primary goals are to greatly increase the precision of
Earth orientation parameters and the International Terrestrial Reference Frame
(ITRF) (Altamimi et al., 2016) by using VGOS, the CRF as an integral part of
the geodetic VLBI is also expected to be — and needs to be — improved in the
VGOS era. Based on 21 actual VGOS sessions, it is demonstrated that the
measurement noise in VGOS group delay observables is at the level of $\sim$2
ps (Xu et al., 2021a). A network of nine VGOS stations has observed a 24-hour
session biweekly in 2019 and 2020 and will start to observe weekly in the near
future; this network is expanding globally as planned (Behrend et al., 2020),
with another eleven stations built and nine stations in the planning stage as
of January 2021. VGOS observations started to contribute to the building of
the ITRF through the geodetic solutions submitted by the IVS analysis centers
in 2020333https://ivscc.gsfc.nasa.gov/IVS_AC/IVS-AC_ITRF2020.htm.

Currently, the VGOS system observes simultaneously at four 512 MHz wide bands
centered at 3.3, 5.5, 6.6, and 10.5 GHz, which are labeled as band A, B, C,
and D, respectively, with 32 recording channels (see Niell et al. (2018) and
Table 2 for the channel frequencies used in the current VGOS observations),
while the legacy S/X system observes at S ($\sim$2.2 GHz) and X ($\sim$8.6
GHz) bands. There is a substantial difference in the observing frequencies and
thus the received radio signals from AGNs between the legacy system and the
VGOS system. The CRF sources have different structure at different bands and
the structure changes over time at angular scales of sub-mas, therefore, we
cannot assume that the locations of the radio emission of a source at
different frequency bands are at the same position or have stable relative
positions between the four bands at angular scales of $\sim$0.1 mas to mas.
The fact that source positions change with frequency is one of the reasons for
which the ICRF3 has three separate catalogs at the three frequency bands
(Charlot et al., 2020). The fundamental question is where the position of a
source determined by VGOS observations with respect to the locations of the
radio emission at the four bands is. We aim to address this question in the
study.

On the other hand, because the impact of source structure on VGOS group delay
observables is about one order of magnitude larger than the random noise level
as shown in Xu et al. (2021a), the IVS is making effort, towards deriving
source-structure corrections for geodetic solutions by collecting information
of antenna system temperatures and gain curves and making images from VGOS
observations. Before VGOS source-structure corrections are generated, it is
necessary that the images at the four bands are aligned with respect to each
other. The impact of the potential errors in that alignment needs to be
studied. This is equivalent to the question that we aim to discuss.

The purposes of this study are: (1) to investigate the potential impacts of
the effects of source structure on the source positions as determined by VGOS
observations if these effects will not be modeled and (2) to demonstrate the
importance of the image alignment over frequency when one wants to correct for
these effects. The paper is structured as follows. We introduce in Sect. 2 the
changes of source positions with respect to frequency based on the ICRF3 and
the images obtained based on actual VGOS images. In Sect. 3 we first describe
how the variations in the channel phases affect the VGOS observables and then
derive the formula of the position estimate from VGOS observations as a
function of the locations of the radio emission at the four bands. We make the
discussion in Sect. 4 and our conclusion in Sect. 5.

##  2 Changes of source positions over frequency

Based on globally absolute astrometric observations by VLBI at S/X, K, and
X/Ka bands, the ICRF3 was established independently at these three frequencies
(Charlot et al., 2020). The two catalogs at K and X/Ka bands were aligned to
the S/X catalog by applying no-net-rotation constraints, leading to non-
significant spin-rotations between them. However, the glide deformation
between the S/X and K catalogs is 10–30 microarcseconds ($\mu$as), and that
between the S/X and X/Ka catalogs is as large as $\sim$300 $\mu$as. These
deformations were believed to be due mainly to the different observing
networks among the three bands, e.g., the X/Ka observations were made from a
network of four stations, instead of the intrinsic source position
differences. After applying transformations including both the first-degree
and the second-degree deformation parameters (in total 16 parameters) fitted
from the position differences between these three catalogs, the median angular
separation of the K-band positions relative to the S/X-band positions is
$\sim$0.2 mas, the same level as that for the X/Ka-band positions; about 6% of
the common sources in the K and S/X catalogs have position differences
significant at the 3$\sigma$ confidence level, and 11% for the common sources
in the X/Ka and S/X catalogs.

Figure 1 shows another example for the source 1157$-$215 with two components.
The ratio of the flux densities between the southeastern component and the
northwestern component largely increases from 15.3 GHz to 8.7 GHz; this shifts
the radio positions towards the southeast direction when frequency decreases.
The northwestern component is more likely to be the core than the brightest
component in these images. These two sources have angular separations of the
positions in the K, X/Ka, and Gaia catalogs relative to that in the S/X
catalog all significant at the 3$\sigma$ confidence level (see Tables 14, 15,
16 in Charlot et al. (2020)). Figure 1 demonstrates that source structure and
source positions change significantly with frequency and for this source the
radio positions move in the directions of the optical positions when the radio
frequency goes higher. The changes of the peak component among the core and
the jet components at various frequency bands will lead to large position
offsets between different frequencies, as the source position referred to by
VLBI observations in general is dominated by the position of the peak.

Figure 1: Three radio positions from the ICRF3 at the S/X, K, and X/Ka bands
and the Gaia position for source 1157$-$215, on its radio images at 8.7 GHz
from Astrogeo (left) and at 15.3 GHz from MOJAVE (right). Overlay contours are
shown at levels of peak percentage specified in the bottom of plot. Since
these radio images do not have information of the absolute positions, the
peaks are formally assumed to be located at the S/X-band positions. Error bars
shown are the uncertainties from the four catalogs. The K- and X/Ka-band
positions are the ones after the full deformation transformations (Charlot et
al., 2020). The angular separations of the K- and X/Ka-band positions and the
Gaia position relative to the S/X-band position for this source are
2.29$\pm$0.31 mas, 3.54$\pm$0.32 mas, and 3.97$\pm$0.13 mas, respectively.

In general cases, however, the cores of geodetic radio sources can be reliably
identified because one can rely on a database with the images of a long time
series and the spectrum index maps. Moreover, these sources tend to have
compact cores with extended but weaker jets. To investigate these general
cases, we use the images constructed from actual VGOS observations by:

  1. 1.

deriving the closure images by using the method in Xu et al. (2021d);

  2. 2.

calibrating VGOS observations based on the closure images;

  3. 3.

performing model fitting by using difmap.

Figure 2 shows the images of source 0016+731 from model fitting in difmap at
the four frequency bands based on the VGOS observations VO0051. The source
0016+731 has a compact core, the northwestern component, and weaker jet
components at all the four bands. Three components are consistently detected
at the highest three frequencies, whereas only two components can be detected
at the lowest frequency band due to much lower angular resolution. At 10.5
GHz, the angular distance between the core and the closest jet component is
about 0.44 mas with a flux density ratio of 0.53, and the angular distance
from the second jet component to the core is about 0.68 mas with a flux
density ratio of 0.25. At 3.3 GHz, the angular distance between the core and
the jet component is about 0.68 mas with a flux density ratio of 0.20. With an
increase in the angular resolution by a factor of twofold to threefold at the
three highest frequency bands, the core at 3.3 GHz is resolved. Based on the
parameters of the Gaussian components at 10.5 GHz, the position of the core
would be shifted towards east by 0.14 mas due to the contribution of the
nearest jet component if it appeared as the same structure at 3.3 GHz and 10.5
GHz. Furthermore, if the effects of source structure are not modeled, the
source position shifts due to the contribution of the jet components will
happen at all the frequency bands with larger magnitudes. It is obvious that
there are source position offsets over frequency, since the angular
resolutions of simultaneous observations at the multiple frequency bands are
significantly different, leading to different contributions of the jet
components to the cores at various frequency bands.

Figure 2: Images of source 0016+731 from VGOS session VO0051 (Feb. 20, 2020)
at the frequencies 3.3 GHz (upper left), 5.5 GHz (upper right), 6.6 GHz
(bottom left), and 10.5 GHz (bottom right). The yellow ellipses indicate the
Gaussian components modeled by difmap. These images are constructed from the
VGOS observations, which are calibrated based on the images derived from
closure phases and closure amplitudes (Xu et al., 2021d). Therefore, the units
of the pixel flux densities are arbitrary. The beam size is shown as a black
ellipse in the bottom left corner of each plot.

Changes of source positions with frequency are due mainly to two factors:
extended source structure and the frequency-dependence of the core position,
the so called core shift (Blandford & Königl, 1979). This is supported by the
ICRF3 multi-frequnecy catalogs and the optical positions from Gaia. It is well
demonstrated that the differences between the radio positions and the optical
positions are parallel to the jet directions (see, e.g., Lambert et al., 2021;
Liu et al., 2021). These two effects in general are larger at the lower
frequency bands (the VGOS frequency bands) than at the higher frequency bands
(K and X/Ka bands). It is expected, however, that the impact of these effects
at S band on the previous ICRFs is significantly reduced. According to the
linear combination of the delay observables at S and X bands to remove the
ionospheric effects, which is done in geodetic data analysis after the fringe
fitting process, the source position determined from VLBI observations at
these two bands, denoted by $\@vec{k}_{\scriptsize\mbox{S/X}}$, is given
approximately by

$\@vec{k}_{\scriptsize\mbox{S/X}}=1.07\@vec{k}_{\scriptsize\mbox{X}}-0.07\@vec{k}_{\scriptsize\mbox{S}},$
(1)

where $\@vec{k}_{\scriptsize\mbox{X}}$ and $\@vec{k}_{\scriptsize\mbox{S}}$
are the group-delay source positions at the two bands (see, e.g., Porcas,
2009). When a source is ideal point-like at S and X band, these two positions
can be located at the same direction of the jet base (Porcas, 2009); in
general cases, they are in different directions. As shown in Eq. 1, the
contribution of the S-band positions and their variations are reduced by a
factor of 14. We should note that the effects of source structure at S band
are also scaled down by that factor. In the legacy VLBI system, the source
positions at the higher band dominate over that at the lower band. In the VGOS
system, however, observations are made at four frequency bands instead of two
bands, and the ionospheric effects are fitted in the fringe fitting process
through which group delay observables are determined. As a consequence, VGOS
source positions need to be studied thoroughly.

##  3 Simulation for VGOS source positions

###  3.1 Changes in broadband observables due to phase variations

Let us recall the model of using 32 channel phases to determine the broadband
observables as done in the VGOS post processing using _fourfit_
444https://www.haystack.mit.edu/wp-content/uploads/2020/07/docs_hops_000_vgos-
data-processing.pdf. Following Cappallo (2016), the visibility phase
$\phi_{\nu_{i}}$ at the channel with frequency ${\nu_{i}}$ can be expressed as

$\phi_{\nu_{i}}=\tau^{g}(\nu_{i}-\nu_{0})+\phi_{0}-\frac{k\leavevmode\nobreak\
\delta\mbox{TEC}}{\nu_{i}},$ (2)

where $\tau^{\scriptsize\mbox{g}}$, $\phi_{0}$, and $\delta\mbox{TEC}$ are the
broadband group delay, the broadband phase at the reference frequency
$\nu_{0}$, which is 6.0 GHz in the current data processing, and the
differential total electron content (TEC) along the ray paths of the radio
emission from a source to the two stations of a baseline, respectively; they
are the broadband observables simultaneously fitted in _fourfit_. Constant $k$
is equal to 1.3445 when phases are in units of turns of a cycle, delays in
units of nanoseconds (ns), frequencies in units of GHz, and $\delta$TEC in
units of TECU5551 TECU $\equiv{10^{16}}$ electrons per square meter.. We can
derive the phase delay by
$\tau^{\scriptsize\mbox{p}}=(\phi_{0}+\mbox{N})/\nu_{0}$, where N is the
integer number of phase turns and $\tau^{\scriptsize\mbox{p}}$ is in the units
of ns.

We denote the observational equation as follows,

$l=Ax+\sigma,$ (3)

where $l$ is the vector of the phases at the 32 channels, $A$ is the design
matrix, $x$ is the vector of the three unknowns, i.e., $\tau^{g}$, $\phi_{0}$,
and $\delta$TEC, and $\sigma$ is the noise vector. The vector $l$ consists of
the simulated channel phases based on delay offsets or source position
offsets. The design matrix $A$, with a dimension of 32 $\times$ 3, consists of
the partial derivatives of phase with respect to the three unknowns, which can
be derived based on Eq. 2. By assuming that the visibility amplitudes over 32
channels are flat — equal weights, the normal matrix can be derived by
$A^{\mbox{T}}A$. By least squares fitting (LSF), the changes in the broadband
observables due to changes in the channel phases can be determined.

The results shown in Table 1 were obtained from LSF for seven possible
combinations of bands with a 1 ps delay offset based on the channel
frequencies listed in Table 2. The four scenarios labeled as “Case 1”
correspond to the cases where only one of the four bands has such a delay
offset causing the variations in the eight channel phases. The normal matrix
in the LSF process of estimating the broadband observables is independent of
channel phases, and the estimates of the broadband observables are linearly
dependent on them. Therefore, there are two features in this broadband fitting
process. First, the coefficients of propagating the delay offsets at
individual bands to the broadband observables are independent of the
magnitudes of those delay offsets: they are invariable for a given set of
channel frequencies. Second, the results of all other possible combinations of
bands with delay offsets can also be derived from the four basic scenarios in
Case 1 through a linear combination. For instance, the first scenario in Case
2 is the summation of the results of the first two scenarios in Case 1, and
the Case 3 is the summation of that of the four scenarios in Case 1.

Table 1: Changes in the broadband observables due to delay offsets of 1 ps at bands of various combinations. | Delay offsets at band |  | Changes in
---|---|---|---
|  A |  B |  C |  D |  | $\tau^{\scriptsize\mbox{g}}$ | $\tau^{\scriptsize\mbox{p}}$ | $\delta\mbox{TEC}$
|  [ps] |  [ps] |  [ps] |  [ps] |  | [ps] | [ps] | [TECU]
Case 1 |  1.0 |  0.0 |  0.0 |  0.0 |  | $0.505$ | $-0.883$ | $-0.024$
0.0 |  1.0 |  0.0 |  0.0 |  | $-1.448$ | $1.729$ | $0.034$
0.0 |  0.0 |  1.0 |  0.0 |  | $-1.458$ | $2.044$ | $0.040$
0.0 |  0.0 |  0.0 |  1.0 |  | $3.401$ | $-1.889$ | $-0.050$
Case 2 |  1.0 |  1.0 |  0.0 |  0.0 |  | $-0.943$ | $0.846$ | $0.010$
0.0 |  0.0 |  1.0 |  1.0 |  | $1.943$ | $0.154$ | $-0.010$
Case 3 |  1.0 |  1.0 |  1.0 |  1.0 |  | $1.0$ | $1.0$ | $0.0$

For general scenarios, where the delay offsets at the four bands are
$\Delta\tau_{\scriptsize\mbox{A}}$, $\Delta\tau_{\scriptsize\mbox{B}}$,
$\Delta\tau_{\scriptsize\mbox{C}}$, and $\Delta\tau_{\scriptsize\mbox{D}}$,
respectively, the change in the broadband group delay, denoted by
$\Delta\tau^{\scriptsize\mbox{g}}$, can be obtained from the results in Table
1 as

$\Delta\tau^{\scriptsize\mbox{g}}=+0.505\Delta\tau_{\scriptsize\mbox{A}}-1.448\Delta\tau_{\scriptsize\mbox{B}}-1.458\Delta\tau_{\scriptsize\mbox{C}}+3.401\Delta\tau_{\scriptsize\mbox{D}},$
(4)

and the change in the phase delay, denoted by
$\Delta\tau^{\scriptsize\mbox{p}}$, as

$\Delta\tau^{\scriptsize\mbox{p}}=-0.883\Delta\tau_{\scriptsize\mbox{A}}+1.729\Delta\tau_{\scriptsize\mbox{B}}+2.044\Delta\tau_{\scriptsize\mbox{C}}-1.889\Delta\tau_{\scriptsize\mbox{D}}.$
(5)

This section is similar to what has been done for the investigation of the
impact of constant instrumental delays between different bands on VGOS
broadband delays by Corey & Himwich (2018). Equation 4 can be used to recover
the results of the fifteen possible scenarios in Fig. 1 and Table 2 in their
VGOS memo666https://www.haystack.mit.edu/wp-
content/uploads/2020/07/memo_VGOS_050.pdf.

###  3.2 Source positions determined by VGOS observations

The delay offset at an individual band can change with time and be baseline-
and source-dependent due to some physical causes, for instance, a source
position offset. We denote the (phase-delay) source position at band A by
$\@vec{k}_{0}+\Delta\@vec{k}_{\scriptsize\mbox{A}}$, where $\@vec{k}_{0}$ is a
reference direction and $\Delta\@vec{k}_{\scriptsize\mbox{A}}$ is a position
offset with respect to the reference direction; and so forth
$\@vec{k}_{0}+\Delta\@vec{k}_{\scriptsize\mbox{B}}$,
$\@vec{k}_{0}+\Delta\@vec{k}_{\scriptsize\mbox{C}}$, and
$\@vec{k}_{0}+\Delta\@vec{k}_{\scriptsize\mbox{D}}$ for the other three bands,
respectively. These source positions are defined at the center frequencies of
individual bands, because source structure and core shift evolve with
frequency even within a band. We remark that in presence of source structure
there is no a unique reference position of a source for different baselines or
a baseline at different observing epochs, but the point here is that there are
position offsets among the four bands due to extended source structure, as we
have discussed in Sect. 2.

The (phase) delay offset at band A due to the source position offset
$\Delta\@vec{k}_{\scriptsize\mbox{A}}$ is computed as

$\Delta\tau_{\scriptsize\mbox{A}}=-\frac{\@vec{B}\cdot\Delta\@vec{k}_{\scriptsize\mbox{A}}}{c},$
(6)

where $\@vec{B}$ is the baseline vector and $c$ is the speed of light. There
is no need need for spherical trigonometry when the position offset is small.
This is always possible because we would expect the differences in the
locations of the radio emission of the CRF sources between the four bands to
be at the milliarcsecond level or even smaller. By applying this common
delay/position-offset relation to the four delay terms in Eq. 4, the VGOS
group delay position with respect to the reference direction, denoted by
$\Delta\@vec{k}^{\scriptsize\mbox{g}}$, is jointly determined by the locations
of the radio emission at the four bands as follows:

$\Delta\@vec{k}^{\scriptsize\mbox{g}}=+0.505\Delta\@vec{k}_{\scriptsize\mbox{A}}-1.448\Delta\@vec{k}_{\scriptsize\mbox{B}}-1.458\Delta\<EMAIL_ADDRESS>(7)

Similarly, the VGOS phase delay position, denoted by
$\Delta\@vec{k}^{\scriptsize\mbox{p}}$, is given by

$\Delta\@vec{k}^{\scriptsize\mbox{p}}=-0.883\Delta\@vec{k}_{\scriptsize\mbox{A}}+1.729\Delta\@vec{k}_{\scriptsize\mbox{B}}+2.044\Delta\<EMAIL_ADDRESS>(8)

The summation of the four coefficients in the right-hand side of Eq. 7 is
equal to unity (roundoff error of the displayed coefficients notwithstanding),
as well as for Eq. 8. Therefore, the position vectors
$\@vec{k}_{0}+\Delta\@vec{k}^{\scriptsize\mbox{g}}$ and
$\@vec{k}_{0}+\Delta\@vec{k}^{\scriptsize\mbox{p}}$ are independent of the
reference direction.

Simulation based on the VGOS session VO0051 (20 February 2020) was done to
demonstrate the results for group delays. Figure 3 shows two cases, where we
assume that the position of source 0016$+$731 at band B is offset by 0.2 mas
in declination with respect to that at the other three bands (in the same
position and formally selected as $\@vec{k}_{0}$) and at band D by 0.1 mas in
declination. By referring to $\@vec{k}_{0}$, the position-offset-induced
phases at the 32 channels of each individual observation were calculated, from
which the broadband observables were fitted using Eq. 2. The calculation was
done for all the observations of the source 0016+731 in the session one by
one. The source positions determined by these simulated broadband group delays
for the 1308 observations of the source in the session are (0.000, $-0.289$)
mas and (0.000, 0.340) mas for the two cases, respectively. They can be
predicted based on Eq. 7; therefore, the results from the simulation based on
actual VGOS observations agree with this equation. In the simulation, the
position offset of 0.2 mas causes delays on a 9000 km baseline with a
magnitude of $\sim$30 ps, which is smaller than the phase-delay ambiguity
spacings at the four bands and thus does not cause an issue of 2$\pi$
ambiguities in channel phases when doing LSF. This issue could happen in the
actual observations if position offsets were larger.

Figure 3: Simulation of broadband group delays (blue dots) of the baseline
GGAO12M–ISHIOKA by assuming the position of source 0016$+$731 at band B to be
offset by $+$0.2 mas in declination (left) and the position at band D to be
offset by $+$0.1 mas in declination (right). In these two cases, the reference
position is selected to be the positions at the other three bands. The red
dots, corresponding to the 47 observations for the source on the baseline in
session VO0051, show the delay offsets at band B for the left panel and at
band D for the right panel due to the assumed position shifts. The blue dots
are the broadband group delays based on the simulation for the 47
observations. The position estimate determined from the simulated broadband
delays for the 1308 actual observations of source 0016$+$731 is (0.0,
$-0.289$) mas for the first case and (0.0, 0.340) mas for the second case. The
models based on these position estimates are shown as blue curves.

Note that source structure and core shift evolve continuously with frequency,
thus leading to additional phase variations within individual bands. These
intraband phase variations, compared to the phase variations from band to band
(much wider frequency ranges), cause a second-order impact on broadband VGOS
observables and are not accounted for in the study.

Two other kinds of simulations were performed the results of which are shown
in the Appendix A and B. It is possible that channel frequencies in VGOS
observations will be changed to a wider bandwidth and a broader frequency
range than the current settings in the future. The results for two other
possible sets of channel frequencies are given in Appendix A. The main
conclusion from this simulation is that with a broader frequency range the
coefficients in Eqs. 7 and 8 will be reduced significantly. This also means if
the VGOS observing frequencies are changed, for instance, going to 14 GHz,
adjusting frequency setups in realtime to avoid RFI at individual stations, or
having one band at a station be missing because of RFI or hardware problems,
the source position estimates may change substantially. This may have a
significant impact on creating and using a VGOS CRF. The results from the
simulation with different values of the reference frequency for phase,
$\nu_{0}$ in Eq. 2, are given in Appendix B. The magnitudes of the four
coefficients can be significantly reduced for VGOS phase delay positions by
increasing the reference frequency for the phase observables in the VGOS data
processing.

##  4 Discussion

###  4.1 Variations in VGOS source positions

The source positions determined by the VGOS observables are linearly dependent
on the locations of the four-band radio emission, as shown in Eq. 7 for group
delays and in Eq. 8 for phase delays. The summations of the four linear
coefficients are unity for both cases of group delay and phase delay source
positions; however, three of them have absolute values larger than 1, and two
have negative values. A major consequence is that if there are position
variations due to structure evolution at the three highest bands, they must
cause larger variations in the estimates of the VGOS source positions than the
actual position variations at individual bands. Figure 4 shows a one-
dimensional diagram of the VGOS group delay positions in the four scenarios of
position offsets between various bands. In three of these four simplest
scenarios, the VGOS group delay positions are located far away from the area
of the actual radio emission — $\@vec{k}_{0}$ to
$\@vec{k}_{\scriptsize\mbox{A|B|C|D}}$. This will add complexities to the
understanding of the radio and optical position differences in the future.

Figure 4: One dimensional diagram of the relation between the VGOS group delay
positions (blue dots) and the locations of the radio emission at individual
bands (red rhombuses). It shows four scenarios: the location of the radio
emission at band A, B, C, or D as marked by
$\@vec{k}_{\scriptsize\mbox{A|B|C|D}}$ is offset by 0.1 mas with respect to
the locations at the other three bands, which are located at the origin marked
by $\@vec{k}_{0}$. The VGOS group delay positions are shown as blue dots: for
example, $\@vec{k}^{\scriptsize\mbox{VGOS}}_{\scriptsize\mbox{D}}$ is where
the VGOS group delay position is located when the location of the radio
emission at band D is offset by 0.1 mas with respect to the other three bands.

Because the core of source 0016+731 at 3.3 GHz as modeled by difmap is shifted
towards east by about 0.14 mas, the corresponding impact on the VGOS source
position can thus be calculated according to Eqs. 7 and 8; it is 0.07 mas
towards east for group delays and 0.12 mas towards west for phase delays.
These impacts may prevent the VGOS from achieving its goal of 1 mm station
position accuracy, and we must emphasize these are the impacts by assuming
that the effects of source structure are modeled in the geodetic data
analysis. When these effects are not modeled, the impacts are expected to be
more complicated and more variable.

Based on the routine geodetic data analysis of the session VO0051 by using
_$\nu$ Solve_777https://sourceforge.net/projects/nusolve/, 25 of the 63
sources with more than three observables available have position adjustments
with respect to their S/X positions with magnitudes larger than three times
the respective uncertainties estimated from the solution. The median angular
separation of the positions from VGOS and S/X observations for these 25
sources is 0.387 mas, which has an uncertainty of 0.056 mas. It is, therefore,
a common strategy in analyzing VGOS observations to estimate source positions,
which is usually unnecessary in the routine daily solutions of the legacy S/X
observations (Sergei Bolotin, private communication).

As demonstrated based on multi-frequency observations for four close CRF
sources by Fomalont et al. (2011), the ICRF position of a source can be
dominated by a jet component displaced from the radio core by $\sim$0.5 mas
and moving with a velocity of 0.2 mas/yr. This offset and the change will be
amplified significantly in VGOS position estimates, leading to much larger
position variations in a VGOS CRF than in the previous ICRFs. However, it
should be noted that this study is for four ICRF sources only.

###  4.2 Aligning the images at the four bands

The simulation so far does not explicitly model the effects of relative source
structure — the apparent two dimensional distribution of emission on the sky
as a function of frequency, as opposed to an ideal point-source per frequency.
Relative source structure affects the phases of individual frequency channels
in a frequency-, baseline length-, and baseline orientation-dependent manner,
and these phases in turn affect the broadband group delay and phase. Given a
model of the relative source structure, such as images of a source at the four
VGOS bands, the frequency channel phases can be corrected to correspond to the
phases of a point source for each band. However, as images derived using self-
calibration (e.g., Wilkinson et al., 1977; Cornwell & Wilkinson, 1981; Pearson
& Readhead, 1984; Thompson et al., 2017) and closure-based imaging (Chael et
al., 2016, 2018) techniques lose absolute source position information, it is
necessary to properly align the images at the four bands in order to generate
coherent source-structure phase corrections across all four bands. Otherwise,
a misalignment of the images at the four bands will introduce a change in the
position estimate from the VGOS observables with these corrections applied.
This has been demonstrated by Xu et al. (2021d). The exact formulas of the
impact of the misalignment are given above as Eq. 7 for group delays and Eq. 8
for phase delays. These results are important for deriving VGOS source-
structure corrections.

The third type of VGOS broadband observables, i.e. $\delta{\mbox{TEC}}$, may
also help to determine the differences in source positions at the four bands
or align the images over frequency, because these observables are affected by
source position offsets as well. Its change due to position offsets among the
four bands, denoted by $\Delta{\mbox{TEC}}$, is given by

$\Delta{\mbox{TEC}}=-\@vec{B}\cdot(-118.7\Delta\@vec{k}_{\scriptsize\mbox{A}}+168.3\Delta\@vec{k}_{\scriptsize\mbox{B}}+200.7\Delta\@vec{k}_{\scriptsize\mbox{C}}-250.3\Delta\@vec{k}_{\scriptsize\mbox{D}})/c,$
(9)

where $\Delta{\mbox{TEC}}$ is in units of TECU given that the baseline vector
is in units of m, the speed of light is in units of m/s, and the position
offsets are in units of mas. This equation is derived from the same way as
those for broadband group and phase delays in Sect. 3. The summation of the
four coefficients in the right-hand side is zero, therefore,
$\Delta{\mbox{TEC}}$ is independent of the reference source position selected.
In the case where the position offsets are a function of the frequency to the
power of $-1$ as discussed in Sect. 4.3, $\Delta{\mbox{TEC}}$ is equal to
zero. With external information about the ionospheric effects for VGOS
observations available, this quantity may help to validate the image alignment
in the presence of core shift.

If there is a position offset at only one of the four bands with a magnitude
of 0.1 mas, $\Delta{\mbox{TEC}}$ for the observations of a 6000 km baseline
will have a pattern of a sinusoidal wave over 24 hours of GMST time with a
magnitude in the range 0.2–0.5 TECU based on Eq. 9, depending on which band
the position offset occurs. Through the comparison of $\delta{\mbox{TECs}}$
from VGOS observations and global TEC maps888See, for instance,
http://ftp.aiub.unibe.ch/ionex/draft/ionex11.pdf derived from global
navigation satellite systems observations, the precision of
$\delta{\mbox{TEC}}$ from global TEC maps is probably on the order of 1–2 TECU
(Zubko et al., in preparation). Therefore, position offsets among the four
bands larger than 0.4 mas may be detectable by comparing $\delta{\mbox{TEC}}$
estimates from VGOS and global TEC maps.

###  4.3 Differences in VGOS group delay and phase delay source positions

Consider the scenario where the position change of a source over frequency
$\nu$ is a power-law function $\Delta\mbox{X}/\nu$, where $\Delta\mbox{X}$ is
a constant position shift, following the studies from Marcaide & Shapiro
(1984) and Lobanov (1998). It corresponds to the core shift effect with the
frequency power-law of $-1$. At the band with a center frequency
$\nu_{\scriptsize\mbox{center}}$, which can be calculated from Table 2, the
source position offset then is
$\Delta\mbox{X}/\nu_{\scriptsize\mbox{center}}$. Applying the position offsets
in this scenario to Eqs. 7 and 8 gives that
$\Delta\@vec{k}^{\scriptsize\mbox{g}}=0$ and
<EMAIL_ADDRESS>The results show
that if core shift is a function of the frequency to the power of $-1$, group
delays refer to the position of the AGN jet base, which is the position at the
finite frequency, but the phase delays refer to the position of the actual
radio emission at the reference frequency. This result was originally derived
by Porcas (2009). We can see that Eqs. 7 and 8 describe the differences in the
reference positions between group delays and phase delays for general
scenarios. This generalization is necessary because astronomical results such
as Fromm et al. (2013) and Plavin et al. (2019b) demonstrate that the position
dependence on frequency is often not the power $-1$, and in fact the power
index can be variable in time. Instead of considering core shift as a function
of frequency, we can describe it directly as position offsets at individual
bands.

As shown in Eqs. 7 and 8, the position offset at one band leads to position
changes in the opposite directions for the broadband group delays and phase
delays. It is likely that these two kinds of broadband observables actually
refer to different directions with a quite large separation on the sky. If
finally the phase observables from VGOS can be made use of in geodetic
solutions as recently did for the very short baselines by Varenius et al.
(2020); Niell et al. (2021); Xu et al. (2021c), the source position
differences need to be addressed. Based on Eqs. 7 and 8, the difference of
source positions between VGOS group delays and phase delays, denoted by
$\Delta\@vec{k}^{\scriptsize\mbox{g-p}}$, is given by

$\Delta\@vec{k}^{\scriptsize\mbox{g-p}}=+1.388\Delta\@vec{k}_{\scriptsize\mbox{A}}-3.177\Delta\@vec{k}_{\scriptsize\mbox{B}}-3.501\Delta\<EMAIL_ADDRESS>(10)

Since the four coefficients have the summation of zero,
$\Delta\@vec{k}^{\scriptsize\mbox{g-p}}$ is a quantity independent of the
reference position $\@vec{k}_{0}$. The difference in source positions between
VGOS group delays and phase delays measures an absolute-scalar product as a
combination of the four-band position offsets, including core shift. The
measurement noise levels of both these two types of VGOS observables are 1–2
ps or even smaller, which allows this position product to be detected at a few
$\mu$as level (Xu et al., in preparation).

##  5 Conclusion

We have derived the formulas of the source position estimates from VGOS
broadband group delays and phase delays as a function of the locations of the
radio emission at the VGOS frequency bands. The resolution across the source
is very different at different VGOS bands (a factor of three within the
current frequency range and nearly a factor of five between 3 and 14 GHz for
the future), and what parts of the jet components contribute to the core flux
and position may vary a lot — this is likely to move the “core” position even
further down the jet at lower frequencies (and resolution), especially if the
additionally included jet components have steeper spectrum. Source position
offsets between various VGOS bands are expected to be common. The variations
in the VGOS source positions will be significantly larger than the actual
changes in the emission locations due to structure evolution if they happen at
the three highest frequency bands. Accordingly to Eqs. 1 and 7, we should
expect VGOS sources to wander around parallel to the jet directions as a
function of time with much larger magnitudes than the S/X positions did
because of variations in source structure and core shift being amplified by
the VGOS fringe-fitting strategy. The only way to mitigate these frequency-
dependent impacts on VGOS source positions is to have source structure and
core shift measured (see, e.g., Kovalev et al., 2008; Fomalont et al., 2011;
Sokolovsky et al., 2011) and then correct these effects. This is critical in
order to make full use of the high quality data from VGOS with a random noise
level of $\sim$2 ps.

If we want to derive source-structure corrections for VGOS observations,
aligning the images at the four bands is very essential: a misalignment can
introduce a larger offset in the position estimate from the VGOS observations
with these corrections applied than the magnitude of the misalignment itself.

###### Acknowledgements.

We are grateful to the IVS VGOS stations at GGAO (MIT Haystack Observatory and
NASA GSFC, USA), Ishioka (Geospatial Information Authority of Japan), Kokee
Park (U.S. Naval Observatory and NASA GSFC, USA), McDonald (McDonald Geodetic
Observatory and NASA GSFC, USA), Onsala (Onsala Space Observatory, Chalmers
University of Technology, Sweden), Westford (MIT Haystack Observatory),
Wettzell (Bundesamt für Kartographie und Geodäsie and Technische Universität
München, Germany), and Yebes (Instituto Geográfico Nacional, Spain), to the
staff at the MPIfR/BKG correlator center and the MIT Haystack Observatory
correlator for performing the correlations and the fringe fitting of the data,
to the NASA GSFC VLBI group for doing the geodetic solutions, and to the IVS
Data Centers at BKG (Leipzig, Germany), Observatoire de Paris (France), and
NASA CDDIS (Greenbelt, MD, USA) for the central data holds.
We would like to thank Bill Petrachenko for his helpful comments on this work.
We used the Astrogeo VLBI FITS image database for our work, and specifically
we thank Bo Zhang for providing the image of source 1157$-$215 at 8.7 GHz.
This research has made use of data from the MOJAVE database that is maintained
by the MOJAVE team (Lister et al., 2018). The research was supported by the
Academy of Finland project No. 315721 and by the German Research Foundation
grants HE5937/2-2 and SCHU1103/7-2.

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##  Appendix A Frequency settings for VGOS observations

The frequencies of the 32 channels in the current VGOS observations are shown
in Table 2.

The IVS proposes to observe with 1 GHz wide bands and to extend to higher
frequencies, as originally planned (Petrachenko et al. 2009).

We first performed the simulation for the frequency range 3.0–11.2 GHz but
with 992 MHz wide bands as listed in Table 3 (Bill Petrachenko, private
communication). The equations equivalent to to Eqs. 7 and 8 are given by

$\Delta\@vec{k}^{\scriptsize\mbox{g}}_{992,11}=+0.399\Delta\@vec{k}_{\scriptsize\mbox{A}}-1.382\Delta\@vec{k}_{\scriptsize\mbox{B}}-1.359\Delta\@vec{k}_{\scriptsize\mbox{C}}+3.342\Delta\@vec{k}_{\scriptsize\mbox{D}},$
(11)

and

$\Delta\@vec{k}^{\scriptsize\mbox{p}}_{992,11}=-0.796\Delta\@vec{k}_{\scriptsize\mbox{A}}+1.7333\Delta\@vec{k}_{\scriptsize\mbox{B}}+2.018\Delta\<EMAIL_ADDRESS>(12)

We then performed the simulation for the frequency range 3.0–14.0 GHz as
listed in Table 4 (Bill Petrachenko, private communication). The results are
given as follows:

$\Delta\@vec{k}^{\scriptsize\mbox{g}}_{992,14}=+0.248\Delta\@vec{k}_{\scriptsize\mbox{A}}-1.008\Delta\@vec{k}_{\scriptsize\mbox{B}}-0.995\Delta\@vec{k}_{\scriptsize\mbox{C}}+2.755\Delta\@vec{k}_{\scriptsize\mbox{D}},$
(13)

and

$\Delta\@vec{k}^{\scriptsize\mbox{p}}_{992,14}=-0.633\Delta\@vec{k}_{\scriptsize\mbox{A}}+1.663\Delta\@vec{k}_{\scriptsize\mbox{B}}+2.078\Delta\<EMAIL_ADDRESS>(14)

There is only a marginal improvement in terms of reducing the magnitudes of
the coefficients by using a larger 960 MHz bandwidth for the current VGOS
frequency range. However, significant improvement happens by extending the
highest frequency to be 14 GHz, especially for group delay positions; the
magnitudes of the four coefficients are decreased by 20% to 50% compared to
those of the current frequency setting in Table 2. This means that the impact
of any position offset at a single band has less influence on the VGOS
group/phase delay position.

Table 2: Channel frequencies in the range 3.0–10.7 GHz currently used in VGOS observations (Units: MHz). | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
Band A | 3032.4 | 3064.4 | 3096.4 | 3224.4 | 3320.4 | 3384.4 | 3448.4 | 3480.4
Band B | 5272.4 | 5304.4 | 5336.4 | 5464.4 | 5560.4 | 5624.4 | 5688.4 | 5720.4
Band C | 6392.4 | 6424.4 | 6456.4 | 6584.4 | 6680.4 | 6744.4 | 6808.4 | 6840.4
Band D | 10232.4 | 10264.4 | 10296.4 | 10424.4 | 10520.4 | 10584.4 | 10648.4 | 10680.4
Table 3: Channel frequencies in the range 3.0–11.2 GHz with 992 MHz wide bands (Units: MHz). | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
Band A | 3000.4 | 3032.4 | 3128.4 | 3288.4 | 3576.4 | 3768.4 | 3896.4 | 3960.4
Band B | 5240.4 | 5272.4 | 5368.4 | 5528.4 | 5816.4 | 6008.4 | 6136.4 | 6200.4
Band C | 6360.4 | 6392.4 | 6488.4 | 6648.4 | 6936.4 | 7128.4 | 7256.4 | 7320.4
Band D | 10200.4 | 10232.4 | 10328.4 | 10488.4 | 10776.4 | 10968.4 | 11096.4 | 11160.4
Table 4: Channel frequencies in the range 3.0–14.0 GHz (Units: MHz). | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
Band A | 3000.4 | 3032.4 | 3128.4 | 3288.4 | 3576.4 | 3768.4 | 3896.4 | 3960.4
Band B | 5688.4 | 5720.4 | 5816.4 | 5976.4 | 6264.4 | 6456.4 | 6584.4 | 6648.4
Band C | 7832.4 | 7864.4 | 7960.4 | 8120.4 | 8408.4 | 8600.4 | 8728.4 | 8792.4
Band D | 13016.4 | 13048.4 | 13144.4 | 13304.4 | 13592.4 | 13784.4 | 13912.4 | 13976.4

##  Appendix B Comparison with different values of $\nu_{0}$

A simulation was performed to compare the results when varying the reference
frequency $\nu_{0}$ in Eq. 2, rather than the reference value of 6.0 GHz used
in the current fringe fitting by _fourfit_. The relation for group delay
positions as Eqs. 4 and 7 does not change when varying $\nu_{0}$, as well as
does that for the ionospheric observable. However, the coefficients in the
relation of VGOS phase delay positions will decrease by increasing $\nu_{0}$.
When setting, for example, $\nu_{0}$=8.5 GHz, the VGOS phase delay position is
given by

<EMAIL_ADDRESS>\scriptsize\mbox{GHz}}=-0.475\Delta\@vec{k}_{\scriptsize\mbox{A}}+0.794\Delta\@vec{k}_{\scriptsize\mbox{B}}+1.014\Delta\<EMAIL_ADDRESS>(15)

By increasing $\nu_{0}$ from 6.0 GHz to 8.5 GHz, $\tau^{\scriptsize\mbox{p}}$,
which is inversely proportional to $\nu_{0}$, will be simply decreased by 30%.
Without taking this expected decrease into account, the magnitudes of the four
coefficients are actually reduced by 16%, 24%, 30%, and 52%, respectively.
However, changing the reference frequency away from the central frequency 6.0
GHz will introduce the errors in group delay estimates into phase estimates.
The optimum reference frequency for phases will have to compromise these two
factors.