∎

11institutetext: Ming H. Xu 22institutetext: Aalto University Metsähovi Radio
Observatory, Metsähovintie 114, 02540 Kylmälä, Finland; Aalto University
Department of Electronics and Nanoengineering, PL15500, FI-00076 Aalto,
Finland; Technische Universität Berlin, Institut für Geodäsie und
Geoinformationstechnik, Fakultät VI, Sekr. KAI 2-2, Kaiserin-Augusta-Allee
104-106, D-10553 Berlin, Germany; Shanghai Astronomical Observatory, Chinese
Academy of Sciences, 200030 Shanghai, China
22email<EMAIL_ADDRESS>33institutetext: James M. Anderson
44institutetext: Technische Universität Berlin, Institut für Geodäsie und
Geoinformationstechnik, Fakultät VI, Sekr. KAI 2-2, Kaiserin-Augusta-Allee
104-106, D-10553 Berlin, Germany; Helmholtz Centre Potsdam, GFZ German
Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany
55institutetext: Robert Heinkelmann, Susanne Lunz 66institutetext: Helmholtz
Centre Potsdam, GFZ German Research Centre for Geosciences, Telegrafenberg,
14473 Potsdam, Germany 77institutetext: Harald Schuh 88institutetext:
Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences,
Telegrafenberg, 14473 Potsdam, Germany; Technische Universität Berlin,
Institut für Geodäsie und Geoinformationstechnik, Fakultät VI, Sekr. KAI 2-2,
Kaiserin-Augusta-Allee 104-106, D-10553 Berlin, Germany 99institutetext:
Guangli Wang 1010institutetext: Shanghai Astronomical Observatory, Chinese
Academy of Sciences, 200030 Shanghai, China

# Observable quality assessment of broadband very long baseline interferometry
system

Ming H. Xu  James M. Anderson  Robert Heinkelmann  Susanne Lunz  Harald Schuh
Guangli Wang

(Received: 29 Nov. 2019 / Accepted: Feb. 2021)

###### Abstract

The next-generation, broadband geodetic very long baseline interferometry
system, named VGOS, is developing its global network, and VGOS networks with a
small size of 3–7 stations have already made broadband observations from 2017
to 2019. We made quality assessments for two kinds of observables in the 21
VGOS sessions currently available: group delay and differential total electron
content ($\delta$TEC). Our study reveals that the random measurement noise of
VGOS group delays is at the level of less than 2 ps (1 ps = 10-12 s), while
the contributions from systematic error sources, mainly source structure
related, are at the level of 20 ps. Due to the significant improvement in
measurement noise, source structure effects with relatively small magnitudes
that are not overwhelming in the S/X VLBI system, for instance 10 ps, are
clearly visible in VGOS observations. Another critical error source in VGOS
observations is discrete delay jumps, for instance, a systematic offset of
about 310 ps or integer multiples of that. The predominant causative factor is
found to be related to source structure. The measurement noise level of
$\delta$TEC observables is about 0.07 TECU, but the systematic effects are
five times larger than that. A strong correlation between group delay and
$\delta$TEC observables is discovered with a trend of 40 ps/TECU for
observations with large structure effects; there is a second trend in the
range 60 ps/TECU to 70 ps/TECU when the measurement noise is dominant.

###### Keywords:

VGOS observations Ionosphere effects VLBI IVS Space geodesy Radio astronomy

##  1 Introduction

Geodetic very long baseline interferometry (VLBI) is a space-geodetic
technique that has regularly made global astrometric/geodetic observations
since 1979, which are the basis for creating the International Celestial
Reference Frame (ICRF2; Fey et al., 2015) and obtaining a full set of Earth
Orientation Parameters. Together with the other three space geodetic
techniques, VLBI plays an important role in establishing the International
Terrestrial Reference Frame (ITRF2014; Altamimi et al., 2016). At the
beginning of this century, the International VLBI Service for Geodesy and
Astrometry (IVS111https://ivscc.gsfc.nasa.gov/index.html; Schuh and Behrend,
2012; Nothnagel et al., 2017) proposed to develop the next-generation geodetic
VLBI system, initially called VLBI2010 (Niell et al., 2006) but subsequently
renamed the VLBI Global Observing System (VGOS). This new VLBI system relies
mainly on the advantages of small ($\sim$12 meters in diameter) and fast-
slewing antennas, ultra-wide observing frequency receivers (from 2 GHz to 14
GHz), and the expectation of continuous operation, 24 hours a day and seven
days a week (Petrachenko et al., 2009). In order to achieve its goal of 1 mm
position accuracy and 0.1 mm/yr velocity stability on global scales, the first
strategy proposed by the VGOS working group was to reduce the random noise
component of the group delays (Niell et al., 2007). Building a global VGOS
network with a sufficient number of stations is in progress, and a small VGOS
network has started to make broadband observations. The technical
implementation of the VGOS system can be found in Niell et al. (2018), and the
data correlation and processing of VGOS observations from a single baseline
can be referred to in Kondo and Takefuji (2016) and Niell et al. (2018).
Analyzing these actual VGOS observations allows us to investigate the
measurement noise level and the systematic behaviors of the VGOS observations.

In this paper we investigate the contribution of random measurement noise and
systematic error sources in VGOS delay and differential total electron content
($\delta$TEC) observables222An observable refers to a specific kind of
quantity, such as amplitude, phase, delay or rate, that has been measured by
maximizing the correlation between the recorded signals of a distant radio
source at the two stations of a baseline; in addition, $\delta$TEC estimate is
included as another kind of observable in VGOS.. The relationship between
these two types of observables is also studied. We use a different method of
assessing the error level of the VGOS system than that in Elosegui et al.
(2018) and Niell et al. (2018), who demonstrated the post-fit residuals from
geodetic VLBI solutions. Furthermore, instead of studying observations of one
short baseline, we present the results of VGOS observations from a global
network. In section 2 we present the VGOS observations currently available and
introduce the method of data analysis that we used. A quality assessment of
group delay observables is given in section 3. In section 4 we demonstrate the
measurement noise level and systematic errors in $\delta$TEC observables,
estimated simultaneously in VGOS observations. The strong correlation between
VGOS group delay and $\delta$TEC observables is studied in section 5. In
section 6 we summarize and discuss the results.

##  2 Broadband VLBI observations and data analysis

The IVS conducted a continuous observing campaign with three VLBI networks
(two legacy S/X networks and one VGOS broadband network) in 2017, called
CONT17 (Behrend et al., 2020). The VGOS broadband network in CONT17 had a
smaller number of stations than the two legacy networks, and it observed only
for one third of the whole CONT17 period. However, it provides the first
public data set of the VGOS broadband system, which was originally proposed
about 20 years ago. As of 15th Nov. 2019, 16 other VGOS sessions carried out
in 2019 were released333Data are available through the NASA CDDIS server:
https://cddis.nasa.gov/archive/vlbi/ivsdata/vgosdb/, as listed in Table 1. On
average, 24-hour VGOS sessions obtain about 2.2 times as many as scans444A
scan consists of simultaneous observations of a radio source by two or more
stations over an interval on the order of 5 seconds to 2 minutes. than the
legacy 24-hour VLBI sessions. These broadband observations were made
simultaneously at four 512-MHz-wide bands centered at 3.2, 5.5, 6.6 and 10.4
GHz. (The detailed technical description of the observing frequency setup is
available in Niell et al. (2018).) The median and mean of the formal errors
for group delay and $\delta$TEC observables in each session are also shown in
the table. The median formal errors of group delay observables for these 21
sessions are in the range of 1.2 ps (1 ps = 10-12 s) to 3.0 ps, and those for
$\delta$TEC observables are in the range of 0.029 TECU (1 TECU = 1016
electrons per square meter) to 0.060 TECU.

We processed the 21 VGOS sessions to determine error contributions in group
delay observables, including measurement noise and source structure effects,
by doing closure analysis (Xu et al., 2016, 2017; Anderson and Xu, 2018). We
adopted the same procedure of closure analysis for the VGOS sessions as was
developed for the CONT14 sessions described in Anderson and Xu (2018). (The
technical description of our closure analysis can be found in the supplemental
information to Anderson and Xu (2018).) In short, the method of closure
analysis statistically determines the baseline equivalent delay error of each
individual observation555In the remainder of this paper, “observation” is used
with the restricted meaning of a pair of two stations — a baseline — observing
a radio source over a short duration, typically on the order of 5 seconds to 2
minutes. from all the available closure delays involving that observation,
called closure-based error estimate; the weighted root-mean-square (WRMS)
delay error of a group of data can then be derived by combining the closure-
based error estimates of the delay observables in the group. The method has
two major advantages: (1) the station-based errors666We refer to effects such
as atmosphere, ionosphere, clock, and geometry as station-based—when there is
a change at one epoch for a station for any of these effects the corresponding
changes with the same magnitude will happen to all the observations on the
baselines of that station within the scan of that epoch—and the errors in
modeling these effects as station-based errors. are canceled out exactly in
closure delays; and (2) complementary to the post-fit residuals from geodetic
solutions, it provides an independent way of assessing the observable quality.
Except for baseline clocks, which in some cases are included in the
parameterization as a constant offset in delays for a specific baseline and
can thus only reduce a constant offset in closure delays, no geodetic
parameters in a routine VLBI solution can absorb nonzero closure delays. They
therefore contribute entirely to the residuals of the VLBI solution and can
bias the estimates of geodetic parameters. In the recent research of Bolotin
et al. (2019), structure model parameters were included in the VLBI solution
of the CONT17 VGOS sessions to reduce the large residual delays of the sources
0552$+$398 and 2229$+$695, which can thus reduce the magnitudes of the delay
misclosures. However, the method has not been demonstrated to be applicable to
general cases of radio sources with structure at different scales or
insufficient numbers of observations. In the paper, closure delays, measuring
intrinsic structure of sources as closure phases and closure amplitudes, are
treated as errors in VGOS broadband delays only because the effects of source
structure bias the geodetic parameters.

Closure analysis was also applied to the estimated ionosphere-like phase
dispersion parameter, called $\delta$TEC, from these VGOS sessions.
$\delta$TEC is the difference of the total electron content (TEC) along the
line of sight from a source to each station of a baseline during a scan.
Closure $\delta$TEC over a triangle of three antennas therefore gives insight
into the errors in $\delta$TEC measurements.

The conditions for the exclusion of an observation, called flagging, are
summarized here: (1) observations with signal-to-noise ratio (SNR) less than
7; (2) station RAEGYEB from the second day to the last day of CONT17 VGOS
observations, that is the sessions B17338, B17339, B17340 and B17341; and (3)
all the observations on the baseline ONSA13NE–ONSA13SW.

Table 1: Observing sessions of the VGOS broadband network Date | Session | Number | Number of | Number | Station list | Delay formal err. |  $\delta$TEC formal err.
---|---|---|---|---|---|---|---
(yyyy/mm/dd) | name | of scans | observations | of sources |  | Median | Mean | Median | Mean
(1) | (2) | (3) | (4) | (5) | (6) | [ps] | [TECU]
2017/12/03 | B17337 | 1180 | 5999 | 67 | GsIsK2YjWfWs | 1.71 | 2.12 | 0.039 | 0.046
2017/12/04 | B17338 | 1170 | 5037 | 66 | GsIsK2YjWfWs | 3.01 | 5.00 | 0.060 | 0.098
2017/12/05 | B17339 | 1180 | 5833 | 65 | GsIsK2YjWfWs | 2.86 | 4.86 | 0.057 | 0.095
2017/12/06 | B17340 | 1130 | 5166 | 66 | GsIsK2YjWfWs | 2.41 | 4.50 | 0.051 | 0.089
2017/12/07 | B17341 | 1246 | 6043 | 66 | GsIsK2YjWfWs | 2.43 | 4.47 | 0.050 | 0.088
2019/01/07 | VT9007 | 1132 | 8310 | 64 | GsK2OeOwYjWfWs | 1.70 | 2.37 | 0.039 | 0.049
2019/01/22 | VT9022 | 1024 | 6070 | 64 | K2OeOwYjWfWs | 1.45 | 2.00 | 0.035 | 0.043
2019/02/04 | VT9035 | 1043 | 4622 | 64 | GsK2OeYjWfWs | 1.39 | 1.72 | 0.036 | 0.042
2019/02/19 | VT9050 | 1115 | 7668 | 62 | GsK2OeYjWfWs | 1.37 | 1.79 | 0.035 | 0.042
2019/03/04 | VT9063 | 1129 | 7645 | 63 | GsK2OeYjWfWs | 1.34 | 1.76 | 0.033 | 0.041
2019/03/18 | VT9077 | 1080 | 5586 | 61 | GsK2OeYjWfWs | 1.29 | 1.77 | 0.032 | 0.041
2019/04/01 | VT9091 | 1121 | 7651 | 62 | GsK2OeYjWfWs | 1.43 | 1.83 | 0.034 | 0.042
2019/04/15 | VT9105 | 1105 | 5102 | 61 | GsK2OeWfWs | 1.44 | 1.85 | 0.035 | 0.043
2019/04/29 | VT9119 | 1126 | 5142 | 63 | GsK2OeWfWs | 1.72 | 2.19 | 0.040 | 0.048
2019/05/13 | VT9133 | 1123 | 4120 | 63 | GsK2OeWfWs | 1.71 | 2.25 | 0.038 | 0.048
2019/05/28 | VT9148 | 676 | 1444 | 60 | GsOeWs | 1.21 | 1.67 | 0.031 | 0.039
2019/06/11 | VT9162 | 1125 | 4891 | 64 | GsK2OeWfWs | 1.43 | 1.84 | 0.034 | 0.041
2019/06/24 | VT9175 | 1110 | 5097 | 66 | GsK2OeWfWs | 1.71 | 2.20 | 0.039 | 0.047
2019/07/08 | VT9189 | 776 | 1860 | 60 | GsOeWs | 1.17 | 1.64 | 0.029 | 0.037
2019/07/22 | VT9203 | 1093 | 6235 | 67 | GsK2OeOwWfWs | 1.94 | 2.44 | 0.042 | 0.050
2019/08/05 | VT9217 | 1174 | 11541 | 74 | GsK2OeOwYjWfWs | 1.63 | 2.29 | 0.039 | 0.050

For completeness, we briefly recall the basic equations of the closure
analysis and describe the terminology used. Closure delay is the sum of delay
observables over a closed triangle of three stations. For a triangle of three
stations, $a$, $b$, and $c$, closure delay is defined by

$\tau_{\mbox{clr}}\equiv\tau_{ab}+\tau_{bc}+\tau_{ca},$ (1)

where, for instance, $\tau_{ab}$ is the delay observable from station $a$ to
station $b$. The reference-time convention in geodetic VLBI defines that the
timestamp of the delay observable as the time of arrival of the wavefront at
the first antenna of a baseline. For instance, delay $\tau_{ab}(t_{0})$ refers
to the delay for a wavefront that arrives at station $a$ at epoch of $t_{0}$.
Therefore, the geodetic delay observables for multiple baselines in a scan,
although they have the same timestamp, do not necessarily refer to the same
wavefront. When these delay observables are used to derive closure delays, a
correction is needed to make the geometry of a triangle completely close;
detailed discussions and dedicated equations can be found in Section 2 of Xu
et al. (2016) and in Section 4.1 of Anderson and Xu (2018). An alternative way
of forming closure delays is to use the delay observables with geocentric
timestamps (the astronomical convention), rather than the delay observables
used in geodetic solutions; the former need no correction.

The uncertainty of a closure delay is calculated from the formal errors of the
three observables forming it by assuming that they are independent.

For the delay observable $\tau_{ab}$ at a single epoch, its closure-based
error estimate, ${\Delta}{\tau}_{ab}$, is statistically determined from all
the closure delays that are formed by $\tau_{ab}$ together with the other un-
flagged observations in the scan at that epoch, written as

$\Delta\tau_{ab}=\frac{\sum^{N}_{i=1}|\tau_{\mbox{\scriptsize
clr}-ab}^{i}|}{\sqrt{3}N},$ (2)

where $N$ is the number of such closure delays and $\tau_{\mbox{\scriptsize
clr}-ab}^{i}$ is the $i$-th one. The number $\sqrt{3}$ in the denominator
scales the mean closure delay to derive a baseline equivalent error by
assuming that the errors in different observations are independent. This
process as defined by equation 2 for the observable $\tau_{ab}$ was repeated
for all observations one by one to derive their closure-based error estimates,
$\Delta\tau$, whenever possible.

The WRMS delay error (not uncertainty), $\delta\tau$, is obtained by combining
the closure-based error estimates as follows:

$\delta\tau=\sqrt{\frac{\sum^{l}_{j=1}w_{j}(\Delta\tau^{j})^{2}}{\sum^{l}_{j=1}w_{j}}},$
(3)

where $l$ is the number of un-flagged observations with closure-based error
estimates available in a data group of interest (e.g., all observations of a
particular source or some selected sources or all observations in one
session), $\Delta\tau^{j}$ is the closure-based error estimate of the $j$-th
observation, and $w_{j}$ is its weight. The weighting is done by setting an
equal weight for all the delay observables, named uniform weighting, or by
using the reciprocal of the square of the uncertainty (formal error) of each
individual delay, named natural weighting. (The uniform and natural weighting
schemes used here have different meanings to those used in the astrophysical
imaging studies.) The same procedure of this closure analysis was applied to
study $\delta$TEC measurements; closure $\delta$TEC, closure-based error
estimate of $\delta$TEC and WRMS $\delta$TEC error are likewise defined.

Note that the closure analysis derives the _baseline equivalent_ error for
each observation from closure quantities. It is obvious that the closure-based
error estimate of an observation is affected (can be enlarged or reduced) by
source structure effects and measurement noise in the observations of the
other baselines in the scan. It is not appropriate to use closure-based error
estimate to quantify the errors at the level of a single observation; however,
the aim of closure analysis is to use closure-based error estimates only to
determine the overall variance of source structure effects and measurement
noise for a given group of data, as defined by equation 3. In this case, it
will work without introducing significant biases when the random measurement
noise, independent between different observations, is the dominant error
source. On the other hand, if the systematic error sources dominate, the mean
of the absolute values of all the closures formed with a common observation
maximizes the possibility of determining these systematic errors in that
observation; it was then scaled by a factor of $\sqrt{3}$ to reduce the
contributions of systematic errors in the other observations forming those
closures. Nevertheless, in the presence of systematic errors, the assumption
for equation 2 is not satisfied. It can lead to biases in interpreting the
derived WRMS delay errors as the magnitudes of source structure effects that
one would expect to have in the post-fit residuals from geodetic solutions. In
order to investigate the potential biases, the median was used in place of the
mean in Equation 2 as an alternative statistic to derive closure-based error
estimates and the corresponding WRMS errors.

In closure analysis, we also directly compare the closure delays for a given
source between various triangles and for a specific triangle between different
sources, which can yield insight into the properties of individual sources,
baselines, and stations.

##  3 Group delay observables

###  3.1 Measurement noise

A closure-based delay error estimate could be derived for 88% of the
observations in the 21 VGOS sessions, while 5 $\%$ did not form any closure
with un-flagged observations, and 7 $\%$ were flagged as described in the
previous section. Based on the uniform weighting and the natural weighting
schemes, the WRMS delay error was calculated from these closure-based error
estimates for each individual session and for all 21 sessions combined. The
results are shown in Table 2.

Table 2: WRMS delay errors determined by closure analysis (in units of picoseconds) Session/Group | $N_{\texttt{obs}\ }$ | $N_{\texttt{CloErr}\ }$ | Uniform Weighting | Natural Weighting
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
B17337 | 5999 | 5620 | 22.5 | 17.9
B17338 | 5037 | 3279 | 26.8 | 19.1
B17339 | 5833 | 3556 | 20.6 | 17.2
B17340 | 5166 | 3042 | 24.4 | 20.3
B17341 | 6043 | 3742 | 24.1 | 21.1
VT9007 | 8310 | 7508 | 36.1 | 33.8
VT9022 | 6070 | 5222 | 23.3 | 18.9
VT9035 | 4622 | 4283 | 18.5 | 14.1
VT9050 | 7668 | 7511 | 18.8 | 17.0
VT9063 | 7645 | 7503 | 20.9 | 19.3
VT9077 | 5586 | 5294 | 20.7 | 17.4
VT9091 | 7651 | 7325 | 21.0 | 21.1
VT9105 | 5102 | 4835 | 19.2 | 18.6
VT9119 | 5142 | 4856 | 21.0 | 25.4
VT9133 | 4120 | 3786 | 21.7 | 17.7
VT9148 | 1444 | 1146 | 22.9 | 25.4
VT9162 | 4891 | 4583 | 21.5 | 19.2
VT9175 | 5097 | 4830 | 21.3 | 16.9
VT9189 | 1860 | 1611 | 23.6 | 28.5
VT9203 | 6235 | 5827 | 20.9 | 26.0
VT9217 | 11541 | 11348 | 22.7 | 20.6
ALL | 121062 | 106707 | 22.9 | 21.0
ALL-19 | 106682 | 93977 | 21.5 | 20.0
CARMS-0.25 | 20998 | 17702 | 6.2 | 2.4

Apart from session VT9007, the WRMS delay errors for the other 20 sessions are
in the range of 18.5 ps to 26.8 ps based on the uniform weighting and in the
range of 14.1 ps to 28.5 ps based on the natural weighting. The WRMS delay
errors for the 21 sessions combined, labelled as “ALL” in the table, are about
23 ps and 21 ps based on the two weighting schemes. This is a significant
improvement compared to the corresponding values of 35.3 ps (uniform) and 25.2
ps (natural) for the CONT14 sessions(Anderson and Xu, 2018), which represent
the best observing campaign of the legacy S/X VLBI system.

For session VT9007 the WRMS delay errors are remarkably high — 36 ps and 34 ps
from the two weighting schemes. This is due to an exceptionally large number
of misclosures of about 310 ps or $-$310 ps in the closure delays, as shown in
Fig. 1. The vast majority of these misclosures involve station ONSA13SW, due
to its phasecal problem at the 6.6-GHz frequency band (Brian Corey, personal
communication, September 7, 2020). After 390 closure delays of station
ONSA13SW with absolute values of about 310 ps were flagged, the WRMS delay
error for session VT9007 was redetermined to be 23.0 ps (uniform) and 18.9 ps
(natural). The WRMS delay errors for the “ALL” group were recalculated from
the 19 sessions excluding VT9007 and VT9022—the latter session undergoes the
same issue but with offsets of around 1100 ps and $-$1100 ps, also related to
station ONSA13SW, but not as many. The WRMS delay errors for the 19 sessions
are 21.5 ps (uniform) and 20.0 ps (natural), labelled as “ALL-19” group in
Table 2. In summary, we argue that the magnitude of the random measurement
noise and the systematic errors in the VGOS observations is in the range of
20.0 ps to 22.9 ps.

Except for sessions like VT9148 and VT9189 with an observing network of three
stations, the natural weighting scheme generally gives significantly smaller
values of the WRMS delay error than the uniform weighting scheme. This is to
be expected when the non-Gaussian delay values due to source structure are
added to the closure delays with an otherwise noise-like distribution. On the
other hand, because source structure effects not only cause structure delays
in delay observables but also reduce observed amplitudes and thus the
observations’ SNR, natural weighting will underestimate the magnitude of their
actual impacts. Thus, while the natural weighting statistics are appropriate
for evaluating the properties of the delay/$\delta$TEC observables, the
uniform weighting statistics can be useful for identifying sources with
systematic errors, such as those due to source structure. Furthermore, the
SNRs of VGOS observations are typically very high, for instance, the median
SNR for the CONT17 VGOS observations is $\sim$90; uniform weighting should be
used to investigate the systematic error levels, especially if these
systematic errors are significantly larger than the random measurement noise
and are correlated with the SNRs, for example, source structure effects.

Figure 1: All closure delays of session VT9007 excluding triangles with
baseline ONSA13NE–ONSA13SW. Closure delay uncertainties are shown as black
bars. There are 7085 closure delays in total. A large number of closure delays
with an absolute offset of about 310 ps is visible. All the closure delays
exceeding the limits of the $Y$ axis are shown on the top or bottom of the
plot as open circles. This convention applies to all of the closure plots in
the paper; plots with no open-circle points on the bottom and top have no
excessively large closure delays. Two solid horizontal lines with an absolute
value of 150 ps are provided as guides.

In order to further investigate the random measurement noise level in VGOS
sessions, we adopted the closure amplitude RMS (CARMS) values based on the
basic weighting scheme777It assumes that the noise floor in log closure
amplitudes is 0.1 and thus adds 0.1 to their formal errors in the quadrature
sense for weighting. from Table 2 in Xu et al. (2019) to identify the sources
with minimum structure in these VGOS sessions. For the definition of CARMS,
please consult equations (2)–(4) and (6)–(8) in Xu et al. (2019). The CARMS
value of each individual source was calculated using all the available closure
amplitudes for X-band only from historical VLBI observations from 1980 to Aug.
2018 (no VGOS broadband observations are included). Apart from thermal noise,
observations of an ideal point source will always give log closure
amplitudes888Note that the natural logarithm was adopted in the definition of
closure amplitude to calculate CARMS values, as shown in the equation (3) in
Xu et al. (2019). equal to zero, while those of radio sources with extended
structure will have log closure amplitudes deviating from zero, leading to
larger CARMS values. Hence, in general, a smaller CARMS value of a source
indicates that it causes less structure effects. Our recent study has
demonstrated the correlation between the magnitudes of the radio-to-optical
source position differences and CARMS values (Xu et al., 2021). Using a
maximum CARMS limit of 0.25 to select sources with minimum structure, 28 low-
structure sources were found in the VGOS measurements, shown in Table 3. The
CARMS value of 0.25 was chosen as a compromise in order to have a sample of
radio sources with both minimum structure and a sufficient number of
observations. These 28 sources are associated with 19.7 percent of the
observations in the 19 VGOS sessions (excluding sessions VT9007 and VT9022).
The WRMS delay error value for these observations, labelled as “CARMS-0.25” in
Table 2, is 6.2 ps for the uniform weighting and only 2.4 ps for the natural
weighting. As we explained already, the uniform weighting indicates the
systematic error contribution and the natural weighting tends to show the
measurement noise level. We therefore conclude that the VGOS measurement noise
is no larger than the 2 ps level as demonstrated by the sources with minimum
structure, and the contributions of systematic errors for these sources are at
the level of 5 ps to 6 ps. Taking source 0529$+$483 as an example, all
available closure delays in the 21 sessions are shown in Fig. 2. If the four
closure delays in VT9007 with an offset of 310 ps and the five closure delays
in VT9022 with offsets of 1100 ps or $-$1100 ps are excluded, the WRMS closure
delay for source 0529$+$483 is only 3.0 ps. However, its closure delays, when
inspecting one specific triangle at scales of a few tens of picoseconds, are
still not randomly distributed, as shown in Fig. 3. Even though the magnitude
of the systematic variations is only about 10 ps, they are visible in the
plot. Similar or even larger systematic variations were detected for other
CARMS-0.25 sources, such as 0716+714 and 0133+476.

Table 3: Source group with CARMS less than 0.25, CARMS-0.25 for short IVS | CARMS | $N_{\texttt{obs}\ }$ |  ICRF3
---|---|---|---
Design. |  |  | category
(1) | (2) | (3) | (4)
0048$-$097 | 0.11 | 95 | D
0054$+$161 | 0.10 | 56 | D
0133$+$476 | 0.23 | 2906 | D
0237$-$027 | 0.15 | 193 | D
0446$+$112 | 0.24 | 589 | O
0529$+$483 | 0.21 | 3120 | D
0536$+$145 | 0.17 | 13 | D
0627$-$199 | 0.15 | 92 | D
0656$+$082 | 0.24 | 36 | O
0716$+$714 | 0.20 | 4865 | D
0723$+$219 | 0.18 | 13 | O
0727$-$115 | 0.24 | 727 | D
0804$+$499 | 0.20 | 211 | D
1040$+$244 | 0.17 | 815 | D
1124$-$186 | 0.21 | 312 | D
1243$-$160 | 0.13 | 288 | D
1300$+$580 | 0.18 | 1386 | D
1417$+$385 | 0.17 | 53 | O
1519$-$273 | 0.18 | 119 | D
1636$+$473 | 0.24 | 141 | D
1749$+$096 | 0.22 | 1163 | D
1908$-$201 | 0.20 | 222 | D
2059$+$034 | 0.24 | 33 | D
2141$+$175 | 0.21 | 730 | O
2215$+$150 | 0.22 | 1656 | D
2227$-$088 | 0.24 | 692 | D
2255$-$282 | 0.18 | 89 | O
2309$+$454 | 0.21 | 383 | O
Figure 2: All closure delays of source 0529+483 in the 21 VGOS sessions with
black bars giving the 1-$\sigma$ measurement uncertainties based on the formal
errors of delay observables. There are four closure delays of about 310 ps
from one scan of session VT9007 showing as one open circle on the top right of
the figure and five closure delays of about 1100 ps or $-$1100 ps from three
scans of session VT9022 shown on the top and bottom of the figure. The WRMS of
all the available closure delays excluding these 9 is only 3.0 ps from natural
weighting. Source 0529+483 demonstrates the measurement noise level in VGOS
delays, which should obviously be below 3 ps. Two solid horizontal lines with
an absolute value of 60 ps are provided as guides. Figure 3: Zoom-in plot of
closure delays of source 0529$+$483 for triangle KOKEE12M–WESTFORD–WETTZ13S.
They are not randomly distributed around zero, suggesting that there are
systematic effects with a magnitude of a few picoseconds for this source.
Three solid horizontal lines are provided to guide the reader.

As discussed at the end of section 2, the median value was also used to derive
closure-based error estimates and then to calculate the corresponding WRMS
delay errors. The differences in WRMS delay error values between the two
techniques are very small for both weighting schemes, no more than 0.5 ps in
most cases.

###  3.2 Source structure effects

As we did for the historical S/X VLBI observations (Xu et al., 2019), it is
beneficial to show a few closure plots for several sources with different
magnitudes of structure effects as examples to understand those effects in the
broadband VLBI system.

#### 0059$+$581

Closure delay plots for source 0059$+$581 are shown in Fig. 4 for two
triangles, GGAO12M–ISHIOKA–WETTZ13S and KOKEE12M–WESTFORD–WETTZ13S. The first
triangle was observed only in CONT17 and has 119 closure delays in total. The
pattern of two peaks with opposite signs separated by a 12-hour GMST period is
a normal behavior of source structure effects. The second triangle, which was
observed in 18 VGOS sessions, produced 329 closure delays. Through it, the
source-structure time evolution is well demonstrated: the peak in the closure
delay pattern changed from $-$30 ps in Dec. 2017 to around 0 ps in early 2019,
increased to $+$60 ps in March and decreased back to $+$30 ps in the middle of
2019\. Source 0059$+$581 is a very typical geodetic source and has been the
most frequently observed source both by the legacy VLBI system and the VGOS
system so far. For the triangle GGAO12M–ISHIOKA–WETTZ13S, it is seen that the
structure effects have a magnitude of as large as 20 ps but the WRMS closure
delay is only 6.9 ps. Source structure effects are more easily visible in VGOS
observations than in the legacy VLBI observations because the measurement
noise in VGOS is well below 3 ps. This is one reason why source structure
effects are so critical for VGOS.

Figure 4: Closure delays for source 0059$+$581 as a function of GMST for two
triangles, GGAO12M–ISHIOKA–WETTZ13S (top) and KOKEE12M–WESTFORD–WETTZ13S
(bottom). The color coding indicates the observation date, and the
corresponding legend is shown on the bottom-right corner of the bottom plot.
The top plot shows a normal pattern of source structure effects, while the
bottom one clearly shows the source-structure time evolution from CONT17 in
Dec. 2017 to 2019 and even within 2019. Two solid horizontal lines with an
absolute value of 30 ps are provided as guides.

#### 0016$+$731

Source 0016$+$731 is another of the important geodetic sources. The closure
delays for source 0016$+$731 are shown in Fig. 5 for triangle
KOKEE12M–WESTFORD–WETTZ13S, which is the same triangle shown in Fig. 3 for
source 0529+483 and in the bottom plot of Fig. 4 for source 0059$+$581\. It
has 460 closure delays in 19 VGOS sessions. The source structure changed
significantly from 2017 to 2019. The magnitudes of structure effects are as
large as 100 ps in 2019.

Figure 5: Plot of closure delays for source 0016$+$731 as a function of GMST
for triangle KOKEE12M–WESTFORD–WETTZ13S, which was shown also for source
0529+483 in Fig. 3 and for source 0059$+$581 in the bottom of Fig. 4. Source
0016$+$731 is another one of the important geodetic sources. However, its
structure effects have significantly larger amplitudes than those of source
0059$+$581\. Two solid horizontal lines with an absolute value of 60 ps are
provided as guides.

#### 3C418

Source 3C418 is a representative of the extremely extended sources in geodetic
VLBI and has been observed frequently in the VGOS sessions. Closure delays for
triangle ISHIOKA–KOKEE12M–WETTZ13S are shown in the bottom plot of Fig. 6.
With replaceable S/X and broadband receivers at the ISHIOKA station and co-
located S/X VLBI stations at the sites of both KOKEE12M and WETTZ13S, it is
possible to have a similar triangle of stations observing in the S/X mode.
Closure delays at X-band from the IVS S/X
observations999https://cddis.nasa.gov/archive/vlbi/ivsdata/vgosdb/ in 2018 and
2019 for triangle ISHIOKA–KOKEE–WETTZELL were calculated and are shown in the
top of the figure. Since the source structure effects in VGOS delays are due
to the structure at the four frequency bands in the range over 3.0 GHz to 10.7
GHz in a complex manner and those in the X-band observations are due to
structure at the frequencies around 8.4 GHz, the variation patterns in these
two plots do not necessarily match with each other. However, the scatters of
the closure delays along the variable curves, indicating the random
measurement noise level, are far smaller for VGOS observations than for the
S/X observations. And even for an extended source like 3C418, those scatters
for VGOS observations are at the level of just a few picoseconds. In the
bottom plot, the closure delays with absolute magnitudes larger than 150 ps
are very likely due to the jumps instead of source structure effects in the
delay observables. The delay jump issue is discussed further in the next
subsection.

Figure 6: Plots of closure delays for source 3C418 as a function of GMST for
two triangles, ISHIOKA–KOKEE–WETTZELL (top, legacy X-band) and
ISHIOKA–KOKEE12M–WETTZ13S (bottom, VGOS). With replaceable S/X and broadband
receivers at station ISHIOKA, the first triangle observed in the S/X mode
while the second one observed in the broadband mode. These two triangles with
a similar geometry allow the direct comparison of structure effects between
the legacy VLBI system and the VGOS system. The VGOS triangle observed only in
CONT17 and the S/X triangle observed in 40 sessions in 2018 and 2019. The
closure delays with absolute magnitudes larger than 150 ps in the VGOS plot
are very likely due to delay jumps instead of source structure effects
directly, which is discussed in subsection 3.3. Two solid horizontal lines
with an absolute value of 150 ps are provided as guides.

###  3.3 Delay jumps

In the S/X VLBI mode, multi-band group delay observables have ambiguities,
typically with spacings of 50 ns (1 ns = 10-9 s) at X-band and 100 ns at
S-band, while the VGOS broadband delays have an ambiguity spacing of 31.25 ns;
they can usually be resolved based on a priori information prior to performing
a geodetic VLBI solution. In the broadband VGOS observations reported here,
jumps in group delays have been found to be at least two orders of magnitude
smaller than the ambiguity spacing of S/X observations, but only 2–3 times the
ambiguity spacing of phase delay at X-band. These delay jumps exist in all of
the VGOS sessions.

Closure delays for 3C418 are shown in Fig. 7 for two triangles,
GGAO12M–ONSA13NE–WESTFORD and KOKEE12M–WESTFORD–WETTZ13S. For the first
triangle, offsets with a magnitude of $\sim$310 ps occurred during the time
period of GMST 22:00 to 05:00 in 13 VGOS sessions. Even more complicated delay
jumps appear in triangle KOKEE12M–WESTFORD–WETTZ13S, but no such jumps show up
in the two bottom plots of Figs. 4 and 5 for 0059$+$581 and 0016$+$731, which
cover the same triangle. These delay jumps are more easily identified in a
plot of closure delays versus closure TEC as shown in Figs. 9 and 10. They
also happen frequently for other extended sources such as 0119$+$115
(CARMS=0.39) and 0229$+$131 (CARMS=0.61). As demonstrated in Figure 3 of
Cappallo (2016), which shows the two-dimensional fringe amplitudes as a
function of $\delta$TEC and group delay, one would expect big jumps in
$\delta$TEC and in group delay if the wrong peak is mistakenly picked up.
Since these jumps tend to happen in the case of extended sources and only a
few tens of closure delays and closure $\delta$TEC for the CARMS-0.25 sources
have jumps, it is likely that the causative factor is source structure.
Nevertheless, other reasons are possible as well, for instance, the phasecal
problem as found in session VT9007. The sizes of the jumps identified in
closure delays seem to be rather stable; however, further studies are
necessary to verify if they have a fixed spacing or at what level they can
change.

Figure 7: Closure delays for source 3C418 as a function of GMST for two
triangles, GGAO12M–ONSA13NE–WESTFORD (top) and KOKEE12M–WESTFORD–WETTZ13S
(bottom). For comparison, closure delays of the second triangle can be seen
for sources 0528+483, 0059$+$581 and 0016$+$731 in Figs. 3, 4 and 5,
respectively.

##  4 Ionospheric effects determined by VGOS

The investigation of $\delta$TEC observables in VGOS is interesting because
(1) unlike the S/X VLBI system, the design of the VGOS system requires that
the dispersion constant in the phase be determined simultaneously with the
group delay, and (2) there is a strong correlation, larger than 0.9, between
$\delta$TEC and group delay estimates based on the current frequency settings,
as shown in the variance-covariance analysis of Cappallo (2014, 2016).
Observations on the single baseline ISHIOKA–KASHIM34 in Kondo and Takefuji
(2016) showed that the standard deviation of the differences between VGOS
$\delta$TEC observables and the global TEC model was 0.25 TECU. Even though
the baseline length of KASHIM34–ISHIOKA (about 50 km) is too short to make a
solid conclusion, the differences are far beyond the formal errors of VGOS
$\delta$TEC observables. The observations of the single baseline
GGAO12M–WESTFORD in Niell et al. (2018) showed a consistency between the VGOS
$\delta$TEC observables and differenced GNSS TEC estimates at co-located sites
at the level of 1 TECU. A bias of GPS relative to VLBI of $-$0.5 $\pm$ 0.1
TECU was found in the observations on this 600 km baseline. However, neither
of these two studies investigated the accuracy of the GNSS-based $\delta$TEC
used for comparison; consequently, it is not clear if these differences come
from the VGOS $\delta$TEC estimates or not. The accuracy of, and the potential
biases in, VGOS $\delta$TEC estimates need to be better understood.

The WRMS $\delta$TEC errors are seen in Table 4 to be in the range 0.24 TECU
to 0.49 TECU for the 20 sessions excluding VT9007, for which the WRMS error
value is 0.73 TECU. Excluding sessions VT9007 and VT9022, the WRMS $\delta$TEC
errors, labelled as “ALL-19”, are 0.31 TECU to 0.34 TECU for the two weighting
schemes. They are about one order of magnitude larger than the uncertainties
of the $\delta$TEC observables, which implies that there are additional error
sources in the $\delta$TEC observables. The closure analysis of observations
of individual sources showed that those additional errors in $\delta$TEC are
source-dependent. The WRMS $\delta$TEC error of the observations for the
sources with minimum structure (the CARMS-0.25 group) is only 0.07 TECU based
on the natural weighting scheme. Source structure must therefore play a
crucial role in the $\delta$TEC measurements.

Table 4: WRMS $\delta$TEC errors determined by closure analysis (in units of TECU) Session/Group | Uniform Weighting | Natural Weighting
---|---|---
(1) | (2) | (3)
B17337 | 0.34 | 0.32
B17338 | 0.45 | 0.34
B17339 | 0.32 | 0.30
B17340 | 0.37 | 0.37
B17341 | 0.35 | 0.34
VT9007 | 0.56 | 0.73
VT9022 | 0.33 | 0.29
VT9035 | 0.28 | 0.24
VT9050 | 0.28 | 0.26
VT9063 | 0.32 | 0.30
VT9077 | 0.33 | 0.28
VT9091 | 0.33 | 0.32
VT9105 | 0.32 | 0.30
VT9119 | 0.32 | 0.38
VT9133 | 0.33 | 0.27
VT9148 | 0.33 | 0.37
VT9162 | 0.36 | 0.28
VT9175 | 0.34 | 0.24
VT9189 | 0.39 | 0.49
VT9203 | 0.34 | 0.37
VT9217 | 0.40 | 0.32
ALL | 0.36 | 0.35
ALL-19 | 0.34 | 0.31
CARMS-0.25 | 0.15 | 0.07

##  5 Correlation between $\delta$TEC and group delay observables from VGOS

A covariance analysis using the VGOS frequency setup predicts a strong
correlation between the group delay and $\delta$TEC estimates (see Cappallo,
2015). It can be more straightforward to understand that correlation and its
influence on VGOS observations by analyzing the actual data. Figures 8 and 9
demonstrate the correlation by showing closure delays and closure TECs for the
sources 0016$+$731 and 3C418 using two plots each. The trends, obtained from
least-square fitting (LSQ), are 68.3 $\pm$ 1.9 ps/TECU and 39.9 $\pm$ 0.2
ps/TECU for the two sources, respectively.

Figure 8: Demonstration of the strong correlation between $\delta$TEC and
group delay observables from VGOS. Closure delays (blue dots) and closure TEC
(red dots) for source 0016$+$731 for triangle GGAO12M–ISHIOKA–KOKEE12M as a
function of GMST are shown in the top plot, whereas these closure delays
versus closure TEC are in the bottom plot. The changing pattern in closure TEC
is the same as that of closure delays. There is a strong correlation between
them, and the linear trend is 68.3 $\pm$ 1.9 ps/TECU.

In the bottom plot of Fig. 9, the points deviating significantly from the red
line form basically four straight lines that are parallel to the red line with
offsets of 133 ps in delay or 3.3 TECU in $\delta$TEC from each other. It
confirms the jumps in either or both the group delay and $\delta$TEC
observables.

Figure 10 shows the closure delays as a function of the closure TEC for all
sources and all triangles in the 21 sessions. The closure quantities in the
upper plot are from un-flagged observations, whereas those in the bottom plot
have at least one of the three observations in a triangle flagged due to the
three cases listed in section 2. Two main linear trends between closure delay
and $\delta$TEC were identified. In the upper plot the data points grouped in
the lines parallel to the red line were used jointly to determine the slope
with a result of 40.5 $\pm$ 0.1 ps/TECU. This estimated value is different
from that derived from the closure quantities shown in Fig. 9 for source 3C418
by three times their derived uncertainties, suggesting that the uncertainties
from LSQ were too optimistic. Based on the remaining data, another linear
trend as indicated by the green line with a slope of 63.8 ps/TECU was
determined and found to be in the range of 59.5 ps/TECU to 68.9 ps/TECU
depending on the flagging and weighting schemes. The results of these two
trends were iteratively determined by excluding the data points larger than
five times the WRMS residual. These two linear trends seem to have different
origins: (1) the trend in the range 59.9 ps/TECU to 68.9 ps/TECU agrees with
the value of $\sim$62 ps/TECU from Cappallo (2016) and is due to the random
measurement noise in the channel phases across the four bands; (2) the trend
of $\sim$40 ps/TECU results from the systematic variations in the channel
phases due to source structure.

Figure 11 is an equivalent plot for CARMS-0.25 sources. Other than the small
isolated groups of closures in the upper right and lower left, which are
associated primarily with only two of the 28 sources in this category, there
are no jumps comparable to those seen in Fig. 10. Were the points for the
CARMS-0.25 sources removed, the jumps would still be prevalent. Since the
closures shown in Fig. 10 are for all sources, removing the points for the
CARMS-0.25 sources would leave the closures for the sources with CARMS greater
than 0.25; these are the sources with nominally the more extended source
structure. Therefore, our findings indicate that the predominant causative
factor of the jumps in delay and $\delta$TEC is source structure, which can
cause large frequency-dependent phase variations across the four bands. This
has been demonstrated by our recent imaging results based on closure phases
and closure amplitudes (figures 11 and 12 in Xu et al., 2020).

Figure 9: Demonstration of the strong correlation between $\delta$TEC and
group delay observables from VGOS and the jumps in them. Closure delays (blue
dots) and closure TEC (red dots) for source 3C418 for triangle
GGAO12M–KOKEE12M–WETTZ13S as a function of GMST are shown in the top plot,
whereas these closure delays versus closure TEC are in the bottom plot. The
linear trend between them is 39.9 $\pm$ 0.2 ps/TECU. The jumps, which can be
two times a certain interval away from the mainstream of points passing the
zero closure delay and zero closure TEC

, are clearly visible in the bottom plot.

Figure 10: Closure delays versus closure TEC with un-flagged observations
(top) and with at least one of the three observations in a triangle flagged
out due to the three cases listed in section 2 (bottom). All sources and all
triangles available in the 21 sessions are included. There are two main linear
trends between them. The slope of the trend indicated by the red line was
determined to be 40.5 $\pm$ 0.1 ps/TECU. That of the trend indicated by the
green line with a slope of 63.8 ps/TECU was found to be in the range of 59.9
ps/TECU to 68.9 ps/TECU depending on the flagging and weighting scheme, which
indicates that it is variable from source to source and from triangle to
triangle. In the bottom plot, the vast majority of the closures are from
observations of station RAEGYEB in the last four sessions in CONT17; while the
closures of the observations flagged due to the other two reasons are nearly
all beyond the limits of the plotting axes. The observations of station
RAEGYEB have a median SNR of 17-23 in these four sessions, while the rest
observations in CONT17 have a median SNR of 92–115. On average, the closures
in the bottom plot are from observations with SNRs smaller by a factor of five
than those in the upper plot. Another difference in the observations between
the two plots is the significant decrease in the channel visibility amplitudes
of station RAEGYEB with increasing frequency due to the antenna pointing issue
since the second day during the CONT17. Figure 11: Equivalent plot to Fig. 10
for the closure quantities of the CARMS-0.25 observations only. Of 20,337
pairs of closure quantities in the plot, there are 17 and 47 pairs in the
upper-right and the bottom-left corners, respectively. Most of them involve
the observations of sources 0133$+$476 and 0716$+$714\. The median value of
the absolute closure delays in the plot is 2.38 ps, and that of the absolute
closure TEC is 0.059 TECU.

##  6 Conclusions and discussions

We processed the 21 VGOS sessions that have been publicly released and made
quality assessments for two kinds of VGOS observables, group delay and
$\delta$TEC, that are determined simultaneously in the process of broadband
bandwidth synthesis. The measurement noise level and the contributions of
systematic error sources in these two types of observables were determined by
running closure analysis for the whole data set and for the selected sources
with minimum structure based on our previous work. By performing closure
analysis, two important features in group delay and $\delta$TEC observables
have been revealed, which are the strong correlation between them and the
jumps in both observables.

The random measurement noise level of VGOS group delays was found to be below
2 ps based on the observations from all the VGOS radio sources that have CARMS
values smaller than 0.25. The estimated random measurement noise level agrees
well with the delay formal errors, as listed in Table 1. However, the
contributions from other systematic error sources, mainly source structure
related, are at the level of 20 ps, as indicated by the WRMS delay errors for
observations of all sources. Due to the significant reduction in measurement
noise over the S/X systems, source structure effects with magnitudes of 10 ps
are clearly visible. In general, source structure evolves at time scales of a
few weeks, which causes the closure delays to change at magnitudes of a few
tens of picoseconds. It thus will be a big challenge to correct source
structure effects in VGOS in order to fulfill its goals. Evidence for another
critical error source in the VGOS system is the presence of discrete jumps in
the closure delays and closure TECs, for instance with a delay offset of about
310 ps or integer multiples of that. The likely cause is found to be source-
structure-induced phase changes across the four bands (Xu et al., 2020).

Closure delays on individual triangles were shown for four sources, 0529$+$483
(CARMS=0.21), 0059$+$581 (CARMS=0.27), 0016+731 (CARMS=0.31), and 3C418
(CARMS=0.61) in figures 3, 4, 5, and 7 to demonstrate the source structure
effects in VGOS delay observables. By showing the closure delays from the same
triangle KOKEE12M–WESTFORD–WETTZ13S in these four figures, the differences in
the magnitudes of these effects can be compared among radio sources with
structure at various scales as indicated by their CARMS values. The magnitudes
of structure effects on the triangle were less than 10 ps for source 0529+483,
about 50 ps for source 0059+581, and about 100 ps for source 0016+731; they
were larger and more complicated for source 3C418. Delay jumps occurred for
the observations of only 3C418 among these four sources.

The random measurement noise level of $\delta$TEC observables was determined
to be below about 0.07 TECU, which is comparable to the formal errors. The
systematic effects are five times larger than that. A strong correlation
between group delay and $\delta$TEC observables is clearly demonstrated, with
two main linear trends. For observations with large structure effects, there
is a dominant slope of $\sim$40 ps/TECU. The slope of the second trend is in
the range 60 ps/TECU to 70 ps/TECU. Due to this strong correlation and the
simultaneous determination of them, group delay and $\delta$TEC observables
need to be studied together and further. The $\delta$TEC estimates from other
sources, such as GPS or global TEC models with a sufficient accuracy, might
improve the determination of the source structure effects in $\delta$TEC
observables; based on the stable linear coefficients between delay and
$\delta$TEC, the source structure effects in group delay observables might be
determined without requiring any model of source structure itself. For
example, external $\delta$TEC estimates can be used to detect the systematic
effects in VGOS $\delta$TEC estimates, which may be able to predict those
effects in delay observables by the linear trends, as discussed in this work.

Delay jumps in the VGOS system need to be understood further. Closure delays
have been demonstrated to be useful, and the correlation between group delay
and $\delta$TEC observables can also be of great help for the delay jump
detection. However, the delay spacing of these jumps will have to be studied
in detail. The exact origins of the two dominant linear trends between
broadband delays and $\delta$TEC, the causes of such jumps, and the method to
fix them are our near-future work.

###### Acknowledgements.

We would like to thank Sergei Bolotin, Arthur Niell, and Brian Corey for their
efforts to review the manuscript and for their helpful comments of high
quality which improved its comprehensibility. The results reported in this
paper were produced using the data owned by the International VLBI Service
(IVS) and its international self-funded member organizations. 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), 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, the VLBA correlator
at Socorro, and the MIT Haystack Observatory correlator for performing the
correlations and the fringe fitting of the data, 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.
This research has made use of the Generic Mapping Tools
package101010https://www.soest.hawaii.edu/gmt/, the pgplot
library111111https://www.astro.caltech.edu/$\sim$tjp/pgplot/, and the SAO/NASA
Astrophysics Data System121212https://ui.adsabs.harvard.edu/.
MHX was supported by the Academy of Finland project No. 315721 and by the
National Natural Science Foundation of China (No. 11973023 and 11873077). JMA,
RH, SL, and HS were supported by the German Research Foundation grants
HE5937/2-2 and SCHU1103/7-2.

## Author contributions

MHX and JMA designed the research; MHX performed the research, analyzed the
data, and wrote the paper; JMA, RH, SL, HS, and GW contributed to the
interpretation of the results and provided suggestions in writing and revising
the paper.

## Data Availability

All the VGOS data in the vgosDB data format are available at the CDDIS server:
https://cddis.nasa.gov/archive/vlbi/ivsdata/vgosdb/. The closure delays,
closure TECs, closure phases and amplitudes from these VGOS sessions are
available upon the request to the corresponding author.

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