Document [original]

Structure Effects for 3417 Celestial Reference Frame Radio Sources

M. H. Xu

1,2,3

, J. M. Anderson

2,4

, R. Heinkelmann

, S. Lunz

, H. Schuh

2,4

, and G. L. Wang

MOE Key Laboratory of Fundamental Physical Quantities Measurement & Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF and School of

Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China; [email protected]

Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, Straβe des 17. Juni 135, D-10623, Berlin, Germany

Shanghai Astronomical Observatory, Chinese Academy of Sciences, No. 80 Nandan Road, 200030, Shanghai, People’s Republic of China

DeutschesGeoForschungsZentrum (GFZ), Potsdam, Telegrafenberg, D-14473 Potsdam, Germany

Received 2018 December 19; revised 2019 March 26; accepted 2019 March 28; published 2019 May 8

Abstract

Geodetic/astrometric very long baseline interferometry (VLBI)has been routinely observing using various global

networks for 40 yr, and it has produced more than 10 million baseline group delay, phase, and amplitude

observables. These group delay observables are analyzed worldwide for geodetic and astrometric applications, for

instance, to create the International Celestial Reference Frame (ICRF). The phase and amplitude observables are

used in this paper, by means of closure analysis, to study intrinsic source structures and their evolution over time.

The closure amplitude rms, CARMS, indicating how far away a source is from being compact in terms of

morphology, is calculated for each individual source. The overall structure-effect magnitudes for 3417 ICRF radio

sources are quantiﬁed. CARMS values larger than 0.3 suggest signiﬁcant source structures and those larger than

0.4 indicate very extended source structures. The 30 most frequently observed sources, which constitute 40% of

current geodetic VLBI observables, are studied in detail. The quality of ICRF sources for astrometry is evaluated

by examining the CARMS values. It is conﬁrmed that sources with CARMS values larger than 0.30 can contribute

residual errors of about 15 ps to geodetic VLBI data analysis and those with the CARMS values larger than 0.4

generally can contribute more than 20 ps. We recommend CARMS values as an indicator of the astrometric quality

for the ICRF sources and the continuous monitoring of the ICRF sources to update CARMS values with new VLBI

observations as they become available.

Key words: astrometry –catalogs –methods: data analysis –quasars: general –reference systems –techniques:

interferometric

Supporting material: machine-readable table

1. Introduction

Extragalactic radio sources have been routinely observed by

geodetic/astrometric

very long baseline interferometry

(VLBI)since 1979 and have been used, for example, to create

the International Celestial Reference Frame (ICRF; e.g.,

Johnston et al. 1995; Ma et al. 1998)adopted by the

International Astronomical Union as the fundamental celestial

reference frame. The newly released, third realization of the

ICRF (ICRF3;

Charlot et al. 2018)contains 4536 radio

sources on the sky. The majority of radio sources in ICRF3

have formal position uncertainties smaller than 1 mas. Since

ICRF radio sources are too distant to exhibit any detectable

proper motion, except apparent proper motions due to the

acceleration of the solar system barycenter (see Titov et al.

2011; Xu et al. 2012; Titov & Lambert 2013), the ICRFs are

considered to be global and quasi-inertial celestial reference

frames accurate at the submilliarcsecond level. For instance, the

second realization of the ICRF (ICRF2; Fey et al. 2015)claims

that its axis stability is at the level of 10 μas.

As pointed out when the ﬁrst realization of the ICRF

(ICRF1)was created, however, the underlying kinematic

physics of ICRF radio sources was not as well understood as

that of stars (Ma et al. 1998), not yet enough to promise such a

high stability of the ICRF for a long term. The radio emissions

of the ICRF sources generally exhibit spatially extended

structure on milliarcsecond scales. More importantly, the

intrinsic structures are variable in time, and it is clear that

many radio sources have changed their reference positions by

far larger than their uncertainties in ICRF because of the

changes in their structures.

Source structure has three negative impacts on geodetic/

astrometric VLBI. First, it leads to variations in the reference

position of a source, which can be caused either by a change in

observing geometry or by a change in intrinsic structure.

Second, as shown, for instance, in Charlot (1990), it gives rise

to structure delay in the group delay observable up to a few

nanoseconds. The ﬁrst impact, the absolute structure effects,

can be absorbed by estimating absolute source position, but the

second impact, the relative structure effects, introduces error

contributions to residuals and can bias estimates of other

geodetic parameters. Third, as studied in Anderson & Xu

(2018a), it reduces the actual signal-to-noise ratio R

and adds

additional thermal noise to VLBI measurements. These three

impacts have led source structure to be a critical issue in

geodetic/astrometric VLBI.

Because geodetic/astrometric VLBI assumes that the ICRF

sources are ideally compact and source-structure effects are not

modeled in data analysis, source structure leads to errors in

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May https://doi.org/10.3847/1538-4365/ab16ea

Original content from this work may be used under the terms

of the Creative Commons Attribution 3.0 licence. Any further

distribution of this work must maintain attribution to the author(s)and the title

of the work, journal citation and DOI.

Astrometry here means the global and absolute positioning of extragalactic

radio sources with VLBI observations made and analyzed in the geode-

tic mode.

http://iers.obspm.fr/icrs-pc/newwww/icrf/index.php

geodetic parameter estimates and causes instabilities of the

ICRF. Source structure and its variability should be invaluable

indicators of source quality for both the creation and the

applications of the ICRFs. For example, Fey et al. (1996)and

Fey & Charlot (1997,2000)derived images of 389 ICRF radio

sources from dual-frequency observations by the Very Long

Baseline Array (VLBA), and based on those images, they

developed so-called structure indices, which are used to

categorize ICRF2 sources and exclude “bad”sources from

routine observations in scheduling. In 2018 April, the Gaia

data release 2 was published (Lindegren et al. 2018).Gaia will

observe for another several years and will thereby improve in

terms of accuracy the positions of radio sources at optical

wavelengths, which do not have to be identical to the positions

at the radio wavelengths at the level of the uncertainties of Gaia

and VLBI (e.g., Petrov & Kovalev 2017; Gaia Collaboration

et al. 2018). Investigating the source structure of each

individual ICRF source over the whole time span of

geodetic/astrometric VLBI observations is therefore critical

both for selecting radio sources with high astrometric qualities

to link optical and radio frames (e.g., Bourda et al. 2008,2011)

and for understanding the position offset between Gaia and

VLBI for individual sources (e.g., Kovalev et al. 2017; Petrov

& Kovalev 2017; Petrov et al. 2018).

On the other hand, for geodetic and geophysical applica-

tions, geodetic VLBI has devised its next-generation system,

the so-called VLBI Global Observing System (VGOS), to meet

the very stringent science requirements for the terrestrial

reference frame (TRF), an accuracy of 1 mm for global scales

and 0.1 mm yr

−1

for long-term stability (Niell et al. 2006;

Petrachenko et al. 2009). Prototype VGOS systems have been

implemented and obtained broadband observations with a

measurement noise of only a few picoseconds (Niell et al.

2018). Based on these prototype VGOS systems, three of the

four proposed strategies that were conceived to achieve its

goals are addressed fully or partly: to reduce the average

source-switching interval, to decrease the random thermal noise

in the delay measurement, and to minimize the susceptibility to

radio frequency interference (RFI). They can be solved mainly

by technical improvements within the VGOS antenna system,

for example, by installing ultra-broadband receivers and

signiﬁcantly increasing the slewing rates of telescopes. As

concluded by Niell et al. (2018), however, the reduction of

systematic errors including source structure, which is the fourth

strategy component of the VGOS system, is not yet addressed

and remains as the main challenge. The initial analysis of

CONT17 observations

from a global VGOS network has

conﬁrmed that several sources have extremely large postﬁt

delay residuals, such as 3C 418, 3C 371, 0229+131, 0552

+398, and 2229+695, and the observations of these sources

have to be deselected in geodetic VLBI data analysis (P.

Elosegui 2018, private communication). The broadband VGOS

systems, the observing frequency of which ranges from 2 to

14 GHz, will bring several beneﬁts for geodesy, but they

certainly will make the impacts of structure effects much worse

than the traditional S/Xsystems. Unlike the ﬁrst three strategies

of the VGOS system, in order to reduce the systematic errors in

VLBI observables, a signiﬁcant investigation of VLBI

observations and a different method of data analysis are

required.

The goal of this paper is to demonstrate the source-structure

effects, to identify the variations in their intrinsic structures

over time, and ﬁnally to quantify the overall structure-effect

magnitudes for as many ICRF3 sources as possible. The entire

data set of VLBI observations used for the creation of the

ICRF3 and geodetic VLBI applications is analyzed, but with

different types of observables, i.e., phase and amplitude, to

provide independent insights into the quality of ICRF3 sources.

This paper is structured as follows. The geometric/astrometric

VLBI observations are described in Section 2. The methodol-

ogy is given in Section 3. The results are presented in

Section 4, with discussions of closure phases in Section 4.1,

closure amplitudes in Section 4.2, and the overall structure-

effect magnitudes in Section 4.3. The usages of these overall

structure-effect magnitudes are discussed in Section 5, while

the summary and future work follow in Section 6. The closure

phase and closure amplitude plots for the sources among the 30

most frequently observed sources that are not addressed in the

main body of this paper are presented in the supplemental

information at [doi:10.11570/19.0007].

2. Data

We analyzed almost 40 yr of dual-frequency, S/Xband,

geodetic/astrometric VLBI observations driven by various

global geodetic VLBI observing campaigns (e.g., the NASA

Crustal Dynamics Project (Coates et al. 1985; Smith &

Baltuck 1993)and the International Radio Interferometric

Survey program (Carter & Robertson 1986)) since 1979, and

coordinated by the International VLBI Service for Geodesy and

Astrometry (IVS; Schuh & Behrend 2012; Nothnagel et al.

2017; please also refer to the IVS website

)since its foundation

in 1999. Geodetic/astrometric VLBI sessions in general have a

duration of either 24 hr or 1 hr. The majority of these 24 hr

sessions are Earth Orientation Parameter (EOP)sessions (twice

per week, R1 and R4), TRF sessions (once every two months),

and CRF sessions (at least once every two months). With other

dedicated types of sessions and regional sessions ﬁlled in, the

24 hr sessions are three to four times per week on average. The

1 hr sessions, called intensives, are operated on a daily basis to

monitor Earth’s phase of rotation. These intensive sessions are

observed by a single baseline seven times per week and by

three or four stations once per week. Refer to the aforemen-

tioned IVS website for a more detailed description of IVS

operations. A complete list of these sessions every year since

1979 is also publicly available.

All 24 hr sessions and the 1 hr sessions

with an observing

network of at least four stations available on 2018 February 26

were used in our analysis. A total of 14,759,830 observations

of 5228 radio sources were obtained in 6533 24 hr sessions and

150 intensive sessions by 191 stations with about 40 yr of

observing history, as listed in Table 1. Even though phase

delays are well known to be more precise than group delays

and have long been used by astronomers to study various

celestial radio sources, they have been rarely used for geodetic

VLBI, due to unresolved ambiguity. The observed amplitudes

are important for the astrophysical imaging, but in geodesy they

https://ivscc.gsfc.nasa.gov/program/cont17/

http://ivscc.gsfc.nasa.gov

https://ivscc.gsfc.nasa.gov/program/master.html

Data for both types of sessions are publicly available from ftp://cddis.gsfc.

nasa.gov/pub/vlbi/ivsdata/.

One observation refers to an ensemble of a single baseline delay, rate,

phase, and amplitude observable.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

are only used empirically to monitor the total ﬂuxes of radio

sources, which are subsequently used for scheduling observa-

tions (Le Bail et al. 2016).Thisistheﬁrst attempt ever to

extensively exploit both phase and amplitude observables in

regular geodetic/astrometric VLBI sessions. Those observables

were extracted from the vgosDB or Goddard databases

searching for the predeﬁned keywords listed in the supplemental

information available at [doi:10.11570/19.0007]. Because about

40% of S-band observations were missing all or parts of the

necessary keywords in the archived databases, the results

reported here are restricted to X-band observations only.

Geodetic/astrometric VLBI schedules observations with an

expected R

to be at least 20 at the Xband and 15 at the Sband

for all baselines in each scan. Due to strong local RFI and

resolved source structures, a signiﬁcant reduction in the R

for

actual measurements can happen. In this study, initial outlier

ﬂagging was done by setting the minimum R

to be 6. After the

initial ﬂagging, the mean and median values of R

for the Xband

are 112 and 47, respectively; about 23% of observables have an

actual R

smallerthan25attheXband. For the cases of R

>6,

the probability functions for the amplitude and phase observables

are approximately Gaussian distributions deﬁned by the indepen-

dent zero-mean thermal noises; therefore, the uncertainty of the

phase observable due to the thermal noise σ

can be obtained

from Thompson et al. (2017, Equation (9.67)),

=

f()

360

1,1

and the uncertainty of the amplitude observable σ

can also be

determined from Thompson et al. (2017, Equation (9.66)),

⎛

⎝

⎜⎞

⎠

⎟()

8,2

sn sn

where νis the correlation amplitude. Thus, the phase and

amplitude observables have their uncertainties calculated based

on these two equations, ν, and R

Another important quantity for our study is the reference

frequency for phase observables, which for these X-band data

ranges from 7741.99 to 8794.99 MHz. The three reference

frequencies that are related to about 90% of observations are

8212.99 MHz (55%), 8210.99 MHz (25%), and 8409.99

MHz (10%).

3. Closure Analysis of Geodetic/Astrometric VLBI Data

The method of closure analysis was applied for this study.

Closure analysis for group delay observables was introduced

and conducted by Xu et al. (2016), who found that source 0642

+449, one of the ICRF2 deﬁning sources, had two compact

cores separated by about 500μas in the R.A. direction in 2014

May. The principle of that method is that group delays on

shorter baselines were used to cancel out the geometric delays,

clocks, and other station-based effects for observables on long

baselines (>7100 km), and the last were used to measure

source-structure effects. The model of closure delay for

geodetic VLBI was given there. Closure analysis was further

applied to geodetic VLBI for phase and amplitude observables

(Xu et al. 2017), where direct model ﬁtting of closure phases

obtained the structure of 3C 371 with a model of three

components. The results were proven to agree with the

structure obtained from the traditional imaging process of

visibility data by comparing the resultant closure phases based

on the two methods. The model of closure phase for geodetic

VLBI observations was developed. A more comprehensive

study of closure analysis was recently done by Anderson & Xu

(2018a). Instead of a speciﬁc case study for a few sources, they

studied all 73 sources in CONT14

in detail both via closure

analysis and imaging. The conclusion was that source structure

is a major contributor to errors in geodetic VLBI. The closure

plots for group delay, phase, and amplitude for all possible

combinations of three or four stations for all sources in

CONT14 during 15 days for both the Sand Xbands can be

found in Anderson & Xu (2018b). The comparison of closures

from observables and from imaging results can be obtained

from those plots or from the tables in the supporting

information of Anderson & Xu (2018a).

Recalling the idea of closure quantities, for a wavefront of a

radio source, the closure phase and delay are the sum of the

phase and delay observables around a closed triangle of three

stations, while for four stations, a,b,c, and d, the closure

amplitude ν

clr

is deﬁned in the study here as

nn nn

⎛

⎝

⎜⎞

⎠

⎟()ln , 3

abcd

ab cd

ac bd

clr

where, for instance, ν

is the amplitude observable on baseline

ab. The sequence of the four stations appearing in the previous

equation matters in nature to form the closure amplitude. For

these four stations, another two closure amplitudes with

different station orders, ν

abdc

and ν

acdb

, can be formed to give

different absolute values of the closure amplitudes deﬁned by

Equation (3), while only two out of the three closure

amplitudes are independent. For clarity in this context, one

closure phase or one closure amplitude refers to an individual

closure value, while a triangle or a quadrangle refers to a

certain combination of three or four stations in a certain

sequence. One must note that calculating the closure phase and

delay needs to meet the requirement of a closed triangle with

moving stations for geodetic VLBI, where the observables in

one scan are not necessarily referring to the same wavefront.

The models and explanations were addressed in our previous

studies (see Xu et al. 2016; Anderson & Xu 2018a).

Based on the assumption of independent thermal noise, the

closure phase uncertainty

clr

was derived from the uncertain-

ties of the phase observables of the three baselines, calculated

from Equation (1), and the closure amplitude uncertainty

nclr,

Table 1

Observational Data Statistics (Data Available on 2018 February 26)

Session Type 24 hr Intensive All

Time period 1979-09 to

2018-02

2016-01 to

2018-02

1979-09 to

2018-02

Sessions 6533 150 6683

Observables 14,741,307 18,523 14,759,830

Stations 191 9 191

Baselines 3186 29 3186

Sources 5228 157 5228

https://ivscc.gsfc.nasa.gov/products-data/index.html

https://ivscc.gsfc.nasa.gov/program/cont14/

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

for example for ν

abcd

, was obtained by

ss s

== + + +

nnnn

⎛

⎝

⎜⎞

⎠

⎟⎛

⎝

⎜⎞

⎠

⎟⎛

⎝

⎜⎞

⎠

⎟⎛

⎝

⎜⎞

⎠

⎟(),4

ab cd ac bd

222 2

abcd

ab cd ac bd

clr

where, for instance,

nab is the uncertainty of the amplitude

observable on baseline ab, which is calculated based on

Equation (2).

A rich variety of stations and networks

was used by

geodetic/astrometric VLBI; the VLBI networks used in

individual sessions had as few as only two stations and had

as many as more than 30 stations, and the antenna sizes ranged

from 3 up to 100 m. The IVS currently organizes the VLBI

observing activities utilizing worldwide available resources in

order to routinely carry out 24 hr sessions three to four times

per week and 1 hr sessions eight times per week, and to keep

the rapid turnaround sessions with their ﬁnal results to be

delivered to the public within a reasonably short time (the

latency of EOP sessions is about two weeks, and it is about one

day for intensives). Therefore, the 40 yr history of geodetic

VLBI observations involves changes in station networks,

observing strategy, frequency setup, sampling rate, hardware

for recording system, correlators, and so on. Compared to the

CONT14 data for our previous studies, the study in this paper

requires a more comprehensive processing of all historical data.

An important and unavoidable process is outlier ﬂagging in

closure analysis.

Two key points for outlier ﬂagging are (1)the closure phase,

closure delay, and closure amplitude deﬁned by Equation (3)

all have an expected value of zero with a thermal noise

(Gaussian)distribution for point-like sources, and (2)for

extended sources, source-structure effects are generally sym-

metric and continuous (except for breaks in closure phases due

to ambiguity)as a function of 24 hr of Greenwich Mean

Sidereal Time (GMST)for each individual baseline. At the

same GMST from different dates, the geometry of a baseline

with respect to a source is identical (by ignoring the nutation

effects on the R.A., polar motion, and station movements).If

the source structure is stable during a period of time, it allows

us to combine the closures from adjacent observations as a

function of wrapped GMST, which helps a great deal for

ﬂagging.

Outlier ﬂagging was done session-wise based on the

behaviors of closure quantities over all sources in one session.

Three main types of phase outliers were detected for these 6533

sessions:

1. Noise-like: the standard deviations of closure phases with

respect to zero and its mean for the whole session, a

speciﬁc station, or a baseline are larger than the threshold;

2. Constant offset: the ratios of the mean closure phase to its

standard deviation for a station or a baseline are larger

than the threshold, by sorting out the closures involved by

the station or the baseline;

3. 180°jump: closure phases of a station or a baseline are

grouped at 0°and 180°, by sorting out the closures

involved by the station or the baseline.

The constant offset outlier detection was also applied to

closure amplitude. Three very short baselines were completely

excluded both for phase and amplitude: WETTZ13N–WETT-

ZELL,YEBES40M–REAGYEB, and HOBART26–HOBART12.

Statistics were then calculated based on the resultant closure

quantities for each source. Because the expected value of

closures for a compact point source is zero, the departures of

closure phase and closure amplitude with respect to zero in

principle indicate how far away the structure deviates from an

ideal point in terms of its effects on phase and amplitude

observables. The statistical quantity of phase, the closure phase

rms (CPRMS),isdeﬁned as

=∣∣ ()

CPRMS , 5

clr

where

clr is the ith closure phase of a source and w

is its

weight. Accordingly, the statistical quantity of amplitude, the

closure amplitude rms (CARMS),isdeﬁned as

=∣∣ ()

CARMS , 6

clr 2

where

clr is the ith closure amplitude of a source and w

is its

weight.

Different weighting schemes were applied. One weighting

scheme naturally uses the uncertainties of closure phases and

closure amplitudes as

() () ()ww

1,1.7

clr clr

The second scheme adds basic noise to prevent extremely high

weights by high R

according to equations

=+=+

() () () ()ww

1.5 ,1

0.1 ,8

22 22

clr clr

where a basic noise of 1°.5 was applied to the closure phase and

that of 0.1 to the closure amplitude. The basic noise for the

phase is determined by assuming that the three observables of a

triangle all have an R

of 60, while a much higher basic noise

is used for the amplitude by assuming that the four observables

of a quadrangle all have an R

of 20, the minimum threshold

for scheduling. As we reported in the previous section, the

mean value of R

for the entire data set is twice as large as the

median value, even though we included the observables with

actual R

lower than 20. This gives a hint that a signiﬁcant

number of observables have much higher R

. These large

differences in R

lead to differences of several orders of

magnitude in the relative weighting from uncertainties, which

have no realistic meaning in the relative accuracies of closure

measurements. The basic-noise weighting can down-weight

those observables with high R

signiﬁcantly by empirically

adding a noise ﬂoor to the uncertainties from Equations (1)

and (2).

The third scheme is a uniform weighting of 1 for all closures.

Uniform weighting has the advantage of taking the source-

structure effects equally into account, whereas the natural

weighting reduces signiﬁcantly the contributions of closures

with strong source-structure effects, because the uncertainties

of closures are derived from R

, which is systematically related

See the IVS station list at ftp://cddis.gsfc.nasa.gov/pub/vlbi/ivscontrol/

ns-codes.txt.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

to source-structure effects—it always decreases when there are

strong structure effects.

The statistics used for our study, CARMS and CPRMS, are

based on the technique of rms, instead of other concepts, such

as mean and median, because of the nature of structure effects.

For an extended structure, it is not necessary to have large

structure effects in phase and amplitude all the time, and

structure effects on closure quantities can be reduced or even

canceled out. But it is always the case that structure effects

from extended sources have large variations in closure

quantities from observations at different epochs of the GMST.

In this special case, the technique of rms works better.

4. Observational Results

Source structures are often represented as images of

brightness distributions derived from phase and amplitude

observables based on various imaging techniques (e.g., hybrid-

mapping algorithms, see Readhead & Wilkinson 1978, Corn-

well & Wilkinson 1981; and the maximum entropy method, see

Narayan & Nityananda 1986, Shevgaonkar 1986). With the

advent of the VLBA, snapshot observations of ICRF sources to

make images with sensitivities obtained previously only with

full-synthesis observations are possible. The largest effort to

image ICRF sources is a series of VLBA astrometric

experiments called Research and Development VLBI sessions

(Fey et al. 1996; Fey & Charlot 1997,2000)and the VLBA

Calibrator Surveys, a series of six campaigns run on the VLBA

from 1994 to 2007 (VCS1–VCS6; Beasley et al. 2002;

Fomalont et al. 2003; Petrov et al. 2005,2006,2008; Kovalev

et al. 2007). As a continuation, the second epoch VLBA

Calibrator Survey campaign (VCS-II)was undertaken to

improve the position estimates of 2400 VCS sources (Gordon

et al. 2016). For ICRF sources in the southern hemisphere, a

joint program of observations with the Australia Telescope

National Facility (ATNF)using geodetic stations and ATNF’s

Long Baseline Array were carried out in order to make images

(Fey et al. 2004a,2004b; Ojha et al. 2004). Images based on

these VLBA observations are publicly available through

websites.

15,16,17

Source structures can also be addressed simply by

demonstrating their effects on phase and amplitude observables

by closures, without calibrations being needed. According to

their deﬁnitions, closure delay and closure phase measure the

summation of source-structure effects on the three baselines of

a triangle, and closure amplitude measures a ratio of structure

effects on the amplitudes of four baselines of a quadrangle.

Even though closure quantities for a speciﬁc triangle or

quadrangle provide signiﬁcantly less knowledge of brightness

distribution compared to images, they do have several

advantages. First, because a change in the pattern (see its

deﬁnition in the next paragraph)of structure effects in a given

triangle or quadrangle necessarily indicates a change in the

intrinsic structure, closure quantities have a particular applica-

tion in monitoring radio sources for changes in intrinsic

structure. These changes in structure can be quantiﬁed and

compared by analyzing closure quantities. Second, closure

quantities directly tell the magnitudes of structure effects on

VLBI observables.

Changes in structure effects of a radio source are caused by

variations in the observing geometry, which depends on the

GMST epoch of the observable for a given baseline, and

changes in its intrinsic structure over time. Because the

timescale of the observing geometry variation is constant—

24 hr of GMST—and the timescales of structure changing are

expected to be signiﬁcantly larger, these two factors for

structure effect changing can be distinguished by using two

time systems for the observing time, the fraction of day by

GMST and the date, for our closure analysis. In this paper, we

refer to the change in structure effects over GMST as pattern

and to the change of the pattern over date as evolution.

Structure effects have different patterns for different triangles

or quadrangles. These patterns also vary both from source to

source and from time to time for a given source. A radio source

can have thousands of triangles or quadrangles for the global

VLBI networks. Because studying the unique instrumental,

source, and triangle/quadrangle dependencies of such a large

number of subarrays in IVS networks is difﬁcult, efforts have

been made to identify and select just a few triangles or

quadrangles out of those many for the best identiﬁcation for a

speciﬁc source. The triangles or quadrangles used for each

source must be determined separately. However, only examin-

ing closure quantities of too few triangles or quadrangles can

lead to biases in understanding source structures. Therefore, the

results are organized in two ways: (1)closure quantities of

several triangles or quadrangles as representatives are shown in

plots together with their statistics, and (2)the statistics for all

available closures for each individual source are calculated and

addressed. The CARMS and CPRMS values from all available

closure phases and amplitudes for each individual source in the

entire observed history are labeled global in Sections 4.1

and 4.2. The CARMS and CPRMS values based on a single

triangle or quadrangle are, on the other hand, shown in the

closure plots and will be addressed in these two sections

as well.

The closure phases will be addressed in Section 4.1 for four

selected sources, and the closure amplitudes will be addressed

in Section 4.2 for seven selected sources. Note that the closure

phase and closure amplitude have different responses to source

structure, but a signiﬁcant change identiﬁed in the pattern of

one type of closure quantity will appear in that of the other

because of the change in intrinsic structure. Source 2229+695

will be discussed with both its closure phases and its closure

amplitudes to demonstrate that effect. The global CARMS and

CPRMS values will be addressed in Section 4.3. The 30 most

frequently observed sources will be discussed there to explore

these statistical values.

Several remarks in order to better understand the results in

the next sections are summarized here. (1)For a given source

and for a speciﬁc triangle or quadrangle, the patterns of closure

phases and closure amplitudes within one session have in

general the feature of continuity, and these patterns remain the

same for observations from other sessions if the structure did

not change. (2)For the same triangle, the patterns of closure

quantities of different sources have no correlation at all, due to

differences in their intrinsic structures. (3)Those patterns

depend on the changes of the uv coordinates of the baselines

(the baseline vectors projected onto the sky plane)involved in

the closure quantities. Therefore, the patterns in general should

have a wave shape. (4)The GMST ranges over which radio

sources are visible differ from source to source; for instance,

http://www.astrophy.u-bordeaux.fr/BVID/

https://www.usno.navy.mil/USNO/astrometry/vlbi-products/rrﬁd

http://astrogeo.org/vlbi_images/

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

sources with high declination can have closures at all 24 hr of

GMST while sources close to the equator tend to have closures

only in limited time zones. (5)Both larger magnitudes of the

peaks and more rapid changes with respect to GMST in those

patterns indicate a more extended structure. (6)Whenever the

pattern of closure quantities, for a speciﬁc triangle/quadrangle

and a given source, evolves, a change in the intrinsic structure

happens. (7)In general, ICRF sources have a core–jet

morphology. Variations in the intrinsic structure result from

changes in the brightness of the core or jet components, the

ejection of a new jet component, and the motion of jet

components along the jet, which cause evolution of the closure

patterns.

4.1. Closure Phases

Source 1357+769, which shows only thermal noise in the

patterns of closure phases for all available triangles, will be

introduced ﬁrst. It is also used to demonstrate the station

performances for several antennas in the geodetic VLBI

networks. Three other sources are chosen as representatives

of sources with structure to be discussed in detail, in order to

present a sample of various sources with different amounts of

structure effects, different variabilities in intrinsic structures

and different categories in ICRF2 and ICRF3. In terms of

categorization, ICRF2 has three categories, deﬁning sources

that deﬁne the reference frame, special-handling sources that

were claimed to have the largest position variations over time,

and other sources, while ICRF3 only maintains the deﬁning

category and treats all the remaining as other sources.

4.1.1. Source 1357+769

Source 1357+769 is close to the north pole and is visible for

most of the northern stations at all times. Based on the three

weighting schemes, its global CPRMS values are 9°.0, 10°.4,

and 13°.9 over 339,062 available closure phases. The upper two

plots in Figure 1show the closure phases of two triangles,

GILCREEK–WESTFORD–WETTZELL and TSUKUB32–

WESTFORD–WETTZELL. The baselines between stations

TSUKUB32,WESTFORD, and WETTZELL are among the

longest baselines of the global VLBI networks and were

frequently scheduled in 24 hr sessions and intensives. The two

plots suggest that 1357+769 is relatively compact. The closure

phases for a smaller triangle, MEDICINA–SVETLOE–WETT-

ZELL, are shown in the bottom of Figure 1. The CPRMS for

this small triangle is 5°.5 and thus signiﬁcantly decreased

compared to the values of 9°.2 and 7°.4 for the larger triangles,

which suggests that it may be slightly resolved on those rather

long baselines. The closure phase patterns for source 1357

+769, which are ﬂat around 0°with small CPRMS values, less

than 10°, for individual triangles, can be applied to detect a

source with a compact structure.

4.1.2. Source 0133+476

In total, 0133+476 has 281,201 closure phases with global

CPRMS values of 8°.7, 11°.1, and 13°.4, based on the three

weighting schemes. The entire data set of its closure phases of

the triangle NYALES20–WESTFORD–WETTZELL is shown in

eight separate subplots of Figure 2to better show the evolution

of structure effects over time. These three stations were

frequently observing 0133+476 for 15 yr from 1999 to 2014.

From its global CPRMS, it is inferred that source 0133+476

was relatively compact or only slightly resolved; however, the

closure phases still show clear patterns of structure effects. For

instance, during the period of 2005-06 to 2006-11 as shown in

plot c of Figure 2, the wave shape with a magnitude of 30°was

repeated in 55 sessions with 444 closure points. Structure

effects became weaker during the period from 2007 to 2009,

and gradually increased again through 2010–2012. Finally,

0133+476 was extremely quiet. By inspecting the plots,

several remarks can be made: (1)structure effects of source

0133+476 on this large triangle have seven stable patterns

during 2001-10 to 2014-08, (2)the magnitudes of the peaks in

those patterns were changing from time to time, and (3)the

GMST epochs of those peaks were rather stable. The most

likely explanation is that the ﬂux density ratio of its jet to the

core changes from time to time, which causes the decreases and

the increases of the peak magnitudes shown in the ﬁgure. For

instance, if that ratio increases, the peak magnitude of closure

phases becomes larger and vice versa.

4.1.3. Source 0552+398

In total, 0552+398 has 447,292 closure phases with global

CPRMS values of 10°.1, 15°.5, and 19°.0 based on the three

weighting schemes. Its closure phases of two triangles GIL-

CREEK–NYALES20–WETTZELL and FORTLEZA–NYALES20–

WETTZELL are shown in Figure 3. The closure phase patterns of

both triangles are relatively stablebutwithanincreasingpeak

magnitude over more than 20 yr. During 10 yr of 332 sessions

from 1994 to 2005, the structure effects on the triangle

GILCREEK–NYALES20–WETTZELL gradually increased with

peaks in closure phase from 30°to 50°as shown in the upper plot

of Figure 3. A slow increase after 2005 is also visible in the lower

plot of Figure 3. 0552+398 is an example of a source with an

extended but relatively stable structure over a long timescale.

4.1.4. Source 2229+695

In total, source 2229+695 has 125,608 closure phases with

global CPRMS values of 14°.2, 17°.9, and 26°.5 based on the

three weighting schemes. The closure phases of two triangles,

KOKEE–NYALES20–WETTZELL and KOKEE–TSUKUB32–

WETTZELL, are displayed in Figure 4. These two plots

demonstrate that 2229+695 was already resolved in 2008, and

its structure continuously and signiﬁcantly increased after that.

Finally, in early 2018, it has structure effects on the closure

phases of the ﬁrst triangle with magnitudes larger than π

radians.

4.2. Closure Amplitude

The study of amplitude observables for geodetic VLBI is as

important as that of phase observables, from which group delay

observables are derived. Amplitude observables are sensitive to

source structure that causes structure delays, and the measure-

ment noises in delay observables are directly correlated with

observed amplitudes. For instance, if the observed amplitude is

only 10% of the ﬂux density used for scheduling, due to

resolved structure, the R

of actual measurements will be one

magnitude smaller than the R

expected from the schedule.

The contribution of thermal noise to the measurement

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

uncertainties will signiﬁcantly increase, sometimes even

leading to a failure in detection.

Because a quadrangle involves four stations, one more than a

triangle, and is sensitive to the station orders, an individual

source generally has many more different quadrangles than

triangles. However, compared to the closure phase, the closure

amplitude is much more vulnerable to the loss of individual

baseline observables in correlation, and because the list of

stations participating in IVS geodetic sessions frequently

changes from session to session, speciﬁc station quadrangles

are formed less frequently than speciﬁc station triangles. Thus,

for a given source, we get many more closure amplitudes than

closure phases, but for each individual triangle or quadrangle,

the number of closure amplitudes is much less than that of

closure phases.

Source 0454−810, which is located close to the south pole,

will be discussed ﬁrst to demonstrate both the behavior of a

compact source and the performances of several typical

southern geodetic stations. Six other sources are addressed as

examples to investigate their structure effects and the

variabilities in intrinsic structure.

4.2.1. Source 0454−810

In total, source 0454−810 has 14,551 closure amplitudes

with global CARMS values of 0.13, 0.14, and 0.15 based on

the three weighting schemes. Closure amplitudes of two

quadrangles, HART15M–HOBART12–KATH12M–YARRA12M

and HART15M–KATH12M–YARRA12M–HOBART12, are dis-

played in Figure 5. The radio source has been heavily observed

by southern stations since 2013. During its observing history, it

showed minimal structure as demonstrated by the plots in

Figure 5and the global CARMS values. This case can be

considered as an investigation of the performance of these

small antennas in the southern hemisphere concerning ampl-

itude observables. In the case of 0454−810, we should notice

that there is no signiﬁcant difference in the CARMS values

from the three weighting schemes.

4.2.2. Source 0016+731

In total, source 0016+731 has 619,226 closure amplitudes

with global CARMS values of 0.19, 0.29, and 0.34 based on

the three weighting schemes. Closure amplitudes of three

quadrangles, GILCREEK–KOKEE–WETTZELL–NRAO85_3,

KOKEE–TSUKUB32–WETTZELL–YEBES40M, and

ISHIOKA–KOKEE–YEBES40M–WETTZELL, are displayed in

Figure 6. Its strongly resolved structure before 2000, which is

identiﬁed by phase observables shown in the supplemental

information at [doi:10.11570/19.0007], can be partly con-

ﬁrmed by amplitude observables in plot a. Note that the two

quadrangles KOKEE–TSUKUB32–WETTZELL–YEBES40M

and ISHIOKA–KOKEE–YEBES40M–WETTZELL have nearly

the same geometry and have the same baseline sequence in

Equation (3)because ISHIOKA and TSUKUB32 are two

stations at one site. Therefore, their closure amplitudes can be

combined together and compared directly. We thus can learn

from plots b and c that the structure effects in amplitude

decreased from 2012 to 2014 and signiﬁcantly increased in

2017. The magnitude of the peak in 2018 increases to more

than 2. The likely reason for these evolutions in structure-effect

patterns is that the brightness of its core or of its jet is very

variable on timescales of several months.

Compared to the global CARMS values of 0014+813, the

global CARMS values of 0016+731 are relatively small. This

indicates that 0016+731 has less structure than 0014+813,

which can be equally addressed by comparing plots b and c in

Figure 1. Plots of closure phases at the Xband for source 1357+769 as a

function of GMST for three triangles, (a)GILCREEK–WESTFORD–WETT-

ZELL,(b)TSUKUB32–WESTFORD–WETTZELL, and (c)MEDICINA–SVE-

TLOE–WETTZELL. Closure phase uncertainties are shown as black bars. The

color coding indicates the observing date of the closure measurement, and the

corresponding legend is shown on the bottom-right corner of each plot. The

station names in the sequence of forming the triangle are shown in the ﬁrst row

of the top. Several statistics for the closure phases of that triangle are shown in

the second row: the starting date and the ending date of observations in the

format of year and month, the CPRMS value using uniform weighting, the

number of closure phases N_clr, and the number of sessions N_sess. For

instance, the triangle GILCREEK–WESTFORD–WETTZELL has 3322 closure

phases in 342 sessions with a CPRMS value of 9°.2 during its entire observing

period of 1994-01 to 2005-09. The thin horizontal black lines at 0°,−60°, and

60°are shown to guide the reading of the variation magnitudes of closure

phases. These three plots demonstrate well the minimal structure of source

1357+769 over 20 yr. In particular, plot c implies the good case of the thermal

noise level in geodetic baseline phase observables, which is about 3°.1.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

Figure 6to the closure plots of 0014+813 for the same

quadrangle in the supplemental information available at

[doi:10.11570/19.0007]. The structure of 0016+731 needs to

be studied carefully for three reasons. First, its structure is

changing at short timescales. Second, it had very extended

structure in the past. Third, it develops to have extended

structure again in 2018. These evolutions should cause changes

in its estimated positions from geodetic VLBI.

4.2.3. Source 0059+581

In total, source 0059+581 has 2,909,197 closure amplitudes

with global CARMS values of 0.18, 0.26, and 0.29 based on the

three weighting schemes. Closure amplitudes of two quadrangles,

NYALES20–TSUKUB32–WETTZELL–WESTFORD and

KATH12M–KOKEE–NYALES20–WETTZELL,aredisplayedin

Figure 7. Plot a demonstrates that its structure is changing from

year to year, indicated, for instance, by the closure amplitudes for

observations made in 2004 shown by green points, 2007 by yellow

points, and 2014 by red points. Much larger structure effects in

amplitude observables were found in the second quadrangle

KATH12M–KOKEE–NYALES20–WETTZELL, which involves a

very long south to north baseline, KATH12M–NYALES20.The

structure-effect evolutions at the timescale of 1 yr are clearly

visible for 0059+581.

4.2.4. Source 0642+449

In total, source 0642+449 has 1,422,456 closure amplitudes

with global CARMS values of 0.31, 0.46, and 0.53 based

on the three weighting schemes. Closure amplitudes of two

quadrangles, KOKEE–NYALES20–TSUKUB32–WETTZELL and

KOKEE–NYALES20–TSUKUB32–WESTFORD, are displayed in

Figure 8. As indicated by the evolution of structure effects shown

in these two plots, the structure of 0642+449 has been extended

since 1999, increased in 2006, and changed several times around

2008. The structure then increased signiﬁcantly in 2014 and again

changed after that. According to our imaging results based on the

CONT14 observations, it had an extended structure along the R.

A. direction. The major jet component was located about 0.5 mas

away from the core with ﬂux density almost equal to the core.

Le Bail et al. (2014)studied in detail the time series of 0642

+449 throughout its observing history. They found out that the

time series of its position had a ﬂicker noise, and signiﬁcant

jumps in its R.A. coordinates happened in 2005–2008 and

2010–2014. Plots a and b in Figure 8provide astrophysical

explanations for those jumps, that is, the evolution of structure.

Figure 2. Plots of closure phases at the Xband for source 0133+476 as a function of GMST for triangle NYALES20–WESTFORD–WETTZELL. The 15 yr history of

closure phase measurements for this triangle is divided into eight segments according to the different well-identiﬁed structure-effect patterns. They are therefore shown

in eight separate plots in order to keep the patterns with small magnitudes from overlapping with each other. See Figure 1for a description of the plot design except the

color coding, which is not used in this ﬁgure. The time evolution of the structure effects for this triangle can be easily seen, which suggests changes in the source

structure of 0133+476 although it was relatively compact. As we can see from these plots, the patterns changed in terms of the magnitudes of the peaks but not the

GMST epochs of the peaks in these eight plots. The peak magnitudes changed, both decreasing and increasing, which suggests that the evolution of the closure

patterns are caused by the changes in the ﬂux density ratio of its jet to the core. The scatter of the closure phases along the pattern in the time period of 2010-12 to

2013-02, shown in plot g, is signiﬁcantly larger than in other periods of time. One can infer that its core was more strongly resolved in 2010-12 to 2013-02.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

Source 0642+449 was one of the ICRF2 deﬁning sources, which

were selected based on observations before 2009, and has been

excluded from that category in the recently released ICRF3. It

demonstrates that a compact source can evolve to have an extended

structure in a time frame of a few years and then keep that extended

structure afterwards. Closure analysis is able to give an

intermediate, reliable warning for this kind of structure evolution.

4.2.5. Source 0851+202 (OJ 287)

In total, source 0851+202 (OJ 287)has 1,291,087 closure

amplitudes with global CARMS values of 0.25, 0.41, and 0.46

based on the three weighting schemes. Closure amplitudes of

two quadrangles, KATH12M–NYALES20–YARRA12M–WETT-

ZELL and KATH12M–WETTZELL–YARRA12M–MATERA,are

displayed in Figure 9. Plots a and b both demonstrate that during

the last 6 yr, the structure effects on closure amplitudes for these

two quadrangles evolved in magnitude from 0.5 to 4 twice and

from 0.5 to 2 several times. The likely reason is that this

equatorial source has an extended structure along the decl. axis,

and the ﬂux density of its jet component or the core is very

variable, on timescales of months. For cases like OJ 287, the

highly variable structure will make correcting structure effects

from geodetic VLBI observables both critical and challenging.

4.2.6. Source 1807+698 (3C 371)

In total, source 1807+698 (3C 371)has 578,885 closure

amplitudes with global CARMS values of 0.54, 0.65, and 0.68

based on the three weighting schemes. Closure amplitudes of two

quadrangles, NYALES20–TSUKUB32–WETTZELL–WESTFORD

and KOKEE–NYALES20–TSUKUB32–WETTZELL,aredis-

played in Figure 10. Plot a shows rapid changes in the

structure-effect pattern with respect to GMST, and plot b shows

Figure 3. Plots of closure phases at the Xband for source 0552+398 as a function of GMST for two triangles, (a)GILCREEK–NYALES20–WETTZELL and (b)

FORTLEZA–NYALES20–WETTZELL. See Figure 1for a description of the plot design. The patterns were stable and were only slightly evolved over more than 20 yr

in 332 sessions for the ﬁrst triangle and in 463 sessions for the second.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

that the peak magnitude is up to 2. Both of them, however,

demonstrate that its structure is reasonably stable over a long

term. All other quadrangles of 1807+698 also show stable

structure-effect patterns. This actually is one of the very few cases

of ICRF3 sources, along with 0552+398, that have extended

structure but remain stable over the entire history of observations.

4.2.7. Source 2229+695

In total, source 2229+695 has 486,119 closure amplitudes

with global CARMS values of 0.33, 0.43, and 0.49 based on

the three weighting schemes. Closure amplitudes of two

quadrangles, KOKEE–NYALES20–WETTZELL–TSUKUB32

and BADARY–KOKEE–YEBES40M–WETTZELL, are displayed

in Figure 11. Plot a demonstrates well the evolution of its

structure during 2008–2016, while plot b demonstrates that it is

a very extended source recently. Although we only present

plots of both closure phases and closure amplitudes for sources

2229+695 and 0607−157 (in the next section)in the main

body of this paper to show the reader an example of the cross-

check between these two kinds of closure quantities, the

extended structure detected by one type of closure is clearly

visible for the other closure type for virtually all sources.

4.3. Statistics of Closure Quantities

General information about the source structure is obtained by

calculating the statistics CARMS and CPRMS over all

available closure quantities for each individual source. Some

sources have already been reported in the previous two

sections, and a full list of 3417 ICRF3 sources that have at

least a certain number of closures—the threshold of that

number will be discussed in the next section—is available in

the machine-readable version of Table 2. These statistics are

then used to classify these sources as “good”or “bad”in terms

of the astrometric quality.

Figure 4. Plots of closure phases at the Xband for source 2229+695 as a function of GMST for two triangles, (a)KOKEE–NYALES20–WETTZELL and (b)KOKEE–

TSUKUB32–WETTZELL. See Figure 1for a description of the plot design. The evolution of its structure effect over 10 yr is clearly visible.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

The 30 most frequently observed sources, the statistics of

which are listed as examples in Table 2, in order of decreasing

number of observations used for the creation of the ICRF3, are

discussed here in detail by exploring their statistics. This is

important and interesting because the observations of these 30

sources make up about 42% of the entire data set of S/X

geodetic VLBI observations. It allows the estimation of how

much structure effects have been embedded in VLBI observa-

tions and the exploration of how useful these statistics can be in

terms of comparing the overall magnitudes of individual source

CARMS and CPRMS values to other sources.

Source 0059+581, having the most observables in geodetic

VLBI, has a CARMS value of 0.27 and a CPRMS value of

10°.8 based on the basic-noise weighting. It is resolved and

undergoes changes in intrinsic structure as demonstrated in

Figure 7. Its statistics, nevertheless, imply that over its

observing history, the impacts of its structure effects on

geodetic VLBI observations are limited. Its variable structure

needs to be monitored in order to guarantee its role as a ﬁducial

mark of the celestial reference frame.

Out of the 30 most frequently observed sources, two sources,

1357+769 and 0727−115, have minimal structure during the

several decades covered by geodetic VLBI observations. This

is demonstrated by the fact that 1357+769 has the smallest

CARMS values and 0727−115 has the smallest CPRMS

values. No signiﬁcant structure effects have been found based

Figure 5. Plots of closure amplitudes at the Xband for source 0454−810 as a function of GMST for two quadrangles, (a)HART15M–HOBART12–KATH12M–

YARRA12M and (b)HART15M–KATH12M–YARRA12M–HOBART12. Closure amplitude uncertainties are shown as black bars. The color coding indicates the

observing date of the closure measurement, and the corresponding legend is shown on the bottom-right corner of each plot. The station names in the sequence of

forming the quadrangle are shown in the ﬁrst row at the top. Several statistics for the closure amplitudes of that triangle are shown in the second row: the starting date

and the ending date of observations in the format of year and month, the CARMS value using uniform weighting, the number of closure amplitudes N_clr, and the

number of sessions N_sess. For instance, quadrangle HART15M–HOBART12–KATH12M–YARRA12M has 968 closure amplitudes in 61 sessions with a CARMS value

of 0.15 during its entire observing history from 2013-11 to 2017-04. The thin horizontal black lines at 0, −1, and 1 are shown to guide the reading of the variation

magnitudes of closure amplitudes. The global CARMS values of 0454−810 are only 0.13, 0.16, and 0.17 based on the three weighting schemes. The small global

CARMS values and these two plots suggest that source 0454−810 has minimal structure. The performance of these four southern stations in amplitude observable can

also be proven.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

Figure 6. Plots of closure amplitudes at the Xband of source 0016+731 as a function of GMST for three quadrangles, (a)GILCREEK–KOKEE–WETTZELL–

NRAO85_3,(b)KOKEE–TSUKUB32–WETTZELL–YEBES40M, and (c)ISHIOKA–KOKEE–YEBES40M–WETTZELL. See Figure 5for a description of the plot

design. The structure-effect patterns in plots b and c can simply be combined because ISHIOKA and TSUKUB32 are two stations at the same site. The evolution of its

structure effects is seen from the three plots.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

on closure phases or closure amplitudes from any available

triangles or quadrangles for these two sources; the plots show a

pattern of thermal noise or minimal structure as illustrated for

1357+769 by Figure 1. They are considered to be compact.

Seven sources, 0133+476, 0955+476, 1749+096, 0454

−234, 1606+106, 1300+580, and 0804+499, have the same

level of source structure as 0059+581. They all have CARMS

values smaller than 0059+581 but have larger CPRMS values,

which potentially means that their cores are more compact than

that of 0059+581 but with signiﬁcant jet components. The

changes in the intrinsic structure of 0133+476 have been

demonstrated by a series of plots in Figure 2. The closure plots

demonstrating slightly resolved structures for the other ﬁve

sources are shown in the supplemental information of this

paper at [doi:10.11570/19.0007].

These are the only 10 sources with the CARMS values

smaller than or equal to 0.30 out of the 30 most frequently

observed sources. Obvious and variable structures are still

identiﬁed for several sources among them, for instance, 0955

+476 and 0059+581. These 10 sources were selected as

deﬁning sources both for ICRF2 and for ICRF3, which is

reasonable because sources with minimal or limited structure

effects in return give rise to having stable positions estimated

from the VLBI.

Three sources, 1803+784, 0552+389, and 1807+698 (3C

371), are discovered to have more extended but stable structure

over the last decades. 1803+784 is resolved and its structure

effects yield constant patterns. The structure of 0552+398 has

increased slowly during the last 20 yr as shown in Figure 3.Of

these three extended but stable sources, the structure effects of

Figure 7. Plots of closure amplitudes at the Xband for source 0059+581 as a function of GMST for two quadrangles, (a)NYALES20–TSUKUB32–WETTZELL–

WESTFORD and (b)KATH12M–KOKEE–NYALES20–WETTZELL. See Figure 5for a description of the plot design. The structure effects shown in the two plots reveal

that 0059+581 was temporarily extended, for instance, in 2007, 2013, and 2015.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

3C 371, as shown in Figure 10, are the largest and change the

most rapidly. 0552+398 apparently has larger structure effects

in amplitude than 1803+784, which suggests that 0552+398

has a stronger jet component relative to its core. Structure

effects of 1803+784 change more rapidly with respect to

GMST than 0552+389, which therefore suggests that the jet of

1803+784 is farther away from its core. 3C 371 has never been

adeﬁning source. 1803+784 was excluded from that category

in ICRF3, and 0552+398 remains a deﬁning source in ICRF3.

The reason that source 0552+398 with such high CARMS

values was classiﬁed as a deﬁning source is likely due to its

stable structure and the need for a uniform geometric

distribution of deﬁning sources.

Six sources, 1741−038, 1739+522, 1638+398

(NRAO512), 0528+134, 0016+731, and 1334−127, have

CARMS values ranging from 0.3 to 0.4 based on basic-noise

weighting. As an example of these, 0016+731 is discussed in

Section 4.2.2 to demonstrate its variable structure. The

evolution in the structures of the other ﬁve sources are

addressed in the supplemental information. Of these six

sources, 1741−038, 1739+522, and 0528+134 are ICRF2

special handling sources primarily due to signiﬁcant variations

in the time series of their estimated positions. 1638+398 was

excluded from the deﬁning category in ICRF3, while 0016

+731 and 1334−127 are still deﬁning sources.

The remaining 11 sources have very large structure effects,

and their structures are highly variable according to our study.

Figure 8. Plots of closure amplitudes at the Xband for source 0642+449 as a function of GMST for two quadrangles, (a)KOKEE–NYALES20–TSUKUB32–

WETTZELL and (b)KOKEE–NYALES20–TSUKUB32–WESTFORD. See Figure 5for a description of the plot design. Its structure has dramatically evolved several

times. The largest magnitude of closure amplitude in plot a is −4, so that the ratio of the observed amplitudes of the four baselines was less than e

−4

(∼0.04). The

closure amplitude patterns change very rapidly with respect to GMST in plot b. A very extended structure is detected.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

All of their CARMS values based on basic-noise weighting and

uniform weighting are larger than or equal to 0.40. Some of

them have been discussed in detail in the previous two sections.

For example, the structure evolution of 0642+449 is discussed

in Section 4.2.4 and 2229+695 in Sections 4.1.4 and 4.2.7.Of

these 11 sources, there are some with structures varying

signiﬁcantly on timescales of months, such as 0851+202 (OJ

287)and 0923+392 (4C 39.25), some with structures varying

from year to year, such as 1611+343, 1044+719, and 1308

+326, and some which were developing to have a very

extended structure in recent years, such as 2037+511 (3C 418),

0642+449, 0229+131, and 2229+695. There are ﬁve sources

already classiﬁed as special handling sources in ICRF2, 0923

+392, 1611+343, 1044+719, 1308+326, and 2145+067.

Two sources, 0851+202 and 0642+449, were excluded from

the group of deﬁning sources in ICRF3, but 2229+695 is still

one of the ICRF3 deﬁning sources.

Another example regarding the categorization of radio

sources in a reverse way is 0607−157, which was one of the

ICRF2 special handling sources but was selected as one of the

deﬁning sources for ICRF3. It has been heavily observed by

southern stations since 2013. In total, source 0607−157 has

156,121 closure amplitudes with CARMS values of 0.23, 0.51,

and 0.57 based on the three weighting schemes. Its closure

phases of the triangle HOBART12–WARK12M–YARRA12M and

closure amplitudes of the quadrangle HOBART12–KATH12M–

WARK12M–YARRA12M are displayed in Figure 12. Plot a

demonstrates that the structure of this equatorial source was

Figure 9. Plots of closure amplitudes at the Xband for source 0851+202 (OJ 287)as a function of GMST for two quadrangles, (a)KATH12M–NYALES20–

YARRA12M–WETTZELL and (b)KATH12M–WETTZELL–YARRA12M–MATERA. See Figure 5for a description of the plot design. The magnitudes of the structure

effect on closure amplitudes evolved within 6 yr from 0.5 to 4 two times and from 0.5 to 2 several times. The timescale of the evolution is less than 1 yr. Very

extended structure is detected.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

evolving during the time period from 2013 to 2017. This is

conﬁrmed by the amplitude observables shown in plot b. 0607

−157 has a very extended and variable structure. There are

three other ICRF2 special handling sources, 0235+164, 0208

−512, and 1448+762, that are selected for ICRF3 to be

deﬁning. The adequacy of being deﬁning is questioned by this

study for 0208−512, 0607−157, and 1448+762, which have

CARMS values of 0.47, 0.51, and 0.66, respectively, using

basic-noise weighting. The fourth source, 0235+164, with a

CARMS value of 0.18 based on basic-noise weighting, can be

conﬁrmed to have minimal structure and thus should be

classiﬁed as deﬁning. These cases prove the importance of

information that is independent of closure phases and closure

amplitudes.

From the study of these 30 most frequently observed

sources, we have learned that the statistics of closure quantities

are highly correlated with the qualities of sources’astrometric

behaviors. This is expected because the CARMS and CPRMS

values tell the overall magnitudes of source-structure effects on

VLBI observations for the creation of ICRF3. They are

independently derived from different types of observables, that

is, phase and amplitude, but they still provide information

about the amount of structure effects on delay observables.

It is not surprising that 20 out of these 30 sources show

extended structures during the time period covered by geodetic

VLBI observations. Those structures have contributed a

signiﬁcant amount of errors to geodetic VLBI data analysis.

Among these 20 extended sources, most are variable in

Figure 10. Plots of closure amplitudes at the Xband for source 1807+698 (3C 371)as a function of GMST for two quadrangles, (a)NYALES20–TSUKUB32–

WETTZELL–WESTFORD and (b)KOKEE–NYALES20–TSUKUB32–WETTZELL. See Figure 5for a description of the plot design. The structure-effect pattern of the

ﬁrst quadrangle repeated for 1454 closure measurements in 132 sessions over 11 yr, with only minor differences. A stable pattern is also shown in plot b. These two

plots demonstrate that its structure remains constant over a long time.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

intrinsic structure, and thus, their apparent positions change on

the sky.

The CARMS value of 0.3 based on basic-noise weighting

and uniform weighting can be referred to as the threshold for

differentiating between sources being reasonably compact or

extended. The sources with CARMS values smaller than

0.3 are considered to be reasonably compact or have limited

structure effects. CARMS values larger than 0.4, however,

indicate that the sources are extremely extended. To determine

how many closure quantities must be measured in order to get a

reliable determination of CARMS and CPRMS, we will

examine a few sources as test cases. There are two ICRF3

deﬁning sources, 1619−680 and 1925−610, that have large

CARMS and CPRMS values but only hundreds of closure

quantities. In total, source 1619−680 has 1225 closure

amplitudes with CARMS values of 0.78, 0.81, and 0.84, and

713 closure phases with CPRMS values of 50°.9, 55°.4, and

61°.9, using the three weighting schemes. Closure amplitudes

of the quadrangle HOBART12–KATH12M–YARRA12M–

WARK12M are displayed in plot a of Figure 13. Even though

the number of closure amplitudes available for this speciﬁc

quadrangle is only 53, the structure-effect pattern in closure

amplitude can be easily seen in this plot. There is only one

point available from 2015 and three points from 2016;

however, they all agree with the clearer pattern determined

from observations in 2013 and 2014. This also shows that

closure quantities can be used to test for evolution in intrinsic

source structure using only a few observations provided that a

clear pattern in structure effects has been detected in advance.

The closure amplitudes of 1925−610 for the quadrangle

Figure 11. Plots of closure amplitudes at the Xband for source 2229+695 as a function of GMST for two quadrangles, (a)KOKEE–NYALES20–WETTZELL–

TSUKUB32 and (b)BADARY–KOKEE–YEBES40M–WETTZELL. See Figure 5for a description of the plot design. The magnitudes of the patterns in both plots

increased, and the changes of the patterns with respect to GMST became more and more rapid.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

HARTRAO–WARK12M–YARRA12M–HOBART26 are shown in

plot b of Figure 13, where its structure effects are convincingly

demonstrated. In total, 1619−680 has only 622 closure

amplitudes with CARMS values of 0.93, 0.99, and 1.07, and

316 closure phases with CPRMS values of 39°.3, 40°.4, and

50°.3, based on the three weighting schemes. Both the CARMS

and CPRMS values, as a cross-check for each other, are

extremely high for these two sources. For radio sources with

extended structures that have only a few closures, these

statistics still work well.

5. Discussions

5.1. Indicator for Astrometric Use

The impact of the three different weighting schemes on the

statistics can be examined in Figure 14. As shown in plot a, the

basic-noise weighting generally gives larger CARMS values

than natural weighting, by up to 60%. As we already discussed

in Section 3, closure uncertainties do not always give realistic

relative weighting, for instance, a measurement with an R

200 is not 100 times more accurate than that with an R

20, and closures with large structure effects often get

down-weighted using natural weighting. A natural weighting

scheme signiﬁcantly underestimates the structure effects on

geodetic VLBI observables. Plot b shows that uniform

weighting gives a slightly larger CARMS value than basic-

noise weighting, but they agree well with each other. Similar

behaviors of these three weighting schemes are found for

CPRMS values. Because the basic-noise weighting takes more

realistic measurement noises into account and reduces the

impacts of unrealistic relative weighting based on R

, basic-

noise weighting is superior to the other two weighting schemes

presented in this article, and we use it for our further analysis.

Both closure amplitudes and closure phases detect structure

effects but utilize independent measurements. We estimate the

correlation between the CARMS and CPRMS values in order

to test how well each method performs for identifying the

magnitude of the source structure, as demonstrated in

Figure 15. All sources with at least 100 closure phases and at

least 100 closure amplitudes are displayed as red dots, whereas

the remaining 848 sources are shown as blue dots. The red dots

generally follow the pattern where the CPRMS values increase

with the increased CARMS values; the blue dots, however,

spread out like noise. This means that with a sufﬁcient number

Table 2

CARMS and CPRMS Values for 3417 ICRF3 Sources

IERS CARMS Number CPRMS [degrees]Number Category

Design. Nat. Bas. Uni. of Nat. Bas. Uni. of in

Weighting clo. amp. Weighting clo. pha. ICRF2 ICRF3

0059+581 0.18 0.27 0.29 2909197 7.7 10.7 14.3 610681 D D

0552+398 0.36 0.57 0.59 1776309 10.1 15.5 19.0 446475 D D

0851+202 0.25 0.44 0.47 1291087 10.0 14.7 19.5 310001 D O

1803+784 0.29 0.35 0.36 2292526 12.6 15.1 17.8 365718 D O

0923+392 0.41 0.80 0.82 1107469 8.5 17.8 22.9 250502 S O

0727−115 0.14 0.24 0.25 668085 5.8 9.0 12.6 203947 D D

1741−038 0.18 0.33 0.35 659623 7.3 12.9 19.7 192577 D D

1357+769 0.13 0.16 0.17 1840583 9.0 10.4 13.9 338961 D D

1739+522 0.23 0.35 0.38 1222766 9.5 12.5 18.3 272840 S O

0955+476 0.18 0.25 0.27 1200368 9.5 10.9 14.6 272489 D D

0133+476 0.16 0.23 0.24 1376813 8.7 11.1 13.4 280323 D D

2037+511 0.37 0.61 0.69 806175 13.0 18.3 28.2 247632 O O

1749+096 0.12 0.22 0.24 867342 7.7 10.8 14.3 214564 D D

0642+449 0.31 0.49 0.53 1422456 7.9 11.8 20.2 262982 D O

1611+343 0.36 0.40 0.42 964008 16.1 17.6 20.1 207158 S O

1807+698 0.54 0.65 0.68 578885 19.5 22.6 27.7 177403 O O

0528+134 0.19 0.37 0.41 509935 7.1 13.8 21.1 121306 S O

1044+719 0.36 0.59 0.66 1232721 10.9 14.9 22.5 189816 S O

1638+398 0.21 0.32 0.34 676370 10.1 12.5 17.4 145161 D O

0454−234 0.17 0.25 0.27 287178 7.7 10.6 15.4 99952 D D

0016+731 0.20 0.31 0.34 619226 7.8 10.2 14.9 163495 D D

1606+106 0.20 0.28 0.30 528235 10.0 13.3 17.5 125453 D D

1300+580 0.14 0.18 0.19 503547 10.4 12.1 15.9 136728 D D

1334−127 0.12 0.31 0.33 279226 7.7 13.6 18.8 86905 D D

1308+326 0.35 0.60 0.64 583789 12.5 19.0 26.8 121829 S O

2145+067 0.37 0.69 0.70 467436 11.1 21.8 25.9 100938 S O

0229+131 0.42 0.61 0.66 337879 14.8 20.9 32.2 88496 D O

0336−019 0.32 0.44 0.47 380810 13.8 18.1 21.8 96359 O O

2229+695 0.34 0.44 0.49 486119 14.2 17.9 26.5 124795 D D

0804+499 0.16 0.20 0.21 862058 11.2 12.4 15.8 137636 D D

Note. Sources are listed in the order of the total number of their observables for the creation of the ICRF3 (Charlot et al. 2018). Columns 2, 3, and 4 are CARMS based

on natural weighting (Equation (7)),basic-noise weighting (Equation (8)) and uniform weighting, respectively, while columns 5, 6, and 7 are CPRMS values.

Columns 11 and 12 are the source categories in ICRF2 and ICRF3 with “D”indicating deﬁning sources, “S”for special handling sources, and “O”for others. This

table is available in its entirety in machine-readable format for 3417 ICRF3 sources; only 30 sources are sorted out and listed here.

(This table is available in its entirety in machine-readable form.)

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

of closure quantities the CARMS and CPRMS values agree

with each other, but with very few closure quantities available

they are subject to outliers in geodetic observables. The two-

dimensional Kolmogorov–Smirnov test (Press et al. 1992,

Chapter 14.7)for these two distributions is done. The

conclusion of this test is that the probability of these two

sample sets being drawn from the same parent distribution is

−58

—a lower value of the probability indicates a smaller

correlation between the two tested distributions, and a

probability value of 1 implies they are statistically identical.

The test detects the most signiﬁcant difference in the two

distributions when the data are divided into quadrants at the

point with a CARMS value of 0.53 and a CPRMS value of

17°.9. As shown at the four corners of Figure 15, the largest

frequency of occurrence of red dots happens in the lower-left

quadrant with a fraction of 71% while that for blue dots in the

upper-left quadrant with a fraction of 40%. A denser

distribution of the blue dots on the upper-left quadrant suggests

that CPRMS values are more vulnerable to outliers than

CARMS values. The main reason is due to the fact that phase

observables are more easily corrupted than amplitude obser-

vables in geodetic VLBI. This will be further investigated in

the future using the imaging process. Based on these

investigations, we suggest the CARMS values from the

basic-noise weighting for the astrometric use, and the CPRMS

values and other weighting schemes for the cross-check. To

maximize the use and to keep the reliability of these statistics,

the minimum numbers of closure quantities for an individual

source are set to be 100.

Figure 12. Plots of structure effects of 0607−157 as a function of GMST on (a)closure phases of the triangle HOBART12–WARK12M–YARRA12M and (b)closure

amplitudes of the quadrangle HOBART12–KATH12M–WARK12M–YARRA12M. For plot a, see Figure 1for a description of the plot design, and for plot b, see Figure 5.

A very extended and variable structure is detected, as shown in the two plots. It was one of the ICRF2 special handling sources but was selected as one of the deﬁning

sources for ICRF3.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

5.2. Connection with Structure Effects on Group Delay

Observables

There are several practical reasons that group delay

observables were not used to investigate source structure in

this study. First, our previous studies (Xu et al. 2016,2017;

Anderson & Xu 2018a)have found out that closure phases and

closure amplitudes always have clearer structure-effect patterns

than closure delays, mainly due to large uncertainties of delay

observables. Second, nonclosing delay offsets exist in most

geodetic VLBI sessions, the impacts of which can be mitigated

by estimating baseline clock offsets in geodetic data analysis

but will make closure-delay analysis unreliable and cause

biases in quantifying structure effects. Third, the outlier

ﬂagging for group delay observables is also unreliable, due

to their large, variable uncertainties in different types of IVS

sessions. Fourth, there are various other technical issues that

make closure-delay analysis relatively difﬁcult, for instance,

the group delay ambiguity, introduced by the bandwidth

synthesis process, of a good X-band delay with a bad S-band

observable is often not properly resolved. In this section,

therefore, we aim to make a connection of CARMS values with

the structure effects on group delay observables, based on our

recent study (Anderson & Xu 2018a), where we have derived

images and done statistical closure analysis for CONT14

observations. The magnitudes of structure effects on group

delay observables for sources with various CARMS values are

determined as test cases, to give insights into the potential

impacts of source structure on geodetic VLBI data analysis.

Figure 13. Plots of closure amplitudes at the Xband as a function of GMST for (a)source 1510−089 for the quadrangle HOBART12–KATH12M–YARRA12M–

WARK12M and (b)source 1925−610 for the quadrangle HARTRAO–WARK12M–YARRA12M–HOBART26. See Figure 5for a description of the plot design. Reliable

structure-effect patterns are detected as shown in the two plots, even with a few tens of closure measurements. Both of these two sources have only several hundreds of

closure measurements in total.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

Figure 14. Comparison of CARMS values (a)between nature weighting and basic-noise weighting and (b)between basic-noise weighting and uniform weighting for

4265 radio sources. The natural weighting often gives lower CARMS values, and the basic-noise weighting and uniform weighting schemes agree well with each

other.

Figure 15. Plot of CPRMS values vs. CARMS values using basic-noise weighting for 4265 sources. The 3417 sources with at least 100 closure phases and at least 100

closure amplitudes are plotted in red dots, while the remaining 848 sources are plotted in blue dots. The difference between these two distributions is investigated by

using the Kolmogorov–Smirnov test, which conﬁrms that they are statistically different from each other. The test identiﬁes the point with a CARMS value of 0.53 and

a CPRMS value of 17°.9, which deﬁnes the four natural quadrants, represented by the two gray dashed lines, to give the largest difference between the distributions of

red dots and blue dots. The frequencies of occurrences of the red dots and blue dots in each of four quadrants are displayed in the four corners with their corresponding

colors. The CARMS values are recommended for astrometric use, and the CPRMS values for the cross-check.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

As shown by Charlot (1990), the structure effects on VLBI

group delay measurements can be modeled based on the spatial

brightness distribution of the observed source, the source

image, and the geometry of the baseline vector projected onto

the plane of the sky. We have derived high-dynamic source

images from the CONT14 observations. The contributions of

intrinsic structures to group delays of two typical geodetic

baselines, TSUKUB32–WETTZELL and NYALES20–TSU-

KUB32, are calculated and shown in Figure 16 for ﬁves

sources with the CARMS values from 0.18 to 0.91. These

CARMS values are recalculated based on the observations in

CONT14 only and, thus, represent their structure-effect

magnitudes at that time. 0016+731 had a very compact

structure in 2014 May with a recalculated CARMS value of

0.19 and had a maximum structure delay of a few picoseconds

on these two baselines. However, the other four sources, with

CARMS values larger than 0.3, had structure delays up to 40 ps

for the shorter baseline NYALES20–TSUKUB32. This magni-

tude can be reduced by removing a sinusoidal signal with the

period of 24 hr, due to the chosen reference source position in

the image, which is not necessarily the same as the position

estimated from geodetic data analysis. However, the variations

will still remain at the level of larger than 10 ps. The structure

delays on the longer baseline TSUKUB32–WETTZELL, the

length of which is also very common for the IVS observations,

increase to several tens of picoseconds and change more

rapidly with respect to GMST for sources with CARMS values

larger than 0.5. They even have a magnitude of 400 ps for the

very extended source 0642+449 in 2014 May. The structure-

effect magnitudes of these ﬁve sources for baseline TSU-

KUB32–WETTZELL are listed in Table 3. The overall

contributions of their source structures to the actual CONT14

Figure 16. Plots of model values of structure effects on group delay observables during 24 hr of GMST for two baselines TSUKUB32–WETTZELL (bottom)and

NYALES20–TSUKUB32 (upper). The sources are 0016+731 (0.19), 0749+540 (0.38), 1739+522 (0.57), 3C 371 (0.71), and 0642+449 (0.91), where the numbers in

parentheses following the source names are the CARMS values calculated from CONT14 observations only. Therefore, these CARMS values should indicate the

structure-effect magnitudes of those sources in 2014 May. Five lines represent the model structure delays as a continuous function of GMST based on the image from

CONT14 observations (Anderson & Xu 2018a), with the colors indicating the ﬁve sources. Filled circles indicate the epochs of actual CONT14 observations. The

reference direction within a source image was chosen to be the position of the peak of emission when calculating the mode values.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

observations, also listed there, are taken from Table C4 of

Anderson & Xu (2018a). The overall contribution for a source

with a CARMS value of 0.38 is already at the level of 15 ps,

and those for extended sources with CARMS values larger than

0.5 are at the level of 20 ps.

5.3. CARMS Distribution in the ICRF Catalogs

The veriﬁcation of categorizations of the ICRF2 and the

newly released ICRF3 is performed by examining the CARMS

and CPRMS distributions over different ICRF categories.

Figure 17 shows the CARMS values, as a function of decl., for

ICRF2 deﬁning sources, ICRF2 special handling sources, and

ICRF3 deﬁning sources. As sources below decl. −45°are not

visible to the VLBA, structure indices are not available for

these sources, and therefore structure index criteria cannot be

applied to select deﬁning sources at lower decl. Unfortunately,

some extremely extended sources have been included as

deﬁning sources in both ICRF2 and ICRF3, as shown on the

upper-left corner of the ﬁgure with blue stars and red circles.

The 39 special handling sources selected in ICRF2 based on

their astrometric qualities all have signiﬁcantly large CARMS

values except 0235+164, which is listed in the deﬁning group

for ICRF3. Compared to ICRF2, ICRF3 excludes some sources

with large CARMS values shown as blue stars in the upper part

of the ﬁgure, and includes tens of sources with small CARMS

values shown as red circles at the bottom. The CARMS values

versus the CPRMS values for the sources in these three

categories are shown in Figure 18, where a good agreement

between these two types of statistics and different features in

the CARMS and CPRMS distributions for these three

categories are addressed. The statistics of CARMS values for

different ICRF groups are listed in Table 4. A big improvement

in selecting deﬁning sources happened from ICRF1 to ICRF2,

mainly because structure index criteria were not used for the

creation of ICRF1. A small improvement also happened in the

creation of ICRF3.

Table 3

Structure Effects in Group Delay for Five Sources with various CARMS values (in units of Picoseconds)

Source CONT14 TSUKUB32–WETTZELL CONT14

CARMS [Min, Max]Median Mean Median Mean

(1)(2)(3)(4)(5)(6)(7)

0016+731 0.19 [−15.5, 15.3]5.4 6.6 4.7 5.1

0749+540 0.38 [−26.5, 21.7]14.9 13.3 14.2 15.1

1739+522 0.57 [−41.9, 58.7]21.8 22.5 25.0 27.4

3C 371 0.75 [−67.7, 68.6]38.9 35.0 18.8 20.8

0642+449 1.01 [−396.1, 111.6]28.0 57.9 23.3 25.9

Note. CONT14 CARMS values in column (2)are determined from CONT14 closure measurements only. The median and mean values in columns (4)and (5)are the

median and mean of the absolute structure effects on baseline TSUKUB32–WETTZELL, while the median and mean values in columns (6)and (7), taken from

Anderson & Xu 2018a), are the variances of structure-effect contributions for all CONT14 observables of individual sources using the statistics median and mean to

determine the contributions of structure effects for each observables.

Figure 17. Plot of CARMS values vs. declinations for 295 ICRF2 deﬁning sources (blue crosses), 39 ICRF2 special handling sources (black triangles), and 296

ICRF3 deﬁning sources (red circles). There are seven ICRF3 deﬁning sources missing because they have fewer than 10 closure amplitudes available. Extremely

extended sources have unfortunately been included as deﬁning sources in both ICRF2 and ICRF3 due to lack of structure indices derived from VLBA observations

and the need to ﬁll deﬁning sources in the southern sky for a uniform distribution, as shown with red circles and blue crosses in the upper-left corner. About 21% of

ICRF3 deﬁning sources have CARMS values larger than 0.4.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

The median and mean of CARMS for all sources are 0.25

and 0.32, respectively. More than half of the sources in ICRF3

are so-called VCS sources that were observed exclusively by

VLBA, whereas approximately only 1000 sources are routinely

observed by the IVS global network. The ICRF3 sources are

thus divided into two groups: VCS sources that have

observations only by VLBA stations, and the remaining

sources named non-VCS sources. The median and mean of

the CARMS values for non-VCS sources are 0.31 and 0.36,

respectively, which are larger than those for ICRF3 deﬁning

sources and suggest that the sources routinely used for geodesy

typically exhibit extended structures and have signiﬁcant

structure effects on geodetic VLBI observations.

The median and mean of CARMS for VCS sources are only

0.23 and 0.30, respectively. Observations from IVS global

networks with higher resolution are subject to structure effects

much worse than those from the VLBA array. This is due to

two reasons: (1)source get more resolved by longer baselines

and show more evidence of structure, and (2)longer baselines

result in the magnitudes of the structure effects being larger.

The difference between the IVS global observations and the

VLBA observations is investigated by separating closure

amplitudes in the following way. A closure amplitude with

all four stations from the VLBA array is classiﬁed as one from

VLBA observations, which then produce VLBA CARMS

values, while the others, with at least one non-VLBA station,

are classiﬁed as ones from non-VLBA observations, giving

non-VLBA CARMS values. This classiﬁcation is relevant

because routine geodetic VLBI observations are mostly made

by the non-VLBA stations. Then, CARMS is recalculated

based on these two sets of closure amplitudes. VLBA CARMS

values are determined for 3788 sources, and non-VLBA

CARMS values are determined for 976 sources. The median

and mean of the non-VLBA CARMS values are 0.37 and 0.43,

and those for the VLBA CARMS values are 0.23 and 0.29,

respectively. Histograms of these CARMS values are displayed

in Figure 19. The overall structure-effect magnitudes of the

geodetic sources, the majority of which are those non-VLBA

sources, therefore are more critical. There are 863 sources that

have both the VLBA and non-VLBA CARMS values. These

CARMS values are displayed in Figure 20, which demonstrates

that VLBA observations nearly always give smaller CARMS

values than the non-VLBA observations. Therefore, a sig-

niﬁcant amount of radio sources that have been observed

exclusively by VLBA, currently having small CARMS values,

are expected to show extended structures if they are observed

by the global IVS networks in the future.

Figure 18. Plot of CPRMS values vs. CARMS values using basic-noise weighting for 295 ICRF2 deﬁning sources (blue crosses), 39 ICRF2 special handling sources

(black triangles), and 296 ICRF3 deﬁning sources (red circles).

Table 4

CARMS Statistics for Different ICRF Categorizations

Source Group Number CARMS

of Sources Median Mean

ICRF1 deﬁning 202 0.41 0.45

ICRF2 deﬁning 295 0.29 0.33

ICRF2 special handling 39 0.60 0.65

ICRF3 deﬁning 288 0.25 0.29

non-VCS sources 966 0.31 0.36

VCS sources 2692 0.23 0.30

All 3890 0.25 0.32

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

It is well known that VLBA antennas have higher sensitivity

and better stability than IVS stations. However, instrumental

phase and amplitude offsets should cancel out exactly in

closure quantities, so that the VLBA instrumental stability does

not account for the lower CARMS values for VLBA-only

observations. By inspecting closure statistics for sources with

little structure, we found that the contributions of measurement

noise to CARMS values are well below 0.2 for IVS networks

and below 0.1 for VLBA. There is a signiﬁcant difference, but

these contributions are still far below the median CARMS

value of 0.37 for IVS networks and the median value of 0.23

for VLBA, reported beforehand for the complete sample of

sources. The difference in the contributions of measurement

noise between IVS network and VLBA does not account for

the discrepancy between VLBA-only and IVS CARMS values,

neither.

On the other hand, one possible reason for that discrepancy,

in particular for the sources that were observed by VLBA only,

might be that VLBA has been used in special sessions to

observe much weaker sources (up to ∼10 times fainter)than

the sources observed in regular IVS sessions. Weaker sources

should have smaller angular sizes, either because they are more

distant or because they are less active. Such weaker sources are

expected to have less structure effects in observations. In fact,

we did not ﬁnd a signiﬁcant difference in the median values of

VLBA-only CARMS between the sources observed by VLBA

only and those observed by both VLBA and IVS networks.

Therefore, this should not be a major reason.

6. Summary and Future Works

We have analyzed phase and amplitude measurements from

the 40 yr history of geodetic/astrometric VLBI observations.

The concept of CARMS based on closure amplitudes is deﬁned

and used to quantify the overall structure-effect magnitude in

the geodetic/astrometric VLBI data set for each individual

source. CARMS values are available for the 3417 ICRF3 radio

sources with at least 100 closure measurements. Almost the

entire ICRF3 catalog is evaluated in terms of source structure.

The main conclusion of the closure analysis in this paper is

that CARMS values smaller than 0.3 suggest relative compact

structures and limited structure effects in geodetic VLBI

observations and that CARMS values larger than 0.4 indicate

very extended source structures. Based on the detailed

investigation of the 30 most frequently observed sources, 28

sources are demonstrated to be resolved at different levels at

the resolutions of the global VLBI networks. Very extended

source structures have been identiﬁed for 19 of these 30

sources, and most of them have structures highly variable on

timescales from years to several weeks, except for three stable

sources, 3C 371, 0552+398, and 1803+784. These highly

variable structures make the corrections of structure effects in

geodetic VLBI data analysis both important and challenging.

For different source categories in the ICRF catalogs, the

CARMS values have dramatic differences. For instance, the

median and mean of the CARMS values for ICRF2 special

handling sources are 0.60 and 0.65 while those for ICRF3

deﬁning sources are 0.25 and 0.29, respectively. However,

there are still a number of radio sources in the ICRF3 deﬁning

group with CARMS values larger than 0.4. We recommend

Figure 19. Histograms of the recalculated CARMS values for 976 sources with closure amplitudes from non-VLBA observations (see its deﬁnition in the text)(left)

and for 3788 sources with closure amplitudes from only VLBA observations (right).

Figure 20. Plot of the recalculated CARMS values based on closure amplitudes

from non-VLBA observations (see its deﬁnition in the text)vs. that based on

closure amplitudes from VLBA observations for 863 sources, which have been

observed both by IVS non-VLBA stations and the VLBA stations. VLBA

observations nearly always give smaller CARMS values.

The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

that CARMS values be used as astrometric quality indicators,

for instance, for the categorization of future ICRFs and in the

weighting of radio sources used for the linking of Gaia and

VLBI catalogs.

In the future, structures of ICRF sources need to be

continuously monitored by updating their CARMS values with

new geodetic VLBI observations while they become available.

As already initially discussed in this paper, the variability of

intrinsic structure with time can be obtained by closure analysis

as well. Changes in intrinsic structure should inevitably lead to

change in derived source positions, which will result in the

deterioration of the stability of the radio celestial reference

frame. Together with the overall structure-effect magnitude

indicated by CARMS, variability of intrinsic structure therefore

should be served as an additional guidance for the use of the

ICRF3 and for the categorization of radio sources, in particular

the deﬁning ones. Identiﬁcation of the epochs and the

magnitudes of structure evolution will be discussed in another

paper. The closure phases and closure amplitudes will be

further used to derive a time series of images with the

techniques, for example, as demonstrated by Chael et al.

(2018). Based on those images, structure effects can be

corrected in geodetic VLBI data analysis. This study sets a

basis for that effort by ﬂagging outliers and examining the

structure-effect pattern and evolution for each individual

source, which is necessary and important for combining

observations from adjacent sessions to make images.

The results published in this paper were based on the data

owned by the International VLBI Service Geodesy and

Astrometry and its international self-funded member organiza-

tions. The original data set can be obtained from internet

servers described in this article. Plots and data of closure

delays, closure phases, and closure amplitudes for all the IVS

data for each individual sources are available via request to the

corresponding author by email. Figures were produced using

the Generic Mapping Tools package (Wessel et al. 2013).We

would like to thank the anonymous reviewer who spent time

and effort to review the long manuscript. M.H.X. was funded

by the National Natural Science Foundation of China

11603060 and 11873077. J.M.A., R.H., and S.L. were

supported by the ECORAS-2 project funded by the German

Research Foundation.

ORCID iDs

M. H. Xu https://orcid.org/0000-0001-9602-9489

J. M. Anderson https://orcid.org/0000-0002-5989-8498

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The Astrophysical Journal Supplement Series, 242:5 (26pp), 2019 May Xu et al.

Erratum: Structure Effects for 3417 Celestial Reference Frame Radio Sources

(2019, ApJS, 242, 5)

M. H. Xu

1,2,3

, J. M. Anderson

2,4

, R. Heinkelmann

, S. Lunz

, H. Schuh

2,4

, and G. L. Wang

MOE Key Laboratory of Fundamental Physical Quantities Measurement & Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF and School of

Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China; minghui.xu@aalto.ﬁ

Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, Straße des 17. Juni 135, D-10623, Berlin, Germany

Shanghai Astronomical Observatory, Chinese Academy of Sciences, No. 80 Nandan Road, 200030, Shanghai, People’s Republic of China

DeutschesGeoForschungsZentrum (GFZ), Potsdam, Telegrafenberg, D-14473 Potsdam, Germany

Received 2020 October 22; published 2021 March 12

Supporting material: machine-readable table

The electronic version of Table 2in the published article is not complete. It includes only 3234 radio sources, but the complete

version should have 3417 radio sources. In this Erratum, we make this complete table available. The complete table is also available

via Vizier (Xu et al. 2019).

The Astrophysical Journal Supplement Series, 253:33 (2pp), 2021 March https://doi.org/10.3847/1538-4365/abd330

Table 2

CARMS and CPRMS Values for 3417 ICRF3 Sources

IERS CARMS Number CPRMS (degrees)Number Category

Design. Nat. Bas. Uni. of Nat. Bas. Uni. of in

Weighting clo. amp. Weighting clo. pha. ICRF2 ICRF3

0059+581 0.18 0.27 0.29 2909197 7.7 10.7 14.3 610681 D D

0552+398 0.36 0.57 0.59 1776309 10.1 15.5 19.0 446475 D D

0851+202 0.25 0.44 0.47 1291087 10.0 14.7 19.5 310001 D O

1803+784 0.29 0.35 0.36 2292526 12.6 15.1 17.8 365718 D O

0923+392 0.41 0.80 0.82 1107469 8.5 17.8 22.9 250502 S O

0727−115 0.14 0.24 0.25 668085 5.8 9.0 12.6 203947 D D

1741−038 0.18 0.33 0.35 659623 7.3 12.9 19.7 192577 D D

1357+769 0.13 0.16 0.17 1840583 9.0 10.4 13.9 338961 D D

1739+522 0.23 0.35 0.38 1222766 9.5 12.5 18.3 272840 S O

0955+476 0.18 0.25 0.27 1200368 9.5 10.9 14.6 272489 D D

0133+476 0.16 0.23 0.24 1376813 8.7 11.1 13.4 280323 D D

2037+511 0.37 0.61 0.69 806175 13.0 18.3 28.2 247632 O O

1749+096 0.12 0.22 0.24 867342 7.7 10.8 14.3 214564 D D

0642+449 0.31 0.49 0.53 1422456 7.9 11.8 20.2 262982 D O

1611+343 0.36 0.40 0.42 964008 16.1 17.6 20.1 207158 S O

1807+698 0.54 0.65 0.68 578885 19.5 22.6 27.7 177403 O O

0528+134 0.19 0.37 0.41 509935 7.1 13.8 21.1 121306 S O

1044+719 0.36 0.59 0.66 1232721 10.9 14.9 22.5 189816 S O

1638+398 0.21 0.32 0.34 676370 10.1 12.5 17.4 145161 D O

0454−234 0.17 0.25 0.27 287178 7.7 10.6 15.4 99952 D D

0016+731 0.20 0.31 0.34 619226 7.8 10.2 14.9 163495 D D

1606+106 0.20 0.28 0.30 528235 10.0 13.3 17.5 125453 D D

1300+580 0.14 0.18 0.19 503547 10.4 12.1 15.9 136728 D D

1334−127 0.12 0.31 0.33 279226 7.7 13.6 18.8 86905 D D

1308+326 0.35 0.60 0.64 583789 12.5 19.0 26.8 121829 S O

2145+067 0.37 0.69 0.70 467436 11.1 21.8 25.9 100938 S O

0229+131 0.42 0.61 0.66 337879 14.8 20.9 32.2 88496 D O

0336−019 0.32 0.44 0.47 380810 13.8 18.1 21.8 96359 O O

2229+695 0.34 0.44 0.49 486119 14.2 17.9 26.5 124795 D D

0804+499 0.16 0.20 0.21 862058 11.2 12.4 15.8 137636 D D

Note. Sources are listed in the order of the total numbers of their observables for the creation of the ICRF3 (Charlot et al. 2018). Columns 2, 3, and 4 are CARMS

based on natural weighting (Equation (7)),basic-noise weighting (Equation (8)) and uniform weighting, respectively, while columns 5, 6, and 7 are CPRMS values.

Columns 11 and 12 are the source categories in ICRF2 and ICRF3 with “D”indicating deﬁning sources, “S”for special handling sources, and “O”for others. This

table is available in its entirety in machine-readable format for 3417 ICRF3 sources and only 30 sources are sorted out and listed here.

(This table is available in its entirety in machine-readable form.)

ORCID iDs

M. H. Xu https://orcid.org/0000-0001-9602-

9489

J. M. Anderson https://orcid.org/0000-0002-

5989-8498

S. Lunz https://orcid.org/0000-0002-0349-2747

H. Schuh https://orcid.org/0000-0001-5443-0370

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