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THE INTERNATIONAL CELESTIAL REFERENCE FRAME REALIZED BY VLBI
Extracted from IERS Technical Note 23, Part II, (Ma et al., 1997)


Introduction

	There exists a large resource of high accuracy, dual frequency VLBI 
data which were acquired from various networks for geodetic and astrometric 
purposes over a span of more than fifteen years and from which various radio 
source catalogues have been constructed. The goal of the work described 
below was to create the definitive catalogue for the ICRF (International 
Celestial Reference Frame) using the best data and methods available at the 
time the work was done. This work is the joint cooperative effort of a 
subgroup of the IAU Working Group on Reference Frames (WGRF) formed expressly 
for this purpose.

	The subgroup has taken an empirical approach in the selection of data, 
analysis, estimation of errors, and categorization of the final results. The 
characterization of a radio source, i.e., its position, how it was treated in 
the analysis, and whether it is suitable for use as a defining object, is 
derived entirely from the VLBI data and analysis and not from any other 
information. This approach leads to a rigorous selection of defining objects 
and a reliable realization of the ICRF as a set of relative positions oriented 
to the axes of the International Celestial Reference System, ICRS (Arias et 
al., 1995).
 
	This realization of the ICRF was considered only one of many, both 
actual and potential, better than preceding ones but by no means attaining 
perfection. 

	The geodetic/astrometric VLBI data set has a rich variety of stations 
and networks. The entire data set was used (except for sessions entirely 
between antennas at a single observatory), pooled cooperatively from all the 
various observing programs. Besides providing the potential for extracting the 
maximum information, the use of the entire data set includes the widest 
variation that the network geometry and station size can impose upon the 
realized ICRF. The ICRF positions and stated uncertainties should then 
represent realistically how confidently the positions can be used in the future
with arbitrary VLBI measurements. The VLBI data for this work were edited 
following the usual procedures of each contributing group.

	It was considered essential that the realization of the ICRF be derived 
from a single analysis, even if imperfect, rather than from a combination 
catalogue made of several VLBI solutions. While the various recent catalogues 
are not inconsistent, except for a few discrepant sources, a combination loses 
certain information. The operational realization of the ICRF is a set of right 
ascensions and declinations, but the actual information is the much larger set 
of relative positions, whose quality is contained in the full covariance matrix.


ICRF analysis

       The underlying conceptual basis of this type of realization of the 
celestial reference frame is that radio source positions are invariant with 
time. A series of solutions was made to ensure that this condition was not 
significantly violated. 

	 The solution for the ICRF was made at Goddard Space Flight Center 
(GSFC).	The configuration of the ICRF analysis was developed as a balance 
between competing goals: the most data and the least systematic error, the 
best models and available options, the largest number of useful estimated 
parameters and computer speed, and the like. As improvements occur in the 
future, the balance will shift and the results will be better still.

	The most important choices are related to data and modeling/estimation.
To extract the most information from the data set both delay and delay rates 
were used, and only observations below 6 degrees were excluded. The troposphere
was modeled using the MTT mapping function (Herring, 1992) and estimates of the
zenith troposphere effects in the form of 1-hr piecewise linear, continuous 
functions with constraints on the size of variations. Time-variable troposphere
gradients were estimated. Because it was not available in the Goddard analysis 
system, no atmosphere structure information was used.

	The primary geodetic parameters, the station positions, were estimated 
separately for each session. Station motions within a day from solid Earth 
tides and ocean loading were derived from unadjusted a priori models (McCarthy,
1992).

	The weighting of the data followed the usual Goddard practice. For each 
session a pair of added noise values was computed for delays and delay rates 
which caused the reduced chi-2 to be close to unity when added to the variance of 
the observations derived from the correlation and fringe-finding process as 
well as the calibration of the ionosphere. Other modifications of the 
observation errors such as elevation-dependent and source-dependent noise were 
not used.

	The unadjusted a priori models for geophysical effects and 
precession/nutation generally followed the IERS Standards (1992) (McCarthy, 
1992). The VLBI theoretical model was the so-called consensus model given in 
the IERS Conventions (1996) (McCarthy, 1996).

	The parameters were estimated using arc-parameter elimination (Ma et 
al., 1990). In the ICRF analysis several classes of parameters were adjusted. 
For each observing session, the adjusted arc parameters included: 

1) positions of "arc" sources with identified excessive apparent motion or 
   random variation,
2) two celestial pole offsets to account for errors in the standard 
   precession/nutation models,
3) positions of the stations,
4) the rate of UT1 relative to a good a priori time series,
5) 1-hour troposphere parameters described above,
6) troposphere gradients in the E-W and N-S directions, linear in time,
7) quadratic clock polynomials for the gross clock behavior,
8) 1-hour clock parameters similar to the 1-hr troposphere parameters, and 
9) necessary nuisance parameters such as clock jumps and baseline clock offsets. 

Certain parameters were adjusted as invariant quantities from the entire data 
set. These global parameters included:

1) invariant source positions,
2) geometric axis offsets for all fixed antennas, and
3) 252 parameters for Earth rotation variations in the diurnal and semidiurnal 
   bands caused by ocean tides.

The axis offset and ocean tide Earth rotation adjustments were all small but 
the estimates were included to eliminate any influence on the source 
positions.

	After a series of test solutions to refine various aspects of the 
analysis, a final solution was run which included 1.6 million pairs of delays 
and rates from August 1979 through July 1995. The postfit weighted root-mean-
square residuals were 32.6 ps for delay and 104.2 fs/s for rates with chi-2 per 
degree of freedom of 1.08. There were 1305 global parameters,  650 000 arc 
parameters, and over 2.5 million degrees of freedom. 


Determination of Realistic Error

	Given the very large number of observations for some sources, the 
error contribution from their observation noise is very small and not a 
meaningful measure of uncertainty. It is necessary to consider several other 
effects to assign realistic errors. One is the statistical validity of the 
formal errors. Another is the cumulative influence of all modeling errors 
and editing decisions. Yet another is the magnitude of specific, identifiable 
systematic errors that could have distorted the results.

	From the consideration of error sources the subgroup concluded that a 
realistic error estimate for the invariant source positions could be made by 
inflating the formal errors by a factor of 1.5 followed by a root-sum-square 
increase of 0.25 mas. For the most frequently observed sources the 0.25 mas is 
the dominant error. The errors of the arc sources were also increased by 
0.25 mas.

	The method adopted at the IERS until 1995 for the realization of the 
extragalactic reference frame consisted of combining individual VLBI frames 
by applying an algorithm based on catalogue comparison. The position 
uncertainties derived from the combination reflected the disagreement between 
individual analyses.

	To address the question of whether adopting a unique "error 
calibration law" for all sources eliminated or at least minimized systematic 
effects, a combined frame was made by adopting a 6-parameter comparison model
(three angles, drifts in right ascension and declination as functions of 
declination, and a bias in declination). Individual VLBI frames submitted by 
GSFC, JPL, NOAA and USNO to the IERS in 1995 (Charlot, 1995) were then 
included in this combination.

	The comparison of the "inflated" uncertainties with those obtained 
from catalogue combination showed that there were still a non-negligible 
number of sources whose inflated uncertainties were smaller than those 
resulting from the comparison of parallel analyses with a deformation 
correction model. For these sources the uncertainty for each coordinate was 
set to be the larger of the inflated or the comparison value. 


Astrophysical Causes for Source Position Variations

	Many extragalactic sources display structure on milliarcsecond scales 
for the strong radio emission associated with their compact cores. Temporal 
variations of the intrinsic structure of these objects may result in apparent 
motion when observations are made at several epochs. Until recently, the 
intrinsic structure of the majority of the sources has been mostly unknown. 
The surveys of Fey, Clegg, and Fomalont (1996) and Fey and Charlot (1997) 
show that most sources, when examined in detail, exhibit spatial structure 
on milliarcsecond scales. Their results show that the variation of intrinsic 
structure from source to source can be quite extreme, ranging from relatively 
compact naked-core objects, to compact double sources, to complex core-jet 
objects. The situation is exacerbated by the fact that compact extragalactic 
radio sources are known to have variable intensity and to have frequency and 
time-dependent intrinsic structure. Consequently, unknown and/or unmodeled 
source structure effects may be introduced into the astrometric solution.  

	Charlot (1990) has modeled the effects of radio source structure on 
measured VLBI group delays and delay rates. Results of this modeling suggest 
that these effects can be significant for extended sources (typically at a 
level of 100 picoseconds [ 3 cm at the surface of the Earth or  1 mas] in the 
group delay). Fey and Charlot (1997) calculated structure corrections based 
on the Charlot analysis using source models derived from Very Long Baseline 
Array (VLBA) observations of 169 extragalactic sources. Results of these 
calculations show that intrinsic structure contributions to the measured 
bandwidth synthesis delay are significant, ranging from maximum corrections 
of only a few picoseconds for the most compact sources to maximum corrections 
of several nanoseconds for the most extended sources. Fey and Charlot found a 
correlation between the compactness of the sources and their position 
uncertainties indicating that the more extended sources have larger position 
formal errors. They also define a source "structure index" based on the 
median of the calculated structure corrections. They suggest that this index 
can be used as an estimate of the astrometric quality of the sources as 
follows. Sources with an X band structure index of 1 may be considered very 
good astrometric sources. Sources with an X band index of 2 may be considered 
good sources while sources with an X band index of 3 should be considered 
marginal (and should only be used with caution). Finally, sources with an X 
band index of 4 should not be used at all for astrometric work. Additionally, 
sources should have an S band structure index of either 1 or 2, with a 
preferred value of 1, regardless of the value of their X band structure index.

	Since even apparently stable sources can suddenly begin exhibiting 
rapid structure variations, the entire ICRF catalogue must be reobserved over 
time to ensure its continued precision on the submilliarcsecond level.  
Structure observations of the entire set of ICRF sources are either planned 
or in progress. This is being accomplished in the Northern Hemisphere with 
the VLBA. It will be more difficult to ascertain the structure of sources in 
the Southern Hemisphere due to the lack of observing facilities capable of 
adequate spatial imaging on the required scales. Almost all of the ICRF 
sources whose detailed structure is known are in the Northern Hemisphere. In 
addition, the ICRF can be improved in the future by enhancing the accuracy of 
the observations and the number of sources defining the frame can be expanded. 
New sources will continually be evaluated for addition to the ICRF catalogue 
and the radio positions of sources already in the catalogue will be revised 
through improved observational techniques. New data will be added to the 
existing database and reanalyzed, producing from first principles an improved 
and updated version of the catalogue whenever necessary.


Categorization of sources

	To be most useful in defining the ICRF a source should ideally show 
no variation in position in the data set, have sufficient data to support the 
absence of variation, and not have shown unexplained differences in position 
between realizations of equivalent validity. Several quality levels can be 
established for each of the 608 sources in the WGRF catalogue. These are 
based on one of three sets of criteria:

	1) quality of data and observation history
	2) consistency of coordinates derived from subsets of data
	3) repercussions of source structure

	To qualify for the list of sources that was used to orient the WGRF 
catalogue with respect to the IERS celestial reference system, a source must 
pass muster in all three categories. 

	We summarize hereafter the somewhat arbitrary criteria used to reject 
sources based on their data, behavior, and structure:

     ------------------------------------------------------------------
     Sources fail to be defining sources for any of the following:
		* Arclength formal error >1 mas
		*  <20 observations
		* <2-year span of data
		* 3 sigma or >0.5 mas discrepancy between catalogues
		* Excessive structure
		* Large, significant apparent motions
		* Arc source, i.e., position adjusted for each session
     ------------------------------------------------------------------

	The sources then fall into three categories: 212 defining sources 
that fail none of the criteria, 294 candidate sources that fail some or all 
of the criteria, and 102 sources with identified excessive position variation, 
either linear or random. Some candidate sources have insufficient observations 
or duration for reliable designation as defining sources while others with 
many observations may have larger than expected differences in position 
between catalogues. Candidate sources potentially could be designated 
defining sources in future realizations of the ICRF as more data become 
available or analysis improves. The third category of other sources does 
include sources that may be useful for purposes such as radio-optical frame 
ties. While only the defining sources have a formal role in the ICRF, the 
positions of all sources are consistent with the ICRF.


Orientation of the ICRF

	The VLBI analysis for the WGRF catalogue described above provided 
accurate relative positions and an overall orientation very near to the ICRS 
(Arias et al., 1995). However, the solution was not designed to obtain 
results directly on the ICRS. The final stage in the ICRF realization was 
the rigid rotation of the relative positions to the ICRS maintained by the 
IERS. The extragalactic radio source catalogue obtained in the WGRF solution 
was aligned to the ICRS by rotating it onto the last realization of the IERS 
celestial reference frame, RSC(IERS) 95 C 02 (IERS, 1996). In the procedure 
applied to rotate the WGRF solution to IERS frame, care was taken not to 
transfer to the former the deformations of the latter. The algorithm used to 
put the WGRF coordinates in ICRS was based on catalogue comparison on the 
basis of common sources (Arias et al., 1988). 

	The extragalactic frame RSC(WGRF) 95 R 01 was obtained by putting the 
radio source coordinates from the WGRF solution in ICRS via comparison to 
RSC(IERS) 95 C 02. Both frames are aligned to better than 0.010 mas. The 
deformation of RSC(IERS) 95 C 02 relative to RSC(WGRF) 95 R 01 is represented 
mainly by a bias of the principal plane. To test the stability of the axes of 
the system, we estimated the relative orientation between RSC(WGRF) 95 R 01 
and RSC(IERS) 95 C 02 on the basis of different subsets of sources. The 
scattering of the rotation parameters obtained in the different comparisons 
indicate that the axes are stable within 0.020 mas. 


Evolution of the ICRF

	The current realization condenses the information from a particular 
VLBI data set spanning a defined interval and reflects a certain state of 
VLBI analysis. As time progresses we expect the realization of the ICRF to 
evolve although changes in the ICRF catalogue will be infrequent compared to 
past practice in VLBI astrometry. 

	There are several features that distinguish this type of realization 
from the conventional stellar catalogues that formerly defined the celestial 
reference frame. First, while we know the positional history of the sources, 
we cannot predict with absolute certainty what future observations will 
reveal. The current positions and velocities are a snapshot (or a movie), 
and continued observations are essential to maintain the viability and 
integrity of the ICRF. New sources must be observed to replenish and expand 
the list of candidates, and their positions in the ICRF must be determined. 
Current sources need to be observed periodically to track their behavior. 
Second, as observations accumulate, it should be possible to move candidate 
sources up or down the scale of usefulness. However, it is conceivable, 
perhaps even probable, that an identical categorization of sources from an 
analysis using twice as long an interval would show sources changing 
categories in unpredictable ways. For example, there is no physical reason to 
expect that linear position changes can continue indefinitely. Such motion 
would call into question the fundamental basis of the extragalactic frame, 
the great distances of the objects. Directed position changes should cease at 
some time. Conversely, a source now stationary could start apparent motion. 
Only the data analysis will show. The problem of position variation may be 
solved in the future if the application of source structure information 
permits the identification and use of truly kinematically stable points in 
the sky. This remains to be demonstrated. Unlike stellar catalogues, however, 
the original VLBI observations should always be accessible for improved 
analysis de novo.

	Despite the burden of maintenance, the ICRF realized by VLBI 
astrometry is a great step forward. Compared to stellar realizations it is 
intrinsically simpler, much more accurate, more stable, and less susceptible 
to systematic deformations. It will serve the purposes of astronomy well.


References

ARIAS, E.F., FEISSEL, M., LESTRADE, J.-F., 1988: Comparison of VLBI celestial 
reference frames, Astron. Astrophys., 199, p. 357-363. (1988A&A...199..357A)

ARIAS, E.F., CHARLOT, P., FEISSEL, M., LESTRADE, J.-F., 1995: The 
Extragalactic Reference System of the International Earth Rotation Service, 
ICRS, Astron. Astrophys., 303, p. 604-608. (1995A&A...303..604A)

CHARLOT, P., 1990: Radio-source structure in astrometric and geodetic very 
long baseline interferometry, Astron. J., 99, p. 1309-1326. 
(1990AJ.....99.1309C)

CHARLOT, P., 1995: IERS Technical Note 19, Observatoire de Paris.

FEY, A.L., CLEGG, A.W., FOMALONT, E.B., 1996: VLBA Observations of radio 
reference frame sources. I., Astrophys. J. Suppl., 105, p. 299-330.
(1996ApJS..105..299F)

FEY, A.L., CHARLOT, P., 1997: VLBA Observations of radio reference frame 
sources. II. Astrometric suitability based on observed structure, Astrophys. 
J. Suppl., 111, in press. (1998ApJS..119...75K)

HERRING, T.A., 1992: Modeling atmospheric delays in the analysis of space 
geodetic data, Symposium on Refraction of Transatmospheric Signals in Geodesy, 
J. C. De Munk and T. A. Spoelstra (eds.), Netherlands Geod. Comm., Delft, p. 
157-164.

IERS, 1996: 1995 IERS Annual Report, Observatoire de Paris, p. II-19.

MA, C., ARIAS, E.F., EUBANKS, T.M., FEY, A.L., GONTIER, A.-M., JACOBS, C.S.,
SOVERS, O.J., ARCHINAL, B.A., CHARLOT, P., 1997: The International Celestial
Reference Frame realized by VLBI, IERS Technical Note 23, Part II, Ma and
Feissel (eds.), Observatoire de Paris.

MA, C., SAUBER, J.M., BELL, L.J., CLARK, T.A., GORDON, D., HIMWICH, W.E., 
RYAN, J.W., 1990: Measurement of horizontal motions in Alaska using very long 
baseline interferometry, J. Geophys. Res., 95, p. 21991-22011.

McCARTHY, D.D.(ed.), 1992: IERS Standards (1992), IERS Technical Note 13, 
Observatoire de Paris.

McCARTHY, D.D.(ed.), 1996: IERS Conventions (1996), IERS Technical Note 21, 
Observatoire de Paris.

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