the contribution of very long baseline interferometry to...

J Geod (2007) 81:553–564DOI 10.1007/s00190-006-0117-x

ORIGINAL ARTICLE

The contribution of Very Long Baseline Interferometryto ITRF2005

Markus Vennebusch · Sarah Böckmann ·Axel Nothnagel

Received: 20 April 2006 / Accepted: 24 October 2006 / Published online: 23 November 2006© Springer-Verlag 2006

Abstract The contribution of the International VLBIService for Geodesy and Astrometry (IVS) to theITRF2005 (International Terrestrial Reference Frame2005) has been computed by the IVS Analysis Coordi-nator’s office at the Geodetic Institute of the Universityof Bonn, Germany. For this purpose the IVS AnalysisCentres (ACs) provided datum-free normal equationmatrices in Solution INdependent EXchange (SINEX)format for each 24 h observing session to be combinedon a session-by-session basis by a stacking procedure. Inthis process, common sets of parameters, transformedto identical reference epochs and a prioris, and espe-cially representative relative weights have been takeninto account for each session. In order to assess the qual-ity of the combined IVS files, Earth orientation parame-ters (EOPs) and scaling factors have been derived fromthe combined normal equation matrices. The agreementof the EOPs of the combined normal equation matriceswith those of the individual ACs in terms of weightedroot mean square (WRMS) is in the range of 50–60 µasfor the two polar motion components and about 3 µs forUT1−UTC. External comparisons with InternationalGNSS Serive (IGS) polar motion components is at thelevel of 130–170 µas and 21 µs/day for length of day(LOD). The scale of the terrestrial reference frame real-ized through the IVS SINEX files agrees with ITRF2000at the level of 0.2 ppb.

Keywords Geodetic VLBI · Solution combination ·Normal equations · ITRF2005

M. Vennebusch (B) · S. Böckmann · A. NothnagelGeodetic Institute of the University of Bonn,Bonn University, Nussallee 17, 53115 Bonn, Germanye-mail: [email protected]

1 Introduction

Modern global terrestrial reference frames (TRFs) formsolid foundations for all kinds of Earth sciences, as wellas for geodetic survey control and navigation. From themiddle of the 1980s, geodetic Very Long Baseline Inter-ferometry (VLBI) observations have contributed to thegeneration and maintenance of TRFs. Starting with theBIH Terrestrial System 1984 (BTS84) (Boucher andAltamimi 1985), the Bureau International de l’Heure(BIH) and its successor, the International Earth Rota-tion and Reference Systems Service (IERS), have beenin charge of the combination of the results of the differ-ent space-geodetic techniques into one common frame,the International Terrestrial Reference Frame (ITRF)as the realisation of the International Terrestrial Refer-ence System (ITRS) (Altamimi et al. 2002).

Up to the last ITRF realisation, which was theITRF2000, individual Analysis Centres (ACs) wereinvited to submit their results directly to the ITRF Prod-uct Centre of the IERS. The inputs consisted of con-solidated TRF solutions with full variance/covariancematrix from each AC. For the ITRF2005, however, onlyone input per technique was requested from the servicesof the International Association of Geodesy (IAG), i.e.,from the International VLBI Service for Geodesy andAstrometry (IVS), from the International GNSS Service(IGS), from the International Laser Ranging Service(ILRS), and from the International DORIS Service(IDS). The official contribution of the IVS has been com-puted by the IVS Analysis Coordinator’s office at theGeodetic Institute of the University of Bonn, Germany.

In the case of IVS, the IERS asked for individual datasets for each VLBI observing session of 24 h durationin Solution INdependent EXchange (SINEX) format

554 M. Vennebusch et al.

(Blewitt et al. 1994). SINEX files permit the transmis-sion of the full variance/covariance information to inter-pret the quality of the solution to its full extent or tofurther combine the results with other solutions. This canbe realized by reporting either the full variance/covari-ance matrix or the normal equation matrix of a solutionsetup (see the latest definition at http://www.tau.fesg.tu-muenchen.de/∼iers/web/sinex/format.php).

The latter option is mainly meant for further com-binations but, if required, variance/covariance informa-tion can easily be extracted through an inversionprocedure, which may have to include a datum defini-tion if necessary. In order to facilitate combination stepsby a procedure that does not require a datum definition,the IVS had decided that IVS ACs report datum-freenormal equation matrices in their SINEX files to theIVS Analysis Coordinator. Consequently, the IVS inputto ITRF2005 is also based on datum-free normal equa-tions.

The generation of combined SINEX files from indi-vidual IVS analyses mainly consists of a stacking of thenormal equation matrices. However, specific aspects,like a common set of parameters, identical referenceepochs and a prioris, and especially a representativerelative weighting have to be taken into account forthe generation of a homogeneous and consistent com-bined VLBI product from the input of several IVS ACs(IVS 2005).

Since the combined data sets refer only to individualobserving sessions, a consistent treatment of any site-specific modelling like (linear) drifts or episodic eventsare left to the ITRF Product Centre, where discontinuityinformation from different techniques for identical sitescan be matched. Another advantage is that if the SIN-EX files contain Earth orientation parameters (EOPs)and station coordinates, EOPs can be determined thatare consistent with the TRF. In this respect, the newapproach is a first step to replace the stand-alone gener-ation of EOPs independently of the TRF (cf. Rothacher2000).

2 Combination of space-geodetic data

2.1 Basics of the IVS combination at the normalequation level

Combination at the level of normal equations (alsoknown as ‘adjustment of groups of observations’; cf.,e.g., Mikhail 1976 or Brockmann 1997) is very close tothe combination at the level of observations, if one con-siders a few basic requirements (e.g., identical modelsand identical a priori values). Thus, it can be regarded

as a compromise that is much easier to realise on anoperational basis than the combination at the observa-tion level.

2.1.1 Adjustment of groups of observations

One way to solve an over-determined system of linearequations in a (weighted) least-squares sense, i.e., byminimizing ‖Ax − b‖2

P, is to compute the normal equa-tions

Nx = y with N = A′PA and y = A′Pb (1)

with A being the design matrix and b being the differ-ence vector of observed and computed observations.N = A′PA describes the ‘normal equation matrix’, whiley = A′Pb is also known as ‘the right-hand side of thenormal equations’. P denotes the weight matrix of theobservations Koch (1999).

Generalisation of Eq. (1) to p independent groups ofobservations (or even to individual observations) leadsto the addition theorem of normal equations:

( p∑i=1

αiA′iPiAi

)x =

p∑i=1

αiA′iPibi (2)

with x being the vector of the estimated parameters andαi denoting weighting factors for the individual contri-butions (see Sect. 2.1.4). Equation (2) can be used forgroups of different observation types, like GPS, VLBI,SLR and DORIS observations, as well as for groups ofobservations from a single technique. In general, apply-ing this method is always possible if the individual obser-vation groups are stochastically independent.

Considering intra-technique combinations, as doneby the IVS, the question arises whether the various ACsreally produce independent data sets. However, the ini-tial observations are processed differently by the ACs inso many ways that the independence of the input normalequations may safely be assumed. The data sets differ inthe number of observations included as well as in theirdimensions (i.e., the number of parameters estimated).

Since every AC treats the observations in a differ-ent way (i.e., application of different stochastic models,corrections, outlier detection and elimination, etc.), nor-mal equations of the same original set of observationsdiffer significantly and can thus be treated as indepen-dent input to Eq. (2). A description of the differentsolutions used can be found in Sect. 2.2.

The contribution of VLBI to ITRF 2005 555

2.1.2 Parameter transformations

One of the basic requirements for using Eq. (2) is thatall normal equation systems have to be based on thesame set of a priori values. Thus, in the case of VLBIobservations, two different transformations have to becarried out:

• Epoch transformation: The reference epoch for thetime-variable parameters to be estimated may dif-fer between the different ACs by up to 15 min dueto differences in defining the reference epoch (mid-dle epoch of the session). Since each AC marks(and excludes) different observations as outliers, therespective first and last observations might be differ-ent and thus the respective mean reference epochsmight differ. Thus, a new mean reference epoch hasto be computed from the individual reference epochsand subsequently the a priori values of the individ-ual solutions have to be transformed to this epoch.Station positions are assumed to be constant duringa 24 h VLBI session.

• A priori value transformation: In order to refer allindividual normal equation systems to the identicalset of a priori values, a priori parameter transforma-tions have to be performed.

Both transformations are special cases of a generalparameter transformation (e.g., Brockmann 1997), whichcan be described as

x = B�x + dx (3)

with B being a u×u transformation matrix and dx being(in general) a u × 1 vector of constants (u is the numberof unknown parameters).

Introducing Eq. (3) into Eq. (1) yields

Nx = y (4)

where

N = B′NB and y = B′ (y − Ndx)

(5)

so that the normal equation system is now referring tothe new parameter set x.

On this basis, a priori value transformation can bedescribed by B = I and dx �= 0, i.e., by setting B to bean identity matrix and dx being the difference betweenthe new and the old set of a priori values. Epoch trans-formation is carried out by expressing the (linearized)model of the specific physical parameter in terms ofthe B matrix and by setting dx = 0. Other examples for

parameter transformations can be found in (Angermannet al. 2004).

2.1.3 Modifications of the normal equation system

In order to reduce the number of parameters of a(normal) equation system (e.g., for removing technique-specific parameters like atmospheric path delays or clockparameters), two different cases have to be distinguished:

• Elimination of parameters describes the deletion ofthe corresponding rows and columns of the normalequations and thus fixing the parameters to their apriori values.

• Reduction of parameters involves a single decompo-sition step for each parameter to be reduced. Thismaintains the original model and transfers the prop-erties of these parameters to the remaining decom-posed normal matrix, e.g., to remove VLBI-specificclock, atmosphere and nutation parameters withoutchanging the solution.

For the reduction of parameters, we consider the nor-mal equation system being subdivided into

[N11 N12N21 N22

] [x1x2

]=

[y1y2

](6)

with x1 being the subvector containing the parameters tobe reduced and x2 being the vector of remaining param-eters of interest (e.g., Mervart 2000). x2 can be computedseparately by solving

(N22 − N′12N−1

11 N12︸︷︷︸N

) x2 = y2 − N′12N−1

11 y1︸︷︷︸y

(7)

with Nx2 = y being the so-called reduced system of nor-mal equations for the parameters x2.

2.1.4 Weighting (variance component estimation)

In order to account for the different qualities of theindividual contributions to the combination process,weighting factors αi should be determined before thefinal accumulation (see Eq. 2).

One way of determining weighting factors is to usevariance component estimation (VCE) methods asdescribed by, e.g., Foerstner (1979), Koch (1999) andKusche (2003). The basic idea of VCE is to computeindividual variance factors σ 2

i for each of p groups ofobservations instead of one common a posteriori vari-ance factor σ 2

0 . Here, a group of observations consists


Fig. 1 Median weightingfactors

of an individual contribution (normal equation for onesession) by one AC to the combination process. The esti-mated variance factors σ 2

i can, thus, be used to computeαi = 1

σ 2i

for weighting each contribution in Eq. (2).

VCE is performed in an iterative way. The iterationconsists of the following steps (with k being the iterationcounter):

1. For each of the p individual ACs an a priori valuefor the noise level σ 2(0)

i has to be chosen (usuallyσ 2(0)

i = 1).2. Accumulation of p normal equation matrices and

computation of a combined solution x(k):

( p∑i=1

1

σ 2(k)

i

A′iPiAi

)x(k) =

p∑i=1

1

σ 2(k)

i

A′iPibi. (8)

Here, only the very basic set of parameters has beenmaintained, which was common to all the individualnormal equations. After accumulation, all stationcoordinates have been fixed to a common datumfor the inversion of the matrix.

3. Computation of p vectors of residuals:

e(k)i = Aix(k) − bi. (9)

Due to the fact that the design matrix is not avail-able, the residuals have to be computed using theweighted square sum of the observations via

e(k)′i Pie(k)

i =m∑

i=1

b′iPibi −

m∑i=1

b′iPiAix(k). (10)

4. Computation of p group redundancy numbers, r(k)i :

r(k)i = ni − ui (11)

where ni denotes the number of observationscontained in the normal equation system of the

individual AC (including the number of pseudo-observations for, e.g., constraining parameter rates);ui denotes the number of original unknowns, i.e., thenumber of unknowns in the original normal equa-tion system.

5. Computation of p variance components:

σ 2(k+1)

i = e(k)′i Pie

(k)i

r(k)i

. (12)

Steps 2–5 have to be repeated until convergence isreached, i.e., until σ 2

i ≈ 1 (e.g., Kusche 2003).The temporal evolution of the internal weighting of

the individual solutions from 1984 until the end of 2005 ispresented in Fig. 1. The dots mark the median value ofthe sessionwise-calculated weighting factors of 1 year.The behaviour of the curves in Fig. 1 is very similarover the whole time-span, with slightly higher weight-ing factors for the GSFC (Goddard Space Flight Center)and USNO (United States Naval Observatory) solutionsthan for the BKG (Bundesamtes für Kartographie undGeodäsie), DGFI (Deutches Geodätisches Forschungs-institut) and SHA (Shanghai Astronomical Observa-tory) solutions.

An exception constitutes the peak in the BKG datain 2003. This peak is caused by the values for the termb′

iPibi, which is calculated during the analysis of eachsession by every AC individually and written into theSINEX file. The mean b′

iPibi values for 2003 for theBKG solution is 700 (ms2) less than half of the meanvalue of the other ACs, with about 1,500 (ms2). Thisleads to smaller variance components and thus to higherweights for the BKG solution in this time span. Thereason for these smaller values for b′

iPibi is not clear yet.However, comparing the results between 2003 and 2005of the combined solution to a combined solution withall weighting factors set to one, the differences betweenboth solutions are within their formal errors.


2.1.5 Including additional information/datum definition

In general, accumulation of normal equations accordingto Eq. (2) can be applied to rank-deficient normal equa-tion matrices. The general terms ‘rank-deficient normalmatrix’ and ‘singular normal matrix’ may be used syn-onymously with the more specific terms ‘unconstrainednormal matrix’ or ‘datum-free normal matrix’. The useof datum-free normal matrices has the advantage thatthe combination process can be carried out without theneed to agree on a datum beforehand.

Depending on the technique-specific characteristicsand the kind of parameters estimated, the system ofnormal equations (Eq. 1) is rank-deficient. In the case ofVLBI, estimation of EOPs (i.e., polar motion, pole coor-dinates xp, yp, and Earth rotation UT1–UTC) togetherwith coordinates of all participating VLBI telescopesleads to a rank-deficiency of six due to the lack of (terres-trial) reference system definition, i.e., three translationsand three rotations.

In order to remove this rank-deficiency, an appropri-ate datum definition has to be applied, e.g., by fixinga sufficient number of site positions or by imposingconditions on parameters (e.g. no-net-translation/no-net-rotation conditions) in the form of constraints forthe datum parameters (see Angermann et al. 2004).The celestial reference system, which would contributeanother three dregrees of freedom (three rotations), hadalready been fixed to the International Celestial Refer-ence Frame (ICRF) (Ma et al. 1998) in each individualVLBI solution.

2.2 VLBI intra-technique combination

Altogether, 4255 VLBI sessions between 1979 and 2005were analysed by five IVS ACs using two different soft-ware packages, CALC/SOLVE Petrov (2002) andOCCAM Titov et al. (2004). Both software packagesuse the least-squares adjustment algorithm applying apriori geophysical models according to the IERS Con-ventions 2003 (McCarthy and Petit 2004). Table 1 liststhese ACs and their analysis software packages.

Each AC generated datum-free normal equations inSINEX format. Unknown parameters were the coordi-nates of all stations and EOPs (UT1–UTC, polar motionand their rates, daily nutation offsets) for each sessionindependently. The source positions were fixed to ICRF.The station clock behaviour was modeled by a sec-ond-order polynomial plus continuous piece-wise lin-ear functions with time spans of 1 h. Atmosphere zenithpath delays were also modeled with continuous piece-wise linear functions.

Table 1 IVS ACs contributing to ITRF2005, together with thesoftware packages used

AC Name Software

BKG Federal Agency for Cartography CALC/SOLVEand Geodesy, Leipzig, Germany

DGFI German Geodetic Research OCCAMInstitute, Munich, Germany

GSFC Goddard Space Flight Centre, CALC/SOLVEWashington DC, USA

SHA Shanghai Astronomical CALC/SOLVEObservatory, China

USNO US Naval Observatory, CALC/SOLVEWashington DC, USA

BKG and DGFI used a time-constant of one hour forthe clock and zenith delay piece-wise linear functions,while 20 min were used by GSFC, SHA and USNO. Fortroposphere gradients, one east and one north offset persession were estimated; the ACs using CALC/SOLVEalso estimated troposphere gradient rates. Troposphericzenith delays were mapped with the Niell (1996) Map-ping Function (NMF) in every solution, a priori tro-pospheric gradients were set to zero in the BKG andDGFI solutions, while GSFC, SHA, USNO used a pri-ori gradients from the GSFC Data Assimilation Office(DAO) weather model. Neither thermal deformationof the VLBI telescopes nor atmospheric loading wasapplied.

For the subsequent combination process, all auxil-iary VLBI-specific parameters had been pre-reducedproducing datum-free normal equations only contain-ing station coordinates and EOPs.

For the combination of the individual contributionsof a single VLBI session (VLBI intra-technique combi-nation) the DOGS-CS software (DGFI Orbit and Geo-detic Parameter Estimation Software - Combination &Solution), developed at the DGFI, Munich, and PERL-scripts, developed at the Geodetic Institute of theUniversity of Bonn (GIUB), have been used. The flow-chart in Fig. 2 shows the procedure of the VLBI intra-technique combination as performed at GIUB for eachindividual observing session separately.

1. All available individual solutions (SINEX files) ofthe session to be combined (taken from the IVSData Centre, ftp://cddisa.gsfc.nasa.gov/vlbi/ivsdata)are converted into a special binary format for theDOGS-CS software.

2. Every solution is transformed to the mean referenceepoch (see Sect. 2.1.2) and to an equal set of a pri-ori values using Eq. (3). Furthermore, reduction ofparameters according to Eq. (7) may be necessary


Yes

No

dogs2snx (accumulated datum−free NEQ)

Submission to IVS

Exclude AC

snx2dogs

Weighting

Combination (accumulation of NEQs)

EOP residuals < threshold ?

− Datum definition (only for solution)

Computation / reconstruction of individual solutions (EOP):

=> EOP for (up to five) Analysis Centers

Combined solution

SINEX data(one session, up to five datum free NEQs)

− apriori transformation− epoch transformation (epochs may differ by up to 15 Min.)− reduction of nutation parameters (use of different models)

Fig. 2 Flowchart of the VLBI intra-technique combination atGIUB

to remove parameters that only apply to individualsessions. In particular, the nutation parameters aretreated in this way since they are not transferred tothe combined system of normal equations (due tothe use of different models).

3. Weighting factors from the VCE (see step 6 below)are applied to the normal equation systems (in thefirst iteration all weight factors are set to unity).

4. The datum-free normal equation matrices are accu-mulated (see Eq. 2) for submission to the IERS andfurther combination with the other techniques.

5. In order to perform quality checks of the individ-ual input contributions as well as of the combineddata set, EOPs are estimated by fixing all stationcomponents to their VTRF2005 Nothnagel (2005)values. Contributions are rejected if EOP resultsfrom individual estimates differ from the results ofthe combined normal matrix by a certain thresh-old (usually 3 · σx). If a contribution is rejected, theprocess is repeated from step 4 onwards.

6. Weighting factors are determined by VCE.7. Steps 3–6 are repeated until convergence is reached

for the variance components, i.e. until σ 2(k+1)

i ≈ 1.8. At last, a new SINEX file is generated containing

the (datum-free) accumulated normal equation sys-tem and further statistical information of the com-bination process. This file is then submitted to the

IVS Data Centre for download by the IERS DataArchive (http://iers1.bkg.bund.de/info).

For the IVS submission to ITRF2005, 4.165 sessionswere finally combined using the strategy describedabove. The remaining sessions were not combined eitherbecause they did not meet the quality criteria or only oneAC contributed a solution. Some of the sessions exhib-ited numerical problems due to the number of VLBIobservations in the session were not sufficient for a sta-ble solution.

3 Quality control

To check the suitability of the individual contributionsfor the use in the combination, two different types ofsolutions have been computed:

1. EOP solution:• for internal comparisons of the individual solu-

tions with respect to the combined solution (Sect.3.1.1), and

• for external comparisons with the EOP of theIGS Final Combined Solution (igs00p02.erp, ftp://cddis.gsfc.nasa.gov/pub/gps/products) as well aswith the IERS C04-EOP series (http://hpiers.ob-spm.fr/eoppc/eop/eopc04/eopc04.62-now) (Sect.3.1.2)

2. coordinate time-series:• for computation of Helmert transformation

parameters with respect to ITRF2000 (Sect. 3.2).

3.1 EOP solution

For 3,468 appropriate sessions (the remaining sessionswere not suitable for reliable EOP determination dueto the small spatial extension of the observing stationnetwork), EOP time-series have been estimated fromthe combined normal equations as well as from the fiveindividual contributions, fixing all station positions toVTRF2005. To get comparable results, the followingcomparisons were carried out only for those sessionsanalysed by every AC.

For the internal comparison, data from 1984 till theend of 2005 has been used, for the external comparison,only data from 2000 on is used due to large system-atic differences with respect to the external series (theIERS-C04 and IGS EOP series before 2000).


3.1.1 Internal comparisons

In a first step, comparisons of the individual EOP serieswith respect to the combined EOP series were carriedout to assess the precision of the combined series andto detect remaining systematics between the individualcontributions. Figure 3 illustrates the EOP differencesexemplarily for polar motion and UT1–UTC betweenthe combined solution and the GSFC solution. The errorbars show the formal errors (1σ ) of the differences, thecontinuous line displays the median of ten values.

Although the concept of combinations on the basis ofdatum-free normal equations allows one to realise the(terrestrial) datum in a consistent way for all data sets,small systematics seem to persist since all biases are sig-nificant (cf. Table 2). However, in absolute magnitude,the biases are generally less than 0.5 mm on the Earth’ssurface.

The WRMS (weighted root mean squared, computedwith offset removed) of the residuals for polar motion(Xp and Yp components) are very similar for every solu-tion, ranging between 44.7 and 54.2 µas, except for theBKG solution with WRMS values about 70 µas. For the

Fig. 3 EOP differences between GSFC solution and the com-bined solution; the error bars display the 1σ formal errors of thedifferences, the continuous line shows the median of ten values

polar motion rates, the WRMS varies between 141.0and 174.8 µas/day, again with the exception of the BKGsolution with more than 230 µas/day. The WRMS forUT1–UTC and LOD are between 2.6 and 3.8 µs andbetween 6.1 and 9.4 µs/day, respectively.

3.1.2 External comparisons

To assess the accuracy, both the combined EOP seriesand each individual EOP series have been compared totwo independent EOP series, the IGS final combinedsolution igs00p02.erp (cf. Table 3) and the IERS-C04series (cf. Table 4). For the comparisons, the EOP valuesof the reference series were interpolated to the VLBIepoch of each session using the Lagrange interpolationmethod (see, e.g., Bronstein et al. 2004).

Before the differences are discussed, it should bementioned that there are three types of VLBI sessions:(a) sessions specifically for EOP determinations, (b)multi-purpose sessions with reasonable sensitivity forEOP variations, and (c) regional network and astrome-tric sessions that are not suitable for high quality EOPdeterminations. Only group (c) is excluded below whilegroup (b) has been kept for reasons of completeness.

The comparison with the IGS EOP series is summa-rized in Table 3. Figure 4 displays the external compari-sons the differences between the combined VLBI EOPseries and the IGS EOP series from 2000 onwards forthe two polar motion components, their respective ratesand LOD. The error bars display the formal errors (1σ )of the difference between both solutions, the continuousline shows the median of ten values.

Regarding the following comparisons, there is obvi-ously a big similarity between the GSFC, SHA andUSNO solutions since their analysis strategies have beenvery similar. Especially for the Yp component with biasesof about −169 µas and WRMS values of about 140 µasfor the GSFC, the SHA and the USNO solution, whilethe BKG and the DGFI solutions have biases of −170.5and −148.5 µas, respectively. The WRMS values for theDGFI solution has nearly the same order of magnitudeas for the GSFC, the SHA and the USNO solution, theBKG WRMS value is significantly higher.

For all ACs, the WRMS residuals are larger by atleast 20 µas for Xp than for Yp, which originates fromthe VLBI network geometry. For the Yp rate, the CALC/SOLVE solutions are all very alike with biases of about85 µas/day and WRMS values between 370 and 400 µas/day, while the OCCAM solution of DGFI shows anoffset of 6.2 µas/day and a WRMS value of the sameorder of magnitude as the CALC/SOLVE solutions of385.2 µas/day. The comparison in LOD shows the same:similar biases for all CALC/SOLVE solutions ranging


Table 2 EOP differences from 1984 on of the individual solution with respect to the combined solution

Xp(µas) Yp(µas) UT1–UTC (µs)

Bias −5.9 ±1.6 −13.0 ±1.5 −0.9 ±0.1BKG WRMS 74.9 70.8 3.8

Bias 4.6 ±1.1 16.4 ±1.1 −0.2 ±0.1DGFI WRMS 54.2 51.7 3.2

Bias −3.5 ±1.2 −10.1 ±1.0 0.4 ±0.1GSFC WRMS 58.5 47.0 2.6

Bias −3.4 ±1.0 −12.9 ±0.9 0.5 ±0.1SHA WRMS 50.1 44.7 2.6

Bias 2.8 ±1.2 −8.0 ±1.0 0.4 ±0.1USNO WRMS 56.9 48.9 2.6

Xprate (µas/d) Yprate (µas/d) LOD (µas/d)Bias 8.4 ±4.9 19.2 ±4.8 1.9 ±0.2

BKG WRMS 234.4 231.0 9.4Bias −13.3 ±3.6 −35.7 ±3.3 −2.7 ±0.2

DGFI WRMS 174.8 160.6 7.8Bias 15.6 ±3.5 26.2 ±3.3 1.4 ±0.1

GSFC WRMS 167.6 159.4 6.4Bias −6.1 ±3.3 17.1 ±2.9 1.7 ±0.1

SHA WRMS 157.8 141.0 6.1Bias 14.0 ±3.6 22.7 ±3.3 1.5 ±0.1

USNO WRMS 172.3 157.2 6.2

Table 3 EOP differences from 2000 onwards with respect to IGS final combined solution

Xp (µas) Yp (µas)

Bias 136.1 ±7.5 −170.5 ±6.0BKG WRMS 187.5 150.5

Bias 137.5 ±6.8 −148.0 ±5.6DGFI WRMS 163.2 138.2

Bias 146.3 ±7.5 −169.8 ±5.6GSFC WRMS 183.1 137.4

Bias 149.9 ±7.4 −168.9 ±5.6SHA WRMS 181.0 139.2

Bias 160.1 ±7.5 −168.7 ±5.7USNO WRMS 185.7 141.5

Bias 149.3 ±7.1 −161.6 ±5.4COMBI WRMS 167.0 131.2

Xprate (µas/d) Yprate (µas/d) LOD (µas/d)Bias −26.8 ±17.5 84.0 ±15.8 2.7 ±0.9

BKG WRMS 438.1 400.2 23.4Bias −92.1 ±15.6 6.2 ±15.4 −2.3 ±1.0

DGFI WRMS 383.2 385.2 22.6Bias −16.4 ±14.6 88.4 ±14.4 2.4 ±0.9

GSFC WRMS 383.1 359.4 21.1Bias −50.0 ±16.2 83.3 ±14.8 2.6 ±0.8

SHA WRMS 402.6 371.0 21.0Bias −15.3 ±16.2 79.7 ±14.8 2.5 ±0.9

USNO WRMS 410.1 376.4 22.0Bias −35.7 ±15.5 62.0 ±14.5 1.2 ±0.9

COMBI WRMS 372.9 348.9 20.8

between 2.4 and 2.7 µs/day, the DGFI solution has anegative offset of −2.3 µs/day.

The results of the comparison with IERS-C04 arepresented in Table 4. Figure 5 displays the differencesbetween the combined VLBI EOP series and IERS-C04from 2000 until 2005 for polar motion, UT1–UTC and

LOD. The error bars show the formal errors (1σ ) of thedifference between both solutions, the continuous lineshows the median of ten values.

Also in the comparison with IERS-C04, the similar-ity between the GSFC, SHA and USNO solutions isdetectable. While these solutions have nearly all the


Fig. 4 EOP differences [µas/µs] between the combined solutionand the IGS EOP series; the error bars display the formal errorsof the differences, the continuous line shows the median of tenvalues

same WRMS values, the WRMS values for the DGFIsolution were significantly smaller for the polar motioncomponents and LOD, but higher for UT1–UTC. TheWRMS values for the BKG solution were slightly higherfor every component compared to the WRMS values forGSFC, SHA and USNO.

Generally, the WRMS values calculated from thedifferences of the combined VLBI series and the refer-ence series show the smallest value for nearly every EOPcomponent. The bias approximately displays a weightedmean of all calculated biases of the particular EOP

Fig. 5 EOP differences [µas/µs] between the combined solutionand IERS-C04; the error bars display the formal errors of thedifferences, the continuous line shows the median of ten values

component. As expected, in contrast to the internal com-parison, the individual VLBI EOP time-series as well asthe combined VLBI EOP time-series differ systemati-cally from the IGS EOP time-series, especially for the xp

component. This could possibly originate from a rota-tion of the TRF used by the IGS (igs00p02.erp) withrespect to the VTRF2005.

To verify the conjecturable similarities a cross-correlation analysis for the EOP series of each singleAC with each other was carried out. The correlationcoefficients r were calculated by

r =∑p

i=1(xiAC1 − xAC1) × (xiAC2 − xAC2)√∑pi=1(xiAC1 − xAC1)

2 × ∑pi=1 (xiAC2 − xAC2)

2

(13)


Table 4 EOP differences from 2000 onwards with respect to IERS-C04

Xp (µas) Yp (µas)

Bias 178.3 ±5.5 −389.9 ±5.0BKG WRMS 141.5 127.2

Bias 179.6 ±3.8 −356.7 ±4.6DGFI WRMS 87.9 102.6

Bias 187.8 ±5.5 −385.2 ±4.5GSFC WRMS 140.3 115.2

Bias 190.9 ±5.2 −386.3 ±4.5SHA WRMS 134.2 116.4

Bias 201.2 ±5.4 −384.7 ±4.6USNO WRMS 138.7 117.8

Bias 188.1 ±4.4 −373.8 ±4.2COMBI WRMS 111.5 104.8

UT1–UTC (µs) LOD (µs)Bias −5.2 ±0.4 1.0 ±1.0

BKG WRMS 9.3 25.1Bias −4.9 ±0.6 −1.6 ±1.0

DGFI WRMS 13.1 21.8Bias −3.8 ±0.4 0.7 ±1.0

GSFC WRMS 8.9 24.8Bias −3.8 ±0.3 0.8 ±1.0

SHA WRMS 8.7 24.7Bias −4.2 ±0.4 0.9 ±1.0

USNO WRMS 9.1 24.5Bias −4.6 ±0.4 −0.3 ±0.9

COMBI WRMS 10.3 22.8

where xiAC1 , xiAC2 are the EOP of the respective AC andxiAC1 , xiAC2 are the mean values.

The results of the cross-correlation are displayed inFig. 6. A remarkable dependency on the analysis soft-ware packages is detectable, with least correlationsbetween the DGFI OCCAM solution and all CALC/SOLVE solutions. The correlations between the GSFC,SHA and USNO solutions are overall very high (r > 0.9for every EOP), while the correlations r between theBKG solution and the GSFC, SHA and USNO solu-tions are slightly lower (between 0.8 and 0.9). A reasonfor the high correlations between the GSFC, SHA andUSNO solutions could be due to a very similar parame-terisations (cf. Sect. 2.2).

3.2 Helmert parameters

Another quality check can be carried out using the scalefactors between the combined VLBI and the ITRF2000polyhedra. For this purpose, Helmert parameters withrespect to ITRF2000 were computed for each sessioncontaining three or more stations. For each VLBI ses-sion, the ITRF2000 station positions have been trans-formed to the corresponding epoch of the VLBI sessionusing the ITRF2000 velocities. Fig. 6 Correlations between the individual EOP time series


Fig. 7 Helmert transformation scale parameters for each VLBIsession in the sense ITRF 2000 minus VLBI TRF

In order to generate an undeformed VLBI TRF solu-tion, no-net-rotation and no-net-translation (NNR/NNT) conditions with respect to the respective a pri-ori coordinates were imposed on the combined nor-mal equation matrices before estimating seven Helmerttransformation parameters by least-squares adjustment(three translations, three rotations and one scale) (seee.g. Altamimi et al. 2002).

Figure 7 shows the results of the scale factors betweenthe VLBI session solutions and the ITRF2000 for eachepoch. A linear regression provides an overall scaledifference at epoch 2000.0 of −(0.24 ± 0.04) parts perbillion (ppb) with a rate of (0.078 ± 0.01) ppb/year and aWRMS of the individual scale factors of 1.1 ppb. Whilethe standard deviations of the scale difference at epoch2000.0 and its time derivative suggest that these val-ues are significant, the WRMS is rather large and mayreduce the level of significance slightly.

The large scatter and some of the systematic varia-tions might be caused by the fact that the VLBI observ-ing networks in each session vary considerably withinone year, as well as over the full period of 20 years.However, as a measure of quality of the sets of nor-mal equations, the individual scale differences indicatethe good level of agreement between the IVS input toITRF2005 and ITRF2000.

4 Conclusion

The generation of the IVS input to ITRF2005 has pro-duced a consistent set of datum-free normal equationsfor about 4,000 24 h observing sessions in SINEX format,combined from input of five IVS ACs. They are currentlybeing combined with official data sets of the other IAGservices, i.e. IGS, ILRS and IDS. The IVS data set isbeing used for the definition of the scale (together withthe ILRS input) and contributes its high geometricalstrength to the global polyhedron of ITRF stations.

The very good quality of the IVS contribution is doc-umented through various checks prior to submission ofthe data sets to the IERS. The internal consistency ofthe EOP time-series derived from the combined SIN-EX files with respect to the individual input series bydatum definition and inversion is much better than thatof the combination of IVS EOP time-series (e.g., Noth-nagel and Steinforth 2002).

In terms of WRMS, the agreement here is in the rangeof 50–60 µas for the two polar motion components andabout 3 µs for UT1–UTC, while the time-series combi-nation provides an agreement of only about 100 µas and6 µs, respectively. External comparisons yield an agree-ment with IGS polar motion components at the level of130–170 µas and 21 µs/day for LOD. Here, consistencyof the combined series is almost always better than thatof the series of individual IVS ACs.

The scale of the TRF realized through the IVS SIN-EX files agrees with ITRF2000 at the level of −0.2ppb. Since its scale difference and its time derivativeof 0.078 ppb/year are significant, the VLBI input shouldhave a noticeable effect in the ITRF2005.

Acknowledgments This paper has made use of data of the Inter-national VLBI Service for Geodesy and Astrometry (IVS). Weare also grateful for the strong efforts of the IVS Analysis Cen-tres to provide their solutions for the combination. This workhas partially been funded by the Geotechnologien Programme ofthe German Bundesministerium für Forschung und Technologie(FKZ 03F0336B and 03F0425B). Furthermore, we would like tothank the reviewers for their insightful comments, which led to asignificantly improved presentation of the results.

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