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556 D. Angermann et al.

general concept of the combination methodology is presented. Sections 4 and 5

focus on the computation of the GGOS-D global terrestrial reference frame, which

is performed in two major steps: (1) The accumulation of the epoch normal equa-

tions per technique, and (2) the computation of the TRF solution by combining the

accumuled VLBI, SLR and GPS normal equations. Finally, conclusions and rec-

ommendations for future realizations of the global terrestrial reference frame are

provided in Sect. 6.

2 Input Data for TRF Computation

The computation of the GGOS-D terrestrial reference frame is based on homoge-

neously processed VLBI, SLR and GPS observation time series. Table 1 gives an

overview about the input data. The time series of the different techniques are pro-

vided as unconstrained datum-free normal equations. The GGOS-D input data also

comprise the local tie information, which is available at the ITRS Product Centre

at http://itrf.ensg.ign.fr/local_survey.php. The geospatial distribution of co-location

sites is shown in Fig. 1.

The GGOS-D input data can be characterized in comparison to ITRF2005 as

follows: (1) The GGOS-D observation time series of the different space techniques

were consistently reprocessed based on uniﬁed standards (see chapter “GGOS-D

Consistent, High-Accuracy Technique-Speciﬁc Solutions” by Steigenberger et al.,

this issue); (2) in case of GPS homogeneously reprocessed observation time series

were used (see Steigenberger et al., 2006; Rülke et al., 2008); (3) the modelling

of the observations was improved (e.g., absolute instead of relative phase centre

corrections, a bug in the implementation of the pole tide model was corrected in

the VLBI software CALC/SOLVE); and (4) the type of GGOS-D input data i s

nearly identical to the original observation equations and is much more appro-

priate for the combination than, e.g., loosely constrained solutions or solutions

with removable minimum constraints. If the latter are used for the combinations

the constraints have to be removed, which was not possible for some of the

ITRF2005 input data. A principle disadvantage is that the necessary inversion

may cause loss of precision by numerical effects (Drewes and Angermann, 2003;

Gerstl, 2003).

Table 1 GGOS-D input data used for the global TRF computation (NEQ = Normal equations)

Technique Institution Software Data Time period

GPS GFZ Bernese Daily NEQ 1994–2007

VLBI DGFI OCCAM 24 h sess. NEQ 1984–2007

VLBI IGG-Bonn CALC/SOLVE 24 h sess. NEQ 1984–2007

SLR DGFI DOGS Weekly NEQ 1993–2007

SLR GFZ EPOS Weekly NEQ 1993–2007

GGOS-D Global Terrestrial Reference Frame 557

Fig. 1 Distribution of co-location sites (triangles: VLBI-SLR; large circles: GPS-SLR; small

circles: GPS-VLBI co-locations)

3 Combination Methodology for TRF Computation

The methodology applied in the GGOS-D project is based on combining datum-

free normal equations of the VLBI, SLR and GPS long observation time s eries.

Target parameters for the combinations are the coordinates of the observing stations

and their temporal variations (terrestrial reference frame), quasar coordinates (celes-

tial reference frame), and the EOP that connect both systems. Reﬁned combination

methodologies have been developed to estimate these parameters consistently in a

common adjustment. The combination has been performed with two different soft-

ware packages, namely DOGS (at DGFI and IGG-Bonn) and ADDNEQ2 of the

Bernese GPS software (a modiﬁed version running at GFZ). The general procedure

of the combination methodology for the terrestrial reference frame computation is

shown in Fig. 2.

As shown in Fig. 2, in principle two different strategies can be applied for the

combination of the different space techniques and the computation of the GGOS-D

terrestrial reference frame:

• (I) This approach comprises the two following major steps: (1) Accumulation of

the time series normal equations per technique and analysis of the time series

solutions; and (2) the inter-technique combination of the accumulated multi-

year normal equations per technique and the computation of the ﬁnal TRF

solution.

558 D. Angermann et al.

Fig. 2 Combination methodology for the terrestrial reference frame computation

• (II) The second approach can be divided in two parts: (1) The inter-technique

combination of the VLBI, SLR and GPS epoch normal equations; and (2) the

generation of a multi-year solution based on the weekly combined inter-technique

normal equations. The ﬁrst step has been applied for computing time series of the

geodetic-geophysical parameters (Thaller et al., 2007; chapter by “Consistent and

Combined Time Series of Geodetic/Geophysical Parameters” by Nothnagel et al.,

this issue).

We applied the ﬁrst approach for the computation of the GGOS-D global ter-

restrial reference frame (see Chaps. 4 and 5). Details on the TRF combination

procedure and the mathematical background are given in various publications (e.g.,

Angermann et al., 2004; Meisel et al., 2005; Drewes et al., 2006; Angermann et al.,

2005, 2007; Krügel and Angermann, 2007; Meisel et al., 2009; Seitz, 2009).

4 Accumulation of Time Series per Space-Technique

In the ﬁrst part of the terrestrial reference frame computation the time series of

daily or weekly normal equations are accumulated to a multi-year normal equation

for each technique. The procedure comprises two major steps: (1) The generation

and analysis of station position time series to detect non-linear behaviors, and (2)

GGOS-D Global Terrestrial Reference Frame 559

the computation of the cumulative multi-year normal equation parameterizing the

individual time dependent variations of the station positions. Four different time

dependent effects have to be considered: constant velocities, discontinuities, post-

seismic effects and seasonal signals in station positions.

In the ﬁrst iteration, constant velocity parameters were set up in the epoch nor-

mal equations to parameterize linear station motions and the epoch station positions

were transformed to positions at the reference epoch 2000.0. From the analysis of

position time series we identiﬁed for many stations discontinuities, which are often

caused by equipment changes or large earthquakes. The discontinuities are param-

eterized by setting up new position and velocity parameters for the corresponding

stations. Figure 3 shows as an example the position time series for the GPS station

HOFN, Iceland, where an antenna and receiver change caused a jump in the height

component of nearly 5 cm.

Within the GGOS-D computations, the total number of discontinuities could sig-

niﬁcantly be reduced compared to the ITRF2005 computation (Krügel et al., 2007).

In ITRF2005 a large number of discontinuities had to be introduced, where many

of them are due to processing and modelling changes. This was the case in partic-

ular for GPS, where 221 discontinuities had been identiﬁed for 332 stations. In the

GGOS-D computations the number of discontinuities could be reduced to 124 for

240 GPS stations. This was mainly achieved due to a consistent processing of the

GGOS-D data, the application of latest absolute antenna phase center corrections as

well as radome calibrations.

A major focus was on investigations concerning temporal variations of station

coordinates and the parameterization of non-linear motions within the TRF com-

putations. It turned out that for many stations seasonal variations are visible,

especially in the height component (see Fig. 4 as an example). A comparison

with geophysical model results, computed from atmospherical, hydrological and

non-tidal oceanic loading variations, shows that a large part of the annual sig-

nal may be caused by loading (e.g., Seitz and Krügel, 2009; Tesmer et al., 2008;

Tesmer et al., 2009). Unfortunately, the geophysical models are not as accurate

as necessary to reduce loading from the variations of the station positions. The

Fig. 3 Time series for the height component of the GPS station HOFN, Iceland

560 D. Angermann et al.

Fig. 4 Seasonal height variation for GPS station Irkutsk, Siberia

analysis of position time series has shown, that for some stations also instru-

mentation effects (rather than geophysical ones) are responsible for the observed

signals.

One possibility to consider the annual variations of station positions is to extend

the current parameterization (i.e., station positions and constant velocities), but it

is necessary to ﬁnd a suitable parameterization for the variations. The investiga-

tions have shown, that the shape of the annual height variations differs between the

stations. Figure 5 shows two examples for the mean average shape of such annual

variations.

While the Brasilia time series clearly shows a maximum and a minimum, Ankara

does not have a clear minimum. The averaged annual motions of both stations can be

mathematically represented rather well by sine/cosine annual and semi-annual func-

tions. We investigated the position time series of all VLBI, SLR and GPS stations

and found that for many stations such a parameterization is a good approximation

for the seasonal variations. On the other hand, the procedure and parameterization

is problematic if the seasonal variations are different over the observation time span.

Another problem is that the additional parameters will affect the stability of the solu-

tions. Thus, handling and parametrization of seasonal variations in station positions

is a challenge for future TRF computations.

Fig. 5 Shape of the “averaged” annual height signal for two GPS stations. The ﬁtted curves

represent the mathematical approximation by annual and semi-annual sine/cosine functions

GGOS-D Global Terrestrial Reference Frame 561

5 Computation of the TRF Solution

The input for t he combination of different techniques are the accumulated intra-

technique VLBI, SLR and GPS normal equations. The NEQs include station

positions, velocities, daily EOPs, and (for VLBI) also quasar coordinates. A key

issue within the inter-technique combination is the connection of the different

techniques’ observations given by local tie measurements between the instru-

ments’ reference points at co-location sites. As shown for example in Krügel and

Angermann (2007), the selection of suitable local ties is difﬁcult because the num-

ber and the spatial distribution of “high quality” co-location sites is not optimal.

Another problem is that there are discrepancies between the coordinate difference

vectors derived from the space geodetic techniques and the local ties for many co-

location sites. Figure 6 shows the discrepancies for some of the VLBI and GPS

co-locations. The results are given for the GGOS-D terrestrial reference frame com-

putation in comparison with the DGFI solution ITRF2005-D. The agreement of

the space geodetic solutions with the local ties is better for most stations within

the GGOS-D computation, which is evidence of the progress compared to the

ITRF2005.

Within the inter-technique combination the local tie selection and implementa-

tion as well as the combination of station velocities at co-location sites was done

in an iterative procedure (see Krügel et al., 2007; Seitz, 2009). The quality of the

selected local ties was analysed by using the following two criteria: (1) The mean

offsets between the x-pole and y-pole estimates of the different techniques are a

measure for the consistency of the inter-technique combination and should be mini-

mized; (2) the deformation of t he networks caused by adding local ties and equating

velocities at co-location sites should be minimized. This is expressed by the RMS

of position and velocity differences between the single-technique solutions and the

combined solution. To identify the best set of co-location sites, different solutions

were computed, varying the co-locations and the assumed accuracy of the intro-

duced local ties. As an example, the results for VLBI and GPS co-locations are

given in Table 2. It shows the mean pole offsets and the network deformation for the

Fig. 6 Comparison of the GGOS-D results with ITRF2005-D. The 3-D difference vectors (mm)

between the VLBI and GPS solutions and the terrestrial difference vectors are given for 21 co-

location sites. The stations located in the southern hemisphere are highlighted

562 D. Angermann et al.

Table 2 Mean pole offsets and network deformation for VLBI and GPS co-locations obtained

from GGOS-D inter-technique combination in comparison with DGFI solution ITRF2005-D

Selected VLBI – GPS

co-locations Mean pole difference

Network deformation

ref. epoch: 2000.0

GGOS-D

ITRF2005-D

0.035 mas

0.041 mas

0.3 mm

1.0 mm

GGOS-D inter-technique combination in comparison with the ITRF2005-D results.

The number of selected co-location sites signiﬁcantly increased for the GGOS-D

computations, accompanied with a better agreement of the VLBI and GPS pole

coordinates and a smaller network deformation.

Other tasks of the inter-technique combination include the weighting of the dif-

ferent techniques, the combination of station velocities of co-located instruments,

the combination of the EOP, and ﬁnally the datum deﬁnition of the combined solu-

tion. The weighting is done by determining variance factors for the normal equations

based on the precision of repeated (daily/weekly) station positions for the differ-

ent techniques. Station velocities at co-location sites were combined only, if the

differences between different techniques are not signiﬁcant.

Concerning the combination of the EOP it has to be considered, that the EOP

derived from VLBI observations are set up as daily offsets and drifts, whereas

the EOP derived from GPS and SLR are parameterized as functional values of a

piece-wise linear representation between 0 and 24 h. Thus, the VLBI EOP were

transformed into values of a piece-wise linear function, too (see chapter “GGOS-D

Consistent, High-Accuracy Technique-Speciﬁc Solutions” by Steigenberger et al.,

this issue).

The inter-technique combination is completed by pseudo-observations for the

selected local ties and for the equated station velocities at co-location sites.

To generate the ﬁnal TRF solution, the datum conditions have to be added to

the free normal equations and the complete normal equation system has to be

inverted.

The geodetic datum is deﬁned as follows: The origin is realized by SLR obser-

vations. As shown for example in Rülke et al. (2008), the network translation

parameters derived from GPS are affected by orbital perturbances and thus GPS

was not used for the datum deﬁnition. The scale of the TRF solution is deﬁned by

SLR, VLBI and GPS observations. This was justiﬁed, since a comparison of the

scale of the technique-speciﬁc station networks showed no signiﬁcant differences.

The orientation of the TRF is deﬁned by No-Net-Rotation (NNR) conditions w.r.t.

ITRF2005. The kinematic datum of the ﬁnal TRF solution is given by an actual plate

kinematic and crustal deformation model (APKIM) derived from observed station

velocities (Drewes, 2009). For the estimation of quasar coordinates NNR conditions

w.r.t. ICRF-Ext.1 were applied (Tesmer et al., 2007).

The GGOS-D combination results comprise the terrestrial reference frame (sta-

tion positions and velocities), daily EOP and the celestial reference frame (quasar

GGOS-D Global Terrestrial Reference Frame 563

coordinates). For the ﬁrst time, these parameters have been estimated in a common

adjustment of VLBI, SLR and GPS observation time series. Thus the results ensure

consistency between the TRF, the CRF and the connecting EOP. Furthermore,

the reference frame results provide the basis for the generation of consistent,

high-quality time series of geodetic-geophysical parameters describing the Earth

system (chapter by “Consistent and Combined Time Series of Geodetic/Geophysical

Parameters” by Nothnagel et al., this issue).

6 Conclusions

The GGOS-D global terrestrial reference frame (station positions and velocities)

has been computed simultaneously with the quasar coordinates and daily EOP to

ensure consistency of the results. The computations have been performed on the

basis of homogeneously processed observation time series for the different space

geodetic techniques. The results of the time series analysis have shown seasonal

variations for most of the stations, which are primarily visible in the height com-

ponents. A large part of the observed signals is caused by geophysical effects (e.g.,

hydrological and atmospheric loading), which are presently not reduced from the

original observations. We investigated the position time series of all VLBI, SLR

and GPS stations and found that for most of the stations the shape of the averaged

seasonal signals can rather well be mathematically approximated by sine/cosine

annual and semi-annual functions. However, there are still some issues that need

to be further investigated regarding the parameterisation of the seasonal signals

within the TRF computation. On the other hand it is clear, that deviations of the

station positions from a linear model will produce errors in the estimated parame-

ters. Furthermore, the alignment of epoch solutions to such a reference frame is also

affected by these non-linear station motions. Therefore the consideration of sea-

sonal signals in station positions is one of the most important tasks for future TRF

realizations.

Acknowledgments This work was partly funded by the project GGOS-D within the

GEOTECHNOLOGIEN programme of the Federal Ministry of Education and Research (BMBF:

Bundesministerium für Forschung und Technologie), FKZ 03F0425.

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GGOS-D Consistent and Combined Time Series

of Geodetic/Geophyical Parameters

A. Nothnagel, T. Artz, S. Böckmann, N. Panaﬁdina, M. Rothacher, M. Seitz,

P. Steigenberger, V. Tesmer, and D. Thaller

1 Introduction

Project GGOS-D, (Chapter “Integration of Space Geodetic Techniques as the Baris

for a Global Geodetic - Geophysical Observing System (GGOS-D): An Overview”

by Rothacher et al., 2009, this issue) and the generation of a GGOS-D terres-

trial reference frame (Chapter “GGOS-D Global Terrestrial Reference Frame” by

Angermann et al., 2009, this issue) provided an ideal foundation for a detailed

analysis of the input data in the form of time series of parameters estimated. At

the current stage, the reference frame itself has to be regarded as a global set

of mean positions with a limited level of kinematic information through veloc-

ity parameters. From time series of the three dimensional coordinate components,

however, a much more detailed knowledge about the kinematic behaviour of the

observing sites can be inferred. In an ideal case, the results of different observing

techniques located close to each other at so-called co-location sites should be very

similar or even identical. Due to the speciﬁc characteristics of the techniques or

of the very location of the measurement platform, this is hardly ever the case to

the full extent. In addition, the accuracy of the observing techniques is still lim-

ited by small systematic effects. Therefore, a certain degree of discrepancy has

still to be expected and provides invaluable information about remaining problem

areas.

In addition to the site coordinates, further information can be retrieved from other

elements of the parameter space and their time series like Earth orientation and

atmosphere parameters. Observing techniques based on observations in the radio

frequency bands are affected by the same refraction effects of the neutral atmo-

sphere. The estimates of the atmospheric parameters should, thus, provide identical

results, if the theoretical bias due to height differences between the reference points

is corrected. Again, the discrepancies give important insights into remaining diver-

gences in modelling. Another aspect beyond individual treatment of the different

A. Nothnagel (B)

Institut für Geodäsie und Geoinformation, Universität Bonn, D-53115 Bonn, Germany

e-mail: nothnagel@uni-bonn.de

565

F. Flechtner et al. (eds.), System Earth via Geodetic-Geophysical Space Techniques,

Advanced Technologies in Earth Sciences, DOI 10.1007/978-3-642-10228-8_47,



Springer-Verlag Berlin Heidelberg 2010