Flechtner F.M., Gruber Th., G?ntner A., Mandea M., Rothacher M., Sch?ne T., Wickert J. (Eds.) System Earth via Geodetic-Geophysical Space Techniques

Подождите немного. Документ загружается.

566 A. Nothnagel et al.

techniques is the combination and its beneﬁt to the quality of the determination of

the parameters. In this article, we describe a few selected examples of time series

with their characteristics and their quality.

2 Time Series of Site Coordinates

All solutions of geometric geodetic space techniques where site positions are treated

as unknown parameters require that a geometric datum is deﬁned. When SLR, GPS

and VLBI station positions are being estimated from weekly, daily and session-wise

blocks of observations, respectively, the datum has to be deﬁned for each of the

weekly, daily or session-wise solutions. In order to avoid distortions of the networks,

no-net-rotation (NNR) and/or no-net-translation (NNT) conditions were applied in

the solutions, either on all sites or a sub-set of sites of the network. In the VLBI solu-

tions, NNR and NNT conditions were set up for all sites of the respective networks

with respect to the ITRF2005 coordinates (Altamimi et al., 2007) of the observ-

ing epochs. The GPS analyses were carried out estimating geocenter coordinates

and applying NNR and NNT conditions w.r.t. IGS05 coordinates (Ferland 2006).

SLR station positions were computed by applying NNR conditions with respect to

ITRF2005 (rescaled), only.

2.1 Seasonal Effects in Site Positions

It has long been known that the vertical components of the positions of some GPS

observing sites exhibit seasonal variations (Dong et al., 2002). The question is

whether similar variations are also detectable by other techniques. If these varia-

tions are of geophysical origin, they should be rather identical at co-located sites. If

not, the behaviour of the differences should give some hints on their origin.

In order to investigate common signals of the different techniques and to detect

systematic and episodic differences, coordinate time series of stations where at least

two techniques are co-located have been compared (see also Tesmer et al., 2009).

As an example, Fig. 1 shows the median-smoothed height component time series

of GPS and VLBI results of the station Wettzell (Germany, upper panel) and the

median-smoothed height time series for GPS and SLR of the station Yarradagee

(Australia, lower panel).

The GPS and VLBI time series of Wettzell show a very similar behaviour with

clear annual variations. Both t echniques exhibit approximately the same ampli-

tude and the same phase. This is not the case for GPS and SLR at the station

Yarragadee. Here, GPS shows annual variations with varying amplitudes while the

SLR variations are more irregular.

Since the annual variations seem to be the strongest systematics, a more detailed

investigation of the annual signal was carried out. A mean annual signal was com-

puted from the whole time span available by averaging all values with identical

GGOS-D Time Series 567

Fig. 1 Time series of the height components (daily estimates smoothed with a 70-day-median

ﬁlter), upper panel: GPS and VLBI time series of station Wettzell (12.9

◦

E.L., 49.1

◦

N.L.); lower

panel: GPS and SLR time series of station Yarragadee (115.3

◦

E.L., -29.0

◦

S.L.)

day-of-year (subtracting a drift ﬁrst). Figure 2 illustrates the results again for the

two stations Wettzell and Yarradagee. The left graph shows that the phases of the

mean annual signal derived from GPS and VLBI are very similar. However, the

amplitude is slightly lower from the VLBI observations. It is conceivable that this

might be caused by the missing atmospheric pressure loading corrections which

have a noticable effect for small VLBI networks (Boehm et al., 2009).

At Yaragadee (right graph of Fig. 2) the GPS data shows a similarly smooth

behaviour which resembles a sinusoid with a slightly larger amplitude than at

Wettzell and with a 180

◦

phase shift for the southern hemisphere as to be expected.

In contrast to this, the SLR data produces a lower level of variation between winter

and summer, and the behaviour of the variations seems to be distorted. Together

with the irregular long term variations as shown in Fig. 1, this is a clear indication

that annual variations are not the main feature of the SLR site variations. In general,

the SLR position time series are much more noisy than those from GPS and VLBI.

In addition, they show rather irregular variations when a mean day-of-year series is

Fig. 2 Mean annual signals of the height components (daily estimates smoothed with a 70-day-

median ﬁlter), left graph: GPS- and VLBI-derived mean annual signal of the station Wettzell, right

graph: GPS- and SLR-derived mean annual signal of the station Yarragadee

568 A. Nothnagel et al.

computed as described above. One of the reasons certainly is that SLR is hampered

by severe irregularities in the ability to track satellites due to overcast periods.

Wettzell and Yaragadee are just two examples which show that in some cases the

techniques provide identical results while at others the variations in station positions

are still technique-dependent.

2.2 Thermal Expansion Effects in VLBI

Owing to thermal expansion, the position of the VLBI reference point of a radio

telescope located at the intersection of the axes depends on the temperature of the

steel construction. Correcting the VLBI observations for thermal expansion effects

in both VLBI software packages was one of the tasks within project GGOS-D. The

validation of modelling thermal expansion effects properly is not quite straight for-

ward. Due to the fact that the network employed in each session consists only of

a small number of observatories, mostly well below 10, and due to the NNR/NNT

datum deﬁnition, the correction at one site has a diluted but signiﬁcant effect on

all other sites. However, this effect is difﬁcult to quantify. It depends on the net-

work conﬁguration and is roughly inverse proportional to the number of network

stations. Nevertheless, the time series of the differences of the vertical site com-

ponent between corrected and uncorrected data shows clear systematic variations

with a period of 1 year. The effect itself can only be depicted if differences between

corrected and uncorrected positions are computed and plotted (Fig. 3).

The example of Algonquin Park in Eastern Canada is particularly informative

since the temperature difference between summer and winter is quite large (Fig. 4).

A clear annual signal is obvious in the time series of the temperatures which ﬁnds a

–5

–4

–3

–2

–1

1994 1996 1998 2000 2002 2004 2006 2008

Height diff [mm]

time [years]

ALGOPARK height differences

Fig. 3 Height differences between solutions without and with thermal expansion modelling of the

Algonquin Park radio telescope in Canada (281.9

◦

E.L., 45.9

◦

N.L.)

–40

–30

–20

–10

1994 1996 1998 2000 2002 2004 2006 2008

Temperature [deg C]

year

ECMWF temperature from VMF1 files

ALGOPARK

Fig. 4 Temperature variations at Algonquin Park in Canada

GGOS-D Time Series 569

Fig. 5 Differences in scale parameters of Helmert transformations w.r.t. ITRF2005, uncorrected

minus corrected for thermal expansion

direct equivalent in the estimated height variations of the VLBI telescope originating

from the thermal expansion modelling. For other sites, this is not as obvious as for

Algonquin, mostly due to the correlations within the estimates of station coordinates

in an NNR/NNT setup.

A composite effect over all VLBI sites can be deduced if the scale components of

the Helmert parameters are computed for each session solution with respect to the

reference solution. The differences between the scale parameters with and without

thermal expansion corrections (Fig. 5) show a clear annual sinusoid. Considering

that the northern hemisphere is over-represented in number of stations and number

of observing sessions, the annual period with a minimum early in the years is a

necessary consequence. This would be quite different if the number of stations and

the number of observations would be balanced between the two hemispheres. In

such a case, the z-translation would be affected most and should also show an annual

period.

The differences in scale of up to 0.4 ppb emphasize the importance of the correc-

tions for thermal expansion effects. Within project GGOS-D, it has been shown that

these corrections do improve the repeatability of baseline lengths by more than 3%.

3 Time Series of Earth Orientation Parameters

3.1 EOP Time Series Resulting from a Consistent Combination

of TRF and EOP

A comparison of the combined EOP time series and the series derived from the

observations of the individual techniques shows to what extent the EOPs ben-

eﬁt from the combination. In Fig. 6 the individual as well as the combined

series are displayed, exemplarily for x-pole, y-pole and UT1-UTC (selected are

3 years of data). In order to illustrate the effect on the EOPs, the time series

570 A. Nothnagel et al.

2000 2000.5 2001 2001.5 2002 2002.5 2003

−0.5

0.5

Year

ypole [mas]

GPS Combination

2000 2000.5 2001 2001.5 2002 2002.5 2003

−0.1

0.1

Year

UT1−UTC [ms]

VLBI Combination

2000 2000.5 2001 2001.5 2002 2002.5 2003

−0.5

0.5

Year

xpole [mas]

GPS Combination

Fig. 6 Earth rotation parameters: combined solution (grey) is compared to the best technique-only

solution (black). The pole coordinates and UT1-UTC are plotted w.r.t IERS EOP 05 C04 (offset

removed)

are reduced by values of a reference series. As reference IERS EOP 05 C04 was

chosen (http://hpiers.obspm.fr/iers/eop/eopc04_05/C04_05.guide.pdf, 10.01.2009).

The corresponding RMS w.r.t. IERS EOP 05 C04 values are given in Table 1. The

IERS EOP 05 C04 series does not necessarily represent the truth. However, it is

reliable to be used here as a reference series to demonstrate better the effect of the

combination. Considering, that the RMS value of the UT1-UTC series is reduced

from 0.0193 ms (VLBI only) to 0.0124 ms (combined) the results show, that all the

EOP parameters are signiﬁcantly improved by the combination.

Table 1 WRMS of pole coordinates and UT1-UTC time series referenced to IERS EOP 05 C04

(offsets removed)

Technique x-pole

mas

y-pole

mas

UT1-UTC

GPS 0.1191 0.1130 –

VLBI 0.1693 0.1627 0.0105

SLR 0.3032 0.3065 –

Combined (only

common epochs)

0.1148 0.1088 0.0126

GGOS-D Time Series 571

3.2 Analysis of Nutation Time Series

Similar to UT1-UTC, the two nutation angles can be determined only by VLBI in

an absolute sense, whereas the satellite-techniques can contribute solely the time-

derivatives, i.e., the nutation rates. Therefore, we ﬁrst look at the time series derived

from a VLBI-only multi-year solution. Two different types of solutions were com-

puted: For the ﬁrst solution, the temporal resolution of the nutation angles was one

set for each session. For the second solution, the session-speciﬁc parameters have

been transformed into a piece-wise linear polygon with an interval length of 28 days.

As the major difference between the estimated nutation angles and the a priori model

IAU2000 is the so-called free core nutation (FCN) with a period of about 432 days,

the representation with such a long interval of 28 days is justiﬁed in order to reduce

the scatter in the time series and to better determine the period of the FCN (Fig. 7).

It can be seen from Fig. 7 that the amplitude of the FCN is not constant. The same

applies to a combined solution from VLBI and GPS observations where the nutation

was estimated as a piece-wise linear polygon of 28 days as well (not shown here).

In order to determine the time dependent amplitudes at each epoch, a ﬁt of a

sinusoid was performed with sliding time windows, each with a length of 865 days

separated by 7 days. According to Roosbeek et al. (1999), the period of the FCN

do not vary very much, therefore, we did the ﬁtting by using a ﬁxed period of 432

days. However, it should be mentioned, that there are other publications that show a

varying period for the FCN (see e.g., Malkin and Miller, 2007). As it is known from

theory that the FCN is a retrograde signal, the ﬁt of the sinusoid was carried out for

1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

−1

−0.5

0.5

Δε [mas]

per session

28−day pwl

1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

−1

−0.5

0.5

Δψ⋅ sin ε

[mas]

per session

28−day pwl

Fig. 7 Nutation in obliquity and longitude estimated from VLBI-only multi-year solutions

(corrections to the IAU2000 model)

572 A. Nothnagel et al.

1994 1996 1998 2000 2002 2004 2006

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Amplitude [mas]

Amplitudes of sinus fit: interval length = 865 d, 7−day steps

VLBI: 28−d pwl

Combined: 28−d pwl

Fig. 8 Amplitude of the FCN estimated by a two-dimensional ﬁt of a sinusoid to the multi-year

solutions using sliding time windows

both nutation angles together so that a phase s hift of 90

◦

is guaranteed automatically.

The estimated amplitudes are shown in Fig. 8 for the solutions with 28-day poly-

gons. The estimated amplitudes of the FCN are around 0.12 mas, except of the

time span 1997 until 2002. During these years, the amplitude rapidly decreased and

increased again, with the minimum value around the middle of 1999. This result

agrees well with earlier analysis, e.g. by Herring et al. (2002) who stated that there

is an indication that the amplitude of the FCN increases again after 2000.

4 Time Series of Atmosphere Parameters

The tropospheric path delay L is represented as a function of elevation angle ε and

azimuth α of the vector between the ground station and the observed GPS satellite

or quasar. The azimuth-independent part of the neutral atmosphere around a station

is described as a sum of a zenith hydrostatic delay (ZHD) L

and a zenith wet delay

(ZWD) L

(Eq. 1). For an observation with elevation angle ε both zenith delays

are mapped to the elevation angle of the observation using the Vienna Mapping

Function 1 (VMF1, Boehm et al., 2006). The azimuth-dependency of the tropo-

spheric delay of the observation is considered with coefﬁcients for gradients in

north-south (G

) and east-west direction (G

), mapped with the gradient mapping

function m

to the elevation of the observation.

L(α,ε) = L

(ε) + L

(ε) + m

(ε) × [G

cos α + G

sin α](1)

GGOS-D Time Series 573

While for each GPS and VLBI station, the ZHD is corrected a priori (with inter-

polated ECMWF grid data), the ZWD is estimated in the least-squares adjustment.

However, the estimated part cannot be interpreted completely as the real wet delay

since it is also affected by differences in atmospheric pressure between in-situ pres-

sure measurements and ECMWF pressure interpolations. For the gradient mapping

function the simple equation by MacMillan (1995) with the wet mapping function

has been used (Eq. 2)

(ε) = m

· cot (ε)(2)

The troposphere zenith delay parameters are estimated as piece-wise linear func-

tions for each station with a temporal resolution of 1 h for VLBI and of two hours

for GPS, as otherwise the number of parameters in the GPS processing is too large.

The GPS-derived gradients are estimated for each site once per day as a linear func-

tion. In the VLBI solutions the gradients are estimated as a linear function for each

24 h-session. A priori values for the gradients are zero. In contrast to GPS, the

VLBI-derived troposphere zenith delay parameters and gradients could not be esti-

mated completely free due to the fact that the observation geometry is poor in some

of the VLBI sessions, especially the older ones. Therefore, the time derivatives of

the troposphere zenith delay parameters had to be constrained to zero with a stan-

dard deviation of 1 cm/h while the troposphere gradients were constrained to zero

with standard deviations of 2.5 mm and 5 mm/day. To be able to better compare

the VLBI- and the GPS-derived troposphere parameters, both were always referred

to intervals of full UTC hours, which is not typical for the VLBI-only solutions.

The GPS troposphere parameters were estimated from weekly solutions providing

continuous results over 1-week periods. The VLBI-derived parameters do not pro-

vide such continuity, as there are only about 3–4 sessions per week with changing

observing networks.

The GPS and VLBI time series of the ZWD estimates for Hartebeesthoek as well

as their differences are displayed in Fig. 9. The differences show an offset of 5 mm

Fig. 9 Time series of zenith wet delays of the station Hartebeesthoek (27.7

◦

E.L., −25.9

◦

S.L.)

forGPSandVLBI(upper panel) and differences of the series, VLBI-GPS (lower panel)

574 A. Nothnagel et al.

Fig. 10 Mean gradients in north–south direction of co-located VLBI and GPS stations

with a standard deviation of 1 mm and a weighted RMS of 8 mm after removing the

offset. A similar result, with an offset of 4.2 ± 6.3 mm, had already been obtained

by Schuh et al. (2004). In addition, a seasonal pattern is visible in the differences.

In summertime, when the water content of the atmosphere as well as the variability

is larger, the ZWD differences show a larger scatter (see Steigenberger et al., 2007).

Beside the zenith wet delays, the tropospheric horizontal gradients from GPS

and VLBI can also be compared. Figure 10 illustrates the mean gradients in north-

south direction for all GPS and VLBI co-located stations sorted from south to north.

For both techniques, the gradients show the expected systematics in direction of

the equator (Chen and Herring, 1997). Furthermore, this comparison shows that the

mean gradients of VLBI are smaller for nearly all stations than the GPS derived

mean gradients. The reason for this offset might be the constraints on the a priori

values for the gradients determined by VLBI as discussed in Steigenberger et al.

2007. In addition, it might play a role that in recent years VLBI observations have

been carried out as far down as 3

◦

elevation. This strengthens the estimates of the

gradients considerably.

5 Conclusions

The time series of site coordinates, Earth orientation parameters and atmosphere

parameters produced within project GGOS-D provide valuable data sets for detailed

studies of these geodetic/geophysical phenomena. The s eries combined from differ-

ent observing techniques are certainly the most stable data sets since they combine

the strengths of the individual techniques and mitigate their weaknesses. However,

differences in the results of the individual techniques are the keys to detect-

ing remaining deﬁciencies in procedures and models. For example, the divergent

behaviour of station coordinates from different observing techniques, though co-

located in close vicinity, indicate that systematic errors in the individual techniques

still exist and that the local ties between the different sensors have to be observed

more often than just once. Estimates of zenith wet delays can only be compared

properly if the height difference of the GPS antenna and the reference point of the

VLBI telescope is taken into account. In addition, technique speciﬁc systematics,

for example, from GPS antenna phase centers and VLBI telescope structure defor-

mations still exist and have to be addressed for further progress in accuracy and

reliability.

GGOS-D Time Series 575

Acknowledgments We are grateful for the use of observation data from IGS (Dow et al.,

2005), ILRS (Pearlman et al., 2002), and IVS (Schlüter et al., 2002). This is publication no.

GEOTECH-1258 of the programme GEOTECHNOLOGIEN of the German Ministry of Education

and Research (BMBF). The work has been carried out in part with support of grants 03F0425A,

03F0425C, 03F0425D of BMBF.

References

Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the

International Terrestrial Reference Frame based on time series of station coordinates and Earth

Orientation Parameters. J. Geophys. Res. 112, doi:10.1029/2007JB004949.

Boehm J, Werl B, Schuh H (2006) Troposphere mapping functions for GPS and very long base-

line interferometry from European Centre for Medium-Range Weather Forecasts operational

analysis data. J. Geophys. Res. 111, B02406, doi:10.1029/2005JB003629.

Boehm V, Heinkelmann R, Mendes Cerveira P, Pany A, Schuh H (2009) Atmospheric load-

ing corrections at the observation level in VLBI analysis. J. Geod. 83(11), 1107–1113,

doi: 10.1007/s00190-009-0329-y.

Chen G, Herring TA (1997) Effects of atmospheric azimuthal asymmetry on the analysis of space

geodetic data. J. Geophys. Res. 102, 20489–20502, doi:10.1029/97JB01739.

Dong D, Fang P, Bock Y, Cheng MK, Miyazaki S (2002) Anatomy of apparent seasonal

variations from gps-derived site position time series. J. Geophys. Res. 107(B4), 2075,

doi:10.1029/2001JB000573.

Dow J, Neilan R, Gendt G (2005) The International GPS Service: Celebrating the 10th anniversary

and looking to the next decade. Adv. Space Res. 36(3) 320–326, doi:10.1016/j.asr.2005.05.125.

Ferland R (2006) Proposed igs05 realization. [IGSMAIL-5447], http://igscb.jpl.nasa.gov/mail/

igsmail/2006/msg00170.html

Herring T, Mathews P, Buffett B (2002) Modeling of nutation-precession: Very long baseline

interferometry results. J. Geophys. Res. 107(B4), doi:10.1029/2001JB000165.

MacMillan D (1995) Atmospheric gradients from very long baseline interferometry observations.

Geophys. Res. Lett. 22(9), 1041–1044, doi:10.1029/95GL00887

Malkin Z, Miller N (2007) An analysis of celestial pole offset observations in the free core

nutation frequency band. In: Böhm J, Schuh H, Wresnik J (eds.), Proceedings of the 18th

European VLBI for Geodesy and Astrometry Working Meeting, Institute for Geodesy and

Geoinformation, TU Vienna, Vienna.

Pearlman MR, Degnan JJ, Bosworth JM (2002) The international laser ranging service. Adv.

Space. Res. 30(2), 125–143.

Roosbeek F, Defraigne P, Feissel M, Dehant V (1999) The free core nutation period stays between

431 and 434 sidereal days. Geophys. Res. Lett. 26(1998GL90022550).

Schlüter W, Himwich E, Nothnagel A, Vandenberg N, Whitney A (2002) IVS and its important

role in the maintenance of the global reference systems. Adv. Space. Res. 30(2), 127–430,

doi:10.1016/S0273-1177(02)00278-8.

Schuh H, Snajdrova K, Boehm J, Willis P, Engelhardt G, Lanotte R, Tomasi P, Negusini M,

MacMillan D, Vereshchagina I, et al. (2004) IVS tropospheric parameters – comparison with

DORIS and GPS for CONT02. In: Vandenberg NR, Baver KD (eds.), International VLBI

Service for Geodesy and Astrometry 2004 General Meeting Proceedings. Ottawa, Canada,

February 9–11, 2004. Edited by Nancy R. Vandenberg and Karen D. Baver, NASA/CP-2004-

212255, 2004. Published online at http://ivscc.gsfc.nasa.gov, p. 461, pp. 461–465.

Steigenberger P, Tesmer V, Krügel M, Thaller D, Schmid R, Vey S, Rothacher M (2007)

Comparisons of homogeneously reprocessed GPS and VLBI long time-series of troposphere

zenith delays and gradients. J. Geod. 81, 503–514, doi:10.1007/s00190-006-0124-y.

Tesmer V, Steigenberger P, Rothacher M, Boehm J, Meisel B (2009) Annual deformation signals

from homogeneously reprocessed VLBI and GPS height time series. J. Geod. 83(10), 973–988.