simultaneous determination of earth orientation parameters … · key words: earth orientation...

SIMULTANEOUS DETERMINATION OF EARTH ORIENTATION PARAMETERS AND STATION COORDINATES FROM COMBINATION OF RESULTS OF DIFFERENT OBSERVATION TECHNIQUES

I. PEŠEK1, J. KOSTELECKÝ1,2

1 Center for Earth Dynamics Research (CEDR), Department of Advanced Geodesy, Faculty of

Civil Engineering, Czech Technical University, Thákurova 7, 166 29 Prague 6, Czech Republic ([email protected])

2 Center for Earth Dynamics Research (CEDR), Research Institute of Geodesy, Topography and Cartography, GO Pecný, 251 65 Ondřejov 244, Czech Republic ([email protected])

Received: March 1, 2005; Revised: February 7, 2006; Accepted: March 31, 2006

ABSTRACT

Described is a method for non-regular combination of different techniques, where the normal equations matrix cannot be restored, to obtain a representative set of Earth orientation parameters and station coordinates. The method is based on combining station position vectors transformed to the celestial reference frame, where they are functions of both the EOP and the station coordinates. Three types of constraints are applied to stabilize the system, separate celestial pole offset from polar motion and, to tie the EOP between individual epochs. VLBI, GPS, SLR and Doris data as collected for the 'IERS SINEX Combination Campaign' was used to check the method. After combination, dispersion of station coordinates decreased from 0.040 to 0.031 m. The effect of the combination on EOP is of the order of 0.2 mas and it can be seen in Figs. 3 and 4 as a difference of the final and a priori values.

K ey word s : Earth orientation parameters, space geodesy techniques

1. INTRODUCTION

The best way for deriving Earth rotation parameters and station coordinates from observations made by various techniques is to solve the original observation equations (or alternatively, normal equations, pre-reduced for “inner” unknowns) in a unique adjustment. At present, Earth orientation parameters, EOP, and station coordinates are combined separately and effort is made to develop methods, which make it possible to derive both the EOP and the station coordinates simultaneously, from observations by various techniques.

The major problem of this task rises from systematic biases between individual techniques or even between results of the same technique as produced by different institutions, due mostly to the differences of mathematical models and standards used.

Stud. Geophys. Geod., 50 (2006), 537−548 537 © 2006 StudiaGeo s.r.o., Prague

I. Pešek and J. Kostelecký

Recently two campaigns have been organized to coordinate this effort, namely the ‘IERS SINEX Combination Campaign’ in 2002 for the purpose of collecting data and to gain experience for the ‘IERS Combination Pilot Project’, currently in progress, whose aim is to prepare and standardize regular combination technologies.

These algorithms need the input data to be in a complete Sinex format, i.e. including the covariance or the normal equation matrix.

The complete SINEX files are not available for some, mostly older data. In this case, the combination can only be derived in a non-regular way, by processing the results, i.e. EOP and station coordinates, of the input solutions. It is still worth doing the combination with this data, because VLBI observations give the EOP a long-term stability, while the short-periodic variations come mainly from GPS.

2. NON-REGULAR COMBINATION

The basic idea of the method is to combine station position vectors, xC, in the celestial reference frame (e.g. Kostelecký and Pešek, 2003), where they are functions of both the Earth orientation parameters and the station coordinates x,

( ) ( ) ( ) ( ) ( )3 1 2C t t GST y x= −P N R R Rx xP P . (1)

P(t) and N(t) are precession and nutation matrices, respectively, GST Greenwich true sidereal time, and Ri matrices of rotation around the axis i. The position vectors xC are treated as fictitious observations.

Input data for the combination consists of M sets of Earth orientation parameters ( px ,

py , UT1 − UTC, dψ, dε)m and corresponding sets of station coordinates (x)m, m = 1, … M, as derived by different analysis centres for individual techniques. One or more independent solutions can be available for the respective technique.

From all stations/instruments only those, which are collocated with other techniques or appear in at least two solutions for the same technique, are selected to enter the combination. Local ties of the collocated instruments have to be known.

There are essentially two approaches on how to combine the station coordinates. It is possible to derive the coordinates of the individual stations. This yields unique values of the station coordinates, but stations not entering the combination cannot be transformed into the new system. Or, the individual input solutions can be considered internally consistent, only differing from each other by slightly shifted origins and coordinate axes. In this case, it is sufficient only to look for systematic differences between the input data sets, i.e. to derive coefficients of the seven-parametric transformation for each input technique/solution. It makes the adjustment more stable and the transformation formulas allow transforming of the remaining stations. This compensates for the disadvantage of the final station coordinates having to be obtained as a weighted mean of transformed coordinates after the adjustment.

Combination needs the Earth orientation parameters from all input solutions to be referred to the same epoch. One-day “normal points” at 12 h UT are currently used.

UT1 − UTC are only provided by VLBI, while other techniques produce length of day, LOD. It has to be converted to UT1 − UTC by

538 Stud. Geophys. Geod., 50 (2006)

Simultaneous Determination of Earth Orientation Parameters and Station Coordinates …

Table 1. List of the data.

Technique Solution - Data Center Institution # of Solution in Figures

Doris doris Institute Geographique National, France #1

GPS igs IGS #2

slr.aus Australian Surv. and Land Inf. Group #3

slr.bkg BKG Frankfurt a/M. #4 slr.csr CSR, Univ. of Texas #5

slr.iaak Institute of Appl. Astronomy, Russia #6

SLR

slr.nerc Natural Envir. Res. Council, UK #7

vlb.bkg BKG Frankfurt a/M. #8 vlb.dgfi DGFI Muenchen #9 VLBI vlb.gsfc NASA, Goddard Space

Flight Center #10

. { }( ) 20 1 2

01 d

tUT UTC t LOD t c c t c t− = + + +∫

Coefficients ci are obtained by the fit to UT1 − UTC from VLBI data. The aim of combination is to produce a “representative” set of the Earth orientation

parameters ( px , py , UT1 − UTC, dψ, dε) for some epochs Ti, and parameters p = p1, … p7 of the seven-parametric transformation for each input set of station coordinates.

The transformation (1) yields observation equations of the form

0C

j C Cobsjj

dξξ

∂= − +

∂∑x

x x v , , 1 , , ,j p px y UT UTC, ξ ψ ε− p , =

where the “observed” vectors C obsx are calculated from the respective input solution,

0Cx are functions of adopted a priori values of the unknowns, and the derivatives are of the form

( ) ( ) ( ) ( ) ( ) ( ) ( )1 3 1 1 3 1 2C

p pt GSTε ψ ε εε

∂= − Δ + Δ −

∂P R R D R R R R

xxy x .

Here Di denotes a matrix operator, ( ) ( ) 1i i id dα α α −⎡ ⎤≡ ⎣ ⎦D R R .

The task leads to a singular system, thus an appropriate constraint, e.g. minimizing mutual shifts and preserving the system as a whole unchanged, has to be introduced,

. (2) minT =∑p p

Stud. Geophys. Geod., 50 (2006) 539


Fig. 1. List of observations entering the combinations.

Fig. 2. Geographical distribution of the observation sites and techniques.

Now the method is suitable for calculating the station coordinates, because one set of transformation parameters per input solution is derived, covering the whole period. On the other hand, orientation parameters are calculated for each individual epoch independently of the others. As a consequence, the systematic differences between the input data, including station coordinates, scatter the EOP substantially. The effect can be reduced by including another constraints, in the form of pseudo-observations, that tie values of the EOP at adjacent epochs, i.e.



1 0i id dξ ξ −− = + v , , , 1 , ,p px y UT UTCξ ψ ε= − . By weighting, the constraints control smoothness of the combined orientation

parameters. In our case the weights 103 were applied. At the same time, these equations make it possible to link time corrections based on the length-of-day (LOD) data to the values from VLBI observations.

Some techniques produce polar motion components as well as celestial pole offset. These cannot be separated from data referred to a single instant. Therefore another two fictitious observations have to be added, separated from the mean instant by ±8 h. Thus, three observation equations, namely at epochs Tn – 8 h, Tn, Tn + 8 h, correspond to the input data of the epoch Tn. The interval of 8 h minimizes correlations between both types of unknowns. Weights of these “triple” equations are reduced by a factor of 3.

Data coming from different institutions is derived using different algorithms and standards. Analyses show that they suffer from mutual biases and periodic variations, which have to be removed as fully as possible, prior to the adjustment. If retained, they would produce false steps in the orientation parameters. To calculate the biases, only epochs containing the respective parameter in all input solutions are taken into account, and the biases are obtained as deflections from a weighted mean of the input data. The LOD-based time corrections are relieved of periodic variations with periods longer than 30 days, which are derived from comparison with VLBI data.

Due to the constraint (2), the result of combination depends on the adopted a priori values of the unknowns. Since all data entering the combination is considered precise, with only minor mutual differences, the weighted mean of all contributing input solutions should be the best estimate of the a priori value of the respective unknown.

3. DATA AND NUMERICAL SOLUTION

The method was tested on data collected at TU Munich for the ‘IERS SINEX Combination Campaign’ (IERS, 2003), which cover the period of 1999 (see Table 1). Time distribution of the data is visible from Fig. 1. Distribution of observing collocation stations is depicted in Fig. 2. Station coordinates and Earth orientation parameters were extracted from Sinex files and weekly solutions were joined to create the monthly data.

Weights of the techniques were derived according to accuracy estimates of the data contributing to IERS (Gambis et. al., 2002). They are 6.8 for GPS, 0.3 for Doris and SLR, and 1.0 for VLBI.

In the early stages of testing the algorithm, the combination was made for the whole one-year interval in one step. Although the input data was corrected for mutual systematic differences, as described above, the dispersion of station coordinates remained practically unchanged and the EOP suffered from false steps and undulations corresponding to changes of contributing techniques/solutions at individual epochs (Pešek and Kostelecký, 2000). The irregularities disappeared when the monthly data were combined independently, nevertheless discontinuities remained in EOP between the successive blocks (Pešek and Kostelecký, 2003).

The presented combination also consists of successive monthly solutions, which were, however, derived from two-monthly data. In this case, the data overlaps substantially reduce the discontinuities.



Fig. 3. Results of the monthly combination from two-monthly data for coordinates of the pole. The values of individual epochs (dots) were smoothed (thick curve) and compared with the smoothed a priori values (thin curve). The IERS c04 solution is substracted from all data to make the graph more readable.

Fig. 4. Results of the monthly combination from two-monthly data for time correction (for explanation see Fig. 3).

The monthly solutions for EOP were merged together to complete a series covering the whole one-year interval. In Figs. 3 and 4, coordinates of the pole px , py and time

corrections UT1 − UTC are reduced by the EOP(IERS) c04 series to highlight their detailed changes. The dots are the estimated EOPs at individual epochs. The graphs are completed by curves, corresponding to the final and a priori values of the EOP, both smoothed by the Vondrak method (Vondrák, 1977), with the smoothing factor ε = 10−3.

It is obvious that the rather large deflections from the c04 series are not caused by instability of the combination, because the effect of the combination, being of the order of 0.1 mas, is given by differences of the estimated (thick curves) and the a priori values (thin curves).



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The celestial pole offset was only derived from VLBI data, which suffered from non-removable systematic mutual biases. This caused unacceptably large deflections from the c04 data, namely dψ = −0.02″, dε = −0.01″.

The seven-parameter transformation parameters were derived for 11 epochs. Its time evolution is visible from Fig. 5. Greater dispersion in z component is probably caused by irregular geographical distribution of the stations.

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Fig. 6. Three examples of time evolution of station coordinates as obtained from successive monthly solutions for individual data sets. Geocentric coordinates were transformed to the local system with following components: northward (SN), eastward (EW) and upward (UP). Zero axes of the SN and UP components are shifted by 150 and 300 mm, respectively. Much bigger dispersion of Doris and SLR (#1 and #2-7) is obvious.



Fig. 7. Evolution of coordinates of stations from Fig. 6 displayed in the local horizontal plane (in cm). Compared are values entering the combination (grey) and results of combination (black). Numbers along the lines correspond to the individual monthly solutions and the hash-marked numbers (see Table 1) denote mean positions of the respective technique.

Time evolution of the coordinates after the transformation is illustrated in Fig. 6 for three of the stations, namely 1003 Toulouse, 14201 Wettzell and 92201 Papeete. Vertical, north-south and west-east components are denoted as UP, NS and EW, respectively. Surprising is the dispersion in the vertical component which is smaller than in the horizontal components. Horizontal movements of the three stations as well as mean positions corresponding to the contributing techniques are shown in Fig. 7.



In most cases, scatter of station coordinates as obtained from the monthly solutions increased after the combination. At the same time though, the combination reduced substantially mutual biases between the techniques and as a result the average dispersion of station coordinates decreased from 0.040 m to 0.031 m, with better improvement of sites present in more input techniques/solutions.

The time evolution of station coordinates is more or less chaotic at some sites. This is caused mainly by the less accurate techniques, especially Doris (#1 in Fig. 7), which affects the a priori as well as adjusted coordinates. The effect is obvious especially at sites equipped with only a few techniques.

It is also worth mentioning the importance of the high accurate local ties. Any of their error would affect the transformation parameters p of the respective technique and thus it would be propagated to coordinates of other sites.

4. CONCLUSIONS

A method is presented for ‘non-regular’ combination of different techniques to derive a representative set of Earth orientation parameters and station coordinates in a unique adjustment. It is based on combining station position vectors in the celestial reference frame, which are functions of both Earth orientation parameters and station coordinates. Three types of constraints are applied to remove singularity of the system, to separate celestial pole offset from pole motion and link the EOP between the adjacent epochs.

The method was designed for combinations of solutions of various techniques, for which the normal equation and/or covariance matrix is not available but it can also be used as an independent check of regular combinations whose algorithms are now in progress.

The method was tested on data collected for the IERS SINEX Combination Campaign. After combination, the dispersion of station coordinates decreases by 25 − 50%, depending of the data used, in the presented case by 22%, and increments to the EOP were of the order of 0.1 mas with respect to the adopted a priori values. Periodic deflections of the orientation parameters from the EOP(IERS) c04 series are not caused by the combination algorithm used, since it is obvious that practically the same holds true for the adopted a priori values, which were obtained as a weighted mean of the input data.

The tentative solutions also proved that it is suitable for solving the combination successively in shorter, e.g. monthly solutions. The discontinuities of the EOP between the successive blocks can be substantially reduced when two-monthly data is combined, taking the middle one-month part as the monthly solution.

In the meantime, the method was sligthly modified and applied to the longer series of data, which became available at the CPP database. It turned out that having high quality data makes the stepwise solution unnecessary and the combination remains stable even when several year´s data is processed in one go (Kostelecký and Pešek, 2006).

Acknowledgements: The authors greatly appreciate the support of grant LC506 awarded by the

Ministry of Education, Youth and Sports of the Czech Republic.



References

Gambis D., Baudouin P., Bizouard C., Bougeard M., Carlucci T., Essaifi E., Francou G. and Jean-Alexis D., 2002. In: B. Richter and W.R. Dick (Eds.), IERS Annual Report 2002, Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt a/M., Germany, 36−46.

IERS, 2003: http://alpha.fesg.tu-muenchen.de/iers/sinex/datapool.html

Kostelecký J. and Pešek I., 2003. Determination of station coordinates and EOP from combination of different techniques. In: B. Richter, W. Schwegmann and W.R. Dick (Eds.), IERS Technical Note No. 30, Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt a/M., Germany, 214−215.

Kostelecký J. and Pešek I., 2006. Determination of Earth orientation parameters and station coordinates from combination of IERS CPP data (internal comparisons). In: IERS Technical Note, Verlag des Bundesamts für Kartographie und Geodäsie, Frankfurt a/M., Germany (in press).

Pešek I. and Kostelecký J., 2000. Simultaneous determination of station coordinates and EOP from combination of different techniques. In: M. Soffel and N. Capitaine (Eds.), Journées 1999 - Systemes de référence spatio-temporels & IX. Lohrmann-Kolloquium, Dresden, September 13−15, 1999. Lohrmann-Observatorium, Technische Universitaet Dresden and Observatoire de Paris, 235, ISBN 2-901057-42-X.

Pešek I. and Kostelecký J., 2003. Simultaneous determination of Earth orientation parameters and station coordinates from combination of results of different observation techniques. In: L. Gerhátová, J. Hefty and M. Mojzeš (Eds.), Significance of Cosmic Methods for Present Geodesy (Význam kozmických metód pre súčasnú geodéziu), Slovak Technical University Bratislava, Slovakia, 89−96 (in Czech).

Vondrák J., 1977. Problem of smoothing observational data II. Bull. Astron. Inst. Czechosl., 28, 84−89.


simultaneous determination of earth orientation parameters … · key words: earth orientation...

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