1. Combination of long time series of tropospheric parameters observed by VLBI R. Heinkelmann, J. Boehm, H. Schuh Institute of Geodesy and Geophysics, TU Vienna 4th IVS General Meeting 2006, Concepción, Chile

2. Introduction 24th IVS General Meeting Aim of this study  Combined long time series of tropospheric parameters for the assessment of climatological trends Approach  Direct combination on result level with scaling factors for the individual AC solutions  Estimation of linear trends using common arrays scaled by variance components (VC)

3. BKG GSFC IAA IGG MAO Data description 3 Long time series of tropospheric parameters from Analysis Centers (ACs) 4th IVS General Meeting

4. Data description 4 Different parameter models Least-squares (LS) estimates of piecewise linear function model (PWLF) Kalman forward running filter (FRF) estimates of random walk stochastic model Square root information filter (SRIF) (forward + backward) estimates of random walk stochastic model * BKG GSFC IGG IAA MAO AC Model * Bierman G.J. 1977 4th IVS General Meeting

5. Data description 5 Different constraints on the parameters Rates of piecewise linear function are constrained using different weights A-priori values for the estimated parameters and the covariance matrices, dynamic model BKG GSFC IGG IAA MAO 15 20 n.a. loose constraint [mm/h] AC Model 4th IVS General Meeting

6. Data description 6 Different epoch and interpolation Linear interpolation of parameters and standard deviations (stdv) @ integer hours Linear interpolation @ integer hours, stdv are from tropospheric offset Average of 1 hour interval centered @ time reference, stdv forwarded by error propagation BKG GSFC IGG IAA MAO AC Model 4th IVS General Meeting

7. Data description 7 Different solution strategies NMF VMF NMF BKG GSFC IGG IAA MAO AC mf cutoff 5° 3° 5° 0° w.r.t. TRF VTRF2003 ITRF2000 VTRF2003 ITRF2000 datum NNT/NNR fixed coord. NNT/NNR 4th IVS General Meeting

8. Data description 8  Summary different functional and stochastic models different a-priori information / constraints different analysis options, geodetic datum different relation of time of reference, interpolations and stdv treatment  Common ground Results characterize the same physical phenomenon Averaging analysts’ noise 4th IVS General Meeting

9. Combination on result level 9 Strategy  Independent analysis of each station and each parameter (wet, hydrostatic zenith delay, gradients)  Elimination of outliers of individual time series  Determination of linear trends using weight factors obtained by variance component (VC) estimation a.with a-priori variances b.without a-priori variances 4th IVS General Meeting

10. Elimination of outliers 10 Strategy  Decomposition of time series by frequency analysis until residuals follow a white noise process, i.e. normal distribution  Detection of outliers w.r.t. the functional model using the BIBER algorithm  Minimal modification of observations to fulfill normal distribution 4th IVS General Meeting

11. Decomposition of time series 11 Characteristics: Begin: 1993/04/21 End: 2004/12/29 17131 data points Irregular sampling Big data gap Begin: 1997/05/27 End: 1998/02/12 4th IVS General Meeting Example: Fortaleza, Brazil, IGG - wet zenith delays

12. : systematic part: offset, trend, seasonal component : vector of observations : vector of residuals: 14  Gauss-Markov model *  Functional model: p1, p2 annual and semiannual periods Functional model of outlier elimination * Koch K.R. 1997 4th IVS General Meeting

13. Characteristics of BIBER outlier elimination: Only one modification per iteration step Correlations are considered Observations are minimally modified 1)compute 2)if 3)where 4)modify observation 15 BIBER algorithm * * Wicki F. 1999 4th IVS General Meeting

14. 16 Outlier cleaned time series # observations linear trend modified observations IGG: 17131 -0.09 mm/year BKG: 19097 -0.27 mm/year GSFC: 18864 -0.12 mm/year IAA: 14952 -0.13 mm/year MAO: 17691 -0.65 mm/year 4th IVS General Meeting Example: Fortaleza, Brazil - wet zenith delays

15. Method: Global best invariant quadratic unbiased estimation (global BIQUE) * applied iteratively ** Minimal computational costs At convergence point independent of approximate values  i 17  Gauss Markov model with unknown VC Variance component estimation * Förstner W. 1979 ** Koch K.R. 1997 4th IVS General Meeting

16. 18 Relative variance components VC considering a-priori stdvneglecting a-priori stdv VC strongly depend on a-priori variance information Example: Fortaleza, Brazil 4th IVS General Meeting

17. 19 Linear trend of common data Example: Fortaleza, Brazil ALL: 87735 without VC linear trend: -0.25 mm/year VC neglecting a-priori stdv -0.26 mm/year VC considering a-priori stdv -0.45 mm/year 4th IVS General Meeting

18. 20 Conclusions  Seasonal signal - must be included in functional model of both outlier elimination and trend determination, trend and sin/cos functions are not orthogonal  A-priori variance information - significantly influence the variance component estimation - stdv from different stochastic models have different level  Linear trends of tropospheric parameters - strongly depend on models, analysis options, combination strategy - from combined time series average the Analyst noise 4th IVS General Meeting

19. 21 Outlook: Combination on NEQ level  Within VLBI: One model for tropospheric parameters Same constraints Same geophysical models and and analysis options Homogeneous meteorological input data Tropospheric parameters estimated at epoch, i.e. no interpolation Output: SINEX files including tropospheric parameters  With other space geodetic techniques Local ties Same meteorological data, models, and height reference Observations at same epoch Output: SINEX files including tropospheric parameters 4th IVS General Meeting

20. contacts: R. Heinkelmann rob@mars.hg.tuwien.ac.at Thank you for your attention end Acknowledgements: All IVS ACs which contribute to this study are greatly acknowledged. project 16992

21. Reference Bierman G.J. 1977 Factorization Methods for Discrete Sequential Estimation, Mathematics in Science and Engineering 128, edited by R. Bellman Foster G. 1996 Wavelets for period analysis of unevenly sampled time series, The Astronomical Journal 112 (4), 1709-1729 Förstner W. 1979 Ein Verfahren zur Schätzung von Varianz- und Kovarianzkomponenten, AVN 11-12, 446-453 Koch K.R. 1997 Parameterschätzung und Hypothesentests, 3rd edition, Dümmler, Bonn Lomb N.R. 1976 Least-Squares Frequency Analysis of unequally spaced data, Astrophysics and Space Science 39, 447-462 Roberts D.H. et al. 1987 Time series analysis with CLEAN. I. Derivation of a spectrum, The Astronomical Journal 93 (4), 968-989 4th IVS General Meeting