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Combination of long time series of tropospheric parameters observed by VLBI

4th IVS General Meeting 2006, Concepci ó n, Chile. Combination of long time series of tropospheric parameters observed by VLBI. R. Heinkelmann, J. Boehm, H. Schuh Institute of Geodesy and Geophysics, TU Vienna. Introduction. Aim of this study

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Combination of long time series of tropospheric parameters observed by VLBI

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

  2. Introduction 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) 4th IVS General Meeting 2

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

  4. Data description Different parameter models Model AC • BKG • GSFC • IGG • IAA • MAO 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 * * Bierman G.J. 1977 4th IVS General Meeting 4

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

  6. Data description Different epoch and interpolation Model AC • BKG • GSFC • IGG • IAA • MAO 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 4th IVS General Meeting 6

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

  8. Data description • 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 8

  9. Combination on result level 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 • with a-priori variances • without a-priori variances 4th IVS General Meeting 9

  10. Elimination of outliers 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 10

  11. Decomposition of time series Example: Fortaleza, Brazil, IGG - wet zenith delays 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 11

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

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

  14. Outlier cleaned time series Example: Fortaleza, Brazil - wet zenith delays 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 # observations linear trend modified observations 4th IVS General Meeting 16

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

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

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

  18. 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 20

  19. 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 21

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

  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

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