Combination of long time series of tropospheric parameters observed by VLBI R. Heinkelmann, J. Boehm, H. Schuh Institute of Geodesy and Geophysics, TU

Combination of long time series of tropospheric

parameters observed by VLBIR. Heinkelmann, J. Boehm, H. Schuh

Institute of Geodesy and Geophysics, TU Vienna

4th IVS General Meeting 2006, Concepción, Chile

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)

• BKG

• GSFC

• IAA

• IGG

• MAO

Data description

3

Long time series of tropospheric parameters from

Analysis Centers (ACs)


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


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

151520

n.a.loose

constraint [mm/h]AC Model


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


Data description

7

Different solution strategies

NMF

NMF

VMF

VMF

NMF

• BKG

• GSFC

• IGG

• IAA

• MAO

AC mf cutoff

5°

3°

5°

0°

0°

w.r.t. TRF

VTRF2003

ITRF2000

VTRF2003

VTRF2003

ITRF2000

datum

NNT/NNR

NNT/NNR

NNT/NNR

fixed coord.

NNT/NNR


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


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) estimationa. with a-priori variancesb. without a-priori variances4th IVS General

Meeting

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


Decomposition of time series

11

Characteristics:Begin:

1993/04/21End: 2004/12/2917131 data

pointsIrregular

samplingBig data gapBegin:

1997/05/27End: 1998/02/124th IVS General

Meeting

Example: Fortaleza, Brazil, IGG - wet zenith delays

: systematic part: offset, trend, seasonal component: vector of observations: vector of residuals:

vlAx

0vE

Ax

14

Gauss-Markov model *

Functional model:

p1, p2 annual and semiannual periods

lv

Functional model of outlier elimination

* Koch K.R. 1997


2222

1111t

p/2sinBp/2cosA

p/2sinBp/2cosAtbaY

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 *

vvmax

vckv

vv

mm1mm

q

kvsignvll

* Wicki F. 1999

(m)v


200300400

200300400

200300400

200300400

1994 1996 1998 2000 2002 2004

200300400

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


Example: Fortaleza, Brazil - wet zenith delays

• Method: Global best invariant quadratic unbiased estimation (global BIQUE) * applied iteratively **

• Minimal computational costs • At convergence point independent of

approximate values i

1m

1nn

2i

2im

2i ˆ

17

Gauss Markov model with unknown VC

Variance component estimation

k

1ii

2i

2i

k

1ii

2i TVK

* Förstner W. 1979 ** Koch K.R. 1997

1ˆ m

2i

vlAx


18

Relative variance components VC

considering a-priori stdv neglecting a-priori stdv

• VC strongly depend on a-priori variance information

Example: Fortaleza, Brazil


1994 1996 1998 2000 2002 2004

150

200

250

300

350

400

19

Linear trend of common data

Example: Fortaleza, Brazil ALL: 87735without VC linear trend:-0.25 mm/year

VC neglectinga-priori stdv-0.26 mm/year

VC consideringa-priori stdv-0.45 mm/year


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


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


contacts:

R. Heinkelmann

[email protected]

Thank you for your attention

end

Acknowledgements:All IVS ACs which contribute to this study are greatly acknowledged.

project 16992

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


Documents

Combination of long time series of tropospheric parameters observed by VLBI R. Heinkelmann, J. Boehm, H. Schuh Institute of Geodesy and Geophysics, TU