Geophysical causes of pole coordinates data prediction errors

1. Geophysical causes of pole coordinates data prediction errors Wiesław Kosek 1) Environmental Engineering and Land Surveying Department, Agriculture University of Krakow, Poland 2) Space Research Centre, Polish Academy of Sciences, Warsaw, Poland European Geosciences Union, General Assembly, Vienna, Austria, 22 – 27 April 2012

2. SUMMARY • introduction • input data • wavelet based comparison of complex-valued time series applied to geodetic and fluid excitation functions and corresponding to them pole coordinates data • prediction of pole coordinates data by the LS+AR method and wavelet based comparison of prediction errors. • conclusions

3. The exact knowledge of the EOP is important for many investigations in astronomy and geodesy. For some tasks the future EOP data are neededto compute real-time transformation between the celestial and terrestrial reference frames. This transformation is important for the NASA Deep Space Network, which is an international network of antennas that supports: - interplanetary spacecraft missions, - radio and radar astronomy observations, - selected Earth-orbiting missions.

4. DATA • x,y pole coordinates data from the IERS: EOPC04_IAU2000.62-now (1962 – 2012.15), Δt = 1 day, http://hpiers.obspm.fr/iers/eop/eopc04/ • Equatorial components of atmospheric angular momentum from NCEP/NCAR (mass+motion), aam.ncep.reanalysis.* (1948 - 2012.15) Δt = 0.25 day, ftp://ftp.aer.com/pub/anon_collaborations/sba/, • Equatorial excitation functions of global ocean angular momentum (mass+motion): ECCO_kf080.chiECCO_JPL (Jan. 1993 - Dec. 2011), Δt = 1 day, c20010701.chiECCO_JPL (Jan. 1980 - Mar. 2002) Δt = 1 day, ECCO_50yr.chi  ECCO_JPL  (Jan 1949 - Dec 2002)   Δt = 10 days http://euler.jpl.nasa.gov/sbo/sbo_data.html,

5. IERS C04_IAU2000 x,y pole coordinates data, their determination errors and recent ratio of the prediction to the determination errors < 3 mm

6. Equatorial components of AAM and connected OAM excitation functions

7. WAVELET TRANSFORM COEFFICIENTS The wavelet transform coefficients of complex-valued signal defined: where - dilation and translation parameters - Discrete Fourier Transforms (DFT) of time series - Continuous Fourier Transform (CFT) of the modified Morlet wavelet function given by the following time domain formula (Schmitz-Hübsch and Schuh 1999): ,

8. WAVELET TRANSFORM SPECTRUM AND POLARISATION SPECTRUM: POLARISATION: retrograde prograde circular elliptic circular the shape of ellipse degenerates to a line

9. WAVELET TRANSFORM SEMBLANCE The wavelet semblance of the order , between and time series is defined as: , where - wavelet coherence, - wavelet phasesynchronization, • wavelet spectrum of - wavelet cross-spectrum - DFT of

10. The wavelet semblance (order=1) between geodetic excitation functions computed from x-iy IERS pole coordinates data and equatorial components of the fluid excitation functions (AAM, OAM and AAM+OAM).

11. x,y pole coordinates model data computed from fluid excitation functions Differential equation of polar motion: - model pole coordinates • equatorial excitation functions corresponding to AAM, OAM, • and AAM+OAM excitation functions - complex-valued frequency, where and Approximate solution of this equation in discrete time moments can be obtained using the trapezoidal rule of numerical integration:

12. IERS, AAM, AAM+OAM, OAM The IERS x,y pole coordinates and the x,y pole coordinates model data computed from fluid excitation functions in 1962 - 2012.

13. The wavelet spectra of x-iy IERS pole coordinates data and pole coordinates model data computed from fluid excitation functions.

14. The wavelet polarization functions of x-iy IERS pole coordinates data and pole coordinates model data computed from fluid excitation functions.

15. The mean (1980-2011) wavelet polarization functions of IERS x-iy pole coordinates data and pole coordinates model data computed from fluid excitation functions (AAM, OAM, AAM+OAM).

16. Prediction of x, y pole coordinates data by the LS+AR method x, y LS model (Chandler circle + annual and semiannual ellipses + linear trend) x, y LSresiduals x, y LS extrapolation AR prediction x, y LS extrapolation Prediction of x, y LSresiduals Prediction of x, y

17. Correlation coefficents IERS/AAM IERS/AAM+OAM 0.56 0.86 0.520.76 0.480.84 0.510.75 Prediction errors of the IERS pole coordinates data and pole coordinates model data computed from fluid (AAM, AAM+OAM) excitation functions for 30 and 60 days in the future, together with the correlation coefficients between these errors in 1990-2012.

18. mas Prediction errors of x IERS pole coordinate and x pole coordinate model data computed from fluid excitation functions from one day to one year in the future

19. mas Prediction errors of y IERS pole coordinate and y pole coordinate model data computed from fluid excitation functions from one day to one year in the future

20. Mean prediction errors of x,y IERS pole coordinates and pole coordinates model data computed from fluid excitation functions (AAM, OAM, AAM+OAM)

21. The wavelet spectrum of complex-valued prediction errors of the IERS x,y pole coordinates data

22. The wavelet spectra of the complex-valued prediction errorsof the IERS x,y pole coordinates data and pole coordinates model data computed from fluid excitation functions (AAM, OAM, AAM+OAM)

23. The wavelet polarizations of complex-valued prediction errors of the IERS pole coordinates data and pole coordinates model data computed from fluid excitation functions (AAM, OAM, AAM+OAM)

24. CONCLUSIONS The wavelet polarization function of complex-valued IERS pole coordinates data and pole coordinates model data computed from fluid excitation functions, show that oscillations in them with periods greater than ~30 days are mostly prograde and become more circular when the period increases. Oscillations with periods less than ~30 days are more retrograde than prograde. The wavelet semblance (order=1) between complex-valued geodetic excitation functions computed from the IERS pole coordinates data and fluid excitation functions are greater for the joint atmospheric-ocean excitation than for the atmospheric or ocean ones. The annual oscillations in geodetic and ocean excitation functionsare out of phase. The contributions of atmosphere and ocean angular momentum excitation functions to the mean prediction errors of pole coordinates data is similar and of the order of 55-60% of the total mean prediction error of pole coordinates data. The contribution of the joint atmospheric-ocean angular momentum excitation to the mean prediction errors of pole coordinates data is of the order of 70 to 80 % of the total mean prediction error of these data. The wavelet spectra and polarization of time series of prediction errors of pole coordinates data show that oscillations in them are mostly prograde. These prediction errors are mostly caused by short period wide band elliptic oscillations in joint ocean atmospheric excitation function and wide band prograde oscillations in the Chandler and annual frequency band caused by mismodelling of the Chandler and annual oscillations in the prediction algorithm.

25. The wavelet coherence between geodetic excitation functions computed from x-iy IERS pole coordinates data and equatorial components of the fluid excitation functions.

26. Skewness (SKE) skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Negative skew indicates that the tail on the left side of the probability density function is longer than the right side. If the distribution is symmetric then skewness is zero. - third moment about the mean - standard deviation error - the expectation operator.

27. Kurtosis (CUR) (Gr. κυρτός, ang. bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable, - fourth moment about the mean - standard deviation error - the expectation operator.

28. Skewness and kurtosis of prediction errorsof the IERS x,y pole coordinates and pole coordinates model data computed from fluid excitation functions (AAM, OAM, AAM+OAM)

29. EOPCPPP (Oct 2010 – Dec 2011) Mean absolute errors (MAE) and standard deviations (SD) computed by different participants of the project. Ensemble prediction (red).

30. Pole coordinates spectra The Morlet wavelet spectra of x,y IERS 08C04 pole coordinates data computed for different σ parameter values. Time span of data: 1962 – 2012.

31. Amplitudes and phases of the most energetic oscillations in x, y pole coordinates data Chandler Annual Amplitudes Semi-annual bold line – prograde thin line - retrograde Chandler Annual Phases Semi-annual

32. EOP prediction – international activity Earth Orientation Parameters Prediction Comparison Campaign(EOPPCC) (Oct. 2005 – Mar. 2008).The main idea of this campaign was to compare the various prediction techniques that can be applied to the EOP prediction. IERS Working Group on Predictions (04. 2006 – EGU). The main idea of the WGP is to investigate the optimum input data sets for the EOP predictions and the strengths and weaknesses of the various prediction algorithms.

33. EOP prediction – international activity Earth Orientation Parameters Prediction Comparison Campaign(EOPPCC) (Oct. 2005 – Mar. 2008) [H. Schuh (Chair), W. Kosek, M. Kalarus] The main idea of this campaign was to compare the various prediction techniques that can be applied to the EOP prediction. During the EOPPCC about 10 participating groups/methods were submitting prediction results of pole coordinates, UT1- UTC, LOD, and celestial pole offsets. IERS Working Group on Predictions (WGP) (04. 2006 – EGU) [W. Wooden (Chair), T. Van Dam (input data) , W. Kosek (algorithms)] The main idea of the WGP is to investigate the optimum input data sets for the EOP predictions and the strengths and weaknesses of the various prediction algorithms. The WGP was formed to investigate the properties of different prediction algorithms, qualities of the input data, and the interactions between input data and prediction algorithms.

34. Future EOP data are neededto compute real-time transformation between the celestial and terrestrial reference frames. This transformation is important for the NASA Deep Space Network, which is an international network of antennas that supports: - interplanetary spacecraft missions, - radio and radar astronomy observations, - selected Earth-orbiting missions. 1989 - IERS Rapid Service Sub-bureau. 2001 - IERS Rapid Service/Prediction Centre (IERS RS/PC)

35. Madrid, Spain Goldstone, California, pustynia Mojave Canberra, Australia. Deep Space Network