2. Comparison of polar motion prediction results supplied by the IERS Sub-bureau for Rapid Service and Predictions and results of other prediction methods W. Kosek 1, D.D. McCarthy 2, T.J. Johnson 2, M. Kalarus 1 1 Space Research Centre, PAS, Warsaw, Poland 2 U.S. Naval Observatory, Washington D.C., USA Journees 2003 “Systemes de Reference Spatio-Temporels”, 22-25 September 2003, St. Petersburg, Russia.

3. Data EOPC01 (1846.0 - 2000.0), Δt =0.05 years http://hpiers.obspm.fr/eop-pc/ EOPC04 (1962.0 - 2003.5), Δt = 1 day http://hpiers.obspm.fr/eop-pc/ USNO (1976.0 - 2003.5), Δt = 1 day (finals.all ) http://maia.usno.navy.mil/bulletin-a.html

4. Accuracy of polar motion prediction depends on: irregular amplitudes and phases of short period oscillations with periods less than 1 year, amplitude variations of the Chandler oscillation, irregular phase and amplitude variations of the annual oscillation, irregular decadal and secular variations. For short period prediction For longer period prediction

5. The Chandler and annual oscillations filtered by the FTBPF from pole coordinates data Chandler Annual

6. Time-frequency FTBPF amplitude spectra (prograde part) with different frequency bandwidths of complex-valued USNO pole coordinate data

7. 2015-06-026 The current polar motion prediction computed by the IERS Sub-Bureau for Rapid Service and Prediction is the LS extrapolation of the circular Chandler and elliptic annual and semiannual oscillations. The LS extrapolation model is fit to the last year of the pole coordinates data and predicted for one year in the future.

8. The amplitude and phase variations of the Chandler circle and annual elliptic oscillations computed by the LS in one year time intervals

9. Two ways of polar motion prediction A prediction method is applied directly to x, y pole coordinates data. Before the prediction is applied the linear trend is removed and the trend extrapolation model is added to the computed forecast. A prediction method is applied in polar coordinate system to the polar motion radius and angular velocity and then their forecasts are transformed to the pole coordinate prediction using linear intersection. The radius and its prediction must be referred to the mean pole and its prediction.

10. The following prediction methods using two ways of prediction were applied: 1) Least-squares (LS) extrapolation (1 and 2 dimensions) 2) Autocovariance (2 dimensions) (Kosek 1997) 3) Autoregressive (AR) (2 dimensions) (Brzeziński 1995) 4) Neural networks (NN) (1 dimension) Different combinations of the two prediction methods that compute the forecast as the sum of the LS extrapolation and the autocovariance, autoregressive and neural networks prediction of the LS extrapolation residuals were also tested.

11. Transformation of pole coordinates data to polar coordinate system radius angular velocity the length of polar motion path (integrated angular velocity) mean pole

12. Transformation of the prediction of radius and angular velocity from the polar to the Cartesian pole coordinate system Linear intersection formula: mean pole prediction

13. The mean pole using Ormsby LPF - filter length, - number of data, - cutoff frequency, - cutoff period, - cutoff frequency – roll off termination frequency. - pole coordinates data, 2003 1849

14. Corr. Coeff. 1900-2003 0.864 1950-2003 0.899

15. The FTBPF time-frequency amplitude spectra of polar motion radius and angular velocity

16. The FTBPF amplitude spectra of polar motion radius and angular velocity

17. Time-frequency FTBPF amplitude spectra of polar motion radius, angular velocity and integrated angular velocity

18. Autocovariance prediction Let be stationary complex-valued time series

19. The absolute values of the difference between x, y pole coordinates data, the radius R and integrated angular velocity L and their autocovariance predictions in the polar coordinate system

20. The absolute value of the difference between x, y pole coordinates data and their IERS and autocovariance predictions computed in the polar coordinate system

21. The mean prediction error of x, y pole coordinates data, the radius R, angular velocity A and integrated angular velocity L in 1984.0-2003.5 computed from the autocovariance predictions in the polar coordinate system

22. The mean prediction error of x, y pole coordinates data in 1984.0-2003.5 computed from the autocovariance predictions in the polar coordinate system and by the IERS Sub-bureau for Rapid Service and Predictions

23. Prediction by combination of the LS and a stochastic method in the Cartesian pole coordinate system x, y LS extrapolation residuals Prediction of x, y LS extrapolation residuals LS extrapolation of x, y Prediction of x, y AUTOCOVARIANCE AR NN x, y pole coordinates data x, y LS model

24. Prediction of x, y pole coordinates data by combination of the LS and a stochastic method in the polar coordinate system x, y pole coordinates data R, A LS model R, A LS extrapolation residuals Prediction of R, A LS extrapolation residuals R, A LS extrapolation Prediction of R, A AUTOCOVARIANCE AR R – radius A – angular velocity Prediction of x, y mean pole + its LS prediction Prediction of R, A

25. The mean prediction error of x, y pole coordinates data, the radius, angular velocity A and integrated angular velocity L in 1984.0-2003.5 computed by the combination of the LS method and the autoregressive prediction of the LS extrapolation residuals in the polar coordinate system

26. The mean prediction error in 1984.0-2003.5 of x, y pole coordinates data computed from the LS predictions of the IERS Sub-Bureau for Rapid Service and Predictions (x - blue and y - red) and from the combination of the LS extrapolation of complex-valued pole coordinate data and the AR prediction of the complex-valued LS extrapolation residuals (x - green, y - yellow) USNO LS + AR

27. The absolute value of the difference between x, y pole coordinates data and their IERS and LS+AR predictions

28. The mean prediction error in 1984.0-2003.5 of x, y pole coordinates data computed from the LS predictions of the IERS Sub-Bureau for Rapid Service and Predictions (x - blue and y - red) and from the combination of the LS extrapolation of complex-valued pole coordinate data and the NN prediction of the real-valued LS extrapolation residuals (x - green, y - yellow) USNO LS + NN

29. The absolute value of the difference between x, y pole coordinates data and their IERS and LS+NN predictions

30. Conclusions The mean prediction errors of x, y pole coordinates data for prediction length less than 50 days in the future of the IERS prediction and the autocovariance prediction in polar coordinate system are of the same order. The problem of any prediction method of pole coordinates data in the polar coordinate system is a significant error in the prediction of the integrated angular velocity. The accuracy of prediction of x, y pole coordinates data by combination of the LS extrapolation and the AR or NN predictions of the LS extrapolation residuals is better than the accuracy of prediction carried out by the IERS Sub-Bureau for Rapid Service and Prediction.