Forecasting of the Earth orientation parameters – comparison of different algorithms

1. Forecasting of the Earth orientation parameters – comparison of different algorithms W. Kosek1, M. Kalarus1, T. Niedzielski1,2 1Space Research Centre, Polish Academy of Sciences, Warsaw, Poland 2Department of Geomorphology, Institute of Geography and Regional Development, University of Wrocław, Poland Journees 2007, Systemes de Reference Spatio-Temporels „The Celestial Reference Frame for the Future” 17-19 September 2007, Meudon, France.

2. Determination errors ofEOPC04 data in 1976-2004 ~1.8 mm ~2.8 mm Prediction errors of EOP data and their ratioto their determination errors in 2000

3. Data IERS • x, y, EOPC01.dat (1846.0 - 2000.0), Δt=0.05 years • x, y, Δ, UT1-UTC, EOPC04_IAU2000.62-now (1962.0 - 2007.6), Δt = 1 day • x, y, Δ, UT1-UTC, Finals.all (1973.0 - 2007.6), Δt = 1 day, USNO • χ3, aam.ncep.reanalysis.* (1948-2007.5) Δt=0.25 day, AER

4. Prediction techniques • Least-squares (LS) • Autocovariance (AC) • Autoregressive (AR) • Multidimensional autoregressive (MAR) Prediction algorithms • 1) Combination of LS and AR (LS+AR), [x, y, Δ, UT1-UTC] • - with autoregressive order computed by AIC • - with empirical autoregressive order • 2) Combination of LS and MAR (LS+MAR), [Δ, UT1-UTC, χ3AAM] • 3) Combination of DWT and AC (DWT+AC), [x, y, Δ, UT1-UTC] • Two ways of x, y data prediction • in the Cartesian coordinate system • in the polar coordinate system

5. Prediction of x, y data by combination of the LS+AR x, y LS model x, y LSresiduals x, y LS extrapolation AR prediction Prediction of x, y LSresiduals Prediction of x, y x, y LS extrapolation

6. Autoregressive method (AR) Autoregressive coefficients: are computed from autocovariance estimate : Autoregressive order:

7. LS and LS+AR prediction errors of x data

8. LS and LS+AR prediction errors of y data

9. Mean prediction errors of the LS (dashed lines) and LS+AR (solid lines) algorithms of x, y data in 1980-2007 (The LS model is fit to 5yr (black), 10yr (blue) and 15yr (red) of x-iy data)

10. Optimum autoregressive order as a function of prediction length for AR prediction of EOP data (Kalarus PhD thesis)

11. Mean LS+AR prediction errors of x, y data in 1980-2007

12. Prediction of x, y data by DWT+AC in polar coordinate system mean pole xm, ym LS extrapolation of xm, ym LS LPF x, y transformation DWT BPF R(ω1), R(ω2) , … , R(ωp) xn, yn R – radius A – angular velocity A(ω1), A(ω2), … , A(ωp) AC prediction Prediction Rn+1, An+1 Rn+1(ω1) + Rn+1(ω2) + … + Rn+1(ωp) linear intersection An+1(ω1) + An+1(ω2) + … + An+1(ωp) Prediction xn+1, yn+1

13. Mean pole, radius and angular velocity 2007

14. Mean prediction errors of x, y data (EOPPCC) 13 predictions 54 predictions

15. Prediction of Δ and UT1-UTC by DWT+AC diff Δ UT1-TAI -- leap seconds UT1-UTC -- Tides DWT BPF Δ-ΔR(ω1), Δ-ΔR(ω2),…, Δ-ΔR(ωp) Δ-ΔR AC Prediction of Δ-ΔR Prediction Δ-ΔR(ω1) + Δ-ΔR(ω2) + … + Δ-ΔR(ωp) + Tides Prediction of Δ Prediction of UT1-TAI Prediction of UT1-UTC int + leap seconds

16. Decomposition of Δ-ΔR by DWT BPF with Meyer wavelet function

17. Mean prediction errors of Δ and UT1-UTC (EOPPCC) 54 predictions

18. Multidimensional prediction - Estimates of Autoregression matrices, - autoregressive order: - Estimate of residual covariance matrix.

19. Prediction of length of day Δ-ΔR data by LS+AR and LS+MAR algorithms (Niedzielski, PhD thesis) εAAMχ3 residuals AAMχ3 LS model Δ-ΔRLSmodel ε(Δ-ΔR) residuals & Δ-ΔR AAMχ3 LS AR MAR Δ-ΔR LS extrapolation Prediction of Δ-ΔR AR prediction ε(Δ-ΔR) MAR prediction ε(Δ-ΔR)

20. Comparison of LS, LS+AR and LS+MAR prediction errors of UT1-UTC and Δ data

21. CONCLUSIONS • The combination of the LS extrapolation and autoregressive prediction of x, y pole coordinates data provides prediction of these data with the highest prediction accuracy. The minimum prediction errors for particular number of days in the future depends on the autoregressive order. • Prediction of x, y pole coordinates data can be done also in the polar coordinate system by forecasting the alternative coordinates: the mean pole, radius and angular velocity. • This problem of forecasting EOP data in different frequency bands can be solved by applying discrete wavelet transform band pass filter to decompose the EOP data into frequency components. The sum of predictions of these frequency components is the prediction of EOP data. • Prediction of UT1-UTC or LOD data can be improved by using combination of the LS and multivariate autoregressive technique, which takes into account axial component of the atmospheric angular momentum. THANK YOU