Forecasting of the Earth orientation parameters – comparison of different algorithms W. Kosek 1, M. Kalarus 1, T. Niedzielski 1,2 1 Space Research Centre, Polish Academy of Sciences, Warsaw, Poland 2 Department 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” September 2007, Meudon, France.

Prediction errors of EOP data and their ratio to their determination errors in 2000 Days in the future x, y [mas] UT1-UTC [ms] Ratio: prediction to determination errors x, y ~7~7~36~85~140~230~340~430 UT1 ~10~58~300~580~1100~2700~5600 YEARS x [mas] y [mas] UT1 [ms] Determination errors of EOPC04 data in ~2.8 mm~1.8 mm

Data x, y, EOPC01.dat ( ), Δt =0.05 years x, y, Δ, UT1-UTC, EOPC04_IAU now ( ), Δt = 1 day x, y, Δ, UT1-UTC, Finals.all ( ), Δt = 1 day, USNO χ 3, aam.ncep.reanalysis.* ( ) Δt=0.25 day, AER IERS

Prediction techniques 1)Least-squares (LS) 2)Autocovariance (AC) 3)Autoregressive (AR) 4)Multidimensional autoregressive (MAR) 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 Prediction algorithms

Prediction of x, y data by combination of the LS+AR x, y LS residuals Prediction of x, y LS residuals x, y LS extrapolation Prediction of x, y AR prediction x, y x, y LS model LS extrapolation

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

LS and LS+AR prediction errors of x data

LS and LS+AR prediction errors of y data

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

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

Mean LS+AR prediction errors of x, y data in

Prediction of x, y data by DWT+AC in polar coordinate system x, y R(ω 1 ), R(ω 2 ), …, R(ω p ) AC R – radius A – angular velocity LS extrapolation of x m, y m Prediction R n+1, A n+1 A(ω 1 ), A(ω 2 ), …, A(ω p ) R n+1 (ω 1 ) + R n+1 (ω 2 ) + … + R n+1 (ω p ) A n+1 (ω 1 ) + A n+1 (ω 2 ) + … + A n+1 (ω p ) LPF mean pole x m, y m LS x n, y n Prediction x n+1, y n+1 DWT BPF prediction

Mean pole, radius and angular velocity 2007

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

Δ-ΔR (ω 1 ) + Δ-ΔR (ω 2 ) + … + Δ-ΔR (ω p ) Prediction of Δ-ΔR Δ-ΔR (ω 1 ), Δ-ΔR (ω 2 ),…, Δ-ΔR (ω p ) UT1-UTC AC Prediction of Δ and UT1-UTC by DWT+AC Prediction of UT1-TAI Prediction of UT1-UTC diff UT1-TAIΔ Prediction of Δ int Prediction DWT BPF

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

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

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

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

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

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