Polar motion prediction and time-frequency relation between polar motion and its excitation

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1 Polar motion prediction and time-frequency relation between polar motion and its excitation Kosek Wiesław Space esearch Centre, Polish Academy of Sciences. Warsaw, Poland Seminar at the U.S. Naval Observatory, Washington D.C., December th 02

1984 1986 1988 1990 1992 1994 1996 1998 00 02 1984 1986 1988 1990 1992 1994 1996 1998 00

03 0.02 0.01 0.03 0.02 0.01 1984 1986 1988 1990 1992 1994 1996 1998 00 02 0.025 1 year 0.

0 5 0 X Y 0 60 70 80 90 0 days in the future Fig. 1.

at different starting prediction epochs and the corresponding mean prediction errors of x

2 days in the future 1 year x y x y year X Y days in the future X Y days in the future Fig. 1. The differences x and y pole coordinate data and their least-squares predictions computed at different starting prediction epochs and the corresponding mean prediction errors of x and y pole coordinates over for the time span of polar motion data going into the least-squares extrapolation model equal to one and three.

3 x/y Chandler x annual y annual 0.0 o y annual 0 2 x/y Chandler x annual x/y Chandler x Annual y Annual 3 o x/y Chandler o C 4 Nino1+2 Nino3 Nino 4 Nino y Annual x Annual Fig. 2. The amplitude and phase variations of the Chandler and annual oscillations computed by the least-squares method in one and three year time intervals and the Nino indices.

period () 1.2 1.0 0.8 0.6 0.4-0.0 - -0.4-0.6-0.8-1.0-1.2 0.6 0.4-0.0 - -0.4-0.6 00 00 0.

1980 00 period (days) 2 180 1 60 0-60 -1-180 -2 1964 1967 1970 1973

0 8 6 4 2 x iy Time-frequency FTBPF amplitude spectra of

4 period () period (days) x iy Time-frequency FTBPF amplitude spectra of complex-valued IES EOPC01/C04 pole coordinate data for filter bandwidth of λ = 1 and λ = 2.

5 x, y t t t t m x, y t m t mean pole x, y t-1 t-1 m 2 m ( x x ) + ( y y ) 2, t = 1,2 n t = t t t t,..., 2 2 ( x x ) + ( y y ), t 2,3 n t t t t t,..., = 1 1 = Fig. 3. Transformation of pole coordinate data into polar coordinate system. n+1 x,y n+1 n+1 n+1 m x, y m x, y n n n n n mean pole x y n+ 1 n+ 1 x y = n n m m yn x n yn cotα + + cot β + m m xn yn xn cotα + cot β ( + ) 4P, cot = ( + ) 4, cotα = β n + 1 n n+ 1 n+ 1 n n+ 1 P n+ 1 + n + n+ 1 P = p( p n+ 1 )( p n )( p n+1 ), p = Fig. 4. inear intersection - computation of the first prediction point from the prediction of the radius and angular distance. 2

6 Butterworth PF (7 ) Ormsby PF (18 ) 6-year mean y 0.0 x y 0 0 x Fig. 5a. Mean pole computed by the boxcar, Butterworth and Ormsby PF from the IES pole coordinate data (blue lines). Three-year least-squares predictions computed at different starting prediction epochs (red lines) of the mean pole computed by the Ormsby PF (black line) and the mean pole computed the IES (green dots).

7 Y year S prediction of Ormsby PF IES mean pole FT PF (18yr) Ormsby PF (18yr) 6-year mean Butterworth PF (7yr) X 0.06 Fig. 5b. Mean pole after year computed the IES (green dots) and computed by the boxcar (orange line), Butterworth (blue line), Fourier Transform (purple line) and Ormsby PF (black line) from the IES EOPC01+C04 pole coordinate data. Three-year least-squares predictions computed at different starting prediction epochs of the mean pole computed by the Ormsby PF (red lines).

1880 1900 19 19 1960 1980 0.4 0.0 1860 1880 1900 19 19 1960 1980 00 Fig. 6. The radius and angular distance computed from the IES C01 pole coordinate data.

8 Fig. 6. The radius and angular distance computed from the IES C01 pole coordinate data. period () Fig. 7. Time-frequency FTBPF amplitude spectra of the polar motion radius and angular distance period () Fig. 8. The beat period of the Chandler and annual oscillations computed from the smoothed minima and maxima of the polar motion radius (green dots) and angular distance (orange d ots) and computed from the least-squares phase variations ϕ of the Chandler and annual oscillations (blue). 2π 2π t + ( ϕ + ϕ ) = + ϕ = const, = T T + T T T + T T + T ( ) beat An An Ch Ch

9 Autocovariance prediction S prediction AMA prediction Fig. 9a. One-year autocovarince, least-squares and autoregressive-moving average (AMA) predictions of polar motion radius and angular distance computed at different starting prediction epochs.

10 Autocovariance prediction S prediction AMA prediction Fig. 9b. One-year autocovarince, least-squares and autoregressive-moving average (AMA) predictions of polar motion angular distance computed at different starting prediction epochs.

11 AUTOCOVAIANCE PEDICTIONS 0.0 x y Fig. a. One-year predictions at different starting prediction epochs of the x and y IES EOPC01 pole coordinate data computed from the autocovariance predictions of the radius and angular distance. AMA PEDICTIONS X Y F ig. b. One-year AMA predictions at different starting prediction epochs of the x and y IES EOPC01 pole coordinate data.

1984 1986 1988 1990 1992 1994 1996 199 8 00 02 1984 1986 1988 1990 1992 1994 1996 1998 00 02 days

predictions computed at different starting prediction epochs. 0.0 0.015 0.

12 days in the future IES Sub-Bureau for apid Service and Predictions x y days in the future Autocovariance prediction Fig.. The differences between x, y pole coordinate data and their least-squares and autocovariance predictions computed at different starting prediction epochs x y x y - autocovariance prediction - IES Sub-Bureau for apid Service and Prediction days in the future F ig. 11. Mean prediction errors of x and y pole coordinates of the autocovariance and leastsquares predictions over x y

13 period () 8 7 S MODE PEDICTION Fig. 9c. One-year least-squares predictions of polar motion radius and angular distance computed at different starting prediction epochs. The following least-squares extrapolation model of the beat period was applied: beat where t is time in. T ( t) = 6.3+ sin[2π ( t )/. + 5o]

14 CONCUSIONS Polar motion least-squares prediction errors depend on irregular phase and amplitude variations of the annual oscillation that had maximum values before the El Niño events in 1982/83 and 1997/98. The increase of the annual oscillation phase and amplitude in 00 indicates that another El Niño is expected in the end of this year. Transformation of pole coordinate data from the Cartesian to a polar coordinate system transforms the Chandler and annual as well as the semi-chandler and the semiannual frequencies into their beat frequencies which helps to solve the frequency resolution problems. Accuracy of polar motion prediction by the method of autocovariance through the transformation to polar coordinate system depends on predicting accurately the mean pole, radius and angular distance. The period of the most energetic oscillation in polar motion radius and angular distance representing the beat period of the Chandler and annual oscillations is variable mainly due to variable phase or period of the annual oscillation. The error of the autocovariance prediction for a few days in the future is less than the prediction error of the current polar motion forecast carried out by the IES apid Service/Prediction Center.

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