Comparison of polar motion prediction results supplied by the IERS Sub-bureau for Rapid Service and Predictions and results of other prediction methods.

Comparison of polar motion prediction Comparison of polar motion prediction results supplied by the IERS Sub-bureau results supplied by the IERS Sub-bureau

for Rapid Service and Predictions and for Rapid Service and Predictions and results of other prediction methodsresults of other prediction methods

W. Kosek1, D.D. McCarthy2, T.J. Johnson2, M. Kalarus1 1Space Research Centre, PAS, Warsaw, Poland 2U.S. Naval Observatory, Washington D.C., USA

Journees 2003 “Systemes de Reference Spatio-Temporels”, 22-25 September 2003, St. Petersburg, Russia.

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

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50

x

arcsec

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-0.40-0.30-0.20-0.100.000.100.200.300.400.500.60

y

Accuracy of polar motion prediction depends on: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 For short period predictionprediction

For longer period For longer period predictionprediction

The Chandler and annual oscillations filtered by the FTBPF The Chandler and annual oscillations filtered by the FTBPF from pole coordinates datafrom pole coordinates data

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-0.30-0.20-0.100.000.100.200.30

Ch x

arcsec

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-0.30-0.20-0.100.000.100.200.30

Ch y

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-0.20-0.100.000.100.20

An x

1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-0.20-0.100.000.100.20

An y

Chandler

Annual

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

1975 1978 1981 1984 1987 1990 1993 1996 1999 2002

10

40

70

100

130

0.5

1.0

1.5

2.0

1975 1978 1981 1984 1987 1990 1993 1996 1999 2002years

10

40

70

100

130

p

erio

d (d

ays)

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0004

0.0014

x - iy (HPF 160 days) m as

18.04.23 6

The current polar motion prediction computed by theThe current polar motion prediction computed by the IERS IERS Sub-Bureau for Rapid Service and Prediction Sub-Bureau for Rapid Service and Prediction is the LS is the LS

extrapolation of the circular Chandler and elliptic annual and extrapolation of the circular Chandler and elliptic annual and semiannual oscillations. The LS extrapolation model is fit to semiannual oscillations. The LS extrapolation model is fit to the last year of the pole coordinates data and predicted for the last year of the pole coordinates data and predicted for

one year in the future. one year in the future.

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

1977 1980 1983 1986 1989 1992 1995 1998 20010.00

0.05

0.10

0.15

0.20

0.25arcsec am plitudes

Ch x/ y

An xAn y

1977 1980 1983 1986 1989 1992 1995 1998 2001years

150

200

250

300

350o

phases

Ch x/ yAn y

An x

Two ways of polar motion Two ways of polar motion predictionprediction

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.

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.

Transformation of pole coordinates data to polar coordinate systemTransformation of pole coordinates data to polar coordinate system

ntt

kkAtL ,...,3,2,

1

ty

tx ,

tR

ntmtyty

mtxtxtR ,...,2,1,

22

nttytytxtxtA ,...,3,2,2

12

1

radius

angular velocity

the length of polar motion path (integrated angular velocity)

mty

mtx ,

tA

1,

1 ty

tx

mean pole

Transformation of the prediction of radius and Transformation of the prediction of radius and angular velocityangular velocity from from the polar to the Cartesian pole coordinate systemthe polar to the Cartesian pole coordinate system

mnym

nx ,

ny

nx ,

1,

1 ny

nx

,4cot,4cot 21

221

21

221 PRRAPARR nnnnnn

cotcot

cotcot

1

1

mn

mn

mn

mn

n

n

n

n

n

n x

y

y

x

x

y

y

x

y

x

2

, 1111

nnn

nnn

ARRpApRpRppP

Linear intersection formula:

1nR1nA

mean pole prediction

nR

The mean pole using Ormsby LPF

LnLtkct

tkck

ktyktx

mty

mtx L

Lk

,...,2,2))((2

)2cos()2cos(

L n- filter length, - number of data,

cc Tt / - cutoff frequency, - cutoff period, yrTc 180415.3 Etc - cutoff frequency – roll off termination frequency.

tt yx , - pole coordinates data,

-0.10.00.10.20.30.4-0.1

0.0

0.1x

yarcsec

arcsec2003

1849

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 20000.0

0.1

0.2

0.3

0.4Radiusarcsec

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 20000.000

0.002

0.004

0.006

0.008Angular velocityarcsec/day

Corr. Coeff.1900-2003

0.8641950-2003

0.899

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

1920 1930 1940 1950 1960 1970 1980years

500

1500

2500

3500

per

iod

(day

s)

0.2

0.4

0.6

0.8

1.0

1 9 20 19 3 0 1 9 4 0 1 95 0 1 9 6 0 1 9 70 1 9 8 0

500

1500

2500

3500

1 02 03 04 05 06 0

3

6

9

yr m asrad ius

angular ve locity m as/day

3

6

9

0.0001

The FTBPF amplitude spectra of polar motion radius and angular velocityThe FTBPF amplitude spectra of polar motion radius and angular velocity

0 100 200 300 400 500 600period (days)

0

1

2

3

4

5mas radius

0 100 200 300 400 500 600period (days)

0.000.020.040.060.080.100.12

mas/ day angular velocity

b )

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

0.0005

m as/day

m as

0.5

1.0

1.5

0.5

1.0

1.5

yrrad ius

angular ve loc ity1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001

100

300

500

2

4

6

8

1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001

100

300

500

perio

d (d

ays)

0.020.040.060.080.100.12

1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001years

100

300

500

2

4

6

8

length o f po lar m otion pa th

1.5

1.0

m as

0.5

Autocovariance prediction

,0)(

1

1

n

n

z

zR

)0(ˆ

)(ˆ

)(1

1)(

1 nzz

kn

kkn

nzz

n c

zkcz

1,...,1,0,1

)(ˆ1

)(

nkforzz

nkc

kn

tktt

nzz

nkforzzn

kckn

tktt

nzz ,...,1,0,

1

1)(ˆ

1

)1(

min,)(ˆ)(ˆ)(1

2)1()(1

kn

k

nzz

nzzn kckczR

?1 nz

Let be stationary complex-valued time seriesnttz ,...,2,1,

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

1985 1987 1989 1991 1993 1995 1997 1999 2001 20030

10203040

1 98 4 1 986 19 88 1 99 0 1 992 1 99 4 1 996 19 98 2 00 0 2 002010203040

Autocovariance prediction

X

Y

1 98 5 1 987 1 98 9 1 991 19 93 1 99 5 1 997 1 99 9 2 001 20 03010203040

d

ays

in t

he

futu

re

0.00

0.01

0.02

0.03

0.04

1985 1987 1989 1991 1993 1995 1997 1999 2001 2003years

010203040

R

L

arcsec

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

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3

010203040

0.01

0.02

0.03

0.04

1985 1987 1989 1991 1993 1995 1997 1999 2001 20030

10203040

X

Y

1985 1987 1989 1991 1993 1995 1997 1999 2001 2003years

010203040

1 9 8 4 1 9 86 19 8 8 1 9 9 0 1 99 2 1 9 9 4 1 9 96 19 9 8 2 0 0 0 2 00 2010203040

d

ays

in t

he

futu

re

LS prediction - IERS SubBureau for Rapid Service and Prediction

Autocovariance prediction

X

Y

arcsec

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

0 50 100 150 200 250 300 350days in the future

0.000.010.020.030.040.050.060.070.080.090.100.110.120.130.14

A

R

xy

Larcsec

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

0 50 100 150 200 250 300 350days in the future

0.00

0.01

0.02

0.03

0.04

0.05

0.06

x

y

arcsec

USNO

autocovarianceprediction

xy

Prediction by combination of the LS and a stochastic methodPrediction by combination of the LS and a stochastic methodin the Cartesian pole coordinate systemin the Cartesian pole coordinate system

x, y LS extrapolation

residuals

Prediction ofx, y

LS extrapolation residuals

LS extrapolation of x, y

Prediction of x, y

AUTOCOVARIANCEARNN

x, y pole coordinates

data

x, y LS model

Prediction of x, y pole coordinates data by combination of the LS and a Prediction of x, y pole coordinates data by combination of the LS and a stochastic methodstochastic method in the polar coordinate systemin 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, ALS extrapolation

Prediction of R, A

AUTOCOVARIANCEAR

R – radius A – angular velocity

Prediction of x, y

mean pole + its LS prediction

Prediction of R, A

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

0 50 100 150 200 250 300 350days in the future

0.000.010.020.030.040.050.060.070.080.090.100.110.120.130.140.15

A

R

xy

L

arcsec

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)

0 50 100 150 200 250 300 350days in the future

0.00

0.01

0.02

0.03

arcsec

xy

xy

USNO

LS + AR

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

1 985 1987 198 9 1991 1993 19 95 1997 1999 2 001 2003

010203040

0.01

0.02

0.03

0.04

1985 1987 1989 1991 1993 1995 1997 1999 2001 20030

10203040

day

s in

th

e fu

ture

x

y

arcsecIERS Sub-bureau for Rapid Service and Predictions

1985 1 987 19 89 1991 1993 1995 1 997 19 99 2001 2003

010203040

0.00

0.01

0.02

0.03

0.04LS prediction + AR prediction of LS extrapolation residuals

1985 1987 1989 1991 1993 1995 1997 1999 2001 20030

10203040

d

ays

in t

he

futu

re x

y

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

0 50 100 150 200 250 300 350days in the future

0.00

0.01

0.02

0.03

arcsec

xy

xy

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

1 985 1987 198 9 1991 1993 19 95 1997 1999 2 001 2003

010203040

0.01

0.02

0.03

0.04

1985 1987 1989 1991 1993 1995 1997 1999 2001 20030

10203040

d

ays

in t

he

futu

re x

y

arcsec

I ER S Sub-bureau for R apid Service and Predictions

19 85 198 7 198 9 199 1 19 93 19 95 19 97 19 99 200 1 200 3

010203040

1985 1987 1989 1991 1993 1995 1997 1999 2001 2003years

010203040

d

ays

in t

he

futu

re

LS + NN

x

y

ConclusionsConclusions 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.