Forecasting of the Earth orientation Forecasting of the Earth orientation parameters – comparison of parameters – comparison of

different algorithmsdifferent 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.

Prediction errors of EOP data and their ratio to their determination errors in 2000

Days in the future 1 7 20 40 80 160 320

x, y [mas] 0.5 2.7 6.3 11 17 25 32

UT1-UTC [ms] 0.12 0.7 3.6 6.9 13 32 67

Ratio: prediction to determination errors

x, y ~7 ~36 ~85 ~140 ~230 ~340 ~430

UT1 ~10 ~58 ~300 ~580 ~1100 ~2700 ~5600

YEARS 1976 1980 1984 1988 1992 1996 2000 2004x [mas] 16.3 2.6 0.72 0.53 0.29 0.12 0.074 0.058

y [mas] 14.3 1.5 0.60 0.47 0.29 0.15 0.074 0.060

UT1 [ms] 0.406 0.238 0.069 0.044 0.016 0.010 0.012 0.006

Determination errors of EOPC04 data in 1976-2004

~2.8 mm~1.8 mm

Data 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

-0.3-0.2-0.10.00.10.20.3 x

arcsec

-0.10.00.10.20.30.40.5 y

1965 1970 1975 1980 1985 1990 1995 2000 2005YEARS

-0.0010.0000.0010.0020.0030.004

s R

IERS

Prediction techniques1) 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 order2) 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+ARPrediction of x, y data by combination of the LS+AR

x, y LS residuals

Prediction of

x, y LS residuals

x, yLS extrapolation

Prediction of

x, y

AR prediction

x, y x, y LS model

LS extrapolation

Autoregressive method (AR)

pntzazazeeE tpptptttp ,...,2,1,ˆ...ˆˆ},|ˆ{|ˆ 1122

}{ˆ kttk zzEc

 

Autoregressive order: min1

1ˆ)( 2

pn

pnpAIC p

0

.

0

ˆ

ˆ

.

ˆ

1

ˆ.ˆˆ

....

ˆ.ˆˆ

ˆ.ˆˆ 2

1

21

21

11 p

popp

po

po

a

a

ccc

ccc

ccc

Autoregressive coefficients:

are computed from autocovariance estimate :

tttpnpnnn iyxzwherezazazaz 11211 ˆ...ˆˆˆ

prar ,...,2,1,ˆ

LS and LS+AR predictionLS and LS+AR prediction errors errors of x data of x data

0

200

0

200

0

200

5yr

10yr

15yr

0

200

d

ays

in t

he

futu

re

0

200

1983 1987 1991 1995 1999 2003 2007

YEARS

0

200

0

0.04

0.08

0.12

0.16

0.2

x, LS (433, 365, 182)

x, LS+AR (433, 365, 182) (AR: 850d)

5yr

10yr

15yr

arcsec

LS and LS+AR prediction errors of y dataLS and LS+AR prediction errors of y data

0

200

0

200

0

200

d

ay

s i

n t

he

fu

ture

0

200

1983 1987 1991 1995 1999 2003 2007

YEARS

0

200

y, LS (434,365,182)

y, LS+AR (434,365,182) (AR: 850d)

5yr

10yr

15yr

5yr

10yr

15yr

0

200

0

0.04

0.08

0.12

0.16

0.2arcsec

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)

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

0.07

5yr

10yr15yr

arcsec y

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

0.07

5yr

10yr

15yr

arcsec x

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

0 100 200 300 400 500days in the future

01234567

p d

t

x , yyears

0 100 200 300 400 500days in the future

01234567

p d

t

years

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

0 50 100 150 200 250 300 350 400days in the future

0.00

0.01

0.02

0.03

0.04arcsec

A I Cx

em p

0 50 100 150 200 250 300 350 400days in the future

0.00

0.01

0.02

0.03

0.04arcsec

A I Cem p

y

Prediction of x, y Prediction of x, y data data by by DWT+ACDWT+AC in polar coordinate system in polar coordinate system

ntytyxtxtR mm ,...,2,1,22

x, y

R(ω1), R(ω2) , … , R(ωp)

AC

R – radius A – angular velocity

LS extrapolation

of xm, ym

Prediction Rn+1, An+1

A(ω1), A(ω2), … , A(ωp)

Rn+1(ω1) + Rn+1(ω2) + … + Rn+1(ωp)

An+1(ω1) + An+1(ω2) + … + An+1(ωp)

LPFmean pole

xm, ym

LS

nttytytxtxtA ,...,3,2,2

12

1

xn, yn

Prediction

xn+1, yn+1

DWT BPF

prediction

1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 20000.0

0.1

0.2

0.3

0.4arcsec

R

1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 20000.0000.0020.0040.0060.0080.010 A

arcsec/day

Mean pole, radius and angular velocity-0 .10.00.10.20.30.4

-0.1

0.0

0.1

2004.3

1849

arcsec

arcsec

y

x2007

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

54 predictions

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

0

20

40

60

80

LS+AR (AIC)

DW T+ACLS+AR (em p)

mas

y

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

0

20

40

60

80

LS+AR (em p)LS+AR (AIC)

DW T+ACmasx

0 5 10 15 20 25 30days in the future

0

2

4

6

8

10

12 LS+AR (AIC)DW T+AC

LS+AR (em p)

masx

0 5 10 15 20 25 30days in the future

0

2

4

6

8

10

12

LS+AR (AIC)

DW T+AC

LS+AR (em p)

m sy

Δ-ΔR(ω1) + Δ-ΔR(ω2) + … + Δ-ΔR(ωp)Prediction of

Δ-ΔR

Δ-ΔR Δ-ΔR(ω1), Δ-ΔR(ω2),…, Δ-ΔR(ωp)

UT1-UTC

AC

Prediction of Δ and UT1-UTC by DWT+AC

Prediction of

UT1-TAIPrediction of

UT1-UTC

diff UT1-TAIΔ

Prediction of

Δ int

Prediction

DWT BPF

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

-0.00080-0.00040 1 3

s

-0.001200.000000.00120

1 2

-0.000220.000000.00022

1 1

-0.000200.000000.00020

1 0

-0.000500.000000.00050

8-0.000400.000000.00040

9

-0.000500.000000.00050

7

-0.000500.000000.00050

6

-0.000500.000000.00050

5

-0.000400.000000.00040

4

-0.000300.000000.00030

3

-0.000200.000000.00020

2

1986 1989 1992 1995 1998 2001 2004 2007-0.000080.000000.00008

1

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

54 predictions

0 5 10 15 20 25 30days in the future

0

1

2

3

4

5UT1- UTC

m s

Gam bis

DW T+AC

Gross

0 5 10 15 20 25 30days in the future

0 . 0

0 . 1

0 . 2

0 . 3ms/day

LS+AR (em p)

DW T+AC

Gross

Gam bis

prAr ,...,2,1,ˆ

min)log()12

1(2

]ˆ)13det[(log)(

pnpn

pCpnpSBC

nttAAMtRYt ,..,2,1

3

)(

C

111ˆ...ˆ

ptptt YAYAY

Multidimensional prediction

- Estimates of Autoregression matrices,

- Estimate of residual covariance matrix.

- autoregressive order:p

ε(Δ-ΔR)residuals

Δ-ΔR LS

extrapolation Prediction

of Δ-ΔRPrediction

of Δ-ΔR

Δ-ΔR Δ-ΔR LS

model

LS

εAAMχ3residuals

AR

AAMχ3 AAMχ3LS 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 Comparison of LS, LS+AR and LS+MAR prediction errors of UT1-UTC and Δ Δ datadata

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

0

20

40

60 UT1-UTCm s LSLS+ARLS+MAR

100200300

1992 1994 1996 1998 2000 2002 2004 2006

YEARS

100200300

LS

LS+MAR

LS+AR

020406080100120140160180

msUT1-UTC

100200300

da

ys

in

th

e f

utu

re

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

0.00

0.10

0.20

0.30

ms/day L SLS+ARLS+MAR

CONCLUSIONSCONCLUSIONS 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