Future improvements in EOP prediction

Future improvements in EOP Future improvements in EOP predictionprediction

Wiesław Kosek

Space Research Centre, Polish Academy of Sciences, Warsaw, Poland

Geodesy for Planet Earth, Buenos Aires , Aug. 31 – Sep. 4, 2009

Summary:Summary: - introduction- introduction - input data - input data - EOP prediction algorithms- EOP prediction algorithms - EOPPCC results- EOPPCC results - possible causes of EOP prediction errors- possible causes of EOP prediction errors - prediction of PM by Kalman filter- prediction of PM by Kalman filter - MAR prediction of UT1-UTC- MAR prediction of UT1-UTC - application of the wavelet transform filter- application of the wavelet transform filter - conclusionsconclusions

EOP mean prediction errors and their ratio to determination errors in 2008

Days in the future 1 7 20 40 80x, y [mas]

UT1-UTC [ms]

0.5

0.12

2.7

0.7

6.3

3.6

11

6.9

17

13

prediction to determination errors ratio x, y _________________________UT1-UTC

~40

~24

~200

~140

~500

~720

~900

~1400

~1400

~2600

Determination errors of x, y and UT1-UTC (EOPC04_IAU2000.62-now) data in 1968-2008

~3÷4 mm

YEARS 1968 1973 1978 1983 1988 1993 1998 2003 2008x [mas] 20.0 15.0 15.0 2.05 0.959 0.232 0.105 0.066 0.011

y [mas] 20.0 15.0 15.0 2.05 0.926 0.192 0.106 0.067 0.014UT1 [ms] 1.50 1.90 1.90 0.400 0.021 0.009 0.007 0.006 0.005

Future EOP data are neededFuture EOP data are needed to compute real-time to compute real-time transformation between the celestial and terrestrial reference transformation between the celestial and terrestrial reference frames. This transformation is important for the NASA frames. This transformation is important for the NASA Deep Deep Space NetworkSpace Network, which is an international network of , which is an international network of antennas that supports: antennas that supports: - interplanetary spacecraft missions, - interplanetary spacecraft missions, - radio and radar astronomy observations, - radio and radar astronomy observations, - selected Earth-orbiting missions.- selected Earth-orbiting missions.

DATA

x, y, UT1-UTC and Δ data from the IERS: EOPC04_IAU2000.62-now (1962 - 2009.6), Δt = 1 day, http://hpiers.obspm.fr/iers/eop/eopc04_05/,

Equatorial and axial components of atmospheric angular momentum from NCEP/NCAR, aam.ncep.reanalysis.* (1948 - 2009.3) Δt = 0.25 day, ftp://ftp.aer.com/pub/anon_collaborations/sba/,

Equatorial components of ocean angular momentum:

c20010701.oam (Jan. 1980 - Mar. 2002) Δt = 1 day, ECCO_kf066b.oam (Jan. 1993 - Dec. 2008), Δt = 1 day, http://euler.jpl.nasa.gov/sbo/sbo_data.html,

Prediction of

x, y

x, y

Prediction of x, y by combination of the LS+AR method

x, yLS residuals

AR prediction ofx, y residuals

LS extrapolation of x, y

ARLS

x, yLS model

Prediction of Δ- δΔ

Δ- δΔ

UT1-UTC

Prediction of UT1-UTC by combination of the LS+AR method

Prediction of

UT1-TAIPrediction of

UT1-UTC

diff UT1-TAIΔ

Prediction of

Δ int

Δ- δΔLS residuals

AR prediction ofΔ- δΔ residuals

LS extrapolation of Δ- δΔ

ARLS

Δ- δΔLS model

Prediction of Δ- δΔ

Δ- δΔ

UT1-UTC

Prediction of UT1-UTC by combination of the DWT+AC method

Prediction of

UT1-TAIPrediction of

UT1-UTC

diff UT1-TAIΔ

Prediction of

Δ int

Δ-δΔ(ω1) + Δ-δΔ(ω2) + … + Δ-δΔ(ωp)

Δ-δΔ(ω1), Δ-δΔ(ω2),…, Δ-δΔ(ωp)

AC

DWT BPF

AC AC

Prediction errors of x, y pole coordinates data computed by the LS and LS+AR methods

arcsec

0

100

200

300

LS (4yr)

0

100

200

300

LS (6yr)

0

100

200

300

day

s in

th

e fu

ture LS (10yr)

0

100

200

300

0

100

200

300

0

100

200

300

1981 1985 1989 1993 1997 2001 2005 2009

years

0

100

200

300

LS (10yr) + AR (850d)

1981 1985 1989 1993 1997 2001 2005 2009

years

0

100

200

300

x y

0

100

200

300

LS (4yr) + AR (850d)

0

100

200

300

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Mean prediction errors of x (thin line), y (dashed line) pole coordinates data computed by the LS and LS+AR methods in 1984-2009

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

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

LS (10yr)

LS (6yr)

LS (4yr)

LS (10yr) + AR (850d)

arcsec

LS (4yr) + AR (850d)

Prediction errors of UT1-UTC data computed by the LS+AR method

0

100

200

300

0

100

200

300

da

ys

in t

he

fru

ture

1982 1985 1988 1991 1994 1997 2000 2003 2006 20090

100

200

300

0

50

100

150

200

250

3000

100

200

300

LS (4yr) + AR(1.5yr)

LS (5yr) + AR (1.5yr)

LS (7yr) + AR (1.5yr)

LS (10yr) + AR (1.5yr)

ms

Mean prediction errors of UT1-UTC data computed by the LS+AR method in 1984-2009

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

0

20

40

60

80

100

120

140

LS (10yr) + AR (1.5yr)

LS (5yr) + AR (1.5yr)

m s

LS (7yr) + AR (1.5yr)LS (15yr) + AR (1.5yr)

The chosen MAE of pole coordinates data from the EOPPCC The chosen MAE of pole coordinates data from the EOPPCC (Kalarus et al., prepared to (Kalarus et al., prepared to J. GeodesyJ. Geodesy))

0 5 10 15 20 25 30days in the future

0

2

4

6

8

10

12

Kosek (LS+AR)Zotov (AR)Kalarus (LS+AR)IERS (LS+AR filtering)

xmas

Kumakshev (SA+LS)Zotov (NN)


0

2

4

6

8

10

12 ymas

IERS (LS+AR filtering)

Zotov (AR)Kosek (LS+AR)

Kalarus (LS+AR)

Kumakshev (SA+LS)

Zotov (NN)

The chosen MAE of UT1-UTC and The chosen MAE of UT1-UTC and ΔΔ data from the EOPPCC data from the EOPPCC (Kalarus et al., prepared to (Kalarus et al., prepared to J. GeodesyJ. Geodesy))


0 . 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5

3 . 0

3 . 5

4 . 0

4 . 5

5 . 0

5 . 5

6 . 0

Gross (Kalman filter)

IERS (LS+AR filtering)

Kosek (DWT+AC)

Zotov (AR)

m s

UT1-UTC


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Gross (Kalman filter)

Kosek (DWT+AC)Kalarus (LS+AR)

IERS (LS+AR filtering)v

ms/day

Zotov (AR)

1900 1920 1940 1960 1980 2000

0.04

0.08

0.12

0.16

0.20

0.24arcsec

1900 1920 1940 1960 1980 2000years

0.00

0.01

0.02

1900 1920 1940 1960 1980 2000years

-240-200-160-120

-80-40

04080

120160200240

o

Amplitudes

Phases

Chandler

Annual

Semi-annual

Chandler

Annual

Semi-annual

Amplitudes and phases of the most energetic oscillations in x, y pole coordinates data

bold line – progradethin line - retrograde

Amplitudes

Phases

AnnualSemi-annual

Annual

Semi-annual

Amplitudes and phases of the most energetic oscillations in Δ-δΔ data

1965 1970 1975 1980 1985 1990 1995 2000 20050.00010

0.00012

0.00014

0.00016

0.00018

0.00020

0.00022s

1965 1970 1975 1980 1985 1990 1995 2000 2005-10

-8-6-4-202468

o

x, y pole coordinates model data computed from fluid excitation functions

)()()( ttmtmich

)()()( tiytxtm

)(2

)(1

)( titt

Qi

Tchch 2

12daysTch 433 170Q

Differential equation of polar motion:

- pole coordinates,

- equatorial fluid excitation functions (AAM, OAM),

- complex-valued Chandler frequency, where and is the quality factor

tittt

tititmttm chch

ch exp)()(2

exp)()(

Approximate solution of this equation in discrete time moments can be obtained using the trapezoidal rule of numerical integration:

LS+AR prediction errors of IERS x, y pole coordinates data and of x, y pole coordinates model data computed from AAM, OAM and AAM+OAM excitation

functions

x (AAM)

x (IERS) y (IERS)

y (AAM)

x (OAM)

arcsec

y (OAM)

1980 1984 1988 1992 1996 2000 2004 20080

100

200

300

1980 1984 1988 1992 1996 2000 2004 20080

100

200

300

0

0.02

0.04

0.06

0.08

0.1

1980 1984 1988 1992 1996 2000 2004 20080

100

200

300

1980 1984 1988 1992 1996 2000 2004 20080

100

200

300

1980 1984 1988 1992 1996 2000 2004 20080

100

200

300

day

s in

th

e fu

ture

1980 1984 1988 1992 1996 2000 2004 20080

100

200

300

1980 1984 1988 1992 1996 2000 2004 2008

years

0

100

200

300

1980 1984 1988 1992 1996 2000 2004 2008

years

0

100

200

300

x (AAM+OAM) y (AAM+OAM)

The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y pole coordinates model data computed from AAM, OAM and AAM+OAM excitation

functions

0 100 200 300days in the future

0.00

0.01

0.02

0.03

yarcsec IERS

AAM+OAM

OAM

AAM

0 100 200 300days in the future

0.00

0.01

0.02

0.03

xarcsec

x, y pole coordinates data prediction by the Kalman filter

TtututttytxtX )](),(),(),(),(),( 2121[)( TkkkykxkZ ])(),(),(),([)( 21

The linear state equation (Gelb 1974):

12

21

12

21

0000

0000

0000

0000

2/2/2/

2/2/2/

bb

bb

aa

aa

QFFQFFQFF

FQFFQFFQF

cccccc

cccccc

F

u

u

a

aT

q

q

q

qEww

00000

00000

00000

00000

000000

000000

100000

010000

001000

000100

000000

000000

G

,435/1cF 100Q

2,1,2,1 bbaa

1w

2w 1uw

2uw

w

- state vector

- observation vector

- constant coefficient matrix,

- zero mean excitation process satisfying:

pole coordinates

equatorial excitationfunctions

residual excitationfunctions

- constant coefficients

variances of white noise processes

1ˆˆ

kXkX

tFexp.

1const

kt

ktt

TkX kukukkkykx )](),(),(),(),(),( 2121[ˆ

prediction of the state vector:

)()()(,)()( tvtHXtZGwtFXdttdX

Prediction errors of x, y pole coordinates computed by Kalman filter and LS+AR method


0.00

0.01

0.02

0.03

0.04

LS + AR

KALMAN(AAM+OAM)

x

y

arcsec

x

y

Kalman filter (AAM + OAM)

1020304050

1985 1989 1993 1997 2001 2005 2009

1020304050

1020304050

d

ays

in t

he

fu

ture

LS + AR

1985 1989 1993 1997 2001 2005 2009

YEARS

1020304050

x

y

x

y

arcsec

00.010.020.030.040.050.060.070.08

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 Δ-ΔR data by LS+AR and LS+MAR algorithms (Niedzielski and Kosek, J. Geodes 2008)

MAR prediction

ε(Δ-ΔR)

AR prediction

ε(Δ-ΔR)

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

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

The frequency components of x (black), y (blue) pole coordinates data computed by the Shannon wavelet decomposition

Ch+An

Sa

0.0000.2000.400 0

arcsec

-0.0400.0000.040 1

-0.0400.0000.040 2

-0.0400.0000.040 3

-0.2000.0000.200 4

-0.0200.0000.020 5

-0.0100.0000.010 6

-0.0070.0000.007 7

-0.0030.0000.003 8

-0.0020.0000.002 9

- 0 . 0 0 20 . 0 0 00 . 0 0 2 1 0

1986 1989 1992 1995 1998 2001 2004 2007 2010

-0.0010.0000.001 11

shorterperiod

longerperiod

The mean LS+AR prediction errors of IERS x, y pole coordinates data, and x, y pole coordinates model data computed by summing the chosen DWTBPF components


0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

arcsec y


0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

arcsec x IERS

Ch + An + short periodCh + An + long periodsum_0^9 freq.comp.

The frequency components of Δ-δΔ data with indices i=1,...,13, computed by the Meyer wavelet decomposition

-0.00080-0.00040 13s

- 0 . 0 0 1 2 00 . 0 0 0 0 00 . 0 0 1 2 0

1 2

- 0 . 0 0 0 2 20 . 0 0 0 0 00 . 0 0 0 2 2

1 1

- 0 . 0 0 0 2 00 . 0 0 0 0 00 . 0 0 0 2 0

1 0

-0.000400.000000.00040

9

-0.000500.000000.00050

8

-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 2010-0.000080.000000.00008

1

An

Sa

longerperiod

shorterperiod

The mean LS+AR prediction errors of IERS UT1-UTC data, and UT1-UTC model data computed by summing the chosen DWTBPF frequency components

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30days in the future

0

1

2

3

4

5UT1-UTC

An + Sa + shorter period

An + Sa + longer period

m s

sum_2^13 freq.comp.

CONCLUSIONSCONCLUSIONS The influence of variable amplitudes and phases of the

most energetic oscillations in EOP data on their short term prediction errors is negligible.

Short term prediction errors of pole coordinates data are caused by wideband short period oscillations in these data. Some big prediction errors of pole coordinates data in 1981-82 are caused by wideband oscillations in ocean excitation functions and in 2006-07 are caused by wideband oscillations in joint atmospheric-ocean excitation functions.

Short term prediction errors of UT1-UTC are caused by short period wideband oscillations in these data.

Recommended prediction method for pole coordinates data is the combination of the least squares and autoregressive prediction.

Recommended prediction method for UT1-UTC data is the Kalman filter.

Longer term variations of UT1-UTC data can be predicted successfully by combination of the LS and multivariate autoregressive method.

To reduced short term EOP prediction errors Wavelet transform low pass filter can be used.

Thank You

AcknowledgementsAcknowledgements

The research was financed by Polish Ministry of Science and Education through the grant no. N N526 160136 under leadership of Dr Tomasz Niedzielski.

Future improvements in EOP prediction

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