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.
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