Prognozowanie współrzędnych bieguna ziemskiego kombinacją … · 2016. 9. 30. · • Barrodale...
Transcript of Prognozowanie współrzędnych bieguna ziemskiego kombinacją … · 2016. 9. 30. · • Barrodale...
Prognozowanie współrzędnych bieguna
ziemskiego kombinacją metody najmniejszych kwadratów i podwójnej autoregresji
Wiesław Kosek
(1) Wydział Inżynierii Środowiska i Geodezji, Uniwersytet Rolniczy w Krakowie
(2) Centrum Badań Kosmicznych PAN, Warszawa
Seminarium
„Współczesne problemy podstawowych osnów geodezyjnych w Polsce”
Grybów, 14-16 września 2016 r.
contents
• Pole coordinates data and their analysis
• LS+AR prediction algorithm
• Analysis and forecast of time series of differences
between pole coordinates data and their
predictions.
• LS+2AR prediction algorithm
• Conclusions
DATA
• x, y from IERS: EOPC04_IAU2000.62-now
(1962.0 – 2016.5), Δt = 1 day, http://hpiers.obspm.fr/iers/eop/eopc04_05/,
• Long term earth orientation data EOP C01
IAU2000 (1890 – 2016.5), Δt = 0.05 years http://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
years
-500
-400
-300
-200
-100
0
100
200
300
400
500
pe
rio
d (
da
ys)
arcsec
FTBPF amplitude spectrum: x-iy lambda=0.0002
-0
0.05
0.1
0.15
0.2
0.25
Time variable amplitude spectrum of complex-valued pole coordinates data computed
by the Fourier transform band pass filter
Amplitudes and phases of the Chandler (green) and Annual (x-blue, y-red) oscillations computed
by combination of complex demodulation and the Fourier transform band pass filter
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
0
0.1
0.2
0.3 Ch x,y
An x
An y
Amplitude
arcsec
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010years
-160-140-120-100-80-60-40-20
020406080
100120140160
Ch x,y
An xAn y
Phaseo
LS+AR prediction
of x, y
x, y
Prediction of x, y by combination of the LS+AR method
x, y
LS residuals
AR prediction of x, y
residuals fit to last
850 days of these residuals
LS extrapolation
of x, y
AR LS
x, y LS model fit
to last 10 years of
PM data
The maps of differences between x,y pole coordinates data and their LS+AR predictions and time
series of these differences for 2 (purple) and 4 (green) weeks in the future.
1980 1985 1990 1995 2000 2005 2010 2015-0.02
-0.01
0
0.01
0.02
arcsec x
1980 1985 1990 1995 2000 2005 2010 2015
years
-0.02
-0.01
0
0.01
0.02
arcsec y
0
100
200
300
1980 1985 1990 1995 2000 2005 2010 20150
100
200
300
da
ys in
th
e f
utu
re
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08x
y
arcsec
Normalized Morlet Wavelet Transform (NMWT)
dibTxTbX )exp()()(2
1),(ˆ
where - translation (or time) parameter and
- dilation (or period) parameter,
- DFT or FFT of complex-valued time series
is the continuous FT of the complex-valued modified Morlet wavelet function,
- the decay parameter which controls the frequency resolution
bT
)(x
)(tx
222222
exp4
2exp
2
2exp)(
22)],(ˆ[)],(ˆ[),( TbXImTbXReTbA
The instantaneous NMWT amplitude:
200
400
600
1980 1985 1990 1995 2000 2005 2010 2015
-600
-400
-200
p
eri
od
(d
ays)
200
400
600
200
400
600
1980 1985 1990 1995 2000 2005 2010 2015
-600
-400
-200
1980 1985 1990 1995 2000 2005 2010 2015
years
-600
-400
-200
p
eri
od
(d
ays)
200
400
600
1980 1985 1990 1995 2000 2005 2010 2015
years
-600
-400
-200
0
5E-005
0.0001
0.00015
0.0002
0
0.0005
0.001
0.0015
0.002
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
arcsec arcsec
arcsecarcsec
1 day in the future
1 week in the future
2 weeks in the future
4 weeks in the future
The NMWT amplitudes as a function of periods T (σ=1) of the differences between
the x-iy pole coordinates data and their LS+AR predictions at 1 day as well as 1, 2
and 4 weeks in the future.
The mean NMWT amplitudes as a function of periods T (σ=1) of the differences between
the x-iy pole coordinates data and their LS+AR predictions at 1, 2, 4 and 8 weeks in the
future.
-800 -600 -400 -200 0 200 400 600 800
period (days)
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
2 weeks 1 week
4 weeks
arcsec
8 weeks
The mean FTBPF amplitude spectra (λ=0.0003) of the differences between the x-iy pole
coordinates data and their LS+AR predictions at 2, 4 and 8 weeks in the future.
-800 -700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700 800period (days)
0
0.001
0.002
0.003
0.004
0.005
0.006
2 weeks
4 weeks
8 weeks
arcsec
100-day autoregressive predictions (red) at different starting prediction
epochs of the differences between x-iy pole coordinates data and their
LS+AR predictions at 2 weeks in the future (grey). The length of these
differences to fit the autoregressive coefficients is equal to 6 years.
49000 50000 51000 52000 53000 54000 55000 56000 57000MJD
-0.02
-0.01
0
0.01
0.02
arcsec y
49000 50000 51000 52000 53000 54000 55000 56000 57000-0.02
-0.01
0
0.01
0.02
arcsec x
100-day autoregressive predictions (red) at different starting prediction
epochs of the differences between x-iy pole coordinates data and their
LS+AR predictions at 8 weeks in the future (grey). The length of these
differences to fit the autoregressive coefficients is equal to 6 years.
49000 50000 51000 52000 53000 54000 55000 56000 57000MJD
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
arcsec y
49000 50000 51000 52000 53000 54000 55000 56000 57000-0.03
-0.02
-0.01
0
0.01
0.02
0.03
arcsec x
The SDE of LS+AR predictions of x (blue) and y (red) pole coordinates data
(dashed line). The SDE of AR predictions of time series of the differences
between x,y pole coordinates data and their LS+AR predictions at 8 weeks in
the future.
0 20 40 60 80 100 120 140 160 180 200days in the future
0
0.01
0.02
0.03
arcsec
x pred diff 8 weeks
y pred diff 8 weeks
x
y
Mean AR prediction error of the differences between x,y data
and their LS+AR prediction for 8 weeks in the future is smaller
than mean LS+AR prediction error of x,y for prediction lengths
greater than L=70 days.
LS+AR prediction
of x, y
x,y pole
coordinates
Prediction of x, y by combination of the LS+2AR method
x,y LS residuals
AR prediction of x, y
LS residuals fit to last
850 days of these residuals
LS extrapolation
of x, y
AR LS
LS model fit to last 10
years of x,y data
DATA
last data point
first prediction point
LS+2AR PREDICTIONS
LS+AR PREDICTIONS
AR prediction of the differences
between x,y pole coordinate and their
LS+AR predictions at 100th day in the
future.
LS+AR prediction at 100th day in
the future
Differences between x,y
and their LS+AR predictions
at 8 weeks in the future
AR prediction of differences
between x,y and their
LS+AR predictions fit to
last 6 years of them AR
LS+2AR prediction
of x, y
+
=
Comparison of the mean LS+AR (thin line) and LS+2AR (dashed
line) prediction errors of x,y pole coordinates data (L=100 days).
0 20 40 60 80 100 120 140 160 180 200days in the future
0
0.01
0.02
0.03
arcsec
x
y
x
y
LS+AR
LS+2AR
The increase of LS+AR mean prediction errors with the prediction
length is caused by mismodelling of the irregular Chandler and annual
oscillations in this forecast model.
Time frequency analysis by the NMWT and FTBPF of the differences
between pole coordinates data and their LS+AR predictions for 1 day
as well as 1, 2, 4 and 8 weeks in the future show wideband signal
corresponding to the residual prograde Chandler and annual
oscillations.
The differences of pole coordinate and their LS+AR predictions at 8
weeks in the future are the most optimum to correct the LS+AR
predictions of pole coordinates data. The mean prediction errors of the
AR prediction of these differences become less than the mean LS+AR
prediction errors for prediction lengths greater than about 70 days.
Conclusions
Acknowledgments
Paper was supported by the Polish Ministry of Science and
Education, project 2012/05/B/ST10/02132 under the
leadership of Prof. A. Brzeziński.
CD+FTBPF and NMWT algorithms were prepared by Kosek
and Popiński.
AR coefficients estimation procedure of Barrodale and
Erickson (1980) was adopted to complex-valued time series
by Brzeziński (1994).
References
• Barrodale I. and Erickson R. E., 1980, Algorithms for least-squares linear
prediction and maximum entropy spectral analysis - Part II: Fortran
program, Geophysics, 45, 433-446.
• Brzeziński A., 1994, Algorithms for estimating maximum entropy
coefficients of the complex valued time series, Allgemeine Vermessungs-
Nachrichten, Heft 3/1994, pp.101-112, Herbert Wichman Verlag GmbH,
Heidelberg.
• Kosek W., 2002, Autocovariance prediction of complex-valued polar
motion time series, Advances of Space Research, 30, 375-380.
*
Comparison of 100-day autoregressive (red) and autocovariance (blue)
predictions at different starting prediction epochs of the differences between
x-iy pole coordinates data and their LS+AR predictions at 8 weeks in the
future. The lengths of these differences to fit the autoregressive coefficients
and to fit the autocovariance prediction model are equal to 6 and 18 years,
respectively.
49000 50000 51000 52000 53000 54000 55000 56000 57000-0.03
-0.02
-0.01
0
0.01
0.02
0.03
arcsec x
49000 50000 51000 52000 53000 54000 55000 56000 57000MJD
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
arcsec y
*
Autoregressive prediction
tMtMttt nxaxaxax ...2211
Autoregressive order:
Akaike godness-of-fit criterion:
MMon cacacacM ˆˆ...ˆˆˆ)(ˆ2211
2
min1
1)()( 2
Mn
MnMMP n
MoMM
Mo
Mo
M c
c
c
ccc
ccc
ccc
a
a
a
ˆ
.
ˆ
ˆ
ˆ.ˆˆ
....
ˆ.ˆˆ
ˆ.ˆˆ
ˆ
.
ˆ
ˆ
2
1
1
21
21
11
2
1
Autoregressive coefficients:
11211ˆ...ˆˆˆ
MnMnnn xaxaxax
22112ˆ...ˆˆˆˆ
MnMnnn xaxaxax
LMnMLnLnLn xaxaxax ...ˆˆˆˆˆ2211
where
1,...,1,0,1
ˆ1
nkforxxn
ckn
t
kttk
*
200
400
600
1980 1985 1990 1995 2000 2005 2010 2015
-600
-400
-200
pe
rio
d (
da
ys)
200
400
600
200
400
600
1980 1985 1990 1995 2000 2005 2010 2015
-600
-400
-200
1980 1985 1990 1995 2000 2005 2010 2015
years
-600
-400
-200
p
eri
od
(d
ays)
200
400
600
1980 1985 1990 1995 2000 2005 2010 2015
years
-600
-400
-200
0
5E-005
0.0001
0.00015
0.0002
0
0.0005
0.001
0.0015
0.002
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
arcsec arcsec
arcsecarcsec
1 day in the future
1 week in the future
2 weeks in the future
4 weeks in the future
200
400
600
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
arcsec
1980 1985 1990 1995 2000 2005 2010 2015
years
-600
-400
-200
p
eri
od
(d
ays)
8 weeks in the future
The NMWT amplitudes as a function of periods T (σ=1) of the differences between the x-iy pole coordinates data and
their LS+AR predictions at 1 day as well as 1, 2, 4 and 8 weeks in the future.
*
The differences between the x,y pole coordinates data and their LS+AR predictions for 1 day (blue)
and 1 week (pink) in the future.
1980 1985 1990 1995 2000 2005 2010 2015
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
arcsec
x
1980 1985 1990 1995 2000 2005 2010 2015
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
arcsec
y
*
100-day autoregressive predictions (red) at different starting prediction
epochs of the differences between x-iy pole coordinates data and their
LS+AR predictions at 4 weeks in the future (grey). The length of these
differences to fit the autoregressive coefficients is equal to 6 years.
49000 50000 51000 52000 53000 54000 55000 56000 57000MJD
-0.02
-0.01
0
0.01
0.02
arcsec y
49000 50000 51000 52000 53000 54000 55000 56000 57000-0.02
-0.01
0
0.01
0.02
arcsec x
*