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

*