w. kosek 1 , a . rzeszótko 1 , w . popiński 2
DESCRIPTION
Contribution of wide-band oscillations excited by the fluid excitation functions to the prediction errors of the pole coordinates data. W. Kosek 1 , A . Rzeszótko 1 , W . Popiński 2 1 Space Research Centre, Polish Academy of Sciences, Warsaw, Poland - PowerPoint PPT PresentationTRANSCRIPT
Contribution of wide-band oscillations Contribution of wide-band oscillations excited by the fluid excitation functions excited by the fluid excitation functions
to the prediction errors of the pole to the prediction errors of the pole coordinates datacoordinates data
W. Kosek1, A. Rzeszótko1 , W. Popiński2
1Space Research Centre, Polish Academy of Sciences, Warsaw, Poland 2Central Statistical Office, Warsaw, Poland
Journées "Systèmes de référence spatio-temporels"and X. Lohrmann-Kolloquium 22, 23, 24 September 2008 - Dresden, Germany
DATADATA
x, y pole coordinates data from the IERS: EOPC04_IAU2000.62-now (1962.0 - 2008.6), Δt = 1 day, http://hpiers.obspm.fr/iers/eop/eopc04_05/,
Equatorial components of atmospheric angular momentum from NCEP/NCAR, aam.ncep.reanalysis.* (1948 - 2008.6) Δt = 0.25 day, ftp://ftp.aer.com/pub/anon_collaborations/sba/,
Equatorial components of ocean angular momentum (mass + motion): 1) c20010701.oam (gross03.oam) (Jan. 1980 - Mar. 2002) Δt = 1 day, 2) ECCO_kf049f.oam (Mar. 2002 - Mar. 2006), Δt = 1 day, http://euler.jpl.nasa.gov/sbo/sbo_data.html,
Equatorial components of effective angular momentum function of the hydrology obtained by numerical integration of water storage data from NCEP: water_ncep_1979.dat, water_ncep_1980.dat, …, water_ncep_2004.dat, Δt = 1 day, ftp://ftp.csr.utexas.edu/pub/ggfc/water/NCEP.
x, y pole coordinates model data computed from fluid excitation functions
)()()( ttmtmich
)()()( tiytxtm
)(2
)(1
)( titt
Qi
Tchch 2
12daysTch 433 170Q
tittt
tititmttm chch
ch exp)()(2
exp)()(
Differential equation of polar motion:
- pole coordinates,
- equatorial excitation functions corresponding to AAM, OAM and HAM,
- complex-valued Chandler frequency, where and
Approximate solution of this equation in discrete time moments can be obtained using the trapezoidal rule of numerical integration:
THE MORLET WAVELET TRANSFORM COHERENCE
,)exp()()(||21),( 2/1 dibaxaabX
2/)2exp()2/exp()( 2 titt
,
),(ˆ),(ˆ
),(ˆ),(ˆ
),(ˆ22
M
Mb
M
Mb
M
Mbxy
abtYabtX
abtYabtXat
)2/)2(exp()( 2
The WT coefficients of complex-valued signal are defined as:
is the CFT of complex-valued Morlet wavelet function:
ba ,0where are dilation and translation parameters, respectively,
Spectro-temporal coherence between and time series is defined as:)(tx )(ty
where M is a positive integer and Δt is the sampling interval.
).()( txx and is the CFT of
)(tx
)12(
),(ˆ
Mt
aaterr xy
The MWT spectro-temporal coherence between IERS x, y pole coordinates data and x, y pole coordinates model data computed from AAM, OAM and HAM excitation functions
200
400
1965 1970 1975 1980 1985 1990 1995 2000 2005
-400
-200
200
400
1980 1985 1990 1995 2000 2005
-400
-200
p
erio
d (
day
s)
200
400
IERS, AAM
IERS, OAM
IERS, HAM
1980 1985 1990 1995 2000 2005
YEARS
-400
-200
x - iy
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
The MWT spectro-temporal coherence between IERS x, y pole coordinates data and x, y pole coordinates model data computed from AAM, AAM+OAM and AAM+OAM+HAM excitation functions
200
400
1965 1970 1975 1980 1985 1990 1995 2000 2005
-400
-200
IERS, AAM
IERS, AAM+OAM
IERS, AAM+OAM+HAM
x - iy
200
400
1980 1985 1990 1995 2000 2005
-400
-200
pe
rio
d (
da
ys)
200
400
1980 1985 1990 1995 2000 2005
YEARS
-400
-200
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Prediction of x, y pole coordinates data Prediction of x, y pole coordinates data by the LS+AR methodby the LS+AR method
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
(Chandler circle + annual and semiannual ellipses + linear trend)
LS extrapolation
LS+AR prediction errors of IERS x, y pole coordinates data and of x, y pole LS+AR prediction errors of IERS x, y pole coordinates data and of x, y pole coordinates model data computed from AAM, OAM and HAM excitation coordinates model data computed from AAM, OAM and HAM excitation
functionsfunctions
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
x (AAM)
x (IERS)
1980 1984 1988 1992 1996 2000 2004 20080
100
200
300
y (IERS)
1980 1984 1988 1992 1996 2000 2004 20080
100
200
300
y (AAM)
x (OAM)
arcsec
y (OAM)
1980 1984 1988 1992 1996 2000 20040
100
200
300
d
ays
in t
he
futu
re
1980 1984 1988 1992 1996 2000 20040
100
200
300
x (HAM) y (HAM)
1980 1984 1988 1992 1996 2000 2004
YEARS
0
100
200
300
1980 1984 1988 1992 1996 2000 2004
YEARS
0
100
200
300
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 (orange), OAM (blue) and HAM (green)
excitation functions
0 100 200 300days in the future
0.00
0.01
0.02
0.03
xarcsecIERS
AAM
OAM
HAM
0 100 200 300days in the future
0.00
0.01
0.02
0.03
yarcsec
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 (red) and AAM+OAM+HAM
(purple) excitation functions
0 100 200 300days in the future
0.00
0.01
0.02
0.03
xarcsecIERS
AAM+OAM
AAM+OAM+HAM
0 100 200 300days in the future
0.00
0.01
0.02
0.03
yarcsec
DISCRETE WAVELET TRANSFORM BAND PASS FILTER
,2,1,...,1,,1,...,1,0)()( 00
112
12,,
pnpjjjntfortStx
j
jkkjkjj
1
0,, )()(
n
tkjkj ttxS
)22/(2)( 2/, knntnt j
jj
kj
- the DWT coefficients,
The DWT j-th frequency component of the complex valued signal x(t) is given by:
,]/)2/(sin[
)1]/)2/(2cos[2](/)2/(2sin[]/)2/(exp[1)(nnt
nntnntnntintjj
j
nn jj /2)2/(
- discrete Shannon wavelets.
1
0
)()(p
jjtxtjx
Signal reconstruction:
20
0 pj 12,...,12,2 000 jjjk
,]/)2/(sin[
]/)2/(2sin[]/)2/(exp[1)(10
0 nntnntnntint
j
j
nn
jj /2)2/(
100
1,...,2,1 00 pjjj 12,...,12,2 111 jjjk
For fixed lowest frequency index and time index
For higher frequency index and time index
-0.040.000.04 j= 0arcsec
-0.040.000.04 j= 1
-0.040.000.04 j= 2
-0.040.000.04 j= 3
-0.300.000.30 j= 4
-0.040.000.04 j= 5
-0.020.000.02 j= 6
-0.010.000.01 j= 7
-0.010.000.01 j= 8
-0.010.000.01 j= 9
-0.010.000.01 j=10
40000 44000 48000 52000MJD
-0.010.000.01 j=11
1962 2008
The DWT frequency components of x pole coordinate data
Chandler + Annual
Semiannual
longer period
shorter period
The mean LS+AR prediction errors of IERS x, y pole coordinates data (black), and of x, y pole coordinates model data computed by summing the chosen DWTBPF components
0 50 100 150 200 250 300 350days in the future
0.00
0.01
0.02
0.03
0.04arcsec
y
0 50 100 150 200 250 300 350days in the future
0.00
0.01
0.02
0.03
0.04IERSarcsecCh + An + Sa
Ch + An + longer periodCh + An + shorter periodx
Ch + An
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 (red) excitation functions as
well as by summing the DWTBPF components corresponding to Chandler, annual and shorter period oscillations (green)
0 100 200 300days in the future
0.00
0.01
0.02
0.03
xarcsecIERS
AAM+OAM
Ch + An + shorter period
0 100 200 300days in the future
0.00
0.01
0.02
0.03
yarcsec
CONCLUSIONSCONCLUSIONS The contributions of atmospheric or ocean angular momentum excitation
functions to the mean prediction errors of x, y pole coordinates data from 1 to about 100 days in the future is similar and of the order of 60% of the total prediction error.
The contribution of ocean angular momentum excitation function to the mean prediction errors of x, y pole coordinates data for prediction lengths greater than 100 days becomes greater than the contribution of the atmospheric excitation function.
The contribution of the joint atmosphere and ocean angular momentum excitation to the mean prediction errors of x, y pole coordinates data is almost equal to the contribution of the sum of Chandler + annual and shorter period frequency components. Both contributions explain about 80÷90% of the total prediction error.
Big prediction errors of IERS x, y pole coordinates data in 1981-1982 and in 2006-2007 are mostly caused by wide-band ocean and atmospheric excitation, respectively.
The contribution of the hydrologic angular momentum excitation to the mean prediction errors of x, y pole coordinates data is negligible.
AcknowledgementsAcknowledgements
This paper was supported by the Polish Ministry of Education and Science, project No 8T12E 039 29 under the leadership of Dr. W. Kosek. The authors of this poster are also supported by the Organizers of Journées "Systemes de référence spatio-temporels" and X. Lohrmann-Kolloquium.
poster available: http://www.cbk.waw.pl/~kosek