wiesław kosek 1,2 , agnieszka wnęk 1 , maria zbylut 1 , waldemar popiński 3
DESCRIPTION
Polarization of signals in Earth centre of mass time series observed by satellite techniques. Wiesław Kosek 1,2 , Agnieszka Wnęk 1 , Maria Zbylut 1 , Waldemar Popiński 3 1) Environmental Engineering and Land Surveying Department, University of Agriculture in Krakow, Poland - PowerPoint PPT PresentationTRANSCRIPT
Wiesław Kosek 1,2, Agnieszka Wnęk 1, Maria Zbylut 1, Waldemar Popiński 3
1) Environmental Engineering and Land Surveying Department, University of Agriculture in Krakow, Poland2) Space Research Centre, Polish Academy of Sciences, Warsaw, Poland3) Central Statistical Office of Poland, Warsaw, Poland
17th International Symposium on Earth Tides „Understand the Earth” 15-19 April, 2013, Warsaw, Poland
Polarization of signals in Earth centre of mass time series observed by satellite
techniques
The accuracy of the ITRF geocenter has a significant impact on the orbit determination and station coordinates accuracy
centre of figurecentre of mass
ITRF
GGOS Requirement (2020):<1 mm TRF
accuracy < 0.1 mm/yr TRF
stability
DATASLR weekly geocenter data computed by Astronomical
Institute of the University of Bern in 1994 – 2013. (Sosnica et al.. 2011)
http://www.bernese.unibe.ch/publist/2011/pres/ks_Geod_Woche.pdf
GNSS weekly combined solutions delivered by International GNSS Service (IGS) in years 1994 – 2013.
ftp://igs-rf.ign.fr/pub/sum/5-4_igs.sum
DORIS geocenter time series available at Crustal Dynamics Data Information System (CDDIS) from 1994.0 to 2013 (Willis et al., 2005)
ftp://cddis.gsfc.nasa.gov/pub/doris/products/geoc/
DORISSLR
GNSS
CoM data
Standard deviations
[mm]SLR x 3.4
y 3.3
z 6.2
GNSS x 3.0
y 4.4
z 5.7
DORIS x 5.5
y 6.9
z 24.8
...5,3,1r )(tx )(ty
1,...,1,0 nt 10 nm
,)),(ˆ(cos),(ˆ),(ˆ atatatxyxyxy
rr
),(ˆ),(ˆ),(ˆ),(ˆ atSatSatS yyxxxyxy at
),(ˆ),(ˆ/]),(ˆ),(ˆ[1
arg),(ˆ2/
2/
abtYabtXabtYabtXm
atm
mb
xy
WAVELET SPECTRO-TEMPORAL SEMBLANCE
, between and
, complex-valued time series is defined for as:
where:
- spectro-temporal coherence,
- spectro-temporal phase synchronization,
- the wavelet spectra and the wavelet cross spectrum of time series
The spectro-temporal semblance of the order
,2
,2
),(ˆ),(ˆ),(ˆ),(ˆ 2/
2/
2/
2/abtYatSabtXatS
m
mbyy
m
mbxx
)(),( tytx
,m/,...,n,m/m/t,nm 2112210
),(ˆ),(ˆ),(ˆ 2/
2/abtYabtXatS
m
mbxy
1,...,1,0,0 nba - dilation and translation parameters.
)/2(exp)/2()(1
),(ˆ
)/2(exp)/2()(1
),(ˆ
2/
12/
2/
12/
nbinayn
aabY
nbinaxn
aabX
n
n
n
n
)(),( yx 2/,12/,...,12/ nnn
)()( tytx
)(
)4/224exp()2/2exp(2)22/2exp()2exp(21)(
tttit
The wavelet transform coefficients of complex-valued time series are defined:
- Discrete Fourier Transforms of and time series
, - Continuous Fourier Transform of the modified Morlet wavelet function given by the time domain formula (Schmitz-Hübsch and Schuh 1999):
)(),( tytx
where
2,
2),(ˆ)(ˆ),(ˆ),(ˆ 2/
2/abXaSabtXatS
mn
mbxx
m
mbxx
0),(ˆ atpxx
)(txThe wavelet polarization and the mean wavelet polarization functions of complex-valued time series are defined as:
1),(ˆ atpxx1),(ˆ atpxx 1),(ˆ0 atpxx0),(ˆ1 atpxx
retrograde prograde
ellipticcircular circular
the shape of ellipse degenerates to a line
- the wavelet spectrum and the mean wavelet spectrum
)(ˆ)(ˆ)(ˆ)(ˆ
|)(|ˆ,),(ˆ),(ˆ),(ˆ),(ˆ
),(ˆaSaS
aSaSap
atSatSatSatSatp
xxxx
xxxxxx
xxxx
xxxxxx
WAVELET POLARIZATION
The mean wavelet spectra of Earth centre of mass complex-valued time series computed from SLR and GNSS observations
ZX
-420-320-220-120 -20 80 180 280 380
period (days)
0.0x10 0
1.0x10 8
2.0x10 8
3.0x10 8
-420-320-220-120 -20 80 180 280 380
period (days)
0.0x10 0
4.0x10 7
8.0x10 7
1.2x10 8
1.6x10 8
2.0x10 8
XY
GNSS
SLR
2),(ˆ)(ˆ abXaS
mn
mbxx
YZ
-420-320-220-120 -20 80 180 280 380
period (days)
0.0x10 0
1.0x10 8
2.0x10 8
3.0x10 8
4.0x10 8
The mean wavelet polarization functions in XY, YZ and ZX planes of complex-valued Earth centre of mass time series determined by satellite techniques
XY
YZ
ZX
SLR GNSS DORIS
Spectro-temporal wavelet polarization functions in XY plane of Earth centre of mass time series determined by SLR, GNSS and DORIS
techniques
1997 1999 2001 2003 2005 2007 2009 2011
100
200
300
400
1997 1999 2001 2003 2005 2007 2009 2011
100
200
300
400
per
iod
(d
ays)
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
1997 1999 2001 2003 2005 2007 2009 2011
years
100
200
300
400
SLR
G N SS
D O R IS
XY plane
prograde
retrograde
Spectro-temporal wavelet polarization functions in YZ plane of Earth centre of mass time series determined by SLR, GNSS and DORIS
techniques
SLR
G NSS
DO RIS
YZ plane
prograde
retrograde
1997 1999 2001 2003 2005 2007 2009 2011
years
100
200
300
400
1996 1998 2000 2002 2004 2006 2008 2010
100
200
300
400
per
iod
(d
ays)
1996 1998 2000 2002 2004 2006 2008 2010
100
200
300
400
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
Wavelet polarization functions in ZX plane of Earth centre of mass time series determined by SLR, GNSS and DORIS techniques
SLR
G NSS
DO RIS
ZX plane
prograde
retrograde
1997 1999 2001 2003 2005 2007 2009 2011
100
200
300
400
1997 1999 2001 2003 2005 2007 2009 2011
100
200
300
400
per
iod
(d
ays)
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
1996 1998 2000 2002 2004 2006 2008 2010 2012
years
100
200
300
400
The mean semblance functions in XY, YZ and ZX planes between Earth centre of mass time series determined by different techniques
SLR - GNSS
GNSS - DORIS
SLR - DORIS
Spectro-temporal semblance functions in XY equatorial plane between Earth centre of mass time series determined by different techniques
100
200
300
400
G NSS-SLR
G N SS-D O R IS
SLR-DO RIS
100
200
300
400
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.0
100
200
300
400
1996 1998 2000 2002 2004 2006 2008 2010
-400
-300
-200
-100
p
eri
od
(d
ay
s)
1996 1998 2000 2002 2004 2006 2008 2010
-400
-300
-200
-100
1996 1998 2000 2002 2004 2006 2008 2010
years
-400
-300
-200
-100
Spectro-temporal semblance in YZ plane between Earth centre of mass time series determined by different techniques
G NSS-SLR
G NSS-DO RIS
SLR-DO RIS
100
200
300
400
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.01996 1998 2000 2002 2004 2006 2008 2010
-400
-300
-200
-100
100
200
300
400
1996 1998 2000 2002 2004 2006 2008 2010
-400
-300
-200
-100
p
erio
d (
day
s)
100
200
300
400
1996 1998 2000 2002 2004 2006 2008 2010
years
-400
-300
-200
-100
Spectro-temporal semblance in ZX plane between Earth centre of mass time series determined by different techniques
G N SS-SLR
G NSS-DO RIS
SLR-DO RIS
100
200
300
400
-1.0
-0.8
-0.6
-0.4
-0.2
-0.0
0.2
0.4
0.6
0.8
1.01996 1998 2000 2002 2004 2006 2008 2010
-400
-300
-200
-100
1996 1998 2000 2002 2004 2006 2008 2010
-400
-300
-200
-100
p
erio
d (
day
s)
100
200
300
400
100
200
300
400
1996 1998 2000 2002 2004 2006 2008 2010
years
-400
-300
-200
-100
THE WAVELET BASED SEMBLANCE FILTERING
)(tx
,1
0,,
1
0,
)()(,)(,
)(
n
tkj
ykj
n
t
xkj
ttyStkj
txS
y
kj
x
kj
y
kj
x
kjkjxykj
SSSSeS,,,,,,
/)cos(function semblance
)(ty
1
0
112
12,,
)()(p
jj
j
jkkj
xkj
tStx
1
0
112
12,,
)()(p
jj
j
jkkjkjtSty y
DWT DWT
wavelet semblance filtering
threshold)cos(0threshold)cos(
,
,,,
kj
kjxkjx
kj ififS
S Thresholding of WT coefficients
threshold)cos(0threshold)cos(
,
,,,
kj
kjykj
kj ififS
S y
WT coefficients
The common oscillations in Earth centre of mass time series computed by the wavelet semblance filtering assuming threshold equal to 0.9
The model center of mass time series computed as the average of GNSS and SLR common oscillations composed of only 6 lower frequency components
ConclusionsThe most energetic oscillation in Earth centre of mass time series
determined by SLR, GNSS and DORIS techniques is the annual one with amplitude of the order of few millimeters.
The spectro-temporal wavelet semblance with application of the modified Morlet wavelet function enables computation of correlation coefficients between two complex-valued time series as a function of time and frequency. The highest positive semblance values occur in the equatorial xy plane for the retrograde annual oscillation in the GNSS and SLR data. The semblance functions between the GNSS and DORIS as well as the SLR -DORIS geocenter data in the annual frequency band are negative in the equatorial XY plane data.
The wavelet based semblance filtering with application of the Shannon wavelet function enables computation of a common signal in GNSS and SLR geocenter time series. This common signal enables determination of the smoothed model geocenter time series as the average of the GNSS and SLR time series reconstructed using lower frequency components.