Earth's pole coordinates determined from Lageos-1/2 laser ranging
The Earth's pole coordinates, obtained from satellite laser ranging data of Lageos-1 and Lageos-2 are presented. The procedure, used models and algorithms are described. Results are compared with similar series from IERS database.
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| Zitieren: | Earth's pole coordinates determined from Lageos-1/2 laser ranging / V.P. Zhaborovskyy, V.Ya. Choliy // Advances in Astronomy and Space Physics. — 2014. — Т. 4., вип. 1-2. — С. 54-57. — Бібліогр.: 8 назв. — англ. |
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Zhaborovskyy, V.P. Choliy, V.Ya. 2017-06-09T20:19:06Z 2017-06-09T20:19:06Z 2014 Earth's pole coordinates determined from Lageos-1/2 laser ranging / V.P. Zhaborovskyy, V.Ya. Choliy // Advances in Astronomy and Space Physics. — 2014. — Т. 4., вип. 1-2. — С. 54-57. — Бібліогр.: 8 назв. — англ. 2227-1481 DOI: 10.17721/2227-1481.4.54-57 https://nasplib.isofts.kiev.ua/handle/123456789/119816 The Earth's pole coordinates, obtained from satellite laser ranging data of Lageos-1 and Lageos-2 are presented. The procedure, used models and algorithms are described. Results are compared with similar series from IERS database. en Головна астрономічна обсерваторія НАН України Advances in Astronomy and Space Physics Earth's pole coordinates determined from Lageos-1/2 laser ranging Article published earlier |
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Earth's pole coordinates determined from Lageos-1/2 laser ranging |
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Earth's pole coordinates determined from Lageos-1/2 laser ranging Zhaborovskyy, V.P. Choliy, V.Ya. |
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Earth's pole coordinates determined from Lageos-1/2 laser ranging |
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Earth's pole coordinates determined from Lageos-1/2 laser ranging |
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Earth's pole coordinates determined from Lageos-1/2 laser ranging |
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Earth's pole coordinates determined from Lageos-1/2 laser ranging |
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earth's pole coordinates determined from lageos-1/2 laser ranging |
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Zhaborovskyy, V.P. Choliy, V.Ya. |
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Zhaborovskyy, V.P. Choliy, V.Ya. |
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2014 |
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English |
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Advances in Astronomy and Space Physics |
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Головна астрономічна обсерваторія НАН України |
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Article |
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The Earth's pole coordinates, obtained from satellite laser ranging data of Lageos-1 and Lageos-2 are presented. The procedure, used models and algorithms are described. Results are compared with similar series from IERS database.
|
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2227-1481 |
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Earth's pole coordinates determined from Lageos-1/2 laser ranging / V.P. Zhaborovskyy, V.Ya. Choliy // Advances in Astronomy and Space Physics. — 2014. — Т. 4., вип. 1-2. — С. 54-57. — Бібліогр.: 8 назв. — англ. |
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AT zhaborovskyyvp earthspolecoordinatesdeterminedfromlageos12laserranging AT choliyvya earthspolecoordinatesdeterminedfromlageos12laserranging |
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2025-11-24T04:21:13Z |
| last_indexed |
2025-11-24T04:21:13Z |
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1850438821444845568 |
| fulltext |
Earth's pole coordinates determined
from Lageos-1/2 laser ranging
V.P. Zhaborovskyy1∗, V.Ya.Choliy1,2
Advances in Astronomy and Space Physics, 4, 54-57 (2014)
© V.P. Zhaborovskyy, V.Ya.Choliy, 2014
1Main Astronomical Observatory of NAS of Ukraine, Akademika Zabolotnoho str., 27, 03680 Kyiv, Ukraine
2Taras Shevchenko National University of Kyiv, Glushkova ave., 4, 03127, Kyiv, Ukraine
The Earth's pole coordinates, obtained from satellite laser ranging data of Lageos-1 and Lageos-2 are presented.
The procedure, used models and algorithms are described. Results are compared with similar series from IERS
database.
Key words: methods: data analysis, reference system, geodesy
introduction
During the past 40 years the precision of Satellite
Laser Ranging (SLR) has changed dramatically. To-
gether with greater computational capabilities and
better algorithms for data storage, searching, and
analysis, the overall precision of the geodynam-
ics and geodetic tasks was substantially improved.
Among the methods substantially improved, the de-
termination of the Earth' pole motion and irregu-
larities of Earth' spin velocity are the central ones.
The Lageos-1 (1976) and Lageos-2 (1992) satellites
were launched speci�cally to accomplish these tasks.
These satellites are modelled as mathematical point-
mass. They are spherically shaped, and their orbits
were designed to diminish the in�uence of di�erent
poorly-modelled accelerations, as to allow relatively
e�ortless laser ranging. The processing of the ob-
servations allows the International Earth Rotation
Service to create and distribute o�cial time correc-
tions and pole coordinate series for users worldwide.
Such a result is not possible without extensive inter-
national collaboration and independent data analy-
sis centres capable of creating independent solutions.
These are the essential components of the work co-
ordinated by IERS.
Since 1987, a data analysis centre of satellite
laser ranging (SLR) has been in operation on a per-
manent basis at the Main Astronomical Observa-
tory of NAS . The previously used software pack-
age KyivGeodynamics is outdated, as over the past
ten years the IERS introduced signi�cant changes
to their Standards [5] three times. That is why
the new version of the code was created [8], called
KyivGeodynamics++ (hereinafter KG++). To validate
the new code the processing of the SLR data for 2001
was conducted.
the procedure of calculation
The algorithm of data processing is based upon
the linearised Least Squares method. The observed
distance between the satellite and the station ~ρ can
be written as:
~ρ(t) = M̂(t)~r(t)− ~R(t) = ~Φ( ~X, t), (1)
where ~r(t) is satellite position in International Celes-
tial Reference Frame (ICRF), ~R is the station posi-
tion in the International Terrestrial Reference Frame
(ITRF), andM is the transformation matrix between
ITRF and ICRF. Another vector ~Φ( ~X, t) is the vec-
tor function (in general case it is highly complicated
with regards to the calculation) of the parameters
~X.
Observation instruments (telescope, equipped
with laser transmitter and receiver) allow us to ob-
serve the absolute value of, say ~ρ, the distance to the
satellite. Having approximate values for ~X0 the (1)
can be linearised, and then
ρo − ρc =
(
∂~Φ( ~X)
∂ ~X
)∣∣∣∣∣
~X0
~x,
where ~x = ~X− ~X0 are corrections to the parameters.
A su�cient number of observations will give us the
possibility to determine a better estimation for ~X.
The observational data is processed using a two-
stage procedure. During the �rst stage the Kepler
∗zhskyy@gmail.com
54
Advances in Astronomy and Space Physics V. P. Zhaborovskyy, V.Ya.Choliy
orbital elements of the satellite are estimated. This
step is repeated until the standard deviation of the
di�erence observed minus calculated stops decreas-
ing. During the second stage, the determined orbit
is used to estimate the pole coordinates along with
empirical acceleration of the satellite. The latter ac-
celeration is widely used to compensate any forgotten
accelerations of di�erent origin not explicitly intro-
duced in the satellite equation of motion.
The elementary processing unit is the arc. It com-
prises the observation of a single satellite on a 7-day
interval (standard IERS value) together with a full
set of vector ~X components derived from it.
All of the observations of Lageos-1 and Lageos-2
satellites for the year 2001 were processed for this ar-
ticle. There were approximately 75000 normal points
in our database. There were 52 seven-day-long arcs:
a end point of an arc served as the starting point for
the next one.
Each arc was processed in two stages. During the
�rst stage only Keplerian orbital elements were esti-
mated until the overall standard deviation of the ob-
served minus calculated values stopped decreasing.
This stage lasted 4�5 iterations. Following this, the
estimation of the empirical acceleration (one value
per day) and pole coordinates (two values per day)
took place. Iterations continued until standard the
deviation decreased to below 10 cm. The speci�ed
10 cm value is su�cient to estimate pole coordinates.
In general, the overall process took 12�15 iterations
per arc. The observed minus calculated residuals
were thoroughly analysed during the iterations, and
their distribution was checked for normality. Rejec-
tion criteria was set to 3σ. Sometimes up to 30% of
the observations were dropped.
New code strictly follows all recommendations
from IERS Conventions 2010 [5]. All software mod-
ules were written from scratch and tested separately
on arti�cial examples before being introduced into
productive code. Processing of the Lageos-1 and
Lageos-2 data is the �rst result from the release ver-
sion code.
A brief outline of the models used is presented
below. If not speci�ed, the models from IERS Con-
ventions were used.
forces:
� Earth's gravity �eld and ocean tide model
EGM96 [6], 20× 20 solution;
� Third body gravity: Moon, Sun and all planets
� DE421 [4];
� Direct solar radiation pressure: re�ectance co-
e�cient a priori 1.14;
� Indirect light pressure: using cylindrical
shadow function;
� Relativistic correction.
reference frame:
� Stations coordinates and stations velocities:
ITRF2005 solution, epoch 2000.0 [1];
� Precession and nutation: full IAU2000A model
[2];
� Polar motion: C04 IERS;
� Tidal e�ects.
other:
� Observations: all normal points for 2001 year
from CDDIS database for Lageos-1 and Lageos-
2;
� Tropospheric refraction: Marini/Murray model
[7];
� Constants (speed of light, gravitational con-
stant etc.) from IERS Conventions.
estimated parameters:
� Estimation criteria: 3σ ' 5 cm for �nal result;
� O-C �lter: 3σ for each iteration;
� Satellite orbital elements are determined as one
set per 7 days arc;
� Pole coordinates determined for each day;
� Empirical acceleration coe�cient determined
for each day.
results
The X and Y pole coordinates were determined
for every day of 2001 from the observations of a sin-
gle satellite. The results are presented in Fig. 1 and
Fig. 2 for Lageos-1 and Lageos-2. The standard de-
viation for the last iteration of each one-day arc does
not exceed 10 cm for Lageos-1 and 7 cm for Lageos-2.
This result was compared with analogous data
obtained in di�erent international data analysis cen-
tres. All of these data are available online1. We
used the series by the Center for Space Research,
USA (crs), Delft University of Technology, Nether-
land (dut), Institute for Applied Astronomy, Russia
(iaa) and standard ILRS solutions (ilrs) (see Fig. 3).
The method of comparison is based upon Helmert
transform and described in [3]. On the �rst stage,
having two di�erent sets of polar data, one can ex-
tract systematic di�erences of these sets by applying
a 6-parameter Helmert transform:
1ftp://hpiers.obspm.fr/iers/series/operational
55
Advances in Astronomy and Space Physics V. P. Zhaborovskyy, V.Ya.Choliy
−0.2 −0.1 0.0 0.1 0.2 0.3
X pole, marcsec
0.1
0.2
0.3
0.4
0.5
Y
p
o
le
,
m
ar
cs
ec
01.01.2001
31.12.2001
Earth X and Y pole by Lageos 1 observations.
Fig. 1: Earth pole from Lageos-1 observation.
−0.2 −0.1 0.0 0.1 0.2 0.3
X pole, marcsec
0.1
0.2
0.3
0.4
0.5
Y
p
o
le
,
m
ar
cs
ec
01.01.2001
31.12.2001
Earth X and Y pole by Lageos 2 observations.
Fig. 2: Earth pole from Lageos-2 observation.
−0.2 −0.1 0.0 0.1 0.2 0.3
X pole, marcsec
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Y
p
o
le
,
m
ar
cs
ec
01.01.2001
31.12.2001
Earth X and Y pole by ILRS
Fig. 3: Earth pole from ILRS.
−0.2 −0.1 0.0 0.1 0.2 0.3
X pole, marcsec
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Y
p
o
le
,
m
ar
cs
ec
01.01.2001
31.12.2001
01.01.2001
31.12.2001
01.01.2001
31.12.2001
Earth X and Y pole by Lageos 1/2 observations.
Fig. 4: Earth pole � all results.
~r2 − ~r1 =
[
a11 a12
a21 a22
]
~r1 +
[
b1
b2
]
,
where ~r is the vector containing two pole coordinates,
where systematics error are described as rotations
(aij , i 6= j) scales (aii) and shifts (bk). Following
this, mutual standard deviations σij of series are ob-
tained:
σ2
ij =
N∑
k=1
(
~r(i)k − ~r∗(j)k
)2
N −M
,
where M is the number of degrees of freedom, and
the symbol ∗ is used to signal the series without sys-
tematic errors. The own standard deviations of the
series may be obtained if a su�cient number of σij
is obtained from the equation: σ2
12 = σ2
1 + σ2
2 − 2ρ12σ1σ2,
σ2
13 = σ2
1 + σ2
3 − 2ρ13σ1σ3,
σ2
23 = σ2
2 + σ2
3 − 2ρ23σ2σ3,
where ρij are the correlations of the corresponding
series.
Varying solutions were used as a base for compar-
ison. Three series were compared. The computed
standard deviations are given in Table 1 where every
cell contains the resulting value found by compar-
ing our satellite solution (row name) and two cen-
tre names given as the column headings. The mean
standard deviation of pole coordinates obtained from
Lageos-1 observations is 3.10 × 10−4, and that ob-
tained from Lageos-2 observations is 1.19× 10−3.
All series of pole coordinates are present in Fig. 4.
56
Advances in Astronomy and Space Physics V. P. Zhaborovskyy, V.Ya.Choliy
Table 1: Standard deviations in mas of some Earth pole coordinates series.
dut � csr csr � iaa iaa � ilrs ilrs � dut mean
Lag1 2.98 · 10−4 3.57 · 10−4 2.44 · 10−4 3.42 · 10−4 3.10 · 10−4
Lag2 1.27 · 10−3 1.22 · 10−3 1.21 · 10−3 1.04 · 10−3 1.19 · 10−3
conclusions
The values presented in Table 1 show two main
points. Firstly, the precision of Lageos-1 based pole
coordinates are nearly one order of magnitude better
than the values from processing Lageos-2 data. The
second point is that in any case both of the preci-
sions agree with values produced by other data anal-
ysis centres. This result demonstrates the correct
work of KG++ on 2 cm precision level and its applica-
bility for the analysis of the SLR data with precision
su�cient for IERS tasks.
references
[1] Altamimi Z., Collilieux X., Legrand J., Garayt B. &
BoucherC. 2007, J. Geophys. Res., 112, B09401
[2] Capitaine N., Chapront J., Lambert S. & Wallace P. T.
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[3] CholiyV.Ya. 2014, Advances in Astronomy and Space
Physics, 4, 15
[4] FolknerW.M., Williams J.G. & BoggsD.H. 2009, The In-
terplanetary Network Progress Report, 178, 1
[5] IERS Technical Note No. 36 (IERS Conventions 2010),
2010, IERS Conventions Center
[6] LemoineF., Kenyon S., Factor J. et al. 1998, `The De-
velopment of the Joint NASA GSFC and National Im-
agery and Mapping Agency (NIMA) Geopotential Model
EGM96', NASA/TP-1998-206861, Goddard Space Flight
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[7] Marini J. & MurrayC. 1973, `Correction of laser range
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above 10 degrees', NASA-TM-X-70555, GSFC, Greenbelt
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tronomy and Space Physics, 1, 96
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