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Advances in Space Research 36 (2005) 376–381
Kinematic positioning of LEO and GPS satellites and IGS stationson the ground
Drazen Svehla *, Markus Rothacher
Forschungseinrichtung Satellitengeodasie, Technical University of Munich, Arcisstrasse 21, D-80333 Munich, Germany
Received 3 November 2004; received in revised form 29 March 2005; accepted 19 April 2005
Abstract
For the first time, we publish results with the kinematic positioning of the GPS satellites and make comparisons with the kine-
matic positioning of LEO satellites and IGS stations on the ground. We show that LEO point-positioning is possible by means of
GPS satellite clocks estimated solely based on phase GPS measurements. In sequel, we introduce a fourth approach in precise orbit
determination, which we call reduced-kinematic POD, where kinematic position differences in time are constrained to corresponding
differences in a priori dynamic orbit.
� 2005 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: LEO; POD; Kinematic; Reduced-kinematic; CHAMP
1. Introduction
In Svehla and Rothacher (2004a), kinematic POD
was presented as a new method in precise orbit determi-
nation of Low Earth Orbiting (LEO) satellites with the
main application in gravity field determination. Further
on, in Svehla and Rothacher (2004b) kinematic and re-
duced-dynamic POD were demonstrated for a period
of 2 years using CHAMP data. A considerable numberof groups already use these kinematic positions to esti-
mate Earth gravity field coefficients and to validate dy-
namic orbits and orbit models. Using the CHAMP
kinematic positions together with the corresponding
variance–covariance information, gravity field coeffi-
cients can be estimated by making use of the energy bal-
ance approach or the boundary value method rather
than the classical numerical integration schemes, seee.g., Gerlach et al. (2003) at TU Munich, Mayer-Gurr
et al. (2005) at TU Bonn, Reubelt et al. (2004) at TU
0273-1177/$30 � 2005 COSPAR. Published by Elsevier Ltd. All rights reser
doi:10.1016/j.asr.2005.04.066
* Corresponding author.
E-mail address: [email protected] (D. Svehla).
Stuttgart and Ditmar et al. (2004) at TU Delft. The val-idation of gravity field models computed in such a way
showed that CHAMP kinematic positions contain
high-resolution gravity field information and the accu-
racy of the derived gravity models is comparable to that
of official CHAMP models, if not better.
Kinematic positions with the corresponding vari-
ance–covariance information are a very attractive inter-
face between the raw GPS data and gravity field modelsor other interesting information that can be derived
from satellite orbits, e.g., air densities or orbit force
model improvements. In this way, the groups that use
kinematic positions do not have to cope with the pro-
cessing and analysis of the GPS observations, including
the adjustment of a huge amount of global parameters
like GPS satellite clocks and orbits, zero- or double-dif-
ference ambiguities, station coordinates, troposphereparameters, Earth rotation parameters, etc.
In Sections 2–4, we present the first kinematic orbit
determination of GPS satellites, the results of the kine-
matic estimation of LEO satellites, and the kinematic
positioning of an IGS station, respectively. At the end,
ved.
D. Svehla, M. Rothacher / Advances in Space Research 36 (2005) 376–381 377
in Section 5, we introduce the reduced-kinematic POD
as a new POD approach.
2. Kinematic positioning of GPS satellites
How accurately can a GPS satellite orbit be estimated
fully kinematically? The basic idea is to fix the coordi-
nates of the IGS GPS points on the ground and to esti-
mate three coordinates of the center-of-mass of the GPS
satellite every epoch using zero or double-difference
phase measurements. The main difference to kinematic
positioning of a ground station or a LEO satellite is
that, due to the very high altitude, the GPS satellites‘‘see’’ all ground stations within a very small range of
nadir angles. A GPS antenna placed on a LEO satellite
or located on the ground can receive signal from the
GPS satellites in elevations ranging from 0� to 90�. Incontrast, the maximal nadir angle of a signal transmitted
from a GPS satellite to a LEO satellite or ground station
is about 14–15�, see Fig. 1. This angle is six times smaller
than the maximum zenith angle of a LEO or groundGPS antenna and thus, the position of the ground sta-
tions in the local orbital system of the GPS satellite var-
ies very little with time.
In the case of a LEO or a ground GPS station the
kinematic positions are computed at the measurement
epoch, which is the same for all GPS satellite tracked.
This is not the case for the kinematic positioning of
GPS satellites where, due to the GPS receiver clock cor-rection and the light-travel time correction, different
ground GPS stations ‘‘see’’ the GPS satellite at different
places along its orbit for nominally the same observa-
tion epoch.
Due to the instability of the GPS receiver clock, the
GPS measurements are not taken exactly at the integer
second in GPS time. Steering of the GPS receiver clock
on the ground or on the LEO satellite can be performed
Fig. 1. Geometry for LEO and GPS satellites and GPS station on the
ground.
using the receiver�s navigation solution based solely on
the code measurements and broadcast GPS orbits and
clocks. In the case of the Blackjack GPS receiver on
board the CHAMP satellite, the clock steering is per-
formed on the level of 0.1 ls. Nevertheless, for some
ground GPS receivers (IGS network) the clock correc-tion w.r.t. GPS time may vary up to 1 ms. In order to
correct for this GPS receiver effect, aiming at an accu-
racy for the GPS orbit of Dx = 1 cm and assuming a
GPS receiver clock correction of Dt = 1 ms, the velocity
of the GPS satellite has to be known with only a very
low accuracy of about Dv = Dx/Dt = 10 m/s. The veloc-
ity of the GPS satellite is required with a higher accu-
racy, however, to correctly apply light-travel time andperiodic relativistic corrections. For the GPS satellites,
the light-travel time correction DLTT and the periodic
relativistic correction DPRC, see Ashby (2003), are given
as:
DLTT ¼ �d~n0~vSc
; DPRC ¼ 2~rS~vSc
; ð1Þ
where d and~n0 denote distance and unit vector between
GPS satellite and ground station,~rS and ~vS are geocen-tric position and velocity vector of the GPS satellite, and
c is the speed of light in vacuum. One can easily see that
the periodic relativistic correction is satellite-specific and
therefore cancels when forming double-differences or
can be absorbed by the GPS satellite clock parameters
when using zero-differences. Following Eq. (1), to com-
pute the light-travel time DLTT with an accuracy of 1 cm,
the velocity of the GPS satellite should be known withan accuracy of Dv � 0.12 m/s. Since not too high
requirements are posed on the velocity in the computa-
tion of the light-travel time correction, the orbits of the
GPS satellites can indeed be determined kinematically.
Nevertheless, an approximate dynamic GPS orbit or
broadcast GPS orbit has to be available to compute
the corrections.
Fig. 2 shows the differences between a kinematicand dynamic orbit for the GPS satellite PRN 20 and
Fig. 3 the corresponding a posteriori RMS values of
the kinematic positions. Both types of orbits were
determined using the same IGS stations, troposphere
parameters, station coordinates and Earth rotation
parameters and the only difference consists in the esti-
mated orbital parameters. Dynamic GPS orbits were
modelled by six Keplerian elements, nine solar radia-tion pressure parameters and one pseudo-stochastic
pulse for the one day arc, whereas three kinematic
coordinates were estimated for PRN 20 (the other sat-
ellites were kept fixed) every epoch (i.e., every 30 s). In
both cases, the ambiguities were kept fixed to their
integer values. One can easily see that the quality of
the estimated kinematic positions is on the level of
10–20 cm. It is to be expected that by replacing thekinematic parameterization by polynomials over a
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5–0.4
–0.3
–0.2
–0.1
0
0.1
0.2
0.3RMS=8.6 cm
Alo
ng–t
rack
in m
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8RMS=22.6 cm
Cro
ss–t
rack
in m
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5–0.4
–0.3
–0.2
–0.1
0
0.1
0.2
0.3RMS=10.2 cm
Time in hours
Rad
ial i
n m
Fig. 2. Differences between the kinematic and dynamic orbit for GPS
satellite PRN 20. In the orbit determination only kinematic and
dynamic parameters were exchanged. Day 200/2002.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 RMS=19.4 cm
Alo
ngtr
ack
in m
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.05
0.1
0.15
0.2
0.25 RMS=11.9 cm
Cro
ss–t
rack
in m
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 RMS=14.7 cm
Time in hours
Rad
ial i
n m
Fig. 3. A posteriori RMS of the kinematic orbit for GPS satellite PRN
20. Day 200 in the year 2002.
378 D. Svehla, M. Rothacher / Advances in Space Research 36 (2005) 376–381
few 10 min the ‘‘kinematic’’ GPS orbits would improve
considerably. We should keep in mind that the dy-namic GPS orbit is usually represented by a polyno-
mial of degree 12 for each step of 1 h in the
numerical integration method used. The rather strong
variations between kinematic and dynamic GPS posi-
tions in Fig. 2 and periodic behavior in the corre-
sponding formal precision displayed in Fig. 3 are
certainly due to the weak, but changing geometry of
ground stations as seen from the satellite.
3. Kinematic positioning of ground IGS stations
The ground GPS baseline from Greenbelt (GODE,
US) to Algonquin Park (ALGO, Canada) with a length
of about 800 km was processed kinematically for a per-
iod of 1 day. One station of the baseline was kept fixed(GODE) and a set of three coordinates was estimated
every 30 s for the second station (ALGO). Ambiguities
were resolved using the Melbourne–Wubbena linear
combination and only phase data were used in the posi-
tioning. Fig. 4 shows the differences between kinematic
positions of the station (ALGO) and the ‘‘true’’ static
coordinates estimated in the global IGS network solu-
tion. One can see that an accuracy of 0.5–1 cm in hori-
zontal position and 2 cm in height can easily be
achieved. Similar results can be obtained whether tropo-
sphere parameters are taken from the global IGS solu-tion or estimated every 1 h. Other GPS baselines
within the IGS network with lengths up to 1000 km
show similar results.
At the moment, the radial accuracy of GPS orbits, as
assessed using SLR measurements, is 2.7 cm, neglecting
a large bias of �5.8 cm, Urschl et al. (2005). Using the
rule of thumb given by Bauersima Dl = lDq/20,000 km
(see, Bauersima (1983)), with a very pessimistic GPS or-bit error of Dq = 5 cm and with a baseline length of
l = 1000 km one can expect errors in the coordinates in-
duced by the GPS orbits of only Dl = 2.5 mm. For a
baseline length of 10,000 km (LEO) we get about
2.5 cm. Therefore, for ground GPS applications the
present accuracy of GPS orbits (2.7 cm) allows a 1-cm
double-difference kinematic positioning for baselines
up to 5000 km. From this analysis it follows that stationmultipath, besides residual troposphere delay errors, is
probably the major error source in ground GPS
positioning.
0 3 6 9 12 15 18 21 24–0.06
–0.04
–0.02
0
0.02
0.04RMS=1.0 cm
Nor
th in
m
0 3 6 9 12 15 18 21 24–0.03
–0.02
–0.01
0
0.01
0.02
0.03RMS=0.6 cm
E
ast i
n m
0 3 6 9 12 15 18 21 24–0.2
–0.15
–0.1
–0.05
0
0.05
0.1
0.15RMS=2.1 cm
Time in hours
Hei
ght i
n m
Fig. 4. Kinematic estimation of the ground IGS point ALGO with
respect to the fixed IGS station GODE. Ambiguity-resolved baseline
with the length of 777 km, day 200 in the year 2002.
0 3 6 9 12 15 18 21 24–0.1
–0.05
0
0.05
0.1RMS=1.6 cm
Alo
ng–t
rack
in m
0 3 6 9 12 15 18 21 24–0.06
–0.04
–0.02
0
0.02
0.04RMS=1.2 cm
Cro
ss–t
rack
in
m
0 3 6 9 12 15 18 21 24–0.1
–0.05
0
0.05
0.1RMS=1.9 cm
Time in hours
Rad
ial i
n m
Fig. 5. Differences between the kinematic and reduced-dynamic orbit
for the CHAMP satellite (based on zero-differences). One can easily
recognize epochs with a small number of GPS satellites tracked and
phase breaks, day 200/2002. Few outliers with very bad variance
properties not displayed.
D. Svehla, M. Rothacher / Advances in Space Research 36 (2005) 376–381 379
4. Kinematic and reduced-dynamic positioning of LEO
satellites using GPS phase clocks
In Svehla and Rothacher (2004a), we showed that
there is no principal difference in results obtained when
the orbit determination of LEO satellites is performed
using zero- or double-differences (with ambiguity resolu-
tion) and an accuracy of 2–3 cm can be achieved. The
main difference between a LEO and a ground station
consists in the tracking time from rising to setting of
the GPS satellites and thus the number of phase ambigu-ity parameters. A ground GPS receiver receives signal
from the same GPS satellite over several hours, whereas
a spaceborne GPS receiver on a LEO satellite can track
the same GPS satellite for only 15–25 min. The huge
number of LEO double-difference ambiguities to be set
up in a short time (compared to that of a ground GPS
station) and the very long baselines between ground
IGS stations and the LEO are the main reasons whyambiguity resolution with the LEO satellites still does
not provide satisfactory results. The zero-difference ap-
proach or precise point-positioning is much faster com-
pared to the double-difference approach, since only the
LEO GPS measurements are involved. It should be
noted though, that the GPS satellite orbits and clocks
have to be determined first.
Fig. 5 shows differences between the kinematic and
reduced-dynamic orbit based on zero-differences for
the CHAMP satellite (day 200/2002). One can easily rec-
ognize epochs with a small number of GPS satellitestracked and phase breaks. Possible reasons to explain
the once-per-rev. variations in the radial and the
along-track component might be deficiencies in the air
drag modelling in the reduced-dynamic POD. More
about kinematic and reduced-dynamic POD methods
can be found in [9–12]. That Kalman filtering also pro-
vides good results in kinematic positioning can be found
in Colombo et al., 2004. Kinematic approach based onestimation of position differences is described in Bock
(2003).
In Svehla and Rothacher (2004b), we showed that
clock parameters for the GPS satellites can be estimated
based solely on the phase GPS measurements using 40
ground stations and one ground hydrogen maser as a
fixed reference clock. The code measurements are a pre-
requisite only to synchronize all receiver clocks at the le-vel of 1 ls. Any epoch-specific bias in the ensemble of
such relative phase clocks will directly propagate into
0 1 2 3 4 5 6–0.08
–0.04
0
0.04
0.08
RMS=1.0 cm
Alo
ng−t
rack
in m
0 1 2 3 4 5 6− 0.08
− 0.04
0
0.04
0.08RMS=0.9 cm
Cro
ss−t
rack
in m
− 0.05
0
0.05
0.1
RMS=1.3 cm
Rad
ial i
n m
380 D. Svehla, M. Rothacher / Advances in Space Research 36 (2005) 376–381
one LEO receiver clock parameter estimated every
epoch. The main advantage of relative phase GPS clocks
rather than GPS clocks estimated by combining phase
and code observations is that the impact of the code
noise can be avoided. By computing GPS satellite phase
clocks and CHAMP kinematic and reduced-dynamic or-bits for a period of 2 years, we demonstrated that such
an approach can easily be performed on a standard
PC with 1 GB of RAM, Svehla and Rothacher, 2004b.
For a 1-day arc, GPS satellite clocks can be estimated
with a sampling of 30 s using the full normal equation
system consisting of phase ambiguities and GPS satel-
lite/receiver clocks as parameters only. The station coor-
dinates ERPs and troposphere parameters were takefrom a global double-difference solution.
What is the impact of the GPS satellite orbits on the
determination of LEO orbits? Following the rule of
thumb given by Bauersima and considering baselines
up to 10,000 km, one can easily draw the conclusion that
to obtain LEO orbits with an accuracy of 1 cm, the GPS
orbits should be determined with an accuracy of 2 cm.
Therefore, if we assume the real accuracy of GPS orbitsto be 2.7 or 5–6 cm including the SLR bias, it might be
possible to further improve LEO orbits by improving
the GPS orbit modelling.
0 1 2 3 4 5 6− 0.1
Time in hours
Fig. 6. Kinematic (black) and reduced-kinematic orbit (grey) for the
CHAMP satellite compared to the best reduced-dynamic orbit, zero-
differences (day 200/2002). In the reduced-kinematic orbit, a smooth-
ing effect can be noticed for epochs with very bad variance–covariance
properties (small number of GPS satellites tracked, phase breaks).
5. Reduced-kinematic precise orbit determination
Compared to dynamic orbits, the main disadvantageof kinematic orbits are the ‘‘jumps’’ between consecutive
kinematic positions that occurs when, e.g., small num-
bers of GPS satellites are tracked or when and phase
breaks happen. Although these ‘‘jumps’’ from epoch
to epoch are fully reflected in the variance–covariance
information, they can be nicely seen in Fig. 5, where
CHAMP kinematic positions are plotted against the re-
duced-dynamic orbit. Typical spikes in kinematic posi-tions, and accordingly in the variance–covariance
information, can be seen around 1.1, 1.3, 2.5 and 4.1 h
and phase breaks can be identified for the isolated arc
from 4.1 to 4.6 h. Compared to the kinematic orbits, dy-
namic orbits are very smooth. In order to reduce the size
of the small jumps in kinematic positions, constraints
can be applied from epoch to epoch to the kinematic po-
sition differences w.r.t. corresponding differences in the apriori dynamic orbit. In this case, we may speak of ‘‘re-
duced-kinematic’’ orbit determination, where the kine-
matic degrees of freedom are reduced by constraints to
the dynamic orbit. It can be shown that the a priori dy-
namic orbit used for constraining can be of very low
accuracy, e.g., defined by only 15 orbital parameters
per day and estimated by means of code measurements
only. The size of the relative constraints applied in thecomputation of reduced-kinematic orbits in Fig. 6 was
5 mm between 30 s epochs. Using the reduced-kinematic
approach, one can get very smooth kinematic orbits
where spikes in the kinematic positions are removed or
considerably reduced. This is illustrated in Fig. 6, where
kinematic and reduced-kinematic orbits are displayed
w.r.t. the best reduced-dynamic orbit. Although the sto-chastic process realized by relative constraints is a ran-
dom walk, the trajectory is not drifting away from the
a priori dynamic orbit. Depending on the strength of
the constraints between consecutive epochs, the esti-
mated reduced-kinematic orbit will be closer either to
the dynamic or the kinematic orbit. The main difference
between reduced-kinematic and reduced-dynamic orbit
determination is, that in the reduced-kinematic PODthe normal equations are set up for the epoch-wise kine-
matic positions (with epoch-wise clocks), whereas in the
reduced-dynamic approach dynamic parameters (like
initial Keplerian state vector, air-drag coefficients,
empirical accelerations, etc.) are determined.
The reduced-kinematic method improves the overall
characteristics of the purely kinematic POD by a consid-
erable reduction of spikes and jumps. Therefore, reduced-kinematic POD can be used for LEO applications that
requires very smooth trajectory such as radio-occultation.
D. Svehla, M. Rothacher / Advances in Space Research 36 (2005) 376–381 381
Since the a priori dynamic orbit used in reduced-kine-
matic POD does not have to be of high accuracy and
can very easily be computed, the reduced-kinematic posi-
tions will not contain significantly a priori gravity field,
butwill allow, e.g., better velocity computation for the en-
ergy balance approach of gravity field determination.
6. Conclusions
Kinematic positioning of the ground IGS points and
LEO satellites can be performed with a similar accuracy
of 1–2 cm. On the other hand, kinematic positioning of
the GPS satellites is more difficult, and is, due to thevery small nadir angle range, limited to an accuracy of
10–15 cm. We may expect that by representing the
GPS orbits by polynomials, the kinematic POD of
GPS satellites could be improved.
Following the rule of thumb given by Bauersima, to
estimate LEO orbits with an accuracy of 1 cm the
GPS orbits should be determined with an accuracy of
2 cm. Therefore, the accuracy of LEO orbits might stillbe improved by improving the quality of the GPS orbits.
The second conclusion that can be draw from the error
analysis is, that the station multipath, apart from tropo-
spheric refraction, is most likely the major error source
in the ground GPS positioning.
GPS satellite clock parameters can be estimated based
on phase GPS measurements using approx. 40 ground
IGS stations and can be used in the next step for thepoint-positioning of the LEO satellites and ground IGS
stations using phase measurements only. In this way,
the code measurements are used only to approximately
synchronize GPS receiver clocks and the negative impact
of the code noise on the results can be avoided.
Besides the dynamic, reduced-dynamic and kinematic
approach, the reduced-kinematic POD can be consid-
ered as a fourth orbit determination approach with themain characteristic, that kinematic positions between
consecutive epochs are smoother. Due to small numbers
of GPS satellites tracked, some kinematic epochs have
worse variance–covariance properties and the reduced-
kinematic approach copes with this problem. Therefore,
reduced-dynamic POD reduces dynamics towards kine-
matics and in the reduced-kinematic case, kinematics is
reduced towards dynamics.
Acknowledgements
We are grateful to GFZ Potsdam for providing the
GPS measurements of the CHAMP satellite. We thank
Peter Steigenberger for kindly making available orbits
of GPS satellites for the period of 2 years computed inthe framework of a global GPS data reprocessing project.
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