6
Kinematic positioning of LEO and GPS satellites and IGS stations on the ground Draz ˇen S ˇ vehla * , Markus Rothacher Forschungseinrichtung Satellitengeoda ¨ sie, 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. Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: LEO; POD; Kinematic; Reduced-kinematic; CHAMP 1. Introduction In S ˇ vehla 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 S ˇ vehla and Rothacher (2004b) kinematic and re- duced-dynamic POD were demonstrated for a period of 2 years using CHAMP data. A considerable number of 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, see e.g., Gerlach et al. (2003) at TU Munich, Mayer-Gu ¨rr et al. (2005) at TU Bonn, Reubelt et al. (2004) at TU 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 models or 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, troposphere parameters, 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, 0273-1177/$30 Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.04.066 * Corresponding author. E-mail address: [email protected] (D. S ˇ vehla). www.elsevier.com/locate/asr Advances in Space Research 36 (2005) 376–381

Kinematic positioning of LEO and GPS satellites and IGS stations on the ground

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www.elsevier.com/locate/asr

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.

Page 2: Kinematic positioning of LEO and GPS satellites and IGS stations on the ground

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

Page 3: Kinematic positioning of LEO and GPS satellites and IGS stations on the ground

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.

Page 4: Kinematic positioning of LEO and GPS satellites and IGS stations on the ground

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

Page 5: Kinematic positioning of LEO and GPS satellites and IGS stations on the ground

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.

Page 6: Kinematic positioning of LEO and GPS satellites and IGS stations on the ground

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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