GRACE-A and GRACE-B together

Kinematic and Dynamic Orbit Determination 

of a Single LEO Satellite 

and 

Formation Flaying Using GPS 

Drazen Svehla, Markus Rothacher 

Forschungseinrichtung Satellitengeodäsie 

Technical University of Munich, Germany 

FGS Workshop 2004, Höllenstein/Wettzell, October 27 - 29, 2004

Content 

• Kinematic positioning of IGS stations, LEOs and GPS satellites 

• Positioning with phase GPS satellite clocks 

• CHAMP and GRACE kinematic & reduced-dynamic POD over 2 years 

• Reduced-kinematic POD 

• POD based on LP linear combination (one-frequency LEO GPS data) 

• CHAMP and Ionosphere 

• POD strategies for LEO formation flying 

• Ambiguity resolution with GRACE 

• Combined GPS/KBR kinematic & dynamic baseline in space 


FGS Workshop 2004, Höllenstein/Wettzell, October 27 - 29, 2004 

Kinematic IGS Station 

Baseline ALGO-GODE (length 777 km) 

Double-differences with fixed ambiguities 

(Melbourne-Wübbena with narrow-laning) 

Differences between kinematic positions 

estimated every 30 s for GODE IGS station 

and weekly GODE coordinates 

All global products kept fixed. Similar results 

with troposphere estimation.


Kinematic GPS Satellite 

PRN 20 

Baselines IGS-GPS (length up to 1000 km) 

Double-differences with flixed ambiguities. 

Differences between kinematic positions 

estimated every 30 s for GPS Satellite 

PRN 20 and dynamic orbit 

All global products were kept fixed, 

except orbit of GPS satellite PRN 20, 

day 200/2002.


Kinematic GRACE-A 

Differences between kinematic and 

reduced-dynamic positions. 

All global products were kept fixed, 

day 200/2003. 

Zero-difference phase measurements.

GPS Clocks 

Estimation of the 5-min and the 30-s phase GPS clocks 

using 40 IGS stations 

Selection of the reference clock (BRUS, USNO or WTZR) 

Alignment of the reference clock to the broadcast GPS clocks 

(daily bias and drift estimation) 

Pre-processing of the phase/code data 

based on Melbourne-Wübbena LC 

Estimation of the 5-min GPS clocks only to clean data 

using code and phase measurements 

Pre-processing of the phase data 

using the 5-min GPS clocks 

Estimation of the 30-s GPS clocks 

using phase measurements and code measurements (only reference station) 

- normal equation matrix with up to 5000 phase ambiguities 


GPS Phase Clocks 


Positioning of the CHAMP Satellite: Two Years 

Daily RMS difference between kinematic and reduced-dynamic orbit 


Kinematic Positioning and 

Gravity Field Recovery 

Users of the CHAMP 2-Year Kinematic Orbits 

Austria (gravity) 

Canada (gravity) 

China (gravity/POD) 

Denmark (gravity) 

Germany (gravity/POD) 

Japan (POD) 

Netherlands (gravity/POD) 

Switzerland (POD) 

Taiwan (gravity/POD) 

UK (gravity) 

US (POD) 


Positioning of the GRACE Satellites: Two Years 


Daily RMS difference between kinematic and reduced-dynamic orbit 

GRACE-A POD over 4 months GRACE-B 


Validation with SLR 

GRACE-A GRACE-B 

Reduced-Dynamic Orbit 


Reduced-Dynamic Orbit 

Kinematic Orbit Kinematic Orbit 

Day of year 

Day of year 

Day of year


Reduced-Kinematic POD 

Kinematic POD with dynamic information in 

order to reduce influence of weak 

geometry and phase breaks (jumps). 

Methods: 

• Absolute constraints to the a priori orbit 

• Relative constraints between epochs 

• Gauss-Markov process 

Normal system solution: 

• Full normal matrix inversion 

(86400 kinematic coordinates for one-day arc) 

• L - U factorization 

(block tridiagonal system), block forward 

elimination and back substitution. 

Characteristics: 

Smoothing effect

Reduced-Kinematic POD 



POD Based on 

LP Linear Combination 

Linear combination of phase and code 

measurements on one frequency. 

LP=(L1+P1)/2 

Ionosphere effect eliminated. 

Wavelength ≈ 95 mm 

Reduction of P-code noise by a factor of 2. 

Pre-processing more difficult.

CHAMP and Ionosphere 


POD Strategies for LEO Formation Flying 

Combined zero & double-difference POD 


Ambiguity resolution with GRACE baseline 

Total number of ambiguities, mean= 416 


Resolved narrow-lane ambiguities using 

kinematic bootstrapping, mean= 93.6% 

Day of year Day of year 

Resolved wide-lane ambiguities 

(Melbourne-Wübbena), mean= 98.4% 

Day of year 

Resolved narrow-lane ambiguities using 

dynamic bootstrapping, mean= 92.8% 

Day of year

Zero-difference orbits 

(like two CHAMP satellites) 

GRACE Formation Flying 

Zero-differences versus 

combined zero&double-difference reduced-dyn. 

orbits, validation with KBR 


Float ambiguities 

(GRACE-A and GRACE-B together) 

RMS=15.7 mm RMS=11.6 mm 

Time in hours RMS=2.8 mm 

Time in hours 

Time in hours 

Fixed ambiguities 

(GRACE-A and 

GRACE-B together)

Baseline Validation with KBR 

Reduced-Dynamic Orbit Kinematic Orbit 

Float ambiguities 

RMS= 15.1 mm RMS= 11.7 mm 

Time in hours Time in hours 

RMS= 3.0 mm 

Time in hours 

pseudo-stochastic 

pulses 

Fixed ambiguities 


RMS= 6.1 mm 

Time in hours

Validation with KBR 

Reduced-Dynamic Orbit Kinematic Orbit 

Zero-differences 

RMS= 12.6 mm 

RMS= 15.0 mm 

Time in hours Time in hours 

Kinematic/Reduced-Dynamic Orbit 

Double-differences 

fixed ambiguities 


Reduced-Dynamic Baseline 

Validation with KBR 

Day of year Day of year

Combined GPS/KBR Baseline 

Kinematic/reduced-dynamic 

GPS Only 

baseline with fixed ambiguities GPS + KBR 


Combined GPS/KBR Baseline 

Impact of the KBR on the 

Kinematic orbit 

kinematic and reduced-dynamic Reduced-dynamic orbit 

GPS baseline with fixed ambiguities 


pseudo-stochastic pulses

Conclusions 

• 1-2 cm kinematic positioning for LEO satellites and IGS stations & 

10 cm for GPS satellites is possible. 

• Kinematic POD can be used as an interface for determination of gravity field 

parameters and other phenomenon that govern the satellite motion. 

• 30-s GPS satellite clocks can be estimated with the full normal equation matrix using 40 

IGS stations. Positioning with the relative phase clocks has been successfuly performed. 

• 90%-100% of all ambiguities can be resolved for GRACE baseline using Melbourne- 

Wübbena wide-lanning with narrow-lane bootstrapping. 

• After ambiguity resolution kinematic and reduced-dynamic baseline results agree on 

the level of 1 cm and 3 mm respectively when validated with KBR measurements. 

• Combined processing of zero- and double-differences changes the orbit by 1-2 cm 

compared to zero-diff. case and is the optimal strategy for the LEO formation flying. 

• KBR and ambiguity resolution clearly shows colored noise in the kinematic positions 

and defficienties in the dynamic modelling. 

• KBR has very strong impact on the kinematic and reduced-dynamic GPS baseline 

even when the ambiguities are fixed.

GRACE-A and GRACE-B together

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