Product Name: SST_PRP_2 Product Name: SST_PRP_2

Orbit arcs (30 hours)Orbit arcs (30 hours)

Final GOCE Orbit Product (IAPG)Final GOCE Orbit Product (IAPG)

(daily files)(daily files)

Var/Cov Product Name: SST_PSV_2Var/Cov Product Name: SST_PSV_2

iapg

3rd International GOCE User Workshop, ESA/ESRIN, 6-8 November 2006

Normal Equation Matrix AlgorithmNormal Equation Matrix Algorithm

Institute of Astronomical and Physical Geodesy Technische Universität München

Generation and Validation of Orbit Products Generation and Validation of Orbit Products

for the GOCE Missionfor the GOCE Mission Drazen Svehla

ABSTRACT: The ESA GOCE Core Explorer Mission will be equipped with the dual-frequency GPS receiver for the precise orbit determination and with the SLR reflector for the orbit validation. The final GOCE orbit will be jointly computed by the AIUB in Bern and the IAPG at the TU München on dai-ly routine basis as integrated part of the GOCE High-Level Processing Facility (HPF). In addition, the IAPG is responsible for the re-processing of the final kinematic and reduced-dynamic GOCE orbit. We present generation and validation of orbit products including the fi-nal GOCE precise kinematic orbit with the variance-covariance matrix and the GOCE precise reduced-dynamic orbits. Both type of orbits will be accompanied with the quality information and orbit validation reports including performance of the GOCE GPS receiver. We present algo-rithm to compute variance-covariance information for the GOCE kine-matic orbit and the way how it is defined as a product. In order to provi-de information about the Earth rotation and transformation between the Earth-fixed and inertial system we present algorithms and the use of the Earth orientation quaternions. At the end, we address the role and pre-sent the chain for the reprocessing of the final GOCE kinematic and re-duced-dynamic orbits. We give an overview of various orbit determina-tion strategies developed for the GOCE mission staring from the purely kinematic, reduced-kinematic, reduced-dynamic to fully dynamic me-thods based on the numerical integration.

Var/Cov Matrix for the Kinematic OrbitVar/Cov Matrix for the Kinematic Orbit

1

12 11 12 22

( 4)

2)

n

Q Q N N −

×

= −

N11 (n×n)

N22 (4×4)

N12 (n×4)

Q21

(4×n)

1 1 1

22 22 22 21 11 12 22(4 4) (4 4) ( )

1)n n

Q N N N Q N N− − −

× × ×

= +

1 1

21 22 12 22(4 4)

(4 )

3)

n

Q N N N− −

××

= −

( )

( )

11 12 1 1

21 22 2 2

1

2

1 1

11 12 22 21 1 1 12 22 2

1

2 22 2 21 1

= ambiguities

= epoch-wise kinematic coordinates & clocks

back substitution:

N N x b

N N x b

x

x

N N N N x b N N b

x N b N x

− −

−

=

− = −

= −

SST_PSO_2

-3h 0h 24 +3h

SST_PSO_2I

-3h 0h 24 +3h

Rapid and Precise Science Orbit ProductsRapid and Precise Science Orbit Products

Fig. 6. The figure on the left shows daily RMS of kinematic orbit vs. reduced-dynamic orbit for a period of 4 months based on the GPS data provided to the GRACE Science Team. Figure above shows differences between the old reduced-dynamic orbit model based on pseudo-stochastic pulses estimated every 6 minutes and the new model based on Fourier series. Epochs where pseudo-stochastic pulses were set up can easily be identified.

Linear combination of phase and code measurements on one frequency. LP=(L1+C/A)/2 Ionosphere effect eliminated. Wavelength ≈ 95 mm Reduction of code noise by a factor of 2. Pre-processing more difficult.

Typical correlations of CHAMP kinematic positi-ons (correlation length of 20-30 min)

KinematicKinematic without constraints

GRACE-A POD over 4 months

(Photo by F. Dilssner)

Absolute Calibration of the GOCE GPS Antenna

GOCE Rapid Science Orbit Reduced-Dynamic + Kinematic Orbit

DEOS

GOCE Precise Science Orbit Reduced-Dynamic + Kinematic Orbit

AIUB + IAPG (re-processing)

ORBIT VALIDATION

IAPG

Fig. 1. DEOS (Department of Earth Observation and Space Systems, Delft University of Technology) and AIUB (Astronomical Institute of the University of Bern) together with IAPG (Institute of Astronomical and Physical Geodesy, Technical University of Munich) are responsible for the precise orbit determination of the GOCE satellite. GOCE POD is divided into a quick-look part or Rapid Science Orbit — RSO, implemented at DEOS and Precise Science Orbit — PSO, implemented at AIUB and IAPG. The final GOCE PSO orbit product is generated at the IAPG (Product name SST_PSO_2) based on 30-hourly orbit arcs provided by AIUB. The IAPG is responsible for the GOCE orbit re-processing and orbit validation. The final GOCE orbit product (SST_PSO_2) generated at the IAPG contains: reduced-dynamic orbit, kinematic orbit, var/cov information for kinematic orbit, Earth orientation quaternions, phase residuals and obit validation report.

Fig. 2. The SST_PSV_2 is the GOCE orbit product containing Var/Cov information for the PSO kinematic orbit. On the left is the algorithm to compute cofactors based on pre-elimination scheme with back substitution. Kinematic POD is based on the zero-difference phase measurements with 4 parameters estimated every epoch (3 kinematic coordinates + 1 clock parameter). Cofactors are provided for 9 measurement epochs before and after the current epoch.

“old“ pseudo-stochastic pulses set up every 6-minutes

Fig. 5. Kinematic (left) and reduced-kinematic orbit (right) vs. reduced-dynamic orbit, day 200/2002. The two figures in the middle show normal equation matrix for kinematic and reduced-kinematic orbit. Compared to the kinematic orbits, dynamic orbits are very smooth. In order to reduce the size of the small jumps in kinematic positions (blue line in the figure on the right), constraints can be applied from epoch to epoch to the kinematic position differences w.r.t. corresponding differences in the a priori dynamic orbit. In this case, we may speak of ‘‘reduced-kinematic’’ orbit determination, where the kinematic degrees of freedom are reduced by constraints to the dynamic orbit (red line in the figure on the right). The a priori dynamic 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. L - U factorization can be applied for such a block tridiagonal system using block forward elimination and back substitution.

Daily RMS of differences between kinematic and reduced-dynamic orbit

Fig. 8. CHAMP reduced-dynamic orbit estimated using

the LP linear combination of the L1 and P1 code measurements, day 200/2002, showing that LEO orbits can be determined with an accuracy of 10 cm based on single-frequency data only. This is possible if code measurements are available with an accuracy of ca 10-20 cm. However, pre-processing of phase measurements is more difficult in that case.

BackBack--up: Single Frequency POD up: Single Frequency POD

GOCE NearGOCE Near--Field MultipathField Multipath

Orbit Validation ReportsOrbit Validation Reports

Page 1: OVERLAPS for PSO: Reduced-dynamic orbit (ETRF, LOREF) Page 2: OVERLAPS for RSO: Reduced-dynamic orbit (ETRF, LOREF)

Page 3: Comparison PSO: Kinematic vs. Reduced-dynamic orbit (ETRF, LOREF) Page 4: Comparison RSO: Kinematic vs. Reduced-dynamic orbit (ETRF, LOREF) Page 5: Comparison RSO/PSO : Kinematic vs. Reduced-dynamic orbit (ETRF, LOREF) Page 6: Comparison RSO/PSO : Red.-dyn. orbit vs. Red.-dyn. orbit (ETRF, LOREF)

Page 7: SLR Residuals for PSO: Reduced-dynamic orbit Page 7: SLR Residuals for PSO: Kinematic orbit Page 8: SLR Residuals for RSO: Reduced-dynamic orbit Page 8: SLR Residuals for RSO: Kinematic orbit

Product Name: SST_PRM_2 Product Name: SST_PRM_2 1 1

2 2

3 3

0

sin2

sin2

sin2

cos2

e

e

e

q

q

q

q

=

=

=

=

Φ

Φ

Φ

Φ

Earth Orientation QuaternionEarth Orientation Quaternion

transformation into inertial frame with only

4 parameters!

Fig. 3. Earth orientation parameters represented by quaternions. Transformation between earth fixed reference frame (EFRF) and inertial reference frame (IFR) can easily be computed using quaternion algebra and computation of the rotation matrix can be avoided. This is very practical way to represent Earth orientation because users do not need to compute models and implement IERS Conventions. In addition, quaternions preserve ortonormality of the rotation during interpolation.

Fig. 4. Orbit validation report generated for RSO and PSO (example).

Kinematic vs. reduced-dynamic orbit

Fig. 7. At the 2nd GOCE User Workshop it was proposed to perform absolute calibration of the GOCE antenna using robot set-up with and without solar wing panel, see figure on the left. Three figures in the middle show impact of the near-field multipath, caused by the solar panel, on antenna phase pattern. One can see that iono-free pattern in considerably modified up to 1 cm. When kinematic POD is performed using such an antenna map we can expect differences up to 10 cm in the kinematic orbit, see figure above.

ReducedReduced--KinematicKinematic with relative constraints

0 1 2 3 0 1 2 3

1 0 3 2 1 0 3 2IRF EFRF

2 3 0 1 2 3 0 1IRF EFRF

3 2 1 0 3 2 1 0IRF EFRF

0 0

q q q q q q q q

q q q q q q q qX X

q q q q q q q qY Y

q q q q q q q qZ Z

− − −

− − −

− − −

− − −

=

DynamicDynamic ReducedReduced--DynamicDynamic