Excess phase computation

1. Excess phase computation S. Casotto, A. Nardo, P. Zoccarato, M. Bardella CISAS, University of Padova

2. Distance between transmitter and receiver antennas Transmitter clock error Receiver clock error Ionospheric delay Tropospheric delay Basic observables Phase observables • Clock errors must be removed from the observations. • Transmitter clock corrections is estimated in the orbit determination task. Ambiguities are not considered, since only the time derivative of the phase delay is needed.

3. Removing clock errors Hence the basic phase observables for the reference link are: • The removal of the receiver clock errors is done by using single differenced phase observables. • A second link is needed, we call this second link the “reference” link, opposite to the leo-occulted GPS link (the “occulted” link). • The main feature of the reference link is that it is not affected by tropospheric delay, since the reference GPS satellite is higher than LEO. The occulted link is affected by tropospheric delay:

4. Ionospheric delay in the reference link The single-differenced phase observables are: We note the presence of the ionospheric delays related to the reference link in the single-differenced phase observables.

5. Removing ionosphere delay in the reference link The removal of the ionosphere delay in the reference link is done by using the L4 combination. This combination is also known as the “geometry-free” combination, since the geometry terms are removed, as the clock error terms and the topospheric delay. The combination is defined by the following equation: Hence the ionospheric delay affecting the reference link can be computed as: so the excesse phase can be defined as:

6. Level 2 data processing flow GPS satellite precise orbits and clocks corrections. LEO precise orbit. Earth rotation parameters. (IERS bull. A) Leo attitude data. (Quaternions, 1 min.) Sun ephemeris (DE405). Leo ephemeris interpolation at the observation epoch. Leo attitude data interpolation and antenna offset correction. Transformation of the GPS satellites and Leo Ephemeris into Inertial frame. (SOFA libraries.) Light time computation for the reference and occulted link. Loop over observation epochs Discrimination between the reference and occulted link Doppler shift correction due to relative motion. RINEX data. (High rate, phase) Phase delay computation

7. Time scale transformations • The times scales used are: • GPS, • TAI. • UTC. • TT/ET. • The conversions between different time scales are defined by: • TAI = GPS + 19.0 s • GPS = UTC + leapseconds - 19. s • TT/ET = TAI + 32.184 s = GPS + 19.0 s + 32.184 s • Conversion between TAI and TT/ET is needed to deal with the JPL DE405 ephemerides, because their time scale is the TT/ET. • Leap-seconds are computed by the SOFA “DAT” subroutine.

8. Reference frame transformation Both the inertial and the terrestrial reference frames are Earth Centred. The transformation in performed by the subroutine “itrf2eci_iau1980”. Position and epoch Earth rotation parameters X_itrf, Y_itrf, Z_itrf, epoch_gps_mjd XP_arcsec, YP_arcsec, ut1_minus_utc, DDP80_arcsec, DDE80_arcsec, time_tag_UTC_mjd Epoch conversion • Earth rotation angle computation (ERA) • Through SOFA subroutines: • iau_NUT80, • iau_OBL80, • iau_EQEQ94, • iau_ANP Earth rotation parameters interpolation at epoch_UTC_mjd GPS > TAI TAI > UTC TAI > TT Define rotation matrices: Transformation matrix Nutation and precession are neglected, since the time span of the occultation event is little in comparison with the caractheristic time of nutation and precession.

9. Ephemeris interpolation Ephemeris needed: • Sun (to compute GPS satellites attitude, DE405 ephemeris). • Leo (CHAMP .sp3, SWOrD .sp3, 1 minute sampled). • GPS satellites (IGS precise orbit, 15 minutes sampled), • The sun position is interpolated by using DE405 native subroutines. • The Leo and GPS satellites ephemeris are interpolated by using a polynomial of degree 9 or 11. • Leo and GPS velocity are computed by differentiating the interpolated positions (time span 0.05 seconds).

10. Antenna offset correction (Leo) The ephemeris of the Leo are referred to the centre of mass. The attitude of the Leo is defined by a time serie of quaternions. Quaternion time series Interpolated quaternion at the epoch of observation Rotation matrix Antenna offset in the inertial reference frame

11. Light time computation The light time computation in done by solving iteratively the following implicit equation : Interpolate GPS satellites ephemeris at the epoch t – dt(0). Compute apparent sun position in the inertial frame at epoch t – dt(0) 3 iterations for the unknown Correct for antenna offset Compute improved light time dt(1) We note that the position vectors are referred to the Antenna Phase Centres. The sun apparent position is needed to define the attitude of the GPS satellite.

12. Antenna offset correction (GPS) GPS satellite-sun direction Solar panel axis unit vector Antenna offset components in the satellite-fixed frame Antenna offset in the inertial frame Antenna position in the inertial frame GPS satellite-geocenter unit vector

13. Doppler shift Nominal wavelengths at the transmitter: Earth potential values at Leo and GPS satellite coordinates Wavelength - frequency relation: Frequency at the receiver: Leo and GPS satellite velocity in the inertial frame [N. Ashby, 2003] Phase observables corrected for Doppler shift Unit vector of the Leo–GPS satellite direction. = 6823.287 km Leo semimajor axis (CHAMP) Where: = -6.96927d-10 Raw phase measurements

14. Discrimination The discrimination between reference and occulted GPS satellite is based on the following condition: Angle between Leo-Earth Centre and reference link directions Earth centre Angle between Leo-Earth Centre and occulted link directions Leo Occulted GPS satellite Reference GPS satellite

15. L1 phase delay Day: 22 Month: January Year: 2004 Best case CHAMP SWOrD Worst case • IGS precise orbit and clocks corrections. • IERS bullettin A ERPs. • DE405 Lunar and Planetary ephemerides. • CHAMP precise orbit. CHAMP SWOrD Test

16. Best case Maximum of residuals: 1 m Worst case Spikes Maximum of residuals: 12 m Presence of drift. Residuals

17. Best case CHAMP SWOrD m/s CHAMP SWOrD m/s seconds m/s s seconds Time derivative, best case

18. CHAMP SWOrD m/s CHAMP SWOrD m/s seconds seconds Time derivative, worst case

19. m/s Best case Residuals are very noisy. Spikes dominates residuals preventing a meaningful investigation of the plots. m/s Worst case seconds seconds Time derivative residuals

20. FILTER: Moving average over 600 data points. Best case CHAMP SWOrD Worst case CHAMP SWOrD Spikes are due to errors related to the electronics of the receiver. These errors affect the L2 tracking and are present in the phase delay observables trough the ionospheric delay removal in the reference link. L4 smoothing

21. Best case Maximum absolute value of the residuals: 1.6 m. Worst case Maximum absolute value of the residuals: 12 m. L4 smoothing, residuals

22. The moving average has eliminated all the spikes m m seconds seconds L4 smoothing, time derivative, best case CHAMP SWOrD

23. CHAMP SWOrD m/s CHAMP SWOrD m/s seconds seconds L4 smoothing, time derivative, worst case

24. BEST CASE Maximum absolute value: 0.1 m/s at the beginning of the occultation, but at the end the residuals are bounded. m/s seconds WORST CASE A bias of 0.12 m/s is present during all the occultation event. m/s Discontinuity due to the filter used. seconds L4 smoothing, time derivative residuals

25. Conclusions • SW modules were developed for level 2 data generations, based on single-differenced observables. • Comparison with CHAMP RO products shows the presence of a drift in the residuals US-CHAMP affecting several occultation events. • Drifts do not depend on errors on position (LEO and GPS satellite). • Drifts seems to be independent from frequency. • Spikes can be easily removed by filtering the L4 observable (moving average filter). • Drifts still remain after data filtering.

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