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

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

Removing clock errors The occulted link is affected by tropospheric delay: 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. Hence the basic phase observables for the reference link are:

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.

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:

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

Time scale transformations The conversions between different time scales are defined by: 1.TAI = GPS s 2.GPS = UTC + leapseconds s 3.TT/ET = TAI s = GPS s 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. The times scales used are: 1.GPS, 2.TAI. 3.UTC. 4.TT/ET.

Reference frame transformation Both the inertial and the terrestrial reference frames are Earth Centred. The transformation in performed by the subroutine “itrf2eci_iau1980”. X_itrf, Y_itrf, Z_itrf, epoch_gps_mjd XP_arcsec, YP_arcsec, ut1_minus_utc, DDP80_arcsec, DDE80_arcsec, time_tag_UTC_mjd GPS > TAI TAI > UTC TAI > TT Position and epoch Epoch conversion Earth rotation parameters Earth rotation parameters interpolation at epoch_UTC_mjd Earth rotation angle computation (ERA) Through SOFA subroutines: 1.iau_NUT80, 2.iau_OBL80, 3.iau_EQEQ94, 4.iau_ANP 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.

Ephemeris interpolation Ephemeris needed: 1.Sun (to compute GPS satellites attitude, DE405 ephemeris). 2.Leo (CHAMP.sp3, SWOrD.sp3, 1 minute sampled). 3.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).

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 seriesInterpolated quaternion at the epoch of observation Rotation matrix Antenna offset in the inertial reference frame

Light time computation Interpolate GPS satellites ephemeris at the epoch t – dt(0). Compute apparent sun position in the inertial frame at epoch t – dt(0) Correct for antenna offset Compute improved light time dt(1) 3 iterations The light time computation in done by solving iteratively the following implicit equation : for the unknown 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.

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

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

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

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

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

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

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

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

Best case 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. CHAMP SWOrD L4 smoothing Worst case CHAMP SWOrD FILTER: Moving average over 600 data points.

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

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

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

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

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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