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Excess phase computation
S. Casotto, A. Nardo, P. Zoccarato, M. Bardella
CISAS, University of Padova
Basic observables
11 NIccL ik
ik
ik
ik
ik
222
21
2 NIf
fccL i
kik
ik
ik
ik
Phase observables
ikI
ik
i
k
Transmitter clock error
Receiver clock error
Ionospheric delay
Tropospheric delay
ik 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
refk
refk
refk
refk IccL 1
refk
refk
refk
refk I
f
fccL
22
21
2
occk
occk
occk
occk
occk IccL 1
occk
occk
occk
occk
occk I
f
fccL
22
21
2
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
refk
occk
occk
refoccrefk
occkrefocc IIccLLL ,1,11
refk
occk
occk
refoccrefk
occkrefocc I
f
fI
f
fccLLL
22
21
22
21
,2,22
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:
ik
ik
ik I
f
fI
f
fILLL
1
22
21
22
21
214
Hence the ionospheric delay affecting the reference link can be computed as:
1
22
21
4
f
f
LI refk
1
22
21
4111
f
f
LccLIccLexc refoccref
kocck
refk
refoccrefk
occkL
1
22
21
422
21
222
21
22
f
f
L
f
fccLI
f
fccLexc refoccref
kocck
refk
refoccrefk
occkL
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 + 19.0 s2. GPS = UTC + leapseconds - 19. s3. 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.
The times scales used are:
1. GPS,
2. TAI.
3. UTC.
4. TT/ET.
Reference frame transformation
)(ERARZ
)( XpRY
)( YpRX
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_ANPDefine rotation matrices:
Transformation matrix
TYXZ XpRYpRERARA )]()()([
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.
qi
qxt i
qyt i
qzt i
qwt i
. i 1, . . . ,n # q
qxt
qyt
qzt
qwt
. #
)(2)(21)()(2)()(2)()(2)()(2
)()(2)()(2)(2)(21)()(2)()(2
)()(2)()(2)()(2)()(2)(2)(21
22
22
22
tqtqtqtqtqtqtqtqtqtq
tqtqtqtqtqtqtqtqtqtq
tqtqtqtqtqtqtqtqtqtq
yxxwzyywzx
zwzyzxzwyx
ywzxzwyxzy
R
fixedbodyT
inertial rRr
Quaternion time series Interpolated 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
...3,2,1)()(1
., ndtttc
dtn
kiipcgpsipcleon
ki rr
The light time computation in done by solving iteratively the following implicit equation :
kidtfor 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)
wvur δwδvδuinertialantenna ,
inertialantennainertialsscenterofmainertialantenna ,,, rrr
u rgps,geo
|rgps,geo |
uvw
usv
s
GPS satellite-geocenter unit vector
Solar panel axis unit vectorGPS satellite-sun direction
uvw
Antenna offset components in the satellite-fixed frame
Antenna offset in the inertial frame
Antenna position in the inertial frame
Doppler shift
cvN
cvN
c
VaGMvVff
GPS
LEOGPSEARTHLEOLEO
/1
/12/22/1
20
2
0
1
.01227600000
.01575420000
0,2
0,1
c
c
c
f
Nominal wavelengths at the transmitter:
Wavelength - frequency relation:
Frequency at the receiver:
LEOV
GPSV
LEOv
GPSv
N
a = 6823.287 km
0 2c = -6.96927d-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)222
111
L
L
Phase observables corrected for Doppler shift
2 Raw phase measurements
Where:
[N. Ashby, 2003]
Discrimination
OCC
Earth centre
Leo
Occulted GPS satellite
Reference GPS satellite
fRe
Occf Re
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
SWOrDWorst 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/ss
Best case
m/s
seconds
CHAMP
SWOrD
m/s
seconds
CHAMP
SWOrD
m/s
seconds
CHAMPSWOrD
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
secondsseconds
m
The moving average has eliminated all the spikes
L4 smoothing, time derivative, best case
CHAMP
SWOrD
m/s
seconds
CHAMPSWOrD
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.
THANK YOU!