excess phase computation s. casotto, a. nardo, p. zoccarato, m. bardella cisas, university of padova

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!