IONOSPHERE PERTURBATIONS IN GPS TIME AND FREQUENCY TRANSFER

S. PIREAUX, P. DEFRAIGNE, N. BERGEOT, Q. BAIRE and C. BRUYNINX

Département 1, Observatoire Royal de Belgique, 3 Av. Circulaire, 1180 Bruxelles, Belgique, sophie.pireaux@oma.be - tel. +32 (0)2 373 67 53 – Fax +32 (0)2 374 98 22

ROBRoyal Observatory of Belgium

[1] Bassiri & Hajj, Man. Geod. 18:280, 1993

[2] Defraigne et al, Int. J. Nav. Obs., Vol. 2008, 2008, Article ID 175468

[3] Fritsche et al, Geophys Res Letters 32, L23311, 2005, formula (14)

[4] Hernandez-Pajares et al, J. Geophys. Res., 112, B08417,2007

[5] Hernandez-Pajares et al, IGS workshop Miami 2008, oral presentation

[6] Solar- Terrestrial Center of Excellence (STCE), http://www.stce.be/index.php]

[7] Steigenberger et al, J. Geophys. Res.,111,B05402,2006

[8] Information Systems and Data Center of GFZ Potsdam: http://isdc.gfz-potsdam.de/gps-pdr

[9] TU Dresden: http://www.tu-dresden.de/ipg/reprocessing.html

ABSTRACT

The stability of time and frequency transfer (i.e. remote atomic clock comparison) with GPS is limited by the fact that GPS signals travel through the ionosphere (Figure 1).

In high precision geodetic time transfer (i.e. based on precise modeling of code and carrier phase GPS data), as dual-frequency GPS data are available, the so-called ionosphere-free combination of the two frequencies (noted P3L3) is used to remove the first order ionosphere effect. In this paper, we investigate the impact of second and higher order ionosphere perturbations on geodetic time transfer solutions.

The time transfer computations presented here have been done using the software ATOMIUM, developed at the Royal Observatory of Belgium, based on a least-square analysis of dual-frequency carrier phase and code measurements, which is able to provide clock solutions in Precise Point Positioning (PPP: zero difference) or single difference mode (Common View –CV-: difference between simultaneous observations of a satellite i by two stations p and q). The implementation of ionosphere higher

order terms in the existing Atomium P3L3-analysis procedure is described and an illustration of the impact of unmodeled second and higher order ionosphere effects on the time transfer solutions is provided. Second order ionosphere effects can reach about 4 picoseconds , which is at the level of precision of most stable atomic clocks, on a quiet day and up to more than 10 picoseconds in case of high ionosphere activity.

III. METHOD TO CORRECT IONOSPHERE PERTURBATIONS FOR FREQUENCY k (bending effect ignored)

A. 1st order ionospheric effect [3]

IV. RESULTS AND PERSPECTIVES

B. 2nd order ionospheric effect (thin shell model) [4]

C. 3rd order ionospheric effect [3]

I. GPS TIME AND FREQUENCY TRANSFER (TFT)

Figure 1 (See Section IIIB

for symbol definition)

22

2

2

1

2

212

2

2

1

2

13

)()(P

ff

fP

ff

fP

−−

−=

22

2

2

1

2

212

2

2

1

2

13

)()(L

ff

fL

ff

fL

−−

−=

214 PPP +−= 2

2

114 Lf

fLL −=

Solar-Terrestrial Center of Excellence [6]

For station p or similarly q, the GPS measurements on the code Pk and phase Lk , for frequency k (1 for

L1 and 2 for L2) and corresponding wavelength λk, can be written in length units as

( )

( ) piL

i

ppp

i

p

i

p

pi

Ppp

i

p

i

p

IINzpdctcL

IIzpdctcP

i

i

3333

333

)(

32)(

32

32

+−++++∆+∆+=

⋅−⋅++++∆+∆+=

εεεελλλλττττρρρρ

εεεεττττρρρρ

I1 is eliminated through the so-called ionosphere-free combination (k=3):

so that the corresponding observation equations contain new factors for 2nd and 3rd order ionosphere effects:

geomagnetic field projectionover Line of Sight direction (LOS)

at the Iono Piercing Point (IPP)

STECBILOSBIPPkk

⋅⋅⋅= −θθθθαααα cos22

The magnitude and sign of I2 depend on the i-p signal direction, the actual STEC and the geomagnetic field B values (Figure 1). STEC is obtained

from L4P4 (see Section IIIA) and BIPP is computed using the accurate International Geomagnetic Reference (IGR) model, as it allows to reduce errors in I2 up to 60% wrt. a dipolar model [4].

( ) ( ) ( ) ( )

⋅−⋅−−−= i

p

slipscyclewithoutarcp

ipi

pi

pi

DCBcDCBcPLLSTEC

444

4

1

1αααα

where P1-P2 Differential Code Biases (DCB), assumed constant during a day, are read from CODE ionex files; and < > means average.

( )

( ) pikkkLk

i

pp

i

p

i

p

i

pk

pi

kkkPp

i

p

i

p

i

pk

IIINzpdctcL

IIIzpdctcP

321

321

−−−++++∆+∆+=

⋅+⋅+++++∆+∆+=

)(

32)(

εεεελλλλττττρρρρ

εεεεττττρρρρ

STECIkk

⋅= 11 αααα

where ρip is the geometric distance i-p ; ∆tp is the station clock synchronization error; ∆τi is the satellite clock

synchronization error; zpdp is the tropospheric path delay for station p; I1k, I2k and I3k are ionosphere 1st, 2nd

and 3rd order delays; Nip are phase ambiguities; εP and εL are the error terms in code and phase, containing

noise and multipath.

4

2,1

max2,1

2437

f

N ηηηηαααα

⋅⋅−=3STECI

kk⋅= 33 αααα

−⋅−=

2

2

2

1

4

113.40

ff1αααα

03 =1αααα

2

2,1

2,1

3.40

f+=1ααααwith

with

Slant Total Electron Content

3

2,1

2,1

7527

f

c⋅−=2αααα

( )2121

32

7527

ffff

c

+⋅⋅

⋅−=2αααα

The 1st order ionosphere effect is used to estimate STEC for each measurement from the geometry-free combination (k=4) (I2 and I3neglected),

according to

with

2

2

2

1

max3

3

2437

ff

N

⋅

⋅⋅−=

ηηηηαααα3

Alternatively, STEC can be computed using

leading to similar results.

( ) ( ) ( ){ }

⋅−⋅−−= i

pphasewithsmoothed

pi

pi

pi

DCBcDCBcPPSTEC 12

4

1

1αααα

In the ionosphere 3rd order contribution, the magnetic field term can be neglected at sub-mm error level, leading to the above formula. The shape factor η is around 0.66 and the peak electron density along the signal propagation path, Nmax, was determined by

a linear interpolation between a typical ionosphere situation and a solar maximum one.The Vertical TEC (VTEC), which is TEC along a vertical trajectory,

is taken as the projection, via the ionosphere Modified Single Layer Model mapping function, of (STEC)i

p from Section IIA with αMSLM=0.9782, R =6371 km, H=506.7km:

( )[ ]( )[ ] ( ) 1218

18

12

max10201055.4

1038.155.4

10620⋅+⋅−⋅

⋅−

⋅−= VTECN

VTECzfSTECMSLM

⋅= )(

( )zHR

Rzf MSLMMSLM ⋅⋅

+−≡

⊕

⊕ αααα2

2

cos11)(

Station clocks

p q

Ionosphere

Troposphere

Empty Space

Disturbed propagation

i

z

A good indicator of the state of the ionosphere is the Total Electron Content (TEC), that is the integrated electron

density inside a cylinder column of unit area along a certain direction between Earth ground and satellite altitude. Slant TEC (STEC) is along i-p direction and 1 TECU=1016 e-/m2 (Figure 3ab). Ionosphere effects in GPS are proportional to STEC and I1 is used to estimate STEC (see section III) .

00

0

The table in Section III illustrates the need to take ionosphere

corrections into account in P and L measurements.

However, to be coherent, in addition to the correction of I2 and I3 on

GPS code and phase data, we should also use satellite orbit and clock

products computed with I2 (and I3) correction(s) in order to estimate

the impact of the ionosphere on station clock synchronization errors

via ATOMIUM. Current IGS products do not take I2 and I3 into

account. But reprocessed orbits [7] taking, among other, higher order

ionosphere into account are available at analysis centers [8] or [9].

Unfortunately, they do not provide satellite clocks. This is why we

present here the impact of ionosphere on clock solutions via

ATOMIUM in CV mode (Figure 7, 8 and 9), as the satellite clock is

eliminated in CV. We choose the link BRUS-ONSA, i.e. Brussels-

Onsala (Sweden) and the day of ionosphere storm (November 30,

2003).

Figure 7 presents the effect of using the reprocessed orbits together

with I2 and I3 corrections. Some variations can be attributed to

differences in the reprocessing other than I2 and I3.

Figure 8 shows the effect of using only the I2 and I3 corrections on

GPS data, without using reprocessed orbits. We see an effect up to 10

picoseconds during the ionosphere storm .

The I3 effect shown in Figure 9 is at the present noise level of GPS

observations.

Note also that, in CV, it is the differential ionosphere effect between

the two stations that influences the solution.

A. Ionosphere corrections in P3L3 estimates

B. Impact of ionosphere correction on station clock estimates

Station: BRUS, ONSA, OPMTYear: 2003Day: 303Sat: GPS 18Method: L4P4+DCB from CODE GIM

mjd

Iono storm

Station: BRUS, ONSA, OPMTYear: 2007Day: 70Sat: GPS 18Method: L4P4+DCB from CODE GIM

mjd

Computed STEC above stations for satellite 18 (TECU)

GPS code

and phaseobservations

L3 and P3

GPS code

and phaseobservations

L3 and P3

SolidEarth tides

+Ocean

Loading

+PhaseCenter

Variation

SolidEarth tides

+Ocean

Loading

+PhaseCenter

Variation

Correctionson the signal

+

Partial Differential

Correctionson the signal

+

Partial Differential

LeastSquare

Inversion

LeastSquare

Inversion

Estimate of :- Station clock

∆tpor ∆tpq

- Station position

(xyz)p or (xyz)pq

- Zenit Path Delayzpdp, zpdq

- AmbiguitiesNi

p or Nipq

Estimate of :- Station clock

∆tpor ∆tpq

- Station position

(xyz)p or (xyz)pq

- Zenit Path Delayzpdp, zpdq

- AmbiguitiesNi

p or Nipq

Ionosphere

I2, I3 correctionsusing STEC from

P1,P2 or L4,P4

Figure 2

B. The ATOMIUM software

The present study is based on the ATOMIUM sofware [2]. It estimates, among other parameters, the station(s) p (and q) clock

synchronization error after a least square adjustment (Figure 2). It uses by default the IGS products for satellite

clocks and orbits.

Time and Frequency Transfer (TFT) means comparison of remote atomic clocks. As GPS satellites are

equipped with atomic clocks, which are synchronized on a same reference time scale, this latter can be used as a common reference in order to determine the synchronization error between two remote clocks on Earth. This technique is currently of major use for the realization of TAI (Temps Atomic International), the basis of the legal

time UTC (Universal Time Coordinated), computed by the “Bureau International des Poids et Mesures” (BIPM) from an ensemble of about 300 atomic clocks distributed in about 40 laboratories around the world.

A. The principle

II. RELEVANCE OF STEC FOR IONOSPHERE PERTURBATIONS

All the terms in above equation are estimated using International GNSS Service (IGS) precise satellite orbits (sp3) and satellite clocks (clk), so that finally, the least square inversion provides the solution for ∆tp, i.e. the clock synchronization error between the atomic clock connected to the GPS receiver and the IGS Time scale

at each epoch. In parallel, the station position and tropospheric zenith delays are estimated as a by product.

Figure 4a

I12 (nanoseconds)

Figures below illustrate I12, I23 and I33 on a quiet (left) versus a ionosphere-stormy day (right).

Iono stormStation: ONSAYear: 2003Day: 303Sat: all GPSMethod: STEC from L4P4

IGR magnetic field model , MSLM mapping fct

0h 24h12h

0h 24h12h

0h 24h12h

Figure 5b

Figure 6b

0h 24h12h

I23 (nanoseconds)

0h 24h12h

Figure 4b

0h 24h12h

Figure 3b

0h 24h12h

Figure 3a

Figure 9

Difference bet. estimated station clock synchro errorwith I2, I3

versus without I3 (seconds)The IGS products are used in both cases

Figure 8

Difference bet. estimated station clock synchro errorwith I2, I3

versus without I2, I3 (seconds) The IGS products are used in both cases

Difference bet. estimated station clocksynchronization error

no I2, no I3, classical igs orbitsversus with Iono2, with Iono 3 and reprocessed products (seconds)

Stations: BRUS-ONSAYear: 2003Day: 303Sat: all GPS

Figure 7

0h 24h12h

0h 24h12h

0h 24h12h

Up to 10 ps peak to peak

during iono storm

STEC is function of the satellite elevation, time of day, time of year, ionosphere particular conditions (as seen in Figure 3b during the ionospheric storm of november 30, 2003) and solar cycle. Hence, GNSS ionosphere induced errors will increase in the next few years due to the increasing solar activity since the beginning of this

24th sunspot cycle.

Orders of magnitude of ionosphere effects I1, I2, I3 in GPS phase

(for code, see converting factor in code measurement formula, Section 1)

I3

I2

I1

Ionosphere

effect

~0 – 3 ps

~0 – 130 ps

~30 ns -100 ns

Delay in L1L2

per 100 TECU

~ 0 – 2 ps

90% of I23~ 0 – 45 ps

99.9% of I123…0

Relevance [5]Delay in L3

per 100 TECU

Figure 5a

Station: ONSAYear: 2007Day: 70Sat: all GPSMethod: STEC from L4P4

IGR magnetic field model , MSLM mapping fct

0h 24h12h

Figure 6a

I33 (nanoseconds)