Ann. Geophys., 29, 1383–1399, 2011
www.ann-geophys.net/29/1383/2011/
doi:10.5194/angeo-29-1383-2011
© Author(s) 2011. CC Attribution 3.0 License.

Annales
Geophysicae

Higher order ionospheric effects in GNSS positioning
in the European region

Z. G. Elmas1,*, M. Aquino1, H. A. Marques2, and J. F. G. Monico2

1Institute of Engineering Surveying and Space Geodesy (IESSG), The University of Nottingham,
Triumph Road, Nottingham, NG7 2TU, UK
2Department of Cartography, Sao Paulo State University, Presidente Prudente, Sao Paulo, Brazil

* Invited contribution by Z. G. Elmas, recipient of the EGU Outstanding Student Poster Award 2010.

Received: 14 February 2011 – Revised: 6 June 2011 – Accepted: 1 August 2011 – Published: 22 August 2011

Abstract. After removal of the Selective Availability in
2000, the ionosphere became the dominant error source for
Global Navigation Satellite Systems (GNSS), especially for
the high-accuracy (cm-mm) demanding applications like the
Precise Point Positioning (PPP) and Real Time Kinematic
(RTK) positioning.

The common practice of eliminating the ionospheric error,
e.g. by the ionosphere free (IF) observable, which is a lin-
ear combination of observables on two frequencies such as
GPS L1 and L2, accounts for about 99 % of the total iono-
spheric effect, known as the first order ionospheric effect
(Ion1). The remaining 1 % residual range errors (RREs) in
the IF observable are due to the higher – second and third,
order ionospheric effects, Ion2 and Ion3, respectively. Both
terms are related with the electron content along the signal
path; moreover Ion2 term is associated with the influence of
the geomagnetic field on the ionospheric refractive index and
Ion3 with the ray bending effect of the ionosphere, which can
cause significant deviation in the ray trajectory (due to strong
electron density gradients in the ionosphere) such that the er-
ror contribution of Ion3 can exceed that of Ion2 (Kim and
Tinin, 2007).

The higher order error terms do not cancel out in the (first
order) ionospherically corrected observable and as such,
when not accounted for, they can degrade the accuracy of
GNSS positioning, depending on the level of the solar activ-
ity and geomagnetic and ionospheric conditions (Hoque and
Jakowski, 2007). Simulation results from early 1990s show
that Ion2 and Ion3 would contribute to the ionospheric error
budget by less than 1 % of the Ion1 term at GPS frequen-
cies (Datta-Barua et al., 2008). Although the IF observable
may provide sufficient accuracy for most GNSS applications,

Correspondence to:Z. G. Elmas
(isxzge1@nottingham.ac.uk)

Ion2 and Ion3 need to be considered for higher accuracy de-
manding applications especially at times of higher solar ac-
tivity.

This paper investigates the higher order ionospheric effects
(Ion2 and Ion3, however excluding the ray bending effects
associated with Ion3) in the European region in the GNSS
positioning considering the precise point positioning (PPP)
method. For this purpose observations from four European
stations were considered. These observations were taken in
four time intervals corresponding to various geophysical con-
ditions: the active and quiet periods of the solar cycle, 2001
and 2006, respectively, excluding the effects of disturbances
in the geomagnetic field (i.e. geomagnetic storms), as well as
the years of 2001 and 2003, this time including the impact of
geomagnetic disturbances. The programRINEXHO (Mar-
ques et al., 2011) was used to calculate the magnitudes of
Ion2 and Ion3 on the range measurements as well as the to-
tal electron content (TEC) observed on each receiver-satellite
link. The program also corrects the GPS observation files
for Ion2 and Ion3; thereafter it is possible to perform PPP
with both the original and corrected GPS observation files
to analyze the impact of the higher order ionospheric error
terms excluding the ray bending effect which may become
significant especially at low elevation angles (Ioannides and
Strangeways, 2002) on the estimated station coordinates.

Keywords. Radio science (Ionospheric propagation)

1 Introduction

After removal of Selective Availability in 2000, the iono-
sphere became the dominant error source in Global Navi-
gation Satellite Systems (GNSS) error budget (El-Rabbany,
2002). The ionosphere is a medium of free electrons and
ions and as such its dispersive nature makes the ionospheric

Published by Copernicus Publications on behalf of the European Geosciences Union.

http://creativecommons.org/licenses/by/3.0/


1384 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

refractive index different from unity – this difference in the
refractive index leads to the group and phase velocities of
the GNSS signals to differ from each other during propaga-
tion through the ionosphere, such that the group velocity de-
creases (leading to the group delay i.e. code measurements
longer than the geometric range) and the phase velocity in-
creases (leading to the carrier phase advance i.e. carrier phase
measurements shorter than the geometric range).

When the phase and group velocities are affected, the ray
direction is also likely to be affectedunlessthe wave is travel-
ling perpendicularly to the gradient in ionospheric refractive
index (Cairo and Cerisier, 1976). This effect, also known as
the ray bending effect, is inversely proportional to the signal
frequency and is highly dependent on the satellite elevation
angle. The error due to the ray bending is orders of magni-
tude smaller than the first order ionospheric error; it is indeed
comparable to that of the higher order error terms (Petrie et
al., 2010).

The diffractive and ray-bending effects of the ionosphere
on GNSS signals are neglected in this work; only the part
of the Ion3 term quadratic in terms of the electron density
is analysed, although the third order error due to ray bend-
ing may, under some conditions, exceed or be comparable
in magnitude to the second order error term (Hoque and
Jakowski, 2008). This negligence of the ray bending effect
assumes that the GNSS signals travel along the straight LoS
path between the receiver and satellite (Hoque and Jakowski,
2007) instead of two slightly different paths (bent and LoS
paths). This assumption (of neglecting a bent path) then im-
plies that the TEC and geomagnetic field effect along the
signal path are the same for different (e.g. L1 and L2) signal
frequencies. The RINEXHO program does not estimate cor-
rections for the bending effect thus thecorrected(for Ion3)
measurements will still be contaminated by the ray bending
effect. It should also be mentioned at this point that for the
estimation of the Ion2 term, the ionosphere is assumed as a
single thin shell at 450 km and the magnetic inductionB is
computed at the ionospheric pierce point (IPP) and not along
the ray path of the signal. This assumption is expected to
introduce some errors to the computation of the Ion2 term
however it has less computational burden than the ray path
approach.

Subsequently when positioning is performed with the cor-
rected and uncorrected measurements an elevation cutoff an-
gle of 10◦ is considered in the PPP. As shown by Hoque
and Jakowski (2007), the ray bending effect on the GNSS
signals becomes significant especially at low satellite eleva-
tion angles – below 10◦; thus this cutoff angle is thought
to be an appropriate threshold for comparing the position-
ing results (considering the corrected and uncorrected ob-
servations) with negligible contribution by the ray bend-
ing effect. More detailed analyses about the impact of ray
bending on the GNSS signal propagation and observations
have been shown, among other researchers, by Hoque and
Jakowski (2008), who provided an empirical formula for

the geometric bending of the GPS signals; Hartmann and
Leitinger (1984), who considered the geometric bending of
the signals in their analysis of the RREs due to the atmo-
sphere; and Petrie et al. (2010), who used the International
Reference Ionosphere (IRI) 2007 model (Bilitza and Rein-
ish, 2008) to estimate the potential size of the ray bending
effect on the estimated GPS parameters and positioning.

The different orders of the ionospheric error terms (Ion1,
Ion2 and Ion3, as denoted in this work for the first, second
and third order terms) can have magnitudes that are observed
to change according to the background solar, ionospheric and
magnetic conditions. At times of high background solar ac-
tivity, as during the peaks of the solar cycle or active days
of an ionospheric storm, the greater amount of solar radia-
tion causes increased levels of electron density in the iono-
sphere. This can cause the slant range delay on the GPS L1
signal link to be as large as 100 m in the uncorrected observ-
able (the error contributed by the Ion1 term is about 10 to
100 m in general as shown by Klobuchar and Kunches, 2003
and Grewal et al., 2007). In the ionospherically corrected
(to the first order as in the dual frequency applications) ob-
servable, RREs can reach tens of centimetres (the Ion2 term
contributes about several centimetres of range error as shown
by Bassiri and Hajj, 1992, 1993; Kedar et al., 2003; Fritsche
et al., 2005; Hoque and Jakowski, 2006; Morton et al.,
2009) and the Ion3 term about 1 cm or less, e.g. during dis-
turbed ionospheric background conditions, which may hap-
pen due to geomagnetic storms, as discussed by Bassiri and
Hajj (1993), Brunner and Gu (1991) and Kedar et al. (2003).

In general, most of the ionospheric range error can be
eliminated depending on the method of positioning per-
formed, i.e. stand-alone or differential.In stand-alone mode,
users with a dual frequency receiver can account for the first
order ionospheric effect by the IF observable, whereas users
with a single frequency receiver can resort to an ionospheric
model like the Klobuchar model (Leick, 2004). The IF ob-
servable can eliminate about 99 % of the total ionospheric
effect (i.e. the Ion1 term) yielding an accuracy sufficient
for most GNSS applications; an ionospheric model like the
Klobuchar model, however, can correct about 50–60 % of the
total ionospheric effect and gives limited performance for the
users outside the mid-latitudes (Orus et al., 2002).In the dif-
ferential mode, for short baselines, the ionospheric error can
be eliminated by ionospheric corrections obtained from a ref-
erence station assuming a spatially and temporally correlated
ionosphere (for such short baseline) between the user and
reference. However, during adverse ionospheric conditions
spatial and temporal correlation of the ionospheric errors can
decrease.

For high accuracy demanding GNSS applications like PPP
and RTK, especially during the peaks of the solar cycle (and
can be worse during geomagnetic storms), Ion2 and Ion3
need to be considered, as they can cause range errors of a
few to tens of centimetres (Wang et al., 2005). The im-
pact of the higher order ionospheric errors on the estimated

Ann. Geophys., 29, 1383–1399, 2011 www.ann-geophys.net/29/1383/2011/



Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning 1385

station coordinates is studied by comparing the coordinates
estimated by PPP performed with the original and corrected
observation files, as done in this work.

Section 1 of this paper gives an introduction of the work
performed; Sect. 2 presents the literature review; Sect. 3 in-
troduces the methodology for calculating the higher order
ionospheric error terms by using the RINEXHO software
and for PPP; Sect. 4 presents the results for the calculated
values of Ion2 and Ion3, whereas Sect. 5 for the PPP results
from both the original and corrected observation files. The
paper concludes with discussion and suggestions for future
work in Sect. 6.

2 Literature review

Previous works related to the higher order ionospheric ef-
fects consider the ionospheric refractive index to derive the
error due to the ionosphere and the Chapman theory for the
layered structure of the ionosphere; they account for the in-
fluence of the geomagnetic field on the refractive index of the
(anisotropic) ionosphere and some may neglect the differen-
tial (frequency and satellite elevation angle dependent) bend-
ing effect on the GNSS signals. Different authors consider
these concepts differently to estimate the contribution of the
higher order ionospheric effects to the GNSS error budget.
In Wang et al. (2005) a multi-GNSS approach is taken to es-
timate the higher order error terms; and the authors focus on
the techniques of eliminating/estimating the ionospheric er-
rors through new linear combinations possible with the new
signal frequencies of the modernized GNSS.

Brunner and Gu (1991) observe that the RREs due to Ion2
and Ion3 in the dual frequency solution (i.e. using the IF ob-
servable) can reach several centimetres at low satellite eleva-
tions when the ionospheric electron density is as high as dur-
ing the active period of the solar cycle. Their proposed model
(using two separate Chapman functions to represent the top
and bottom sides of the ionospheric electron density profiles)
can eliminate the RREs to better than 1 mm by considering
a series expansion of the ionospheric refractive index, an ac-
curate ionosphere model (that provides the electron density
as a function of height in the ionosphere), the International
Geomagnetic Reference Field (IGRF) and also by account-
ing for the differential bending effect (important especially
at low elevation angles) of the GPS signals (along with the
tropospheric effect on the curvature of the GPS signals).

Bassiri and Hajj (1993) propose an approach which can
eliminate the RREs to the millimetre level by considering a
series expansion of the ionospheric refractive index, a thin
shell model for the ionosphere (as a sum ofE, F1 andF2
Chapman layers), a tilted dipole model for the geomagnetic
field and by neglecting the bending effect on the GPS signals
(since they assume that the bending effect is insignificant for
the satellite elevation angles greater than 30◦).

Ioannides and Strangeways (2002) show an analytical per-
turbation technique to determine the Ion2 and Ion3 terms for
which they account for the ray bending effect, and the au-
thors compare these results with those obtained from precise
ray-tracing calculations for the GPS frequencies. They con-
clude that the refracted geometrical path increases compared
with the LoS and there is a corresponding increase in the
TEC with an associated phase advance. If the influence of
the magnetic field for the L band signals is neglected then the
total curvature error is of comparable magnitude with the in-
crease in the geometrical path length related with the longer
curved path but of opposite sign; this represents the phase
advance. The authors thereafter suggest that both terms do
not need to be determined since the total curvature error is of
the same magnitude but opposite sign of the increase in the
geometrical path.

Kedar et al. (2003) focus on the impact of Ion2 on PPP by
considering a co-centric tilted magnetic dipole and the GIM
software (Global Ionospheric Mapping software from the Jet
Propulsion Laboratory – JPL, 2010) which provides two-
dimensional electron density maps for the ionosphere taken
as a thin layer at 400 km altitude. They use the satellite clock
and orbit productsnotcorrected for Ion2. In their comparison
of the coordinate time series corrected for Ion2 with the orig-
inal uncorrected coordinates, they find sub-centimeter level
error contribution by the Ion2 term to the GNSS positioning
error.

Wang et al. (2005) present a triple-frequency method for
correcting Ion2 and propose an ionosphere-free linear com-
bination method based on three frequencies, claiming that
their triple-frequency method can correct the effects to the
millimetre level. Moreover, they derive a formula for Ion3
for which they apply the semi-empirical ionospheric model
developed by Anderson et al. (1987) who define TEC as a lin-
ear function of the maximum electron density (Nmax) in the
ionosphere (Eq. 1) which givesNmax as 4.405×10−6 TEC;
this agrees well with the linear interpolationNmax≈ 4.415×

10−6 TEC applied by Fritsche et al. (2005). After obtaining
TEC from pseudorange (PR) measurements with L1 and L2,
Wang et al. (2005) can estimate Ion3 with an accuracy of
±1 mm.

Fritsche et al. (2005) investigate the impact of correcting
the satellite orbits and Earth rotation parameters while esti-
mating the station coordinates in a non-fiducial PPP approach
using the Bernese GPS Software V5.0 (Bernese, 2007). Fol-
lowing the mathematical model of Bassiri and Hajj (1993)
for Ion2 and Ion3 and using a thin shell model for the iono-
sphere, they apply GIMs for TEC data and a co-centric
tilted magnetic dipole for the geomagnetic field. They show
that both the station coordinates and the satellite orbits can
change at the centimeter level when the corrections for Ion2
and Ion3 are applied. They emphasize that a consistent cor-
rection method for RREs should use the corrected GPS ob-
servations and products rather than the corrected observa-
tions without taking into account the RREs for the products.

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1386 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

Hoque and Jakowski (2007) quantify the residual “phase”
error due to Ion2 and neglect that due to Ion3 (differential
bending of the GNSS signals also neglected) claiming that on
a disturbed day (e.g. about 100 TECU) the RRE due to Ion3
is at the sub-millimetre level. Their model, which can pro-
vide better than 2 mm accuracy for GNSS users in Germany,
does not require knowledge of the instantaneous geomag-
netic field since they take the geomagnetic field component
for a reference user position in central Germany. Knowledge
about the electron distribution along the propagation path is
also not required. These assumptions make the model appli-
cable for real time GNSS applications in central Germany.

Kim and Tinin (2007) use perturbation theory to study
the residual error in the dual frequency ionosphere free ob-
servable; they investigate in particular the Ion3 term associ-
ated with ray bending effect on the GNSS signals penetrat-
ing through an inhomogeneous ionosphere. Taking into ac-
count that Ion3 term includes not only the quadratic correc-
tion due to the refractive index but also the correction for the
ray bending effect, they show that the ray bending effect may
dominate the Ion2 error contribution. They consider both the
regular large scale and random small scale irregularities in
the ionosphere such that the latter can, at times, cause resid-
ual error comparable to or greater than that of the Ion2 term,
dominating the contribution to the residual error in the IF ob-
servable.

Pajares et al. (2007) consider the impact of Ion2 on the
satellites clocks and show that the estimates of the RREs
on the receiver coordinates, satellite positions and clocks are
correlated. Regarding the receiver positions, they infer that
Ion2 has a sub-daily impact of less than 1 mm during March
in 2001 – a year during the peak of the solar cycle. As for the
satellite positions, they show that Ion2 causes a daily mean
global southward displacement of several millimetres, de-
pending on the ionization level in the ionosphere. Regarding
the satellite clocks, which are most affected by the higher
order ionospheric effects according to their results, RREs
can cause deviations even larger than 30 picoseconds (corre-
sponding to about 1 cm) depending on the latitude and local
time of the receiver position.

Datta-Brua et al. (2008) show that, unlike the Ion1 term
which has the same magnitude but opposite signs for the
group and carrier phase measurements, the Ion2 and Ion3
have different magnitudes and signs for these two types of
measurements. For this reason, the authors claim that the
higher order errors accumulate in the carrier smoothing of
the IF (to the first order) code observable; they authors show
that the errors in the carrier-smoothed code measurements
are mostly due to Ion2 and Ion3. In other words theunac-
countedhigher order group errors contribute to the error in
the carrier smoothing. Although can be neglected in many
applications, these residual errors can be crucial in high pre-
cision applications.

Pajares et al. (2008b) focus on Ion2 and different methods
to obtain slant TEC (STEC) and the geomagnetic field com-

ponent projected onto the receiver-satellite path. Considering
the error due to Ion2 on the positioning, they emphasize that
correction for Ion2 must be applied to all fiducial coordinates
– application only to the unknown station (user) can lead to
errors in the estimated coordinates that can be worse than
applying no correction at all at the any receiver involved.

Kim and Tinin (2011), in a more recent study, explore the
possible ways of eliminating the higher order ionospheric er-
ror terms considering a multi-frequency GNSS approach and
show how the GNSS accuracy can be improved considering
the propagation of the signals through an inhomogeneous and
random structure of the ionosphere. Through numerical sim-
ulation they show that the systematic, residual ionospheric
error can be significantly reduced (under certain ionospheric
conditions) through triple frequency GNSS.

Moore and Morton (2011) focus on the Ion2 term intro-
duced by the interaction between the GNSS signal and the
magnetic field of the Earth. The anisotropy of the ionosphere
causes the right hand circularly polarized (RHCP) GPS sig-
nals propagate in two (ordinary and extraordinary) modes,
as a linear combination of them, depending on the angle
between the GPS signal wave normal and the Earth’s mag-
netic field. These two modes correspond to two different
magneto-ionic polarizations each with a particular refractive
index that needs to be considered in the Ion2 term. The au-
thors show that near the geomagnetic equator signals arriving
from the north propagate with the ordinary polarization (as-
sociated with left hand circularly polarized wave) yielding a
“positive” Ion2 term for the carrier phase; and those arriving
from the south propagate in the extraordinary mode polariza-
tion (associated with right-hand circularly polarized wave)
making the Ion2 term “negative” for the carrier phase. A
positive Ion2 term corresponds to the presence of error still
to be accounted for in the (first order) ionospherically cor-
rected measurements. The authors also point out a miscon-
ception in the work of Bassiri and Hajj (1993) who assume
that the left hand circularly polarized (LHCP) component of
GPS signals propagates in the ordinary mode and do not re-
alize the fact that the RHCP signal component may indeed
travel in either of the propagation modes. Moore and Mor-
ton (2011) show that the magneto-ionic polarization of the
predominantlyRHCP GPS signal depends on the direction
of the GPS signal wave vector with respect to the magnetic
field line. Considering three different geographic locations to
show the influence of this fact on the Ion2 term, Moore and
Morton (2011) show that Ion2 is asymmetric with respect to
the geomagnetic equator such that depending on the magni-
tude of the angle between the wave vector and the magnetic
field line, a RHCP wave has different propagation modes thus
considering only one propagation mode is expected to lead to
mismodelling inaccuracies in estimating the error due to the
Ion2 term.

From the literature review it can be understood that, while
estimating the magnitudes of the errors due to the Ion2
and Ion3, higher accuracy can be achieved by (1) using a

Ann. Geophys., 29, 1383–1399, 2011 www.ann-geophys.net/29/1383/2011/



Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning 1387

 

TRO1

ONSA

HERS 

MATE

Fig. 1. The four IGS stations considered for this work in Europe. Station coordinates are 
provided in Table 1.  

1 

2 
3 

4 

5 

6 

 

 

 

312 313 314 315 316
0

100

200
STEC (TECU)

 

 

312 313 314 315 316
0

100

200

 

 

312 313 314 315 316
0

100

200

 

 

312 313 314 315 316
0

100

200

TRO1

ONSA

HERS

 

 
MATE

7 

8 

2001 DOY 312‐316

Fig.2. STEC (TECU) for the observation stations on DOY 312‐316 in 2001. 

 

1 
 

Fig. 1. The four IGS stations considered for this work in Europe.
Station coordinates are provided in Table 1.

more precise geomagnetic field like the IGRM instead of the
dipole model; (2) obtaining accurate estimates for the elec-
tron content along the signal link (STEC values) which can
be either retrieved from Global Ionospheric Maps (GIMs)
or estimated from PR measurements; (3) using the satellite
and orbit products estimated by applying corrections for the
higher order ionospheric error terms (this is particularly im-
portant for a systematic and accurate analysis of the impact
of the higher order terms on PPP). Regarding point 2, ver-
tical TEC (VTEC) data from GIMs can be converted into
STEC by making use of a single layer mapping function in
RINEX HO. Alternatively STEC can also be obtained from
the PR measurements on the L1 and L2 frequencies (Eq. 2);
this, however, requires inputting the receiver and satellite
Differential Code Biases (DCBrec and DCBsat, respectively)
which were obtained from CODE for use in RINEXHO in
this work. The uncertainty in any of the terms on the right
hand side of (Eq. 2) propagates into the calculation of STEC
on a particular receiver-satellite link, according to the error
propagation law (Eq. 3). Although STEC can be estimated
with a comparable accuracy using either GIMs or PRs (Mar-
ques et al., 2007), the non-availability of the DCB values
at some instances hinders estimation of STEC from PRs.
Therefore STEC is obtained from GIMs in RINEXHO in
this work. Regarding the third point, since such corrected
products were not available during the progress of this work,
the satellite and orbit products estimated without corrections
for the higher order error terms were used.

Table 1. Coordinates of the IGS stations considered for analyses in
this work.

TRO1 ONSA HERS MATE

Latitude (deg), N 69.6627 57.3953 50.8673 40.6491
Longitude (deg), E 18.9396 11.9255 0.3362 16.7045
Height (m), U 138.0000 45.5000 76.4990 534.5000

3 Methodology

The observation stations in this work are selected from the
International GNSS Service (IGS, 2010) network, aiming for
a reasonably good latitudinal (mid and high latitudes, includ-
ing the auroral region) and longitudinal coverage in Europe
(Fig. 1); the stations coordinates are provided in Table 1.

Four sets of days (Table 2) are selected for analysis. In
order to investigate the impact of the solar activity devoid
of disturbances in the geomagnetic field, day-of-year (DOY)
312–316 in 2001 and 321–326 in 2006 were selected; for
these periods, the planetary geomagnetic index,Kp, is <4,
which is a good threshold to exclude the influence of ge-
omagnetic storms (NOAA, 2010). In order to include the
influence of a more disturbed geomagnetic field, DOY 294–
296 in 2001 and 301–307 in 2003 were selected, whenKp
was ≥4. Other geomagnetic indices like the AE (Auro-
ral Electrojet index) or Dst (Disturbance Storm Time index)
could also be considered (World Data Center for Geomag-
netism – WDC, 2011) while selecting the days for analysis.
However theKp index was deemed adequate, given the focus
of this work on the mid-to-high latitudes.

The range errors in the GPS observables due to Ion2 and
Ion3 (on GPS L1 and L2 frequencies) were estimated and
corrected using the software tool RINEXHO (Marques et
al., 2011), developed at the Sao Paulo State University in
Presidente Prudente, Brazil. The program requires as input
the observation and navigation files (in the receiver indepen-
dent exchange, RINEX, format), and GIM files or DCB in-
formation (according to the user’s choice of the method for
TEC calculation). An input text file is used, with the rele-
vant file names and execution specifications (in this case the
choice of the method to calculate TEC). The program applies
the corrections to the GPS code and phase observables, cor-
rects the input observation file accordingly and returns the
output files (corrected observation file, corrections for Ion2
and Ion3 on L1 and L2 frequencies for code observations
and STEC for each receiver-satellite link).

The corrected observation file allows to perform position-
ing in order to compare the station coordinates estimated us-
ing the corrected and original observation files. This allows
assessment of the impact of the higher order ionospheric ef-
fects on PPP, which is accomplished using the Bernese V5.0
(Bernese, 2007) software in this work.

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1388 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

Table 2. Days used in the analyses (given as day-of-year, DOY)
and the corresponding 3-hourlyKp values for each day such that
the firstKp value corresponds to midnight 00:00 LT.

DOY Kp indices (3 hourly)

2001 8 November 312 2 1 2 2 1 2 2 1
9 November 313 1 0 0 0 1 2 2 2
10 November 314 1 1 0 0 2 2 3 3
11 November 315 2 1 0 2 1 2 2 1
12 November 316 1 0 0 0 1 2 1 1

2001 21 October 294 2 3 3 2 3 6 6 7
22 October 295 6 5 4 6 5 7 6 5
23 October 296 4 5 3 2 2 2 2 1

2003 28 October 301 3 4 4 4 3 4 3 4
29 October 302 4 3 9 8 7 7 9 8
30 October 303 8 7 6 5 5 8 9 9
31 October 304 8 7 7 6 6 5 4 4
1 November 305 4 5 4 3 3 3 3 3
2 November 306 3 4 3 3 3 4 4 3
3 November 307 3 3 2 3 2 3 2 3

2006 17 November 321 2 2 1 1 2 1 0 1
18 November 322 0 0 0 0 1 0 0 1
19 November 323 1 1 1 1 1 0 0 0
20 November 324 0 0 0 0 1 1 0 1
21 November 325 0 0 0 0 0 0 0 1
22 November 326 0 0 0 0 1 1 3 3

The code and the carrier phase GNSS observation equa-
tions are given in Eqs. (4) and (5), respectively. The iono-
spheric delay term (If 1) appears with a “+” sign for the code
delay (Eq. 4) and with a “−” sign for the advance of the
carrier phase (Eq. 5). Focusing on the PR measurements
(Eq. 4), the ionospheric delay effect (If 1) can be represented
more explicitly as a series in inverse powers of frequency
within the geometric optics (GO) approach i.e. considering
the refraction of the GNSS signals penetrating through (large
scale) electron density irregularities in the ionosphere. A se-
ries representation of the delay effect on the PR measure-
ments,δρg in Eq. (6), can be derived when the Appleton-
Hartree equation (Budden, 1966) is considered for the iono-
spheric refractive index that is different than unity. Based on
the GO approach, the Appleton-Hartree formula leads to the
three terms on the right hand side of Eq. (7) which are the
first (Ion1), second (Ion2) and third (Ion3) order ionospheric
effects that delay the code measurement, respectively. The
same order effects for the carrier phase measurements are
given in Eq. (8). Hereafter the discussion considers the iono-
spheric effects on the PR measurements (Eq. 7); the argu-
ments are however applicable for the carrier phase range
measurements taking into account the correct sign notation
and coefficients for these three terms.

 

TRO1

ONSA

HERS 

MATE

Fig. 1. The four IGS stations considered for this work in Europe. Station coordinates are 
provided in Table 1.  

1 

2 
3 

4 

5 

6 

 

 

 

312 313 314 315 316
0

100

200
STEC (TECU)

 

 

312 313 314 315 316
0

100

200

 

 

312 313 314 315 316
0

100

200

 

 

312 313 314 315 316
0

100

200

TRO1

ONSA

HERS

 

 
MATE

7 

8 

2001 DOY 312‐316

Fig.2. STEC (TECU) for the observation stations on DOY 312‐316 in 2001. 

 

1 
 

Fig. 2. STEC (TECU) for the observation stations on DOY 312–316
in 2001.

∫
Nds term in Eq. (6) is the integral of the electron den-

sity (N ) along LoS between the receiver and satellite inside
a columnar cylinder of unit cross sectional area such that the
integration gives the (slant) total electron content, along LoS,
STEC (1 TEC unit, 1 TECU, is 1016 e−/m2. At times of low
background solar activity, as during the quiet periods of the
solar cycle, STEC is usually around 20–30 TECU at mid lat-
itudes, corresponding to about 3–8 m range delay on GPS L1
frequency giving negligible RREs due to Ion2 and Ion3 terms
under these conditions (1 TECU has about 0.16 m delay ef-
fect on GPS L1, Kintner Jr., 2006). STEC depends on the ge-
ometry of the receiver and satellite link, time of day (with a
diurnal variation that attains a peak around local noon), time
of year (seasonal dependency) and the solar cycle (greater
STEC during peak of the solar cycle). The diurnal variation
of STEC on the receiver-satellite links for GPS L1 can be
seen for the observation stations in Figs. 2, 5, 8 and 11, for
the different levels of solar and geomagnetic conditions.

B0cosθ term in the Ion2 (Eq. 6) can be taken out of the
integral assuming that it is LoS-independent, leaving

∫
Nds,

i.e. STEC. Ion3 term contains the integral of the square of the
electron density,

∫
N2ds (Eq. 6), which can be estimated us-

ing the shape parameterη (Hartmann and Leitinger, 1984).
This helps to approximate the ionospheric electron density
profile in terms of the maximum electron density,Nmax, and
the shape parameter,η, giving ηNmaxSTEC for this integral.
The shape parameterη can be taken as 0.66 which is valid for
different satellite elevations and maximum electron densities
(Hartmann and Leitinger, 1984). Based on these arguments,
Eq. (6) can be written as Eq. (7). For the carrier phase mea-
surements, Eq. (8) represents the ionospheric range errors (in
meters) up to the third order, neglecting the bending effect
that is associated with the third order error term. Following

Ann. Geophys., 29, 1383–1399, 2011 www.ann-geophys.net/29/1383/2011/



Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning 1389

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030

TRO1

ONSA

HERS

MATE

9 

10 

2001 DOY 312‐316  

Fig. 3. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations on DOY 312‐316 in 2001. 

312 313 314 315 316
0

2.5

5
x 10-3 Ion3L1 (m)

 

 

312 313 314 315 316
0

2.5

5
x 10-3

 

 

312 313 314 315 316
0

2.5

5
x 10-3

 

 

312 313 314 315 316
0

2.5

5
x 10-3

TRO1

ONSA

HERS

 

 
MATE

11 

12 

2001 DOY 312‐316  

Fig. 4. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations on DOY 312‐316 in 2001. 

2 
 

Fig. 3. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations
on DOY 312–316 in 2001.

Eq. (7), the magnitudes of Ion2 and Ion3 in the code-based
measurements can be expressed as in Eq. (9) and Eq. (10),
respectively.

From Eqs. (7) and (8), it can be seen that the higher order
ionospheric error terms for code measurements can be esti-
mated from the carrier phase measurements, and vice versa,
applying appropriate multiplicative terms (e.g. magnitude of
Ion2 for code observations is twice as large as that for carrier
phase observations) and sign notation for each order term.
Due to the multiplicative terms it can be understood that the
first order linear combination of PR observations does not
eliminate the higher order error terms. It can also be seen
(Eqs. 7 and 8) that TEC along LoS is important for all the
error terms (it should be reminded that the bending effect
in Ion3 is neglected here). Moreover, due to the LoS de-
pendency of TEC the magnitudes of Ion2 and Ion3 change
according to the receiver-satellite geometry. These residual
error terms become more important in the differential posi-
tioning mode (especially for long baselines when signal links
pierce through comparably different parts of the ionosphere)
and at low latitudes, in particular during high solar activity
(when ionization in the ionosphere is expected to be greater).

Accurate values of STEC for Ion2 and Ion3 can be ob-
tained from (a) Global Ionospheric Maps, GIMs, which con-
tain VTEC data accurate to about 2–8 TECU (Feltens and
Schaer, 1998) or (b) PR measurements according to Eq. (2).
These two methods show comparable accuracy (2–8 TECU)
in the estimated STEC values (Marques et al., 2007). For the
former method, VTEC from GIMs is converted into STEC
using a mapping function, considering the IPP, given by the
intersection of the receiver satellite path with the ionosphere
assumed as a single thin shell at an altitude of 450 km (same
value as taken in RINEXHO) according to the receiver-

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030

312 313 314 315 316
-0.030

-0.015

0

0.015

0.030

TRO1

ONSA

HERS

MATE

9 

10 

2001 DOY 312‐316  

Fig. 3. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations on DOY 312‐316 in 2001. 

312 313 314 315 316
0

2.5

5
x 10-3 Ion3L1 (m)

 

 

312 313 314 315 316
0

2.5

5
x 10-3

 

 

312 313 314 315 316
0

2.5

5
x 10-3

 

 

312 313 314 315 316
0

2.5

5
x 10-3

TRO1

ONSA

HERS

 

 
MATE

11 

12 

2001 DOY 312‐316  

Fig. 4. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations on DOY 312‐316 in 2001. 

2 
 

Fig. 4. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations
on DOY 312–316 in 2001.

satellite LoS geometry. As stated by Pajares et al. (2008b),
GIMs can provide less accurate STEC values at the low lati-
tudes for low elevation satellites; yet, since in this work mid-
latitude stations and a cutoff angle of 10◦ are considered, this
is not of concern here. For the latter, there is the need for
the receiver and satellite interfrequency biases (also referred
to as Differential Code Biases, DCBrec and DCBsat, respec-
tively). These frequency-dependent biases are relatively con-
stant in time and must be input in RINEXHO. Within this
work they were provided from the Center for Orbit Determi-
nation in Europe (CODE, 2010). Non-availability of these
biases may halt the process of STEC estimation from PRs
(Eq. 2). Thus for continuity of calculations STEC values
were obtained from GIMs (a user option in the program).

As can be seen in Eq. (9), Ion2 depends on the projec-
tion of the geomagnetic field (B) onto the receiver-satellite
link, (B0cosθ ), which can be more accurately calculated if
a precise geomagnetic field like the International Geomag-
netic Reference Model (IGRM, 2010) is used instead of a
dipole model. The GEOPACK library (Geopack subroutines,
2011) contains FORTRAN subroutines for computing the ge-
omagnetic field in the Earth’s magnetosphere, transforming
between various coordinate systems and tracing along field
lines (Tsyganenko, 2001). IGRM is used in RINEXHO for
a physically more realistic and accurate modelling of the ge-
omagnetic field.

In Eq. (10) it can be seen that Ion3 depends onNmax for
which a linear interpolation can be used to approximateNmax
in terms of TEC (Fritsche et al., 2005). A modified version of
this interpolation is also available (Piraux et al., 2010) where
Nmax is redefined in terms of TEC (Eq. 11).

The RREs (due to Ion2 and Ion3) are calculated for GPS
L1 and L2 carrier frequencies in RINEXHO; however the

www.ann-geophys.net/29/1383/2011/ Ann. Geophys., 29, 1383–1399, 2011



1390 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

294 295 296
0

100

200
STEC (TECU)

 

 

294 295 296
0

100

200

 

 

294 295 296
0

100

200

 

 

294 295 296
0

100

200

TRO1

ONSA

HERS

 

 
MATE

13 

14 

 2001 DOY 294‐296 

Fig. 5. STEC (TECU) for the observation stations on DOY 294‐296 in 2001. 

294 295 296
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

 

 

294 295 296
-0.030

-0.015

0

0.015

0.030

 

 

294 295 296
-0.030

-0.015

0

0.015

0.030

 

 

294 295 296
-0.030

-0.015

0

0.015

0.030

TRO1

ONSA

HERS

 

 
MATE

15 

16 

2001 DOY 294‐296   

Fig. 6. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations on DOY 294‐296 in 2001. 

3 
 

Fig. 5. STEC (TECU) for the observation stations on DOY 294–296
in 2001.

results are shown for GPS L1 frequency (Figs. 2–13); for the
L2 frequency the plotted results would have greater magni-
tudes due to the smaller frequency of the L2 carrier and the
inverse frequency dependency of Ion2 and Ion3.

Based on the equations for the higher order range er-
rors (Eqs. 7 and 8), a straightforward calculation (taking
150 TECU along line of sight,B0cosθ about 2.7× 10−5

Tesla at IPP takingB0 as 3.12× 10−5 and θ as 30◦ and
Nmax as 4.416× 10−6 TEC) gives about 24 m, 2.3 cm and
sub-millimeter (negligible which may be due to the fact that
bending effect is excluded in this calculation) level range er-
rors for Ion1, Ion2 and Ion3, respectively, for GPS L1. For
such conditions, for instance, Ion2 is about 0.09 % of Ion1
for the code based range measurements (equivalently 0.05 %
for carrier phase range measurements). In this case, using
the IF observable to remove the Ion1 term would be accu-
rate up to about 99.9 %; this agrees well with the estimation
of Klobuchar (1987) that the IF observable is accurate up to
about 0.1 %. However the crude values assigned to the pa-
rameters involved in Eqs. (7) and (8) should be kept in mind
in this estimation.

The final stage of the work presented here is to analyze
the estimated station coordinates when processing the data
in PPP, using the Bernese software (Bernese, 2007). This
part of the work focuses on the impact of using the corrected
(for Ion2 and Ion3) observation files in PPP in order to in-
fer the significance of the contribution from the higher or-
der ionospheric effects in GNSS positioning in the European
region during different background physical (solar, geomag-
netic, ionospheric) conditions.

In the coordinates estimation process, Ion2 and Ion3 cor-
rections are applied only to the receiver observations; the
orbit and clock products (used in the Bernese software) in-

294 295 296
0

100

200
STEC (TECU)

 

 

294 295 296
0

100

200

 

 

294 295 296
0

100

200

 

 

294 295 296
0

100

200

TRO1

ONSA

HERS

 

 
MATE

13 

14 

 2001 DOY 294‐296 

Fig. 5. STEC (TECU) for the observation stations on DOY 294‐296 in 2001. 

294 295 296
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

 

 

294 295 296
-0.030

-0.015

0

0.015

0.030

 

 

294 295 296
-0.030

-0.015

0

0.015

0.030

 

 

294 295 296
-0.030

-0.015

0

0.015

0.030

TRO1

ONSA

HERS

 

 
MATE

15 

16 

2001 DOY 294‐296   

Fig. 6. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations on DOY 294‐296 in 2001. 

3 
 

Fig. 6. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations
on DOY 294–296 in 2001.

294 295 296
0

2.5

5
x 10-3 Ion3L1 (m)

 

 

294 295 296
0

2.5

5
x 10-3

 

 

294 295 296
0

2.5

5
x 10-3

 

 

294 295 296
0

2.5

5
x 10-3

TRO1

ONSA

HERS

 

 
MATE

17 

18 

2001 DOY 294‐296   

Fig. 7. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations on DOY 294‐296 in 2001. 

301 302 303 304 305 306 307
0

100

200
STEC (TECU)

 

 

301 302 303 304 305 306 307
0

100

200

 

 

301 302 303 304 305 306 307
0

100

200

TRO1

HERS

 

 
MATE

19 

20 

 2003 DOY 301‐307

Fig. 8. STEC (TECU) for the observation stations on DOY 301‐307 in 2003. 

4 
 

Fig. 7. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations
on DOY 294–296 in 2001.

volved in PPP are computed from a global network that does
not apply corrections for Ion2 and Ion3. It must be noted that
a systematic and accurate investigation of the impact of the
higher order terms on PPP requires the use of satellite and or-
bit products estimated while accounting for these higher or-
der error terms. As shown by Pajares et al. (2008b), a more
consistent and correct approach to consider (e.g. Ion2 in PPP)
can be to perform dual frequency, carrier phase differential
positioning where both the orbits (and other satellite prod-
ucts) and user coordinates are estimated considering the Ion2
correction. According to Pajares et al. (2007) the effect of
Ion2 on the satellite clocks can be larger than 30 picoseconds
(1 cm in range equivalent units) and several millimetres on
the satellite positions. These products can be obtained from

Ann. Geophys., 29, 1383–1399, 2011 www.ann-geophys.net/29/1383/2011/



Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning 1391

294 295 296
0

2.5

5
x 10-3 Ion3L1 (m)

 

 

294 295 296
0

2.5

5
x 10-3

 

 

294 295 296
0

2.5

5
x 10-3

 

 

294 295 296
0

2.5

5
x 10-3

TRO1

ONSA

HERS

 

 
MATE

17 

18 

2001 DOY 294‐296   

Fig. 7. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations on DOY 294‐296 in 2001. 

301 302 303 304 305 306 307
0

100

200
STEC (TECU)

 

 

301 302 303 304 305 306 307
0

100

200

 

 

301 302 303 304 305 306 307
0

100

200

TRO1

HERS

 

 
MATE

19 

20 

 2003 DOY 301‐307

Fig. 8. STEC (TECU) for the observation stations on DOY 301‐307 in 2003. 

4 
 

Fig. 8. STEC (TECU) for the observation stations on DOY 301–307
in 2003.

different analysis centres which may consider either Ion2 or
Ion3 or both. For instance, as of 2009 JPL has been consider-
ing only the Ion2 in their ionospheric model, whereas CODE
considers both Ion2 and Ion3 for their products. As stated by
Pajares et al. (2008a), using the standard products, which are
not corrected for the higher order ionospheric effects, with
the corrected GPS observations, blurs the net impact of cor-
rections. However, since a set of standard satellite orbit and
clock products are yet unavailable (Piraux et al., 2010), the
JPL satellite and orbit products estimated without accounting
for Ion2 and Ion3 were used in PPP in this work, where only
the receiver positions are estimated based on data corrected
for the higher order ionospheric effects.

4 Results for the higher order error terms and the
STEC values

The results (RREs for Ion2 and Ion3, PPP station coordinate
differences) are shown for Ion2 and Ion3 on the code obser-
vations for the GPS L1 frequency since this applies to a wider
user community using the civil code on GPS L1. Due to the
inverse frequency dependency of both Ion2 and Ion3, the cal-
culated Ion2 values for GPS L2 are about 2.11 times and the
Ion3 values for GPS L2 are about 2.71 times those obtained
for the GPS L1.

4.1 DOY 312–316 in 2001

These results are presented in Figs. 2 to 4. The diurnal varia-
tion in STEC values can be seen in Fig. 2 for all stations. The
midday peak values are greater at the mid-latitudes (e.g. at
MATE) than at the high latitudes (e.g. TRO1). For the high
latitude station TRO1, the night-time enhancement in TEC
values can be as large as half the noon-time values. Although
ionization due to solar radiation is absent at night and free

301 302 303 304 305 306 307
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

 

 

301 302 303 304 305 306 307
-0.00

-0.015

0

0.015

0.030

 

 

301 302 303 304 305 306 307
-0.030

-0.015

0

0.015

0.030

TRO1

HERS

 

 
MATE

21 

22 

2003 DOY 301‐307   

Fig. 9. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations on DOY 301‐307 in 2003. 

301 302 303 304 305 306 307
0

2.5

5
x 10

-3

 

 

301 302 303 304 305 306 307
0

2.5

5
x 10

-3

2003 DOY 301‐307 

 

 

301 302 303 304 305 306

5
0

-3

307
0

x 1 Ion3L1 (m)

 

 
TRO1

2.5

HERS

MATE

23 

24 

 

Fig. 10. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations on DOY 301‐307 in 2003. 

5 
 

Fig. 9. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations
on DOY 301–307 in 2003.

electrons and ions tend to recombine reducing the amount
of ionization, at the high latitudes night time ionization can
continue due to the movement of the ionization from the day-
time to the night-time part of the Earth as well as due to en-
ergetic particles arriving with the solar winds to the vicinity
of the Earth and penetrating into lower altitudes of the iono-
sphere along the almost vertical geomagnetic field lines at
these high latitudes. In the latter case, the particles moving
vertically downward collide with the ionospheric particles
causingcollisional ionization. Such effect can be observed
especially at the high latitudes where the geomagnetic field
lines can route the particles and during the active period of
the solar cycle when the solar radiation is stronger (Buon-
santo, 1999).

As seen in Eq. (9) Ion2 has a LoS dependency; thus for sta-
tions at different latitudes the values for Ion2 (Fig. 3) change
from being more confined to about−1 cm (for TRO1) to
scattering between±2 cm (for MATE). The negative values
for Ion2 (in all plots) are due to theB0cosθ term, which can
attain positive or negative values depending on the satellite-
receiver geometry. A mid-latitude station (e.g. MATE) can
track the satellites with a wider range of elevation angles
whereas a high latitude station as TRO1 tracks with a more
confined range of elevation angles; thus the LoS dependency
and thereafter the Ion2 errors are different for the receivers at
different latitudes.

In Fig. 4, it can be seen that the estimated Ion3 is sig-
nificant during the noon-hours for all stations, ranging from
about 2 to 3 mm from north (TRO1) to south (MATE). It can
be observed that for the high latitudes, there can be signifi-
cant correction for Ion3 at the night-time hours (see station
TRO1 in Fig. 4).

www.ann-geophys.net/29/1383/2011/ Ann. Geophys., 29, 1383–1399, 2011



1392 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

301 302 303 304 305 306 307
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

 

 

301 302 303 304 305 306 307
-0.00

-0.015

0

0.015

0.030

 

 

301 302 303 304 305 306 307
-0.030

-0.015

0

0.015

0.030

TRO1

HERS

 

 
MATE

21 

22 

2003 DOY 301‐307   

Fig. 9. Ion2 (m) for GPS L1 (Ion2L1) for the observation stations on DOY 301‐307 in 2003. 

301 302 303 304 305 306 307
0

2.5

5
x 10

-3

 

 

301 302 303 304 305 306 307
0

2.5

5
x 10

-3

2003 DOY 301‐307 

 

 

301 302 303 304 305 306

5
0

-3

307
0

x 1 Ion3L1 (m)

 

 
TRO1

2.5

HERS

MATE

23 

24 

 

Fig. 10. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations on DOY 301‐307 in 2003. 

5 
 

Fig. 10. Ion3 (m) for GPS L1 (Ion3L1) for the observation stations
on DOY 301–307 in 2003.

4.2 DOY 294–296 in 2001

These results are presented in Figs. 5 to 7. Comparing the
results in Fig. 5 with those in Fig. 2, it can be noticed that the
night time enhancement in TEC values can be as large as the
noon-time values (e.g. station TRO1 in Fig. 5) when there
are geomagnetic storms in addition to high background solar
activity. This causes almost two peaks for the diurnal TEC
values, especially for the high latitude stations. Movement
of the ionization from the day side to the night side of the
Earth at the high latitudes can be enhanced by the geomag-
netic storms (Ho et al., 1997). As seen in Fig. 5 for ONSA,
enhancement in the auroral TEC at the night time can be due
to the expansion of the auroral oval by the influence of the
geomagnetic storms during this peak year of the solar cycle
as also evident in the highKp values. Such night-time en-
hancement in TEC is not apparent in Fig. 2 for ONSA. The
mid-latitude stations are observed not to have enhanced lev-
els of night time TEC, which means that the equatorward
expansion of the auroral oval was not significant enough as
to influence the ionosphere above these stations during these
geomagnetic conditions.

It can be seen in Fig. 6 that the error due to Ion2 is over-
all within 1–2 cm, which agrees to those observed in Fig. 3.
However, due to the geomagnetic storms – which may cause
enhancements in the ionization levels (i.e. higher TEC val-
ues), Ion2 values for ONSA can be noticeably large during
the night-time hours as well – such enhancement is not appar-
ent in Fig. 3 for ONSA. Also, the night-time values for Ion2
at TRO1 can be as large as the noon-time values (TRO1 and
ONSA in Fig. 6). Since theB0cosθ term in Ion2 calculated
by IGRM does not in general consider the actual geomag-
netic disturbances, the enhanced values of Ion2 can be more
correctly related with the enhancement in TEC. However it
should be noted that TEC can increase due to the geomag-
netic storms during such conditions. The similarity between

321 322 323 324 325 326
0

100

200
STEC (TECU)

 

 

321 322 323 324 325 326
0

100

200

TRO1

2006 DOY 321‐326

 

 

321 322 323 324 325 326
0

100

200

 

 

MATE

HERS

25 

26 

 

Fig. 11. STEC (TECU) for the three observation stations on 321‐326 in 2006. 

321 322 323 324 325 326
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

 

 

321 322 323 324 325 326
-0.030

-0.015

0

0.015

0.030

 

 

321 322 323 324 325 326
-0.030

-0.015

0

0.015

0.030

TRO1

HERS

 

 
MATE

27 

28 

2006 DOY 321‐326  

Fig.12. Ion2 (m) for GPS L1 (Ion2L1) for the three observation stations on 321‐326 in 2006. 

6 
 

Fig. 11. STEC (TECU) for the three observation stations on 321–
326 in 2006.

the Ion2 error profile (Fig. 6) and the corresponding TEC
values along the signal links (Fig. 5) suggests a strong de-
pendency of Ion2 on STEC.

Compared with Fig. 4, it can be seen in Fig. 7 that when
geomagnetic disturbances are also considered the error due
to Ion3 becomes significant during the midday hours for all
stations, however this time the magnitude of the error ranges
from about 1 to 5 mm from north (TRO1) to south (MATE).
The noticeable night time peaks in Fig. 7 for TRO1 and
ONSA suggest that for the high latitude stations, Ion3 may
need to be considered during night-time hours as well, Over-
all it can be seen in both Figs. 4 and 7 that Ion3 is important
during daytime, amounting to a few millimetres in the range
errors at these stations during the peak of the solar cycle.

4.3 DOY 301–307 in 2003

These results are presented in Figs. 8 to 10. Although within
a post-peak year of solar cycle 23, the period DOY 301–307
in 2003 coincides with the so-called Halloween Storm, dur-
ing which the solar radio flux index,F10.7 (measure of ra-
dio emission from the sun at 10.7 cm wavelength correlating
well with the sunspot number and used as an indicator of
solar activity (Ionospheric Prediction Service – IPS, 2011)),
went up to as high as 270/275 solar flux units (Space Weather
Prediction Center – SWPC, 2011). It should be mentioned,
however, that the change in the geomagnetic field inductance
due to geomagnetic disturbances and the estimation of its in-
fluence on the Ion2 term is not straightforward.

During this period, increase in the Ion2 error term is ex-
pected to be due to higher levels of solar activity as well as to
the disturbances in the geomagnetic field. Similarly, increase
in Ion3 can be associated with higher levels of solar activ-
ity during this period. Results show high levels of ionization

Ann. Geophys., 29, 1383–1399, 2011 www.ann-geophys.net/29/1383/2011/



Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning 1393

321 322 323 324 325 326
0

100

200
STEC (TECU)

 

 

321 322 323 324 325 326
0

100

200

TRO1

2006 DOY 321‐326

 

 

321 322 323 324 325 326
0

100

200

 

 

MATE

HERS

25 

26 

 

Fig. 11. STEC (TECU) for the three observation stations on 321‐326 in 2006. 

321 322 323 324 325 326
-0.030

-0.015

0

0.015

0.030
Ion2L1 (m)

 

 

321 322 323 324 325 326
-0.030

-0.015

0

0.015

0.030

 

 

321 322 323 324 325 326
-0.030

-0.015

0

0.015

0.030

TRO1

HERS

 

 
MATE

27 

28 

2006 DOY 321‐326  

Fig.12. Ion2 (m) for GPS L1 (Ion2L1) for the three observation stations on 321‐326 in 2006. 

6 
 

Fig. 12. Ion2 (m) for GPS L1 (Ion2L1) for the three observation
stations on 321–326 in 2006.

in the ionosphere on DOY 301–307 in 2003. Such levels of
ionization can influence both Ion2 and Ion3 terms.

During this period Ion2 ranges from about 1 to 2 cm
(Fig. 9) and Ion3 from sub-millimetre to about 2 mm
(Fig. 10) from TRO1 to MATE in both cases. Thus, even
slightly outside the highest phase of the solar cycle, the so-
lar activity can be strong enough to enhance the ionization
in the ionosphere. During the high phase of the geomagnetic
storm (DOY 301–302, 2003), Ion2 and Ion3 have significant
enhancement; whereas during the absence of such storms
(e.g. DOY 312–316, 2001) the higher order error terms are
more predictable.

4.4 DOY 321–326 in 2006

These results are presented in Figs. 11 to 13. In 2006, it can
be seen that during the quiet period of the solar cycle, ion-
ization in the ionosphere is low, about 20–40 TECU (Fig. 11)
and the corresponding higher order range errors are also less
significant than those during the active period of the solar cy-
cle: Ion2 is observed to be one order of magnitude smaller
(at millimetre level as seen in Fig. 12 as opposed to centime-
tre level as observed during other analysis periods) and Ion3
is at sub-millimetre level (Fig. 13) during this quiet period of
the solar cycle in absence of geomagnetic disturbances.

Between the high and low periods of the solar cycle
(November 2001 and November 2006, respectively, with-
out the influence of geomagnetic disturbances in both cases)
there is a significant difference in the calculated STEC val-
ues (e.g. as high as about 160 TECU during the peak of the
solar cycle in November 2001 at the mid-latitudes, and about
40 TECU during the quiet period in November 2006 at the
same latitudes). Significant differences are also observed
in the Ion2 and Ion3 values – they change by an order of
magnitude between the two periods, therefore indicating that

321 322 323 324 325 326
0

2.5

5
x 10

-3 Ion3L1 (m)

 

 

321 322 323 324 325 326
0

2.5

5
x 10

-3

 

 

321 322 323 324 325 326
0

2.5

5
x 10

-3

TRO1

HERS

 

 
MATE

29 

30 

31 

32 

2006 DOY 321‐326   

Fig. 13. Ion3 (m) for GPS L1 signal for the three observation stations on DOY 321‐326 in 2006. 

 

 

7 
 

Fig. 13. Ion3 (m) for GPS L1 signal for the three observation sta-
tions on DOY 321–326 in 2006.

the significantly different TEC values along the signal paths
cause the different values for the higher order range errors
during these two periods. This highlights the importance of
considering the higher order range errors during the upcom-
ing solar maximum, predicted for around 2013.

During this quiet period of the solar cycle (i.e. charac-
terised by low levels of ionization in the ionosphere), when
geomagnetic field disturbances are negligible the high order
ionospheric error terms should not degrade the measurement
accuracy significantly.

5 PPP results

PPP is a high accuracy positioning method which can be per-
formed with a dual frequency receiver (so that the IF observ-
able can be used) and that exploits the use of highly accurate
externally provided (e.g. by the IGS) satellite orbit and clock
corrections (Bernese, 2007). Due to its high (potentially cen-
timetre level) positioning accuracy, PPP was performed in
this work with the Bernese software to investigate the impact
of correcting the GPS range observations for the errors due
Ion2 and Ion3. With the Bernese software, a free network
solution can be carried out, i.e. a solution where the satellites
orbits define the coordinates system to which the estimated
positions refer to.

As detailed before, RINEXHO applies corrections for
Ion2 and Ion3 to the GPS observation files in the RINEX for-
mat and outputs a corresponding corrected observation file in
the same format. In this work, PPP was performed respec-
tively with both observation files, i.e. a file with the “uncor-
rected” and another with the “corrected” (for the higher or-
der terms) observations. Figures 14 to 17 show how much
the stations coordinates differ in both cases (in latitude, lon-
gitude and ellipsoidal height) for all four sets of days and

www.ann-geophys.net/29/1383/2011/ Ann. Geophys., 29, 1383–1399, 2011



1394 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

Table 3. Differences in the calculated station coordinates (delta height/latitude/ longitude, in meters) when PPP is performed with the
corrected and uncorrected observation files. Positive differences in height, latitude and longitude are upward, northward and eastward,
respectively.

CORRECTED – UNCORRECTED

delta height (m) delta latitude (m) delta longitude (m)

HERS MATE ONSA TRO1 HERS MATE ONSA TRO1 HERS MATE ONSA TRO1

312 0.0035 0.0180 0.0130 –0.0060 –0.0470 –0.0054 –0.0013 0.0013 –0.0160 –0.0150 –0.0122 –0.0054
313 –0.0018 0.0245 0.0110 –0.0080 –0.0040 –0.0072 –0.0013 0.0040 –0.0145 –0.0048 –0.0181 –0.0100
314 0.0120 0.0147 0.0160 0.0012 –0.0052 –0.0033 –0.0014 0.0037 –0.0170 –0.0210 –0.0194 –0.0100
315 0.0206 0.0251 0.0207 –0.0123 –0.0033 0.0012 –0.0017 0.0019 –0.0260 –0.0165 –0.0225 –0.0090
316 0.0240 0.0167 0.0235 0.0250 –0.0011 –0.0012 –0.0010 0.0039 –0.0215 –0.0146 –0.0208 –0.0104

294 0.0100 –0.0072 0.0057 0.0120 0.0054 0.0010 –0.0014 0.0028 –0.0040 0.0040 0.0018 –0.0019
295 0.0033 –0.0088 –0.0020 0.0032 –0.0018 –0.0020 –0.0030 0.0027 0.0110 0.0088 0.0021 0.0014
296 0.0120 –0.0017 –0.0017 0.0089 –0.0052 –0.0015 –0.0026 –0.0010 –0.0037 0.0089 –0.0016 –0.0030

301 0.0064 0.0240 0.0060 –0.0124 –0.0025 0.0022 –0.0046 –0.0017 –0.0006 –0.0001 –0.0001 0.0001
302 –0.0089 0.0044 –0.0090 –0.0080 –0.0040 –0.0050 0.0053 –0.0042 0.0001 0.0015 0.0015 –0.0015
303 0.0014 –0.0010 0.0015 –0.0050 –0.0039 –0.0050 0.0207 –0.0030 0.0001 0.0001 0.0001 –0.0001
304 –0.0035 0.0018 –0.0040 –0.0140 –0.0041 0.0014 –0.0042 –0.0014 –0.0001 –0.0001 –0.0001 0.0001
305 0.0075 0.0080 –0.0040 –0.0110 –0.0038 –0.0052 –0.0037 –0.0013 –0.0001 –0.0001 –0.0001 0.0001
306 0.0022 0.0056 –0.0037 –0.0105 –0.0050 –0.0034 –0.0080 0.0012 –0.0001 –0.0001 –0.0001 0.0001
307 0.0059 0.0182 0.0114 –0.0280 –0.0060 –0.0090 –0.0075 0.0030 –0.0001 –0.0001 –0.0001 0.0001

321 –0.0050 –0.0050 0.0020 –0.0152 0.0012 –0.0088 0.0110 0.0021 0.0008 0.0013 0.0007 0.0012
322 –0.0090 –0.0220 –0.0092 –0.0120 0.0018 –0.0020 0.0019 0.0029 0.0010 –0.0020 0.0009 0.0011
323 –0.0018 –0.0150 –0.0013 –0.0120 0.0011 –0.0014 0.0089 0.0009 0.0010 –0.0009 0.0008 0.0011
324 –0.0017 –0.0216 –0.0256 –0.0158 0.0046 –0.0021 0.0055 0.0010 0.0007 0.0007 0.0007 0.0007
325 –0.0030 –0.0120 –0.0194 –0.0180 0.0044 –0.0045 0.0050 0.0022 0.0007 –0.0006 0.0007 0.0007
326 –0.0020 –0.0010 0.0042 –0.0256 0.0011 –0.0026 0.0110 –0.0008 0.0009 –0.0004 0.0009 0.0009

stations being analysed. The differences in all three co-
ordinate components are computed by subtracting the PPP
results obtained with the original observation (uncorrected)
files from those obtained with the corrected files (see Table 3
for the numerical values of the differences). The results are
discussed below:

– Considering the high solar activity period with negligi-
ble disturbances in the geomagnetic field (DOY 312–
316 in 2001, Fig. 14) when the corrections for Ion2 and
Ion3 account majorly for the impact of the solar activity,
it can be observed (Fig. 14, top plot) that the high lati-
tude stations get northward corrections (about 2–3 mm)
and mid-latitude stations southward (about 1 cm). Dur-
ing this period all stations are observed to have west-
ward corrections (about 1–2 cm) in general (Fig. 14,
middle plot) and the vertical component of the station
coordinates were greater (by about 2–3 cm) in general
for the mid-latitudes and smaller for the high latitudes
(Fig. 14, bottom plot) when the corrected observation
files were used in PPP. Pajares et al. (2007) who focus
only on the Ion2 term and its impact on the geodetic
estimates show that applying Ion2 correction to sub-
daily differential positioning (using IGS data network)
changes the receiver positions at submillimeter level

which isnorthwardfor thehigh latitudes andsouthward
for the low latitudes.

– Considering the high solar activity period with dis-
turbed geomagnetic conditions whenKp was as large
as 7 (DOY 294–296 in 2001, Fig. 15), it is difficult to
decide on a general common trend for the latitudinal
corrections (of few millimetres, see Fig. 15, top plot) in
general (Fig. 15, top plot). During this period, the pre-
viously (Fig. 14, middle plot) observed westward cor-
rection seems to be suppressed. The estimated vertical
corrections (Fig. 15, bottom plot) are overall upward
however the mid-latitudes are corrected downward (at
sub-cm level) on average. It should be pointed out that
the short observation period considered here may hinder
a more conclusive analysis.

– During the post-peak period of the solar cycle with dis-
turbances in the geomagnetic field, during the so-called
Halloween Storm, (DOY 301–307 in 2003, Fig. 16),
overall a southward correction (Fig. 16, top plot) can
be deduced with magnitudes mostly at millimetre level
but at times a few centimetres. A distinguishable fea-
ture during this post-peak period can be observed in
the longitudinal corrections (Fig. 16, middle plot): the
geomagnetic activity seems to suppress changes in the

Ann. Geophys., 29, 1383–1399, 2011 www.ann-geophys.net/29/1383/2011/



Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning 1395

312 313 314 315 316
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
at
 (m

)

312 313 314 315 316
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
on

 (m
)

312 313 314 315 316
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dh
 (m

)

TRO1 HERS ONSA

DOY

MATE

33 

34 
35 

36 

Fig. 14. “Corrected – Uncorrected” PPP results for the geodetic coordinates of the stations on 
DOY 312‐316 in 2001.  

 

 

8 
 

Fig. 14. “Corrected – Uncorrected” PPP results for the geodetic
coordinates of the stations on DOY 312–316 in 2001.

longitude component of the station coordinates. Re-
garding the estimated heights of the stations during
this period, 1–2 cm level corrections can be observed
(Fig. 16, bottom plot) such that the high latitudes are
corrected downward and mid-latitudes upward.

– Considering the period of low background solar activ-
ity with quiet geomagnetic conditions (DOY 321–326
in 2006, Fig. 17), it can be seen in the horizontal sta-
tion components that PPP results do not show signifi-
cant differences when the observation files are corrected
for Ion2 and Ion3 (Fig. 17, top and middle plots). It is
difficult to observe a general trend in the direction for
the vertical corrections (Fig. 17, bottom plot).

Considering that PPP can potentially provide centimetre
level accuracy for the estimated station coordinates and that
the corrections for Ion2 and Ion3 are about centimetre and
millimetre levels, respectively, during adverse ionospheric
and geomagnetic conditions, it can be expected that impact
of Ion3 in PPP may be unnoticeable due to the noise level of
the positioning. It can be expected that the differences in the
estimated station coordinates (using the corrected and origi-
nal observation files) would be mostly due to corrections for
Ion2 and then for Ion3 and the noise level of the positioning
solution.

6 Discussion and suggestions for future work

Based on the analysis involved in this work the following
points can be drawn:

– Enhancement in TEC values can occur due to greater
solar activity, as during the peak of the solar cycle.

294 295 296
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
at
 (m

)

294 295 296
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
on

 (m
)

294 295 296
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dh
 (m

)

TRO1 HERS ONSA

DOY

MATE

37 

38 
39 

 

Fig. 15. “Corrected – Uncorrected” PPP results for the geodetic coordinates of the stations on 
DOY 294‐296 in 2001. 

9 
 

Fig. 15. “Corrected – Uncorrected” PPP results for the geodetic
coordinates of the stations on DOY 294–296 in 2001.

Similarly the geomagnetic field disturbances can also
drive mechanisms that enhance ionization in the iono-
sphere. As TEC is an important parameter while calcu-
lating the range errors due to the ionosphere, more con-
tribution from the ionospheric error terms is expected
when the TEC values are higher. During the post-peak
period of the solar cycle, if geomagnetic field distur-
bances are present (e.g. DOY 301–307 in 2003), the
higher order ionospheric error terms were observed to
contribute to the overall measurement accuracy at mag-
nitudes comparable to those occurring during the peak
of the solar cycle without the presence of such distur-
bances (e.g. DOY 312–316 in 2001). During the quiet
period of the solar cycle (low ionization levels in the
ionosphere), when the geomagnetic field disturbances
were negligible (e.g. DOY 321–326 in 2006), the higher
order ionospheric error terms were observed to be very
small. Even during the post-peak years of the solar cy-
cle, high TEC values may be observed which can be
explained by the geomagnetic activity that can route the
incoming solar particles to lower altitudes in the iono-
sphere, especially at the high latitudes where the geo-
magnetic field lines are mostly directed towards the sur-
face of the Earth. Thus, enhancement in TEC should
not be expected only during the peak years of the so-
lar cycles; it can be observed in general correlated with
increased levels of geomagnetic activity.

– In terms of the effects of the Ion2 and Ion3 in longi-
tude, a general westward correction was observed in
this work (during active period of the solar cycle) where
the mid-latitude stations were observed to be affected
more than the high latitude ones. Latitudinal correc-
tions were in general southward for the mid-latitudes

www.ann-geophys.net/29/1383/2011/ Ann. Geophys., 29, 1383–1399, 2011



1396 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

301 302 303 304 305 306 307
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
at
 (m

)

301 302 303 304 305 306 307
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
on

 (m
)

301 302 303 304 305 306 307
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dh
 (m

)

TRO1 HERS ONSA

DOY

MATE

40 

41 
42 

 

Fig. 16. “Corrected – Uncorrected” PPP results for the geodetic coordinates for the stations on 
DOY 301‐307 in 2003. 

10 
 

Fig. 16. “Corrected – Uncorrected” PPP results for the geodetic
coordinates for the stations on DOY 301–307 in 2003.

and northward for the high latitudes. For the height
component, the mid latitude stations are observed to be
corrected upward in general and the high latitude ones
downward.

– Diurnal variation in RREs was such that a minimum was
observed before sunrise and after sunset, with a maxi-
mum around noon for the stations analysed in Europe.
The strong diurnal variation in TEC is expected to be a
reason for this.

– Had the corrected satellite orbit and clock products been
used in PPP, a more systematic and realistic analysis
of the differences in the positioning results could have
been carried out. In the approach followed in this work,
the net effect of correcting the observation files is ex-
pected to be obscured since the corrections were applied
only to the receiver observations.

– Comparing the Ion2 and Ion3 values presented in this
work, the diurnal variation in Ion3 is stronger than that
in Ion2 (i.e. the relative difference between the mini-
mum and maximum for Ion3 and greater than that for
Ion2). This can be explained by the fact that in addi-
tion to the dependence on STEC, Ion2 depends on the
projection of the geomagnetic field onto the signal path
and Ion3 on the maximum electron density in the iono-
sphere; in the former no diurnal variation is anticipated
(IGRM consideration of TEC for the magnetic field is
not there thus a clear relation between this magnetic
term and TEC cannot be established), whereas in the
latter diurnal variation can be anticipated since the max-
imum electron density normally reaches a maximum at
about the local afternoon (Ratcliffe, 1972).

301 302 303 304 305 306 307
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
at
 (m

)

301 302 303 304 305 306 307
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dl
on

 (m
)

301 302 303 304 305 306 307
-0.03

-0.02

-0.01

0

0.01

0.02

0.03

dh
 (m

)

TRO1 HERS ONSA

DOY

MATE

40 

41 
42 

 

Fig. 16. “Corrected – Uncorrected” PPP results for the geodetic coordinates for the stations on 
DOY 301‐307 in 2003. 

10 
 

Fig. 17. “Corrected – Uncorrected” PPP results for the geodetic
coordinates for the stations on DOY 321–326 in 2006.

– It was observed that both the horizontal and vertical
components of the estimated station coordinates in PPP
can differ at cm-mm level when the corrected (for Ion2
and Ion3) observation files are used. Differences in the
estimated horizontal components can be influenced by
not only the presence but also the strength of the geo-
magnetic field disturbances. It was also observed that
when PPP is performed respectively with both the cor-
rected and original observation files, the differences in
the estimated station coordinates during the non-peak
period of the solar cycle can be comparable to those dur-
ing the peak period if the former has contribution from
significant levels of geomagnetic field disturbances.

– This work describes a method to analyze the higher or-
der ionospheric effects on the GNSS observations and
on positioning; future work will be carried out on the
basis of this methodology, further accounting for the
bending effect in the Ion3 term. Longer term periods
will also be analyzed – monthly or yearly analyses for
the higher order ionospheric effects can be performed
with open-sky observations during the upcoming solar
maximum, expected around 2013. Future work can also
consider the new GNSS signals like GPS L2C, L5 and
Galileo L1 and E5 which may be available by then. This
can allow an assessment of the advantage of these new
signals for the GNSS user community to correct range
errors related to the ionosphere, especially its higher or-
der terms that have so far been mostly ignored.

Ann. Geophys., 29, 1383–1399, 2011 www.ann-geophys.net/29/1383/2011/



Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning 1397

Appendix A

List of equations

TEC≈ 2.27×105
×Nmax (A1)

Nmax is the maximum electron density.

STEC=
f 2

1 f 2
2

40.3(f 2
2 −f 2

1 )
{PR1−PR2−c(DBDrec+DCBsat)+ε1,2} (A2)

STEC is the Total Electron Content (TEC) along line of
sight – i.e. Slant TEC.fi is the signal frequency and PRi

the pseudorange fori-th frequency wherei = 1 for GPS
L1 = 1575.42 MHz andi = 2 for GPS L2= 1227.6 MHz. c
is the speed of light. DCBrec and DCBsat are the Differential
Code Biases for the receiver and satellite, respectively. The
termε1,2 contains noise in the PR measurement.

σ 2
STEC=

{
f 2

1 f 2
2

40.3(f 2
2 −f 2

1 )

}2(
σ 2

PR1
+σ 2

PR2
+c2σ 2

DCBrec
+c2σ 2

DCBsat

)
(A3)

σ 2
STEC is the variance in the estimation for STEC.σ 2

PR1
is the

variance of the PR measurement fori-th frequency.σ 2
DCBrec

andσ 2
DCBsat

are the variances of the DCBs for the receiver
and satellites, respectively.

PR1 = ρ +c(dtr −dts)+If 1 +T +MPR,f 1 +εPR,f 1 (A4)

PR1 is the pseudorange for GPS L1 signal.ρ is the geometric
(true) distance between the receiver and satellite.

dtr is the receiver clock offset, anddts is the satellite clock
offset.If 1 is the ionospheric error in the PR measurement for
f1 frequency.T is the tropospheric delay.MPR,f 1 is the mul-
tipath effect andεPR,f1 is the noise on the PR measurement
for f1 frequency.

L1=ρ +c(dtr−dts)−If 1+T +ML,f 1+εL,f 1+λf 1·Nλ (A5)

L1 is the carrier phase pseudorange for GPS L1 signal.If 1

is the ionospheric error in the carrier phase measurement
for f1 frequency. ML,f 1 is the multipath effect andεL,f 1

is the noise on the carrier phase range measurement forf1
frequency.Nλ is the ambiguity term whereλf 1 is the wave-
length of the signal atf1 frequency andNλ is the integer
ambiguity in the carrier phase measurement.

δρg =
κ

f 2

∫
Nds +

κe
∫

NB0cosθds

πmf 3
+

3κ2

2f 4

∫
N2ds (A6)

δρg is the total delay due to the ionosphere on the code
(group) observation (in other wordsIf 1 term in Eq. 4). κ

is a constant (40.3, unitless) andf is the signal frequency.N
is the electron density (distribution) along the signal path and
the integral

∫
Nds is taken along this path in increments of

ds. e is the electron charge,B0 is the magnitude of the geo-
magnetic field at the ionospheric pierce point (IPP) where the
signal from the satellite penetrates the ionosphere (e.g. about

3.12×10−5 Tesla at the equator).θ is the angle between the
signal wave vector and the geomagnetic field vector at the
IPP.m is the electron mass.

δρg=
κ

f 2
STEC+

κeB0cosθ

πmf 3
STEC+

3κ2

2f 4
ηNmaxSTEC (A7)

Equation (7) follows from the Eq. (6) by replacing the inte-
gral

∫
Nds with STEC; by taking theB0cosθ term out of the

integral (assuming that this product is LoS independent); and
by approximating the integral

∫
N2ds based on the shape pa-

rameterη, which is quite independent of satellite elevation,
and the maximum electron density in the vertical electron
density profile (Nmax), where all integrals are assumed along
LoS between the receiver (rec) and satellite (sat) (Hartmann
and Leitinger, 1984):

η =

∫ sat
recN

2ds

Nmax
∫ sat

recNds

The first, second and third terms on the right hand side (RHS)
of Eq. (7) are the first (Ion1), second (Ion2) and third (Ion3)
order ionospheric error terms, respectively. In this case,
Eq. (7) can be written more compactly as:

δρg = Ion1+ Ion2+ Ion3

δρg=
κ

f 2
STEC−

κeB0cosθ

2πmf 3
STEC−

κ2

2f 4
ηNmaxSTEC (A8)

Equation (8) represents the advance on the carrier phase
range measurement due to the ionospheric effect (in other
wordsIf 1 term in Eq. 5). The parameters are as defined for
Eq. (7).

As can be seen, the ionospheric error terms for carrier
phase measurements can be obtained from those for the PR
measurements (or vice versa) after applying appropriate scal-
ing factors for each error term with a sign change:

δρP = −Ion1−
Ion2

2
−

Ion3

3

Ion2g,i =
κeB0cosθ

πmf 3
i

STEC (A9)

Following Eq. (7) for the code (group) based observation
for the range, Eq. (9) represents the second order error term
for i-th signal frequency. The parameters are as defined for
Eq. (6).

Ion3g,i =
3κ2

2f 4
i

ηNmaxSTEC (A10)

Following Eq. (7) for the code (group) based observation for
the range, Eq. (9) represents the third order error term fori-th

www.ann-geophys.net/29/1383/2011/ Ann. Geophys., 29, 1383–1399, 2011



1398 Z. G. Elmas et al.: Higher order ionospheric effects in GNSS positioning

signal frequency. The parameters are as defined for Eqs. (6)
and (7).

Nmax=
(20−6)×1012

(4.55−1.38)×1018

{
STEC×

[
1−(

RE

RE+H
)2

cos2(α ·z)
] 1

2
−4.55×1018

}
+20×1012 (A11)

H is the altitude of the ionospheric single layer;RE is the
radius of the earth;α is correction factor andz is the zenith
angle for the signal path piercing the ionospheric single layer
with respect to the local vertical. For a zenith angle of 80◦

andH = 506.7 km,α = 0.9782 (Piraux et al., 2010).

Acknowledgements.As the main author all those that I would like
to acknowledge are already the co-authors, so I will not add any-
thing to the editor’s note.

Topical Editor C. Jacobi thanks M. M. Hoque and M. Tinin for
their help in evaluating this paper.

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