REVIEW ARTICLE
Review of code and phase biases in multi-GNSS positioning
Martin Ha
˚
kansson
1,2
•
Anna B. O. Jensen
1
•
Milan Horemuz
1
•
Gunnar Hedling
2
Received: 18 March 2016 / Accepted: 11 October 2016 / Published online: 21 October 2016
The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract A review of the research conducted until present
on the subject of Global Navigation Satellite System
(GNSS) hardware-induced phase and code biases is here
provided. Biases in GNSS positioning occur because of
imperfections and/or physical limitations in the GNSS
hardware. The biases are a result of small delays between
events that idea lly should be simultaneous in the trans-
mission of the signal from a satellite or in the reception of
the signal in a GNSS receiver. Consequently, these biases
will also be present in the GNSS code and phase mea-
surements and may there affect the accuracy of positions
and other quantitie s derived from the observations. For
instance, biases affect the ability to resolve the integer
ambiguities in Precise Point Positioning (PPP), and in
relative carrier phase positioning when measurements from
multiple GNSSs are used. In addition, code biases affect
ionospheric modeling when the Total Electron Content is
estimated from GNSS measurements. The paper illustrates
how satelli te phase biases inhibit the resolution of the
phase ambiguity to an integer in PPP, while receiver phase
biases affect multi-GNSS positioning. It is also discussed
how biases in the receiver channels affect relative GLO-
NASS positioning with baselines of mixed receiver types.
In addition, the importance of code biases between signals
modulated onto different carriers as is required for mod-
eling the ionosphere from GNSS measurements is dis-
cussed. The origin of biases is discussed along with their
effect on GNSS positioning, and descriptions of how biases
can be estimated or in other ways handled in the posi-
tioning process are provided.
Keywords Hardware biases GNSS positioning
Multi-GNSS
Introduction
Today, Global Navigation Satellite Systems (GNSSs) are
used for a multitude of applications around the world, and
there is a general quest for better positioning accuracy and
reliability, as well as faster position acquisition from both
user groups and the GNSS research community. Combin-
ing observations from multiple GNSSs in one positioning
process and/or using multiple frequencies from one or
more GNSSs is important step toward reaching these goals
(Gleason and Gebre-Egziabher 2009). Accounting for all
error sources in the positioning process, including hard-
ware biases, is a prerequisite for accurate results.
GNSS hardware biases occur because of imper fections
and/or physical limitations in GNSS hardware. The biases
are a result of small delays between events that ideally
should be simultaneous in the transm ission of the signal
from a satellite or in the reception of the signal in a GNSS
receiver. Consequently, these biases will also be present in
& Martin Ha
˚
kansson
martin.hakansson@lm.se
Anna B. O. Jensen
anna.jensen@abe.kth.se
Milan Horemuz
milan.horemuz@abe.kth.se
Gunnar Hedling
gunnar.hedling@lm.se
1
Division of Geodesy and Satellite Positioning, KTH - Royal
Institute of Technology, Drottning Kristinas va
¨
g 30,
100 44 Stockholm, Sweden
2
Department of Geodesy, Lantma
¨
teriet - The Swedish
Mapping, Cadastral and Land Registration Authority,
801 82 Ga
¨
vle, Sweden
123
GPS Solut (2017) 21:849–860
DOI 10.1007/s10291-016-0572-7
the GNSS code and phase measurements. Moreover,
hardware-induced biases differ between different signals,
e.g., P1 and P2, and between different carrier waves, e.g.,
L1 and L2. Hardware-induced biases will cause degrada-
tion in the accuracy of the positioning solution if not
handled properly. This is especially important in high-ac-
curacy positioning with multiple GNSSs (Odijk and Teu-
nissen 2012; Paziewski and Wielgosz 2014; Tegedor et al.
2014), in Precise Point Positioning (PPP) for the resolution
of the integer ambiguities (Teunissen and Khodabandeh
2014), and when using GNSS observations for estimation
of the Total Electron Content (TEC) in the ionosphere
(Jensen et al. 2007; Lanyi and Roth 1988; Sardon and
Zarraoa 1997).
The topic of GNSS hardware biases has received a great
deal of attention in recent years. The introduction of
GLONASS besides GPS in precise positioning requires
knowledge of biases in the receiver hardware that tend to
be specific to the receiver model (Leick et al. 1998; Raby
and Daly 1993; Wanninger and Wallstab-Freitag 2007).
The emergence of new GNSSs, such as the European
Galileo (OS-SIS-ICD-1.2 2015) and the Chinese BeiDou
(BDS-SIS-ICD-2.0 2013), further increases the need of
understanding about GNSS hardware biases, as such
knowledge can lead to both an increase in the accuracy of
the positioning solution, as well as a reduction in the
solution convergence time. The Interna tional GNSS Ser-
vice (IGS) (Dow et al. 2009) arranged bias workshops in
2012 and 2015 to address this issue. In addition, a new data
format with the purpose to store and exchange bias infor-
mation has been developed recently. The format is called
SINEX BIAS, and it is based on the Solution (Software/
technique) INdependent EXchange Format (SINEX). It
supports storage of code and phase biases specific to a
particular GNSS, satellite, receiver, or satellite–receiver
combination (Schaer 2016).
As it turns out, code and phase biases are difficult to
estimate in their undifferenced form, as they are highly
correlated with other terms, e.g., clock errors. Thus, only
differences between biases are possible to estimate directly
from code and phase observations. However, very often, it
is sufficient to know only the differences between certain
biases, as common offsets to the absolute biases might be
absorbed by other terms (e.g., the receiver clock error) in
the positioning process and thereby not influencing the
calculated positions. Bias differences can be formed in
various ways, relevant for different applications. Here, a
review is performed of various phase and code bias dif-
ferences, and a special emphasis is given to biases that
have relevance for precise positioning. The term bias will
be used exclusively for delays that are induced either in the
satellite or in the receiver hardware.
Theoretical description of various biases
The observation equations have the following form for the
code and phase observables, respectively (Hoffman-Wel-
lenhof et al. 2008). They are slightly modified to also
include the receiver and satellite phase and code biases.
/
sys;s
f ;r
¼ q
s
r
þ c d
r
d
s
þ b
sys
f ;r
b
s
f
þ s
sys
þ T
s
r
I
s
f ;r
þ m
s
f ;r
þ k
f
N
s
f ;r
þ e
/
ð1Þ
R
sys;s
sig;r
¼ q
s
r
þ c d
r
d
s
þ B
sys
sig;r
B
s
sig
þ s
sys
þ T
s
r
þ I
s
f
sig
;r
þ M
s
sig;r
þ e
R
ð2Þ
The notation ðÞ
sys;s
sig=f ;r
is henceforth used for a term
associated with a signal sig or carrier wave frequency f,
recorded by a receiver r, and which is transmitted by
satellites, belonging to a GNSS system sys. Absence of
either of these notations means that the term represents a
contribution that is independent of that notation, only
limited to the context in which the equation appea rs. Here
the term ‘‘signal’’ depicts a ranging code modulated on a
particular carrier frequency.
In (1) and (2), the terms are defined in the following
way: q
r
s
true geometrical distance between receiver r and
satellite s, d
r
receiver clock error, d
s
satellite clock error,
B
sig,r
sys
receiver hardware code bias for signal sig, B
sig
s
satellite hardware code bias for signal sig, b
f,r
sys
receiver
hardware phase bias for carrier wave frequency f, b
f
s
satellite hardware phase bias for carrier wave frequency f,
s
sys
time offset for the system time of GNSS system sys
with respect to a chosen reference, T tropospheric delay,
I ionospheric delay, M code multipath, m phase mul tipath,
k
f
wavelength of the carrier wave with frequency f, N phase
ambiguity term, e
/
phase noise, and e
R
code noise.
In (1) and (2), some error sources have been omitted for
the sake of brevity. These error sources include antenna
phase center variations, earth tides, ocean loading, and for
phase observations also the phase windup effect. The time
dependence of the terms has been omitted for the same
reason. In addition, extra care has to be taken with the
receiver clock error as the observation time tags also
depend on this error. It can be corrected with an additional
term
_
q
s
r
d
r
, where
_
q
s
r
is the time derivative of the geomet-
rical distance between receiver r and satellite s.
It is here assumed that the receiver hardware delays are
the same for satellites belonging to the same constellation
and broadcasting the same signal. As will be shown, this
assumption holds true most often for GNSSs using code
division multiple access (CDMA) to distinguish between
signals transmitted by different satellites. It is, however,
850 GPS Solut (2017) 21:849–860
123
not true for GLONASS biases, as GLONASS employs
frequency division multiple access (FDMA) instead of
CDMA. A consequence of FDMA is that the receiver
hardware bias will vary depending on the satellite tracked,
as the channels for different carrier wave frequencies will
cause different delays in the receiver. These GLONASS-
related biases apply both for phase and code measurements,
and they will be discussed later.
Table 1 gives a summary of the biases that will be
treated in the following sections. GNSS hardware biases
appear both in the receiver and in the satellite hardware,
and this is reflected in the second column in Table 1. For
completeness, the absol ute biases as given in (1) and (2)
are also included in the table even though these biases are
not estimable directly from GNSS observations; thus, the
third column indicates whether the bias is an absolute value
or a relative value (most often the product of combinations
of observations). A bias will here also be defined as relative
if it is biased by other error sources. The fourth column
refers to the symbols used for the biases in this paper, and
the fifth column lists the temporal variation of the biases. In
general, GNSS hardware biases have been shown to be
stable over time, and this is reflected for most of the biases
estimated for practical applications. However, in some
cases, the estimated bias might contain residues from other
error sources that will affect its long-term stability. The last
two columns list how the biases are normally treated on the
user side in the positioning process. Here, we distinguish
Table 1 GNSS Hardware biases
Bias type Origin Absolute/
relative
Symbol used Temporal
variation
PPP user Relative user
Receiver phase bias Receiver HW Absolute b
f,r
sys
Long term Eliminate Eliminate
a
(CDMA)
Calibrate/
Eliminate
(FDMA)
Satellite phase bias Satellite HW Absolute b
f
s
Long term – Eliminate
Satellite phase bias Satellite HW Relative – Short term Correction
b
–
Intersystem bias (ISB) phase Receiver HW Relative
b
sys
1
sys
2
f ;r
1
r
2
Long term Estimate TC
c
:
Estimate/
Calibrate
LC
d
:
Eliminate
GLONASS inter-frequency bias
(IFB) phase
Receiver HW Relative
bc
GLO
r
1
r
2
þ k
f
s
bv
GLO
r
1
r
2
Long term Estimate
e
/
Calibrate
Calibrate/
Eliminate
Receiver code bias Receiver HW Absolute B
sig,r
sys
Long term Eliminate Eliminate
(CDMA)
Calibrate
(FDMA)
Satellite code bias Satellite HW Absolute B
sig
s
Long term – Eliminate
Differential code bias (DCB) Satellite and
receiver HW
Relative DCB
s
sig
1
sig
2
;r
f
Long term Calibrate Calibrate
Intersystem bias (ISB) code Receiver HW Relative – Long term Estimate TC:
Estimate/
Calibrate
LC:
Eliminate
GLONASS inter-frequency bias
(IFB) code bias
Receiver HW Relative – Long term Calibrate Calibrate
a
In single constellation positioning
b
The form of the satellite phase bias correction depends on the PPP model used
c
TC Tight combining
d
LC Loose combining
e
In a float solution, they can be merged with the phase ambiguities
f
Total satellite–receiver DCB
GPS Solut (2017) 21:849–860 851
123
between four different ways of dealing with biases on the
user side:
• Eliminate—the bias cancels out in the positioning
model used, usually by between satellites or between
receivers differencing
• Estimate—the bias is estimated as an unknow n param-
eter in the positioning process
• Correction—the bias is estimated by other sources and
broadcasted to the user in real-time as the bias only
have a short-term stability
• Calibrate—the bias is pre-estimated by other sources
and used for the more stable biases
When applying these methods, the bias in question
might be merged with other error sourc es, i.e., for elimi-
nation, the bias might be eliminated together with other
error sourc es, and for estimation, the bias does not need to
be explicitly expressed in the model and might be merged
with other parameters.
In Table 1, the symbols for the relative satellite phase
bias, the code intersystem bias (ISB), and for the GLO-
NASS code inter-frequency bias (IFB) have been omitted,
as they are not described by any equat ions in this paper.
Further, in the user columns, as mentioned earlier, none of
the absolute biases can actually be estimated, either in the
estimation process, or as a correction. However, they can
be eliminated by forming either between receivers or
between satellites differences.
Phase biases
Precise positioning techniques rely on measuring the phase
of the carrier wave on which the GNSS signals are mod-
ulated. In comparison with the code observable, the phase
observable has a much lower noise level, which allows for
a higher positioning accuracy. However, the phase
observable is ambiguous by an unknown number of
wavelengths, which also has to be resolved in the posi-
tioning process. In addition, as is apparent by (1), the phase
observable is biased by delays induced by the receiver and
satellite hardware. These delays prevent integer ambiguity
resolution if n ot accounted for properly.
Relative precise positioning
Relative precise positioning techniq ues often employ
double differencing, even thoug h relative positioning can
be performed through an undifferenced approach (De
Jonge 1998). The process of forming double differences is
described in Hoffman-Wellenhof et al. (2008). Double
differencing (1) gives
/
sys
1
sys
2
;s
1
s
2
f ;r
1
r
2
¼ q
s
1
s
2
r
1
r
2
þ cb
sys
1
sys
2
f ;r
1
r
2
þ T
s
1
s
2
r
1
r
2
I
s
1
s
2
f ;r
1
r
2
þ m
s
1
s
2
f ;r
1
r
2
þ k
f
N
s
1
s
2
f ;r
1
r
2
þ e
/
DD
ð3Þ
where :ðÞ
sys
1
sys
2
;s
1
s
2
f ;r
1
r
2
¼ :ðÞ
sys
2
;s
2
f ;r
2
:ðÞ
sys
2
;s
2
f ;r
1
:ðÞ
sys
1
;s
1
f ;r
2
:ðÞ
sys
1
;s
1
f ;r
1
, and satellite s
i
belongs to system sys
i
.
It is here apparent that the satellite bias term cancels out
along with the satellite clock term when differencing
between receivers. This is not true for the receiver bias
when differencing between satellites, as these may belong
to different GNSS constellations. The remaining receiver
bias term is the so-called ISB, which will be discussed later
on. In addition, the carrier wave frequencies may differ if
the satellites belong to different constellations even if that
is not reflected in the formula above. However, it should be
noted that while the double-differenced ambiguity is an
integer in the single frequency case, different frequencies
would mean that the integer nature of the double-differ-
enced phase ambiguity is lost.
PPP
PPP is an absolute precise positio ning technique, where
undifferenced or between satellites single differenced
observations are used (Kouba and He
´
roux 2001; Zumberge
et al. 1997). In contrast to relative positioning, where most
biases that inhibit the integer ambiguity resolution cancel
out in the double differencing process, these biases remain
as an error source in PPP. Their presence will affect the
quality of the positioning solution if not dealt with
accordingly. As error sources need to be handled explicitly
to a greater degree in PPP, rank deficiencies might appear
in the positioning model as a consequence of an increasing
number of parameters to estimate. Various ways to deal
with these rank deficiencies have been developed over the
years, and they can be summarized as either lumping dif-
ferent parameters together, or assuming some of their
values. This is a reason why the hardware biases often
appear in the equations as merged with other error sources
to which they are highly correlated. In the following two
sections, about satellite phase biases and phase ISBs, the
usage of constellations employing CDMA is assumed. The
last section about phase biases will treat the FDMA case
with GLONASS inter-frequency biases.
Satellite phase biases
Unlike relative positioning, where the biases cancel out
when forming the double differences as in (3), the receiver
and satellite hardware biases remain in the PPP model,
described by (1). Because of these biases, the resolution of
852 GPS Solut (2017) 21:849–860
123
phase ambiguities to integers cannot be performed the
same way in PPP as in relative positioning.
An advantage of integer ambiguity resolution in PPP is
that the convergence time for real-time applications is
reduced, at the same time as the accuracy of the solution is
increased, especially in the longitudinal direction (Collins
et al. 2008). Unfortunately, a rank deficiency of the system
of observation equations for this positioning model makes
it impossible to unambiguously and simultaneously esti-
mate both the phase bias terms and the ambiguity term in
(1) (Teunissen and Khodabandeh 2014). The size of typical
phase biases does not allow the resolution of the integer
ambiguities when the ambiguity term and the phase bias
terms are lumped together (Ge et al. 2008). It is thereby
only possible to resolve the integer ambiguities in PPP
when the phase biases are known beforehand.
In (1) and (2), it was assumed that the receiver hardware
biases were the same for all satellites in the same constel-
lation. The assumption that receiver phase biases are similar
for different satellites is proven correct by the fact that the
phase ambiguities of double-differenced phase observations
can be resolved as integers. Because of this similarity of the
receiver phase bias with respect to the tracked satellite, this
term is not correlated with the ambiguity term in (1), and it
might even cancel out together with the receiver clock error
if single differences between sat ellites are formed. For this
reason, the main error sources that inhibit the resolution of
the integer ambiguities in PPP are the satellite clock error
together with the satellite phase biases.
Consequently, PPP with integer ambiguity resolution
needs satellite phase bias corrections to counteract the
presence of phase biases in the observations. When it
comes to estimating these bias corrections by the service
provider, as mentioned above, they are impossible to esti-
mate to their true undifferenced value due to the system of
observation equations being rank deficient. The satellite
phase bias term is highly correlated with the phase ambi-
guity term. On the user side, phase bias corrections are
needed to restore the integer nature of the phase ambigui-
ties. Fortunately, it is not necessary to know the true
undifferenced phase biases at the user in order to restore
the integer ness of the ambiguities. Even corrections that
are biased by an unknown integer value will achieve this
goal. For this reason, this bias is marked as relative in
Table 1. It should not be confused with the absolute
satellite phase bias in the same table, which can be elimi-
nated in relative positioning.
Several models for PPP with resolution of the integer
ambiguities have been presented in recent years. These
models include for instance Geng et al. (2012), Bertiger
et al. (2010), Collins et al. (2010), Laurichesse et al.
(2009), and Ge et al. (2008). It was shown by Teunissen
and Khodabandeh (2014) that the various models for
integer ambiguity resolved PPP differ in their choice of S-
basis (Teunissen 1985) and in the way they are para-
metrized. To choose a S-basis means to assume values of
certain terms in the equation system to remove rank defi-
ciencies, and the assumed terms compose the S-basis. This
is similar to setting minimal constraints as described in
Leick et al. (2015). The difference in parametrization and
the choice of S-basis between the models affects the way
the satellite phase bias corrections are p rovided to the user.
They can roughly be divided into either providing frac-
tional cycle biases (FCBs) (Ge et al. 2008; Geng et al.
2012), where the fractional parts of the wide- and narrow-
lane biases are distributed to the user, or as the satellite
phase biases being merged with the satellite clock correc-
tions (Collins et al. 2010
; Laurichesse et al. 2009). It is
therefore crucial that the same PPP model is employed both
at the service provider side and at the user side. The
satellite bias correction correspo nds to the relative satellite
phase bias in Table 1, as it is lumped either with an
unknown integer number of cycles or with the satellite
clocks. Since the satellite bias correction in this case is
lumped with other error sources, it only has a short-term
stability. Acco rding to Ge et al. (2008), the narrow-lane
bias correction needs to be supplied to the user at least
every 15 min.
Phase ISB
When multiple GNSSs are used for one positioning solu-
tion, care has to be taken of timescale and reference frame
differences between the GNSSs. In addition, there exists an
intersystem delay due to receiver and satellite hardw are
biases. Assuming that the system-related satellite biases are
handled appropriately (such as the differences in timescales
between GNSSs and satellite biases related to the GNSS
own system time), the remaining delay can be attributed to
the receiver hardware alone, and it is commonly referred to
as an ISB. The ISB appears in the receiver hardware as a
consequence of the various signal structures used by
satellites belonging to different GNSS constellations (He-
garty et al. 2004). The ISB is thereby also present in cases
where identical carrier frequencies are used by the systems.
The following discussion about ISBs will be divided into
separate parts about relative carrier phase-based position-
ing and PPP, respectively.
ISBs in relative positioning
As obvious from (3), the ISB is a bias that persists even
after forming the double differences if these are formed
between satellites belonging to different systems. This bias
will therefore be of relevance also in relative positioning.
The role of ISBs in relative positioning can be divided
GPS Solut (2017) 21:849–860 853
123