Satellite and Receiver Phase Bias Calibration for

Undifferenced Ambiguity Resolution

Simon Banville2, Rock Santerre1, Marc Cocard1 and Richard B. Langley2
1
Center for Research in Geomatics, Department of Geomatics Sciences
Laval University, Quebec City, Canada
2
Geodetic Research Laboratory, Department of Geodesy and Geomatics Engineering,
University of New Brunswick, Fredericton, Canada

BIOGRAPHY increased number of applications. Its integration into

many practical areas is, however, slowed down by the
Simon Banville is a Ph.D. student in the Department of long convergence time required in order to obtain cm-
Geodesy and Geomatics Engineering at the University of level accuracy. This drawback is caused by the difficulty
New Brunswick. He holds an M.Sc. in GPS from Laval in fixing carrier-phase ambiguities to integers. As
University, Quebec and a B.Sc. in geomatics engineering opposed to the differential mode, where many error
from the same institution. sources are eliminated or greatly reduced, PPP has to
properly account for all of them. Some of these error
Rock Santerre is a full professor in the Department of sources, such as code and phase biases, are complex to
Geomatics Sciences and a member of the Centre for model as they tend to merge with the ambiguity
Research in Geomatics at Laval University. He is the parameters during the estimation process, leading to
director of the bachelor’s degree program in geomatics unsuccessful ambiguity resolution.
engineering. Since 1983, his research activities have been
mainly related to high precision GPS. This paper focuses on the receiver and satellite phase bias
calibration required to recover the integer nature of
Marc Cocard is a professor of geodesy in the Department carrier-phase ambiguities. A proper estimation of these
of Geomatics Sciences at Laval University and a member biases would allow correcting the measurements and
of the Centre for Research in Geomatics. His research using the ambiguity resolution techniques developed for
interests relate to the efficient exploitation and future differential positioning. In this way, instantaneous cm-
applications of global navigation satellite systems level accuracy could be something conceivable even with
including, among others, mitigation of the different error a single GPS receiver, considering that the other error
sources. sources have been reduced to a significant level.

Richard B. Langley is a professor in the Department of The first step taken to achieve this objective is to clearly
Geodesy and Geomatics Engineering at UNB, where he understand PPP’s functional model representing the code
has been teaching and conducting research since 1981. He and phase measurements made by the GPS receiver.
has a B.Sc. in applied physics from the University of Special attention is paid to hardware delays, such as code
Waterloo and a Ph.D. in experimental space science from and phase biases, which play a crucial role in the
York University, Toronto. Prof. Langley has been active estimation process using undifferenced measurements.
in the development of GPS error models since the early The impact of these quantities on some estimated
1980s and has been a contributing editor and columnist parameters is described in order to have a better
for GPS World magazine since its inception. He is a understanding of the concepts presented throughout this
fellow of The Institute of Navigation (ION) and the Royal paper.
Institute of Navigation. He was a co-recipient of the ION
Burka Award for 2003. In the second step, a receiver phase-bias calibration
technique using a GPS signal simulator is introduced. A
ABSTRACT simulator has been used to generate errorless signals
which are ideal to isolate the biases inherent to the
Precise point positioning (PPP), considered an alternative receiver. Results show that this calibration process is
to differential positioning, is used in a significantly complex due mainly to the correlation between the
 Pi = ρ i + c(dT − dt ) + T + I i + b P + b
receiver clock and the ambiguity parameters. As such, Pi
+εP (2)
between-satellite single differencing still seems to be the i i
best way to eliminate receiver phase biases.
where
The last part of this paper concerns satellite phase-bias
calibration. To properly calibrate the satellite phase φi
biases, the impact of code biases has to be carefully taken N i = N i + bφ + b (3)
i
into consideration. For this purpose, an alternate widelane
phase-bias calibration method is proposed and is shown to and
be coherent with PPP’s functional model.
i identifies the frequency-dependant terms
INTRODUCTION
Φi is the carrier-phase measurement (m)
While cm-level positioning accuracy with a single GPS Pi is the code measurement (m)
receiver once seemed like a hardly achievable task, the ρ is the instantaneous range between the phase
objective is now finding the quickest method to reach this center of the satellite and receiver’s antennas
threshold. This problem is of a lesser concern in including earth tides, ocean loading and
differential positioning where the phase ambiguity relativistic effects (m)
parameters can be treated as integer values. By
constraining those parameters to integers, one can usually c is the vacuum speed of light (m/s)
obtain high accuracy within seconds to minutes
dT is the satellite clock bias (s)
depending mainly on the baseline length.
dt is the receiver clock bias (s)
On the other hand, in precise point positioning (PPP),
fixing ambiguity parameters to integers is a much more T is the tropospheric delay (m)
complex task. There are two major concerns to consider
when addressing this issue: 1) the error budget affecting Ii is the ionospheric delay (m)
the observations must be kept to a reasonably low level λi is the wavelength of the carrier (m)
(usually within quarter of a carrier wavelength), and 2)
the hardware biases affecting the observations need to be Ni is the integer carrier phase ambiguity (m)
adequately handled. The former concern has been wi is the phase windup effect (m)
investigated by several authors and is beyond the scope of
this paper. Instead, a light will be shone on how the phase bφi is the receiver carrier phase bias (cy)
and code biases become such a nuisance to ambiguity φi
b is the satellite carrier phase bias (cy)
resolution in PPP.
b Pi is the receiver code bias (m)
This paper focuses on three aspects aiming at recovering Pi
the integer nature of phase ambiguity parameters in PPP. b is the satellite code bias (m)
First, the functional model describing the GPS ε Φ , ε P are the measurement noise components,
measurements is briefly recalled in order to demonstrate i i
including multipath (m)
the propagation of unmodeled hardware biases into the
estimation process. Following this discussion, a receiver
phase-bias calibration method based on the use of a GPS Impact of Satellite and Receiver Hardware Biases
signal simulator is presented. Finally, while keeping in When combining code and phase observations in point
mind PPP’s functional model, an alternative to existing positioning, hardware biases (other than clock biases)
methods of satellite phase-bias calibration is introduced. become a major concern. Each satellite contains an
oscillator having a fundamental frequency (f0) of 10.23
ANALYSIS OF PPP’S FUNCTIONAL MODEL MHz, used to generate the carriers and the modulations
[IS-GPS-200D, 2004]. When combining those
Observation Equations components together, several delays can occur, as
Code observations are mandatory in PPP due to a linear illustrated in Figure 1. A similar phenomenon can also be
dependency relating the receiver clock and the ambiguity observed in the receiver when it generates the signal
parameters. Hence, PPP’s functional model can be replica.
described by the following equations:

Φ i = ρi + c ( dT − dt ) + T − I i + λi N i + wi + ε Φ (1)
i
 - Receiver phase biases are expected to be the same
for each satellite but dependant on frequency. They
will then merge into several parameters, such as the
receiver clock, the phase ambiguities and potentially
coordinate estimates.

The abovementioned facts will be used throughout this

paper to support the development of the calibration
methods. Remember that accurate knowledge of these
delays would allow the correction of the phase and code
observations in order to obtain not only unbiased receiver
time estimation but integer ambiguity parameters as well.
For this purpose, the following sections present
methodologies for phase bias calibration.

Figure 1: Satellite hardware biases (based on Wells et al. RECEIVER PHASE BIAS CALIBRATION
[1987] and IS-GPS-200D [2004])
Until now, few attempts have been made to calibrate the
While different techniques have been developed to receiver phase biases since they are subject to important
estimate the intra-frequency and inter-frequency variations that are due mainly to the instability of the
differential delays of signal paths (see, for example, receiver’s oscillator. A zero-baseline test already
Schaer [1999] and Gao et al. [2001]), the absolute delay confirmed that a receiver restart changes the value of the
associated with a particular signal or modulation is much biases [Wang and Gao, 2007], which makes calibration an
more complex to determine. This is due to the linearity of extremely complex process.
many parameters in Equations (1) and (2), such as the
hardware biases, the clock offsets, and the ambiguity On the other hand, receiver bias calibration offers the
parameters. advantage of controlling the environment in which the
tests are performed. For instance, using a GPS signal
In the context of GPS data processing, the biases that simulator allows generating (almost) errorless signals,
cannot be eliminated or modeled usually tend to merge free from satellite biases. From this perspective, a new
with other parameters, thus altering the estimated values. calibration method has been investigated to learn about
Here is a summary of the way each bias affects the the behavior and the characteristics of receiver hardware
estimation process: delays.

- Satellite code biases are mostly eliminated from the Methodology

code observations by using the satellite clock In the process of isolating receiver phase biases, a GPS
corrections (from the broadcast message or the signal simulator has been used to generate phase and code
International GNSS Service (IGS)) along with the observations free from the following error sources:
appropriate differential code delay corrections ephemeris, satellite clock offsets and hardware delays,
[Collins et al., 2005]. On the other hand, code biases troposphere, ionosphere, earth tides, ocean loading, phase
are introduced in phase observations when using windup, multipath and antenna phase-center variations.
satellite clock corrections (refer to the later section This scenario can be described by simplifying Equations
“Satellite Phase Bias Calibration”). (1) and (2), that is:

- The receiver clock parameter absorbs the common  

Φ i = ρ + cdT + λi  N i + bφ 
 + εΦ (4)
part of receiver code biases and the non common  i  i
part is expected to propagate into the code residuals Pi = ρ + cdT + b P + ε P (5)
and estimates of other parameters such as the receiver i i
coordinates.
The satellite and station coordinates being known, the
- Satellite phase biases are different for each satellite only unknown parameters are the receiver clock offset,
on each carrier frequency and they tend to merge into the ambiguities and the receiver’s code and phase biases.
the ambiguity parameters. This is not a problem when Furthermore, the noise level is greatly reduced in this
using the ionosphere-free combination because the scenario and depends primarily on signal resolution. For
ambiguities are no longer integers. For ambiguity the sake of simplicity, hardware simulator delays have
resolution, this aspect becomes a major concern. been omitted.
To obtain the receiver phase biases, the ambiguity To highlight this effect, a simple test has been carried out.
parameters are estimated as real numbers at every epoch Using the methodology described above (Equations (4)
using a least-squares adjustment technique. The fractional and (5)), two scenarios were performed subsequently:
part of the resulting ambiguity values is simply first, the unknown parameters (dT and N) have been
considered to be the bias sought. Also, note that the code estimated using L1 phase observations along with P2 code
biases will be estimated as an intrinsic part of the clock observations. Then, the same phase observations were
offset (refer to the “Preliminary Discussions” subsection used along with the C/A code instead of the P2 code.
hereafter).
Figure 3 presents the receiver clock parameter estimated
Test Description in both scenarios, as well as the corresponding
In order to verify the validity of the proposed ambiguities estimated independently at each epoch for
methodology, a test has been performed using the Spirent satellite PRN 1. A different clock bias value can be
STR4760 GPS signal simulator at the University of New observed in each case, which is a direct consequence of
Brunswick and a NovAtel ProPack V3 receiver (Figure the differential code bias between both code observations.
2). Also, note that the ambiguity values obtained are
different.

Figure 2: Receiver phase-bias calibration setup

Figure 3: Propagation of receiver code biases in
Two sessions lasting approximately three hours each and ambiguity parameters
using the same satellite configuration have been
performed. Between each session, the receiver and the This shows that the receiver code biases alter the
simulator have been turned off to observe the behavior of estimation of the ambiguity parameters, resulting in an
the biases in the context of a receiver reset. inadequate phase bias.

The receiver used outputs phase measurements on L1 and The ideal solution to this problem would be to calibrate
L2 as well as the C/A and P2 code measurements. Since independently the code biases. This issue has already
the P2 code resolution is superior to the one from the C/A been tackled for time transfer purposes [Petit et al., 2001],
code, only the former has been used to compute the clock but the approximated error budget is still around 1 ns (≈
bias in all cases (except where indicated below). 30 cm) [Plumb et al., 2005], which is too large for our
purpose (it is even greater than the wavelength of the
Preliminary Discussions signals).
Before going any further, an important discussion is
required. Even though this methodology has several Results
advantages, it also contains some important drawbacks. Even with the constraint previously mentioned, some
As mentioned previously, it is impossible to estimate information can be deduced on the variability of the
independently the receiver clock offset and the receiver phase/code biases. Figure 4 presents the L1 phase biases
code biases because they are both linear terms and they obtained for the first calibration session along with the
affect identically all simultaneous code observations of a clock offset estimate. Even if the fractional part of a float
given type. The receiver clock parameter will then absorb number is within a range of [0, 1], the results have been
a great part of the receiver code biases. This has a direct transformed into a range of [0.5, -0.5]. This approach was
consequence on the values of the ambiguities estimated initially used in Gabor [1999] and the established
because of the strong correlation between the receiver “convention” has been kept.
clock and phase ambiguity parameters.
Figure 4: Receiver phase biases estimated from the first Figure 5: Receiver phase biases estimated from the
calibration session second calibration session

Each series of a distinct color represents the receiver Additional tests would be needed in order to get a better
phase bias of a particular satellite. As one can see, all comprehension of the characteristics of the biases
satellites have almost identical biases, which is logical observed. For instance, using an external oscillator in a
since it has been known for a long time that differential temperature-controlled environment could reveal valuable
techniques (between satellites) greatly reduce receiver information. Currently, the most efficient solution is still
hardware delay effects. A slight difference can be noticed to perform satellite-satellite single differencing (SSSD) to
between the series which could possibly be caused by eliminate the receiver biases. This approach will then be
channel dependant delays. Moreover, a drift is visible: used in calibrating the satellite phase biases, which is the
initial values around -0.15 cycles are observed while they topic of the next section.
end at approximately -0.05 cycles after a three-hour
period, which could be caused by thermal effects. Finally, SATELLITE PHASE-BIAS CALIBRATION
the correlation between the receiver clock and the
ambiguity parameters is noticeable. When a clock slew Satellite phase biases are certainly the most complicated
happens, even if the ambiguities preserve the same integer delays to handle in ambiguity resolution for PPP.
value, all fractional parts are subject to an identical jump However, because of a possible long-term stability of
and then converge back to the mean value. Further these biases, several calibration methods have been
investigations are needed to understand more adequately proposed recently [Gabor, 1999; Ge et al., 2006; Leandro
this behavior. and Santos, 2006; Laurichesse and Mercier, 2007]. This
section will first review the basics on which most of the
Figure 5 shows the L1 phase biases of the second existing approaches rely to get a perspective of the
calibration session made a day later. The mean value of potential problems that could be encountered. Then, an
the biases is clearly different as compared to the one from alternate calibration method will be presented to
the previous session, which confirms the results obtained overcome some of the limitations discovered.
by Wang and Gao [2007] indicating that the biases are
different when the receiver is turned off and back on. On The “Melbourne-Wübbena” approach
the other hand, it is not possible to conclude with Most of the existing methods for satellite phase-bias
certitude that the phase (only) biases are modified, as it calibration rely on the “Melbourne-Wübbena” signal
has previously been shown that code biases are not combination [Melbourne, 1985; Wübbena, 1985] because
constant either without a stable external oscillator [Petit et it allows reducing considerably the error budget affecting
al., 2001]. the resulting observation. Expressed in satellite-satellite
single difference (SSSD), denoted as ∇ in the following
Figure 5 also shows a much more accentuated drift at the equations, this combination can be formed using the
beginning of the session. In the first session, the receiver widelane carrier phase combination (wl):
had been powered on for a certain period of time before
the data recording, while for the second session, both f 1 ∇Φ 1 − f 2 ∇Φ 2
events occurred almost simultaneously. Thermal effects ∇Φ wl =
(receiver warming up) then become a plausible f1 − f 2
(6)
( )
explanation for this behavior. f1 φ wl
= ∇ρ + I 1 + λ wl ∇N wl + ∇b
f2
and the narrowlane code combination (nl): biases in the observables [Collins et al., 2005]; that is to
say:
f1∇P1 + f 2 ∇P2
∇Pnl = Pif f1 P1 − f 2 P2
2 2
f1 + f 2 dt = dt + b = dt + (10)
2 2
P P
(7) f1 − f 2
f1 f1 ∇b 1 + f2 ∇b 2
= ∇ρ + I1 +
f2 f1 + f 2 Unlike with code observables, applying the group delay
correction included in the broadcast message or estimated
Using Equations (6) and (7), the Melbourne-Wübbena by the IGS will not completely remove the contribution of
combination (mw) can be formed as: the biases introduced in the phase measurements. The
code biases being unique to each satellite, they will merge
∇Φ mw = ∇Φ wl − ∇Pnl with the ambiguity parameters and add a contribution to
the values estimated.
( )−
P1 P2
φ wl f 1 ∇b + f 2 ∇b (8)
= λ wl ∇N wl + ∇b By combining Equations (6) and (10), the SSSD widelane
f1 + f 2
ambiguities estimated with PPP become:
where the newly introduced terms are
φ wl 1 Pif
∇N PPP = ∇N wl + ∇b − ∇b (11)
ρ the geometric range combined with all non- λ wl
frequency-dependant terms (m)
fi the carrier frequency i (Hz) It seems logical then that, in order to recover the integer
nature of the ambiguities in a PPP positioning context,
Equation (8) has often been used in the past to compute one would have to remove the biases included in Equation
the widelane ambiguity and to get an estimate of the (11) or, at least, their fractional contribution. It is also
widelane phase bias. It is important to note though that obvious that the ambiguities obtained from both
the narrowlane code biases also present in the equation approaches (Equation (9) and (11)) will be different. This
will not only change the fractional part of the ambiguity difference can be expressed as:
estimated, but will also contribute to an integer portion of
the computed ambiguity. Using Equation (8), the resulting
ambiguity can then be expressed as:
∇N mw − ∇N PPP =
1
λ wl
(∇b Pnl
− ∇b
Pif
)
(∇b )
(12)
1 f1 f 2 P2 P1
φwl 1 Pnl = − ∇b
∇N mw = ∇N wl + ∇b − ∇b (9) λ wl f 12 − f 22
λ wl
The term between the parentheses corresponds to the
Combining, at this stage, the estimated ambiguities with
SSSD differential code bias (DCB) between P1 and P2.
the ambiguities estimated using the ionosphere-free
combination to obtain the L1 ambiguities (as suggested in
According to the previous results, two calibration
some of the aforementioned references) introduces further
scenarios are conceivable to obtain satellite phase biases
biases. This causes the situation to become even more
(that obviously contain code biases as well) that would
complex. For the sake of simplicity, only the widelane
allow recovering of the integer nature of ambiguities with
case will be analyzed in this paper. The other cases are
respect to the hardware delays present in the observations:
discussed by Banville [2007].
1) Directly use the PPP functional model without
Derivation of an Alternate Method
any explicit code/phase combination. This
Can the phase biases computed using Equation (9) be
method will however be more sensitive to
used in PPP to recover the integer nature of the
observational errors such as atmospheric effects,
ambiguities? To answer this question, one must first know
orbital errors, etc.
the nature of the biases present in the SSSD widelane
2) Use the Melbourne-Wübbena combination of
observable. According to Equation (6), it seems like the
Equation (8) and apply the correction described
only bias present is the widelane phase bias. However, the
by Equation (12).
satellite clock corrections ( dt ), currently estimated using
the ionosphere-free combination by the IGS or the GPS Practical Comparison of the Methods
control segment, introduce the ionosphere-free (if) code The mathematical proof of the preceding subsection can
be validated using a simple empirical test. The GPS
observations collected at four IGS stations (NRC1, symbol on the graph represents a single pass difference
CAGS, GODE and USNO) from January 8th to 10th 2007 between the PPP-estimated widelane ambiguity and the
have been used for this purpose (see Figure 6). one computed from the Melbourne-Wübbena combination
for each station. One can clearly see that the differences
can reach several widelane cycles (1 cycle ≈ 86 cm).

Figure 7: Difference between the PPP-estimated

widelane ambiguities and the ones computed from the
Figure 6: IGS stations used for the satellite phase-bias Melbourne-Wübbena combination
calibration test
In order to confirm that the differences observed match
First, using the PPP software developed by the first author Equation (12), the DCB values were taken from the
at Laval University, the following parameters have been IONEX files of January 8th 2007 [IGS products, 2007]
estimated independently at every station for each day of and then scaled using the previously mentioned equation.
the test: The results are shown in Figure 8.
- constrained coordinates
- wet tropospheric zenith delay
- stochastic ionospheric delays (1/satellite/epoch
using constraints from Global Ionospheric Maps
(GIM) [IGS products, 2007])
- receiver clock bias
- L1 and L2 ambiguities (combined later on to form
the widelane ambiguities)

The IGS stations were chosen in pairs forming baselines

of about 20 km each. This strategy allowed exploiting the
use of integer double differenced ambiguities as
constraints on the undifferenced ambiguities, as well as
constraining the relative atmospheric delays between the
stations. However, for the test described in this paper, this Figure 8: SSSD DCB from January 8th 2007, scaled
approach is not explored in further detail. according to Equation (12)

Then, the same data has been reprocessed using the The two figures show good agreement, which leads us to
Melbourne-Wübbena linear combination of Equation (8). believe that the biases affecting each estimation technique
This approach allowed us to compute for each station a have been correctly identified in Equations 8 and 11.
value of the widelane ambiguity for each satellite at each
epoch. Then, an average of all ambiguity values computed CONCLUSION AND FUTURE WORK
for a particular station for a single satellite pass has been
performed in order to reduce the noise. Receiver and satellite phase biases are in a great part
responsible for the problems related to ambiguity
In both cases, the SSSD ambiguities were formed with resolution in PPP. Even though several techniques can be
respect to satellite PRN 14. Then, the ambiguity values used to deal with those biases, this paper opted for the
coming from both methods were differenced as suggested calibration route. Thus, calibration methods for both types
by Equation (12). The results are shown in Figure 7. Each of biases were presented and a special attention has been
paid to correctly handle the impact of code biases on the Ph.D. Thesis, University of Texas at Austin, Texas. 198
estimated values. pp.

Receiver phase-bias calibration has been performed using Gao, Y., F. Layahe, P. Héroux, X. Liao, N. Beck and M.
a GPS signal simulator. However, the receiver code biases Olynik (2001). “Modeling and Estimation of C1-P1 Bias
contaminated the receiver clock estimation which, in turn, in GPS Receivers”. Journal of Geodesy, Vol. 74, No. 9,
affected the estimated ambiguities. The tests also allowed pp. 621-626.
the confirmation of the results of previous studies
showing that the receiver phase bias is not constant after a Ge, M., G. Gendt and M. Rothacher (2006). “Integer
receiver re-initialization. Additional tests using an Ambiguity Resolution for Precise Point Positioning”.
external oscillator in a temperature-controlled Proceedings of VI Hotine–Marussi Symposium of
environment could allow valuable information to be Theoretical and Computational Geodesy: Challenge and
obtained on the receiver biases. One should also keep in Role of Modern Geodesy, Wuhan, China, May 29 – June
mind that the signal simulator introduces further biases 2.
which are hard to quantify.
IS-GPS-200D (2004). Navstar GPS Space Segment /
Finally, the use of the Melbourne-Wübbena combination Navigation User Interfaces. IS-GPS-200, Revision D,.
has been discussed for the satellite phase-bias calibration. ARINC Research Corporation, El Segundo, California.
It has been shown that an additional correction would be
needed in order to use the bias computed with this IGS products (2007). International GNSS Service data
combination with PPP’s functional model. The alternate products. http://igscb.jpl.nasa.gov/components/prods.html
calibration method presented in this paper considers code
biases with a special care in order to estimate coherent Laurichesse D. and F. Mercier (2007). “Integer ambiguity
phase biases. Only the case of the widelane has been resolution on undifferenced GPS phase measurements and
presented in this paper, but the rationale behind this its application to PPP”. Proceedings of ION GNSS 2007,
method can be used to estimate the L1 and L2 phase biases Forth Worth, Texas, 25-28 September, pp. 839-848.
as well. In the future, tests should be performed to assess
the performance of this method on ambiguity resolution Leandro, R. and M. Santos (2006). “Wide Area Based
success rate. Precise Point Positioning”. Proceedings of ION GNSS
2006, Fort Worth, Texas, 26-29 September, pp. 2272–
ACKNOWLEDGMENTS 2278.

Special thanks are due to the Natural Sciences and Melbourne, W.G. (1985). “The Case for Ranging in GPS
Engineering Research Council of Canada (NSERC), the Based Geodetic Systems”. Proceedings of 1st
Canadian Space Agency (CSA) and the Geomatics for International Symposium on Precise Positioning with the
Informed Decisions (GEOIDE) Network of Centres of Global Positioning System, edited by Clyde Goad, U.S.
Excellence for the financial support accorded to the first Department of Commerce, Rockville, Maryland, 15-19
author during his M.Sc. research at Laval University. April, pp. 373-386.

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