# GPS Carrier-Phase Time Transfer Using Single-Difference Integer Ambiguity Resolution

Authors: J. Delporte; F. Mercier; D. Laurichesse; O. Galy
Journal: International Journal of Navigation and Observation
 (2008-01-27)

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## Abstract

GPS is widely used for time and frequency transfer. To estimate the clock difference between two ground
stations, a single baseline solution can be performed using external products for the modelling of the geometrical effects (constellation ephemeris and station coordinates). The baseline solution relies on a single-difference formulation, using code and phase ionosphere-free measurements. The phase ambiguities are usually adjusted as floating parameters. Such solutions give very good results when modelling hypotheses are consistent between the external solution (e.g., GPS orbits) and the baseline solution.
However, the frequency bias in the computed clock is very sensitive to discrepancies in the models, and is only observed thanks to the code measurement, with limitations due to the noise. Here, we propose to solve the integer ambiguities on single-difference phase measurements. The advantage is the complete elimination of the clock drifts observed in floating ambiguities solutions. This formulation allows also a reliable continuous connection between overlapping clock solutions (jumps between such solutions can be completely eliminated). Several time transfer results are analyzed and compared to TWSTFT. The methodology has been extended to a network of stations using integer ambiguities on zero-difference measurements. The corresponding results are given for a few European stations.


---

## Body

## 1. Introduction

GPS
time transfer is known to be an accurate way to compare ground clocks. An
extensive analysis of the various techniques is given in [1]. The
discontinuities at day boundaries observed in the current IGS precise time and
geodesy solutions degrade the precision of their time and frequency solutions,
but also provide valuable information about their timing accuracy [2]. In [1],
two methods are proposed to deal with this problem: clock handover and
ambiguity stacking. The drawback of these methods is that many internal
parameters must be kept with each individual daily solution to compute a global
continuous clock solution (e.g., normal equations and ambiguities of the
overlapping passes). These methods are, therefore, not usable by external users
who only have access to the daily ephemeris and clocks solutions. Independent
time and frequency transfer applications need a method not so strongly related
to the internal characteristics of a given global solution.Here,
we will first focus on single baseline solutions (time and frequency transfer
between two stations). The GPS code measurement
provides information on the clock difference between the two stations. The performance is
limited for short durations by the noise of the code. The results can be
improved by adding phase measurements that are much less noisy but whose use
has some drawbacks. This is interesting for short
durations (typically below ten days).The
single baseline solution is performed using external
products for the modeling of the geometrical effects (constellation ephemeris
and stations coordinates, e.g., IGS products [3]). These precise products rely
generally on double difference processing with integer ambiguity
fixing.The
single difference ionosphere-free code and phase measurements are processed to
obtain the clock difference at each epoch. The clock difference between the two
receivers is solved together with one floating-phase ambiguity for each pass [4].
The frequency bias of the solution is very sensitive to the model precision.
Such biases appear even in very good global solutions, which use
double-difference ambiguity fixing (see [2, Figure 8]). These frequency biases are
due to small errors accumulated on the floating zero-difference ambiguities. It
is shown here that when the integer nature of the ambiguity is taken into
account on single- or zero-difference measurements, these frequency biases are
totally suppressed. Such properties were shown in [5] on short baselines and
single-difference measurements. Simulations on long baselines [6] have also
shown that taking into account integer ambiguities minimizes the magnitude of
frequency biases in the solutions.Integer
ambiguity fixing is a well-developed method in precise geodetic applications [1].
In these applications, the integer ambiguity fixing is performed on double-difference
measurements. This process has the advantage of eliminating clocks from the
solution leading to smaller problems to solve. Other propagation common errors
are also reduced in the process depending on the relative geometry. When an integer
constraint on double-difference ambiguities is added to the global clock
solution, the solution is stabilized making it more robust to potential mismodeling
[5, 7].Different
methods have already been developed for integer ambiguity searching like the
LAMBDA method [8], but the latter is more focused on real-time data processing for
precise positioning. However, in these geodetic applications, the integer
ambiguity property is only considered on double-difference data.The
approach presented here uses a new formulation allowing integer ambiguity fixing
on single-difference measurements while solving for clock offsets at each
epoch. It is well adapted for post-processed least-square analysis on a single
baseline.The
method has been extended to more than two stations, giving global clock
solutions with integer ambiguity fixing on a station network. In this case, the
ambiguities are defined at zero-difference level [9, 10]. Several time-transfer
results using the single baseline [11] and the network methods are presented in
this paper and compared to TWSTFT and to IGS clock products.


## 2. Ambiguity Resolution Method

### 2.1. Methodology

For a given baseline, the general methodology to estimate the time offset between
the two ground stations is as follows:(i)
absolute positioning for both stations using IGS precise ephemeris
and clocks (precise point positioning (PPP) or “all-in-view” technique).(ii)
relative solution (“common-view” technique) with estimation of the
integer ambiguities and clock difference.The
absolute positioning solution is a floating-ambiguity solution using ionosphere-free
code and phase combinations. The troposphere propagation delays, the station
coordinates (if necessary), and the receiver clock offset relative to the time
of the IGS constellation solution are computed. The purpose of this solution is
to obtain a precise model for the propagation distance of the ionosphere-free
signal.The
relative solution uses a single-difference formulation. Integer ambiguities (one
value per pass) and clock differences are solved simultaneously. The station coordinates
and the troposphere delays are held fixed to the values resulting from the
absolute positioning step.


### 2.2. Model Equations

Initial
preprocessing is carried out by sequential finite differences on the
measurements. Passes are defined as continuous phase measurements on the two
frequencies. Phase measurements are then roughly aligned on code measurements
by adding an integer number of cycles constant per pass. Pseudorange
measurements are notedp1 and p2 for the two
frequencies f1 and f2, respectively; L1 and L2 denote the corresponding aligned
phase measurements.By
combining these data, the ionosphere-free combinations for code (pc)
and phase (Qc) can be constructed:(1)Pc=P2−γP11−γ,Qc=λ2L2−γλ1L11−γ,where λ1 and λ2 are the wavelengths corresponding
to f1 and f2, and γ is the ratio f12/f22.To
model measurements and to compute partial derivatives with respect to station
coordinates and vertical troposphere delays, it is necessary to know the geometry
of the problem. IGS ephemeris and clocks are used to
compute this geometry. The model equations for the ionosphere-free data are(2)Pc=Dgeo+Dtropo+(hrec−hemi)+μrecemi,Qc=Dgeo+Dtropo+Dwind+(hrec−hemi)−λ2N2−γλ1N11−γ,where Dgeo is the geometrical distance between
the ionosphere-free phase centers of the transmitter and the receiver antennas, Dwind
corresponds to the effect of the relative-phase
rotation between the antennas of the transmitter and receiver (wind-up effect),
and Dtropo
is
the troposphere propagation delay. N1 and N2 are the unknown integer ambiguities on
each frequency, constant over a pass. hrec is
the receiver clock offset and hemi
the
transmitter clock offset.μrecemi is an unknown bias which
reflects the fact that code and phase may be misaligned [9]. We use phase as
the reference for our clocks, so μrecemi represents the code bias. This contributes to
the code-phase bias identified in [1].


### 2.3. Absolute Positioning

This
step consists in estimating the station coordinates and the zenith troposphere delays
while identifying a floating ambiguity per pass. The receiver clock offset with
respect to the reference time (here IGS time) at each epoch is also adjusted. No
constraint is applied on the coordinates or on the clock. Zenith troposphere
delays are adjusted every two hours assuming a linear evolution in-between. The
delays are constrained according to(Tri+1−Tri)2/σ2, where Tri is the ith zenith troposphere delay and σ is equal to 5 mm.


### 2.4. Relative Solution

It is shown in [9, 12] that the difference between the ambiguities N2 and N1 (widelane ambiguity) can be represented as(3)Nw=N2−N1=f(P1,P2,L1,L2)+(ai−aj),where ai and aj are biases related, respectively, to the
receiver i and transmitter j. f(P1,P2,L1,L2) is the four observables Melbourne-Wübenna linear
combination. When computing the station-station single difference, the
satellite contribution aj vanishes and the equation becomes (Δ is the single-difference operator)(4)ΔNw=ΔN2−ΔN1=f(ΔP1,ΔP2,ΔL1,ΔL2)+ai1i2,where ai1i2 is the difference between receivers i1 and i2 biases. For most geodetic receivers, ai biases are slowly varying and their excursion
remains small (typically below 0.1 widelane cycle) and thus can be estimated
over the whole dataset [9] up to an arbitrary integer constant. It is then possible
to compute the ΔNW values for each pass using (4).The
model equations for the single-difference problem are derived from (2). Using
(4), ΔN2 can be expressed as the sum of ΔNW and ΔN1 leading to(5)ΔPc=Δ(Dgeo+Dtropo)+Δhrec,ΔQc+λ21−γΔNw=Δ(Dgeo+Dtropo+Dwind)+Δhrec−λcΔN1,λc=λ2−γλ11−γ, where Δhrec is the difference between
the receiver clocks and λc is the ionosphere-free
wavelength (10.7 cm). The outputs of the absolute positioning step are used to
compute the geometrical quantities on the right-hand side. The troposphere is held
fixed at the values previously adjusted (this provides a better observability,
since all satellites in view are used.) Given the size of the wavelength and
the precision of the geometrical models (a few centimeters), it is then possible
to solve for integer values of ΔN1.The
computation of the integer ambiguities is done sequentially by aligning each pass
with the preceding ones (using only the ionosphere-free phase equation). The
phase clock differenceΔhphase is then obtained using the computed integer ΔN1 values, (6)Δhphase=〈ΔQc+λ21−γΔNw+λcΔN1−Δ(Dgeo+Dtropo+Dwind)〉epoch(mean of the contribution of all satellites at a given epoch).Because
of the uncertainty in the integer ambiguity for the first pass, theΔhphase clock differences are also
defined up to an integer number of λc. Clock solutions computed in
different batches may differ by an integer number of λc, however, when datasets overlap,
it is possible to realign the solutions. This means that continuous long-duration
frequency transfer solutions can be constructed simply by solving the integer λc jumps at overlaps.Becauseai biases are only known up to an arbitrary
integer constant, a bias is left in Δhphase. This bias can be
adjusted using code measurements in order to achieve time transfer (and not
only frequency transfer). We will refer to this clock difference as GPS carrier
phase with integer ambiguity, CP(A). The advantage of our approach is that this
final bias can be adjusted over the length of the complete data set (if there
are no phase interruptions), leading to a significant reduction in the uncertainties.


## 2.1. Methodology

For a given baseline, the general methodology to estimate the time offset between
the two ground stations is as follows:(i)
absolute positioning for both stations using IGS precise ephemeris
and clocks (precise point positioning (PPP) or “all-in-view” technique).(ii)
relative solution (“common-view” technique) with estimation of the
integer ambiguities and clock difference.The
absolute positioning solution is a floating-ambiguity solution using ionosphere-free
code and phase combinations. The troposphere propagation delays, the station
coordinates (if necessary), and the receiver clock offset relative to the time
of the IGS constellation solution are computed. The purpose of this solution is
to obtain a precise model for the propagation distance of the ionosphere-free
signal.The
relative solution uses a single-difference formulation. Integer ambiguities (one
value per pass) and clock differences are solved simultaneously. The station coordinates
and the troposphere delays are held fixed to the values resulting from the
absolute positioning step.


## 2.2. Model Equations

Initial
preprocessing is carried out by sequential finite differences on the
measurements. Passes are defined as continuous phase measurements on the two
frequencies. Phase measurements are then roughly aligned on code measurements
by adding an integer number of cycles constant per pass. Pseudorange
measurements are notedp1 and p2 for the two
frequencies f1 and f2, respectively; L1 and L2 denote the corresponding aligned
phase measurements.By
combining these data, the ionosphere-free combinations for code (pc)
and phase (Qc) can be constructed:(1)Pc=P2−γP11−γ,Qc=λ2L2−γλ1L11−γ,where λ1 and λ2 are the wavelengths corresponding
to f1 and f2, and γ is the ratio f12/f22.To
model measurements and to compute partial derivatives with respect to station
coordinates and vertical troposphere delays, it is necessary to know the geometry
of the problem. IGS ephemeris and clocks are used to
compute this geometry. The model equations for the ionosphere-free data are(2)Pc=Dgeo+Dtropo+(hrec−hemi)+μrecemi,Qc=Dgeo+Dtropo+Dwind+(hrec−hemi)−λ2N2−γλ1N11−γ,where Dgeo is the geometrical distance between
the ionosphere-free phase centers of the transmitter and the receiver antennas, Dwind
corresponds to the effect of the relative-phase
rotation between the antennas of the transmitter and receiver (wind-up effect),
and Dtropo
is
the troposphere propagation delay. N1 and N2 are the unknown integer ambiguities on
each frequency, constant over a pass. hrec is
the receiver clock offset and hemi
the
transmitter clock offset.μrecemi is an unknown bias which
reflects the fact that code and phase may be misaligned [9]. We use phase as
the reference for our clocks, so μrecemi represents the code bias. This contributes to
the code-phase bias identified in [1].


## 2.3. Absolute Positioning

This
step consists in estimating the station coordinates and the zenith troposphere delays
while identifying a floating ambiguity per pass. The receiver clock offset with
respect to the reference time (here IGS time) at each epoch is also adjusted. No
constraint is applied on the coordinates or on the clock. Zenith troposphere
delays are adjusted every two hours assuming a linear evolution in-between. The
delays are constrained according to(Tri+1−Tri)2/σ2, where Tri is the ith zenith troposphere delay and σ is equal to 5 mm.


## 2.4. Relative Solution

It is shown in [9, 12] that the difference between the ambiguities N2 and N1 (widelane ambiguity) can be represented as(3)Nw=N2−N1=f(P1,P2,L1,L2)+(ai−aj),where ai and aj are biases related, respectively, to the
receiver i and transmitter j. f(P1,P2,L1,L2) is the four observables Melbourne-Wübenna linear
combination. When computing the station-station single difference, the
satellite contribution aj vanishes and the equation becomes (Δ is the single-difference operator)(4)ΔNw=ΔN2−ΔN1=f(ΔP1,ΔP2,ΔL1,ΔL2)+ai1i2,where ai1i2 is the difference between receivers i1 and i2 biases. For most geodetic receivers, ai biases are slowly varying and their excursion
remains small (typically below 0.1 widelane cycle) and thus can be estimated
over the whole dataset [9] up to an arbitrary integer constant. It is then possible
to compute the ΔNW values for each pass using (4).The
model equations for the single-difference problem are derived from (2). Using
(4), ΔN2 can be expressed as the sum of ΔNW and ΔN1 leading to(5)ΔPc=Δ(Dgeo+Dtropo)+Δhrec,ΔQc+λ21−γΔNw=Δ(Dgeo+Dtropo+Dwind)+Δhrec−λcΔN1,λc=λ2−γλ11−γ, where Δhrec is the difference between
the receiver clocks and λc is the ionosphere-free
wavelength (10.7 cm). The outputs of the absolute positioning step are used to
compute the geometrical quantities on the right-hand side. The troposphere is held
fixed at the values previously adjusted (this provides a better observability,
since all satellites in view are used.) Given the size of the wavelength and
the precision of the geometrical models (a few centimeters), it is then possible
to solve for integer values of ΔN1.The
computation of the integer ambiguities is done sequentially by aligning each pass
with the preceding ones (using only the ionosphere-free phase equation). The
phase clock differenceΔhphase is then obtained using the computed integer ΔN1 values, (6)Δhphase=〈ΔQc+λ21−γΔNw+λcΔN1−Δ(Dgeo+Dtropo+Dwind)〉epoch(mean of the contribution of all satellites at a given epoch).Because
of the uncertainty in the integer ambiguity for the first pass, theΔhphase clock differences are also
defined up to an integer number of λc. Clock solutions computed in
different batches may differ by an integer number of λc, however, when datasets overlap,
it is possible to realign the solutions. This means that continuous long-duration
frequency transfer solutions can be constructed simply by solving the integer λc jumps at overlaps.Becauseai biases are only known up to an arbitrary
integer constant, a bias is left in Δhphase. This bias can be
adjusted using code measurements in order to achieve time transfer (and not
only frequency transfer). We will refer to this clock difference as GPS carrier
phase with integer ambiguity, CP(A). The advantage of our approach is that this
final bias can be adjusted over the length of the complete data set (if there
are no phase interruptions), leading to a significant reduction in the uncertainties.


## 3. Time Transfer Results

### 3.1. Continental Baseline (OPMT/PTBB,∼700 km)

OPMT is an IGS station located in Paris Observatory; it is driven by a Hydrogen
Maser which is compared to UTC(OP) on an hourly basis (these corrections were
provided by Paris Observatory). The receiver is an Ashtech Z-12T. The station
is calibrated so that we can relate the measurements to UTC(OP).PTBB
is also an IGS station located in PTB (Braunschweig, Germany); it is directly
driven by UTC(PTB). Calibration delays for both stations have been used for the
computation of GPS CP(A). Therefore, a GPS time transfer between these stations
allows to compute UTC(OP)-UTC(PTB). This offset is also computed by BIPM
using TWSTFT and GPS P3 CV (common-view) using Vondrak smoothing. The BIPM data have been retrieved from their
ftp site [13].In
Figure1, we compare the TWSTFT and GPS P3 CV from BIPM data to our GPS CP(A)
time transfer.Figure 1
Time offset UTC(OP)-UTC(PTB) computed by three different
techniques using the single baseline approach.We
observe a 5-nanosecond bias between the TWSTFT and the two GPS solutions. Short-term
variations are very similar to TWSTFT and GPS CP(A), while they are invisible
with GPS P3 CV with Vondrak smoothing.In
Figure2, the difference between BIPM TWSTFT and GPS CP(A) is presented in the
same period.Figure 2
Difference between BIPM TWSTFT and GPS CP(A).The standard deviation of the difference between TWSTFT and GPS CP(A) is 0.25 nanosecond,
with no drift. A constant bias of 5 nanoseconds has been removed from the data.Figure3 shows the stability of UTC(OP)-UTC(PTB) computed with TWSTFT, GPS P3 CV from
BIPM data, and CNES GPS CP(A).Figure 3
Allan deviation of the time offset UTC(OP)-UTC(PTB) for the three different techniques.The
frequency stabilities of the time offset UTC(OP)-UTC(PTB) computed by TWSTFT and GPS CP(A)
agree very well. The apparent good short-term stability of the BIPM GPS P3 CV
is only due to the smoothing and is not deemed to be representative of the
clock difference behavior.


### 3.2. Transatlantic Baseline (USN3/PTBB,∼6300 km)

USN3 is an IGS station located at the US Naval Observatory; it is driven by
UTC(USNO) with no corrections to apply (see USN3 IGS log file [14]).The
goal is to perform the GPS time transfer between the USN3 and PTBB stations to
compute UTC(USNO)-UTC(PTB) and compare results to the BIPM data.
For such a long baseline, a direct ambiguity solution is not possible because
modeling errors are large (in particular troposphere modeling at low elevation).
In addition single difference passes are shorter, and there are few
simultaneous passes. In order to apply our ambiguity resolution method, we
introduce two bridge stations (STJO and REYK). This leads to three short
baselines (~2500 km).The three
baselines are processed independently (but with the same geometrical parameters),
giving estimates for the three clock differences (6 days solution). Table1
summarizes the results.Table 1
N1 ambiguities on the three baselines.
BaselineLength (km)Total N1Integer N1 (%)PTBB-REYK2265231191 (83%)REYK-STJO2597230199 (87%)STJO-USN32179152145 (95%)During the
identification of the integer ambiguities, some passes have been rejected due
to inconsistent overlaps (threshold used 0.25 cycle on the overlap between two
passes).The
clock difference for the baseline PTBB/USN3 is obtained by adding the three
contributions. There are eleven days of data that have been split into two
batches of 6 days with one-day overlap. These results (with and without integer
fixing) are shown in Figure4.Figure 4
PTBB/USN3 clock difference for the
two 6-day batches for integer (blue) and floating (red) solutions (units in cycles).As
observed in [4, 5], floating solutions sometimes exhibit important drifts,
which are not present in the integer resolution. In addition, the integer
clocks differ exactly by one cycle on the central overlapping day. A continuous
clock covering the whole period is then easily obtained by shifting the second solution by one cycle to
align it with the first one. The remaining alignment error is then below one
cycle (about 0.3 nanosecond).In
Figure5, we compare the two BIPM techniques to the GPS CP(A) time transfer.Figure 5
Time offset UTC(USNO)-UTC(PTB) computed by three different
techniques using the single baseline approach.The same conclusions as for Figure1 can be drawn. The standard deviation of the
difference between the TWSTFT and GPS CP(A) on this baseline is 0.3 nanosecond.


### 3.3. Network Solution

The method has been generalized to a network solution, which is detailed in [9, 10].
The global solution approach is similar to the single baseline formulation
extended to zero-difference measurements (identification of ground stations and
GPS satellite clocks). This means that all receiver clocks are computed relative to a reference clock chosen in the network, modulo an integer number of λc wavelengths. The corresponding CP(A) clock differences are
obtained after alignment to code measurements. Figure 6 shows a map of the
stations used in the solution. BRUS was chosen as time reference.Figure 6
European network used in this study.The
station coordinates and zenith troposphere delays are forced to the values
obtained from the absolute positioning that is performed independently for each station as for the single baseline solution.Clocks and ambiguities are adjusted in the integer solution; 98% of N1 integer ambiguities are identified (statistics are computed only for passes with at
least one common view with another station).Figure7 presents the time offset UTC(OP)-UTC(PTB) from BIPM TWSTFT data (blue circles),
from IGS clock products (red dots), and from our network solution. We have
subtracted the same arbitrary value for the two GPS solutions, because the receiver
internal calibration delays were not available.Figure 7
Time offset UTC(OP)-UTC(PTB) from BIPM TWSTFT, IGS clock products, and GPS CP(A).We notice a good agreement between
the three solutions. Differences between
BIPM TWSTFT and CP(A), and between IGS and CP(A), are shown in Figure8 (TWSTFT
has been interpolated every hour).Figure 8
Difference between the solutions for UTC(OP)-UTC(PTB)—referenced to GPS CP(A).The
standard deviation of the difference between GPS CP(A) and BIPM TWSTFT is 0.23 nanosecond. The standard deviation between the two GPS solutions is lower,
however, we observe bias variations up to 1 nanosecond in the IGS clocks at day
boundaries. Thus, the CP(A) solution noise is close to the IGS noise, while
offering the same long term coherency as TWSTFT. Some errors below one cycle
(0.3 nanosecond) may remain between the two GPS solutions because the current
processing does not take into account possible code/phase biases.


## 3.1. Continental Baseline (OPMT/PTBB,∼700 km)

OPMT is an IGS station located in Paris Observatory; it is driven by a Hydrogen
Maser which is compared to UTC(OP) on an hourly basis (these corrections were
provided by Paris Observatory). The receiver is an Ashtech Z-12T. The station
is calibrated so that we can relate the measurements to UTC(OP).PTBB
is also an IGS station located in PTB (Braunschweig, Germany); it is directly
driven by UTC(PTB). Calibration delays for both stations have been used for the
computation of GPS CP(A). Therefore, a GPS time transfer between these stations
allows to compute UTC(OP)-UTC(PTB). This offset is also computed by BIPM
using TWSTFT and GPS P3 CV (common-view) using Vondrak smoothing. The BIPM data have been retrieved from their
ftp site [13].In
Figure1, we compare the TWSTFT and GPS P3 CV from BIPM data to our GPS CP(A)
time transfer.Figure 1
Time offset UTC(OP)-UTC(PTB) computed by three different
techniques using the single baseline approach.We
observe a 5-nanosecond bias between the TWSTFT and the two GPS solutions. Short-term
variations are very similar to TWSTFT and GPS CP(A), while they are invisible
with GPS P3 CV with Vondrak smoothing.In
Figure2, the difference between BIPM TWSTFT and GPS CP(A) is presented in the
same period.Figure 2
Difference between BIPM TWSTFT and GPS CP(A).The standard deviation of the difference between TWSTFT and GPS CP(A) is 0.25 nanosecond,
with no drift. A constant bias of 5 nanoseconds has been removed from the data.Figure3 shows the stability of UTC(OP)-UTC(PTB) computed with TWSTFT, GPS P3 CV from
BIPM data, and CNES GPS CP(A).Figure 3
Allan deviation of the time offset UTC(OP)-UTC(PTB) for the three different techniques.The
frequency stabilities of the time offset UTC(OP)-UTC(PTB) computed by TWSTFT and GPS CP(A)
agree very well. The apparent good short-term stability of the BIPM GPS P3 CV
is only due to the smoothing and is not deemed to be representative of the
clock difference behavior.


## 3.2. Transatlantic Baseline (USN3/PTBB,∼6300 km)

USN3 is an IGS station located at the US Naval Observatory; it is driven by
UTC(USNO) with no corrections to apply (see USN3 IGS log file [14]).The
goal is to perform the GPS time transfer between the USN3 and PTBB stations to
compute UTC(USNO)-UTC(PTB) and compare results to the BIPM data.
For such a long baseline, a direct ambiguity solution is not possible because
modeling errors are large (in particular troposphere modeling at low elevation).
In addition single difference passes are shorter, and there are few
simultaneous passes. In order to apply our ambiguity resolution method, we
introduce two bridge stations (STJO and REYK). This leads to three short
baselines (~2500 km).The three
baselines are processed independently (but with the same geometrical parameters),
giving estimates for the three clock differences (6 days solution). Table1
summarizes the results.Table 1
N1 ambiguities on the three baselines.
BaselineLength (km)Total N1Integer N1 (%)PTBB-REYK2265231191 (83%)REYK-STJO2597230199 (87%)STJO-USN32179152145 (95%)During the
identification of the integer ambiguities, some passes have been rejected due
to inconsistent overlaps (threshold used 0.25 cycle on the overlap between two
passes).The
clock difference for the baseline PTBB/USN3 is obtained by adding the three
contributions. There are eleven days of data that have been split into two
batches of 6 days with one-day overlap. These results (with and without integer
fixing) are shown in Figure4.Figure 4
PTBB/USN3 clock difference for the
two 6-day batches for integer (blue) and floating (red) solutions (units in cycles).As
observed in [4, 5], floating solutions sometimes exhibit important drifts,
which are not present in the integer resolution. In addition, the integer
clocks differ exactly by one cycle on the central overlapping day. A continuous
clock covering the whole period is then easily obtained by shifting the second solution by one cycle to
align it with the first one. The remaining alignment error is then below one
cycle (about 0.3 nanosecond).In
Figure5, we compare the two BIPM techniques to the GPS CP(A) time transfer.Figure 5
Time offset UTC(USNO)-UTC(PTB) computed by three different
techniques using the single baseline approach.The same conclusions as for Figure1 can be drawn. The standard deviation of the
difference between the TWSTFT and GPS CP(A) on this baseline is 0.3 nanosecond.


## 3.3. Network Solution

The method has been generalized to a network solution, which is detailed in [9, 10].
The global solution approach is similar to the single baseline formulation
extended to zero-difference measurements (identification of ground stations and
GPS satellite clocks). This means that all receiver clocks are computed relative to a reference clock chosen in the network, modulo an integer number of λc wavelengths. The corresponding CP(A) clock differences are
obtained after alignment to code measurements. Figure 6 shows a map of the
stations used in the solution. BRUS was chosen as time reference.Figure 6
European network used in this study.The
station coordinates and zenith troposphere delays are forced to the values
obtained from the absolute positioning that is performed independently for each station as for the single baseline solution.Clocks and ambiguities are adjusted in the integer solution; 98% of N1 integer ambiguities are identified (statistics are computed only for passes with at
least one common view with another station).Figure7 presents the time offset UTC(OP)-UTC(PTB) from BIPM TWSTFT data (blue circles),
from IGS clock products (red dots), and from our network solution. We have
subtracted the same arbitrary value for the two GPS solutions, because the receiver
internal calibration delays were not available.Figure 7
Time offset UTC(OP)-UTC(PTB) from BIPM TWSTFT, IGS clock products, and GPS CP(A).We notice a good agreement between
the three solutions. Differences between
BIPM TWSTFT and CP(A), and between IGS and CP(A), are shown in Figure8 (TWSTFT
has been interpolated every hour).Figure 8
Difference between the solutions for UTC(OP)-UTC(PTB)—referenced to GPS CP(A).The
standard deviation of the difference between GPS CP(A) and BIPM TWSTFT is 0.23 nanosecond. The standard deviation between the two GPS solutions is lower,
however, we observe bias variations up to 1 nanosecond in the IGS clocks at day
boundaries. Thus, the CP(A) solution noise is close to the IGS noise, while
offering the same long term coherency as TWSTFT. Some errors below one cycle
(0.3 nanosecond) may remain between the two GPS solutions because the current
processing does not take into account possible code/phase biases.


## 4. Conclusion

A simple method for the resolution of integer ambiguities on zero- or
single-difference measurements and the computation of the associated clocks has
been presented and  validated.The
main advantages are as follows.(i)
The phase clocks estimated with integer ambiguities are very
stable over the long term: the drifts present in similar floating solutions
disappear completely, and it is easy to handle day-boundary discontinuities.(ii)
The clock solution is self-sufficient; it uses only IGS precise
ephemeris as external modeling input.The
results have been successfully compared to the BIPM TWSTFT on the OP-PTB and
USNO-PTB baselines. Similar results are also shown in a European network of 6 stations.Agreement
with BIPM TWSTFT at the level of 0.3 nanosecond (standard deviation over 11 days) has been achieved even on transatlantic baselines. It would be of
particular interest to repeat such a comparison over a longer time period in
order to confirm this good agreement and to investigate possible seasonal
effects.The
level of precision which has been obtained now allows the observation of receiver
specific delays (“code/phase” biases). Further studies are necessary to characterize
the code behavior and to determine new calibration procedures for the time
receivers.



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*Source: 273785-2008-03-18.xml*