|Título/s:||Three-cornered hat method via GPS common-view comparisons|
|Autor/es:||Luna, Diego A.; Pérez, Daniel N.; Cifuentes, Alejandro; Gómez, Demián|
|Institución:||Instituto Nacional de Tecnología Industrial. INTI-Física y Metrología. Buenos Aires, AR|
|Palabras clave:||Relojes; Cesio; Energía nuclear; Tiempo; Mediciones acústicas; Frecuencia|
| Ver+/- |
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 8, AUGUST 2017 2143
Three-Cornered Hat Method via GPS
Diego Luna, Daniel Pérez, Alejandro Cifuentes, and Demián Gómez
Abstract— This paper evaluates the stability of three industrial
cesium clocks located in different sites, comparing them via the
common-view technique and using the three-cornered hat (TCH)
method. Validation of the implementation is obtained by compar-
ing results with Coordinated Universal Time (UTC) and Rapid
UTC values reported for the three cesium clocks involved. An
enhanced TCH method is presented and implemented, yielding
corrections up to 30% in the Allan deviation estimates. The main
feature of the developed method is the possibility of computing
the absolute stability of remote clocks at averaging times of
about 2 h.
Index Terms— Atomic clocks, frequency stability, global
positioning system (GPS), noise measurement, time-domain
ATOMIC clocks are a key component in differenttechnological fields. Besides their use in time-keeping
laboratories, several applications like navigation systems,
telecommunications, astronomy, and radar use them as time
and frequency references. Therefore, assessment of high-
quality clocks performance is of great importance. In particu-
lar, phase and frequency instabilities are the most significant
Global navigation satellite systems (GNSSs) are designed
as a satellite-based radio navigation system. In geospatial
applications, the GNSS are used to find the instantaneous
position of the antenna of the receiver. In addition, GNSS
may also be used for timing applications. For example, remote
standards can be compared using the signals broadcasted
by the GNSS. Common view (CV) is one of the available
techniques for this purpose .
The CV method is based on the observation of the signal
transmitted by a source and received at different locations. The
concept of global positioning system (GPS) CV is as follows.
Consider two stations (i and j ), which simultaneously
measure the time difference between their local clocks and
a received time signal tk , from a GPS satellite k. The time
differences between the station i and the satellite k and
Manuscript received February 16, 2017; accepted February 17, 2017.
Date of publication April 3, 2017; date of current version July 12, 2017.
The Associate Editor coordinating the review process was Dr. Dario Petri.
D. Luna and D. Pérez are with the INTI-Fisica y Metrologia Center, Buenos
Aires, Argentina (e-mail: email@example.com).
A. Cifuentes is with the Observatorio Naval de Buenos Aires.
D. Gómez is with the Instituto Geográfico Nacional and The Ohio
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIM.2017.2684918
between the station j and the satellite k are thus expressed
(after several delay corrections) according to
δtki = ti − t
δtkj = t j − tk . (2)
With this in mind, the time difference (ti − t j ) between
clocks can be calculated as
δti j = δtki − δt
j = ti − t j . (3)
As previously stated, (3) is a simplified expression of the
GPS CV principle. In practice, measurements are corrected by
several biases : ionospheric and tropospheric delay, internal
delays of the receiver, and so on.
Allan variance is the most common way to characterize the
frequency stability of an oscillator in the time domain.
When measuring the phase difference between two oscil-
lators x(t), the results are related to the fractional frequency
x(tk + τ )− x(tk)
where the bar denotes the average over the measurement
(sampling) interval τ .
The Allan variance  σ 2y (τ ) is defined as
σ 2y (τ ) =
〈(y¯k+1 − y¯k)2〉 (5)
where 〈. . .〉 denotes the average over a large number of
samples. In terms of phase differences, (5) can be expressed
σ 2y (τ ) =
〈(x(tk + 2τ )− 2x(tk + τ )+ x(tk))2〉. (6)
This original definition of the Allan variance (5) or (6) has
been improved by its overlapping version, which makes a more
efficient use of the data set. This modern version has a lower
dispersion in the results and is the one implemented in this
In practice, frequency stability measurements include noise
contributions from both the device under test and the reference.
In an optimal scenario, the reference noise is low enough to
neglect its contribution in the characterization of the device
under test. Another possibility is the so-called “three-cornered
hat” (TCH) method for computing the individual variances of
the clocks , .
Given a set of three pairs of measurements for three
independent frequency sources 1–3, it is possible to calculate
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2144 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 8, AUGUST 2017
the individual variances as follows: the first step is to take a
time series corresponding to the difference between each of
the three clock pair combination and to form an estimate of the
Allan deviation for each of the three time series. The variance
of each pair will have contributions of both oscillators, so we
can consider them as follows:
σ 212 = σ
1 + σ
σ 213 = σ
1 + σ
σ 223 = σ
2 + σ
where σ1, σ2, and σ3 correspond to the variance of each
individual clock and no correlations between variances are
considered. The original source variances  can be obtained
by solving the system
σ 21 =
σ 212 + σ
13 − σ
σ 22 =
σ 212 + σ
23 − σ
σ 23 =
σ 213 + σ
23 − σ
The phase differences between clock pairs in TCH imple-
mentations are usually carried in single locations, i.e., clocks
are compared in situ. Recently, a remote comparison to
estimate TCH variances has been implemented over fiber
networks . In this paper, CV phase differences are used
as data sources for TCH computation. To the best of our
knowledge, there is no record of the combination of both
This paper evaluates the performance of three cesium clocks
located in different facilities, combining the CV and the
TCH techniques. An enhanced version of TCH method is
implemented to account for the noise added by the remote
This paper is organized as follows. Section II proposes a
new version of TCH, which considers a noise contribution
independent of the clocks. Section III describes and validates
the CV algorithm implemented in this analysis. The stability
of the three clocks under test is discussed in Section IV.
Transfer noise is computed in Section V by implementing the
proposal of Section II to measured data. Finally, conclusions
are presented in Section VI.
II. TRANSFER NOISE MODEL
This section proposes an enhanced version of the TCH
method. This extended version of TCH (eTCH) performs
correction for the noise introduced in CV comparisons.
Equations (7)–(9) model the measured variances in the
absence of correlations and measurement noise. If we consider
noise sources, independent of the clocks, that are present in the
three CV measurements, we can model this new contribution
σ 212 = σ
1 + σ
2 + σ
σ 213 = σ
1 + σ
3 + σ
σ 223 = σ
2 + σ
3 + σ
The intercomparison of a set of three clocks yields the mea-
sured time differences δt12, δt13, and δt23. Ideally, tclosure ≡
δt13−δt12−δt23 = 0 should hold for the measurements, since
in principle, the variances of the individual clocks do not affect
tclosure (the closure involves first differences, so oscillations
in the clocks outputs are compensated in this estimator).
Possible biases in the calibrations of the receivers also cancel
out for the same reason. Generally, measurement noise avoids
these time differences to perfectly compensate. In the partic-
ular case of CV comparisons, each of the time differences
between clocks is in fact an average of the clocks differences
over different sets of satellites, and the dispersion in the values
of δt12, δt13, and δt23 avoids tclosure from being identically
zero. These dispersions are generated mainly by residual
errors from the ionospheric and tropospheric corrections,
which are known to have an uncertainty  of the order
of 2 ns, consistent with the typical standard deviations
obtained in Fig. 3.
As the baselines involved in this work (∼20 km) are much
shorter than the orbits of the GPS satellites , we can
consider that the contribution of atmospheric noise is common
to the three receivers. To account for this noise source, we
define σlink(τ ) as the Allan variance of the residual values
tclosure and use it as a correction in the measurements to
compensate for atmospheric noise.
It is straightforward to show that if we consider σ 2link_12 ≈
σ 2link_13 ≈ σ
link_23 ≈ σ
link, the individual variances are now
σ 21 =
(σ 212 + σ 213 − σ 223 − σ 2link)
σ 22 =
(σ 212 + σ 223 − σ 213 − σ 2link)
σ 23 =
(σ 213 + σ 223 − σ 212 − σ 2link). (12)
A similar approach was tested  in the 1980s and more
recently in . However, closures and closures variances
were used only as stability and accuracy limitations due to
GPS CV measurement noise over long baselines. They were
not implemented as corrections in TCH computations of clocks
III. IMPLEMENTATION OF THE COMMON-VIEW
Phase differences between clocks were measured using
satellites of the GPS in CV. The oscillators are Symmetri-
com/HP 5071A cesium clocks. The three clocks belong to
different institutions in Argentina, all of them separated by
less than 20 km. The institutes are the Instituto Nacional de
Tecnología Industrial (INTI), Observatorio Naval de Buenos
Aires (ONBA), and the Instituto Geográfico Nacional (IGN).
The INTI uses its clock to contribute to Coordinated Univer-
sal Time (UTC), Rapid UTC (UTCr), and the Inter-American
Metrology System (SIM) Time Scale, a multinational time
scale . The ONBA contributes to UTC and is in charge of
the legal time in Argentina . Finally, the IGN contributes
to UTC and UTCr .
LUNA et al.: TCH METHOD VIA GPS CV COMPARISONS 2145
Fig. 1. Time differences between the INTI and the IGN calculated using
UTCr and CV. The full black symbols are the results obtained by the CV
algorithm, and the gray lines indicate the intervals of plus–minus one standard
deviation. The open square symbols correspond to time differences extracted
from the UTCr results. (For the sake of visibility, a linear trend was removed.)
Fig. 2. Number of satellites in CV accumulated over 2.4 h for a 15-day
period. Top: time differences between clocks maintained at the INTI and the
IGN, computed for 2 months.
The GPS measurements of this work have been analyzed
with an implementation of the CV algorithm described in
A 3-MAD filter was applied to raw measurements data for
the removal of outliers. This filtering procedure is performed
with the aim of removing any outliers caused mainly by an
unhealthy satellite. After this procedure, no further filters were
implemented in the algorithm.
The mean values of time differences between the three
laboratories were obtained each 2.4 h (i.e., a maximum of nine
measurements of 16 min). Measurement data were collected
over 2 months and missing data were reconstructed with
interpolated values .
The good agreement between UTCr and CV values is shown
in Fig.1. Here, time differences between the INTI and the IGN
during 15 days are depicted.
Fig. 3. Time differences, standard deviations, and satellites in CV for the
INTI and IGN CV comparisons.
Fig. 2 shows the number of satellites in CV for the three
different pairs of laboratories, during the same 15-day period
as in Fig. 1. On days 56879 and 56886, there were short-
term decreases in the number of CV measurements. In the
first case (indicated by blue arrows in Fig. 2), the reason for
this decrease was an issue in the receiver of the INTI, since
there is no simultaneous anomaly in the IGN–ONBA number
of measurements. On the contrary, during modified Julian date
(MJD) 56886 (indicated by red arrows in Fig. 2), the decrease
appears on the three pairs of comparisons, indicating an issue
external to the laboratories’ references and/or receivers. These
short-term decreases in the number of satellites in CV did not
have any impact on the means or the standard deviations of
the time differences (see Fig. 1).
Long-term analysis of the CV measurements is shown
in Fig. 3. After the removal of the linear trend in the phase
differences (i.e., a frequency offset), the range of the values
is 52 ns. The red segments in Fig. 3 indicate interpolated
values due to the lack of CV measurements. The middle and
bottom panels in Fig. 3 show the standard deviations of phase
differences and the number of satellites in CV, respectively.
Zero values are assigned to standard deviations during periods
in which there were no CV measurements. Most of the time,
the dispersions remained below 5 ns (see the dashed line
in Fig. 3). Between MJDs 56895 and 56905, the number
2146 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 66, NO. 8, AUGUST 2017
Fig. 4. Allan deviations of the three clocks analyzed. The dotted lines indicate
the maximum instability specified by the manufacturer (τ−1/2 dependence for
τ < 5 days). An arbitrary τ−1 regime is plotted as reference.
of CV dropped below 30, causing an increase in the standard
deviations. Nevertheless, no clear anomalies are present in the
time differences (upper panel) during this 10-day period.
IV. STABILITY EVALUATION
Fig. 4 shows the instabilities obtained for the three clocks.
According to TCH estimates, all the clocks show stabili-
ties according to their specifications (see the dotted lines
in Fig. 4). As the ONBA contributes only to UTC, the only
possible analysis for averaging times shorter than 7 days is
the present work. At about τ = 0.5 days, a small rise in
the Allan deviation is obtained. This feature was already
observed in the other CV analysis . The likely source of
this semidiurnal perturbation is the sensitivity of the antenna
and cable with the outside temperature .
The variances obtained with TCH and through UTCr for
the IGN are consistent between 1 and 3 days. Nonphysical
results were obtained (negative Allan variance) for long aver-
aging times in the IGN clock. This feature can appear in
TCH estimates because of several reasons. One of them are
correlations in the outputs of the clocks , . In the
present study, clocks are not placed in the same environment,
so this seems not to be the limitation of the method in this case.
Riley  suggests that negative variances arise when there is
Fig. 5. Transfer noise in TCH via CV measurements: INTI–IGN–ONBA
closure and its Allan deviation. An arbitrary τ−1 regime is plotted as
Fig. 6. Allan deviations obtained through TCH (solid lines) and corrected
under the eTCH scheme (dashed lines).
a clock with significative better stability than the other two.
Dispersions in the values can also be an issue, in particular
for the longer times, where the number of averaged phase
differences is smaller.
Deviations obtained through TCH and UTCr for the INTI
are in full agreement for τ = 1 to 12 days. According to the
SIM Time Network reports, results for τ < 0.5 days slightly
exceed the specification of the clock.
V. ESTIMATION OF TRANSFER NOISE
Following the proposal of Section II, the obtained val-
ues for σlink and tclosure are shown in Fig. 5. Closures are
centered around 0 ns and only two measurements of the 600
exceeded 5 ns.
As expected, noise link levels are always below the
variances of the individual clocks. For averaging times longer
than 0.5 days, fluctuations in the closure behave like a τ−1
process, consistent with white noise phase modulation .
Fig. 6 shows the clock variances corrected under the
eTCH scheme for τ = 0.1 to 1 day. The proposed method
LUNA et al.: TCH METHOD VIA GPS CV COMPARISONS 2147
improves the estimates up to about a 30% for τ = 0.1 days
on IGN’s clock.
Given the steeper slope in comparison with the dependence
of the three clock instabilities (Fig. 4), corrections for link
noise are more significant at shorter averaging times than at
The possibility of making stability estimates combining
the CV and the TCH techniques has been shown. The results
are consistent with the ones obtained through the BIPMs
This new approach allows the computation of absolute
Allan variances with time averages down to 2.4 h, using GNSS
signals. This may be of great importance for laboratories
owning only one clock and need to evaluate their references
at shorter intervals than UTC or UTCr reports.
An extended scheme for the TCH method has been proposed
and tested. This enhanced version allowed the estimation of
the noise added by the link in the comparison of clocks. The
three clocks studied showed instabilities according to their
In the future, analyses over longer baselines should be
carried on, in order to explore the robustness of the method.
Uncertainties in the variances obtained by the eTCH method
must be evaluated, in particular the degrees of freedom remain-
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Diego Luna received the Ph.D. degree in physics from the University of
Buenos Aires, Buenos Aires, Argentina, in 2014.
He joined the Instituto Nacional de Tecnología Industrial, Buenos Aires,
in 2009. His current research interests include the time scales generation, time
transfer, and remote calibrations.
Daniel Pérez joined the Instituto Nacional de Tecnología Industrial, Buenos
Aires, Argentina, in 2003. He has participated actively in the contribution of
Argentina to the SIM time and frequency network.
Alejandro Cifuentes received the Degree in astronomy from the University
of La Plata, La Plata, Argentina.
Since 2006, he has been the Scientific Director with the Buenos Aires
Naval Observatory. His current research interests include the astrometry and
keeping legal time in Argentina.
Demián D. Gómez received the Ph.D. degree in geo-
physics from the University of Memphis, Memphis,
TN, USA, in 2016.
He is currently a Post-Doctoral Researcher with
Ohio State University, Columbus, OH, USA, and is
also affiliated with the International Time Service,
National Geographic Institute of Argentina. He is
involved in GPS reference frame realizations as well
as various geophysical projects.