SUBMILLIMETRIC GNSS DISTANCE DETERMINATION FOR METROLOGICAL
PURPOSES
Baselga S1 , Garcı́a-Asenjo L2 , and Garrigues P3
1
Cartographic Engineering, Geodesy and Photogrammetry Department, Universitat Politècnica de València, Camino de
Vera s/n 46022 Valencia, Spain, Email: [email protected]
2
Cartographic Engineering, Geodesy and Photogrammetry Department, Universitat Politècnica de València, Camino de
Vera s/n 46022 Valencia, Spain, Email: [email protected]
3
Cartographic Engineering, Geodesy and Photogrammetry Department, Universitat Politècnica de València, Camino de
Vera s/n 46022 Valencia, Spain, Email: [email protected]
ABSTRACT
Absolute length measurement in the open air with an uncertainty of few tenths of a millimetre has been the focus of considerable interest in the last few years in many
fields such as metrology, macro-engineering projects or
deformation monitoring at critical sites.
In this contribution we present our approach to investigate the capabilities of GNSS for determining distances
up to 1 km with submillimetric accuracy. Our approach
is specifically tailored to the problem of determining only
the slant distance that substantially differs from the traditional geodetic approach, where ambiguities are usually
determined and the baseline spatial orientation is also required.
Some preliminary results were obtained after an automated processing of different baselines in 1h timespans
along entire years. We found results to be consistent in
the long term below the millimeter level. Comparison
with the absolute SI meter was conducted in the UPV
calibration baseline whose distances are traced to the SI
meter.
Key words: submillimetric length determination; GNSS;
metrology.
1.
INTRODUCTION
Accurate positioning with satellite navigation systems
and international time correlation would not be possible
without metrology, the science of measurement [1]. The
basis of metrology is the definition, realisation and dissemination of units of measurement like the metre, which
is the base unit of length in the International System of
Units (SI). The metre is currently defined as the length of
the path travelled by light in vacuum during a time interval of 1/299792458 of a second [2].
The most accurate length measurements in metres can
reach a relative standard uncertainty u = 10−12 by using
laser interferometers with stabilised frequencies, though
such a high accuracy can only be obtained for short distances, i.e. several metres, under very well controlled laboratory conditions. This type of indoor measurements are
used by the National Metrology Institutes (NMIs) to provide the reference or primary standard for the metre in
their corresponding countries.
Following the core concept of metrological traceability,
those primary standards must be transferred from laboratories to outdoor metrological facilities or calibration
baselines. These calibration baselines, whose lengths
usually range from several hundred metres to several
kilometres, are then used as secondary and working standards for practical purposes [3, 4, 5, 6].
Internationally acknowledged as a length standard in
geodetic metrology, the Nummela Standard Baseline in
Finland (864 m) is the unique calibration baseline that
has proved stable with a relative standard uncertainty near
u = 10−8 for more than 70 years [7, 8]. However, since
their distances are based on the technically demanding
Väisälä interferometry [9], they are only measured every five years and subsequently transferred to other calibration baselines of reference by using sub-millimetric
electronic distance meters (EDM) like the Kern ME5000
[10, 11, 12]. Since the transferring process has to cope
with problems such as the determination of the actual
air refractive index, the instability of the EDM carrier
wave or possible displacement of the baseline’s pillars,
the attainable relative standard uncertainty is diminished
to u = 10−7 or even u = 10−6 for the shortest baselines.
Consequently, absolute length measurement in the open
air with uncertainties of few tenths of a millimetre has
been the focus of considerable metrological interest in
recent years [13, 14, 15]. At present, new measurement techniques are being developed to go beyond the
now out-of-production and scarce Kern ME5000 such
as laser trackers [16], new frequency comb-based EDMs
[17, 18, 19] or GNSS techniques [20, 21].
Previous research conducted at the Department of Cartographic Engineering, Geodesy and Photogrammetry of
the Universitat Politècnica de València (UPV) has shown
the potential of GNSS techniques for measuring distances
over several hundred meters with uncertainties below one
millimetre [14, 22, 23, 24].
The initial assumptions of the intended computing approach are described in section 2. The used functional
model, which differs from the traditional geodetic approach, is given in section 3. Section 4 summarizes the
results obtained in regard with distance reproducibility.
Finally, the accuracy of the determined distances is analysed in section 5 by contrasting them with a length pattern traced to the definition of the SI-metre.
2.
INITIAL ASSUMPTIONS
Our approach to investigate the capabilities of GNSS for
length determination with submillimetric accuracy relies
on four initial assumptions: modern receivers provide
carrier phase observations with submillimetre level noise,
GNSS distances are proved highly stable at global scale,
the module of a GNSS vector may be more precisely determined than their corresponding components, and finally, the increasing number of operational GNSS satellites allows to optimize the selection of satellites as to
improve length determination. Following, the four assumptions are further discussed.
Figure 1. GNSS measurement at the UPV calibration
baseline using a 3D choke-ring antenna equipped with
a signal splitter and two receivers collecting the same
signals.
Regarding the noise of GNSS signal, it used to be assumed as a random error of 1% of the carrier wavelength,
but more recent literature shows that modern GNSS receivers can provide enhanced performance [25, 26]. The
random noise of a GNSS signal depends on the electronics of the receivers, but also on the quality of the signal. Therefore, a different noise should be expected for
each constellation and even for each individual satellite.
Figure 2. Zero-baseline double difference residuals for a
pair of multi-constellation Leica GS10 in July 2015.
For instance, Galileo signal is supposed to be transmitted
with a better signal-noise ratio than those corresponding
to GPS or GLONASS. A standard method to assess the
overall random noise of a GNSS receiver is to perform
a zero-baseline, which consists in processing the double
differences of phase observations of two receivers that are
connected to the same antenna by means of a signal splitter (see Fig. 1).
Fig. 2 shows the overall double difference residuals obtained for a couple of multi-constelation Leica GS10 in
July 2015. The overall value obtained for L1/E1 double
difference residuals was 0.7 mm thus confirming our first
initial assumption that they are currently below 1 mm ×2.
Concerning the GNSS scale stability, it is worldwide
maintained at the level of 1 ppb = 10−9 , which is to say
1 mm over 1000 km. Nonetheless, some open questions
still remain as to whether this high absolute scale accuracy is also applicable to short distance determination in
the existing metrological infrastructures. Moreover, the
GNSS scale stability relies on the accuracy of the precise ephemeris and ultimately on the International Reference Frame (ITRF) which is used for their determination. Since the ITRF scale is determined by a combination of SLR and VLBI observations whose scale differs
by about 10−9 [27], the linkage between the center of
the instruments used by SLR, VLBI and GNSS at Fundamental Geodetic Observatories (FGO), also known as
local-ties, should be performed with submillimetric accuracy [28]. Thus, an additional application of submillimetric length determination with an appropriate metrological
uncertainty evaluation based on traceable instruments and
methods would be to improve the ITRF scale and consequently the accuracy of the precise ephemeris of Galileo
satellites.
The third assumption is that GNSS distance may be better determined than the individual components of the corresponding vector. Double diference processing usually
yields precise vectors whose components ∆X, ∆Y, ∆Z
are subsequently used to compute the distance
D=
p
∆X 2 + ∆Y 2 + ∆Z 2
(1)
Fortunately,
correlation usually makes that σD <
p
2
2
2
σ∆X
+ σ∆Y
+ σ∆Z
(and even σD < σ∆X and σD <
σ∆Y , σD < σ∆Z ) because in the expression
coordinates of both ends with a few cm accuracy, which
can be achieved by means of a PPP static processing, thus
(0)
(0)
(0)
(0)
(0)
(0)
obtaining (Xi , Yi , Zi ) and (Xj , Yj , Zj )
Given the double difference equation
2
σD
=
+
+
∆X 2 2
∆X 2 2
∆X 2 2
) σ∆X + (
) σ∆Y + (
) σ∆Z
D
D
D
∆X ∆Z
∆X ∆Y
σ∆X∆Y + 2
σ∆X∆Z
2
D D
D D
∆Y ∆Z
σ∆Y ∆Z
(2)
2
D D
(
kl
kl
kl
λϕkl
ij = ρij + λNij + ǫij
(3)
for substraction with the order
l
l
k
k
l
l
k
k
ϕkl
ij = (ϕj − ϕi ) − (ϕj − ϕi ) = ϕj − ϕi − ϕj + ϕi (4)
the covariances may be positive or negative depending on
the relative geometry between the baseline and the observed satellites. Thus, a proper selection of observables
may lead to a reduction in the standard error of the obtained GNSS distance.
Finally, a multi-constellation approach has clear advantages for length determination if compared to a singleconstellation one i.e. only GPS. A large number of satellites in view allows to include in the system of normal
equations only those double diferences whose impact on
the baseline is lower than an assumed threshold as well as
to eliminate those satellites whose phase observables contain a strong multipath error. As an illustration, in the last
GNSS campaign that was carried out at the UPV calibration baseline in July 2015, the number of satellites registered in a time spam of eight hours was the following: 21
GPS, 19 GLONASS and 4 Galileo. Thus, the inclusion
of the new and emerging constellations gives rise to new
possibilities for precise GNSS length determination.
and
expanding
(0)
(0)
(0)
(Xi , Yi , Zi )
FUNCTIONAL MODEL
The functional model used in our approach is specifically tailored to length determination and differs from the
corresponding geodetic approach in four features: only
L1/E1 double differences are processed, ambiguity determination is not required, and both ionospheric and tropospheric corrections are not included.
Since L1/E1 carrier phase observables are less noisy than
L2, L5, E5a or E5b observables, or any combination of
them [29], double differences of L1/E1 carrier phase are
the preferred equations to obtain the final solution. Additionly, the power transmission of L1/E1 is usually higher
than the other transmitted signals and consequently those
signals can be collected with lesser noise-signal ratio
even in pre-existing metrological facilities or local-ties at
FGOs, which tend to be not very favourable environments
for GNSS measurements.
A free ambiguity approach reduces the number of unknowns to determine thus strengthening the model [30].
An additional advantage is that the solution is not affected
by cycle slips. The only requirement for using the freeambiguity method is a prior determination of the absolute
(0)
(0)
(0)
(Xj , Yj , Zj )
held fixed, Eq.
with
3 can be rewrit-
ten as
kl
λϕkl
ij − ρij
λ
=
1 ∂ρkl
1 ∂ρkl
ij
ij
dXj +
dYj
λ ∂Xj
λ ∂Yj
+
1
1 ∂ρkl
ij
dZj + Nijkl + ǫkl
λ ∂Zj
λ ij
(5)
which can be subsequently arranged as
kl
kl
λϕkl
λϕkl
ij − ρij
ij − ρij
− int(
) =
λ
λ
+
3.
around
1 ∂ρkl
1 ∂ρkl
ij
ij
dYj +
dZj
λ ∂Yj
λ ∂Zj
+
1 ∂ρkl
ij
dXj +
λ ∂Xj
1 kl
ǫ
λ ij
(6)
where the only unknowns to be solved are dXj , dYj , dZj .
Regarding tropospheric delays, our current model assumes that they are negligible for distances below 1 km.
Note that existing metrological facilities are short and
quite horizontal and consequently the path traveled by a
GNSS signal to reach both ends of the baseline is fairly
the same, specially for those satellites with higher elevation. In addition, we collected GNSS data in the nighttime. Thus, only residual atmospheric delay errors with
minor influence on the distance are expected.
For longer distances, ionospheric delay can be determined using a combination of carrier waves and subsequently use them to correct L1/E1 carrier phase observables. On the contrary, the influence of residual tropospheric delays should be taken into account when both
ends of the measured baseline have a substantial height
difference which can happen in some local-ties or geodetic control networks. Since the tropospheric errors expected in the double difference equations are very small
and also dependant on local meteorological parameters,
global models may not be accurate enough for submillimetric length determination.
4.
STUDY OF REPRODUCIBILITY
Since the scarce studies on the consistency of GNSSderived lengths over short baselines are usually limited
to hours or few days [21], we decided to process GNSS
data for a longer period of time i.e. one year [24].
The processing data were selected from IGS, EPN and
CORS stations as to conform short, approximately horizontal, baselines with clear surroundings and antennas
mounted on pillars or masts with concrete foundations.
Solutions were obtained for every 1 hour of observations (120 epochs) during the year 2011, amounting to
24 × 365 = 8760 baselines per year. The main conclusions can be summarized as follows.
Figure 4. BAY5BAY6 baseline length variation for two 24
h periods: day 139 (dotted line) follows the same pattern
as day 124 (solid line) with a 1 h advance.
A clearly repeated pattern was observed in a Fourier analysis (see Fig. 3) as well as in individual baseline comparisons (see Fig. 4). It has a period of approximately
23 hours 56 minutes, which suggest a possible relationship with the apparent GPS constellation repetition every
sideral day.
Figure 5. CAGL-CAGZ baseline experimental standard
deviations for different timespan data.
5.
STUDY OF ACCURACY
In order to assess their accuracy, the obtained GNSS distances have to be compared with distances accurately
traced to SI-metre.
Figure 3. Single-sided amplitude spectrum of the measured distance obtained for the baseline CAGL-CAGZ.
Fulfilling most of the technical requirements for a metrological facility, the existing calibration baseline at UPV
was chosen for practical reasons.
The obtained results seem to confirm preceding explanations concerning additional periodic effects that can be
expected due to orbit mismodelling [31], multipath playing a major role in short baselines [32] or periodic effects
that may also arise due to errors or approximations in the
antenna/radome calibration [33]. At the case at hand, we
expect this error pattern to be mainly caused by multipath
and possible deficiencies in antenna calibration models.
In 2012, the Finnish Geodetic Institute (FGI) transferred
scale to the UPV calibration baseline. They used an operational Mekometer ME5000 that was calibrated at Nummela before and after the transferring campaign. Therefore, the traceability chain for calibration of the UPV
baseline was: SI definition of the meter, quartz gauge bar,
Väisälä interferometer, FGI Nummela Standard Baseline,
FGI Mekometer ME5000 and UPV calibration baseline.
The resulting nominal distances were obtained with expanded total uncertainties Uk=2 of 0.1 − 0.3 mm. Additionaly, a long term agreement between UPV and the
Universidad Complutense de Madrid (UCM) has enabled us to perform subsequent annual monitoring campaigns (2013, 2014 and 2015) using the only Mekometer
Although 24 hours would be the optimal timespan to
obtain a precise solution, little improvement can be expected after 10 hours (see Fig. 5) once the bulk of the
above-mentioned periodical effects has been removed.
The difference between the computed GNSS distances and their corresponding FGI-certified and UCMMekometer ME5000 were respectively +0.12 mm and
+0.09 mm. Taking into account their respective uncertainty (standard deviation for GNSS), the difference can
be considered statistically negligible. In addition, there is
no evidence of pillar displacements from 2012 to 2013.
The GNSS measurements in 2014 were collected using
two single-constellation Leica GS10 and they were performed in the nighttime whereas the corresponding monitoring campaign using the UCM Mekometer ME5000
was carried out during the same days in the daylight. The
results obtained for the line 1 − 3 are shown in Fig. 8.
Figure 6. Geometric configuration of the UPV 330 m calibration baseline. The line measured line 1-3 is depicted
in blue colour.
ME5000 available in Spain.
Since the UPV calibration baseline was originally
planned in 2007 to evaluate the uncertainty of geodetic instruments and their ancillary equipment according
to ISO 17123 series, it shows the typical not favourable
GNSS measuring conditions inherent to the majority of
the existing outdoor metrological facilities. Accordingly,
only some pillars (1, 2, 3) are suitable for GNSS measurements (see Fig.6). Since the GNSS measurements were
planned to be carried out coincident with each deformation monitoring campaign, which needs at least four days,
only three measurement campaigns have been performed
so far.
The GNSS measurements in 2013 were collected using
four single-constallation Trimble 5700 and two signal
splitters. They were performed in the daytime shortly
before the corresponding monitoring campaign using the
UCM Mekometer ME5000. The results obtained for the
line 1 − 3 are shown in Fig. 7.
Figure 7. Results obtained in the 2013 campaign where
FGI is the SI-traced distance obtained in 2012 (Uk=2 ),
UCM is the distance obtained in 2013 using the UCM
Kern ME5000 (Uk=2 ), and GNSS represents the computed distance using the GNSS data collected in 2013(2σ
level of precision).
Figure 8. Results obtained in the 2014 campaign where
FGI is the SI-traced distance obtained in 2012 (Uk=2 ),
UCM is the distance obtained in 2014 using the UCM
Kern ME5000 (Uk=2 ), and GNSS represents the computed distance using the GNSS data collected in 2014(2σ
level of precision).
In 2014, the difference between the computed GNSS
distances and the corresponding FGI-certified was
−0.59 mm, thus suggesting that the pillars had moved
from 2013 to 2014. When the GNSS distance is compared to the EDM-derived one for the same epoch the
difference decreases up to −0.23 mm which can be considered statistically negligible taking into account their
respective uncertainties (standard deviation for GNSS).
Consequently, a displacement around 0.6 mm between
pillars 1 and 3 might have happened from 2013 to 2014.
The GNSS measurements in 2015 were collected in the
nighttime using two multi-constellation Leica GS10. The
spantime was eight hours and the receivers measured carrier phase observables for the following number of satellites: 21 GPS, 19 GLONASS and 4 Galileo. The corresponding monitoring campaign using the UCM Mekometer ME5000 was carried out during the same days in the
dayligth. The results obtained for the line 1-3 are shown
in Fig. 9.
Since both EDM based distances, i.e. FGI-certified
and the actual obtained by using the UCM Mekometer
ME5000, agreed in 2015 as they did in 2013, the possible
displacement detected in 2014 seems to have been temporary. Once again, the difference between the obtained
provement. Some aspects that deserve more attention are
the use of individual absolute antenna calibrations, optimum selection of measurements and the study of most
favourable geometries for each particular baseline length
determination.
For the sake of completeness, our methodology for submillimetric GNSS length determination should be tested
in other baselines of reference.
ACKNOWLEDGMENTS
Figure 9. Results obtained in the 2015 campaign where
FGI is the SI-traced distance obtained in 2012 (Uk=2 ),
UCM is the distance obtained in 2015 using the UCM
Kern ME5000 (Uk=2 ), and GNSS represents the computed distance using the GNSS data collected in 2015(2σ
level of precision).
GNSS distance and the corresponding SI-traced is satisfactorily small (over 0.1 mm).
Nonetheless, the computed distances should not be considered a significant sample to come to definite conclusions and further research have to be conducted in order
to assess the validity of this degree of agreement and possibly to reduce the standard deviation of the computed
GNSS distances.
6.
CONCLUSIONS
In the light of the obtained results, our approach for
processing GNSS phase observables has proved potentially valid to obtain short distances with sumbmillimetric repeateability. The computed distances have a reproducibility of 0.2 − 0.3 mm after some hours of observations and they are consistent in the long term within this
precision level.
In order to obtain submillimetric results, similar antennas and radomes have to be used in both ends. Optimal results are obtained using high quality antennas, e.g.
3D choke-ring antennas, with absolute individual calibrations.
Accuracy assessment of the resulting GNSS distances is
likely the most challenging problem to study because it
requires very stable outdoor metrological infrastructures
with known distances consistent with the absolute meter
at the level of few tenths of a mm. In addition, the nominal distances of those calibration baselines are valid as
long as their pillars prove stable, thus periodical deformation monitoring is also required.
Although the traceability of GNSS lengths to the SI metre is still an open question that requires further investigation, the size of residuals for real field baseline determination shows that there still may be some room for im-
This research is founded by the Spanish Ministry of Science and Innovation (AYA2011-23232). The authors
are grateful to the UCM which granted the use of its
Mekometer ME5000 through a cooperation agreement,
to José Marı́a Grima for his support to calibrate the UCM
Kern ME5000, thermometers and barometers at the Calibration Laboratory of the UPV, and to the Centro Español
de Metrologı́a (CEM) for acting as an observer of the research project.
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