ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-35-97-2017Scintillation measurements at Bahir Dar during the high solar activity phase of solar cycle 24KriegelMartinmartin.kriegel@dlr.deJakowskiNorbertBerdermannJenshttps://orcid.org/0000-0002-3308-4584SatoHiroatsuhttps://orcid.org/0000-0002-5886-2768MershaMogese WassaieGerman Aerospace Center (DLR), Institute of Communications and Navigation, Kalkhorstweg 53, 17235
Neustrelitz, GermanyWashera Geospace and Radar Science
Laboratory, Bahir Dar University, Bahir Dar, EthiopiaMartin Kriegel (martin.kriegel@dlr.de)13January20173519710622July201629November20162December2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/35/97/2017/angeo-35-97-2017.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/35/97/2017/angeo-35-97-2017.pdf
Small-scale ionospheric disturbances may cause severe radio scintillations of
signals transmitted from global navigation satellite systems (GNSSs).
Consequently, small-scale plasma irregularities may heavily degrade the
performance of current GNSSs such as GPS, GLONASS or Galileo. This paper
presents analysis results obtained primarily from two high-rate GNSS receiver
stations designed and operated by the German Aerospace Center (DLR) in
cooperation with Bahir Dar University (BDU) at 11.6∘ N,
37.4∘ E. Both receivers collect raw data sampled at up to 50 Hz, from
which characteristic scintillation parameters such as the S4 index are
deduced.
This paper gives a first overview of the measurement set-up and the observed
scintillation events over Bahir Dar in 2015. Both stations are located close
to one another and aligned in an east–west, direction which allows us to
estimate the zonal drift velocity and spatial dimension of equatorial
ionospheric plasma irregularities. Therefore, the lag times of moving
electron density irregularities and scintillation patterns are derived by
applying cross-correlation analysis to high-rate measurements of the slant
total electron content (sTEC) along radio links between a GPS satellite and
both receivers and to the associated signal power,
respectively. Finally, the drift velocity is derived from the estimated lag
time, taking into account the geometric constellation of both receiving
antennas and the observed GPS satellites.
Ionosphere (ionospheric disturbances; ionospheric irregularities; instruments and techniques)Introduction
It is generally agreed that localized depletions of the low-latitude F region
electron density may be generated due to the Rayleigh–Taylor plasma
instability after sunset . The plasma density inside these
irregularity regions is strongly reduced, thus forming equatorial plasma
bubbles (EPBs) that together with electron density irregularities cause
diffraction and forward-scattering of transionospheric radio signals; this is
known as scintillation .
When the plasma irregularities are fully developed, the irregular structures,
causing scintillation of satellite signals, are expected to flow along the
ambient zonal drift motion. The E×B zonal drift moves
the plasma perpendicularly to the ambient horizontal magnetic field and the
downward-directed electric field that is mainly generated by thermospheric
winds in the low-latitude ionosphere . The drift motions
have diurnal variations in flow direction and magnitude, but in general the
zonal drift flows westward in the daytime and eastward in the nighttime. The
peak eastward drift speed in the low-latitude pre-midnight sector has a
typical magnitude of 100 m s-1, while daytime drifts usually peak at
40 m s-1 in the quiet-time ionosphere . From GPS L1 signals,
derived eastward velocities of ionospheric
irregularities of 50–100 m s-1 at midnight in the South American
sector for magnetically quiet days.
Although the occurrence of eastward drift irregularities is expected in the
low-latitude African sector, their existence has been shown only in a few
cases (e.g. ). Using a single global
navigation satellite system (GNSS) station,
have recently shown statistics of
scintillation events occurring over the Bahir Dar region for GPS frequencies
L1, L2C, and L5, GLONASS L1 and L2, and Galileo E1 and E5, covering the year
2013. These frequencies range between 1176.45 MHz (Galileo E5) and
1610.485 MHz (GLONASS L1). The obtained results are based on high-rate
real-time GNSS measurements recorded at the DLR's msbd01 station (see
Fig. ) in Bahir Dar, Ethiopia. It has been
shown that the L5/E5a frequency is much more strongly affected by
scintillations compared to the L1/E1 frequency. Overall GLONASS L2 shows the
largest sensitivity with respect to scintillation events. Additionally, these
studies also confirm the daily and seasonal variability of scintillation
events in low-latitude regions reported by
and .
In this work we apply a multi-station analysis to study the characteristics
of plasma irregularities, including their zonal drift characteristics.
Therefore, this paper first describes both station set-ups and their
capabilities to detect scintillation events. Afterwards, we give a short
review of simultaneously observed scintillation events in 2015. This includes
the analysis of our scintillation processor output compared to external data
from a nearby GNSS station operated by the Technical University of Berlin
(TUB). Finally, we estimate the zonal drift velocity and spatial dimension of
plasma irregularities over Bahir Dar using a cross-correlation analysis
method, which is applied to high-rate data from DLR's GNSS network.
GNSS station set-up
Since 2010 DLR has established two high-rate GNSS receiver stations in close
cooperation with Bahir Dar University for monitoring and analysing
amplitude scintillations in equatorial regions.
The first station, msbd01, was initially deployed in 2010 at Bahir Dar
University (Peda Campus). It operates a Javad Delta G3T receiver combined
with an external Temex LPFRS (low-profile frequency Rb standard) rubidium clock and a Javad RingAnt-G choke ring
antenna to minimize multipath effects. To intensify scintillation research in
equatorial regions and as a result of the cooperation with Bahir Dar
University the second station, msbd02, was deployed 2014 at the Yibab
Campus (see Fig. ), which is located around
7 km westward of msbd01. The slightly improved hardware and software
set-up operates a JAVAD Delta 3G GNSS receiver in combination with an
external Temex LPFRS rubidium clock and a Leica AR25 choke ring antenna.
After the ISEA14 conference in 2015 also msbd01 was modernized to harmonize
the level of performance and robustness. Both JAVAD receivers are configured
to track GPS (L1, L2, L5), GLONASS (L1, L2), Galileo (E1, E5a) and Beidou as
well as geostationary satellites of space-based augmentation systems like the
Wide Area Augmentation System (WAAS) and the European Geostationary
Navigation Overlay Service (EGNOS) with a sampling rate of 50 Hz. Since
power outages occur frequently in Ethiopia, both stations are plugged to
uninterruptible power supply (UPS) devices in order to bridge power outages
of up to 7 h which is adequate for 95 % of all outages.
To simplify the station maintenance and to enable distributed real time and
post processing of the high-rate GNSS data for scintillation research, both
stations are integrated into the EVnet (Experimentation and Verification Network) architecture of DLR
. Over the past years both systems were able to
continuously record high-rate GNSS raw data, providing an important dataset
for ongoing studies on amplitude scintillations in equatorial regions.
In collaboration with the Technical University of Berlin and the French
company IEEA, two additional high-rate GNSS station were deployed. The TUB
station uses a Septentrio PolaRx GNSS receiver and is located at the Peda
Campus near the msbd01 station . The IEEA station
operates a Novatel GSV4004B GISTM and is located at the Poly Engineering
Campus, roughly 2.6 km north of the Peda Campus. All together these four
stations, as illustrated in Fig. , form an
unique local network to study amplitude scintillations.
Geographic distribution of high-rate GNSS receivers in Bahir Dar,
Ethiopia.
Monitoring of amplitude scintillations
Both DLR stations were designed to monitor total electron content (TEC) and
signal strength scintillations on GNSS signals in near real time. Amplitude
scintillations are rapid irregular variations in the signal amplitude (e.g.
signal fading and scattering) due to the spatially and temporally varying
density of free electrons on a very small scale throughout the different
layers of the ionosphere. To detect whether GNSS signal is affected by
amplitude scintillations, the variation in the signal intensity I is
observed and quantified by the widely used S4 index, which is the standard
deviation of the received signal intensity I divided by its mean value (for
normalization). In general the S4 index is calculated per GNSS link by
using a moving window in the time domain with a window width of 60 s
S4=〈I2〉-〈I〉2〈I〉2,
where 〈〉 denotes the corresponding average value. We derive
S4 measurements according to Eq. () from high-rate GNSS
data streams (50 Hz) in real-time (1 min update rate) and in
post-processing mode, since the operational JAVAD receivers including
JAVAD's software tools do
not provide S4 directly. The signal intensity I can be derived from the
signal amplitude A (I∝A2). The scintillation-processing software
calculates the signal amplitude A according to
A2=i2+q2.
The high-rate inphase (i=A⋅cosΦ) and
quadrature (q=A⋅sinΦ) components are provided by the
GNSS receiver p. 88. The magnitude of S4 is limited
within the range {S4ϵR∣0≤S4≤2} when assuming a Nakagami distribution as a fading model for GNSS
signal fading . In order to rate multiple
S4 measurements, we classify scintillation events by their magnitude as
listed in Table .
Classification of amplitude scintillation events.
Magnitude of S4Classification of scintillation eventS4>0.3moderateS4>0.6strongS4>0.9extreme
The monitoring of S4 derived from the DLR's worldwide
distributed EVnet stations showed that S4 values up to 0.3 are mostly
caused by the natural noise of the signal intensity.
The interpretation of the magnitude of S4 should incorporate the spatial
geometry of the GNSS link, since it is correlated with the elevation angle of
the observed GNSS satellite. If the elevation is very low, the propagation
time through potentially perturbed regions of the ionosphere increases, with
the consequence that the occurrence of scintillation events during
low-elevation scenarios is naturally higher than at high-elevation scenarios.
Also, multipath effects due to very low-elevation scenarios or physical
obstacles may degrade the signal quality and can lead to false assumptions on
ionosphere-generated amplitude scintillations. We did not quantify these
effects but systematically try to exclude them by only considering
observations at elevations greater than 20∘. Based on the
geographical receiver location, there might also be a dependency
on the azimuth of the observed GNSS link. The receivers in Bahir Dar
are located between the northern and southern crest anomaly regions. As
Fig. shows, GNSS measurements with azimuth angles
approximately in the north–south direction and
vice versa are naturally much more affected by amplitude scintillation events
than measurements in azimuthal east–west direction. The strong scintillation
activity around 14–19∘ N obviously fits well with the steep TEC
gradient associated with the northward crest of the equatorial anomaly (e.g.
).
In order to select appropriate scintillation events for the estimation of
equatorial plasma bubble characteristics over Bahir Dar, all available
datasets obtained in 2015 were scanned for identifying and classifying
scintillation events by applying Table . Although there
were many events at each DLR station, only a few events having small data
gaps could be simultaneously recorded at both receivers, as shown in
Table . As is well known (e.g. ;
; ), signal
scintillations could be observed in the evening hours after sunset.
Scintillation events recorded by both DLR stations in 2015.
The gap from May to December is due to the outage of msbd01
before its modernization in October 2015.
S4 estimated from 50 Hz GNSS data recorded between 16:00 and
00:00 UTC (LT -3 h) on 28 February 2015 for msbd02.
To prove the plausibility of the detected scintillation events we compared
DLR data with data collected by TUB in the same time periods at the most
significant events.
Figure shows an example of a typical scintillation
event which was simultaneously recorded independently of the stations
msbd01, msbd02 and tubbd01 (see Fig. ) on
28 February 2015. It shows enhanced signal intensity variations on multiple
GNSS links within the L1 band (1575.42 MHz). Also, the link-based signature
of S4 evolving over time derived by DLR and TUB software reveals a high
similarity as shown for GPS satellite G24 in Fig. .
This proves the fact that both systems are capable of properly detecting
amplitude scintillation events for correlation studies.
Typical amplitude scintillation event after sunset recorded at both
DLR stations and the TUB station on 28 February 2015 over Bahir Dar,
Ethiopia.
Signature (red line, exponentially-weighted moving average, window
size: 3 min) of S4 indices (grey dots) calculated for satellite G24 from
different scintillation processors in comparison to the averaged elevation
(below).
Characterization of plasma irregularities
Both DLR stations, msbd01 and msbd02, are located at different longitudes
but at nearly the same latitude and simultaneously record signals from the
same GPS satellite at a sampling rate of 50 Hz.
Figure illustrates the overall measurement
principle for characterizing plasma irregularities from high-rate GPS
measurements.
After sunset, equatorial plasma irregularities begin to evolve, thus causing
radio scintillations associated with, e.g., enhanced signal power variability
and deep fading effects when the ray path enters the first Fresnel zone
p. 33. Assuming that electron density irregularities
are well established at a height of about hI=350 km, the first
Fresnel (n=1) zone of GPS L1 signals (λ=19.042 cm) in zenith
direction (d1≃hI, d2≃20 850 km) has a radius of
rF≈256 m:
rF=nλd1d2d1+d2,
where d1 is the distance between the ionospheric pierce point (IPP) and
the GPS satellite and d2 is the distance between the GNSS receiver and the
related IPP according to Fig. . Scintillation
and TEC depletion patterns can be observed by monitoring the signal power and
the slant total electron content (sTEC) over time. In
Figs. and , measurements of the satellite G24 for both DLR stations obtained on
28 February 2015 are shown. The depletion signature starts developing when
the irregularity region enters the first Fresnel zone, reaches its maximum
depletion at the highest Fresnel zone coverage and decreases when the bubble
region leaves the first Fresnel zone.
Scheme of GNSS link-based estimation of plasma bubble
characteristics from multiple high-rate GNSS stations in Bahir Dar.
Since the horizontal component of the plasma irregularity drift is a zonal
motion, irregularity effects should be visible at both links with a similar
signature but shifted in time. Analogously to , the
overall irregularity drift velocity, vλ, can be estimated from the irregularity pattern velocity, vλPATTERN, and the scanning
velocity, vλIPP:
vλ=vλPATTERN+vλIPP.vλ can be estimated by determining the time lag of the
irregularity pattern observed on the same links from both stations and its
distance. vλIPP is derived from the dynamically changing
geometry of both observing GNSS links. According to
and Figs. and
, the estimated plasma drift velocity
vλ can be used to derive the zonal dimension dλ of the
irregularity region by multiplying the velocity with the temporal width of
the depletion signature according to
dλ=vλ⋅(te-ts),
with ts as starting time and te as end time of the
depletion signature in seconds.
Observed sTEC depletion moving eastward within a time span of 1 h
and a maximum depletion of 10 TEC units for satellite G24 over Bahir Dar,
Ethiopia.
Signal power, SI, derived from inphase and quadrature
components. SI increases with the decreasing distance to the rising
satellite and decreases due to the growing distance when the satellite sets.
To estimate the drift velocity of the plasma irregularity patterns, the cross-correlation analysis is applied on two different link-based time series in
order to estimate the time lag of the irregularity patterns between both
stations as initially developed by and
. The first series involves the scintillation index S4,
calculated according to Eq. () for a window length of 1 min, but the window is advanced by the native data rate of 50 Hz. The
second series involves the relative sTEC derived from 50 Hz carrier phase
measurements Φn, where fn and λn denote the signal's
frequency and wavelength and K=40.3 m3 s-2:
sTEC=f12f22K(f12-f22)(λ1Φ1-λ2Φ2)-B.
Both datasets need to be preprocessed before applying them to the
cross-correlation analysis. Figure shows a
typical example of the estimated relative sTEC derived from
Eq. () for msbd01 and msbd02. Both time series show a typical
depletion scenario of several TEC units over 1 h with similar sTEC
variation, but an offset, ΔB=Bmsbd01-Bmsbd02, of
both time series can be observed. The offset ΔB is composed of the
sum of differential phase biases and the thermal noise of both receivers and
the observed satellite, integer ambiguities for both signal measurements
including possible cycle slips, and multipath effects in low-elevation
scenarios. To get optimal results for the cross-correlation analysis, ΔB is minimized as follows.
Strong scintillations might lead to a loss of lock and cycle slips. Within the
sTEC time series, the effect of cycle slips can be found as large jumps
between consecutive epochs
p. 186. These jumps are
detected by analysing the temporal variation in the sTEC time series. In
cases of detected cycle slips, the slip-induced offset is determined and the
time series of sTEC are connected by reducing the time series after the slip
by this offset.
After cycle slip correction the impact of unknown phase ambiguities has the
highest contribution to ΔB. This is due to the different lock times
when tracking the satellite. A common practice to approximate the
differential phase ambiguities is to adopt a Hatch filter and apply it to low-noise sTEC derived from phase measurements by introducing additional sTEC
estimations from pseudorange measurements
p. 79. One disadvantage of this
procedure is that the P-code-filtered sTEC measurements are affected by a
much higher noise level than the initial estimations. The noise introduced strongly depends on the enhanced noise level of the pseudorange measurements
and on the length of the smoothing window of the Hatch filter. The
differential P code biases can be estimated by using a model-based TEC
calibration as used for generating TEC maps provided by the DLR service SWACI
in near real time . In the end, the
techniques mentioned to minimize ΔB are intentionally not applied for
this analysis because they will degrade the precision of the initial low-noise sTEC derived from carrier phase measurements. They might also change
the shape of the irregularity pattern and as a consequence sophisticate the
results of the cross-correlation analysis. Since we are predominantly
interested in the estimation of the lag of a pattern occurring in two
different time series, the knowledge of the absolute sTEC value is not
mandatory. To minimize ΔB, both sTEC time series are standardized by
simply subtracting their mean behaviour 〈sTEC〉:
sTEC‾=sTEC-〈sTEC〉.
The mean behaviour 〈sTEC〉 is modelled by using the
exponentially weighted moving average with the smoothing factor α=0.133×10-3 that corresponds to a span size s of 5 min for 50 Hz data
(α=2s+1):
Normalized sTEC and S4 measurements for G24 as input for cross-correlation analysis.
Both time series show high cross-correlations for G24 for a lag time
around 99 s.
〈sTEC〉=(1-α)⋅sTECt-1+α⋅sTECt.
The S4 series of both stations already shows a similar mean behaviour and
variation as plotted in Fig. , since
the S4 index is already a normalized observable. Finally, the lag of the
scintillation pattern, τS4, and the lag of the depletion pattern, τsTEC, can be estimated by
determining the argument of the maximum of the cross-correlation applied to
both time series by replacing S with S4 or sTEC‾:
τS=arg maxtk∑kSmsbd01(tk)Smsbd02(τ-tk)∑kSmsbd012(t)∑kSmsbd022(tk).
Due to the normalization, the resulting vector of the calculated cross-correlation is limited to {cϵR∣-1≤c≤1}. Deriving the drift velocities vλPATTERNS4 and
vλPATTERNsTEC is straightforward when the
distance of both measurement locations, sIPP, is known:
vλPATTERNS4=sIPPτS4vλPATTERNsTEC=sIPPτsTEC.
To reconstruct the link geometry, the satellite coordinates are calculated
according to by using the received broadcast ephemeris data
and the antenna positions of both stations as given in
Table . In this analysis the observation
location is defined to be the location of the IPP by using a single-shell
approximation for the ionosphere at the assumed plasma irregularity height,
hI. The latitude, φIPPi, and longitude, λIPPi, of the IPP from both receivers to the same GNSS
satellite link can be derived for every time step from the GNSS link geometry
by applying the following equations according to :
Ψi=90∘-Ei-arcsinRERE+hIcosEi,φIPPi=arcsin(sinφRCVcosΨi+cosφRCVisinΨicosAi),λIPPi=λRCVi+arcsinsinΨisinAicosφIPPi,
where i denotes the link from the receivers msbd01 and msbd02 to the
same GPS satellite, E and A are the link-based elevation and azimuth
angles, λRCV and φRCV are the receiver
antennas longitude and latitude, and Ψ is the so-called Earth angle. The spherical distances
between both IPPs at the height hI are calculated by (RE=6378.1 km):
sIPP=(RE+hI)⋅arccos(a+b),a=cosφIPP1cosφIPP2cos(λIPP1-λIPP2),b=sinφIPP1sinφIPP2.
The mean zonal speed of both IPPs can be calculated using the
zonal IPP velocity of both stations:
vλIPP=(RE+hI)⋅12(ΔλIPP1ΔtcosφIPP1+ΔλIPP2ΔtcosφIPP2).
Locations of DLR's high-rate GNSS stations in Bahir Dar, Ethiopia.
For the observed GPS satellite G24 the mean distance sIPP according
to Eq. () is about 7 km. For a lag time of
approximately 99 s, the estimated approximative pattern speed is vλPATTERN=71 m s-1. Since G24 was mostly moving from north to
south, a very low mean zonal velocity of both IPPs was observed; vλIPP=10 m s-1. Considering the IPP movement according to
Eq. () and neglecting effects of upward drifts, we have
computed a plasma irregularity velocity of vλ=81 m s-1.
These drift velocities agree quite well with those derived by
. The zonal dimension of the irregularity
region is assumed to be 292 km.
Normalized sTEC and S4 measurements for G29 used as input for the
cross-correlation analysis showing a reversal of the drift direction after
midnight local time (21:00 UTC).
As shown in Fig. , we observed an
interesting behaviour for GPS link G29. Beginning at 23:10 LT (20:10 UTC), we observed a depletion signature moving eastward with a velocity of
approximately 80 m s-1. After midnight the eastward propagation
decreased and finally rapidly changed to a depletion signature moving
westward with a velocity of approximately 102 m s-1.
, and
observed similar characteristics while
analysing GPS observations, geostationary satellite observations at Ancón,
Peru, and optical measurements (630 nm) at Cachoeira Paulista, Brazil. The
authors concluded that these results point to the dominant role of a
disturbance-dynamo-associated westward thermospheric wind. This explanation
might be applicable to our observations when looking at the enhanced Kp index
plotted in Fig. . Under geomagnetic quiet conditions a
reversal of the drift was found to be much smoother with a reversal time
during early morning around 04:00 to 05:00 LT (e.g.
).
The presented results of plasma irregularity characteristics at low latitudes
(Table ) underline previous studies of
, ,
, and
that reported plasma drift velocities around
midnight between 50 and 150 m s-1 and a zonal extension up to
approximately 500 km.
Conclusions
This paper presents a preliminary investigation of scintillation measurements
during the high solar activity phase of solar cycle 24 in the low-latitude
region at Bahir Dar, Ethiopia. To capture the small-scale irregularities over
Bahir Dar's ionosphere, we have implemented the spaced-receiver method by
employing two high-rate GNSS receiver stations (msbd01 and msbd01).
Small-scale ionospheric irregularities are common phenomena in the
low-latitude ionosphere after sunset and could be properly recorded by
high-rate GPS measurements. We observed eastward drifting plasma
irregularities after sunset by monitoring sTEC depletions and enhancements of
radio scintillation activity. By applying the cross-correlation analysis
method we could derive zonal irregularity drift velocities of about
80 m s-1 eastward and under enhanced geomagnetic disturbances even a
westward reversal. This supports results of prior investigations when
analysing data from geostationary satellites and optical instruments. Several
research institutions in this research field such as Boston College, DLR,
IEEA, Kyoto University, TUB and the United States Air Force (USAF), have been
deploying ground receivers/sensors in Bahir Dar, Ethiopia. Thus, coordinated
measurements of these different receivers will enable studying spatial
characteristics and dynamics of the irregularity pattern in more detail.
Future studies will utilize an extended network of high-rate GNSS stations
with its multi-constellation links. Combining the ground-based GNSS
measurements with spaceborne observations such as ESA's Swarm constellation
mission or regional beacon measurements, the increasing scientific
opportunities in Africa can significantly improve our understanding of the
spatial and temporal characteristics of plasma irregularities at low
latitudes.
The night on 1 March belongs to the five
most disturbed days in March 2015 with a Kp Index > 5 .
Plasma irregularity characterization over Bahir Dar, Ethiopia.
DateLTPRNDirectionVelocitySize28 Feb 201523:30G24eastward81 m s-1292 km28 Feb 201523:10G29eastward80 m s-1144 km28 Feb 201524:10G29westward102 m s-1312 km8 Apr 201523:00G21eastward80 m s-158 km8 Apr 201523:00G26eastward84 m s-1151 km8 Apr 201523:30G26eastward78 m s-1187 kmData availability
The high-rate GNSS raw data used in this study cover the Ethiopian region
around Bahir Dar and are stored in an internal DLR archive. Direct open
access to this archive is not possible, since data stored there are used for
other research activities too. However, access to the DLR GNSS raw data used
within this study can be requested by contacting the first author of this
paper.
Acknowledgements
The authors appreciate the support from Bahir Dar University for hosting
and maintaining both GNSS stations within the framework agreement between DLR
and BDU. The authors also thank Roman Galas and Maria Cokrlic from the
Technical University Berlin for providing S4 data used within this
study. The article processing charges for this
open-access publication were covered by a Research
Centre of the Helmholtz
Association.The topical editor, C.
Stolle, thanks Y. Beniguel and two anonymous referees for their help in
evaluating this paper.
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