An analysis of processing settings impacts on estimated
tropospheric gradients is presented. The study is based on the benchmark
data set collected within the COST GNSS4SWEC action with observations from
430 Global Navigation Satellite Systems (GNSS) reference stations in central Europe for May and June 2013.
Tropospheric gradients were estimated in eight different variants of GNSS
data processing using precise point positioning (PPP) with the G-Nut/Tefnut
software. The impacts of the gradient mapping function, elevation cut-off
angle, GNSS constellation, observation elevation-dependent weighting and
real-time versus post-processing mode were assessed by comparing the
variants by each to other and by evaluating them with respect to
tropospheric gradients derived from two numerical weather models (NWMs).
Tropospheric gradients estimated in post-processing GNSS solutions using
final products were in good agreement with NWM outputs. The quality of
high-resolution gradients estimated in (near-)real-time PPP analysis still
remains a challenging task due to the quality of the real-time orbit and clock
corrections. Comparisons of GNSS and NWM gradients suggest the 3∘
elevation angle cut-off and GPS+GLONASS constellation for obtaining
optimal gradient estimates provided precise models for antenna-phase centre
offsets and variations, and tropospheric mapping functions are applied for
low-elevation observations. Finally, systematic errors can affect the
gradient components solely due to the use of different gradient mapping
functions, and still depending on observation elevation-dependent weighting.
A latitudinal tilting of the troposphere in a global scale causes a
systematic difference of up to 0.3 mm in the north-gradient component, while
large local gradients, usually pointing in a direction of increasing
humidity, can cause differences of up to 1.0 mm (or even more in extreme cases)
in any component depending on the actual direction of the gradient. Although
the Bar-Sever gradient mapping function provided slightly better results in
some aspects, it is not possible to give any strong recommendation on the
gradient mapping function selection.
Introduction
When processing data from Global Navigation Satellite Systems (GNSS), a
total signal delay due to the troposphere is modelled by epoch- and
station-wise zenith total delay (ZTD) parameters, and, optimally, together
with tropospheric gradients representing the first-order asymmetry of the
total delay. ZTDs, which are closely related to integrated water vapour
(IWV), are operationally assimilated into numerical weather models (NWMs) and
have been proven to improve precipitation forecasts (Vedel and Huang, 2004;
Guerova et al., 2006; Shoji et al., 2009). Previous studies demonstrated
that the estimation of tropospheric gradients improves GNSS data processing
mainly in terms of receiver position and ZTDs (Chen and Herring, 1997;
Bar-Sever et al., 1998; Rothacher and Beutler, 1998; Iwabuchi et al., 2003;
Meindl et al., 2004). Nowadays, tropospheric gradients are not assimilated
into NWMs; however, they could be assimilated in future (see Zus et al.,
2019) and they are essential for reconstructing slant total delays (STDs).
The STDs represent the signal travel time delay between the satellite and
the station due to neutral atmosphere and they are considered useful in
numerical weather prediction (Järvinen et al., 2007; Kawabata et al.,
2013; Bender et al., 2016) and reconstruction of 3-D water vapour fields using
the GNSS tomography method (Flores et al., 2000; Bender et al., 2011).
Brenot et al. (2013) showed a significant improvement in IWV-interpolated 2-D
fields when tropospheric gradients are taken into account. With the improved
IWV fields, the authors studied small-scale tropospheric features related to
thunderstorms. Douša et al. (2018a) demonstrated the advantage of using
tropospheric gradients in the two-stage troposphere model combining NWM and
GNSS data. Morel et al. (2015) presented a comparison study on zenith delays
and tropospheric gradients from 13 stations at Corsica in the year
2011. Despite good agreement in the ZTD, they found notable discrepancies
in tropospheric gradients when estimated by using two different GNSS
processing software packages, two different gradient mapping functions, and two
different processing methods: (1) double-differenced network solution, and (2) precise point positioning, PPP (Zumberge et al., 1997), solution. Douša et al. (2017) indicated a problem with systematic errors in tropospheric
gradients due to absorbing instrumentation errors. Few attempts were made to
compare the tropospheric gradients with independent estimates, i.e. those
derived from water vapour radiometer (WVR) or NWM data. For a selected number
of stations such a comparison was made in Walpersdorf et al. (2001), where
ZTDs and tropospheric gradients from GPS were compared with those derived
from a high-resolution NWM, ALADIN. A good correlation between GPS and NWM
gradients was found for inland stations but not for coastal ones. More
recently Li et al. (2015) and Lu et al. (2016) showed that with the upcoming
finalization of new systems such as Galileo and BeiDou the improved
observation geometry yields more robust tropospheric gradient estimates. Li
et al. (2015) found an improvement of about 20 %–35 % for the
multi-GNSS processing when compared with NWMs and 21 %–28 %
when compared to WVR. Another multi-GNSS study on tropospheric gradients
(Zhou et al., 2017) used data from a global network of 134 GNSS stations
processed in six different constellation combinations in July 2016. An
impact of the gradients' estimation interval (from 1 to 24 h) and cut-off
elevation angle (between 3 and 20∘) on a repeatability
of receiver coordinates was examined. Better results were found for
solutions where a shorter time interval of tropospheric gradient estimation
was used and where the elevation cut-off angle of 7 or
10∘ was applied. However, strategies were not compared from the
point of view of actually obtained gradient values. Finally, systematic
differences and impacts of a gradient mapping function or observation
elevation weighting on estimated gradients have not been studied yet.
In this work, we systematically evaluate the quality of tropospheric
gradients estimated from a regional GNSS dense network under different
atmospheric conditions. Using a unique data set, we study the impact of
several approaches. ZTDs and tropospheric gradients are then compared with
the ones estimated from two NWMs – ERA5, which is a global atmospheric
reanalysis, and a limited-area short-range forecast utilizing the Weather
Research and Forecasting (WRF) model. Finally, we quantified systematic
differences in tropospheric gradients coming from the gradient mapping
function and the method of observation weighting during a local event with
strong wet gradients.
Data and methodsBenchmark data set
The benchmark campaign was realized within the European COST Action ES1206
GNSS4SWEC to support development and validation of a variety of GNSS
tropospheric products. An area in central Europe covering Germany, the Czech
Republic and part of Poland and Austria was selected as a domain and May
and June 2013 as a suitable time period due to the occurrence of severe weather
events including extensive floods. Data from 430 GNSS stations were
collected together with meteorological observations from various instruments
(synoptic, radiosonde, WVR, meteorological radar, etc.). In addition,
tropospheric parameters from two global and one regional NWMs were
generated. Detailed information about the benchmark campaign can be found in
Douša et al. (2016). Although the presented study is based on the GNSS
data collected within the benchmark campaign, all the presented GNSS and NWM
solutions were newly prepared for this study.
Estimation of tropospheric gradients from GNSS
The STD as a function of the azimuth (a) and elevation (e) angle can be
written as follows:
STD(ae)=mfh(e)⋅ZHD+mfw(e)⋅ZWD+mfg(e)⋅(Gn⋅cos(a)+Ge⋅sin(a)),
where ZHD denotes the zenith hydrostatic delay and ZWD denotes the zenith
wet delay. The elevation angle dependency is given by mapping functions,
which are different for the hydrostatic (mfh), wet (mfw) and gradient (mfg) parts. The
tropospheric horizontal gradient vector is defined in the local horizontal
plane with two components, one for the north–south direction (Gn) and one for
the east–west direction (Ge). The GNSS gradient modelled by Eq. (1) represents
a total gradient (the hydrostatic and wet components are not explicit in
this formulation).
During GNSS data processing, the ZHD is commonly taken from an a priori
model, e.g. Saastamoinen (1972) or Global Pressure and Temperature (GPT,
Boehm et al., 2007) based on climatological data, or it can be derived from
NWM data. The ZWD, or a correction to the modelled ZHD, and tropospheric
gradients are estimated as unknown parameters using a deterministic or
stochastic model.
Current mapping functions for hydrostatic (mfh) and wet (mfw) delay components are
based either on climatological data, e.g. the Global Mapping Function, GMF
(Boehm et al., 2006a), or NWM data, e.g. the Vienna Mapping Function, VMF (Boehm
et al., 2006b). An advantage of the first approach is its independence of
external data. Several mapping functions for tropospheric gradients have
also been developed in the past, e.g. by Bar-Sever et al. (1998), by Chen
and Herring (1997), or the tilting mapping function introduced by Meindl et
al. (2004). The gradient mapping function (mfg) by Bar-Sever (BS) is given as
mfg=mfw⋅cot(e),
and from the formula it is apparent that it depends on the selected mfw. The Chen
and Herring (CH) mfg reads as
mfg=1/(sin(e)⋅tan(e)+c),
where c=0.0032. Since c is related to the scale height, it experiences
spatio-temporal variations. Nevertheless, based on Balidakis et al. (2018), a
variable c does not yield a statistically significant improvement in
describing the atmospheric state over a constant c. Finally, the tilting
mapping function is defined in a generic way as a derivative of the mfw with
respect to the elevation angle:
mfg=-∂(mfw)/∂e.
Figure 1 illustrates the variability of the gradient contribution term
(Gn⋅cos(a)+Ge⋅sin(a)) in Eq. (1) and the size of
the mapping factors represented by actual values of the three mfg. We included
gradient contributions corresponding to all GNSS observations in the
benchmark campaign during a single day (31 May 2013). Obviously, an actual
magnitude of the gradient depends on the mapping factor. While the BS mfg
generates higher mapping factors and thus smaller gradient contribution
terms (scatters in the y axis), the CH mfg provides lower mapping factors and thus
higher gradient contribution terms. The tilting mfg then gives factors in
between BS and CH mfg and results in gradient contributions in between the two.
In the following we focus on BS and CH mfg only, as these can be considered
two extreme cases.
Variability of gradient mapping factors and tropospheric gradient
contributions expressed in azimuths of individual satellites. Three mfg were
studied on 31 May 2013: Chen and Herring mfg (blue), Bar-Sever mfg (red) and
tilting mfg (green).
We use the G-Nut/Tefnut software (Václavovic et al., 2014) for GNSS data
processing of the benchmark campaign. This software utilizes the PPP method
and is capable of multi-GNSS processing in real-time (RT), near-real-time
(NRT) and post-processing (PP) modes with a focus on all the tropospheric
parameters' estimation: ZTDs, tropospheric gradients and slant delays
(Douša et al., 2018b). Stochastic modelling of the troposphere allows an
epoch-wise parameter estimation by extended Kalman filter in RT solutions
(FLT) or its combination with a backward smoother which is used for NRT and
post-processing solutions (FLT+SMT); see Václavovic and Douša (2015).
Table 1 describes all eight variants of solutions for the benchmark campaign
produced using the G-Nut/Tefnut which differ in (a) elevation cut-off angle
(3 or 7∘), (b) gradient mapping function (Chen and
Herring: CH or Bar-Sever: BS), (c) constellations (GPS only: Gx or
GPS+GLONASS: GR) and (d) processing mode (post-processing using the
FLT+SMT processing or simulated real time using the FLT processing only).
Five variants based on the post-processing mode used the backward smoother
and the ESA final orbit and clock products (http://navigation-office.esa.int/GNSS_based_products.html, last access: 14 June 2019). Three variants, abbreviated as RT1GxCH3, RT3GxCH3 and
RTEGxCH3, were used to test the performance of the Kalman filter and RT
orbit and clock corrections using the IGS01 (RT1GxCH3) and IGS03 (RT3GxCH3)
corrections from the IGS Real-Time Service (RTS, http://rts.igs.org, last access: 14 June 2019). The IGS01 RTS product is a GPS-only single-epoch
solution produced using software developed by ESA/ESOC. The IGS03 product is
a GPS+GLONASS solution based on the Kalman filter and the BKG's BNC
software. The last solution, RTEGxCH3, applying the ESA final product is
used to test a benefit of the backward smoothing on the one hand and an
impact of the quality of RT corrections on the other hand. Unfortunately,
the solution based on the processing of GPS+GLONASS data in the simulated
RT mode had to be rejected due to a highly variable quality of RT
corrections in 2013 affecting mainly the GLONASS contribution (and we noted
temporal problems in GPS solutions too; see Fig. 4).
The GPT model was used for calculating a priori ZHDs and the GMF was used
for mapping hydrostatic and wet delays to the zenith. Estimated tropospheric
parameters are thus independent of any meteorological information. GNSS
observations were processed using 30 h data batches when starting 6 h before the midnight of a given day in order to eliminate the PPP
convergence. In all variants, the observation sampling of 300 s was used
with ZTDs and tropospheric gradients estimated for every epoch. The station
coordinates were estimated on a daily basis. The random walk of 6 mm/sqrt(h) was applied for the ZTD and 1.5 mm/sqrt(h) for the
gradients. Absolute IGS model IGS08.ATX was used for the antenna-phase
centre offsets and variations. All variants used the elevation observation
weighting of 1/sin2(e).
Processing parameters of individual variants from the G-Nut/Tefnut
software. Mode FLT denotes a simulated real-time solution using a Kalman
filter only and FLT+SMT a post-processing solution using the Kalman filter
and the backward smoother.
Solution nameElevation cut-offConstellationGradient mapping functionProductsModeGxCH33GPSChen and HerringESA finalFLT+SMTGRCH33GPS+GLONASSChen and HerringESA finalFLT+SMTGRBS33GPS+GLONASSBar-SeverESA finalFLT+SMTGxCH77GPSChen and HerringESA finalFLT+SMTGRCH77GPS+GLONASSChen and HerringESA finalFLT+SMTRT1GxCH33GPSChen and HerringIGS01 RTFLTRT3GxCH33GPSChen and HerringIGS03 RTFLTRTEGxCH33GPSChen and HerringESA finalFLTEstimation of tropospheric gradients from NWM
Tropospheric gradients and zenith delays were derived from the output of two
different numerical weather models: the ERA5 (https://www.ecmwf.int/en/forecasts/datasets/archive-datasets/reanalysis-datasets/era5, last access: 14 June 2019)
and a simulation utilizing the Weather Research and Forecasting (WRF) model
(Skamarock et al., 2008). The ERA5 is a reanalysis produced at the European
Centre for Medium-Range Weather Forecasts (ECMWF). The pressure, temperature
and specific humidity fields are provided with a horizontal resolution of
approximately 31 km (T639 spectral triangular truncation) on 137 vertical
model levels (up to 0.01 hPa) every hour. The WRF simulations are performed
at GFZ Potsdam. The initial and boundary conditions for the limited-area
24 h free forecasts (starting every day at 00:00 UTC) stem from the analysis
of the Global Forecast System (GFS) of the National Centers for
Environmental Prediction (NCEP). The pressure, temperature and specific
humidity fields are available every hour with a horizontal resolution of 10 km on 49 vertical model levels (up to 50 hPa).
The ray-trace algorithm by Zus et al. (2012) is used to compute STDs. The
tropospheric gradients are derived from STDs as follows. At first, 120 STDs
are computed at elevation angles 3, 5, 7, 10, 15, 20, 30,
50, 70, and 90∘ and all azimuths between
0 and 360∘ with an interval of 30∘. Second,
we compute azimuth-independent STDs from the local vertical refractivity
profile. Third, the differences between the azimuth-dependent STDs and the
azimuth-independent STDs are computed. Finally, the gradient components are
determined by a least-square fitting. For details the reader is referred to
the Appendix in Zus et al. (2015).
Using 10 years of ERA5 global data, we tested different observation
elevation-weighting schemes (equal versus the elevation-dependent weighting
of 1/sin2(e)) and two mfg (BS and CH) in the least-square parameter fitting.
While using different observation elevation-weighting schemes led to
negligible differences in the tropospheric gradients, we found a significant
systematic difference in the north-gradient component between tropospheric
gradients derived with BS and CH mfg (see Appendix A). In this regard it is
important to note that NWM-derived tropospheric gradients presented in this
study were computed using CH mfg.
We note that tropospheric gradients can be computed with the closed-form
expression depending on the north–south and east–west horizontal gradients of
refractivity (Davis et al., 1993). We compared the ERA5 tropospheric
gradients derived with our method and the closed-form method with GNSS
tropospheric gradients from the GRCH3 solution. We find that for the
considered stations (over the entire benchmark period), the root-mean-square
deviation between NWM and GNSS tropospheric gradients is 10 % smaller if
we apply our method instead of the closed-form method. This can be explained
by the fact that our method is closer to the method actually applied in the
GNSS analysis (parameter estimation).
We also compared our NWM tropospheric gradients with NWM tropospheric
gradients provided by the TU Vienna (see Appendix B). We found good
agreement between the estimates, in particular between our tropospheric
gradients and the so-called refined horizontal gradients (Landskron and
Boehm, 2018).
Comparison of gradient estimates
Absolute values of tropospheric-gradient components stay typically below 1–2 mm under standard atmospheric conditions and can reach 4–6 mm during severe
weather conditions. The gradient of 1 (6) mm corresponds to about 55 (330) mm slant delay correction when projected to 7∘ elevation angle.
For an illustration, an example time series of tropospheric gradients at
station LDB2 (Brandenburg, Germany) for a period between 15 May and 15 June 2013 is given in Fig. 2.
Tropospheric gradients retrieved from GNSS data processing (GRCH3,
RT1GxCH3) and from NWM ERA5 at station LDB2 (52.209∘ N,
14.121∘ E, Germany) for a period from 15 May till 15 June 2013.
In the presented study, ZTDs and tropospheric gradients from all eight GNSS
variants were compared to each other and also to the tropospheric parameters
from ERA5 and WRF to evaluate the impact of various settings in GNSS data
processing. Although about 430 GNSS stations are available in the benchmark
data set, statistical results given in Sect. 3 are based on a subset
of 243 stations. Firstly, 84 stations without the capability of receiving
GLONASS signals were excluded. Secondly, stations which did not have at
least 5 % of all the observations in the range of elevation angles
between 3 and 7∘ were excluded as well. This rule was
applied to allow a systematic evaluation of elevation cut-off angle impact
on tropospheric parameters. The majority of the stations (103) had to be
excluded because of an inability to provide a sufficient number of observations
at very low-elevation angles.
Statistics presented in Tables 3, 4 and 5 were computed directly from ZTD and
tropospheric gradient differences from 243 GNSS stations over 55 d with
288 estimates per day, i.e. a total of ∼3.4 million
differences. During their computation a standard data screening was
applied to exclude outlier values identified as differences exceeding a
given threshold value. Moreover, epochs were RT GNSS variant of solution
RT3GxCH3 provided unrealistic tropospheric gradients (see Sect. 3.3) were
also excluded from all the statistics computations for all compared GNSS
(NWM) solutions except from the coordinates' repeatability evaluation.
Identification of these unrealistic epochs was realized by a visual
inspection of gradient maps (see Sect. 3.3). Actual numbers of differences
used for computation of presented statistics for the two compared GNSS
(NWM) solutions are provided in Tables 4 and 5.
Tropospheric parameters from the G-Nut/Tefnut software were provided every 5 min, while the output from both NWM models was available every hour.
Therefore, comparisons between GNSS solutions (Sect. 3.2) are based on a
5 min interval, while comparisons between GNSS and NWM solutions (Sect. 3.3) are based on a 1 h interval.
Impact of applied processing settings on GNSS tropospheric gradient
estimation
The section starts with an introductory evaluation of mean tropospheric
gradients and formal errors of their estimates. This is followed by
comparisons between individual GNSS solutions and comparisons between GNSS
and NWM solutions.
Comparison of mean tropospheric gradients and formal errors of their estimates
Mean gradient magnitudes and azimuth angles (direction of gradient) over the
whole benchmark period were computed for 243 GNSS stations and are presented
in Table 2. Mean magnitudes of tropospheric gradients from all
post-processing GNSS variants oscillated around 0.85 and 0.67 mm when
using the CH mfg and the BS mfg, respectively. The latter shows about 17 %
smaller gradients compared to the former if all the processing aspects
remained identical. Both RT solutions also resulted with higher gradient
magnitudes, namely +14 % for RT1GxCH3 and +42 % for RT3GxCH3 when
compared to the corresponding GxCH3 post-processing variant. A mean gradient
magnitude of about 0.7 mm was found for both NWM solutions, i.e. of about
0.1 mm smaller than for the GRCH3 solution. This can be mainly explained by
the limited horizontal resolution of the NWMs.
Mean magnitudes and azimuth angles of tropospheric gradients from
all individual GNSS variants of processing and NWMs ERA5 and WRF.
SolutionMeanMeanPercentage ofPercentage ofNumber ofmagnitudeazimuthstations with meanstations with meanoutlier(mm)(∘)azimuth =azimuth =stationstotal_mean ±15∘total_mean ±30∘GRCH30.82170.089.799.22GRBS30.67170.292.698.83GxCH30.83168.288.597.56GxCH70.86168.073.795.511GRCH70.84170.279.097.17RT1GxCH30.95151.992.698.75RT3GxCH31.18162.796.398.83RTEGxCH30.75168.385.697.56ERA50.68169.396.3100.00WRF0.73170.9100.0100.00
Table 2 shows that mean tropospheric gradients point towards the Equator,
which is in agreement with Meindl et al. (2004). Such a mean gradient
direction does not depend on the gradient mapping function. By adding
GLONASS observations the mean gradient direction was changed by
+2∘; however, actual effects were found to be highly
station-dependent with a typical range of ±5∘ for
individual stations. The direction of the mean gradient in both NWM solutions
was in very good agreement with all GNSS post-processing variants.
Directions of the mean gradient over individual stations were mostly within
±15∘ when compared to the total mean gradient estimated for
the stations and the solution variant. On the other hand, the performance
was not identical for the individual solutions. A change in cut-off
elevation angle from 7 to 3∘ led to an increased number
of stations with the mean gradient direction within ±15∘ of
the total mean direction and to a decreased number of stations with a mean
gradient direction differing for more than 30∘ (regarded as
outlier stations in Table 2). Two GNSS stations were marked as outliers by
all processed variants, with their mean gradient direction differing by more
than 50∘ from the total variant mean. Both of them are located in
an urban area in south-western Germany and use the same receiver and
antenna type from Leica, which is however used by many other stations in the
same region where no issues with gradient mean angle were identified. Still,
the reason for their different behaviour can be of instrumental or
environmental origin.
Mean position repeatability and formal errors and their standard
deviation for tropospheric parameters from individual GNSS processing
variants.
Table 3 summarizes mean repeatability of daily coordinates as well as
statistical comparison of formal errors of estimated ZTDs and tropospheric
gradients from different GNSS-processing variants. The station coordinates'
repeatability is improved when using combined GPS+GLONASS solutions
compared to GPS-only solutions, namely by factors of 2 and 1.2 in
horizontal components and the height, respectively. The number of available
satellites and their geometry play a significant role in this context. An
increase in the elevation angle cut-off (from 3 to 7∘)
resulted in improved height repeatability, which is consistent with the
results of Zhou et al. (2017) suggesting an optimal 7∘ cut-off for
the height repeatability when comparing results of a different elevation angle
cut-off (3–15∘). However, it should be noted that
GPT+GMF models and the PPP method were used in both cases. In contrast,
Douša et al. (2017) observed an improvement in the height repeatability
even when using the elevation angle cut-off 3∘ (compared to
7 and 10∘) when exploiting double-difference
observations, the VMF1 mapping function (Boehm et al., 2006b) and the
Bernese GNSS software (Dach et al., 2015). Douša et al. (2017) also indicated
worse results when using GPT+GMF compared to VMF1, which can be
attributed to modelling errors in the former, particularly when applied in PPP
(Kouba, 2009). We also notice a slightly better performance in the case of the
BS mfg when compared to the CH mfg, while this difference was found to be
statistically significant in the north and up component by the Wilcoxon
signed-rank test at the 5 % significance level. The results of the forward
filter processing did not show any degradation when using the ESA final
products (RTEGxCH3). When using the IGS real-time product, the repeatability
of all coordinates became worse by factors of 2–3 and 4–5 for the RT1GxCH3 and
RT3GxCH3 variants, respectively. The latter is attributed to a lower quality
of the IGS03 RT product during some periods; see Fig. 4.
Formal error of the parameter can be generally regarded as an estimation
uncertainty. Formal errors increase when the number of observations and/or
the geometry decrease. This can be observed in Table 3 when the elevation
cut-off is increased. Formal errors are about 17 % and 11 % smaller when
using the 3∘ cut-off (GRCH3) compared to the 7∘ cut-off
(GRCH7) for horizontal gradients and ZTDs, respectively, thus indicating a
higher impact on the former. A decrease in formal errors of tropospheric
gradients estimated with a 3∘ cut-off compared to a 10∘
cut-off was previously reported also by Meindl et al. (2004). Interestingly,
using the BS mfg resulted in smaller formal errors of tropospheric gradients,
but we have not observed any change in formal errors of other estimated
parameters. The smaller formal errors may suggest an improvement in
estimated parameters using BS mfg, as also found from the coordinates'
repeatability.
Comparison of individual GNSS variants with each other
Results for individual GNSS variant comparison are presented in Table 4. We
notice good agreement among all the post-processing variants (top part of
Table 4). The mean differences stayed below 0.2 mm for ZTD and ±0.02 mm for tropospheric gradients with one exception for the latter parameter.
This was a comparison between results provided by CH and BS mfg where the mean
differences reached -0.05 and 0.03 mm for the north- and east-gradient
components, respectively. These small systematic effects can be attributed to
the average difference between tropospheric gradients computed with BS mfg
compared to CH mfg. The standard deviation (SD) indicates the smallest impact
due to the change in mfg for both ZTD estimates (0.2 mm) and tropospheric
gradients (∼0.14 mm). The impact increases then for both ZTD
and gradients when comparing results of single and dual constellations (1.2 mm for ZTD, ∼0.17 mm for gradients). It should be noted that
GLONASS observations were down-weighted by a factor of 1.5 in
dual-constellation variants of the solution to reflect both a lower quality of
precise products and observations. The gradients estimated with improved
geometry and using more observations are expected to be more accurate and
reliable. It is notable in the comparisons of single or dual constellations at
different elevation cut-off angles (the impact is larger for a higher
cut-off). The largest impact is eventually observed due to the elevation
cut-off angle, i.e. 2.2 and ∼0.20 mm for ZTD and
tropospheric gradients, respectively. Linear correlation coefficients
(CorCoef) reach a value of ∼1 in all cases for the ZTD
comparisons. The ZTDs were thus practically unaffected by different gradient
models. For the gradient comparisons, the correlation coefficients are
progressively decreasing from 0.99 to 0.95, while values of SD are
increasing.
An increased scatter of RT processing is visible in significant mean
differences and in the standard deviation values of ZTD and tropospheric
gradients increased by a factor of 3. These are also emphasized by the
reduction of correlation coefficients mainly for tropospheric gradients. The
two RT solutions can still be considered of good quality if we take into
consideration results found in Ahmed et al. (2016) or Kačmařík (2018), where mean biases and SD values of up to 12 mm were reported for
comparisons between RT ZTD solutions based on IGS01 and IGS03 streams and
post-processing solutions based on final products. Since virtually zero mean
differences for both ZTD and tropospheric gradients are found in the
RTEGxCH3 variant, when using the Kalman filter too, the degraded quality of
RT tropospheric parameters is mainly a consequence of the poorer quality of
the IGS01 and IGS03 RT products (Douša et al., 2018b).
The differences of ZTDs and tropospheric gradients from all compared
variants of solutions were also statistically tested. And in all cases, the
differences were found to be statistically significant at the 5 %
significance level while using the Wilcoxon signed-rank test
(https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wilcoxon.html, last access: 14 June 2019).
This non-parametric test was used since none of the processed variants of
solutions evinced a normal distribution of their ZTDs and tropospheric
gradients.
Comparison of individual variants of the GNSS data processing run in
post-processing mode (top) and in simulated real-time mode (bottom). Units:
mean and SD in millimetres; CorCoef represents a linear correlation coefficient.
Compared post-processing solutionsZTD N-gradient E-gradient Number of pairsMeanSDCorCoefMeanSDCorCoefMeanSDCorCoefGRCH3 – GRBS30.00.21.000-0.050.140.9950.030.130.9963 439 426GRCH3 – GxCH30.11.11.0000.000.160.973-0.020.150.9763 438 678GRCH7 – GxCH70.11.21.000-0.010.190.963-0.020.170.9683 438 453GRCH3 – GRCH70.12.11.0000.010.200.9610.000.180.9663 439 042GxCH3 – GxCH70.22.21.0000.010.230.949-0.010.200.9573 438 617Compared RT solutionsZTD N–S gradient E–W gradient Number of pairsMeanSDCorCoefMeanSDCorCoefMeanSDCorCoefRT1GxCH3 – GxCH33.45.70.996-0.100.540.7160.180.550.6693 414 572RT3GxCH3 – GxCH32.76.20.996-0.050.660.6990.090.680.6513 355 457RTEGxCH3 – GxCH30.14.40.998-0.000.390.833-0.010.430.7763 428 621RT1GxCH3 – RT3GxCH30.85.00.997-0.030.650.7180.090.630.7123 366 450Comparison of individual GNSS variants with NWM
The statistics for the GNSS and NWM comparisons are summarized in Table 5.
For ZTDs a mean difference of about 1 (4) mm is visible between GNSS and
ERA5 with standard deviations around 9 (10) mm and correlation coefficients
around 0.99 (0.99) for individual post-processing (RT) GNSS solutions. The
negative mean difference of -3 mm in ZTD between GNSS and WRF might be due
to the global NCEP GFS analysis which is used for the initial and boundary
conditions for the WRF solution. A negative mean difference of -5 mm in ZTD
between two GNSS reference solutions and a solution based on the NCEP GFS
was already reported in the past (Douša et al., 2016). The standard
deviations of differences are about 2 mm larger when GNSS and WRF are
compared. This is probably due to the fact that the solution from WRF is
based on a 24 h forecast (errors are supposed to grow with increasing
forecast length), whereas the solution from ERA5 is based on a reanalysis.
Comparison of individual variants of the GNSS data processing run in
post-processing mode (top) and in simulated real-time mode (bottom) with NWM
solutions. Units: mean and SD in millimetres; CorCoef represents a linear correlation
coefficient.
With regards to the tropospheric gradients, the mean differences between
post-processed GNSS and NWM stayed within a range from -0.05 to 0.03 mm. The
existing differences between two GNSS variants of solutions based on
different mfg can be attributed to usage of CH mfg for derivation of NWM
tropospheric gradients and to the existing systematic difference between
tropospheric gradients estimated using these two mfg (see Sect. 2.2). The
standard deviations between GNSS and NWM were approximately doubled or
tripled when compared to standard deviations between individual variants of
GNSS solutions (Table 4). They were also found to be higher for the WRF than
for ERA5. Again, this can be probably explained by the fact that the
solution from WRF is based on a 24 h free forecast, whereas ERA5 is based
on a reanalysis.
Tropospheric gradient maps from the GNSS GRCH3 solution (a, d), NWM
ERA5 solution (b, e) and NWM WRF solution (c, f) on 31 May 2013, 18:00 UTC (a, b, c) and on 3 June 2013 00:00 UTC (d, e, f).
Tropospheric gradient maps from the GNSS GxCH3 solution (a, d), GNSS
RT1GxCH3 solution (b, e) and GNSS RT3GxCH3 solution (c, f) on 31 May 2013, 18:00 UTC (a, b, c) and on 6 May 2013, 18:00 UTC (d, e, f).
Both NWMs lead to consistent results: standard deviations are smaller and
correlation coefficients higher for GNSS solutions using a lower cut-off
elevation angle (3∘ instead of 7∘) and/or more
observations (GPS+GLONASS). For example, the SD for the north-gradient
component between GNSS and ERA5 is 0.54 mm for the GxCH7 variant and 0.46 mm for the GRCH3 variant. This represents a decrease of 15 %. In this
regard we also derived tropospheric parameters from both NWMs using a
7∘ cut-off elevation angle and repeated the comparisons to test whether
GNSS variants of a solution with a 7∘ cut-off would be closer to NWM
solutions based also on the 7∘ cut-off angle. And we always found
better agreement between any evaluated GNSS variant of the solution and the
NWM solution based on the 3∘ cut-off angle – in terms of mean
difference, standard deviation and correlation coefficient. From two GNSS
variants differing only in the mfg, the solution applying the BS mapping
function is closer to the NWMs in terms of standard deviation. Since the CH
mfg was used to derive tropospheric gradients from NWMs, the opposite situation
could be expected, and we generally note that presented results of
comparisons between tropospheric gradients from the GNSS GRBS3 solution and
NWMs should be taken only as informative. The lower values of standard
deviation can be partly understood as the magnitudes computed as Gn2+Ge2 of GNSS tropospheric gradients using the BS mfg are
smaller compared to the CH mfg (see Sect. 2.2), and the magnitudes of NWM
tropospheric gradients are more smoothed compared to the GNSS tropospheric
gradients.
In order to evaluate the statistical significance of differences of ZTDs and
tropospheric gradients from all variants of GNSS solution and both NWMs, we
again applied the Wilcoxon signed-rank test. Again, the differences were
found to be statistically significant at the 5 % significance level in all
cases.
Maps showing tropospheric gradients were generated for all the variants of
GNSS solutions and both NWM solutions and were visually evaluated for the whole
benchmark period. For better visualization we included all the GNSS stations
of the benchmark campaign, i.e. not just the subset of 243 stations used for
the presented statistics. Generally, GNSS provided homogenous fields of
tropospheric gradients without noisy behaviour at the level of individual
stations, and very good agreement in gradient directions and usually also
in gradient magnitudes was found between GNSS and NWM gradient maps. In
Fig. 3, two examples are shown for different events when weather fronts
were passing over the studied area. For a description of meteorological
conditions prevailing during these events the reader is referred to
Douša et al. (2016). Tropospheric gradients derived from NWM provided
more smoothed gradient fields, but were somehow limited to render local
structures mainly due to the spatial resolutions of both NWMs. As the ERA5
model has coarser spatial resolution than the WRF model, such behaviour was
a little bit more apparent in its results. On the other hand, when compared
to results of the 1∘× 1∘ resolution global
models ERA-Interim and NCEP GFS (Douša et al., 2016), the presented NWM
tropospheric gradients have larger magnitudes.
Comparing GNSS to NWM products in Table 5 indicated that the RTEGxCH3
solution driven by the Kalman filter and the ESA final product shows a
comparable performance to the GxCH3 solution driven by the Kalman filter and
the backward smoother. An increase in mean difference and standard deviation
values for other solutions based on RT mode indicates that the quality of
the RT tropospheric solution is dominated by an actual quality of RT orbit
and clock corrections. In this regard, we examined systematically all
tropospheric gradient maps and found that gradients from the RTEGxCH3
solution are always in very good agreement with post-processing solutions.
Although there were imperfections in matching RT1GxCH3 gradients and
post-processing solutions, the performance can still be considered
generally good and stable. This was however not the case of the RT3GxCH3
solution, where we observed a varying quality of estimated tropospheric
gradients. For the majority of epochs, in particular during the periods with
strong gradients, the tropospheric gradients could be evaluated as
acceptable. However, situations when gradients from all the stations point
to the same direction occurred from time to time, obviously without a
physical relation to the actual weather situation. An example of this
behaviour is presented in Fig. 4, where tropospheric gradients from the
RT3GxCH3 solution behave normally on 31 May 2013, 18:00 UTC, and became
unrealistic on 6 May 2013, 18:00 UTC, where all the stations point to the
south-westerly direction and reveal high-gradient magnitudes. Such issues
occurred occasionally for a limited period of time in the RT3GxCH3 solution
only. The reason is an instability of the RT3 stream during the initial
period (the first half of 2013) affected by many interruptions, and data gaps
thus caused frequent parameter re-initialization in PPP.
Impact of different gradient mapping functions and elevation-dependent
weighting
Impacts of mapping functions on estimated ZHD (ZWD) and gradient parameters
are different, though both represent something of an elevation-dependent
parameter scaling. The latter is more sensitive to the mapping function
compared to the former, additionally considering their relative magnitudes.
The gradient mapping function is strongly driven by the cot(e) approximation (Eq. 2), which is growing quickly for low-elevation angles. Because gradients
represent the second-order effect of the tropospheric delay, quickly growing with
the distance from the station, they are practically estimated using
low-elevation observations and, consequently, the impact of mfg becomes
significant.
Post-fit-phase residuals' distribution when using different
gradient mapping functions, Bar-Sever (red) and Chen and Herring (blue), and
observation weighting: SINEL2 (a) and EQUAL (b).
In this section, we focus on studying systematic differences induced purely
by different mfg and observation elevation-dependent weighting (OEW) during
8 days from 25 May to 1 June 2013. For two solutions defined in Sect. 2.2 and utilizing CH mfg (GRCH3) and BS mfg (GRBS3), we additionally generated
four variants using various OEW schemes: (1) EQUAL, equal weighting, (2) SINEL1, 1/sin(e) , (3) SINEL2, 1/sin2(e), and (4) SINEL4,
1/sin4(e). Generally, in the SINEL OEW schemes, the contribution of
low-elevation observations to all estimated parameters decreases with
increasing power y in 1/siny(e).
Figure 5 displays example distributions of carrier-phase post-fit residuals
with respect to the elevation for the SINEL2 observation weighting (panel a), and without any weighting, i.e. EQUAL (panel b). While the
residuals from the former are affected by the mfg only below 15∘
elevation, the residuals in the latter are affected at any elevation angles
even close to the zenith direction. Above a 30∘ elevation angle, the
distribution of residuals is smoother for the SINEL2 compared to the EQUAL
and more stable according to our experience with many other stations. This
is particularly visible when comparing the distribution of residuals at the
lowest and highest elevation angles between variants, though both
generally follow the expected behaviour when considering errors in GNSS
observations and models. These errors include contributions from the
atmosphere, multipath, uncertainty of receiver antenna-phase centre
variations, a lower signal-to-noise ratio, and cycle slips, all usually increasing
with the decrease in the observation elevation angle and with the smallest
errors in the zenith direction. Using a weak or no elevation-dependent
weighting, the hydrostatic/wet delay mapping separation errors can introduce
significant errors in both ZTD and the height coordinate component (Kouba,
2009). Though we generally recommend the use of SINEL2 elevation weighting,
we also show below the impact of other weighting schemes on estimated gradients.
Tropospheric gradient maps on 31 May 2013 (18:00 UTC) from GNSS
solutions using the SINEL2 observation weighting scheme: Chen and Herring
mfg (a); Bar-Sever mfg (b).
Mean differences (calculated over the full day of 31 May 2013) of
the tropospheric north-gradient component (a) and east-gradient component (b) due to different mfg: Chen and Herring (CH) and Bar-Sever (BS) when
using the SINEL2 observation weighting schemes.
Figure 6 displays maps of situations with large tropospheric gradients
observed on 31 May 2013 at 18:00 UTC when using GRCH3 (panel a) and
GRBS3 (panel b) solutions and applying the SINEL2 OEW scheme. The day
is interesting due to the presence of an occlusion front over Germany clearly
captured by strong tropospheric gradients achievable from both GNSS and NWM
analyses. Such events with significant gradients captured in a dense network
can help to evaluate differences between mfg and other processing parameters,
while they could easily remain hidden in most of the other cases. The impact of
mfg
on estimated gradients shows systematic changes in gradient magnitudes –
the gradients estimated with CH mfg (panel a) are always larger than with BS
mfg (panel b), independent of the OEW scheme (not shown). It should be
also noticed here that the magnitudes of gradients estimated using the
SINEL4 scheme were significantly reduced compared to any other OEW scheme.
Figure 7 shows mean differences, calculated over all epochs on 31 May 2013,
in the (panel a) north- and (panel b) east-gradient components between
the two mfg (BS minus CH) when using the SINEL2 scheme. Although the magnitudes
of CH gradients are always larger compared to BS gradients, the sign of the
component differences depends on the gradient direction (north–south for
Gn and east–west for Ge). Positive differences in the north- and east-gradient components
appear when the estimated gradients point south and west, respectively,
and negative differences occur when the gradients point in opposite
directions.
Differences in tropospheric gradients between Chen and Herring and
Bar-Sever mfg for four observation weighting schemes: EQUAL (EQ), SINEL (S1),
SINEL2 (S2), and SINEL4 (S4).
Figure 8 shows histograms of tropospheric gradient differences of all the
stations in the network when using different mfg and OEW schemes on 31 May 2013. Obviously, the impact of the mfg on estimated gradients is significantly
reduced for SINEL4 (well below 0.2 mm), while it is higher for all the other
schemes. This corresponds to the fact that large gradients are related to a
horizontal anisotropy of the troposphere affecting more significantly
low-elevation observations. The strongest effect can be observed for the
EQUAL scheme, reaching systematic differences of 1.0 mm or even higher. Such
systematic differences reached 2-fold values of the SD obtained from
comparisons of gradients using independent sources such as GNSS and NWM; see
Sect. 3.3 or Douša et al. (2017).
Eastern tropospheric horizontal gradients (a) estimated
using Chen and Herring (light columns) and Bar-Sever (dark columns) mfg and the
differences (b) in gradient magnitudes between them. The SINEL2 OEW
scheme was applied over 8 days in May/June 2013.
Figure 9 compares magnitudes of estimated gradients (east component only)
and corresponding histograms of total gradient differences over all stations
in the network on 8 consecutive days (25 May–1 June 2013) when using
CH and BS mfg and the SINEL2 OEW scheme. We can notice the days with a stronger
tropospheric anisotropy (27–28 May, 31 May, 1 June), identifiable by the
presence of gradients larger than 1.0 mm. The histograms systematically
deviate from the zero on some days; prevailing negative and positive east
components indicate that gradients in the network point westwards and
eastwards, respectively. Differences in gradient magnitudes are then shown
in panel b. The impact due to utilizing different mfg clearly
corresponds to the original gradient magnitudes. Both are high during the
days with a strong tropospheric anisotropy, while differences due to the
mfg choice demonstrate systematic effects of up to 1 mm or more in such extreme
cases.
Conclusions
We presented an impact assessment of selected GNSS processing settings on
estimated tropospheric gradients together with an evaluation of differences
resulting from the gradient mapping function and observation elevation
weighting. We exploited the GNSS4SWEC benchmark campaign covering May and
June in 2013 with prevailing wet weather. Although the time period covered
some severe weather events, it also contained a lot of days with standard
weather conditions with tropospheric gradients close to zero. Presented
results could therefore be considered representative of European conditions
during the warmer part of the year.
ZTD values and tropospheric gradients were estimated in eight variants of
GNSS data processing and derived from two NWMs (a global reanalysis and a
limited-area short-range forecast). All solutions gave tropospheric
parameters in high temporal resolution (5 min). Since no meteorological
data providing any information about prevailing atmospheric conditions
during the evaluated time period entered the GNSS data processing (because
we used empirical mapping functions and a priori tropospheric delays),
estimated tropospheric gradients can be regarded as fully independent and
therefore can provide additional interesting information, along with the
ZTD, in support of NWMs (see Douša et al., 2016; Guerova et al., 2016).
When lowering elevation angle cut-off (from 7 to 3∘),
more accurate tropospheric gradient estimates were obtained. The standard
deviations of differences of GNSS gradients to NWM gradients were reduced by
10 %, formal errors of tropospheric gradients were reduced, and
station-wise mean gradient directions were also more stable. On the other
hand, the usage of a lower cut-off angle led to a slightly worse station
height repeatability (10 %), which is partly in contradiction with the
results of Douša et al. (2017) but in agreement with Zhou et al. (2017). The discrepancy is attributed to the use of the PPP method with
simplified modelling (GPT+GMF) for low-elevation observations. The
3∘ elevation angle cut-off can nevertheless be recommended for an
optimal gradient estimation from GNSS data.
A small decrease in the standard deviation of the estimated gradients (2 %) was
observed when using GPS+GLONASS instead of GPS only and compared to NWM
gradients. This indicates that the post-processing tropospheric gradients
can be reliably estimated solely with a GPS constellation. However, it may
still depend on applied software, strategy, products and processing, e.g.
(near) real time. In this regard, Li et al. (2015) and Lu et al. (2016)
demonstrated that tropospheric gradients from multi-GNSS PPP processing
improved their agreement with those estimated from NWM and WVR when compared
to stand-alone GPS processing.
Using a simulated real-time processing mode, the agreement of GNSS versus
NWM tropospheric gradients revealed an increase in standard deviation of
about 19 % (53 %) for IGS01 (IGS03) RT products when compared to the
corresponding GNSS post-processing gradients. We also show that the quality
of real-time tropospheric parameters is dominated by the quality of
real-time orbit and clock corrections and to a much lesser extent by the
processing mode, i.e. a Kalman filter without backward smoothing. Tropospheric
gradients from the RT solution using the IGS03 RT product showed
occasionally a large misbehaviour of tropospheric gradients at all GNSS
stations obviously not related to weather conditions. This was caused by
frequent PPP re-initializations due to interruptions and worse quality of
the IGS03 RT product, while normal results were achieved by using the IGS01
RT product. Thus, providing high-resolution gradients in a (near-)real-time
solution still remains challenging, which would require optimally a
multi-GNSS constellation and high-accuracy RT products.
We studied systematic differences in estimated tropospheric gradients.
Unlike for ZTDs, average systematic differences of up to 0.5 mm over a day, and
of up to 1.0 mm or even more for individual gradient components during extreme
cases, can affect the magnitude of estimated tropospheric gradients solely
due to utilizing different gradient mapping functions or observation
elevation-dependent weightings. While the mfg choice affects the magnitude of
the estimated gradient, it does not affect the direction of the gradient.
However, any difference in the magnitude causes systematic errors in
gradient components which depend on the gradient direction too. At a global
scale, the long-term mean gradient pointing to the Equator causes systematic
differences of up to 0.3 mm in the north-gradient component between Bar-Sever
and Chen and Herring mfg (see Appendix A).
Both smaller gradient formal errors and slightly improved height
repeatability, which was found to be statistically significant, suggest more
accurate modelling when using the Bar-Sever mfg. Without an accurate and
independent gradient product, it is still difficult to make a strong
recommendation among different mfg, i.e. resulting in different absolute
gradient values. More work therefore needs to be done in order to find an
optimal gradient mapping function, and it will require high-resolution and
highly accurate NWM data sets. In any case, we could strongly recommend using
the same mfg implemented in the same form whenever comparing or combining
tropospheric gradients derived from different sources (GNSS, WVR or NWM). On
the other hand, if tropospheric gradients are used solely for reconstructing
slant total delays, different mfg should provide very similar results.
Data availability
GNSS data from the EUREF Permanent Network (EPN) stations are freely
available through the anonymous FTP, e.g. from the EPN historical data
centre at ftp://epncb.oma.be/pub/obs/ (last access: 14 June 2019) maintained by the Royal
Observatory of Belgium. Other GNSS data were primarily collected for the
purpose of COST Action ES1206 (GNSS4SWEC project; see Douša et al.,
2016) and cannot be distributed. The ECMWF is acknowledged for making
publicly available ERA5 reanalysis fields that were generated using
Copernicus Climate Change Service Information 2018
(https://www.ecmwf.int/en/forecasts/datasets/archive-datasets/reanalysis-datasets/era5, last access: 14 June 2019).
The Global Forecast System data were provided by the National Centers for
Environmental Prediction (http://nomads.ncdc.noaa.gov/data/gfsanl, last access: 14 June 2019). All the validation results in
the form of figures and tables for all types of presented comparisons and
stations can be provided on request to michal.kacmarik@vsb.cz.
In Fig. 10a and b the systematic difference in the derived
tropospheric gradients based on ERA5 data (average over 10 years) is shown
for any point on Earth's surface between tropospheric gradients estimated
utilizing the BS mfg and tropospheric gradients estimated utilizing the CH
mfg, whereas there is no considerable systematic difference in the east-gradient component: it reaches up to 0.3 mm in the north-gradient component
(positive in the Northern Hemisphere and negative in the Southern Hemisphere). If we
exclude oceans, the maximum values can be found in north-eastern America and
north-eastern Asia. In the region of the benchmark campaign, the difference is
around 0.15 mm. We note that the mean tropospheric gradients point to the
Equator, i.e. the north-gradient component is negative in the Northern
Hemisphere and positive in the Southern Hemisphere. This can be seen in Fig. 10c and d, showing the mean north- and east-gradient
components utilizing the CH mfg, and can be explained by the fact that the mean
zenith delays increase towards the Equator. The systematic difference
between these two mfg is due to the fact that for the same slant total delays
the magnitudes of tropospheric gradients which are estimated utilizing a
smaller mfg are larger than the magnitudes of tropospheric gradients which
are estimated utilizing a larger mfg. The product of the mfg and the tropospheric
gradients, i.e. the azimuth-dependent part of the tropospheric delay,
remains approximately the same.
(a, b) Systematic difference (average over 10 years) for
any point on Earth's surface between tropospheric gradients estimated
utilizing the gradient mapping function of Bar-Sever and tropospheric
gradients estimated utilizing the gradient mapping function of Chen and
Herring. (c, d) Mean north- and east-gradient components (average over
10 years) for any point on Earth's surface utilizing the mapping function of
Chen and Herring. Panels (a) and (c) show the north-gradient component, (b) and (d) the east-gradient component. The results are based on ERA5 data.
NWM tropospheric gradients presented in this paper were also compared with
NWM tropospheric gradients provided by the TU Vienna (see
http://vmf.geo.tuwien.ac.at/, last access: 14 June 2019). Specifically, we compared the NWM
tropospheric gradients based on ERA5 with the so-called linearized
horizontal gradients (LHGs) (Boehm and Schuh, 2007). We note that the LHGs are
based on the closed-form expression depending on the north–south and
east–west horizontal gradients of refractivity (Davis et al., 1993). The LHGs
are solely available for several stations, and they are no longer supported
(their provision ended in 2017). Recently, Landskron and Boehm (2018)
provided refined horizontal gradients based on a least-square adjustment
which are currently recommended for use. We decided to look at three
stations available in all data sets, ONSA, POTS and WTZR, and we provide the
comparisons in Fig. 11. As expected, we find better agreement between
ERA5 tropospheric gradients and the refined horizontal gradients. We also
find that the magnitude of the ERA5 tropospheric gradients is larger than
the magnitude of the refined horizontal gradients. This is not surprising
since the NWM that is used in the generation of the refined horizontal
gradients has a horizontal resolution of 1∘ only (ERA-Interim
provided by the ECMWF). For example, Zus et al. (2016) showed how an
increased horizontal resolution of the NWM amplifies the tropospheric
gradient components under severe weather conditions.
Panels (a), (c) and (e) show the time series (1 May–30 June 2013) of
the east-gradient component for stations ONSA, WTZR and POTS,
respectively. Panels (b), (d) and (f) show the time series of the north-gradient
component for the same stations. The black line corresponds to the ERA5
tropospheric gradients (GFZ, regarded in the paper as NWM ERA5), the red
line corresponds to the refined horizontal gradients provided by the TU Vienna
(VIE) and the blue line corresponds to the so-called linearized horizontal
gradients provided by the TU Vienna (LHGs). The red numbers represent the mean
and standard deviation between VIE and GFZ. The blue numbers are the mean
and standard deviation between LHG and GFZ.
Author contributions
MK, FZ, JD, GD and JW designed the whole study. MK calculated
statistics and evaluated all the results, prepared gradient maps and wrote major
parts of the paper draft. JD and PV processed all the GNSS solutions in
the G-Nut/Tefnut software, and JD prepared Sect. 4 and revised the paper.
FZ ran NWM WRF, estimated the tropospheric parameters from NWM ERA5 and
NWM WRF and prepared Sect. 2.3, Appendices A and B. KB prepared
NWM ERA5 fields for the study and prepared NWM ERA5 outputs for Appendix A.
GD and JW provided insights and contributed with careful reading and
improvement of the text.
Competing interests
The authors declare that they have no conflict of interest.
Special issue statement
This article is part of the special issue “Advanced Global Navigation Satellite Systems tropospheric products for monitoring severe weather events and climate (GNSS4SWEC) (AMT/ACP/ANGEO inter-journal SI)”. It is not associated with a conference.
Acknowledgements
The authors thank all the institutions which provided GNSS observations for
the COST ES1206 benchmark campaign (Douša et al., 2016). Florian Zus wants to
thank Thomas Schwitalla (Institute of Physics and Meteorology,
University Hohenheim) for the introduction to the WRF system. The ECMWF is
acknowledged for making publicly available ERA5 reanalysis fields that were
generated using Copernicus Climate Change Service Information 2018
(https://www.ecmwf.int/en/forecasts/datasets/archive-datasets/reanalysis-datasets/era5, last access: 14 June 2019).
The GFS analysis fields are provided by the National Centers for
Environmental Prediction
(http://www.ftp.ncep.noaa.gov/data/nccf/com/gfs/prod, last access: 14 June 2019).
Financial support
The study was
realized during a research stay of Michal Kačmařík at GFZ Potsdam funded by EU ESIF
project no. CZ.02.2.69/0.0/0.0/16_027/0008463. Jan Douša and Pavel Václavovic
acknowledge the Ministry of the Education, Youth and Science of the Czech
Republic for financing the study with project no. LO1506 and supporting
benchmark data with project no. LM2015079.
Review statement
This paper was edited by Olivier Bock and reviewed by two anonymous referees.
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