We have studied linear horizontal gradients in the atmospheric propagation delay above
ground-based stations receiving signals from the Global Positioning System (GPS).
Gradients were estimated from 11 years of observations from five sites in Sweden.
Comparing these gradients with the corresponding ones from the
European Centre for Medium-Range Weather Forecasts (ECMWF) analyses
shows that GPS gradients detect effects over different timescales caused
by the hydrostatic and the wet components.
The two stations equipped with microwave-absorbing material below the antenna,
in general, show higher correlation coefficients with the ECMWF gradients
compared to the other three stations.
We also estimated gradients using 4 years of GPS data from two co-located
antenna installations at the Onsala Space Observatory.
Correlation coefficients for the east and the north wet gradients, estimated
with a temporal resolution of 15 min from GPS data, can reach
up to 0.8 for specific months when compared to simultaneously estimated wet gradients
from microwave radiometry.
The best agreement is obtained when an elevation cut-off angle
of 3

Space geodetic techniques, where the fundamental observable is a radio signal's
time of arrival at a station on the surface of the Earth,
are affected by variations in the propagation velocity in the atmosphere.
Because time measurements avoid problems related to accurate calibration,
which are common for systems measuring different types of emissions,
it is a common view that Global Navigation Satellite Systems (GNSSs)
have a long-term stability
and are well suited for climate monitoring, e.g.
in terms of the atmospheric water vapour content.
Estimates of the total propagation delay above a GNSS station can be used to
determine the integrated amount of water vapour.
It is also common practice to estimate two-dimensional horizontal linear gradients
for each station in the GNSS data processing
because they improve the reproducibility of estimated geodetic parameters;
see, e.g.

We have studied estimated gradients primarily from Global Positioning System (GPS) data
from Swedish GNSS stations
by comparing these gradients to independent measurements.
An important site is the Onsala Space Observatory where
a geodetic very-long-baseline interferometry (VLBI) telescope and a water vapour
radiometer (WVR) are installed
and co-located with GNSS receiver stations.
The overall goal was to study the usefulness of GPS-derived gradients
in atmospheric and climate research.
Previous studies have been carried out using GNSS data from Onsala.
Comparing the gradients derived from VLBI, GPS, and a WVR,

There are some interesting questions actualized by previous work,
which we tried to take further.
Of specific interest in our study was to investigate if there is any systematic
seasonal behaviour in
the estimated gradients in Sweden
and if they can be explained by the influence of regional-scale weather systems.
The question about the seasonal changes of gradients was previously
studied by

In Sect.

The delay of space geodetic signals propagating through the atmosphere
depends on the refractive index.
For space geodetic applications it is meaningful to define one hydrostatic
and one wet component

Hydrostatic gradients are determined by pressure and temperature gradients
and exist mainly over regional scales (e.g. persistent high- and low-pressure systems)
and synoptic scales (e.g. weather systems).
Using a European and a global GPS network,
including three of the GPS stations used in this study,

Temperature and especially water vapour can show strong horizontal gradients over
small (kilometre) scales and temporal variability is typically
also much higher than that of the hydrostatic gradients; see, e.g.

We note that none of the known processes is expected to be strictly horizontally linear, but the strength in the geometry, the distribution of the observations in the sky, and the GNSS data quality makes it difficult to determine additional atmospheric parameters of a higher order.

The atmospheric parameters that are normally estimated
when processing space geodesy data
are an equivalent zenith wet delay and linear horizontal delay gradients in the east
and the north directions.
The uncertainties of the estimates depend on the geometry of the observations
and the accuracy of the so-called mapping functions,
used to describe the estimated parameters dependence on the elevation angle,
given the specific weather conditions at the site, at the time;
see, e.g.

In addition to the east and the north gradient components, we also studied the gradient amplitude,
defined as follows:

We compared gradients estimated from GPS observations acquired at five sites and
six antenna and receiver installations:
Kiruna (KIR0), Mårtsbo (MAR6), Borås (SPT0), Visby (VIS0),
and Onsala (ONSA and ONS1)
with respect to VLBI, WVR, and ECMWF estimates.
These stations are also part of the EUREF network

The five sites used in the study. Two antenna installations, ONSA and ONS1, are co-located together with the VLBI telescope and the WVR at the Onsala site. An antenna installation is referred to as a station.

We used 11 years of GPS data (2006–2016) from the five Swedish GNSS
sites mentioned above.
Gradients in the east and the north directions were estimated with a temporal
resolution of 5 min.
Two GNSS stations are operating continuously at the Onsala Space Observatory
on the western coast of Sweden.
The primary station, ONSA, was established in 1987
and the other station, ONS1, was taken into operation in 2011.
The six antenna installations are shown in Fig.

The analysis of the GPS data followed the same lines as described by

In order to investigate the impact of different constraints on the estimated gradients
we also reprocessed two days of GPS data for ONSA, where large changes (2–3 mm) in both the east
and the north gradient components were observed over a couple of hours by both the GPS and the WVR data.
In addition to the constraint value of 0.3 mm

Recent work by

Based on the 5 min gradients, we calculated mean values over 15 min, 6 h, 1 d, and 1 month in order to match the temporal resolution of the comparison data and to study the variability of the wet and the hydrostatic gradients over different timescales.

The six antenna installations used to acquire the GPS data.
See Fig.

Processing of GPS data.

Examples of the sky coverage of the GPS observations are shown in Fig.

Sky plots of GPS observations at Onsala from 06:00 to 12:00 UT

The microwave radiometer, shown in Fig.

During the time period 2013–2016 the WVR was observing in a sky mapping mode,
as is illustrated in Fig.

The water vapour radiometer (WVR) Konrad at the Onsala Space Observatory.

Specifications for the Konrad WVR.

A measurement cycle of the WVR begins with two azimuth scans.
In order to avoid emission from the ground, the lowest elevation angle observed was 20

Number of data points per day observed by the WVR. During days without data loss, e.g. due to rain,
each estimated gradient was based on

In order to avoid ground-noise pickup the WVR provided observations of the wet delay
in the different directions above 20

We used the VLBI data from the CONT14 campaign coordinated
by the International VLBI Service

The CONT14 campaign was observed during 6–20 May 2014.
The VLBI data were analysed with the Calc/Solve analysis software

Figure

The directions of the VLBI observations for the time period from 06:00 to 12:00 UT

The Technical University of Vienna provides hydrostatic
and wet gradients based on ECMWF data
for many space geodetic sites globally.
The product used here is usually referred to as LHGs (linear horizontal gradients)
and is described by

The ECMWF gradients for the Onsala (ONSA) site during the 4-year time period studied in
Sect.

There are alternative methods of deriving gradients from Numerical Weather Model data
using ray tracing methods; see, e.g.

In this study we used the LHG data from 2006 to 2016,
resulting in a time series of 11 years.
As an introduction, examples of the ECMWF hydrostatic and wet gradients
are illustrated in Fig.

The results of comparisons between the gradients from these datasets are presented in the next two sections.
The usage is defined in Table

Summary of used datasets.

Formal errors of the remote-sensing techniques.

We start by investigating the characteristics of the gradients over the year.
In Fig.

We can clearly see negative north gradients in the winter,
with a mean value around

The results for the other four stations (KIR0, MAR6, SPT0, and VIS0) show
similar systematic features.
One exception is KIR0, which is at a higher latitude and has a less humid climate.
At KIR0 the average monthly wet gradients are much smaller
except during the summer months.
Furthermore, the influence of the Icelandic low pressure in the winter
is not as large as it is at the other four stations.
Another exception is seen in the ECMWF wet gradients for ONSA in
Fig.

Monthly means of estimated gradients at the ONSA station for the period 2006–2016.
The top graphs show the total gradients from ECMWF

We study the agreement, in terms of correlation coefficients,
between the total GPS and ECMWF gradients from five GPS stations using data from 2006 to 2016.
These are shown in Table

Correlation coefficients for the total east and north gradients estimated from GPS data and compared to ECMWF data.

The correlations seen in all cases confirm that a consistent atmospheric signal in terms of gradients is detected by the GPS observations and ECMWF analyses. We note that the correlation coefficients increase for longer-averaging time periods. Our interpretation is that by long-term averaging we compare a larger fraction of the gradient that is caused by large-scale temperature and pressure gradients. Unfortunately, the temporal resolution of 6 h in the ECMWF data is not sufficient to resolve either rapid changes in the pressure related to moving weather systems or many of the short-lived small-scale gradients associated with the variability in the water vapour.

Another result worth noting is that the two stations
with the highest correlation coefficients, especially for the
monthly averages, are ONSA and SPT0.
The 95 % confidence interval is

The mean values and the SD of the gradients,
for the three different temporal resolutions, are presented in
Tables

We note that the SD obtained for the KIR0 station for 6 h and 1 d is smaller.
This is likely a consequence of the lower humidity at the station.
For monthly averages, these differences are reduced
and the SD for all stations is in the range 0.13–0.18 mm indicating
that the hydrostatic gradients and other effects,
e.g. signal multipath effects, become relatively more important.
Variations in the electromagnetic environment that change the impact of the signal
multipath at a station may be due to, e.g. snow, rain, vegetation, and soil moisture.
The relative importance of hydrostatic and wet gradients
was illustrated in Fig.

Mean values and standard deviations (SDs) over the 11 years of estimated total gradients from GPS data for different temporal resolutions.

Mean values and standard deviations (SDs) over the 11 years of estimated total gradients from ECMWF data for different temporal resolutions.

For the Onsala site we study total gradients from the two GPS stations
and one VLBI station and wet gradients from the WVR for the time period 2013–2016.
We use the hydrostatic gradients from ECMWF to calculate wet gradients from GPS and
VLBI total gradients.
The designs of the two GPS stations are different (see Fig.

Gradients in the east and the north directions are estimated from
the GPS data for five different solutions.
We use three different elevation cut-off angles for the VMF1
zenith delay mapping functions.
One additional solution is carried out with elevation-dependent weighting
(

The GPS wet gradients for ONSA and ONS1 are computed by
subtracting the hydrostatic gradients from ECMWF
(see Fig.

The results for the different GPS solutions are summarized in
Tables

The solution giving the best agreement, when comparing gradients
from ONSA and ONS1 data with each other,
is the one with elevation-dependent weighting,
whereas the comparisons with the WVR, for both ONSA
and ONS1, give the best agreement without weighting.
The choice of elevation cut-off angle is a compromise between having
a good geometry and avoiding effects of signal multipath.
Our interpretation is that the gradients from ONSA and ONS1 are estimated based
on very similar observational directions and have common error sources,
such as orbit errors, resulting in correlations around 0.9.
In order to increase an already high correlation, the observations at the
lowest elevation angles are not that important, since multipath effects will be increasingly different the closer to the horizon the observations are made.
When ONSA and ONS1 gradients are compared to those from the WVR
the situation is different because these gradients are independent
and the geometry of the GPS observations becomes more important in
order to estimate a more accurate gradient.
Although we note that the correlation coefficients are reduced here
to 0.68 and 0.64 for the east and the north component, respectively.
Since the WVR provides independent gradients,
we will in the following focus on the VMF 3

Assessment of the different GPS solutions comparing total gradients from the two GPS stations ONSA and ONS1.

Assessment of the different GPS solutions for the wet gradients from the two GPS stations ONSA and ONS1 relative to the WVR data.

An overview of the data in terms of monthly means of the wet gradient amplitude and the ZWD
is presented in Fig.

Time series of

(1) The WVR is sensitive to liquid water in the atmosphere. This is a cause for positive systematic errors in the ZWD, as well as occasional overestimates of gradient amplitudes. We investigated this possibility by deleting all WVR observations implying a liquid water content larger than 0.3 mm. However, the impact was insignificant. The average gradient amplitude decreased by 0.01 mm. The reason being that large liquid contents are rather infrequent, given that already data acquired during rain (which was assumed to occur when liquid water content was larger than 0.7 mm) have been removed.

(2) The WVR gradients for one 15 min period do not depend on earlier or later estimates, whereas the GPS gradients are estimated using constraints on the variability. A constraint has a similar impact as a low-pass filter (peaks with a short duration will be reduced).

(3) The fact that the WVR and the GPS gradients are computed for different
elevation cut-off angles has two possible impacts:
(i) the larger volume sensed by GPS (with a 3

We conclude that the constraints and the sampling of different air masses are the likely explanations for the differences in gradient amplitudes estimated from GPS and WVR data but cannot, based on these results, determine their relative importance.

The impact of the elevation cut-off angle on the estimated 15 min GPS gradient amplitude

A correlation plot for the total gradients from ONSA and ONS1 for the VMF1 solution
with a 3

Correlations between estimated total gradients from the GPS stations ONSA and ONS1 using all data with a 5 min resolution from the period 2013–2016.

Correlation plots for the wet gradients from ONSA, ONS1, and the WVR are presented
in Fig.

The reasons for the lower correlation coefficients between the WVR and the GPS gradients are almost identical
to the reasons listed above of why the WVR gradient amplitudes are higher:
(1) they do not have common sources of errors;
(2) the WVR data suffer both from white noise and algorithm errors, especially when liquid water is present;
(3) the WVR data for each 15 min period are independent of the successive periods,
whereas there are temporal constraints on the gradients estimated from the GPS data;
(4) the sampling of the sky also agrees much better between the two GPS stations, assuming that, in general,
the directions of the observations are towards the same satellites,
whereas the WVR observations are evenly spread over the sky and above an elevation angle of 20

Concerning the sampling of the atmosphere, the use of a multi-GNSS constellation has been shown to improve
the agreement between GNSS gradients with those estimated from a WVR

Correlations between estimated wet gradients from the WVR, ONSA, and ONS1 using all data from the period 2013–2016.
The data in the graphs with ONSA and ONS1

Correlations between estimated wet gradients from the WVR data and the GPS data from ONSA (solid lines) and ONS1 (dotted lines), averaged over 15 min when the hydrostatic gradients are removed from the total GPS gradients for each month of the 4 years. The east gradients are presented with red lines and the north gradients with blue lines.

We investigated if an average of the wet gradients from both GPS stations, ONSA and ONS1, estimated at the same time epoch, will improve the agreement with the WVR. We see an overall small improvement. For the east gradient the individual correlation coefficients were improved from 0.678 (ONSA) and 0.682 (ONS1) to 0.698. The corresponding values for the north gradient were increased from 0.639 (ONSA) and 0.635 (ONS1) to 0.666. Our interpretation is that by averaging the GPS gradients from ONSA and ONS1 the stochastic noise is reduced.

Correlation plots are shown in Fig.

Comparing the results obtained for ONSA with those from ONS1 they are almost identical
(in both Figs.

Time series of ECMWF hydrostatic and wet gradients during the CONT14 campaign.

The wet gradients and the ZWD during the VLBI CONT14 campaign 6–20 May (days 126–140). The temporal resolution for the VLBI (blue circles) gradients is 6 h and the ZWD 30 min, 5 min for the GPS gradients for ONSA (red dots) and ONS1 (black dots), and 15 min for the WVR (green pluses).

The wet gradients from the two space geodetic techniques GPS
and VLBI are compared to each other
and to the WVR during the CONT14 campaign.
Observations from several earlier CONT campaigns have been analysed
in terms of gradients, with different results depending on the station
and the time of the campaign

Zoom in on the time series in Fig.

Again we note that the size of the WVR gradients is larger compared to all other instruments. The VLBI gradients correlate with the gradients from the other instruments but their amplitudes are smaller. Given that the sampling of the atmosphere is much more sparse with the VLBI telescope, a short-lived gradient in combination with the assumption of linear functions in 6 h segments will probably reduce the variability in the estimated amplitude.

Table

We note that the correlation coefficients are lower for the north component for all three comparisons, whereas the SDs are similar. The reason is that the size of the east gradients are larger compared to the north gradients during this 15 d period. Scatter plots (not shown) confirm what is indicated by the SDs: that the quality of the east and north components is similar.

Comparison of estimated wet gradients from VLBI relative to GPS and WVR data.

We attribute the lower correlation coefficients obtained between VLBI-GPS
and VLBI-WVR, using 6 h averages during the CONT14 campaign
compared to GPS-WVR 15 min averages for the month of May 2014 in
Fig.

Finally, we like to use this 15 d long time series for a discussion on gradient variability.
At the end of day 135 (see the ZWD plot in Fig.

We have estimated linear horizontal gradients from GPS data acquired at
five sites in Sweden.
Averaging gradients in the east and the north direction over 1 month
gives correlation coefficients of up to 0.9 when compared to gradients calculated from
meteorological analyses of the ECMWF

We studied wet gradients estimated with a temporal resolution of 15 min
from GPS and WVR data.
We found that an elevation cut-off angle of 3

We also note that when using a 3

Correlation coefficients between wet gradients estimated from GPS and the WVR data can reach up to 0.8 for specific months. We note a strong seasonal dependence, from 0.3 during months with smaller gradients to 0.8 during months with larger gradients, typically during the warmer and more humid part of the year. Related to this we suggest further studies of large wet gradients estimated from GPS, again in combination with meteorological high-resolution models for verification of the quality of the gradients.

In general, we also note slightly higher correlation coefficients for the GPS-derived gradients in the east compared to the north direction. We interpret this difference to be caused by an inhomogeneous spatial sampling of the sky, which is important when we assume that the linear model describing horizontal gradients has deficiencies. The different sampling of the sky is an important issue for any comparison between different techniques. This question remains unresolved and is also recommended for further study.

Additional issues that deserve attention in future studies, in addition to similar studies in different climates, e.g. the tropics, include multi-GNSS observations. At latitudes similar to those in this study, the use of GNSS satellites with a higher orbit inclination will reduce the part of the sky not sampled by GPS.

For VLBI the use of VGOS (twin) telescopes will also dramatically improve the sampling of the atmosphere. When WVR data are used to evaluate gradients from the space geodetic techniques one may consider also applying different constraints on the temporal variability of these estimates. We suggest that future work on gradients should focus on the interplay between the elevation cut-off angle, the temporal resolution, and the constraint value using both single and multi-GNSS. Furthermore, we believe that the outcome of such studies will be weather- and site-dependent, which will make it difficult to arrive at just one optimal recommendation that is globally valid.

The input GNSS data, in RINEX format, are available from EUREF:

GE coordinated and wrote the major part of the manuscript and together with TN planned the different GNSS data analyses during the COST Action ES1206. TN performed the GNSS data analyses, resulting in the estimated gradients. PF and RH carried out the same task for WVR and VLBI data, respectively. All authors contributed in the writing process, to the sections presenting the results produced by each author in particular, and approved the entire paper before submission.

The authors declare that they have no conflict of interest.

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.

We appreciate the constructive comments and suggestions from the editor and the three referees.
They resulted in a significantly improved paper, offering additional possible explanations for the obtained results
as well as additional studies.
For example, the use of a 3

This paper was edited by Olivier Bock and reviewed by three anonymous referees.