Within the Global Climate Observing System (GCOS) Reference Upper-Air Network (GRUAN) there is a need for an assessment of the uncertainty in the integrated water vapour (IWV) in the atmosphere estimated from ground-based global navigation satellite system (GNSS) observations. All relevant error sources in GNSS-derived IWV are therefore essential to be investigated. We present two approaches, a statistical and a theoretical analysis, for the assessment of the uncertainty of the IWV. The method is valuable for all applications of GNSS IWV data in atmospheric research and weather forecast. It will be implemented to the GNSS IWV data stream for GRUAN in order to assign a specific uncertainty to each data point. In addition, specific recommendations are made to GRUAN on hardware, software, and data processing practices to minimise the IWV uncertainty. By combining the uncertainties associated with the input variables in the estimations of the IWV, we calculated the IWV uncertainties for several GRUAN sites with different weather conditions. The results show a similar relative importance of all uncertainty contributions where the uncertainties in the zenith total delay (ZTD) dominate the error budget of the IWV, contributing over 75 % of the total IWV uncertainty. The impact of the uncertainty associated with the conversion factor between the IWV and the zenith wet delay (ZWD) is proportional to the amount of water vapour and increases slightly for moist weather conditions. The GRUAN GNSS IWV uncertainty data will provide a quantified confidence to be used for the validation of other measurement techniques.

In the hydrological cycle, water vapour is an important variable for
transferring heat energy from the Earth's surface to its atmosphere
and in moving heat around the Earth. Meanwhile, water vapour is a
very important greenhouse gas due to its ability to absorb long-wave
thermal radiation emitted from the Earth's surface. Hence, the
atmospheric water vapour is very important for the Earth's climate
system, and its variability is a key to understanding the hydrological
cycle. A variety of systems exist for measuring the atmospheric
integrated water vapour (IWV), e.g. radiosondes

The procedure for the estimation of the atmospheric IWV from GNSS
measurements begins with the refraction of the GNSS signals
(e.g. GPS signals transmitted at frequencies

All GNSS measurements are subject to error sources that influence
the uncertainty of the estimated ZTD which are converted from the
slant delays using mapping functions

Depending on different applications, the requirement on the accuracy of the estimated IWV varies. For the forecasting application the demand is mainly the timing of moving air masses, while the accuracy of individual IWV estimate and the IWV biases is of less importance. Therefore, less accurate real-time orbit estimation is accepted in forecasting applications. For climate research, it is crucial to have a high accuracy with the smallest biases possible in order to obtain the absolute value over a long timescale. In this case, the final GNSS orbit estimation with the highest accuracy is necessary. The focus of this study is to discuss and assess the accuracy of the IWV derived from ground-based GNSS measurements obtained from post-processed data and mainly used for climate research.

Many studies have investigated the uncertainty of the GNSS-derived
IWV either by comparing the GNSS IWV with data obtained from other
independent techniques (e.g.

The paper is structured as follows. Sections 2 and 3 discuss the two uncertainty analyses: a statistical and a theoretical one. A sequence of subsections is given in Sect. 3 in order to describe each of the errors which contributes to the final total uncertainty of the GNSS-derived IWV. In the last subsection, the individual error sources are then summarised into an overall error budget in order to obtain the final uncertainty of the GNSS-derived IWV. The conclusions are given in Sect. 4.

ZWD and corresponding IWV uncertainties given by a statistical analysis using observations from Onsala on the Swedish west coast.

A statistical analysis

For a long time period of measurements giving a zero mean of random
errors (

After including the third technique

Since the

From Eq. (

One example of a statistical analysis can be found
in

Since all known biases must be removed before derivation of the uncertainty, an assumption is made about no bias in another measurement technique. For most of the GRAUN sites, measurements simultaneously acquired from at least three co-located independent techniques are difficult. In this case, the next approach to calculate GNSS-derived IWV uncertainty is more useful.

A theoretical analysis is desired where the total uncertainty of
the GNSS-derived IWV (

The GNSS-derived IWV, denoted by

Combination of Eqs. (

Insertion of Eq. (

In order to calculate

The fundamental observable of the GNSS technique is the propagation time of a signal, transmitted from the GNSS satellites, passing through the Earth's atmosphere to the receivers (ground-based). When we consider the error budget of the GNSS-derived ZTD, errors from several parts need to be taken into account, which will be discussed in this section.

Errors in the estimates of the satellite coordinates will
propagate directly to the estimates of the GNSS parameters. If we
use the precise point positioning (PPP) strategy to process the data
obtained from a permanent GNSS site where the site coordinates are
usually kept fixed (one estimate per day), eliminate the ionosphere
delay to the first order, and use the final clock product from the
International GNSS Service (IGS), the orbit error is compensated by the
ZTD, receiver clock, and ambiguity parameters

Figure

The impact of the radial and the tangential orbit errors in the estimated ZTD.

Table

Accuracy of IGS reprocessed orbits for one day

The ZTD error due to orbit errors for three GRUAN sites:

Figure

The Earth's ionosphere contains electrons in sufficient
quantity to significantly delay the propagation of GNSS signals.
The ionospheric delay is dependent on the total amount of free
electrons along the propagation path, named total electron
content (TEC), and on the carrier frequency of GNSS signals. Normally,
in order to remove the ionospheric impact, an ionosphere-free linear
combination is used:

Differences in the ZTD caused by signal multipath depend
strongly on the elevation angle of the observation and are different
from site to site due to a changing electromagnetic environment. In
order to demonstrate the impact of multipath effects, we carried out
analyses using different elevation cutoff angles in the PPP data
processing for three GRUAN sites (LDB0, LDRZ, and NYA2) together
with two other sites: LDB2 (14.1

The impact of the elevation cutoff angle on the estimated ZTD for
five GRUAN sites. The result obtained from the 5

A similar investigation on the multipath effects was carried out
by ^{®}, attached below the
antenna plane.

Signal multipath is highly dependent on the local environment and
can vary in time, e.g. due to changes in soil
moisture

In order to obtain the highest accuracy in the ZTD
estimates, antenna-related errors, i.e. phase centre variations
(PCVs) and radome effects, need to be removed. Therefore, an absolute
calibration of the PCV for all the GNSS satellite-transmitting
antennas and the ground antenna

Photographs of two GRUAN sites:

To avoid the accumulation of snow and for general protection, many
GNSS antennas are equipped with radomes. Different shapes of radomes
yield different impacts on the phase of the GNSS
signal.

Mean and standard deviation (

In GNSS data processing, the slant path delay is converted
to the equivalent ZTD (sum of the ZHD and the ZWD) using MFs:

Values calculated from Eqs. (

Currently the most popular and accurate MF is the Vienna Mapping
Function 1 (VMF1) since it can capture the short-term variability of
the atmosphere by utilising data from a numerical weather
model

All GRUAN GNSS data will be processed by GFZ using its
inhouse GNSS software package, Earth Parameter and Orbit
determination System (EPOS)

Corrections for the second order of the ionospheric delay are applied.

Final orbit/clock products from IGS or equivalent are used.

Absolute satellite and ground antenna PCV models and radome calibrations are implemented. A hemispheric radome and an ECCOSORB plate are recommended.

Signal multipath effects need to be minimised either by implementing microwave-absorbing material to the antenna or by locating the GNSS antenna in an favourable place.

Use an elevation cutoff angle of 10

The time series of the estimated ZTD and the corresponding total ZTD uncertainty, for one day (1 June 2014) and for the three GRUAN sites: LDB0, LDRZ, and NYA2.

The procedure to determine the total ZTD uncertainty for each time
epoch is described as follows. For each GRUAN site, the daily GPS
data will be processed first using a PPP strategy in order to obtain
the ZTD estimates and corresponding formal errors together with
receiver clock errors. Thereafter, the estimated ZTD and receiver
clock errors will be used in a simulation in order to estimate the
ZTD errors due to the orbit errors. Then the root sum square (RSS)
of the simulated ZTD error and the corresponding formal error gives
total ZTD uncertainty for each time epoch. Figure

The ZHD for a given GNSS site can be calculated using the ground pressure

Values of

Differences in the surface pressure measured by three different barometers.

All GRUAN sites should be equipped with surface barometers which
provide accuracies much better than 0.5 hPa. One example is shown in
Fig.

For the GNSS sites which are not equipped with barometers, other
methods have been investigated.

The uncertainty of the conversion factor

The conversion factor

In order to evaluate the impact of the uncertainty in each variable
on the total uncertainty of

The parameter

Due to the fact that the ECMWF data provides a temporal resolution
of 6 h and a horizontal resolution of about 50 km and there is
normally a difference between the model height and the GPS height, a
temporal, horizontal, and vertical interpolation of the ECMWF data to
the time and position of the GPS site is necessary. Details for the
interpolation of the ECMWF data can be found in

We now can calculate the total uncertainty of the
GNSS-derived IWV after substituting Eqs. (

Table 4 summarises the calculated total uncertainties of the
GNSS-derived IWV for three GRUAN sites: LDBO, LDRZ, and NYA2. For
each site, the GPS data acquired from the year 2014 were processed
using a PPP strategy to obtain ZTD time series. The corresponding
total ZTD uncertainties were then determined using the approach
discussed in Sect.

As shown in Table 4, the uncertainties in the ZTD dominate the error
budget of the resulting IWV, contributing over 75 % of the
total IWV uncertainty. The impact of the uncertainty associated with
the conversion factor

The IWV uncertainty is calculated for each data point in the GRUAN
data products. Therefore, we first calculated the time series of the
total ZTD uncertainty using the method discussed in
Sect.

The estimated total IWV uncertainty for the month of June 2014 and
for three GRUAN sites:

Uncertainties in the GNSS-derived IWV calculated from the uncertainties associated with input variables.

As discussed above, currently the total IWV uncertainty obtained
from a theoretical analysis is calculated by ignoring the
site-dependent effects, i.e. signal multipath. If the GRUAN site is
co-located with other techniques, the total IWV uncertainty can be
estimated from the statistical method. One example is seen for the
IGS station ONSA at the Onsala Space Observatory. The mean total
uncertainty obtained from the theoretical analysis is 0.59 kg m

Two methods were discussed in order to determine the total uncertainty of the GNSS-derived IWV. When there are at least three co-located techniques available, measuring the variability of the IWV at the same time, a statistical analysis is applied. This method is, however, difficult to apply on the present observational network because three independent methods for IWV measurement are not available. Therefore, a theoretical analysis, where the total uncertainty of the IWV is calculated from each one of the input variables according to the rule of uncertainty propagation for uncorrelated errors, is used.

In order to minimise the ZTD uncertainty caused by the factors
discussed in Sects.

The theoretical method will be implemented in the GRUAN GNSS central
data processing. In summary, the following steps will be taken to
calculate IWV uncertainty for each data point. Firstly, the ZTD
uncertainty, including the systematic satellite orbit error and the
formal error, is calculated. Secondly, the ZHD uncertainty is
obtained using the uncertainty of ground pressure, estimated using
the method presented by

For sites where two additional independent techniques are available, the IWV uncertainty estimated from the statistical method can be used to assess the stability of the data quality and potentially improve the operational theoretical method. For example, the statistical method can be used to quantify the ZTD uncertainty, which may change due to site-dependent effects such as signal multipath.

Although the method presented in the work is based on a PPP analysis, it can be applied to calculate uncertainties of the ZTD obtained from a network solution, but with some modifications. In PPP, the receiver clock parameters can partly compensate for the orbit errors. It is, however, not the case for a network solution where both satellite and receiver clock errors are cancelled out. Meanwhile, the baseline length and orientation in a network solution have a significant impact on the resultant ZTD uncertainties.

This research was supported by the Deutscher Wetterdienst through the project “Development of a data product for the GNSS integrated precipitable water vapor column (IPW) following the requirements of the Guide for the GCOS Reference Upper Air Network (GRUAN)”. We would also like to thank NIWA for providing the GNSS measurements at Lauder which are conducted as part of NIWA's government-funded, core research programme. The article processing charges for this open-access publication were covered by the German Research Centre for Geosciences (GFZ). Edited by: I. Moradi