Tropospheric delay is an important error source in space geodetic
techniques. The temporal and spatial variations of the zenith wet delay (ZWD)
are very large and thus limit the accuracy of tropospheric delay modelling.
Thus, it is worthwhile undertaking research aimed at constructing a precise
ZWD model. Based on the analysis of vertical variations of ZWD, we divided
the troposphere into three height intervals (below 2 km, 2 to 5 km, and 5
to 10 km) and determined the fitting functions for the ZWD within these
height intervals. The global empirical ZWD model HZWD, which considers the
periodic variations of ZWD with a spatial resolution of

The radio waves experience propagation delays when passing through the neutral atmosphere (primarily the troposphere), which are known as the tropospheric delays. The tropospheric delay is one of the main error sources in space geodetic techniques. In the processing of the space geodetic data, the tropospheric delay along the propagation path is generally expressed as the product of zenith tropospheric delay (ZTD) and mapping function (MF). The ZTD is divided into a zenith hydrostatic delay (ZHD) and a zenith wet delay (ZWD) (Davis et al., 1985), and the ZHD can be accurately determined using pressure observations. Unlike the ZHD, the ZWD is difficult to calculate accurately due to the high spatio-temporal variation in water vapour. Its spatial distribution is characterized with a near-zonal dependency, with values varying from about 2 cm at high latitudes to about 35 cm near the Equator (Fernandes et al., 2013). The temporal variation pattern of ZWD is mainly characterized by the seasonal variability, including annual and semi-annual components (Jin et al., 2007; Nilsson et al., 2008). The high variabilities in ZWD make it the main factor influencing tropospheric delay correction.

Various methods and models are developed to estimate the ZWD. Ray-tracing
uses the observations from radiosonde profiles (Davis et al., 1985; Niell,
1996) or numerical weather models (Hobiger et al., 2008; Nafisi et al., 2012)
to calculate the ZWD. It can provide the most accurate ZWD corrections.
Models such as those developed by Bevis et al. (1992, 1994) make use of
single-layer parameters from atmospheric models, such as total column water
vapour (TCWV) and temperature. While Stum et al. (2011) proposed a model that
only uses TCWV. These models provide similar results to the study of Davis et
al. (1985) (that uses 3-D parameters) but only at the level of the model
orography to which the meteorological parameters refer to. As this orography
may depart significantly from the actual surface, and the vertical variation
of the ZWD is not well known, at a different elevation they possess errors
associated with the uncertainty in the modelling of the ZWD height variation
(Fernandes et al., 2013, 2014; Vieira et al., 2018). The traditional
Saastamoinen model (1972) and Hopfield model (1971) approximate the ZWD with
surface observations as temperature and water vapour pressure observations.
Without the information about the vertical distribution of water vapour, the
stability and reliability of their ZWD estimates are poor. Moreover, both
models are highly dependent on meteorological data. The aforementioned models
have the limitations of application in wide area augmentation and real-time
navigation and positioning. Therefore, the empirical climatological models
were proposed as required practical conditions. The RTCA-MOPS (2016),
designed by the US Wide Area Augmentation System (Collins et al., 1996),
estimates ZWD by using the latitude band parameters table. The modified
RTCA-MOPS model – called UNB3m (Leandro, 2006) – uses relative humidity as
a parameter instead of the water vapour pressure to calculate the ZWD,
effectively improving the precision of ZWD estimation to 5.5 cm (Möller
et al., 2014), but the model deviation is increased when the height exceeds
2 km (Leandro, 2006). The TropGrid model (Krueger et al., 2004, 2005)
provides the meteorological parameters needed to calculate tropospheric delay
in the form of a

The water vapour changes rapidly with respect to height, and the trends in water vapour at different heights vary, so the wet delay with direct relation to water vapour has complex spatio-temporal variations in the vertical direction. Kouba (2008) proposed an empirical exponential model to account for the height dependency of ZWD, but it will only be applicable within the height below 1000 m. The aforementioned empirical models are all based on a fixed height (average sea level or surface height) and use only a single decrease factor to describe the variation of water vapour or wet delay with respect to height, which makes it difficult to allow for the vertical distribution differences in water vapour (or wet delay) in the upper troposphere. In the course of aircraft dynamic navigation and positioning, the zenith delay error will result in twice as many errors in the station height estimate (Böhm and Schuh, 2013). Thus, it is necessary to correct the wet delay at different heights, which is clearly difficult for the aforementioned models. Based on the analysis of the characteristics of the ZWD profile, an empirical ZWD model, named HZWD, is established based on three functions applicable within corresponding height intervals, and the model precision is verified by European Centre for Medium-Range Weather Forecast (ECMWF) reanalysis data as well as radiosonde data.

Water vapour pressure profile

ZWD is defined as the integral of the wet refractivity along the vertical
profile above the station.

ZWD vertical gradients profile

ERA-Interim can provide data at 00:00, 06:00, 12:00, and 18:00 UTC daily
with a spatial resolution of not more than

ZWD gradients profiles at grid points in different latitude bands (12:00 UTC, 1 January 2010).

ZWD gradients profiles at grid points in different latitude bands (12:00 UTC, 1 July 2010).

Figure 3 shows the ZWD vertical gradients with respect to height at grid points in different latitude bands. Figure 4 shows the similar ZWD vertical gradients as Fig. 3 but for a different season. The variations are similar to those in Fig. 2a, which show trend changes at about 2 and 5 km. It is worth noting that the ZWD gradients at low latitudes are much larger and water vapour is more variable than at high latitudes, resulting from the fact that the water vapour at low latitude is more variable. In addition, the ZWD gradient trends in the Southern Hemisphere are significant. In contrast, the ZWD gradients in the Northern Hemisphere are slightly complicated with respect to height: the reason for this may be that the Southern Hemisphere is mostly oceanic while the Northern Hemisphere has many sea coasts. The terrain complexity in the Northern Hemisphere contributes to the disturbances in the ZWD gradient in specific areas. According to the vertical variation characteristics of ZWD, we divided the troposphere into three height intervals (below 2 km, 2 to 5 km, and 5 to 10 km) and assumed 10 km to be the empirical tropopause beyond which the ZWD is assumed to be zero. For ZWD fitting with respect to height, TropGrid2 and GPT2w use exponential functions, while some scholars have also used a polynomial to describe the tropospheric delay with respect to height (Song et al., 2011). We used both polynomial and exponential functions to fit the variation trend of the ZWD with respect to height in the three selected intervals, respectively. The results showed that the quadratic polynomial used under 2 km and the exponential functions between 2 and 5 km and between 5 and 10 km gave the best fits. The combination of the quadratic polynomial and exponential functions for different height intervals are termed piecewise height functions. Table 1 summarises the global fitting statistics of different fit functions, demonstrating the superiority of piecewise height functions to the single polynomial function and single exponential function used for the whole troposphere.

Decadal time series and cycle fitting results of function
coefficients

Fitting RMSE of piecewise height functions, single quadratic polynomial function, and single exponential function (unit: mm).

From the above analysis of ZWD vertical variation and fitting, the piecewise
height functions of the proposed HZWD model are:

Therefore, taking the annual and semi-annual cycles into consideration, we
used Eq. (4) to fit the function coefficients derived from Eq. (3) to
temporal parameters for each grid point (Böhm et al., 2015).

Figure 6 shows the global distributions of annual means of model coefficients

Global distributions of annual means of HZWD model coefficients

After the fitting processes involving Eqs. (3) and (4), the global ZWD model
HZWD, using piecewise height functions, is established. The spatial
resolution of the HZWD model is

To test the precision of the HZWD model and analyse the model correction
performance compared to other troposphere models, we used the ERA-Interim
pressure level data and radiosonde data from the year 2015 as external data
sources and compared the results with the commonly used models UNB3m and
GPT2w. The parameters used for the validation are bias and root mean square error
(RMSE), expressed as follows.

For the UNB3m model, the ZWD at mean sea level (m.s.l.) is first calculated,
then a vertical correction is applied to transform the ZWD to the target
height. The formulae are as follows (Leandro et al., 2006):

Modelling of the HZWD model is based on the monthly mean profiles of ERA-Interim pressure level data from 2001 to 2010, while we used the ERA-Interim pressure level data with the full time resolution of 6 h in 2015 for the model validation. This is to validate the model performance on the daily scale. Regarding the ZWD profiles calculated from these data as reference values, we calculated the global annual average bias and RMSE of the ZWD for three models (HZWD, GPT2w, and UNB3m) within the three height intervals: below 2 km, 2 to 5 km, and 5 to 10 km (Table 2).

Error statistics for the three models compared to the 2015 ECMWF data (unit: mm).

From Table 2, it can be seen that the HZWD model is the most accurate model
across all three intervals, followed by the GPT2w model, and the UNB3m model
has the worst performance. The annual average biases of the HZWD model are
lower than that of the GPT2w model and the UNB3m model, except below 2 km.
Compared with the RMSEs in the GPT2w model, those of the HZWD model are
decreased by 1.4, 0.9, and 1.2 mm within the three height intervals,
corresponding to improvements of about 6 %, 6 %, and 32 %, respectively. The
improvements of the HZWD model over GPT2w model will result in precision
improvements of 2.8, 1.8, and 2.4 mm respectively in height estimates in
real-time aircraft positioning. The correction performance improvement from 5
to 10 km height is particularly evident. Figure 7a shows the ECMWF ZWD
profile and the ZWD profiles of the three models at 12:00 UTC on 1 January 2015 at a representative grid point (0

The ZWD profiles

The variation of the troposphere has a strong correlation with latitude. To
analyse the correction performances of the three models in different regions
around the world, we calculated the three models' errors in different
latitude bands (10

Bias comparisons between the three models (HZWD, GPT2w, and UNB3m) in different latitude bands over the year 2015.

Figure 9 shows the RMSEs of the three models. It can be seen from Fig. 9
that the precision of the HZWD model is significantly better than that of the
UNB3m model across the three height intervals and all latitude bands, which
is better than the GPT2w model in general. The precision of the three models
declines with decreasing latitude, because the active change in water vapour
in these areas limits the precision of the model. Corresponding to Fig. 8,
the errors in UNB3m are asymmetric: the main reason for this is that the
meteorological parameters of UNB3m are interpolated from the coarse look-up
table with a latitude interval of 15

RMSE comparisons between the three models (HZWD, GPT2w, and UNB3m) in different latitude bands over the year 2015.

Error statistics for the three models validated by 2015 radiosonde data (unit: mm).

Summarising the distributions of bias and RMSE across different latitude
bands, we can see that the HZWD model performs best with the ECMWF data as
reference values. Compared with the models GPT2w and UNB3m, the HZWD model
basically eliminates systematic error in the 5 to 10 km height interval and
the correction performance is stable at all heights and regions. To
investigate the model's performance over time, Fig. 10 shows the time series
of RMSE for the three models at 6 h intervals throughout the year 2015 at
grid point (0

RMSEs in ZWD estimates of the three models (HZWD, GPT2w, and
UNB3m) compared to the ECMWF data over the year 2015 at grid point
(0

Uncertainty of ZWD with respect to height at station 01241
(63.70

Global distributions of bias for the three models (HZWD, GPT2w, and UNB3m) compared to 2015 radiosonde data.

A radiosonde is used in a sounding technique that regularly releases balloons
to collect atmospheric meteorological data at different heights: it can
obtain profiles of various meteorological data with high accuracy. At
present, the Integrated Global Radiosonde Archive (IGRA) website
(

Global distributions of RMSE for the three models (HZWD, GPT2w, and UNB3m) compared to 2015 radiosonde data.

RMSEs in ZWD estimates of the three models (HZWD, GPT2w, and UNB3m) for radiosonde station 01241 over the year 2015.

Taking the radiosonde ZWDs as reference ZWD values, we validated the ZWDs
from the models HZWD, GPT2w, and UNB3m. Table 3 shows the statistical results of
the three models. It can be seen from Table 3 that the HZWD model has the
best overall stability of the average bias and RMSE, indicating the best
precision, and the UNB3m model is the worst. Compared with the GPT2w model,
the RMSEs in HZWD in the three height intervals are reduced by 0.6, 0.9,
and 1.7 mm, which equates to precision improvements of 2 %, 5 %, and 33 %,
respectively. Moreover, these improvements correspond to an error reduction
of 1.2, 1.8, and 3.4 mm respectively in height estimates in geodetic
techniques. Taking the uncertainty of radiosonde ZWD into account, the
improvement of the HZWD model over GPT2w model below 2 km seems to be
insignificant. Nevertheless, we can reasonably think that the ZWD predicted
by HZWD is closer to true ZWD due to its smaller RMSE. It is worth
noting that the bias and RMSE of the HZWD model and the GPT2w model are
both larger than those of the results from ECMWF data in Table 2. The reason
is that the HZWD model and the GPT2w model are based on ECMWF data, and thus the
test results with radiosonde data are slightly worse than those using ECMWF
data. On the contrary, the bias of the UNB3m model decreases, and the RMSE between 2 and 5 km and between 5 and 10 km are less than those in Table 2.
It may be due to the fact that most of the radiosonde stations are in the
Northern Hemisphere, accounting for more than 60 % (

Figure 12 shows the global distributions of bias for the three models within the three height intervals, and Fig. 13 shows the global distributions of RMSE for the three models. As can be seen from Fig. 12, the three models show a poorer performance in low-latitude areas than in mid- and high-latitude areas for all height intervals, similar to the results in Sect. 4.1. Within the 5 to 10 km interval, the bias of the GPT2w model is large and positive in the equatorial region, indicating that the ZWD of the GPT2w at this height is significantly overestimated, and the global bias of the UNB3m model in this height interval is positive, also indicating an overestimate of the ZWD in the UNB3m model. The bias of the HZWD model does not show obvious regional differences with respect to height, and the overall distribution of HZWD model bias has no tendency to either the positive or negative. Figure 13 further illustrates the precision of the HZWD model. The global RMSE distributions of the HZWD model are similar to that of GPT2w model below 2 km and between 2 and 5 km, but the precision of the HZWD model is slightly better. Combining this with the bias distribution of the GPT2w model in Fig. 12, the GPT2w model also has a large RMSE near the Equator in the 5 to 10 km interval, which shows that the GPT2w model is unstable at high height in low-latitude areas. The precision of the UNB3m model is poorer than that of both the HZWD and GPT2w models. Below 2 km, the UNB3m model reaches decimetre-level precision near the Equator, and even exceeds 12 cm in some areas: the distribution of north–south heterogeneity remains obvious.

These results validate the spatial stability of the precision of the HZWD model; furthermore the temporal stability of the model precision is verified next. Figure 14 shows the results of ZWD corrections of the three models for the radiosonde station 01241 for the whole of 2015. It can be seen from Fig. 14 that the HZWD model and the GPT2w model are relatively stable throughout the year, while the correction performance of the UNB3m model in 2015 is worse than those of the HZWD and GPT2w models. The probable reason for this is that the UNB3m model only takes into account the annual variations in the metrological elements with a fixed phase, resulting in precision instability throughout the year. The improved performance arising from use of the HZWD model, compared to that arising from use of the GPT2w model, is more apparent with increasing height: this shows that modelling ZWD piecewise with height can effectively approximate the real ZWD profile and improve the precision of ZWD estimation.

The complexity of spatio-temporal variations makes the modelling of
tropospheric ZWD difficult. In this paper, the characteristics of vertical
variation of wet delay are analysed. The troposphere is divided into three
height intervals: below 2 km, 2 to 5 km, and 5 to 10 km according to
different trends (10 km is assumed to represent the empirical tropopause). A
quadratic polynomial and two exponential functions are used to describe the
variation of wet delay within each of the three intervals. Based on the
monthly mean data of ECMWF ZWD from 2001 to 2010, a global ZWD model with
spatial resolution of

The HZWD model offers good precision stability in the vertical direction and can meet the requirements of ZWD correction at different heights within the troposphere; however, it can be seen that neither the HZWD nor the GPT2w models, i.e. those non-meteorological parameter-based models, performed well in the lowest region of the troposphere. In addition, compared with GPT2w, HZWD model is a closed model with a limitation to facilitate on-site meteorological observations. Further research is required to assess the variation and factors influencing the wet delay and to explore the possibility of incorporation of on-site meteorological data.

The ERA-Interim data can be downloaded from ECMWF (Dee et
al., 2011). The radiosonde data can be downloaded from the IGRA
(

YY and YH contributed to the conception of the study. YY contributed significantly to the data analysis and manuscript preparation. YH performed the model validation and wrote the manuscript.

The authors declare that they have no conflict of interest.

The authors would like to thank the ECMWF and IGRA for providing relevant data. We thank the reviewers for their insightful comments and constructive suggestions. This research was supported by the National Key Research and Development Program of China (2016YFB0501803) and the National Natural Science Foundation of China (41574028) and Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University (16-02-03). Edited by: Marc Salzmann Reviewed by: Jakub Kalita and M. Joana Fernandes