Troposphere tomography measurement using a global navigation
satellite system (GNSS) generally consists of several types of input
information including the observation equation, horizontal
constraint equation, vertical constraint equation, and a priori
constraint equation. The reasonable weightings of input information
are a prerequisite for ensuring the reliability of the adjustment
of the parameters. This forms the focus of this research, which
tries to determine the weightings, including the observations, for the
same type of equation and the optimal weightings for
different types of equations. The optimal weightings of the proposed
method are realized on the basis of the stable equilibrium
relationship between different types of a posteriori unit weight
variances, which are capable of adaptively adjusting the weightings
for different types of equations and enables the ratio between the
two arbitrary a posteriori unit weight variances to tend to
unity. A troposphere tomography experiment, which was used to
consider these weightings, was implemented using global positioning
system (GPS) data from the Hong Kong Satellite Positioning Reference
Station Network (SatRef). Numerical results show the applicability
and stability of the proposed method for GPS troposphere tomography
assessment under different weather conditions. In addition, the root mean square (RMS) error in the
water vapor density differences between tomography-radiosonde and tomography-ECMWF (European Centre for
Medium-Range Weather Forecasts)
are 0.91 and 1.63 g m

Tomography is one of the most cost-effective means of obtaining the spatio-temporal three-dimensional distribution of integral measurements within the study region. Since the use of tomography was first publicized (Bramlet, 1978), it has developed as a powerful and practical tool to retrieve information of interest in many application areas, such as geology (Bourjot and Romanowicz, 1992), earthquakes (Kissling et al., 1994) and ionosphere and troposphere modeling (Hajj et al., 1994; Rius et al., 1997; Braun et al., 1999; Flores et al., 2000), wind (Gao et al., 1999).

Global navigation satellite system (GNSS) troposphere tomography usually refers to the three-dimensional reconstruction of water vapor information based on the GNSS signal rays that cross the tomographic modeling area (Flores et al., 2000). The tomographic area is usually discretized into many voxels in the latitudinal, longitudinal, and vertical directions, by which integral observations can be used to calculate the wet refractivity value and water vapor density value of each voxel (Hirahara, 2000; Champollion et al., 2005; Rohm, 2013; Rohm et al., 2014); however, for voxels without signals crossing within the area of interest, which is mainly caused by the unfavorable geometry of ground-based GNSS receivers and the constellation of GNSS satellites, the parameter information value is unavailable. This drawback also leads to a rank deficiency in the tomographic observation modeling. To overcome this ill-posed problem, some constraint conditions were mentioned in previous studies for superposition onto the tomographic modeling (Flores et al., 2000; Hirahara, 2000; Troller, et al., 2002; Skone and Hoyle, 2005; Nilsson and Gradinarsky, 2006; Bender and Raabe, 2007; Perler et al., 2011; Brenot et al., 2014). The most commonly used constraints include a horizontal constraint, vertical constraint, and a priori constraint.

Regardless of the type of constraint condition imposed on the tomographic
modeling, these different types of constraint information need to be
considered as the independent input information, which means that (1) the
reasonable weightings among the observations for the same type of
input information should be considered, (2) and the reasonable unit weight
variances for different types of input equations should also be given
before the tomographic modeling resolution. Flores et al. (2000)
first used the weightings of three constraints (horizontal, vertical,
and boundary constraints) to a constant

The aforementioned methods of selecting the weights of different kinds of input equations usually include a determination of the weights by using an iterative method until the final unit weight variances are equal on the basis of statistics or mathematics; however, some points still need to be discussed as follows: (1) as for the same type of equation, how to determine the weightings among observations was seldom mentioned in the results of previous published studies, and this is important when trying to obtain an accurate tomographic result; (2) the scheme used to select equal weights was unable to embody a reasonable correlation among different types of equations; (3) even although iterative methods were applied to calculate the weights, it may be difficult to obtain a reasonable result due to the various practical conditions in regions subject to tomography; (4) and it may be unreasonable and time-consuming merely from the perspective of mathematics or statistics when determining the optimal weights for various equations. Consequently, in this work, (1) the determination of weights for the same type of input information is presented based on the characteristics of their practical relationship, (2) and for the different types of equations, a weighting algorithm is proposed based on the concept of a stable equilibrium relationship for their a posteriori unit weight variances.

This paper is organized as follows: the basic theory of GNSS troposphere tomography modeling is introduced in Sect. 2. The determination of the weightings is described in Sect. 3. In Sect. 4, the test results from tomographic experiments under various weather conditions (21-day duration) are validated by comparison with radiosonde and ECMWF data. Conclusions and recommendations for future research are provided in Sect. 5.

Previous studies have shown that two kinds of input information derived from GNSS observations are usually considered to build the tropospheric observation equation. On the one hand, the slant wet delay (SWD) was introduced to obtain the troposphere wet refractivity information (Flores et al., 2000; Skone and Hoyle, 2005; Troller et al., 2006; Rohm and Bosy, 2009; Notarpietro et al., 2011; Guo et al., 2016). On the other hand, the slant-integrated water vapor (SIWV) was applied to reconstruct the three-dimensional atmospheric water vapor field (Champollion et al., 2005; Bi et al., 2006; Xia et al., 2013; Jiang et al., 2014; Yao et al., 2016). In our project, we are trying to assimilate the three-dimensional water vapor data into a weather research and forecasting model (WRF), and this paper thus focuses on GNSS water vapor field reconstruction using SIWV data.

Generally, two types of processing modes are adopted for GNSS observations when retrieving tropospheric parameters: the first is the un-differencing mode, which only needs one GNSS receiver to estimate the absolute troposphere parameter, also known as precise point positioning (PPP; Zumbergre et al., 1997). Secondly, GNSS observations can also be processed by using a double-differencing approach, which requires at least two receivers and only the relative troposphere parameter can be obtained for a regional network if additional receivers are not considered (Duan et al., 1996).

In our study, the latter method is applied and the global positioning
system (GPS) observations are processed using GAMIT/GLOBK (v. 10.5; Herring et al., 2010) with the following configuration: the sampling
rate is 30

Therefore, the final SIWV can be obtained based on the following
formula (Bevis et al., 1992):

Following the principle mentioned in previous studies, the tomography
area is discretized into a number of voxels, in which the water vapor
density is assumed to be the same during a given period. Subsequently,
the tomographic observation equation of a single signal ray can be
established:

Geometric illustration of a signal ray crossing the discretized voxels.

Subject to the constellation of GNSS satellites and the specified
distribution of GNSS receivers in a regional network, it is possible that
many voxels are not crossed by any rays during the given period. Therefore,
the observation equation established above is a large sparse matrix that will
lead to a rank deficiency upon inversion of the equation. To overcome this
problem, methods of imposing constraints or using the singular value
decomposition (SVD) can be adopted. In this paper, the former method is used.
A Gauss-weighted functional method (Song, 2004) is applied for the horizontal
inner voxels, while the negative exponential function (Flores et al., 2000)
is established to describe the relationship between vertical voxels. In
addition, the initial value of water vapor density derived from radiosonde
data over the first 3 days in the tomography area is also considered as
a prior constraint on the location of the radiosonde station and the voxels
vertically above it. Therefore, four kinds of input information are used in
our tomographic modeling:

As mentioned above, it is necessary to apply the appropriate weightings to guarantee the reliability and accuracy of the estimated parameters. There are two kinds of weightings to be considered for troposphere tomographic modeling. On the one hand, the weightings for the same type of input information should be determined, which will affect the accuracy of tomographic modeling if the improper weightings are applied. On the other hand, the weightings for different types of equations are also important in the final tomographic result. In the present study, a method considering both of the weightings mentioned above is proposed.

For various types of equations, the methods for determining the
weightings among the same type of input are different. For the
tomographic observation equation, the SIWV value is closely related to
satellite elevation angle, and therefore the weighted function of the cutoff
elevation angle should be considered:

Usually, constraint equations are selected empirically, and the
accurate unit weight variances of different constraint conditions are
unavailable for most cases. Consequently, the appropriate unit
weight variances should be given before the tomographic modeling
resolution. Here, a method is proposed with which to determine the
weightings for different types of equations. The flowchart of the
proposed method is shown in Fig. 2 and the specified steps are as
follows.

Initialize the unit weight variances for various types of
equations as 1, and set initial weight matrix to

Calculate the a posteriori residuals for all types of input equations

Update the unit weight variances of all types of equations

In addition, the updated mean square error of the unit weight is also
exploited to remove outliers from the tomography observation equation
if

The co-integration test (Engle and Granger, 1987) is introduced
to judge whether or not the estimated posterior unit weight
variances are acceptable. This test is based primarily on whether
or not the linear combinations of those unit weight variances are
in a stable equilibrium relationship. The corresponding
co-integration test procedure is as follows.

Establish the relationship for the estimated unit weight
variances. Here, the first-order auto-regression variable sequence
is selected for those variances, and the relationship can be
established as

Calculate the test statistic

Give the null hypothesis and the alternative; due to the accurate weightings used for all types of equations being unavailable on the first time of use, the hypothesis is determined as follows.

Original hypothesis

Alternative hypothesis

Select a proper threshold for

Accept a hypothesis based on the calculated

Update the weight matrices for all types of input equations
based on the given equation if the hypothesis

The final unit weight variances for all types of equations are
determined if the hypothesis

Flowchart of the proposed method considering the weightings for different types of equations.

As mentioned in Sect. 2.2, a tomographic experiment was carried out
based on the data from 12 SatRef stations (Fig. 3) over
21 days (DoY 84–104, 2014). Different weather conditions (sunny
days and rainy days) are included during the selected period. The time
span for each step in the tomographic model solution is
0.5

To analyze the method proposed above, four schemes are designed. Only
one step of the tomography experiment is presented at 00:00–30:00 UTC
for DoY 87 in 2014 (Scheme 1); the whole day tomography experiment is presented for
DoY 87 in 2014 (Scheme 2), the whole day tomography experiment is presented for
DoY 89 in 2014 (Scheme 3), and a 21-day analysis of the tomography
experiment is presented from DoY 84–104 in 2014 (Scheme 4). Two of the days mentioned above
(DoY 87 and 89) are selected as they correspond to the two different
weather conditions assessed; DoY 87 was a sunny day and DoY 89 was
a rainy day with a total precipitation of 115.6

Geographic distribution of the Hong Kong Satellite Positioning Reference Station Network (SatRef) and radiosonde station.

The a posteriori unit weight variances for all types of equations are
first analyzed for Scheme 1. Table 1 lists the numerical results of
troposphere tomographic modeling resolved by using the proposed method,
which includes the number of iterations, the posterior unit weight
variances for four kinds of input information, and the
statistics from the co-integration test. It can be seen from Table 1
that the corresponding statistic is 1.777 after the first iteration,
which means that the unit weight variances for four types of inputs do
not show a stable equilibrium relationship. Therefore, the weightings
of horizontal, vertical, and a priori equations are tuned, while

Posterior unit variances for different input equations and test statistics calculated by using the proposed method for Scheme 1.

Tomography experiments for Schemes 2 and 3 are performed to evaluate
the proposed method on a sunny day (DoY 87) and a rainy day (DoY
89). According to the statistical results from the two days, the
maximum number of iterations is 11 and the minimum is 4 when
the alternative hypothesis

The change in SIWV residuals with cutoff elevation angle for Schemes 2 and 3.

The comparison of SD for Schemes 2 and 3.

Some studies have shown that accurate vertical water vapor
information can be derived from radiosonde data (Niell, 2001; Adeyemi
and Joerg, 2012; Liu et al., 2013); therefore, radiosonde data are
selected as a reference to validate the water vapor density derived
from other methods. Fortunately, there is a radiosonde station in the
research area (red circle, Fig. 3). In addition, ECMWF also provides
meteorological data that can be used to calculate water vapor
density (Böhm et al., 2015; Wang et al., 2016). In our study, the
spatial resolution of ECMWF data (ERA-Interim reanalysis data) for our
study is

Water vapor density profile derived from radiosonde, ECMWF,
and troposphere tomographic results. Time periods in panels

Water vapor density profile derived from radiosonde, ECMWF,
and troposphere tomographic results. Time periods in panels

Statistical results of water vapor density between radiosonde,
ECMWF,
and tomography for Schemes 2 and 3 (unit:

The integrated water vapor (IWV) profiles derived from tomographic
modeling, radiosonde, and ECMWF data were first compared for the
period DoY 84 to 104 in 2014. The IWV value for this radiosonde station
was calculated by vertical integration (Brenot et al., 2006) using the
water vapor information derived from the tomographic result at
a specified epoch: 00:00–00:30 and 12:00–12:30 UTC, at which the sounding balloon is launched
daily. Figure 8 shows a direct IWV comparison between tomography and
radiosonde data. For ECMWF data, only the data at the epoch of 00:00
and 12:00 UTC are selected to unify the temporal resolution with
radiosonde data. Consequently, there are two epochs compared on each
day, and the total number of compared epochs is 42. The IWV derived
from the tomographic modeling using the proposed method provided good
agreement with that from the radiosonde and ECMWF. Numerical results
over 21 days (Table 3) were analyzed: the bias, RMS, and SD are

Comparison of IWV time series derived from radiosonde, ECMWF, and tomography modeling for Scheme 4.

Statistical results of integrated water vapor (IWV) between radiosonde, ECMWF, and tomography for Scheme 4 (unit: mm).

Although the experimental comparison above indicates that the IWV time
series derived from troposphere tomography conforms to that from
radiosonde data, more tests are needed to validate the proposed method
(as such, the compared result may not guarantee that the
three-dimensional water vapor distribution is correctly
modeled; Chen and Liu, 2014). Here, the radiosonde and ECMWF data
are both considered to validate the water vapor distribution at
different altitudes. For ECMWF data, the water vapor information
at different heights for the location of this radiosonde station is
interpolated using the calculated water vapor density across the
12 grid points. The relative error is defined as follows to
evaluate the relationship between the tomographic result error and
height:

Figure 9 shows the water vapor density profiles and the RMS
and relative error change with height between tomography, radiosonde,
and ECMWF data. Figure 9 shows that the mean water vapor density
decreases with altitude, and the same trend can also be seen with
regard to the RMS error. In contrast, the relative error increases
with height (especially above 2

Water vapor density, RMS, and relative error change with height for Scheme 4.

Statistical results of water vapor density between radiosonde,
ECMWF,
and tomography for Scheme 4 (unit:

For the comparisons above, the proposed method with variance component
analysis (VCA) for GNSS tomography has been validated and the
tomographic result is good. To further validate the performance of VCA
proposed in this paper, another comparison is performed between the
results of a GNSS tomography solution based on the VCA and one without VCA
methods for the period of 21 days. The two schemes are defined as the VCA
scheme and the N-VCA scheme (without considering the weightings of
different input information). The reconstructed water vapor density
of voxels over the radiosonde stations using the two schemes are compared
with that from the radiosonde data for the selected period. Figure 10
presents the RMS comparison between the VCA and N-VCA schemes during
the selected 21 days: for most cases, the RMS error derived from the VCA
method proposed in this paper is smaller than that from the N-VCA
method because the VCA can be applied to find a realistic
scenario for different observation types if the accurate information
is unavailable; however, it should be handled with caution since it
tends to increase data scatter in the tomographic solution, which
probably leads to some RMS errors in the VCA scheme being greater than
those of the N-VCA scheme in our experiment (Möller,
2017). According to the statistical results, the bias, RMS, and SD of water
vapor density differences between the VCA scheme and radiosonde data
are 1.28, 1.33, and 0.38

A comparison of RMS for the VCA and N-VCA schemes during the period of 21 days.

For GNSS troposphere tomography, some external constraints are required for the reasons mentioned in Sect. 2. Therefore, reasonable weightings among the observations for the same type of input information and weightings for different types of equations are a prerequisite for obtaining an accurate tomographic result. Consequently, this paper presents a method with which to determine the two types of weightings mentioned above. For the determination of the first type of weightings, some a priori knowledge was used, e.g., the functional relationship between the cutoff elevation angle and time span are implied for the observation equation; the SD of the different vertical heights derived from radiosonde data was introduced to determine the weightings for vertical and a priori constraints. For the weightings for different types of equations in the tomographic modeling, the method ensures that the posterior unit weight variances are in a stable equilibrium based on an iterative process.

A tomographic experiment was carried out to validate the proposed
method using data from 12 stations in the SatRef network and
data from radiosonde and ECMWF. Troposphere parameters were
estimated using a double-differenced method, and the input SIWV for
tomography experiments was calculated as described in
Sect. 2. Different weather conditions (a sunny day and a rainy day)
were both included during tomographic testing and the numerical results
indicated that the proposed method could tune the unit weight
variances of various types of inputs. A comparison of water vapor
densities derived from tomography, radiosonde, and ECMWF showed that
the applicability of the proposed method and the bias, RMS, and SD were
0.090, 1.401, and 1.398

GNSS troposphere tomography considering two kinds of weightings is proposed before tomographic modeling resolution. In this study, although only the GPS-derived SIWVs were used to validate the effectiveness and applicability of the proposed method, it also can be used for other systems (Galileo, GLONASS, and BDS). In addition, more observations can also be used for water vapor tomography, such as data from a radio occultation apparatus, solar spectrometers, a water vapor radiometer (WVR), an interferometric synthetic aperture radar (InSAR), and the Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) radio occultation (Liu et al., 2013; Xia et al., 2013; Alshawaf, 2013; Heublein et al., 2015; Benevides et al., 2015), which can also be regarded as a new type of input information in the proposed method. Hence, more voxels are expected to be crossed and the geometric structure of the observation equation will be enhanced.

The Radiosonde and ECMWF data can be download from the IGRA
(

The authors declare that they have no conflict of interest.

The authors would like to thank IGAR and ECMWF for providing access to the web-based IGAR and layered meteorological data, respectively. The Lands Department of HKSAR is also acknowledge for providing GPS data from the Hong Kong Satellite Positioning Reference Station Network (SatRef) and the corresponding meteorological data. This research was supported by the National Key Research and Development Program of China (2016YFB0501803) and the National Natural Science Foundation of China (41574028). The topical editor, Marc Salzmann, thanks two anonymous referees for help in evaluating this paper.