The determination of the distribution of water vapor in the atmosphere plays an important role in the atmospheric monitoring. Global Navigation Satellite Systems (GNSS) tomography can be used to construct 3-D distribution of water vapor over the field covered by a GNSS network with high temporal and spatial resolutions. In current tomographic approaches, a pre-set fixed rectangular field that roughly covers the area of the distribution of the GNSS signals on the top plane of the tomographic field is commonly used for all tomographic epochs. Due to too many unknown parameters needing to be estimated, the accuracy of the tomographic solution degrades. Another issue of these approaches is their unsuitability for GNSS networks with a low number of stations, as the shape of the field covered by the GNSS signals is, in fact, roughly that of an upside-down cone rather than the rectangular cube as the pre-set. In this study, a new approach for determination of tomographic fields fitting the real distribution of GNSS signals on different tomographic planes at different tomographic epochs and also for discretization of the tomographic fields based on the perimeter of the tomographic boundary on the plane and meshing techniques is proposed. The new approach was tested using three stations from the Hong Kong GNSS network and validated by comparing the tomographic results against radiosonde data from King's Park Meteorological Station (HKKP) during the one month period of May 2015. Results indicated that the new approach is feasible for a three-station GNSS network tomography. This is significant due to the fact that the conventional approaches cannot even solve a network tomography from a few stations.

Information of the distribution and variation of atmospheric water vapor is essential for meteorological applications. Nowadays, the most commonly used technology for measuring atmospheric water vapor is radiosonde due to its high vertical resolution and high accuracy, even though its horizontal resolution is very low – several hundreds of kilometers, and its temporal resolution is also low – twice daily. With the development of Global Navigation Satellite Systems (GNSS), using GNSS measurements to remotely sense water vapor in the atmosphere has attracted significant attention due to their 24 h availability, global coverage and low cost. Based on GNSS measurements collected from a regional or global GNSS reference network, a regional or a global tomographic model, which is three-dimensional (3-D), can be constructed. The tomographic model reflects the spatial variation of water vapor in the time period investigated, thus it has the potential to be used to investigate the evolution of heavy rain events for severe weather forecast (Wang et al., 2017; Chen et al., 2017; Zhang et al., 2015).

Using the slant wet delays (SWDs) estimated from the GNSS signals of a GNSS
network to construct a tomographic model is called GNSS tomography. Flores
et al. (2000) built the first GNSS tomographic model using 4

In all the above tomographic approaches, the tomographic fields are all assumed rectangular cubes. The size and location of the rectangular cubes are determined based on the distribution of GNSS signals only on the top boundary of the tomographic field – the rectangular cube that best fits the top boundary is adopted (Bastin et al., 2005; Bender et al., 2009; Champollion et al., 2005; Ding et al., 2017; Gradinarsky and Jarlemark, 2004; Hoyle, 2005; Rohm et al., 2014; Seko et al., 2000; Troller et al., 2006; Xia et al., 2013; Ye et al., 2016). In fact, the field that GNSS signals cover has roughly the shape of an upside-down cone, meaning that in the part near the edge of the cube, especially in the lower part, none of the GNSS signals cross through. This region is named the empty spatial region (ESR) in this paper merely for convenience. In fact, the inclusion of those voxels or nodes in the ESR in the discretization of the model not only contributes little to the improvement on the accuracy of the model solution but also adds extra meaningless unknown parameters to be estimated. More parameters mean more horizontal constraints are needed and also degradation of the accuracy and stability of the solution, especially in the case the network consists of a few stations, e.g., only three stations. This is because the difference in the sizes covered by the GNSS signals in the bottom and top planes of the tomographic field is large, meaning a large number of voxels or nodes in the ESR and far away from the observed signals, especially in the lower part of the tomographic field. In the estimation process of the model, the horizontal constraints imposed on these nodes or voxels are usually from extrapolated results based on their nearest observations. If these voxels or nodes are far away from the observed signals, the constraints are too weak and will cause difficulty in the solving of the tomographic equations. The large number of nodes or voxels contained in ESRs, stemming from a small number of GNSS stations is the main reason for the unsuitable of the current GNSS tomographic approaches to using a-few-station networks.

In this study, a new node parameterization approach for dynamic determination of tomographic fields and the discretization of the fields at each tomographic epoch was proposed. It is adaptive node parameterization for varying density on different tomographic planes. This differs from all current approaches in which the same pre-set rectangular cube roughly determined by the distribution of the signals only on the top tomographic plane is adopted for all planes and all epochs of the tomography. In addition, for the discretization of the tomographic field determined for each plane at each epoch, the location and number of all the nodes on the plane are determined according to the size of the tomographic field. As a result, the tomographic model is tailor-made for all planes and all epochs. Moreover, the new approach is applicable to GNSS networks with any number of stations, i.e., equal to or larger than three.

GNSS signals are bent and delayed when they propagate through the
atmosphere. The atmosphere can be divided into the ionosphere and
troposphere. The first order ionospheric delay was eliminated using the
so-called ionosphere-free linear combination of dual-frequency observations.
The tropospheric delay can be divided into two components – the dry delay
and the wet delay. The wet component is the SWD and can be expressed by

In GNSS tomographic modeling, the SWDs of GNSS signals in a tomographic field are used as the observations for the estimation of water vapor parameters in the field.

Voxel and node parameterization are the two common GNSS tomographic approaches. In the former, the tomographic field, which is usually assumed as a rectangular cube, is divided into many voxels (small rectangular cubes) and in the latter, and the field is discretized by nodes, as all the black and circle nodes shown in Fig. 1. In this study, the node parameterization approach was adopted due to its better fitting of the spatial correlation of water vapor.

In the current node parameterization approaches, if the GNSS network is very small, e.g., a three-station network from the Hong Kong Satellite Positioning Reference Station Network (SatRef) as shown in Fig. 1, a large number of nodes are in the ESR (see the hollow circles) within the rectangular cube that is the tomographic field. These nodes, as part of the unknown parameters, need to be estimated. The inclusion of these unknown parameters in the estimation process does not only add more “redundant” parameters but also degrades the accuracy of the solution.

A three-station GNSS network from the Hong Kong Satellite Positioning Reference Station Network (SatRef) as an example for GNSS tomography – the rectangular cube is the tomographic field adopted in current node parameterization approaches, the solid nodes are those near GNSS signals and the hollow nodes are those in the ESR.

Distributions of GNSS signals at the three stations shown in Fig. 1 on the top plane of the tomographic field with the sampling rate of 30 s at 00:00 UTC on 1, 16 and 31 May 2015.

In addition, a fixed rectangular cube is used as the tomographic field for all time in the current approaches. In fact, the spatial region that the signals travel through varies with time, as shown in Fig. 2 for the different distributions of the signals at the three stations shown in Fig. 1 on the top plane of the tomographic field at 00:00 UTC on 1 (day of year, DOY) 121), 16 (DOY 136) and 31 (DOY 151) in May 2015.

To address the above issues, a new node parameterization approach that dynamically adjusts the tomographic field based on the spatial distribution of the GNSS signals at the tomographic epoch and also dynamically adjusts the location and number of all the nodes based on the size of the tomographic field is proposed. Its procedure is elaborated in the next section.

The procedure for the new approach mainly includes two steps: determination of tomographic field and determination of node position, which are introduced below.

A tomographic field is regarded to be comprised of many layers in the vertical dimension and these layers with the same or different thickness, depending on the distribution of water vapor at the height of the layer, as shown in Fig. 3a, each layer is formed by two neighboring horizontal planes. After all these planes are determined, the next task is to determine the tomographic boundary for each plane, according to the distribution of the GNSS signals on the plane. Figure 3b shows the tomographic boundary on each of the planes shown in Fig. 3a, which is determined from the following three steps that were used in the Graham scan (Graham, 1972), determining all the intersections (the blue points) of the GNSS signal paths on the plane (referred to as pierce points in this paper); (2) using a stack of the pierce points to detect and remove all those pierce points that are in concavities and (3) connecting the rest pierce points to form a convex hull, which is the tomographic boundary (black polygon).

Since the shape of the tomographic boundary determined using the new approach is irregular, it is difficult to generate equidistant nodes within the boundary. This differs from current node parameterization approaches in which uniformly distributed nodes can be easily pre-set. In this study meshing techniques are used to adjust the position of nodes for each plane and each tomographic epoch, and their procedure is discussed in the next section.

Meshing techniques for the generation of equidistant nodes of a GNSS
tomographic model include three steps and each of the steps are introduced
below.

A mesh background in a desired size with nodes is used to provide initial nodes for each plane (see Fig. 4a), in which the polygon is obtained from the last section for the tomographic boundary on the plane and at all the vertices of the polygon a new set of nodes are also attached to the initial nodes; see Fig. 4b for the final initial nodes.

Delaunay triangulation (Delaunay, 1934) is used to establish a topology for the above initial nodes on each plane. It determines non-overlapping triangles that fill the region in a polygon such that every edge is shared by at most two triangles and none of the vertices is inside the circumcircle of any of the triangles. Delaunay triangulations maximize the minimum angle of all the triangles to avoid sliver triangles which has undesirable properties during some interpolation or rasterization processes (Edelsbrunner et al., 2000). Several methods have been developed to compute the Delaunay triangulation such as the commonly used flipping edges and conversing a Voronoi diagram. In this study, the flipping edges method is adopted to connect the initial nodes shown in Fig. 4b, by the edges of Delaunay triangles on each plane and the topology formed is shown in Fig. 4c.

The force displacement algorithm (Persson, 2005) is applied to the above
topology for the adjustment of the initial nodes into equidistance with a
reasonable length fitting the size of the tomographic boundary on each
plane. This method is based on the assumption that each edge in the topology
has a force value (let it be

where

It is noted that the sizes of the tomographic boundaries on different planes
are different (Fig. 3b) while the numbers of the signals on different
planes are the same, so the densities of the signals on different planes are
different, and the densities of the nodes on different planes better be
different through using different

After equidistant nodes for all planes are determined (like Fig. 4d), the next step is to estimate water vapor parameters at these nodes from observation equations of GNSS-derived SWDs. The derivation of the observation equations is described below.

Theoretically, SWD is defined as the integral of wet refractivity

In GNSS tomography, in each of the piecewise integrals expressed in Eq. (5),
i.e., SWD

The methods for obtaining wet refractivity at each of the points are as follows.

Five equally spaced points (black solid squares) for an
approximation of wet refractivity for the

Wet refractivity at points P

Wet refractivity at points P

Tomographic boundary and nodes on three planes

RMSE of model-derived water vapor density values at all RS sampling points of the RS profile below 10 800 m at tomographic epochs 00:00 and 12:00 UTC on each day of the month (DOYs 121–151).

After the above procedures are carried out, SWD

The final GNSS tomographic observation equations of all SWDs from the GNSS
network for the tomographic modeling is expressed as follows:

The

The ART has been successfully applied to reconstruction of water vapor field
(Chen and Liu, 2014; Bender et al., 2011). Its main advantage is the high
numerical stability, even under adverse conditions and also relatively easy
to incorporate prior knowledge into the reconstruction process. The ART used
to solve Eq. (9) is (Kaczmarz, 1937) as follows:

It is noted that Eq. (10) needs to be sorted in a certain sequence for Eq. (10). This is different from the commonly used observation equation system in which the order of the observation equations is not a matter. In this study, an access order scheme based on prime number decomposition (PND), proposed in Ding et al. (2017), was used for the ordering of the observation equations such that the observation equations between two consecutive iterations are largely uncorrelated.

The unknown parameters

Tomographic field and signal distribution at tomographic
epoch 00:00 UTC on DOY 121

Monthly statistics of new approach and ANP.

Graphic presentation for the distribution of the
tomographic results at the two epochs on every day during the month:

Test data used in this study were from three stations in the Hong Kong
Satellite Positioning Reference Station Network (SatRef), and the horizontal
and vertical distributions of the three stations are presented in Fig. 6a
and b, respectively. The area of our interest ranges from
113.749 to 114.474

The test data were from the whole month of May 2015 (day of year (DOY)
121–151) with the sampling rate of 30 s, and the GAMIT software was
used to obtain SWDs at the same rate in the data processing. For the
tomographic modeling, a 5 min sampling rate for SWDs and a 30 min
interval for a tomographic epoch were adopted, meaning that SWD observations
from seven epochs stacked to one tomographic modeling interval were used
– including the two sample data at the two ends of the interval. The
reason for the selection of data from May 2015 is that its monthly total
rainfall was 513.0 mm, 68 % larger than the normal level of 304.7 mm.
The weather in Hong Kong was hot on the first few days of the month. After a
cloudy but relatively rain-free day on 8 May, another trough of low pressure
brought heavier showers and thunderstorms to Hong Kong on 9–10 May. Two
rainstorm episodes on 20 and 23 May brought rain to most parts of the Hong
Kong. Another rapidly developed rainstorm occurred on 26 May. The weather
improved gradually with sunny periods on 28–30 May. However, the weather
turned cloudy again with isolated showers and thunderstorms on 31 May. The
tomographic scheme for testing is as follows. The first step is to determine
the vertical planes/layers for the tomographic field. Non-uniform vertical
intervals from 300 to 3800 m (Fig. 6b) were selected for adaption to the
inherent characteristic of water vapor spatial distribution – it
exponentially decreases with the increase of height. The use of this
structure can also avoid too many unknown parameters overfitting the SWD
observations. The next step is to determine the tomographic polygon/boundary
on each of the above planes using the methods in Sect. 2.2.1 and based on
the GNSS signals in the tomographic interval, then according to the
polygon's perimeter, a

Figure 7 shows the boundary and nodes on three tomographic planes at tomographic epoch 00:00 UTC on DOY 121, 2015 for an example. Tomographic model results are presented in the next section.

Characteristic values of the box plots in Fig. 10b.

Water vapor density values obtained from the tomographic models at tomographic epochs 00:00 and 12:00 UTC on each day of the month (DOYs 121–151) were compared against radiosonde (RS) data for evaluation of the model's accuracy. The values of the tomographic results at all RS sampling points were calculated first using the interpolation method mentioned in Sect. 2.2.1, then the root-mean-square-error (RMSE) of the differences between the interpolated values and RS observations at all the sampling points of the RS profile from the ground surface to 10 800 m at each epoch was calculated for the accuracy of the profile. All the results at the 62 epochs during the 31 day period are shown in Fig. 8.

The maximum RMSEs, i.e., the worst results, at 00:00 (red) and 12:00 UTC (blue) are on DOY 143 and DOY 150, respectively; while the best result (the minimum RMSEs) at the two epochs are on DOYs 121 and 146. In order to find the reason for the large difference between the worst and best results, the tomographic field, the distribution of the signals and the nodes at these four epochs are given in Fig. 9, where Fig. 9a and b correspond to the best results at 00:00 and 12:00 UTC, respectively, both of which show uniform distributions of the GNSS signals. However, the distributions of the GNSS signals corresponding to the worst results at 00:00 (Fig. 9c) and 12:00 UTC (Fig. 9d) are different in the sparse signals shown in blue lines, which is one of possible reasons for the poor accuracy of the model results.

The results shown in Fig. 8 are the statistics of the model results for each epoch on each day. In Table 1, the statistics of the model results at both epochs together in the whole month are compared with that of the adaptive node parameterization approach (ANP) (Ding et al., 2018) during the same periods. Unlike the results of new approach are based on three stations of SatRef, 17 stations of SatRef are used to estimate the results of the ANP. The RMSE and IQR values of new approach are similar to that of the ANP, meaning that the new approach for a few GNSS stations, such as three stations, is feasible. But in terms of the Bias, the new approach has a poor performance.

To indicate the spread of all the errors (the ones used to calculate the
above monthly statistics), scatter plots shown in Fig. 10a are used to
analyze the characteristics of these errors in different intervals. The

How well all the hollow circles “fit” the red line indicates the overall
quality of the model results. It is clear that the hollow circles have a
cigar-shaped (fusiform) distribution. The hollow circles in both ending
intervals ([0–5] and [20–25] g m

The box plot is mainly for the indication of those large errors at all
sampling points. Q1 and Q3, which are the first and third quartiles,
respectively, determine the IQR value in Table 1; Q2, the second quartile,
roughly reflects the bias of all the errors; the whiskers, i.e., the two
black bars, located at Q1

In the last section, the RMSE of model-derived water vapor density values at all sampling points for each profile (Fig. 8) and the errors at all the sampling points and two epochs on each day during the month (Fig. 9) are analyzed for the assessment of the overall performance of the models. In this section, the monthly RMSE at all the sampling points but in 11 different tomographic layers and the monthly mean of the relative errors in these layers are investigated, see Fig. 11.

In those layers below 1500 m, the two lines in both subfigures show the same tendency of variation with height – the error value increases with the increase of height. This is because the higher the layer, the more the spread of the GNSS signals, the worse the accuracy of the result. However, in the layers above 1500 m, the two lines show opposite tendencies of variation with height because the higher, the smaller the water vapor density. The smaller water vapor density values in these high layers lead to the small RMSE and large relative errors.

In this study a new node parameterization approach for determination of a tomographic field based on the distribution of GNSS signals at the tomographic epoch and also for discretization of the tomographic field is proposed. The number and the position of the nodes on each tomographic plane are determined based on the perimeter of the tomographic boundary on the plane and meshing techniques, respectively. Since the tomographic model is tailor-made for the tomographic field at the epoch, the new approach is applicable to not only GNSS networks with several stations, but also GNSS networks with few stations, e.g., three stations, which cannot be solved by conventional approaches. The new approach was tested using GNSS data from three stations in the Hong Kong Satellite Positioning Reference Station Network during the period of May 2015 and its model results were validated by comparing them against radiosonde data at 00:00 and 12:00 UTC from HKKP. Results suggest that the new approach is feasible for a three-station GNSS network. In addition, monthly statistics of the tomographic results on each tomographic layer indicated that the size of the tomographic boundary and the magnitude of water vapor are two critical factors affecting the accuracy of the tomographic result of the layer.

Our future work will be focusing on using unevenly distributed nodes that fit the density of the GNSS signals.

GNSS data in the RINEX format used for this study can be downloaded from
website (

ND contributed to conceptualization, data curation, formal analysis, investigation, methodology, resources, software, validation, visualization, writing the original draft and reviewing and editing.

SZ contributes to conceptualization, formal analysis, funding acquisition, methodology, project administration, resources, supervision, writing the original draft, reviewing and editing.

SW contributes to formal analysis, supervision, writing the original draft, reviewing and editing.

XW contributes to conceptualization, formal analysis, investigation, methodology, resources, validation, writing the original draft, reviewing and editing.

AK contributes to formal analysis, methodology, resources, supervision, writing the original draft.

KZ contributes to conceptualization, formal analysis, funding acquisition, investigation, project administration, resources, supervision, writing the original draft, reviewing and editing.

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 does not belong to a conference.

This study is supported by Double World-class Construction of Independent Innovation Project of China (No. 2018ZZCX08). The authors acknowledge the Survey and Mapping Office (SMO) of Lands Department, Hong Kong for provision of GNSS data from the Hong Kong Satellite Positioning Reference Station Network (SatRef). We also thank King's Park Observatory for provision of radiosonde data, and the Department of Earth Atmospheric and Planetary Sciences, MIT for the GAMIT/GLOBK software. The editor and reviewer team is also highly appreciated for their valuable comments, which makes great improvements in the quality of the paper. Edited by: Dietrich G. Feist Reviewed by: two anonymous referees