In Global Navigation Satellite System (GNSS) tomography, precise information about the tropospheric water vapor
distribution is derived from integral measurements like ground-based GNSS
slant wet delays (SWDs). Therefore, the functional relation between
observations and unknowns, i.e., the signal paths through the atmosphere, have
to be accurately known for each station–satellite pair involved. For GNSS
signals observed above a 15

For the conversion of precise integral measurements into 2- or 3-D structures, a technique called tomography has been
invented. In the field of GNSS meteorology, the principle of tomography
became applicable with the increasing number of Global Navigation Satellite
System (GNSS) satellites and the build-up of densified ground-based GNSS
networks in the 1990s

While in most tomography approaches, observations gathered at low elevation
angles are discarded

Based on the existing studies, in the following, a more detailed discussion of possible error sources in signal path reconstruction is provided. Therefore, Sect. 2 describes the effect of atmospheric bending and its handling in GNSS signal processing. Section 3 describes the principles of GNSS tomography and how the basic equation of tomography is solved for wet refractivity. Section 4 introduces the concept of the reconstruction of signal paths using ray-tracing techniques. Here, the modified piecewise linear ray-tracing approach is described – including its ability for reconstruction of the GNSS signal geometry. In Sect. 5, the defined ray-tracing approach is applied to real slant wet delays (SWDs) and its impact on the tomography solution is assessed and validated against radiosonde data. Section 6 concludes the major findings.

The effect of atmospheric bending on GNSS signals is related to the
propagation properties of electromagnetic waves. In a vacuum, GNSS signals
travel at the velocity of light. When entering into the atmosphere, the
electromagnetic wave velocity changes, dependent on the electric permittivity
(

In GNSS signal processing, the integral along the signal path is usually
replaced by the zenith delay and a mapping function. Therefore, Eq. (4) is rewritten as follows:

According to

Assuming that the geometrical optics approximation is valid and that the
atmospheric conditions change only inappreciably within one wavelength, the
signal path is well reconstructible by means of ray-tracing shooting
techniques

The main difference between both observation types is related to the
observation geometry. While for space-based GNSS observations derived from limb
sounding, the bending angle is usually described as a function of impact
parameter

In order to find an optimal approach for the operational analysis of ground-based
measurements,

The starting point for the 2-D piecewise linear ray tracer is the receiver
position in ellipsoidal coordinates

Geometry of the ray-tracing approach with the geocentric coordinates

Therefore, the initial parameters for ray tracing (see Fig. 1), i.e., the
geocentric coordinates

In the ray-tracing loop, for each height layer

The ray-tracing loop stops when the ray reaches the top layer

The ray-tracing loop is repeated until

The quality of the ray-traced signal paths depends primarily on the quality of the refractivity field. Especially if no good a priori data can be made available, e.g., if standard atmosphere (StdAtm) is used instead of numerical weather model data (ALARO), the reconstructed signal path might deviate significantly from the true signal path.

Ray-traced signal path differences

Figure 2 shows the impact of the refractivity field on the signal geometry
as an example for a GNSS signal observed at Jenbach station, Austria

In order to reduce the impact of possible refractivity errors on the
reconstructed ray paths and in further consequence on the tomography
solution, ray tracing was carried out iteratively. Therefore, the
refractivity field obtained from the first tomography solution replaces the
initial refractivity field for ray tracing for the next iteration and so on.
The processing is repeated until

Convergence behavior

Figure 3a shows the convergence behavior assuming standard atmosphere
(StdAtm) and ALARO model data as input. In both cases, the standard
deviations
of the differences in path length between two consecutive epochs (

Point error at the voxel model top (

In addition to the refractivity field, the quality of the reconstructed ray paths
might also be affected by errors in the bending model as defined by Eq. (32).
Comparisons of the bending model with ray-traced bending angles on a global

Beyond, the ionosphere also influences GNSS signal propagation. In order to
assess the impact of free electrons in the ionosphere (above 80

Profiles of ionospheric refractivity

Following the approach by

In the following, the differences between straight-line and bended
ray tracing are further analyzed. For a high degree of consistency, the ray tracing
approach defined in Sect. 4 was used for both straight-line and bended
ray tracing. The only difference is that in the case of straight-line ray tracing
the ratio

In the beginning, the ray position is equal for both methods but diverges
with increasing height. Thereby, in most cases, the bended ray is traveling
“above” the straight ray; i.e., the straight ray enters the voxel model top
“earlier” than the bended ray. This leads to the effect that the straight ray
remains in the voxel model longer than the bended ray; i.e., the straight ray
path within the voxel model

Additional ray path caused by the straight-line assumption

The additional ray path decreases rapidly with increasing elevation angle.
Thus, a mixed ray-tracing approach can be defined, which considers
ray bending only for

Figure 6a also shows that in some cases, even for low elevation angles, the difference in path length is small (below 0.1

Both drying effects have to be considered as additive and are strongly connected to the current atmospheric conditions as well as to the parametrization applied for interpolation of the refractivity field. In our analysis we assumed an exponential decrease of refractivity between the vertical layers of the voxel model and applied a bilinear interpolation method for horizontal interpolation between the grid points.

In order to study the impact of bended ray tracing on the tomography solution, a GNSS tomography test case was defined. The corresponding settings are summarized in Table 1.

Summary of GNSS tomography test case settings.

Figure 7a shows the differences in wet refractivity between Sol1 and
Sol2 (as defined in Table 1). Even though on average over all voxels no bias
in wet refractivity is observed, specific voxels show differences in wet
refractivity of up to 10

Figure 7b shows the differences in wet refractivity between the first
two iterations of the mixed ray-tracing approach (Sol2). In this particular
case, refractivity differences are smaller than 0.05

Error in wet refractivity caused by the straight-line assumption

From all differences in wet refractivity over 248 epochs in May 2013, a
maximum of 14.2

For validation of the mixed ray-tracing approach against straight-line
ray tracing, the tomography-derived wet refractivity fields were compared
with radiosonde data at the airport of Innsbruck (

Differences in wet refractivity between radiosonde, ALARO and the
two tomography solutions based on straight-line (blue) and bended ray tracing
(red) as an example for 1 May 2013 at 03:00 UTC

GNSS signals which enter the neutral atmosphere at low elevation angles (

Nevertheless, if reliable a priori data are not available or if the quality
is unknown, iterative ray tracing helps to reduce the impact of wet
refractivity errors on the tomography solution. Therefore, the wet
refractivity field obtained from an initial tomography solution is used for
reconstruction of the signal paths for the next iteration. The processing is
repeated until the tomography solution converges. This ensues usually after
two iterations. Further, a bending model, like the one provided by

In contrast, ionospheric bending effects have less impact on the GNSS
tomography solution. Even during periods of solar maximum, ray path deviation
caused by ionospheric bending is negligible for signals in the L band (1–2

In addition, comparisons with radiosonde data revealed that if atmospheric
bending effects are considered in GNSS tomography, the quality of the
tomography solution can be improved by 1–2

The 2-D piecewise linear ray tracer for GNSS tomography as well as
the RADIATE ray tracer are part of the Vienna VLBI and Satellite Software (VieVS).
The code of the RADIATE ray tracer is available at

In the case of VMF1

The unmodeled geometric bending effect in VMF1
hydrostatic mapping function (

In the case of

GM, as main author, did most of the analysis, drafted the manuscript and designed the figures. DL provided the ray-traced delays based on the RADIATE ray tracer, contributed to the definition of the ray-tracing approach for GNSS tomography and gave feedback on the draft.

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.

Open access funding was provided by the Austrian Science Fund (FWF). The authors would like to thank the Austrian Science Fund (FWF) for financial support of this study within the project RADIATE ORD (ORD 86) and the Austrian Research Promotion Agency (FFG) for financial support within the project GNSS-ATom (840098). Edited by: Jonathan Jones Reviewed by: two anonymous referees