AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-12-23-2019Atmospheric bending effects in GNSS tomographyAtmospheric bending effects in GNSS tomographyMöllerGregorgregor.moeller@tuwien.ac.athttps://orcid.org/0000-0002-6153-3084LandskronDanielDepartment of Geodesy and Geoinformation, Vienna University of Technology, Vienna, AustriaGregor Möller (gregor.moeller@tuwien.ac.at)3January201912123346July201816July20183December201811December2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://amt.copernicus.org/articles/12/23/2019/amt-12-23-2019.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/12/23/2019/amt-12-23-2019.pdf
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∘ elevation angle, the signal path is well
approximated by a straight line. However, since electromagnetic waves are
prone to atmospheric bending effects, this assumption is not sufficient
anymore for lower elevation angles. Thus, in the following, a mixed 2-D
piecewise linear ray-tracing approach is introduced and possible error
sources in the reconstruction of the bended signal paths are analyzed in more
detail. Especially if low elevation observations are considered, unmodeled
bending effects can introduce a systematic error of up to 10–20 ppm, on
average 1–2 ppm, into the tomography solution. Thereby, not only the
ray-tracing method but also the quality of the a priori field can have a
significant impact on the reconstructed signal paths, if not reduced by
iterative processing. In order to keep the processing time within acceptable
limits, a bending model is applied for the upper part of the neutral
atmosphere. It helps to reduce the number of processing steps by up to 85 %
without significant degradation in accuracy. Therefore, the developed
mixed ray-tracing approach allows not only for the correct treatment of low
elevation observations but is also fast and applicable for near-real-time
applications.
Introduction
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 . Since then, a
variety of tomography approaches based on raw GNSS phase measurements
, double difference residuals , slant
delays or slant integrated water vapor
have been developed for the accurate reconstruction
of the water vapor distribution in the lower atmosphere. An overview about
the major developments within this field of research since
is provided by .
While in most tomography approaches, observations gathered at low elevation
angles are discarded ,
straight-line signal path reconstruction is sufficient for the determination
of the path lengths. However, showed that
especially low elevation observations can be a very useful source of
information in GNSS tomography. In addition to their information content about the
lower troposphere, the additional observations strengthen the observation
geometry and therewith contribute to a more reliable tomography solution.
However, the correct treatment of low elevation observations requires more
advanced ray-tracing algorithms. The first paper which deals with bended ray
path reconstruction in GNSS tomography was published by ,
with a main focus on the reconstruction of the signal paths for delay
estimation but also for the assimilation of GNSS slant delays into numerical
weather prediction systems. Most recently, published
their results about 3-D ray tracing in water vapor tomography and briefly
analyzed its impact on the tomography solution.
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.
Atmospheric bending effects in GNSS signal processing
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
(ϵ) and magnetic permeability (μ) of the atmospheric
constituents and the frequency of the electromagnetic wave. The ratio between
the velocity of light c in a vacuum and the velocity ν in a medium
defines the refractive index n.
n=cν=ϵ⋅μϵ0⋅μ0
For signals in the microwave frequency band, n ranges from 0.9996 to 1.0004.
Thus, n is usually replaced by refractivity N, expressed in
mmkm-1 (ppm).
N=106⋅(n-1)
The GNSS signal delay in the lower atmosphere, also known as slant total
delay (STD), is related to refractivity by the following equation
:
STD=10-6⋅∫RN⋅ds+∫Rds-∫Sds.
The first term of Eq. (3) describes the change in travel time due to velocity
changes along the true ray path R. The second term (about 3 orders of
magnitude smaller than the first term) is related to the difference in
geometrical path length between the true (R) and the chord signal path (S).
According to Dalton's law, the refractivity of air can be split up into a
hydrostatic and a wet component: N=Nh+Nw. Therefore, the GNSS
signal delay reads
STD=SHD+SWD=10-6⋅∫RNh⋅ds+10-6⋅∫RNw⋅ds+∫Rds-∫Sds.
The slant wet delay (SWD) depends on the wet refractivity along the true ray path R.
SWD=10-6⋅∫RNw⋅ds
The slant hydrostatic delay (SHD) results from the hydrostatic
refractivity along R and, by definition, from the additional path length
due to atmospheric bending.
SHD=10-6⋅∫RNh⋅ds+∫Rds-∫Sds
While the signal path S follows from the straight-line geometry between
the satellite and the receiver, the true signal path R depends in addition on the
hydrostatic and the wet refractivity distribution along the signal path (see
Sect. 3 for more details).
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:
STD(ε,α)=SHD+SWD=ZHD⋅mfh(ε)+ZWD⋅mfw(ε)+G(ε,α),
where ZHD is the zenith hydrostatic delay, ZWD is the zenith wet
delay and mfh and mfw are the corresponding mapping functions,
which describe the elevation (ε) dependency of the signal
delay. The elevation-dependent and azimuth (α)-dependent first-order
horizontally asymmetric term G(ε,α) reflects local
variations in the atmospheric conditions; see ,
or . In practice, e.g., when using
the VMF1 mapping function or similar mapping concepts, the
tropospheric delay due to atmospheric bending is absorbed by the hydrostatic
mapping function term mfh. Comparisons between ray-traced SHD(ε) and “mapped” SHD(ε)=ZHD⋅mfh(ε) slant hydrostatic delays reveal that about 97 % of the
atmospheric bending effect is compensated by the VMF1 hydrostatic mapping
function (see Appendix A for further details).
The principles of GNSS tomography
According to , the general principle of tomography is described as follows:
fs=∫Rgs⋅ds,
where fs is the projection function, gs is the object property
function and ds is a small element of the ray path R along which the
integration takes place. In GNSS tomography, gs is usually replaced by
wet refractivity Nw, and integral measure fs by SWD (the
prefactor of 106 vanishes if ds is provided in kilometers and SWD
in millimeters).
SWD=∫RNw⋅ds
A full nonlinear solution of Eq. (9) for wet refractivity is not of
practical relevance since according to Fermat's principle, first-order
changes of the ray path lead to second-order changes in travel time. In
consequence, by ignoring the path dependency in the inversion of Nw
along ds and by assuming the ray path as a straight line, a linear
tomography approach can be defined which is well applicable to SWDs above
a 15∘ elevation angle . However, with decreasing
elevation angle, the true signal path deviates significantly from a straight
line. In consequence, by ignoring atmospheric bending, a systematic error is
introduced in the tomography solution. In order to overcome this limitation,
in the following, an iterative tomography approach is defined in
which the bended signal path is approximated by small line segments. It is similar
to the linear tomography approach in that the neutral atmosphere or parts of
it are discretized in volume elements (voxels) in which the refractivity Nw,k
in each voxel k is assumed as constant. Consequently, Eq. (9)
can be replaced by
SWD=∑k=1mNw,k⋅dk,
where dk is the traveled distance in each voxel. Assuming l
observations and m voxels, a linear equation system can be set up. In
matrix notation it reads
SWD=A⋅Nw,
where SWD is the observation vector of size (l,1),
Nw is the vector of unknowns of size (m,1) and A is a matrix of
size (l,m) which contains the partial derivatives of the slant wet
delays with respect to the unknowns, i.e., the traveled distances dk in
each voxel.
A=δSWD1δNw,1⋯δSWD1δNw,m⋮⋱⋮δSWDlδNw,1⋯δSWDlδNw,m
Solving Eq. (11) for Nw requires the inversion of matrix A.
Nw=A-1⋅SWD
The inverse A-1 exists if A is squared and if the
determinant of A is nonzero, otherwise matrix A is
called singular. Unfortunately, singularity appears in GNSS tomography in
most cases since the observation data are “incomplete” and matrix A
is not of full rank. Therefore, Eq. (13) becomes ill-posed, i.e., not uniquely
solvable. In order to find a solution which preserves most properties of an
inverse, in the following, matrix A is replaced by the pseudo inverse A+.
According to the pseudo inverse is defined
as follows:
A+=V⋅S-1⋅UT,
where U and V are orthogonal, normalized left and right singular matrices of
A
and matrix S is a diagonal matrix, which contains the singular values in
descending order. In case a priori information (Nw0) can be made
available, it enters the tomography solution as first guess as follows:
Nw=Nw0+V⋅S-1⋅UT⋅AT⋅P⋅(SWD-A⋅Nw0),
where matrices U, V and S are obtained by
singular value decomposition of matrix AT⋅P⋅A+Pc. The weighting matrices P and Pc are
defined as the inverse of the variance–covariance matrix C for the
observations and Cc for the first guess, respectively. Assuming
that the observations are uncorrelated, the non-diagonal elements of C
and Cc are zero and the diagonal elements are defined as
follows:
σC2=sin2ε⋅σZWD2σCc2=∂Nw∂T⋅σT2+∂Nw∂q⋅σq2+∂Nw∂p⋅σp2,
whereby σZWD=2.5mm reflects the uncertainty of the ZWD.
The values for σT, σq and σp were taken from
height-dependent error curves for pressure (p), temperature (T) and
specific humidity (q) as provided by for the ECMWF
(European Centre for Medium-Range Weather Forecasts) analysis data. For
further details, the reader is referred to .
Reconstruction of GNSS signal paths
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 . Thereby, the basic
equation for ray tracing, the so-called eikonal equation, has to be solved
for obtaining optical path length L.
||▿L||2=n(r)2
From Eq. (18), a number of 3-D and 2-D ray-tracing approaches have been derived
for the reconstruction of ground-based and space-based GNSS measurements and
of their signal paths through the atmosphere .
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 a, for ground-based observations, elevation and azimuth angles
are used for characterizing the signal geometry. In consequence, the optimal
ray-tracing approach will be significantly different for various observation
geometries.
In order to find an optimal approach for the operational analysis of ground-based
measurements, carried out a number of exploratory
comparisons. Based on the outcome, the 2-D piecewise linear ray tracer was
defined as the optimal reconstruction tool for the iterative reconstruction
of the atmospheric signal delays including atmospheric bending. It is limited
to positive elevation angles but it is fast and almost as accurate as the 3-D
ray tracer. However, for use in GNSS tomography, the ray-tracing approach
had to be further modified. In the following, the developed ray-tracing
approach but also its impact on the GNSS tomography solution are discussed in
more detail.
Piecewise linear ray tracer
The starting point for the 2-D piecewise linear ray tracer is the receiver
position in ellipsoidal coordinates (φ1,λ1,h1), the
“vacuum” elevation angle εk (see Fig. 1) and the azimuth angle
α under which the satellite is observed. In the case of GNSS tomography,
these parameters can be determined with sufficient accuracy from satellite
ephemerides and the receiver position – assuming straight-line geometry.
Geometry of the ray-tracing approach with the geocentric coordinates
(y,z), the geocentric angles (η,θ), elevation angle ε and
d as the distance between two consecutive ray points.
Therefore, the initial parameters for ray tracing (see Fig. 1), i.e., the
geocentric coordinates (y1,z1) and the corresponding geocentric angles
(η1,θ1), read
y1=0z1=RG+h1η1=0θ1=εk,
where RG is the Gaussian radius, an adequate approximation of the Earth
radius:
RG=a2⋅b(a⋅cosφ1)2+(b⋅sinφ1)2,
with a and b as the semi-axes of the reference ellipsoid (e.g., GRS80). The z axis connects the geocenter with the starting point; the y axis
is defined perpendicular to the z axis in direction (azimuth angle) of the
GNSS satellite in view. After setting the initial parameters, the “true” ray
path is reconstructed iteratively by making use of ray-tracing shooting
techniques. Therefore, total refractivity derived from an a priori field is
read in and preprocessed for ray tracing. Here, the input data are
interpolated vertically and horizontally to the vertical plane, spanned by
the y and z axis.
In the ray-tracing loop, for each height layer hi+1 with i=1:(t-1),
whereby t defines the top layer of the voxel model, the
geocentric coordinates and the corresponding angles are computed as follows:
yi+1=yi+di⋅cosεizi+1=zi+di⋅sinεiηi+1=arctanyi+1zi+1θi+1=arccosnini+1⋅cos(θi+ηi+1-ηi)di=-(RG+hi)⋅sinθi+(RG+hi+1)2-(RG+hi)2⋅cos2θiεi+1=θi+1-ηi+1,
where di is the reconstructed path
length between the height layer hi and hi+1 (hi+1>hi). It
depends on the observation geometry but also on the atmospheric conditions
(refractive indices ni and ni+1). By default, for our analysis,
the spacing between two height layers hi and hi+1 was set to 5 m,
which corresponds to a maximum path length di of 100 m – assuming an
elevation angle of 3∘ (5m/sin3∘).
The ray-tracing loop stops when the ray reaches the top layer t of the
voxel model. Assuming spherical trigonometry, the spherical coordinates (φi+1,λi+1)
of the ray path segments are defined as follows:
φi+1=arcsinsinφ1⋅cos(ηi+1-η1)+cosφ1⋅sin(ηi+1-η1)⋅cosαλi+1=λ1+arctansinαcot(ηi+1-η1)⋅cosφ1-sinφ1⋅cosα,
where φi+1 and λi+1 are defined in the range [-π/2,π/2] and [-π,π],
respectively. The ray coordinates are necessary for interpolation of the
refractive indices ni and ni+1 for the next processing step i
but also for computation of the intersection points with the voxel model
boundaries.
The ray-tracing loop is repeated until εt-εk+gbend is smaller than a predefined threshold (e.g., 10-6∘).
While the elevation angle εt is obtained by Eq. (29) for i=t-1,
the correction term gbend accounts for the additional bending
above the voxel model. Since the atmosphere is almost in a state of hydrostatic
equilibrium, gbend can be well approximated by a bending model, like
the one of :
gbend[∘]=0.02⋅exp-h6000tanεk,
where h is replaced by ht, the height of the voxel top layer. After
convergence of the ray-tracing loop, the path length in each voxel is
obtained by summing up the distances di in each voxel. Thereby,
allocation of the ray parts is carried out by comparison of the ray
coordinates (φi,λi,hi) with the coordinates of the voxel
model. The obtained ray paths in each voxel – for each station and each
satellite in view - are used for setting up design matrix A (see
Eq. 12).
Quality of reconstructed ray pathsThe refractivity field
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 (b) caused by differences in the a priori refractivity field (a).
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 (φ=47.4∘, λ=11.8∘, h=545m), with ε=5∘ and α=230∘. At this particular epoch (4 May 2013 at 15:00 UTC),
standard atmosphere deviates by about 30 ppm from the ALARO
model data. Assuming ALARO as the reference, ray tracing through the standard
atmosphere causes a ray deviation of 100–200 m (see Fig. 2b).
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 Nw converges.
Convergence behavior (a) and ray-traced signal path differences
after convergence (b). All iterations are based on the same first guess
(standard atmosphere or ALARO numerical weather model data) but differ with
respect to the refractivity field used for reconstruction of the bended
signal paths.
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 (dk,i+1-dk,i)
were selected as convergence criteria. Both solutions
converge after two iterations, thereby “improving” the path lengths within each voxel by about 22 m in the case of the standard atmosphere and by 11 m in the case of ALARO data. This result was expected, since ALARO data are closer to
the true atmospheric conditions. By comparison of Fig. 3b with
Fig. 2b, it is clearly visible that the two additional iterations
help to reduce the ray offset caused by errors in the standard atmosphere
from 100–200 to 30–40 m. In Sect. 5 the resulting effect on the
tomography solution is assessed.
Point error at the voxel model top (h=13.6km) caused by the
bending model of – computed on a global 10∘×10∘ grid over the period of one year, 2014, by comparison with
ray-traced bending angles based on ECMWF analysis data.
The empirical ray-bending model
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 10∘×10∘
grid over the period of 1 year reveal that the
error in bending is usually kept below 0.8 arcsec. Assuming a GNSS site near
sea level and an elevation angle of 5∘, an error in bending angle
of ±0.8 arcsec causes an error in path length of up to ±10m;
i.e., the reconstructed GNSS signal enters the voxel model slightly earlier or
later than the observed GNSS signal. In Fig. 4, the bending error is
visualized as a pointing error at the voxel model top. However, for the tomography
solution, this effect is too small to be significant. Thus, it can be
concluded that the bending model of is well applicable
for the reconstruction of the bending angle above the voxel model, in particular
if the voxel model height ht is set to 12 km or higher.
Ionospheric bending effects
Beyond, the ionosphere also influences GNSS signal propagation. In order to
assess the impact of free electrons in the ionosphere (above 80 km altitude)
on the signal path, the electron density model by was
executed in three scenarios, assuming a vertical total electron content
(VTEC=∫Ne⋅dh) of 34 TECU (average daytime), 120 TECU (solar
maximum) and 455 TECU (maximum possible; see ), respectively.
N(f)=106⋅-40.2993⋅Nef2
By making use of Eq. (33), the obtained electron density profiles were
converted into profiles of refractivity (N), assuming signal frequency f1=1575.42MHz
(GPS L1) and f2=1227.60MHz (GPS L2). Figure 5 shows
the obtained vertical profiles of ionospheric refractivity as an example for
frequency f1. The higher the signal frequency f, the lower the phase
velocity through the ionosphere and the less its refraction.
Profiles of ionospheric refractivity N(f) assuming signal frequency f=1575.42MHz.
Following the approach by , the ray paths in the
ionosphere were reconstructed separately for GPS L1 and L2. The analysis
revealed significant path differences between the true ray path and its
chord line but also between the two signal frequencies. Assuming a VTEC of
455 TECU and an elevation angle of 3∘, the maximum deviation from
the straight-line signal path is 800 m for L1 and 550 m for L2, respectively,
at h=400km, slightly below the layer of peak electron density.
Fortunately, ray path deviation decreases significantly with decreasing VTEC
and altitude to a few tens of meters at h=13.6km (the upper rim of the
troposphere at which the top of the voxel model was defined). In consequence,
the impact of free electrons on the signal path in the lower atmosphere is
negligible under moderate and low ionospheric conditions.
Impact of atmospheric bending on the tomography solution
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 ni/ni+1 in Eq. (27) was set to 1, thereby guaranteeing that only the impact of atmospheric refraction is assessed.
Expected drying effect
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 (h<13.6km) is systematically longer than
the bended ray path. The differences between both ray paths are plotted in
Fig. 6a as a function of elevation angle. Therefore, ALARO model data were
selected as input for the bended ray tracer.
Additional ray path caused by the straight-line assumption (a), the
resulting drying effect due to the additional ray path (b) and the
resulting drying effect caused by the fact that the straight-line ray travels
through lower atmospheric levels than the true bended ray (c).
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 ε≤15∘. At higher elevation angles, the additional
ray path is below 0.1 km, and straight-line ray tracing is sufficient for ray
path reconstruction.
Figure 6a also shows that in some cases, even for low elevation angles, the difference in path length is small (below 0.1 km). This appears when the
ray enters the voxel model not through the top layer but through the lateral surface of the voxel model. In this particular case, the difference in path
length between both ray-tracing approaches is negligible (be aware that only
the entire distance through all voxels is comparable for both ray-tracing
approaches, not the individual distances in each voxel). Figure 6b and c
show the expected drying effects in the tomography solution caused by errors
in the reconstructed signal paths assuming straight-line geometry. Here, it
is distinguished between the drying effect caused by the additional ray path
(dNw1; panel b) and the drying effect caused by the fact that the straight line
travels through lower layers of the voxel model (dNw2; panel c). Both effects
were assessed as follows:
dNw1=SWDb⋅(dk,s-dk,b)dNw2=(SWDb-SWDs)⋅dk,s,
where SWDb and SWDs are the slant wet delays obtained by
ray tracing through ALARO model data along the bended and the straight-line
ray path, respectively. The variables dk,b and dk,s are the
corresponding path lengths within the voxel model. The sum of their
differences along the ray paths is identical to the additional ray paths
plotted in Fig. 6a.
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.
Results from the Austrian GNSS tomography test case
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.
ParameterSettingsPeriodMay 2013, eight epochs per dayVoxel domainWestern Austria (46.4–48.0∘ lat, 10.4–13.4∘ long.; h=0–13.6 km)Voxel size0.4∘ lat×0.6∘ long. (4×5 ground voxels); 15 height layersGNSS data30 s dual-frequency GPS and GLONASS observations – obtained from sixEPOSA reference sites: SEEF, MATR, JENB, KIBG, ROET, SILLA priori modelALARO analysis data of temperature and specific humidity– provided on 18 pressure levels in grib1 format for eight epochs per dayObservationsSWDs for all GPS and GLONASS satellites in view above a 3∘elevation angle – derived from 1 h ZTD and 2 h gradient estimatesRay tracerSol1: straight-line ray tracing for all observations up to h=13.6kmSol2: straight-line ray tracing for ε>15∘ and bendedray tracing for ε≤15∘ (mixed approach) up to h=13.6km
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 ppm, particularly if different
voxels than in the straight-line solution are traversed due to bending.
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 ppm, which implies
that the a priori model used for ray tracing is already close to the true
atmospheric conditions; i.e., in this particular case no further iteration was
necessary.
Error in wet refractivity caused by the straight-line assumption (a)
and the differences in wet refractivity between first and second iterations
(b). Here, voxel number 1 is dedicated to the southwest corner and
number 20 to the northeast corner of the voxel model. For visualization, a
bilinear interpolation method was applied between the grid points. Analyzed
period: 4 May 2013 at 15:00 UTC.
From all differences in wet refractivity over 248 epochs in May 2013, a
maximum of 14.2 ppm, a bias of 0.12 ppm and a standard deviation of 0.24 ppm
were obtained. Although the bias and the standard deviation over all
voxels are small, differences of about 1 ppm were observed on average at
each epoch, especially when observations below a 10∘ elevation angle
enter the tomography solution.
Validation with radiosonde data
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 (φi=47.3∘, λ=11.4∘, h=579m). First, the radiosonde data
obtained once a day between 02:00 and 03:00 UTC were preprocessed; i.e., outliers in
temperature were removed and dew point temperature was converted to water
vapor pressure and further to wet refractivity. Finally, the radiosonde
profiles were vertically interpolated to the height layers of the voxel model
and the tomography-derived wet refractivity fields were horizontally
interpolated to the ground position of the radiosonde launching site,
respectively. As an example, Fig. 8 shows the differences in wet refractivity as a function
of height above surface for two epochs in May 2013. In both
cases, the bended ray-tracing approach helps to reduce the tomography error
by about 1–2 ppm, especially in the lower 4 km of the atmosphere. Largest
differences are visible when the bended ray traverses other voxels than its
chord line. This appeared in about 2 % of the test cases, especially if
observations below a 10∘ elevation angle enter the tomography solution.
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 (a) and 31 May at 03:00 UTC (b).
Conclusions
GNSS signals which enter the neutral atmosphere at low elevation angles (ε<15∘) are significantly affected by atmospheric
bending. In the case that the bending is neglected when setting up design matrix A,
a systematic error of up to 10–20 ppm, on average 1–2 ppm, is introduced into the GNSS tomography solution. This error can be widely
reduced if atmospheric bending is considered in the reconstruction of the signal
paths. Therefore, a 2-D piecewise linear ray-tracing approach was defined,
which describes the bended GNSS signal path by small line segments. By
limiting the length of the line segments to 100 m in the case of ε=3∘
or even shorter for higher elevation angles, the true signal
path can be widely reconstructed. However, the quality of the reconstructed
signal paths depends primarily on the quality of the a priori refractivity
field. Comparisons between refractivity fields derived from standard
atmosphere and ALARO weather model data reveal that a refractivity error of
30 ppm can cause a ray deviation of up to several hundred meters; i.e., the
distance traveled in each voxel but also the number of traversed voxels are
prone to misallocations. In consequence, reliable a priori data, e.g., derived
from numerical weather model data, are recommended for GNSS tomography.
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
, helps to significantly reduce computational cost by
describing the remaining bending in the higher atmosphere (above the voxel
model). In consequence, the ray tracer can be stopped right after the
reconstructed signal leaves the voxel model. In the case of ht=13.6km, the
number of processing steps is reduced by 85 %, which is a tremendous
reduction in processing time without a significant loss of accuracy.
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 GHz).
However, even if ionospheric bending has no impact on the tomography
solution, first and higher order ionospheric effects should be taken into
account when processing GNSS phase observations.
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 ppm. Within the defined test
case, especially voxels in the lower 4 km of the atmosphere benefitted
from the applied mixed ray-tracing approach. Due to significant optimization,
the mixed ray-tracing approach ensures processing of large tomography test
cases in adequate time. A test case with 72 GNSS sites and 7×9×15 voxels
can be processed in less than 2 min. Thus, the developed mixed
ray-tracing approach is also applicable in near-real time and therefore well
suited for operational purposes.
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
https://github.com/TUW-VieVS/RADIATE (last access: 19 December 2018).
For more details on VieVS, the reader is referred to
http://vievswiki.geo.tuwien.ac.at (last access: 19 December 2018).
Unmodeled bending effects in the Vienna hydrostatic mapping function
In the case of VMF1 or similar mapping concepts, azimuthal
asymmetry is not considered, and for convenience, only a single hydrostatic
mapping coefficient per site (ah) is determined as follows:
ah=-mfh(ε)⋅sinε-1mfh(ε)sinε+bhsinε+ch-11+bh1+ch,
where bh is 0.0029, ch depends on the day of year and latitude and
mfh(ε) is defined as the ratio between SHD(3∘) and
ZHD, obtained by ray tracing through numerical weather model data. For
assessing the remaining unmodeled geometric bending dgbend(ε,α), ray-traced slant hydrostatic delays were compared with mapped
slant hydrostatic delays as follows:
dgbend(ε,α)[m]=ZHD[m]⋅mfh(ε)-ZHD[m]⋅mfh0(ε)-gbend(ε,α)[m],
where ZHD is the zenith hydrostatic delay obtained by vertical
integration, gbend(ε,α) is the geometric bending effect
as obtained by ray tracing, mfh(ε) is the VMF1 hydrostatic
mapping function determined by SHD(3∘)/ZHD and mfh0(ε)
is the hydrostatic mapping function determined by SHD0(3∘)/ZHD,
where SHD(3∘) and SHD0(3∘)
are the slant hydrostatic delays obtained by ray tracing for a vacuum
elevation angle εk=3∘ with and without geometric
bending, respectively. Figure A1 shows the remaining unmodeled geometric
bending as obtained for six elevation angles (and 16 equidistant azimuth
angles) as an example for the two VLBI sites in Fortaleza, Brazil (φ=-3.9∘, λ=321.6∘; h=23m), and Wettzell, Germany
(φ=49.1∘, λ=12.9∘; h=669m).
The unmodeled geometric bending effect in VMF1
hydrostatic mapping function (dgbend),
as an example for VLBI sites in Fortaleza, Brazil, and Wettzell, Germany. Analyzed period: January–February 2014.
In the case of ε=3∘, almost no bending error is visible
since mfh(ε) was tuned for this elevation angle. However, for
other elevation angles, the unmodeled geometric bending is about 3 % of the
slant hydrostatic delay, e.g., up to ±5mm at a 5∘ elevation
angle. In the case of Wettzell, dgbend(ε,α) is mostly
negative; i.e., the mapped SHD is smaller than the observed SHD, and
vice versa for Fortaleza. So far, these small variations have been neglected when
using the VMF1 hydrostatic mapping function in GNSS signal processing.
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.
Acknowledgements
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
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