remote sensing
Article
Impact of Tropospheric Mismodelling in GNSS Precise Point
Positioning: A Simulation Study Utilizing Ray-Traced
Tropospheric Delays from a High-Resolution NWM
Florian Zus 1,*, Kyriakos Balidakis 1, Galina Dick 1, Karina Wilgan 1,2 and Jens Wickert 1,2
Citation: Zus, F.; Balidakis, K.; Dick,
G.; Wilgan, K.; Wickert, J. Impact of
Tropospheric Mismodelling in GNSS
Precise Point Positioning: A
Simulation Study Utilizing
Ray-Traced Tropospheric Delays from
a High-Resolution NWM. Remote
Sens. 2021,13, 3944. https://doi.org/
10.3390/rs13193944
Academic Editor: Chung-yen Kuo
Received: 12 August 2021
Accepted: 29 September 2021
Published: 2 October 2021
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1GFZ German Research Centre for Geosciences, 14473 Potsdam, Germany;
[email protected] (J.W.)
2Institute of Geodesy and Geoinformation Science, Technische Universität Berlin, 10623 Berlin, Germany
*Correspondence: [email protected]
Abstract:
In GNSS analysis, the tropospheric delay is parameterized by applying mapping functions
(MFs), zenith delays, and tropospheric gradients. Thereby, the wet and hydrostatic MF are derived
under the assumption of a spherically layered atmosphere. The coefficients of the closed-form
expression are computed utilizing a climatology or numerical weather model (NWM) data. In this
study, we analyze the impact of tropospheric mismodelling on estimated parameters in precise
point positioning (PPP). To do so, we mimic PPP in an artificial environment, i.e., we make use
of a linearized observation equation, where the observed minus modelled term equals ray-traced
tropospheric delays from a high-resolution NWM. The estimated parameters (station coordinates,
clocks, zenith delays, and tropospheric gradients) are then compared with the known values. The
simulation study utilized a cut-off elevation angle of 3
◦
and the standard downweighting of low
elevation angle observations. The results are representative of a station located in central Europe
and the warm season. In essence, when climatology is utilized in GNSS analysis, the root mean
square error (RMSE) of the estimated zenith delay and station up-component equal about 2.9 mm
and 5.7 mm, respectively. The error of the GNSS estimates can be reduced significantly if the correct
zenith hydrostatic delay and the correct hydrostatic MF are utilized in the GNSS analysis. In this case,
the RMSE of the estimated zenith delay and station up-component is reduced to about 2.0 mm and
2.9 mm, respectively. The simulation study revealed that the choice of wet MF, when calculated under
the assumption of a spherically layered troposphere, does not matter too much. In essence, when
the ‘correct’ wet MF is utilized in the GNSS analysis, the RMSE of the estimated zenith delay and
station up-component remain at about 1.8 mm and 2.4 mm, respectively. Finally, as a by-product of
the simulation study, we developed a modified wet MF, which is no longer based on the assumption
of a spherically layered atmosphere. We show that with this modified wet MF in the GNSS analysis,
the RMSE of the estimated zenith delay and station up-component can be reduced to about 0.5 mm
and 1.0 mm, respectively. In practice, its success depends on the ability of current (future) NWM to
predict the fourth coefficient of the developed closed-form expression. We provide some evidence
that current NWMs are able to do so.
Keywords:
GNSS precise point positioning; atmospheric remote sensing; numerical weather model;
simulation study
1. Introduction
In GNSS analysis the signal travel time delay induced by the neutral atmosphere
between the satellite and the station, known as tropospheric delay, is approximated by uti-
lizing mapping functions (MFs), zenith delays, and tropospheric gradient components [
1
].
The so-called hydrostatic and wet MF (the ratio of slant and zenith delays) are derived
under the assumption of a spherically layered atmosphere. In order to take into account
Remote Sens. 2021,13, 3944. https://doi.org/10.3390/rs13193944 https://www.mdpi.com/journal/remotesensing
Remote Sens. 2021,13, 3944 2 of 18
the deviation from a spherically layered atmosphere, a dedicated gradient MF and gradient
components are introduced into the tropospheric delay model. The parameters of the tropo-
spheric delay model are derived from a climatology or numerical weather model (NWM).
In one of the most demanding GNSS applications, known as precise point positioning
(PPP) [2], the station coordinates are estimated with cm-level accuracy. To reach this level
of accuracy in the positioning domain, the inaccuracy of the underlying climatology or
NWM must be taken into account. To do so, some of the parameters in the tropospheric
delay model, i.e., the zenith delay and gradient components, are estimated together with
the station coordinates (the coefficients of the MFs are kept fixed). Thus, the output of PPP
is two products: station coordinates and respective tropospheric parameters. This implies
that PPP can be regarded as a tool for atmospheric remote sensing. The zenith delays can
be related to the precipitable water vapor (PWV) above the considered station [
3
], and the
gradient components can be roughly related to the (first-order) horizontal PWV gradient at
the respective stations [
4
]. These relations explain the interest of meteorology and climate
research in GNSS-based atmospheric data [5].
The GNSS is all-weather; i.e., the station coordinates and the respective tropospheric
parameters are obtained under all weather conditions. This is a clear advantage over other
atmospheric remote sensing instruments. However, the accuracy of GNSS estimates must
be weather dependent to some extent; the reason being the inaccuracies in the tropospheric
delay model. First, the functional form of the tropospheric delay model is inaccurate. This
can be recognized when ray-traced tropospheric delays, especially those derived from a
high-resolution NWM, are compared with parameterized tropospheric delays. Second, let
us assume for a moment that the functional form of a tropospheric delays is correct; the
parameters of the tropospheric delay model are derived from climatology or a NWM, and
neither of them can be regarded as error free. As mentioned before, this error source is
mitigated, since the zenith delays and gradient components are estimated together with
the station coordinates in the GNSS analysis. Nevertheless, a priori zenith delays and the
MFs are important error sources [
6
]. The most commonly used a priori zenith delay to
date is probably the one from the global pressure and temperature model (GPT) [
7
], and
the most commonly used MF to date is probably the global mapping function (GMF) [
8
].
The drawback of this MF is that it is based on climatology, and thus, limited in its ability
to predict short-term variability of the atmosphere. In contrast, the NWMs capture the
short-term variability of the atmosphere and, thus, are considered a valuable data source
for MFs [
9
]. The most prominent example of an MF based on NWM data is the Vienna
Mapping Functions 1 (VMF1) [
10
]. Recently, various other MFs based on NWM data have
been successfully applied in practice [
11
–
14
]. This includes those MFs based on NWM data
that are no longer based on the assumption of a spherically layered atmosphere, such as
the Vienna Mapping Functions 3 (VMF3) [15] and variants thereof [16].
The larger the deviation of the climatology or the NWM from the true state of the
atmosphere, the larger the errors in the GNSS estimates. This is obvious and suggested in
many studies; however, it is not trivial to provide concrete numbers. For example, we may
exchange one MF, say an MF based on a climatology, with another MF, say an MF based on
an NWM, and study the impact of this exchange on the GNSS estimates. However, this
exercise will not provide us with the absolute error of the tropospheric mismodelling of
GNSS estimates. The purpose of our study is to estimate such numbers. To do so we mimic
PPP in an artificial environment; i.e., we make use of a linearized observation equation,
where the observed minus modelled term equals ray-traced tropospheric delays from a
high-resolution NWM. The estimated parameters, station coordinates, zenith delays, and
tropospheric gradients are then compared with the perfectly known values.
Remote Sens. 2021,13, 3944 3 of 18
2. Materials and Methods
2.1. Tropospheric Delay
The tropospheric delay Tis given through:
T=Zn ds −g(1)
where ndenotes the index of refraction, sdenotes the arclength of the ray-path, and g
denotes the geometric distance. The ray-path follows Fermat’s principle. The index of
refraction nis related to the refractivity Ψthrough:
n=10−6Ψ+1 (2)
The refractivity field depends on the pressure, temperature, and humidity field [
17
].
The pressure, temperature, and humidity fields are unknown, and, hence, the true refrac-
tivity field is also unknown. In this study we assume that the output of a mesoscale NWM
gives the true refractivity field. Specifically, we utilize the weather research and forecast-
ing (WRF) model [
18
] to simulate the refractivity field of the atmosphere. We consider a
limited area covering the central part of Europe (for details see below). The initial and
boundary conditions for the limited area were the global forecast system (GFS) analysis of
the National Centers for Environmental Prediction (NCEP). The 24-h free forecasts start
every day at 0 UTC. The following models were applied: the Thompson scheme for the
microphysics [
19
], the Kain–Fritsch scheme for the cumulus parameterization [
20
], the
Yonsei University scheme for the planetary boundary layer [
21
], the RRTMG Short and
Longwave scheme for the radiation [
22
], the Unified Noah Land Surface Model for the
land surface [
23
], and the revised mm5 scheme for the surface layer [
24
]. The refractivity
was calculated from the pressure, temperature, and humidity and was available every
hour with a horizontal resolution of 10 km on 50 vertical model levels up to 50 hPa. The
refractivity at an arbitrary point was obtained by interpolation [
25
]. Then the algorithm
by [
26
] allowed us to compute tropospheric delays with high speed and precision for any
station–satellite link.
The artificial environment that we made use of in the following simulations has some
limitations. We ran the WRF model with a limited horizontal resolution of 10 km. The
simulation would be made more realistic by increasing the horizontal resolution. However,
even with an horizontal resolution of say 2 km the mesoscale model would not resolve
all tropospheric features. In particular, the mesoscale model does not explicitly resolve
turbulence in the planetary boundary layer (the turbulence is parameterized).
2.2. Zenith Delays and Tropospheric Gradient
The tropospheric delay can be regarded as a function of the elevation angle eand the
azimuth angle a. Thus, the tropospheric delay can be written as:
T=T(e,a)(3)
For any station location, we consider k= 120 tropospheric delays, where the elevation
angles are 3
◦
, 5
◦
, 7
◦
, 10
◦
, 15
◦
, 20
◦
, 30
◦
, 50
◦
, 70
◦
, and 90
◦
and the spacing in azimuth is 30
◦
.
With this set of tropospheric delays we are prepared to define the ZTD and the gradient
components. Both the ZTD and the gradient components can be understood as specific
linear combinations of tropospheric delays. Specifically, the ZTD is defined as:
ZTD =T90◦, 0◦(4)
Remote Sens. 2021,13, 3944 4 of 18
The north gradient component Nand the east gradient component Eare defined as:
N=∑k
i=1mg(ei)sin(ei)2cos(ai)T(ei,ai)
∑k
i=1mg(ei)2sin(ei)2cos(ai)2
E=∑k
i=1mg(ei)sin(ei)2sin(ai)T(ei,ai)
∑k
i=1mg(ei)2sin(ei)2sin(ai)2
(5)
Here the indices indicate the specific elevation and azimuth angle and m
g
denotes the
gradient MF.
mg(e)=1
sin(e)tan(e)+C(6)
where C= 0.0031 [1].
The ZTD is defined as the tropospheric delay in the zenith direction and no further
explanation is required. The definition for the north and east gradient components deserves
some further explanation. To understand this definition let us assume that the tropospheric
delay is approximated by:
T(e,a)∼T0(e)+mg(e)·[N cos(a)+E sin(a)] (7)
where T
0
denotes the tropospheric delay under the assumption of a spherically layered
atmosphere. Given the set of tropospheric delays the gradient components are determined
by a weighted least square fit [27]
[N,E]=ΓTWΓ−1ΓTW(T−T0)(8)
The entries of the design matrix
Γ
are given by the partial derivatives of the right-
hand side of Equation (7) with respect to the gradient coefficients and the weight matrix
W
reads as:
Wkl =sin(ek)sin(el)δkl (9)
where the indices denote the specific elevation angles. After some algebra it is not difficult
to verify that for the set of tropospheric delays Equation (8) reduces to Equation (5). For
the details, the reader is referred to [28].
The definition of the ZTD, and in particular the definition of the tropospheric gradient
components, appears arbitrary. However, as we will demonstrate in the following, these
quantities can be estimated with a high accuracy using the GNSS.
2.3. Tropospheric Delays in the GNSS Analysis
In the GNSS analysis, the tropospheric delay is parameterized
T(e,a)∼mh(e)·ZHD +mw(e)·ZWD +mg(e)·[N cos(a)+E sin(a)] (10)
where ZHD denotes the Zenith Hydrostatic Delay, ZWD denotes the Zenith Wet Delay, m
h
denotes the hydrostatic MF, and mwdenotes the wet MF. The ZTD is given by:
ZTD =ZHD +ZWD (11)
Typically, the hydrostatic and wet MF are derived under the assumption of a spheri-
cally layered atmosphere. Under this assumption, it is recognized that the elevation angle
dependency of the hydrostatic and wet MF is accurately described by the continued fraction
form [29]
m(e;a,b,c)=
1+a
1+b
1+c
sin(e)+a
sin(e)+b
sin(e)+c
(12)
Remote Sens. 2021,13, 3944 5 of 18
where a,b, and care called MF coefficients. The MF coefficients are computed utilizing
a climatology or an NWM. Specifically, for some refractivity profiles (the assumption is
that the refractivity is a function of the altitude only) the ratios of slant and zenith delays
(mapping factors) are computed (the elevation angles are 3
◦
, 5
◦
, 7
◦
, 10
◦
, 15
◦
, 20
◦
, 30
◦
,
50
◦
, 70
◦
, and 90
◦
) and the MF coefficients are determined by least-squares fitting. We can
confirm that under the assumption of a spherically layered atmosphere the continued
fraction form with three MF coefficients yields an exquisite level of precision. We call our
realization of the MF, where all three MF coefficients are determined by least-squares fitting,
the Potsdam mapping function (PMF). The most prominent example for a MF based on a
climatology is the new mapping function (NMF) [
29
]. The most prominent example of a
MF based on NWM data is the VMF1 [
10
]. The VMF1 is of particular interest because it is
based on an efficient concept; the MF coefficients b and c come from a climatology, and the
MF coefficient a is determined from a single mapping factor by inverting the continued
fraction form. This is also why several other MFs based on the VMF1 concept exist, e.g.,
the UNB-VMF1 [
30
] and the GFZ-VMF1 [
31
]. The main difference with the original VMF1
is that they are based on different NWMs.
In all the above mentioned concepts, the assumption is that the atmosphere is spher-
ically layered. Recently, attempts have been made to move away from this concept. For
example, instead of utilizing a single refractivity profile above the station, we utilize the
refractivity field above the station and one can compute 120 mapping factors for various
elevation and azimuth angles (the elevation angles are 3
◦
, 5
◦
, 7
◦
, 10
◦
, 15
◦
, 20
◦
, 30
◦
, 50
◦
, 70
◦
,
and 90
◦
and the spacing in azimuth is 30
◦
). Then, by averaging over the azimuth angles,
10 mapping factors are computed, and the three coefficients of the MF are determined by
least-squares fitting. However, some caution is required here as the continued fraction form
no longer gives a perfect fit with the mapping factors. The continued fraction form gives a
perfect fit with the mapping factors provided that they are calculated under the assumption
of a spherically layered atmosphere. This was also recognized in the recently developed
VMF3 [
15
], and variants thereof [
16
]. In the present work, as a by-product of the simulation
study, we will also develop a new MF concept. The details will be provided below.
2.4. Precise Point Positioning Simulation
We simulated PPP in an artificial environment, i.e., the refractivity field of the mesoscale
NWM. The simulator that we implemented is very similar to the simulators utilized to
study various other effects, such as multipath and geometry effects [
32
] or higher order-
ionospheric effects [
33
]. The simulation was simplified by ignoring carrier-phase ambi-
guities in the observation equation. In essence, we utilized the following version of the
linearized observation equation
T(e,a)−mh(e)·ZHD =−u(e,a)·δr+c0·δt+mw(e)·ZWD +mg(e)·[N cos(a)+E sin(a)] (13)
The left hand side of the equation represents the measured minus modelled term
and the right hand side of the equation includes the parameters to be estimated. Here
u
denotes the tangent-unit vector of the station–satellite link,
δr
denotes the coordinate
residual,
δ
tdenotes the clock residual, and c
0
denotes the vacuum speed of light. In the
present study we consider 120 station satellite links for a single epoch, where azimuth
angles are selected randomly and elevation angles are obtained through e= 90
◦−
87
◦√
w,
where w
∈
[0,1] is obtained from a random number generator. The set of station–satellite
links mimics a realistic observation geometry with a cut-off elevation angle of 3
◦
. The
observation geometry is realistic in so far as the elevation angle dependency reflects the
linearly increasing number of observations with decreasing elevation angles [
25
]. However,
the simple azimuth angle dependency does not reflect the lack of observations to the North
(South) for a station located in the Northern (Southern) Hemisphere. This might have
an impact on the results, in particular the tropospheric gradients, and must be studied
in future. The tropospheric delays that enter the left hand side of the equation are ray-
traced tropospheric delays. For each day, we consider 24 epochs, i.e., every hour, we
consider 120 station–satellite links. Then, we combine the 24 times 120 equations and
Remote Sens. 2021,13, 3944 6 of 18
obtain, using the least-square adjustment, the coordinate residual on a daily basis and
the clock- and tropospheric parameter residuals epoch-wise. Standard elevation angle
dependent weighting is applied in the least-square fit. In short, let Tdenote the ray-traced
tropospheric delays and Adenote the a priori tropospheric delays, i.e., the product of the
hydrostatic MF and the ZHD, then the solution, which includes the coordinate residual
(daily) and the clock- and tropospheric parameter residuals (epoch wise), is obtained
through:
[δr,ZWD1, . . . , ZWD24,N1, . . . , N24,E1, . . . , E24,δt1, . . . , δt24]=QTWQ−1QTW(T−A)(14)
The design matrix
Q
is defined by the partial derivatives of the right-hand side of
Equation (13), with respect to the coordinate residual, the clock residual, the ZWD, and the
gradient components. The weight matrix equals the weight matrix defined in Equation (9)
and, therefore, downweights the low elevation angle observations. One of the reasons
to do so in practice is the known deficiencies of the tropospheric model at low elevation
angles. The final ZTD is given by the sum of the a priori ZHD and the estimated ZWD.
Next, let ZTD
WRF
denote the ZTD calculated from the NWM, following Equation (4),
and let N
WRF
and E
WRF
denote the north and east gradient component calculated from the
NWM, following Equation (5); then, the errors of the GNSS estimates are given by:
∆ZTD =(ZHD +ZWD)−ZTDWRF
∆N=N−NWRF
∆E=E−EWRF
∆r=δr
∆t=δt
(15)
and quantify the impact of the tropospheric mismodelling in the GNSS analysis. Note that
ZTD
WRF
,N
WRF
, and E
WRF
stand for the true ZTD, the true north, and the true east gradient
component, since in the context of our simulation study the NWM represents the true state
of the atmosphere.
2.5. Experiment Setup
The error of the GNSS estimates depends on the chosen ZHD, the chosen hydrostatic
MF, and the chosen wet MF. In total, we ran five experiments, as summarized in
Table 1.
In the first experiment, the ZHD came from the GPT [
7
] and the hydrostatic and the wet
MF were taken from the GMF [
8
]. In essence all tropospheric parameters that enter the
observation equation came from climatology. This approach is the most widely used in
practice, due to its simplicity. There are no external data required in this approach. In
the second experiment, the ZHD came from the NWM, but the hydrostatic and wet MF
were based on the climatology. This approach is also considered practical, as it solely
requires the pressure at the station, due to an existing relationship between the pressure
and ZHD [
34
]. In the third experiment, the ZHD and the hydrostatic MF were derived from
the NWM, whereas the wet MF was still based on the climatology. In the fourth experiment
the ZHD, the hydrostatic and the wet MF were derived from the NWM. It is important to
note that up to this point, both the hydrostatic and the wet MF were calculated under the
assumption of a spherically layered atmosphere. Finally, in the fifth experiment, the ZHD,
the hydrostatic MF and the wet MF were again derived from the NWM. The difference,
however, was that the wet MF was no longer calculated on the assumption of a spherically
layered atmosphere. The details on the derivation of this modified wet MF are provided in
the next section.
In the simulation study we considered a limited area covering Germany, the Czech
Republic, and parts of Poland and Austria, and a period of two months (May and June)
in 2013. We utilized data from more than 400 stations. The locations of the stations
are shown in Figure 1. This choice was motivated by an existing dense station network,
which was utilized in the Benchmark campaign within the European COST Action ES1206
GNSS4SWEC (Advanced GNSS tropospheric products for monitoring severe weather and
Remote Sens. 2021,13, 3944 7 of 18
climate) [
35
]. The station locations and time period considered in the simulation study
represent stations located in central Europe and the warm season. The station POTS
(Potsdam, Germany), located in the center of the limited area model, was considered
representative of the bulk of the stations. Thus, we will provide some more details for this
station. For the rest of the stations we solely provide some statistics.
Table 1.
Summary of the simulation experiments. The simulation experiments differed in the chosen
ZHD, hydrostatic and wet MF. GMF stands for the global mapping function, GPT stands for the
global pressure and temperature model, and PMF stands for the Potsdam mapping function, i.e., the
MF calculated from the NWM. The subscripts h and w indicate hydrostatic and wet MF, respectively.
The subscript z indicates modified wet MF. For details refer to the text.
Experiment ZHD mhmw
#1 GPT GMFhGMFw
#2 NWM GMFhGMFw
#3 NWM PMFhGMFw
#4 NWM PMFhPMFw
#5 NWM PMFhPMFz
Remote Sens. 2021, 13, x FOR PEER REVIEW 7 of 18
GNSS4SWEC (Advanced GNSS tropospheric products for monitoring severe weather and
climate) [35]. The station locations and time period considered in the simulation study
represent stations located in central Europe and the warm season. The station POTS (Pots-
dam, Germany), located in the center of the limited area model, was considered repre-
sentative of the bulk of the stations. Thus, we will provide some more details for this sta-
tion. For the rest of the stations we solely provide some statistics.
Table 1. Summary of the simulation experiments. The simulation experiments differed in the chosen
ZHD, hydrostatic and wet MF. GMF stands for the global mapping function, GPT stands for the
global pressure and temperature model, and PMF stands for the Potsdam mapping function, i.e.,
the MF calculated from the NWM. The subscripts h and w indicate hydrostatic and wet MF, respec-
tively. The subscript z indicates modified wet MF. For details refer to the text.
Experiment ZHD mh mw
#1 GPT GMFh GMFw
#2 NWM GMFh GMFw
#3 NWM PMFh GMFw
#4 NWM PMFh PMFw
#5 NWM PMFh PMFz
Figure 1. Map showing the locations of the stations in the simulation study. In total 431 stations
cover the area of interest. The red dot indicates the location of the station POTS (Potsdam, Germany).
2.6. Modified Wet Mapping Function
The tropospheric delay is approximated as:
𝑇(𝑒,𝑎) ~ 𝑚(𝑒;𝑎,𝑏,𝑐)∙𝑍𝐻𝐷 + 𝑚(𝑒;𝑎,𝑏,𝑐)∙𝑍𝑊𝐷+⋯
𝑚(𝑒)∙𝑍+⋯
𝑚(𝑒)∙[𝑁 𝑐𝑜𝑠(𝑎)+𝐸 𝑠𝑖𝑛(𝑎)]+⋯
𝑚(𝑒)∙[𝑍 𝑐𝑜𝑠(2∙𝑎)+𝑍 𝑠𝑖𝑛(2∙𝑎)]+⋯
𝑚(𝑒)∙[𝑍 𝑐𝑜𝑠(3∙𝑎)+𝑍 𝑠𝑖𝑛(3∙𝑎)]+⋯
(16)
where the hydrostatic and wet MF are calculated as usual, under the assumption of a
spherically layered atmosphere and the coefficients Z can be understood as higher-order
tropospheric parameters. The proposed approximation for the tropospheric delay can be
considered as follows: The difference between tropospheric delays and tropospheric de-
lays calculated under the assumption of a spherically layered atmosphere can be regarded
as a function of the elevation and azimuth angle. Given a set of tropospheric delays and a
set of tropospheric delays calculated under the assumption of a spherically layered atmos-
phere (the elevation angles are 3°, 5°, 7°, 10°, 15°, 20°, 30°, 50°, 70°, and 90° and the spacing
in azimuth is 30°), then, for a specific elevation angle, the difference between the
Figure 1.
Map showing the locations of the stations in the simulation study. In total 431 stations
cover the area of interest. The red dot indicates the location of the station POTS (Potsdam, Germany).
2.6. Modified Wet Mapping Function
The tropospheric delay is approximated as:
T(e,a)∼mh(e;ah,bh,ch)·ZHD +mw(e;aw,bw,cw)·ZWD +. . .
mg(e)·Z0+. . .
mg(e)·[N cos(a)+E sin(a)] +. . .
mg(e)·[Z1cos(2·a)+Z2sin(2·a)] +. . .
mg(e)·[Z3cos(3·a)+Z4sin(3·a)] +. . .
(16)
where the hydrostatic and wet MF are calculated as usual, under the assumption of a
spherically layered atmosphere and the coefficients Zcan be understood as higher-order
tropospheric parameters. The proposed approximation for the tropospheric delay can be
considered as follows: The difference between tropospheric delays and tropospheric delays
calculated under the assumption of a spherically layered atmosphere can be regarded as a
function of the elevation and azimuth angle. Given a set of tropospheric delays and a set of
tropospheric delays calculated under the assumption of a spherically layered atmosphere
(the elevation angles are 3
◦
, 5
◦
, 7
◦
, 10
◦
, 15
◦
, 20
◦
, 30
◦
, 50
◦
, 70
◦
, and 90
◦
and the spacing in
azimuth is 30
◦
), then, for a specific elevation angle, the difference between the tropospheric
delays can be expanded in a Fourier series. In general, the coefficients of the Fourier series
Remote Sens. 2021,13, 3944 8 of 18
expansion will be different for different elevation angles. Suppose we assume that the
elevation angle dependency of these coefficients of the Fourier series expansion follows a
simple law, namely that this elevation angle dependency can be represented by the gradient
MF. In that case, we end up with the expression for the tropospheric delay provided above.
In this interpretation, the north and east gradient components can be considered the second
and third coefficients of the Fourier series expansion. The Zcoefficients are discarded in the
GNSS analysis. Obviously, the danger in the estimation of the ZTD, station up-component,
and clock lies in the presence of Z
0
; as Z
0
appears in a term that depends on the elevation
angle only, and this term is present on the left-hand side of the observation equation, but
there is no corresponding term on the right-hand side of the observation equation, and
it will be absorbed by the estimated ZTD, station up-component, and clock (also see the
linearized observation equation). One possible solution is to introduce a modified wet MF,
which takes into account the coefficient Z0. Thus, we rewrite the tropospheric delay as:
T(e,a)∼mh(e;ah,bh,ch)·ZHD +m∗
w(e;aw,bw,cw,zw)·ZWD +. . .
mg(e)·[N cos(a)+E sin(a)] +. . .
mg(e)·[Z1cos(2·a)+Z2sin(2·a)] +. . .
mg(e)·[Z3cos(3·a)+Z4sin(3·a)] +. . .
(17)
where the modified wet MF is defined as:
m∗
w(e;aw,bw,cw,zw)=mw(e;aw,bw,cw)+mg(e)·zw(18)
and the fourth coefficient zwis given by:
zw=Z0
ZWD (19)
The coefficient Z
0
is determined by a least-square fit. For the determination of the
three coefficients a
w
,b
w
, and c
w
, solely the refractivity profile above the station and 10 map-
ping factors are required, whereas for the determination of the fourth coefficient z
w
the
refractivity field above the station and 120 additional mapping factors are required. Clearly,
this makes the determination of the modified wet MF more expensive. Adding the term
containing Z
0
to the wet MF, and not to the hydrostatic MF, is motivated by the origin of
this extra term, which is more likely to be the wet than the hydrostatic refractivity field.
We make no attempt to estimate Z
1
to Z
4
in the GNSS analysis. We think that with the
increasing number of parameters to be estimated besides the coordinates and clocks, the
GNSS solution would probably get worse.
Equation (16) appears similar to the approximation proposed by [
36
], except for the
extra term containing Z
0
. This can be explained by the fact that in [
36
] the isotropic part of
the tropospheric delay, i.e., the part of the tropospheric delay which depends solely on the
elevation angle, already contains the average over various azimuth angles. The isotropic
part in [
36
] is not calculated under the assumption of a spherically layered atmosphere. The
hydrostatic and wet MF which enter our Equation (16) are calculated under the assumption
of a spherically layered atmosphere.
3. Results
In this section we show the results of the simulation experiments listed in Table 1.
The estimation of the station clock is part of the PPP simulation. However, we will only
show and discuss the impact of the tropospheric mismodelling on the estimated station
coordinates and tropospheric parameters, as they are the parameters of interest in precise
positioning and atmospheric remote sensing.
3.1. Experiment 1
The error in the ZTD and the error in the station up-component for the station POTS
as a function of time are shown in the left panel of Figure 2. The error in the ZTD reached
Remote Sens. 2021,13, 3944 9 of 18
15 mm, and the error in the up-component reached 10 mm. The time series shows the
known correlation between the ZTD and station up-component error. The step-like struc-
ture in the time series for the error of the station up-component is due to the fact that
the coordinate residual was estimated on a daily basis. The noise-like structure in the
time series for the error in the ZTD is due to the zenith delay residual being estimated
epoch-wise. The error in the ZTD is the composite of two error sources; the climatology
is not able to follow either the slow change in the hydrostatic portion of the refractivity
field or the rapid change in the wet portion of the refractivity field. The right panel in
Figure 2shows the station-specific root mean square error (RMSE) for the ZTD, the gradient
components, and the station up-component. With only a few exceptions, the RMSE is the
same for the considered stations and, hence, it is meaningful to provide the average RMSE
for the respective parameters. This is also why we considered the POTS station to represent
the bulk of the stations. We obtained 2.9 mm for the ZTD, 0.11 mm for the north (east)
gradient component, and 5.7 mm for the station up-component. The significant RMSE
for the ZTD and the station up-component was owing to the hydrostatic MF; the wet MF
and the ZHD are inaccurate because they are based on a climatology. The RMSE for the
gradient components is not regarded as significant, as a deviation of 0.1 mm in the gradient
component converts into a tropospheric delay deviation of about 10 mm at an elevation
angle as low as 5
◦
. The small deviations can be explained by the fact that the way the
gradient components are defined and the way gradient components are estimated in the
GNSS analysis are very similar. The remaining deviations were mainly due to the different
geometries in their determination.
Remote Sens. 2021, 13, x FOR PEER REVIEW 9 of 18
3.1. Experiment 1
The error in the ZTD and the error in the station up-component for the station POTS
as a function of time are shown in the left panel of Figure 2. The error in the ZTD reached
15 mm, and the error in the up-component reached 10 mm. The time series shows the
known correlation between the ZTD and station up-component error. The step-like struc-
ture in the time series for the error of the station up-component is due to the fact that the
coordinate residual was estimated on a daily basis. The noise-like structure in the time
series for the error in the ZTD is due to the zenith delay residual being estimated epoch-
wise. The error in the ZTD is the composite of two error sources; the climatology is not
able to follow either the slow change in the hydrostatic portion of the refractivity field or
the rapid change in the wet portion of the refractivity field. The right panel in Figure 2
shows the station-specific root mean square error (RMSE) for the ZTD, the gradient com-
ponents, and the station up-component. With only a few exceptions, the RMSE is the same
for the considered stations and, hence, it is meaningful to provide the average RMSE for
the respective parameters. This is also why we considered the POTS station to represent
the bulk of the stations. We obtained 2.9 mm for the ZTD, 0.11 mm for the north (east)
gradient component, and 5.7 mm for the station up-component. The significant RMSE for
the ZTD and the station up-component was owing to the hydrostatic MF; the wet MF and
the ZHD are inaccurate because they are based on a climatology. The RMSE for the gra-
dient components is not regarded as significant, as a deviation of 0.1 mm in the gradient
component converts into a tropospheric delay deviation of about 10 mm at an elevation
angle as low as 5°. The small deviations can be explained by the fact that the way the
gradient components are defined and the way gradient components are estimated in the
GNSS analysis are very similar. The remaining deviations were mainly due to the different
geometries in their determination.
Figure 2. Results for the first simulation experiment. Left panel: the error in the ZTD and the error
in the station up-component for the POTS station as a function of time. Right panel: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
3.2. Experiment 2
The left panel in Figure 3 shows the error in the ZTD and the error in the station up-
component for the station POTS as a function of time. It is not obvious that the introduc-
tion of the ZHD from the NWM yielded a significant reduction in the ZTD and station up-
component error. Again, the error in the ZTD reached 15 mm and the error in the up-
component reached 10 mm. However, a closer look reveals that the introduction of the
ZHD from the NWM yielded a somewhat smaller deviation around zero. Due to the ZHD
from the NWM, the slow change in the hydrostatic portion of the refractivity field was, to
some extent, taken into account. The right panel in Figure 3 shows the station-specific
RMSE for the ZTD, the gradient components, and the station up-component; we obtained
Figure 2.
Results for the first simulation experiment.
Left panel
: the error in the ZTD and the error in
the station up-component for the POTS station as a function of time.
Right panel
: the station-specific
RMSE for the ZTD (top), the gradient components (middle), and the station up-component (bottom).
The number with yellow background corresponds to the averaged RMSE of the respective parameter.
3.2. Experiment 2
The left panel in Figure 3shows the error in the ZTD and the error in the station
up-component for the station POTS as a function of time. It is not obvious that the
introduction of the ZHD from the NWM yielded a significant reduction in the ZTD and
station up-component error. Again, the error in the ZTD reached 15 mm and the error in
the up-component reached 10 mm. However, a closer look reveals that the introduction
of the ZHD from the NWM yielded a somewhat smaller deviation around zero. Due to
the ZHD from the NWM, the slow change in the hydrostatic portion of the refractivity
field was, to some extent, taken into account. The right panel in Figure 3shows the station-
specific RMSE for the ZTD, the gradient components, and the station up-component; we
obtained 2.7 mm for the ZTD, 0.11 mm for the north (east) gradient component, and 5.2 mm
for the station up-component. We can state that introducing the ZHD from the NWM
yielded a small but consistent reduction of the error in the ZTD and station up-component.
Regarding the gradient components, it appears that the chosen ZHD did not have an
Remote Sens. 2021,13, 3944 10 of 18
impact, since if we replace the ZHD from the climatology with the ZHD from the NWM
the error for the gradient components remains unchanged.
Remote Sens. 2021, 13, x FOR PEER REVIEW 10 of 18
2.7 mm for the ZTD, 0.11 mm for the north (east) gradient component, and 5.2 mm for the
station up-component. We can state that introducing the ZHD from the NWM yielded a
small but consistent reduction of the error in the ZTD and station up-component. Regard-
ing the gradient components, it appears that the chosen ZHD did not have an impact,
since if we replace the ZHD from the climatology with the ZHD from the NWM the error
for the gradient components remains unchanged.
Figure 3. Results for the second simulation experiment. Left panel: the error in the ZTD and the
error in the station up-component for the POTS station as a function of time. Right panel: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
3.3. Experiment 3
The error in the ZTD and the error in the station up-component for the POTS station
as a function of time are shown in the left panel of Figure 4. The introduction of the ZHD
and hydrostatic MF from the NWM yielded a significant reduction in the ZTD and station
up-component error. The error in the station up-component remained below 6 mm and
the error in the ZTD was typically below 10 mm. A large ZTD error, of about 15 mm
around day 30, is still visible in the time series. Since the ZHD and hydrostatic MF come
from the NWM, the slow change in the hydrostatic portion of the refractivity field was
taken into account. The wet MF came from the climatology, and, hence, the rapid change
in the wet portion of the refractivity field was not considered. The right panel in Figure 4
shows the station-specific RMSE for the ZTD, the gradient components, and the station
up-component; we obtained 2.0 mm for the ZTD, 0.1 mm for the north (east) gradient
component, and 2.9 mm for the station up-component. Regarding the gradient compo-
nent, it appears that the chosen hydrostatic MF was not significant, as replacing the hy-
drostatic MF from the climatology with the hydrostatic MF from the NWM resulted in a
difference in the error for the gradient components of only 0.01 mm.
Figure 3.
Results for the second simulation experiment.
Left panel
: the error in the ZTD and the
error in the station up-component for the POTS station as a function of time.
Right panel
: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
3.3. Experiment 3
The error in the ZTD and the error in the station up-component for the POTS station
as a function of time are shown in the left panel of Figure 4. The introduction of the ZHD
and hydrostatic MF from the NWM yielded a significant reduction in the ZTD and station
up-component error. The error in the station up-component remained below 6 mm and the
error in the ZTD was typically below 10 mm. A large ZTD error, of about 15 mm around
day 30, is still visible in the time series. Since the ZHD and hydrostatic MF come from
the NWM, the slow change in the hydrostatic portion of the refractivity field was taken
into account. The wet MF came from the climatology, and, hence, the rapid change in
the wet portion of the refractivity field was not considered. The right panel in Figure 4
shows the station-specific RMSE for the ZTD, the gradient components, and the station
up-component; we obtained 2.0 mm for the ZTD, 0.1 mm for the north (east) gradient
component, and 2.9 mm for the station up-component. Regarding the gradient component,
it appears that the chosen hydrostatic MF was not significant, as replacing the hydrostatic
MF from the climatology with the hydrostatic MF from the NWM resulted in a difference
in the error for the gradient components of only 0.01 mm.
Remote Sens. 2021, 13, x FOR PEER REVIEW 10 of 18
2.7 mm for the ZTD, 0.11 mm for the north (east) gradient component, and 5.2 mm for the
station up-component. We can state that introducing the ZHD from the NWM yielded a
small but consistent reduction of the error in the ZTD and station up-component. Regard-
ing the gradient components, it appears that the chosen ZHD did not have an impact,
since if we replace the ZHD from the climatology with the ZHD from the NWM the error
for the gradient components remains unchanged.
Figure 3. Results for the second simulation experiment. Left panel: the error in the ZTD and the
error in the station up-component for the POTS station as a function of time. Right panel: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
3.3. Experiment 3
The error in the ZTD and the error in the station up-component for the POTS station
as a function of time are shown in the left panel of Figure 4. The introduction of the ZHD
and hydrostatic MF from the NWM yielded a significant reduction in the ZTD and station
up-component error. The error in the station up-component remained below 6 mm and
the error in the ZTD was typically below 10 mm. A large ZTD error, of about 15 mm
around day 30, is still visible in the time series. Since the ZHD and hydrostatic MF come
from the NWM, the slow change in the hydrostatic portion of the refractivity field was
taken into account. The wet MF came from the climatology, and, hence, the rapid change
in the wet portion of the refractivity field was not considered. The right panel in Figure 4
shows the station-specific RMSE for the ZTD, the gradient components, and the station
up-component; we obtained 2.0 mm for the ZTD, 0.1 mm for the north (east) gradient
component, and 2.9 mm for the station up-component. Regarding the gradient compo-
nent, it appears that the chosen hydrostatic MF was not significant, as replacing the hy-
drostatic MF from the climatology with the hydrostatic MF from the NWM resulted in a
difference in the error for the gradient components of only 0.01 mm.
Figure 4.
Results for the third simulation experiment.
Left panel
: the error in the ZTD and the error
in the station up-component for the POTS station as a function of time.
Right panel
: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
Remote Sens. 2021,13, 3944 11 of 18
3.4. Experiment 4
The left panel in Figure 5shows the error in the ZTD and the error in the station
up-component for the POTS station as a function of time. The introduction of the ZHD,
hydrostatic, and wet MF from the NWM significantly reduced the ZTD and station up-
component error. The error in the station up-component remained below 6 mm, and the
error in the ZTD was typically below 10 mm. Again, a large ZTD error of about 15 mm
around day 30 is visible in the time series. The ZTD and station up-component error
reduction mainly came from the introduction of the NWM, ZHD, and NWM hydrostatic
MF. The introduction of the NWM wet MF brought almost no reduction in the ZTD and
station up-component error, and we can state that the errors of the GNSS estimates were
insensitive to the chosen wet MF. However, we must be more specific, and better yet, we
state that the errors of the GNSS estimates were insensitive to the chosen wet MF, as long
as it was calculated under the assumption of a spherically layered atmosphere. The right
panel in Figure 5shows the station-specific RMSE for the ZTD, the gradient components,
and the station up-component; we obtained 1.8 mm for the ZTD, 0.1 mm for the north (east)
gradient component, and 2.4 mm for the station up-component. The statistics confirm
that as long as the wet MF is calculated under the assumption of a spherically layered
atmosphere, it does not have too much impact. In essence, if we replace the wet MF, which
is based on the climatology, with the wet MF, which is based on the NWM, the RMSE is
reduced by only 0.2 mm for the ZTD and 0.5 mm for the station up-component. Regarding
the gradient components, it appears that the chosen wet MF did not have an impact. If we
replace the wet MF from the climatology with the wet MF from the NWM, the error for the
gradient components remains unchanged.
Remote Sens. 2021, 13, x FOR PEER REVIEW 11 of 18
Figure 4. Results for the third simulation experiment. Left panel: the error in the ZTD and the error
in the station up-component for the POTS station as a function of time. Right panel: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
3.4. Experiment 4
The left panel in Figure 5 shows the error in the ZTD and the error in the station up-
component for the POTS station as a function of time. The introduction of the ZHD, hy-
drostatic, and wet MF from the NWM significantly reduced the ZTD and station up-com-
ponent error. The error in the station up-component remained below 6 mm, and the error
in the ZTD was typically below 10 mm. Again, a large ZTD error of about 15 mm around
day 30 is visible in the time series. The ZTD and station up-component error reduction
mainly came from the introduction of the NWM, ZHD, and NWM hydrostatic MF. The
introduction of the NWM wet MF brought almost no reduction in the ZTD and station
up-component error, and we can state that the errors of the GNSS estimates were insensi-
tive to the chosen wet MF. However, we must be more specific, and better yet, we state
that the errors of the GNSS estimates were insensitive to the chosen wet MF, as long as it
was calculated under the assumption of a spherically layered atmosphere. The right panel
in Figure 5 shows the station-specific RMSE for the ZTD, the gradient components, and
the station up-component; we obtained 1.8 mm for the ZTD, 0.1 mm for the north (east)
gradient component, and 2.4 mm for the station up-component. The statistics confirm that
as long as the wet MF is calculated under the assumption of a spherically layered atmos-
phere, it does not have too much impact. In essence, if we replace the wet MF, which is
based on the climatology, with the wet MF, which is based on the NWM, the RMSE is
reduced by only 0.2 mm for the ZTD and 0.5 mm for the station up-component. Regarding
the gradient components, it appears that the chosen wet MF did not have an impact. If we
replace the wet MF from the climatology with the wet MF from the NWM, the error for
the gradient components remains unchanged.
Figure 5. Results for the fourth simulation experiment. Left panel: the error in the ZTD and the error
in the station up-component for the station POTS as a function of time. Right panel: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
3.5. Experiment 5
The error in the ZTD and the error in the station up-component for the POTS station
as a function of time are shown in the left panel of Figure 6. The introduction of the ZHD,
hydrostatic, and modified wet MF from the NWM yielded a significant reduction in the
ZTD and station up-component error. The error in the station up-component remained
below 3 mm, and the error in the ZTD was typically below 2 mm. Since the ZHD and the
Figure 5.
Results for the fourth simulation experiment.
Left panel
: the error in the ZTD and the error
in the station up-component for the station POTS as a function of time.
Right panel
: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
3.5. Experiment 5
The error in the ZTD and the error in the station up-component for the POTS station
as a function of time are shown in the left panel of Figure 6. The introduction of the ZHD,
hydrostatic, and modified wet MF from the NWM yielded a significant reduction in the
ZTD and station up-component error. The error in the station up-component remained
below 3 mm, and the error in the ZTD was typically below 2 mm. Since the ZHD and
the hydrostatic MF came from the NWM, the slow change in the hydrostatic portion of
the refractivity field was taken into account. In addition, the modified wet MF came from
the NWM, so that the rapid change in the wet portion of the refractivity field was taken
into account as well. In the previous section, we assumed that the errors of the GNSS
estimates are not very sensitive to the chosen wet MF when the wet MF is calculated under
the assumption of a spherically layered atmosphere. Here, we support this assertation;
Remote Sens. 2021,13, 3944 12 of 18
provided that the wet MF is no longer calculated under the assumption of a spherically
layered atmosphere, the errors of the GNSS estimates can be reduced significantly. The right
panel in Figure 6shows the station-specific RMSE for the ZTD, the gradient components
and the station up-component; we obtained 0.5 mm for the ZTD, 0.09 mm for the north
(east) gradient component, and 1 mm for the station up-component. Regarding the gradient
components, it appears that the chosen ZHD, hydrostatic, and modified wet MF from the
NWM did not have a significant impact.
Remote Sens. 2021, 13, x FOR PEER REVIEW 12 of 18
hydrostatic MF came from the NWM, the slow change in the hydrostatic portion of the
refractivity field was taken into account. In addition, the modified wet MF came from the
NWM, so that the rapid change in the wet portion of the refractivity field was taken into
account as well. In the previous section, we assumed that the errors of the GNSS estimates
are not very sensitive to the chosen wet MF when the wet MF is calculated under the
assumption of a spherically layered atmosphere. Here, we support this assertation; pro-
vided that the wet MF is no longer calculated under the assumption of a spherically lay-
ered atmosphere, the errors of the GNSS estimates can be reduced significantly. The right
panel in Figure 6 shows the station-specific RMSE for the ZTD, the gradient components
and the station up-component; we obtained 0.5 mm for the ZTD, 0.09 mm for the north
(east) gradient component, and 1 mm for the station up-component. Regarding the gradi-
ent components, it appears that the chosen ZHD, hydrostatic, and modified wet MF from
the NWM did not have a significant impact.
Figure 6. Results for the fifth simulation experiment. Left panel: the error in the ZTD and the error
in the station up-component for the POTS station as a function of time. Right panel: the station-
specific RMSE for the ZTD (top), the gradient components (middle), and the station up-component
(bottom). The number with yellow background corresponds to the averaged RMSE of the respective
parameter.
4. Discussion
4.1. On the Relation between the Tropospheric Parameter Z0 and the Errors in the GNSS
Estimates
The GNSS estimates are not very sensitive to the wet MF when the wet MF is calcu-
lated under the assumption of a spherically layered atmosphere. In other words, it does
not matter whether the three coefficients of the wet MF are derived from the climatology
or the NWM. The situation changes if the wet MF is not calculated under the assumption
of a spherically layered atmosphere. The fourth wet MF coefficient zw is more important
than the three wet MF coefficients aw, bw, and cw taken together. It is also important to note
that the modified wet MF is reduced to the standard wet MF if the fourth coefficient zw,
and hence the tropospheric parameter Z0, vanishes. Thus, it is worth studying the relation
between the tropospheric parameter Z0 and the errors in the GNSS estimates in detail. The
left panel in Figure 7 shows the tropospheric parameter Z0 for the Potsdam station as a
function of time. A one-to-one comparison of the left panel in Figure 7 and the left panel
in Figure 5 reveals that the error in the ZTD and the appearance of Z0 are correlated. The
error in the ZTD appears to be proportional to Z0. This is confirmed when we take a look
at all the stations and epochs analyzed. The right panel in Figure 7 shows the error in the
ZTD as obtained from the fourth simulation experiment versus the tropospheric parame-
ter Z0 for all stations and epochs analyzed. With the linear fit (indicated by the red line)
we obtain the following relation:
𝛥𝑍𝑇𝐷 ~ 6.5 ∙ 𝑍 (20)
Figure 6.
Results for the fifth simulation experiment.
Left panel
: the error in the ZTD and the error in
the station up-component for the POTS station as a function of time.
Right panel
: the station-specific
RMSE for the ZTD (top), the gradient components (middle), and the station up-component (bottom).
The number with yellow background corresponds to the averaged RMSE of the respective parameter.
4. Discussion
4.1. On the Relation between the Tropospheric Parameter Z
0
and the Errors in the GNSS Estimates
The GNSS estimates are not very sensitive to the wet MF when the wet MF is calculated
under the assumption of a spherically layered atmosphere. In other words, it does not
matter whether the three coefficients of the wet MF are derived from the climatology or
the NWM. The situation changes if the wet MF is not calculated under the assumption
of a spherically layered atmosphere. The fourth wet MF coefficient z
w
is more important
than the three wet MF coefficients a
w
,b
w
, and c
w
taken together. It is also important to note
that the modified wet MF is reduced to the standard wet MF if the fourth coefficient z
w
,
and hence the tropospheric parameter Z
0
, vanishes. Thus, it is worth studying the relation
between the tropospheric parameter Z
0
and the errors in the GNSS estimates in detail. The
left panel in Figure 7shows the tropospheric parameter Z
0
for the Potsdam station as a
function of time. A one-to-one comparison of the left panel in Figure 7and the left panel
in Figure 5reveals that the error in the ZTD and the appearance of Z
0
are correlated. The
error in the ZTD appears to be proportional to Z
0
. This is confirmed when we take a look
at all the stations and epochs analyzed. The right panel in Figure 7shows the error in the
ZTD as obtained from the fourth simulation experiment versus the tropospheric parameter
Z
0
for all stations and epochs analyzed. With the linear fit (indicated by the red line) we
obtain the following relation:
∆ZTD ∼6.5·Z0(20)
Another question is whether the tropospheric parameter Z
0
introduces a systematic
or random error. We find that for the POTS station and the bulk of stations, the tropo-
spheric parameter Z
0
introduces a random error into the GNSS estimates. There are some
stations where small systematic errors were introduced and those stations are located in
complex terrain. Our interpretation is that in complex terrain, the topography disturbs
the stratification of the atmosphere systematically, and this cannot be described with the
first-order gradient in the PWV field alone, and, therefore, the second-order gradient in
the PWV field is required. For example, the moist atmospheric boundary layer follows, to
some extent, the topography, so that for a station located on a mountain top or in a valley
Remote Sens. 2021,13, 3944 13 of 18
the atmosphere (on average) will not appear spherically layered. From the station’s per-
spective, the surrounding PWV field looks either convex or concave (the second derivative
in the PWV does not vanish). However, some caution is needed with this interpretation,
as the limited horizontal resolution of 10 km that we utilized does not draw conclusive
results for stations in complex terrain. Those stations should be analyzed case by case with
high-resolution weather model data.
Remote Sens. 2021, 13, x FOR PEER REVIEW 13 of 18
Another question is whether the tropospheric parameter Z0 introduces a systematic
or random error. We find that for the POTS station and the bulk of stations, the tropo-
spheric parameter Z0 introduces a random error into the GNSS estimates. There are some
stations where small systematic errors were introduced and those stations are located in
complex terrain. Our interpretation is that in complex terrain, the topography disturbs the
stratification of the atmosphere systematically, and this cannot be described with the first-
order gradient in the PWV field alone, and, therefore, the second-order gradient in the
PWV field is required. For example, the moist atmospheric boundary layer follows, to
some extent, the topography, so that for a station located on a mountain top or in a valley
the atmosphere (on average) will not appear spherically layered. From the station’s per-
spective, the surrounding PWV field looks either convex or concave (the second derivative
in the PWV does not vanish). However, some caution is needed with this interpretation,
as the limited horizontal resolution of 10 km that we utilized does not draw conclusive
results for stations in complex terrain. Those stations should be analyzed case by case with
high-resolution weather model data.
Figure 7. Left panel: the tropospheric parameter Z0 for the POTS station as a function of time. Right
panel: the error in the ZTD as obtained from the fourth simulation experiment versus the tropo-
spheric parameter Z0 for all stations and epochs analyzed. The red line indicates the linear fit.
In short, the simulation study suggests that we have two options to reduce errors in
the GNSS ZTD estimates: (1) replace the standard wet MF with the modified wet MF in
the GNSS analysis, or (2) correct the estimated ZTD a posteriori by utilizing the correction
provided in Equation (20). Clearly, in practice, success will depend on the ability of the
current (future) NWM to predict the additional tropospheric parameter Z0. We have some
evidence that current NWMs can predict the tropospheric parameter Z0. This is discussed
in the next section.
4.2. Preliminary Results from GNSS Meteorology in Central Europe
In Europe the EUMETNET EIG GNSS water vapor programme (E-GVAP) is in charge
of collecting operational GNSS tropospheric products for numerical weather prediction.
GFZ is one of the operational E-GVAP Analysis Centers and processes more than 500 sta-
tions in near real-time (NRT). About 300 stations are located in central Europe (the area of
interest). The Earth Parameter and Orbit determination System (EPOS) software, devel-
oped at the GFZ, is used to estimate tropospheric products from the GNSS carrier phase
and code measurements in PPP mode [37]. The precise satellite orbits and clocks, as well
as earth rotation parameters, are available from the International GNSS Service (IGS) anal-
ysis center at the GFZ. The station coordinates are estimated in a sliding window mode
(24 h), station clock errors are estimated epoch by epoch, and ZTDs and tropospheric gra-
dients are estimated every 15 min. The a prior ZHD comes from the GPT, and the hydro-
static and the wet MF is taken from the GMF. For details, the reader is referred to [38].
Notably, the ZTDs provided by GFZ are used by several European weather services, such
Figure 7. Left panel
: the tropospheric parameter Z
0
for the POTS station as a function of time.
Right panel
: the error in the ZTD as obtained from the fourth simulation experiment versus the
tropospheric parameter Z
0
for all stations and epochs analyzed. The red line indicates the linear fit.
In short, the simulation study suggests that we have two options to reduce errors in
the GNSS ZTD estimates: (1) replace the standard wet MF with the modified wet MF in
the GNSS analysis, or (2) correct the estimated ZTD a posteriori by utilizing the correction
provided in Equation (20). Clearly, in practice, success will depend on the ability of the
current (future) NWM to predict the additional tropospheric parameter Z
0
. We have some
evidence that current NWMs can predict the tropospheric parameter Z
0
. This is discussed
in the next section.
4.2. Preliminary Results from GNSS Meteorology in Central Europe
In Europe the EUMETNET EIG GNSS water vapor programme (E-GVAP) is in charge
of collecting operational GNSS tropospheric products for numerical weather prediction.
GFZ is one of the operational E-GVAP Analysis Centers and processes more than 500 sta-
tions in near real-time (NRT). About 300 stations are located in central Europe (the area of
interest). The Earth Parameter and Orbit determination System (EPOS) software, devel-
oped at the GFZ, is used to estimate tropospheric products from the GNSS carrier phase
and code measurements in PPP mode [
37
]. The precise satellite orbits and clocks, as well as
earth rotation parameters, are available from the International GNSS Service (IGS) analysis
center at the GFZ. The station coordinates are estimated in a sliding window mode (24 h),
station clock errors are estimated epoch by epoch, and ZTDs and tropospheric gradients
are estimated every 15 min. The a prior ZHD comes from the GPT, and the hydrostatic
and the wet MF is taken from the GMF. For details, the reader is referred to [
38
]. Notably,
the ZTDs provided by GFZ are used by several European weather services, such as the
Met Office, the United Kingdom’s national weather service; Météo-France, the French na-
tional meteorological service; and DWD, the German Weather service, for their day-by-day
weather forecasts. Thus, the high quality of the tropospheric products must be ensured.
As a part of the quality control at the GFZ, the tropospheric products are compared daily
to the corresponding tropospheric parameters derived from the GFS of the NCEP. We
utilized short range forecasts (the forecast length ranges from 6 to 11 h), available four
times per day, to obtain refractivity fields valid for every hour. These refractivity fields are
available with a horizontal resolution of 0.25
◦
. Given these refractivity fields, we computed
the station-specific ZHDs, ZWDs, the three hydrostatic (wet) MF coefficients, the two
Remote Sens. 2021,13, 3944 14 of 18
tropospheric gradient components, and, recently, also the tropospheric parameter Z
0
. As
an example, Figure 8shows a one-to-one comparison of GNSS and NWM ZWDs for the
area of interest on the 15 May 2021 at 11 UTC. An inspection by eye indicates that the GNSS
and NWM ZWDs agree fairly well. We can also measure this agreement; e.g., the mean
and standard deviation between the GNSS and NWM ZWDs. We find that these mean
and standard deviations equal
−
5.0 mm and 6.2 mm, respectively. The deviations are a
composite of the error of the GNSS ZWDs and the error of the NWM ZWDs. For example,
we think that the origin of the negative bias is the NWM ZWDs [
35
]. The GNSS ZWDs
include errors caused by the tropospheric mismodelling. For example, our simulation
study suggested that the error in the GNSS ZWD caused by ignoring the tropospheric
parameter Z
0
in the GNSS analysis is about 1 mm. Hence, we corrected the GNSS ZWDs
a posteriori, utilizing the correction provided in Equation (20) and found that the mean
and standard deviations equal
−
5.1 mm and 5.8 mm, respectively. The mean deviation is
hardly affected. As mentioned before, the tropospheric parameter Z
0
introduces a random,
and not systematic, error. The interesting point is the reduction in the standard deviation,
which is small; however, for this particular epoch, it reached several percent. In order
to show that the reduction in the standard deviation was not accidental, we show in the
upper panel of Figure 9the time series of the reduction in the standard deviation with
a temporal resolution of 1 h for the month of May in 2021. The standard deviations are
reduced by a small amount, but they are reduced systematically. For the considered period
we found a reduction of the standard deviation for every hour. On average, the reduction
in the standard deviation is about 5%. The lower panel of Figure 9puts some weight on
this statistic, as it shows the sample number that enters the statistic. There are some epochs
with missing data, due to the NRT processing of the GNSS data, but in general the sample
number per epoch is above 200. In conclusion, the small but systematic reduction in the
standard deviation gives some evidence that current NWMs can predict the tropospheric
parameter Z
0
. This is not too surprising. A one-to-one comparison in Figure 8shows that
the NWM can predict the ZWD and, hence, the water vapor field fairly well; therefore, we
have good reason to believe that the NWM can predict the tropospheric parameter Z0.
Remote Sens. 2021, 13, x FOR PEER REVIEW 14 of 18
as the Met Office, the United Kingdom’s national weather service; Météo-France, the
French national meteorological service; and DWD, the German Weather service, for their
day-by-day weather forecasts. Thus, the high quality of the tropospheric products must
be ensured. As a part of the quality control at the GFZ, the tropospheric products are
compared daily to the corresponding tropospheric parameters derived from the GFS of
the NCEP. We utilized short range forecasts (the forecast length ranges from 6 to 11 h),
available four times per day, to obtain refractivity fields valid for every hour. These re-
fractivity fields are available with a horizontal resolution of 0.25°. Given these refractivity
fields, we computed the station-specific ZHDs, ZWDs, the three hydrostatic (wet) MF co-
efficients, the two tropospheric gradient components, and, recently, also the tropospheric
parameter Z0. As an example, Figure 8 shows a one-to-one comparison of GNSS and
NWM ZWDs for the area of interest on the 15 May 2021 at 11 UTC. An inspection by eye
indicates that the GNSS and NWM ZWDs agree fairly well. We can also measure this
agreement; e.g., the mean and standard deviation between the GNSS and NWM ZWDs.
We find that these mean and standard deviations equal −5.0 mm and 6.2 mm, respectively.
The deviations are a composite of the error of the GNSS ZWDs and the error of the NWM
ZWDs. For example, we think that the origin of the negative bias is the NWM ZWDs [35].
The GNSS ZWDs include errors caused by the tropospheric mismodelling. For example,
our simulation study suggested that the error in the GNSS ZWD caused by ignoring the
tropospheric parameter Z0 in the GNSS analysis is about 1 mm. Hence, we corrected the
GNSS ZWDs a posteriori, utilizing the correction provided in Equation (20) and found
that the mean and standard deviations equal −5.1 mm and 5.8 mm, respectively. The mean
deviation is hardly affected. As mentioned before, the tropospheric parameter Z0 intro-
duces a random, and not systematic, error. The interesting point is the reduction in the
standard deviation, which is small; however, for this particular epoch, it reached several
percent. In order to show that the reduction in the standard deviation was not accidental,
we show in the upper panel of Figure 9 the time series of the reduction in the standard
deviation with a temporal resolution of 1 h for the month of May in 2021. The standard
deviations are reduced by a small amount, but they are reduced systematically. For the
considered period we found a reduction of the standard deviation for every hour. On av-
erage, the reduction in the standard deviation is about 5%. The lower panel of Figure 9
puts some weight on this statistic, as it shows the sample number that enters the statistic.
There are some epochs with missing data, due to the NRT processing of the GNSS data,
but in general the sample number per epoch is above 200. In conclusion, the small but
systematic reduction in the standard deviation gives some evidence that current NWMs
can predict the tropospheric parameter Z0. This is not too surprising. A one-to-one com-
parison in Figure 8 shows that the NWM can predict the ZWD and, hence, the water vapor
field fairly well; therefore, we have good reason to believe that the NWM can predict the
tropospheric parameter Z0.
Figure 8. A one-to-one comparison of GNSS and NWM ZWDs in meters on the 15 May 2021 at 11
UTC. Left panel: the ZWDs are estimated with the GNSS. Right panel: the ZWDs are derived from
the NWM. For the considered epoch about 300 stations provide ZTDs. The numbers in the left panel
Figure 8.
A one-to-one comparison of GNSS and NWM ZWDs in meters on the 15 May 2021 at
11 UTC.
Left panel
: the ZWDs are estimated with the GNSS.
Right panel
: the ZWDs are derived
from the NWM. For the considered epoch about 300 stations provide ZTDs. The numbers in the
left panel correspond to the mean and standard deviation between the GNSS and NWM ZTDs. The
numbers with the yellow background correspond to the case when we corrected the GNSS ZTDs a
posteriori. For details, refer to the text.
Remote Sens. 2021,13, 3944 15 of 18
Remote Sens. 2021, 13, x FOR PEER REVIEW 15 of 18
correspond to the mean and standard deviation between the GNSS and NWM ZTDs. The numbers
with the yellow background correspond to the case when we corrected the GNSS ZTDs a posteriori.
For details, refer to the text.
Figure 9. Upper panel: the reduction in the standard deviation in percentage, as a function of time,
when GNSS ZWD estimates are corrected a posterior for the month of May in 2021. The number
with the yellow background corresponds to the average reduction in the standard deviation. Lower
panel: the number of samples (corresponding to the number of stations that provide ZTDs), as a
function of the time.
5. Conclusions
We analyzed the impact of tropospheric mismodelling on the estimated parameters
in PPP: station coordinates, clocks, zenith delays, and tropospheric gradients. The true
state of the atmosphere is unknown; hence, we performed a simulation study. This was
done by mimicking PPP in an artificial environment; i.e., we used a linearized observation
equation, where the observed minus modelled term equals the tropospheric delays de-
rived from a NWM. The model that we utilized had a horizontal resolution of 10 km and
was thus on the edge between meso-beta and meso-gamma scale models. Thus, it can
resolve some, but not all, small-scale atmospheric features. A more realistic description of
the true state of the atmosphere is possible by increasing the horizontal resolution, and
this is what we suggest for future studies. We utilized a limited area and time; i.e., our
results are representative for central Europe and the warm season. Different setups, in
particular in the tropics, are recommended for future simulation experiments. Stations in
complex terrain should be carefully analyzed, with the respective high-resolution NWM
data. Having these limitations in mind, we can draw the following conclusions:
1. The quality of GNSS estimates is weather dependent. The reason being that the trop-
ospheric delay model is inaccurate. This tropospheric delay model is based on a cli-
matology or a NWM, and neither of these can be regarded as error free. The larger
the deviation of the climatology or the NWM from the true state of the atmosphere,
the larger the errors in the GNSS estimates. This is obvious and claimed in many
studies; however, it is not trivial to provide some numbers. We provide such num-
bers. In essence, for the considered area and period, when the climatology was uti-
lized in the tropospheric delay model, the error in the estimated ZTD and station up-
component was about 2.9 mm and 5.7 mm, respectively. This error must be under-
stood in a statistical sense; the individual errors can be roughly five times larger.
2. The error in the GNSS estimates can be reduced significantly if the true ZHD and the
true hydrostatic MF are utilized in the GNSS analysis; in this case the error in the
estimated ZTD and station up-component will be reduced to about 2.0 mm and 2.9
mm respectively. The true ZHD and the true hydrostatic MF are unknown. However,
Figure 9. Upper panel
: the reduction in the standard deviation in percentage, as a function of time,
when GNSS ZWD estimates are corrected a posterior for the month of May in 2021. The number
with the yellow background corresponds to the average reduction in the standard deviation.
Lower
panel
: the number of samples (corresponding to the number of stations that provide ZTDs), as a
function of the time.
5. Conclusions
We analyzed the impact of tropospheric mismodelling on the estimated parameters
in PPP: station coordinates, clocks, zenith delays, and tropospheric gradients. The true
state of the atmosphere is unknown; hence, we performed a simulation study. This was
done by mimicking PPP in an artificial environment; i.e., we used a linearized observation
equation, where the observed minus modelled term equals the tropospheric delays derived
from a NWM. The model that we utilized had a horizontal resolution of 10 km and was
thus on the edge between meso-beta and meso-gamma scale models. Thus, it can resolve
some, but not all, small-scale atmospheric features. A more realistic description of the
true state of the atmosphere is possible by increasing the horizontal resolution, and this is
what we suggest for future studies. We utilized a limited area and time; i.e., our results are
representative for central Europe and the warm season. Different setups, in particular in the
tropics, are recommended for future simulation experiments. Stations in complex terrain
should be carefully analyzed, with the respective high-resolution NWM data. Having these
limitations in mind, we can draw the following conclusions:
1.
The quality of GNSS estimates is weather dependent. The reason being that the
tropospheric delay model is inaccurate. This tropospheric delay model is based on a
climatology or a NWM, and neither of these can be regarded as error free. The larger
the deviation of the climatology or the NWM from the true state of the atmosphere, the
larger the errors in the GNSS estimates. This is obvious and claimed in many studies;
however, it is not trivial to provide some numbers. We provide such numbers. In
essence, for the considered area and period, when the climatology was utilized in the
tropospheric delay model, the error in the estimated ZTD and station up-component
was about 2.9 mm and 5.7 mm, respectively. This error must be understood in a
statistical sense; the individual errors can be roughly five times larger.
2.
The error in the GNSS estimates can be reduced significantly if the true ZHD and
the true hydrostatic MF are utilized in the GNSS analysis; in this case the error
in the estimated ZTD and station up-component will be reduced to about 2.0 mm
and 2.9 mm respectively. The true ZHD and the true hydrostatic MF are unknown.
However, there is good reason to believe that the ZHD and the hydrostatic MF
from a NWM (analysis or short range forecast) are close to the true ZHD and the
true hydrostatic MF. This also explains the success of the NWM based MFs that are
currently recommended for analyzing space geodetic data [
10
]. An indication of
Remote Sens. 2021,13, 3944 16 of 18
the high quality of the NWMs in this respect is the agreement between them. For
example, the two state of the art NWM based tropospheric delay models, VMF1 and
UNB-VMF1, gave almost the same results in a geodetic analysis [
14
]. There were
some differences for specific locations and times, but a clear advantage of one NWM
over the other was not demonstrated in a statistical sense.
3.
The simulation study showed that the wet MF, when calculated under the assumption
of a spherically layered troposphere, did not matter too much. In essence, when
the true wet MF (calculated from the true refractivity profile above the station in
question) is utilized in the GNSS analysis, the error in the estimated ZTD and station
up-component remains about 1.8 mm and 2.4 mm, respectively. This is somewhat
surprising; however, it is in line with the results from [
9
]. This study demonstrates
that the improvement that may be realized for the wet MF is significantly less than
for the hydrostatic MF. Reference [
9
] stated that: “
. . .
Contrary to the expectation
that water vapor is the major source of error, the height error is dominated by the
hydrostatic component, except possibly near the equator. Thus it is clear that an
improvement in the hydrostatic component will have the larger impact on improving
the measurement of height and atmosphere delay by VLBI and GPS”. However,
the assumption is that the atmosphere is spherically layered. This brings us to the
next item.
4.
We developed a modified wet MF, which is no longer based on the assumption of a
spherically layered atmosphere. We have shown that with this modified wet MF the
error in the estimated ZTD and station up-component can be reduced to about 0.5 mm
and 1.0 mm, respectively. Its success, in practice, depends on the current (future)
NWMs predicting the fourth coefficient of the developed closed form expression. We
provided some evidence that current NWMs are able to do so. The determination of
the modified wet MF is more expensive than the standard wet MF; the determination
of the standard wet MF requires 10 tropospheric delays, whereas the determination of
the modified wet MF requires an additional 120 tropospheric delays. The algorithm
by [
25
], which calculates tropospheric delays with high speed and precision on an off-
the-shelf PC also makes this feasible for real-time applications. We routinely calculate
the additional 120 tropospheric delays, because they are required to determine the
tropospheric gradients. The modified wet MF is a by-product of the simulation
study (the primary purpose of our study was to quantify the tropospheric delay
mismodelling). A more sophisticated MF concept could certainly be developed. In
this respect, the newly developed VMF3 [
15
] sets a new standard, and we are going
to analyze if we should follow this concept or refine our current concept in the future.
Author Contributions:
Conceptualization, F.Z. and K.B.; methodology, F.Z. and K.B.; software, F.Z.
and G.D.; validation, F.Z.; formal analysis, F.Z.; investigation, F.Z. and K.B.; resources, F.Z. and G.D.;
data curation, F.Z. and G.D.; writing—original draft preparation, F.Z., K.W. and J.W.; writing—review
and editing, F.Z., K.B., K.W., G.D. and J.W.; visualization, F.Z. and K.W.; All authors have read and
agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The operational GNSS tropospheric products and the correspond-
ing NWM tropospheric products are available via ftp://ftp.gfz-potsdam.de/pub/home/GNSS/
products/nrttrop/ (accessed on 12 August 2021).
Acknowledgments:
The Global Forecast System data are provided by the National Centres for Envi-
ronmental Prediction (https://www.nco.ncep.noaa.gov/pmb/products/gfs/, accessed on 12 August
2021). This study was performed under the framework of the Deutsche Forschungsgemeinschaft
(DFG) project Advanced MUlti-GNSS Array for Monitoring Severe Weather Events (AMUSE) number
41887048 and the DFG project Exploitation of GNSS tropospheric gradients for severe weather Moni-
Remote Sens. 2021,13, 3944 17 of 18
toring And Prediction (EGMAP) number 443676585. The three reviewers are gratefully acknowledged
for their comments, which helped to improve the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
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