Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12
https://doi.org/10.1186/s40645-024-00613-2

RESEARCH ARTICLE

Progress in Earth andPlanetary Science

Open AccessTroposphere delay modeling in SLR basedon PMF, VMF3o, and meteorological data
Mateusz Drożdżewski1*   and Krzysztof Sośnica1   

Abstract
Satellite laser ranging (SLR) requires accurate troposphere delay models to properly correct the observed distances
to satellites and derive fundamental geodetic and geodynamic parameters. The currently used models for the tropospheric delay employ in situ meteorological data collected simultaneously with laser measurements. However,the standard models assume full symmetry of the atmosphere above the SLR stations because all meteorological data
come from one sensor. In this study, we evaluate various methods of troposphere delay modeling based on numerical weather models, such as the Potsdam Mapping Function (PMF) and Vienna Mapping Function for optical frequencies (VMF3o), in situ measurements, the Wrocław gradient model (WGM) and the combination of different models.
We found large discrepancies between pressure, temperature, and humidity records between in situ measurements
and numerical models. The best results for the zenith delays are obtained when using in situ meteorological datawith the estimation of tropospheric biases. For stations with some deficiencies in proper humidity measurements, e.g.,
Zimmerwald in Switzerland, the best results are obtained when using hydrostatic zenith delays based on in situ data
and wet delays based on numerical weather models. Finally, we found that using horizontal gradients of the tropospheric delay is indispensable to avoid biases in the SLR-based Earth rotation parameters of approximately 20 µas
for polar motion. The horizontal gradients successfully account for the asymmetry of the troposphere above SLR stations and can be derived from PMF, VMF3o, or a parameterized WGM model with similar accuracies.
Keywords Troposphere delay, Satellite laser ranging, Geocenter coordinates, Earth rotation parameters, Horizontal
gradients
1 Introduction
The integrated space geodetic techniques contribute to
the International Terrestrial Reference Frame (ITRF)(Altamimi et al. 2023) realizations by deriving crucial
information about the origin, orientation, and geometric scale datum. The influence of the lowest layer of theatmosphere, the troposphere, is one of the major sourcesof errors in data processing of space geodetic techniqueobservations. In recent years, we observed a vast number*Correspondence:
Mateusz Drożdżewski
mateusz.drozdzewski@igig.up.wroc.pl
1
Institute of Geodesy and Geoinformatics, Wrocław Universityof Environmental and Life Sciences, Grunwaldzka 53, 50‑375 Wrocław,
Lower Silesia, Polandof studies related to tropospheric modeling for microwave-based geodetic techniques, such as Global Navigation Satellite Systems (GNSS) (Tregoning and Herring2006) and Very Long Baseline Interferometry (VLBI)(Boehm et al. 2006; Zus et al. 2014; Landskron and Böhm
2018; Wang et al. 2022). Moreover, the tropospheric delay
is also a subject of scientific interest in other techniques,
such as satellite altimetry or Synthetic Aperture Radar
Interferometry (InSAR), where local atmospheric variations significantly deteriorate interferograms (Wilgan
et al. 2019). One of the main factors limiting the accuracy of standard Satellite Laser Ranging (SLR) solutions
is neglecting some atmospheric effects, such as atmospheric asymmetry above the stations. The impact of horizontal refractivity on SLR was introduced by Abshire andGardner (1985), who proposed implementing two-color© The Author(s) 2024. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to theoriginal author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images orother third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.

Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

laser systems at several stations located on different
continents and tracking radiosondes equipped with retroreflectors and comparing the results to atmosphericmodels. Degnan (1993) reported that although measurements are conducted in a clear sky in many SLR observations, the emitted photons pass through the invisible
parts of cirrus clouds, introducing unconsidered atmospheric delays.
The currently used tropospheric delay model in SLR
consists of a common mapping function for the sumof hydrostatic and non-hydrostatic delays. The modelis based on one year of radiosonde data (Mendes et al.
2002). The Mendes–Pavlis model for the mapping function (Mendes et al. 2002) requires a temperature record
derived from the meteorological station at the epoch ofobservation. Hulley and Pavlis (2007) proposed the raytracing method to compute the slant total delay for eachobservation at the site. The simulation proposed by Hulley and Pavlis (2007) improved the residuals of observations, especially for low elevation angles at 10 degrees;
additionally, the proposed model considers the atmospheric asymmetry above the station. Hulley and Pavlis(2007) demonstrated that the variance of SLR observation residuals can be reduced from 10–43% dependingon the source of the numerical weather model (NWM)
used. Wijaya and Brunner (2011) derived a formula thatcan be used to determine the total troposphere delay forstations equipped with two laser frequencies (2F-SLR)
by considering the impact of the non-hydrostatic delay.
In that case, the water vapor pressure could be delivered from the GNSS station co-located with the SLR
station. However, the accuracy of current photodetectors is beyond the range to render this formula useful inthe operational mode in two-color ranging. The further
proposals of troposphere delay models, mapping functions, and horizontal gradient models for SLR are basedon NWM. These models are called the Potsdam Mapping
Function (PMF) (Zus et al. 2015) and the Vienna Mapping Function 3 for optical frequencies (VMF3o) (Boisits
et al. 2020) and employ ERA5 data and operational
products of the European Center of Medium Weather
Forecast (ECMWF) for PMF and VMF3o, respectively.
However, none of the above-mentioned atmosphericmodels can fully handle the atmospheric turbulence that
occurs above the stations. In PMF, due to the low sensitivity of SLR observations to the wet part of the troposphere delay, the separation of the mapping function for
laser frequencies is neglected. This approach is consistent
with the statement from the International Earth Rotation and Reference Systems Service (IERS) Conventions
2010 (Petit and Luzum 2010) concerning the delays oflaser techniques in the troposphere. In contrast, VMF3o
proposed by Boisits (2018) introduced the concept of the

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separated mapping function to the hydrostatic and nonhydrostatic model of tropospheric delay, as well as separated zenith total delay (ZTD) and horizontal gradients.
The VMF3o approach allows additional experiments tobe conducted with a mixed approach of zenith wet delay(ZWD). In almost all of the analyzed cases in this paper,
we use the zenith hydrostatic delay (ZHD) and ZWD
based on meteorological measurements derived from theco-located meteorological stations at the SLR site.
Drożdżewski et al. (2019) noticed that the Mendes–
Pavlis model for the ZTD and ZWD extended by PMFbased horizontal gradients improves the consistency
between SLR and microwave-based techniques (GNSS,
VLBI) by reducing the mean long-term offset at thelevel of 20 µas for the X and Y components of the polar
motion. Drożdżewski and Sośnica (2021) introduced
the troposphere delay correction in SLR in the form of
estimated tropospheric biases. The authors also mentioned that other errors, which have similar nature to
the troposphere delay correction, such as intensity
biases, are also absorbed by estimated troposphericbiases. The authors described the impact of tropospheric biases, such as station coordinates, geocenter
coordinates, Earth rotation parameters, and low-degree
spherical harmonics of Earth’s gravity field based on
SLR observations to LAGEOS satellites. Strugarek et al.
(2022) confirmed that the estimation of tropospherecorrection removes the skewness from the observation
residuals when processing SLR data to low Earth orbiters. Strugarek et al. (2022) confirmed that using theestimation of troposphere delay removes other elevation-dependent errors. However, one should be carefulwith estimating tropospheric corrections due to strong
correlations with other parameters, such as horizontal components of station coordinates and horizontal gradients. Moreover, for a limited number of SLR
observations, constraining tropospheric parameters is
indispensable. In this paper, we provide a comparisonof the recently developed tropospheric models for SLR,
i.e., PMF and VMF3o, and the impact of the mapping
function differences, horizontal gradients, and ZTD.
These models are shown first as a comparison of input
data, and second, as the impact on the crucial geodetic
parameters, such as station coordinates and geocenter
coordinates, on an example of SLR observations to passive geodetic satellites LAGEOS 1 and LAGEOS 2. PMF
and VMF3o are compared to the standard Mendes-Pavlis model that employs meteorological records for tropospheric delays. The corrections to the troposphericdelays are estimated for each model to verify the levelof tropospheric biases. Finally, the mixed approach istested with selected data from meteorological sensors
and some from NWM.

Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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2 Methods/experimental
The currently recommended IERS mapping function
FCULa (Mendes et al. 2002) is based on one year ofradiosonde data from 1999 and in situ temperature, pressure, and humidity records derived from meteorologicalstations. The new approaches to determining mapping
functions for SLR, such as PMF and VMF3o, are based
only on NWM ERA5 and ECMWF operational forecasts, respectively. Table 1 provides the selected differences between the ERA5 and ECMWF operationalmodels (Hersbach et al. 2020). Drożdżewski et al. (2019)
showed the benefits of applying PMF based on the ERAinterim reanalysis. The new release of ECMWF reanalysis the so-called ERA5 distinguishes better spatial
and temporal resolution with respect to its predecessor, ERA-interim. Both NWMs are also based on in situ
measurements of temperature, pressure, and humidity.
Therefore, the mixed approach of mapping function andgradients based on NWM and ZTD based on in situ data,as tested in this paper, should still be consistent. Huang
et al. (2023) emphasize that in ERA 5, the annual pressure biases are reaching 0.07 hPa on the surface of 31 km
squared. McGarry et al. (2019) defined that the in situ
atmospheric pressure measured in NASA’s Next-Generation Satellite Laser Ranging (NGSLR) system shouldmeasure the pressure with accuracy at the level of 0.08
hPa. However, in tests conducted by Drożdżewski et al.

(2019) the ZTD derived from NWM resulted in worse
SLR results than when using in situ pressure data.
Even though both models, PMF and VMF3o, are basedon the NWM meteorological analysis, the reader should
be aware of some crucial differences. VMF3o proposes a
mapping function dedicated to laser observations basedon the VMF3 concept (Landskron and Böhm 2018)
which splits the hydrostatic and non-hydrostatic part
for: (1) ZTD, (2) the mapping function, and (3) horizontal gradients. Splitting the ZTD and mapping function
is commonly used in GNSS. Due to differences in thephysical properties of the microwave and optical signals,
dedicated mapping functions for SLR are expected (seeTable 2). The PMF model assumes that due to low sensitivity to the wet part of the troposphere delay, the separation of the mapping function for laser frequencies could
be neglected for observations provided at an elevation
angle above 10 degrees. This concept is consistent with
the IERS 2010 Conventions that recommend the Mendes
et al. (2002) model for the mapping function which combines the hydrostatic and wet parts of the delay. In PMF,
all coefficients are tabularized, whereas VMF3o uses the
expansion into spherical harmonics for selected coefficients. Finally, PMF applies nonlinear gradients using
7 coefficients to describe the atmospheric asymmetry,
whereas VMF3o provides 2 linear gradient coefficientsfor each hydrostatic and wet delay.

Table 1 The main differences between ERA5 and ERA
operational reanalysis (Hersbach et al. 2020)

2.1 Standard solution based on a common mapping
function for ZTD

Model

ERA5

ECMWF-op

Period1979–present

1979–present

Horizontal resolution

0.3° × 0.3°

1° × 1°

Vertical resolution

137 levels to 0.01 hPa

25 levels to 0.01 hPa

Temporal resolution

1h

6h

The standard approach for SLR tropospheric delay modeling (Mendes and Pavlis 2004) consists of a common
mapping function (m(e)) depending on the elevation
angle e for the projection of the sum of ZHD and ZWD(dh , dw ). The mapping function coefficients (a1 , a2 , a3 ) as
well as ZHD and ZWD are based on the in situ records ofTable 2 Characteristics of signal propagation in the atmosphere for GNSS and SLR solutions with respect to the IERS 2010
Conventions
Microwave (e.g., GNSS)

Optical (SLR)

Troposphere

Non-dispersive medium

Dispersive medium

ZHD

Dominated by air pressure approximately
equal to 2.3 m

Dominated by air pressure approximately equal to 2.4 m

ZWD

Dominated by water vapor pressure and dynamicallyvariable up to  0.4 m in the zenith direction

The sensitivity to non-hydrostatic delay up to 0.01 m
in the zenith direction

Mapping function

Based on NWM

Troposphere asymmetry modeling Estimation of horizontal gradients, possibly with a priori
values derived from NWM

Based on Mendes et al. (2002)—radiosonde data
Not considered

Geometry of observations

Good geometry and dense distribution of observations

A limited number of observations

Ionosphere

Dispersive medium

No ionosphere delayDrożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

pressure, temperature, and humidity resulting in the slantdelay (datm):(1)

datm =m(e) · (dh + dw ).
a1
a2
1+ 1+a3m(e) =
a1
sin e +a2
sin e+ sin e+a

1+

(2)

.

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aw
bw
1+ 1+cwm(e)VMF3ow =.
aw
sin e +w
sin e+ sin be+cw

1+

mgh (e) =1.
sin(e) · tan(e) + kh

(10)

1
.
sin(e) · tan(e) + kw

(11)

3

mgw (e) =2.2 PMF solution

The PMF in the original version included 2 linear coefficients for the gradients and 3 additional coefficients fornonlinear dependencies (Drożdżewski et al. 2019). The
updated formula of PMF is consistent with that of Zus
et al. (2021), who extended the gradient model by highorder terms. The PMF dedicated to SLR is adopted for
SLR frequencies equal to 532 nm and reads as:

where datm h describes the hydrostatic delay considering horizontal refractivity projected on elevation anglee, and GNh and GEh describe the hydrostatic part of horizontal gradients for the north and east components. Thea(h,w) , b(h,w) c(h,w) describe the mapping function coefficient for the hydrostatic part (h) and for the wet part (w).
GN w, GE w represent the wet part of the horizontal gra-z
ldatm = datm
· mPMF (e) + (GZ0 + GN · cos A + GE · sin A+ GZ1 · cos 2A + GZ2 · sin 2A + GZ3 · cos 3A + GZ4 · sin 3A) · mg (e).

where mPMF (e) is the mapping function, A describes theazimuth of each observation, GN and GE represent thehorizontal gradients for the north and east components,
respectively, and GZ1–Z4 represent the higher-orderhorizontal gradient terms. The mapping function for thegradients considers both the hydrostatic and wet part ofthe delay with k = 0.0032 (Chen and Herring 1997) being
consistent with the dominating hydrostatic part:mg (e) =1.
sin(e) · tan(e) + k(4)

2.3 VMF3o solution

The VMF3o consists of a separated mapping function as
well as horizontal gradients to the hydrostatic and nonhydrostatic parts (Boisits et al. 2020):(5)

datm =datmh + datmw .

1+m(e)VMF3oh =sin e +

(9)(3)

dients for the north and east components. The mgh , mgw
are functions for projecting horizontal gradients, wherekh is equal to 0.0031 and kw is equal to 0.0007 (Chen andHerring 1997). In the other approaches without separation into hydrostatic and non-hydrostatic parts, the samefunction mg is typically used with a k coefficient equal
to 0.0032. VMF3o provides tabular values for the mapping function: a wet and hydrostatic coefficients andzenith delays dh and dw, whereas b and c are provided
as an expansion into spherical harmonics separately forthe mean values and seasonal signals and with a separation between the hydrostatic and wet parts. Moreover,VMF3o provides tropospheric data from NWM, such astemperature, pressure, and water vapor, interpolated forthe location of the SLR station, allowing for direct comparison with in situ data. The latency of the operational
VMF3o products is 24 h; the new data are available atapproximately 18:00 UTC every day for the previous day.

datmh =dh · m(e)VMF3oh + mgh (e) · (GNh · cosA + GEh · sinA).

(6)

datmw =dw · m(e)VMF3ow + mgw (e) · (GNw · cosA + GEw · sinA).

(7)

ah
b

1+ 1+ch

h

.

ah

b

h
sin e+ sin e+ch(8)Predictions for the next day are generated at 9:00 UTC;
however, they are not publicly available on the VMF website. Therefore, VMF3o can be applied for operational SLR
products, as the latency of the product is approximately
one week.

Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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Fig. 1 Stacked amplitudes of horizontal gradients for the northcomponent of 43 SLR stattions (mm). The x-axis is scaled in days

2.4 WGM solution

The Wrocław Gradient Model (WGM), derived in thisstudy, is based on time series of linear gradients derived
from NWM fitted into an empirical function. The WGM
employs the standard FCULa mapping function derived
by Mendes et al. (2002). For gradients, WGM is based ona fit of the first-degree polynomial with the annual andsemi-annual signals into 16 years of data for the east andnorth components of the total delay, i.e., the sum of thehydrostatic and wet parts. This solution allows for thereconstruction of most of the major signals included in
the horizontal gradients. Figure 1 shows the stacked
amplitudes for the north component of 43 SLR stations.
For most stations, characteristic annual and semi-annual
signals are observed, as well as daily and monthly fluctuations, which were described by Boisits et al. (2020).
Nevertheless, the daily and monthly fluctuations havevariable and small amplitudes; therefore, annual andsemi-annual signals with stable amplitudes were selected
for the simplified models of gradients.

2π2πt) + as1 sin( t)
T
T
4π
4π
+ac2 cos( t) + as2 sin( t).
T
T

Fig. 2 Time series of horizontal gradients (mm) for the northand east components for Arequipa (7403) and Herstmonceux (7840)
with the fit of the WGM for the period 2007–2018. The black solidline represents the annual signal fitted by least squares adjustment
process

f (t) = a0 + a1 t + ac1 cos((12)
where a0 describes the offset, a1 is the drift, as1 and ac1
are the annual components, ac2 , as2 describe the semiannual components, t describes the elapsed days fromthe beginning of a year and T is the length of the year indays. Finally, the slant delay can be estimated as a sum of
the ZTD multiplied by a mapping function and gradients
multiplied by the gradient mapping function developedby Chen and Herring (1997):zdatm = datm
· m(e) + (GN · cosA + GE · sinA) · mg (e).
(13)
To determine this model, the discrete gradients dedicatedto SLR optical observations from both PMF and VMF3o
sources can be used with no significant differences.

However, for WGM derivation in this study, PMF was
employed. The WGM can also be applied for both operational and postprocessing products. The polynomial consists of the annual and semiannual signals that retrieve
only the systematic, deterministic part of the gradients(see Fig. 2). The mean value a0 for the north componentof gradients has a negative value in the Northern Hemisphere, with the maximum value at the level of − 0.5 mm
for Japanese stations (see Fig. 3). The mean value of thenorth gradient in the Southern Hemisphere is positive.
For the east component, the values of the mean offset arein the range from − 0.15 to 0.15 mm; only for a few stations located in South America the values are at the levelof ±0.3 mm. In central Asia and South America, larger
fluctuations in the mean east component occur due tolocal atmospheric conditions and orography.

Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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3.1 Meteorological data records comparison

Fig. 3 The mean offset of the WGM gradient model (mm). Top—
north component, bottom—east component

3 Results and discussion
In this section, we analyze the impact of different
approaches to troposphere delay modeling in mixed
configurations with respect to the standard approach
recommended by the 2010 IERS Conventions (Petit
and Luzum 2010). Figure 4 shows the differences indelays projected at low elevation angles of 7, 15, and 20
degrees for two high-performing SLR stations locatedin the Northern and Southern hemispheres, Wettzell(8834) and Yarragadee (7090). The top figure illustrates
the differences between the PMF and Mendes–Pavlis (MP) mapping function (FCULa), the middle figure
shows the differences between VMF3o and Mendes–
Pavlis, and finally, the bottom figure shows the differences between PMF and VMF3o. The largest differences
are obtained for observations provided at an elevation
angle of 7 degrees. The PMF and VMF3o, which areboth based on NWM, are consistent even for 7 degrees
of elevation angle at the level of 1–2 mm. In the caseof the comparison with the Mendes–Pavlis model, themapping functions generate differences at the levelof ± 8 mm. Figure 5 shows the horizontal gradients
derived from PMF and VMF3o projected at 10 degrees
elevation angle. We observe the highest differences forstation Mt Stromlo at the level of 3 mm. In general,
we can perceive very good agreements between PMF
and VMF3o solutions, which confirm that the separation of mapping function in SLR measurements can beneglected.

The major difference between PMF, VMF3o, andFCULa is the source of input data. PMF and VMF3o
are based on the pressure, temperature and humidity derived from NWM, but only VMF3o products
make publicly available the discrete values of temperature, pressure, and water vapor pressure with a 6-hourtime span (Boisits et al. 2020). Hence, we decided tocompare the in situ meteorological data with discrete
data derived by VMF3o interpolated to the epochs ofactual SLR observations. Figure 6 shows differencesin the time series of pressure records for selected stations. Drożdżewski and Sośnica (2021), introduced
the estimation of troposphere delay corrections as amethod to detect and handle barometer malfunctioning. The estimated ZTD values can detect pressure
jumps that occurred recently, e.g., at the Wettzell andGraz stations. Boisits et al. (2020) showed the differences between in situ pressure and temperature measurements and NWM. However, the question of theseinconsistencies and the application of this informationby the SLR community still remains open. Although thepressure derived from NWM is still beyond the accuracy of the in situ pressure measurements, the analysis
of pressure differences with an assessment of estimatedstation coordinates may help expose the undetected
and unmodeled pressure jumps in the Inernational
Laser Ranging Service (ILRS) data time series. The
SLR products are very sensitive to pressure biases; as aresult, undetected pressure issues may deteriorate crucial geodetic parameters, such as station coordinates,
the scale of the reference frame, or geocenter coordinates (Drożdżewski and Sośnica 2021).
Figures 7 and 8 show the comparison of in situ temperature and humidity measurements and VMF3o values for selected stations with an approximate separationof the seasons of the year. For station Haleakala (7119),
we observe a correlation between in situ measurements
and VMF3o of 0.58 and an interquartile range (IQR) ofdifferences between records at the level of 3.6 ◦C (seeTable 3). This value indicates a rather poor agreement
and a substantial discrepancy between in situ measurements and VMF3o, which will be investigated in the further part of this article. Station San Juan (7406) is locatedin Argentina at a height of 727 m above sea level. For thisSLR site, the lowest temperature is noticed in the middleof the year (red dots in Fig. 7); for that period, the highest discrepancies can be noticed. Figure 8 illustrates thecomparison of water vapor pressure (WVP) between
in situ and VMF3o records. The WVP impact in SLRDrożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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Fig. 4 Differences between mapping function values in 2008 for stations Wettzell and Yarragadee located in the Northern and Southern
Hemispheres, respectively. The differences are projected at the levels of 7, 15, and 20 degrees and shown in red, green, and blue, respectively

measurements contributes to approximately 1% of theZTD values, which is approximately equal to subcentimeter values in zenith. However, even several millimetersof the discrepancies in the zenith direction may resultin a few centimeters of errors for observations projected
at low elevation angles. For stations Matera (7941) andZimmerwald (7810), the in situ measurements agree
quite well with VMF3o in the winter, spring, and autumn,
whereas large discrepancies occur in the summer, where
VMF3o is remarkably lower than the in situ records, evenby one third.

3.2 Troposphere delay bias as a tool for mapping function
validation

We test different approaches to zenith delay modelingwith and without separation between wet and hydrostaticdelays for zenith delays and mapping functions, considering horizontal gradients from VMF3o, PMF, and simplified model WGM, as well as mixed approaches, in whichsome information is taken from NWM and some is basedon in situ measurements. The tested solutions are shownin Table 4. Moreover, ZTD corrections to a priori values are also estimated in selected solutions to identifyDrożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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Fig. 5 Comparison of PMF and VMF3o Horizontal gradients projected onto 10◦ elevation angle for selected stations: Herstmonceux, Mt Stromlo,
Graz and Yarragadee. O1 denotes the linear gradients from PMF; O2 denotes the higher-order gradient terms in PMF

tropospheric biases (TRP). The different approaches tomodeling the troposphere delay in SLR may be assessedby the estimated troposphere delay corrections for eachstation (Landskron and Böhm 2018). The same constraint for ZTD correction equal to 0.01 m with respect
to the a priori value was applied in all solutions. The firsttwo solutions from Table 4, Mendes–Pavlis (M–P), correspond to the approach currently used by the ILRS in
operational SLR products, except for the ZTD bias estimation in the second solution (M–P TRP). In the ILRS
approach, all tropospheric delay components are derived
from in situ measurements. In other solutions from
Table 4, some components were replaced by corresponding values based on NWM, e.g., ZWD or ZHD.
Figure 9 shows the box plots of estimated ZTD corrections from solutions using different a priori delays andmodeling. For some stations, the estimated corrections
agree between solutions using in situ and NWM data;
for some other stations, the corrections are different,

especially for those where atmospheric pressure records
are employed from NWM. For station Simosato (7838),
the models based on atmospheric pressure derived from
NWM, i.e., VMF3o ZTD and PMF ZTD, lead to a positive
estimated ZTD bias, as opposed to all other approaches,in which ZHD is based on in situ atmospheric pressurerecords (M–P, VMF3o ZWD, VMF3o, M–P model). The
atmospheric pressure records from NWM can thus bebiased because solutions using NWM pressure lead to amuch larger bias for Simosato than when based on in situ
pressure data. The pressure differences together withestimated troposphere delay correction are reliable methods to identify and handle such kinds of pressure errors.
For Greenbelt and Concepcion, the solutions based on
NWM lead to large negative tropospheric biases of several millimeters. For Wettzell, most of the solutions
provide similar results, suggesting that the in situ atmospheric pressure measurements are incorrect. In Concepcion, using in situ atmospheric pressure measurementsDrożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

Fig. 6 Differences between pressure records derived
from in situ records at SLR stations and VMF3o based on NWM
with discontinuities detected in the period from 2008 to 2019
for Grasse, France (7845), Simosato, Japan (7838), Wettzell, Germany(8834) and Matera, Italy (7941). The green and red solid line represent
the annual signal fitted by least squares adjustment

leads to a positive bias of approximately 2.5 mm, whereas
the NWM pressure values lead to estimates of large negative biases out of range of Fig. 9. This implies that both
NWM and in situ data are inconsistent with actual tropospheric states in Concepcion. For Potsdam (7841) andYarragadee (7090), differences between estimated tropospheric biases are below 1 mm for different approaches.
Although PMF and VMF3o are based on the ECMWF
products, the results slightly differ from each other forthe same stations. For Potsdam (7841), Grasse (7845),
and Tahiti (7124), the median value of tropospheric bias

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is the smallest in VMF3o ZTD, which uses NWM forthe a priori atmospheric pressure.The complete PMF
model (PMF ZTD) provides the smallest biases for
Hartebesthoek (7501), Wettzell (8834), and Yarragadee(7090); however, for the mean biases, PMF-based biases
do not change much for these stations. In the case of utilizing the wet part of the troposphere delay from NWM(VMF3o ZWD), a ZTD bias reduction at the level of 0.2
mm is noticed for station Matera (7941). In the case ofsolutions PMF and VMF3o, the small estimated ZTD
bias is related to improvement emerging from using the
mapping function based on NWM data instead of in situ
measurements. Using PMF leads to reduced biases for
Tahiti (7124), Greenbelt (7105), Matera (7941), and Zimmerwald (7810). The mean bias reduction introduced
by VMF3o is at the levels of 0.1, 0.2, 0.1, and 0.2 mm in
the zenith direction for stations Graz (7839), Potsdam(7841), and Matera (7941), respectively. For stations Zimmerwald (7810) and Mt Stromlo (7825), the reduction of
the interquartile range (IQR) in solution VMF3o ZWD
has been noticed as well as a reduction of the range of
the whiskers. For Zimmerwald, discrepancies between
in situ measurements and NWM were detected in thesummertime (see Fig. 8). The smallest IQR values fromthe VMF3o ZWD TRP solutions mean that the best solution can be obtained when the hydrostatic zenith delay
is based on in situ data, whereas the wet delay is derived
on the basis of NWM for Zimmerwald. Therefore, theunique feature of VMF3o with the separation between
the wet and hydrostatic delay may be useful to accountfor the errors of WVP recording. For station Matera(7941), where the inconsistency between WVP derived
from in situ measurements and VMF3o has been noticed,
we do not observe any improvements in the solutions
with ZHD based on NWM.
3.3 Station coordinate repeatability

Figure 10 illustrates the 3d RMS of station coordinates’
repeatability based on observations from the period
2010.0–2019.0 for selected representative solutions.
Only the M–P solution does not include the estimationof tropospheric corrections, whereas all other solutions
use ZTD values from different sources with zenith troposphere corrections estimated. Figure 10 shows that thedifference in the station coordinate repeatability is minor
irrespective of the a priori ZTD values used when estimating tropospheric corrections. Figure 11 shows the difference in station coordinate repeatability with respect tothe M–P solution with no troposphere corrections estimated for the north, east, and up components. The horizontal gradients from NWM or a simple WGM model
especially improve the horizontal components of station
coordinates, which is visible for the north component.

Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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Table 3 Statistics of temperature and humidity differences between in situ measurements and VMF3o for selected SLR stations
StationTemperature
Mean Δ°C

IQR Δ%

Correlation

McDonald (7080)

1.9

1.6

0.92

Haleakala (7119)

− 2.0

3.6

0.58

Herstmonceux (7840)

2.0

4.1

0.90

San Juan (7406)

4.8

5.5

0.87

Station

Humidity
IQR Δ%

Correlation

Mean Δ%
Mt Stromlo (7825)

− 0.3

1.3

0.87

Herstmonceux (7840)

0.4

1.4

0.90

Matera (7941)

0.3

2.4

0.81

Zimmerwald (7810)

− 1.2

2.1

0.89

The estimated ZTD correction mainly impacts the upcomponent of the station coordinates. Improvements
in station coordinate repeatability above 1 mm wereobserved for stations Shanghai (7821), Simosato (7838),
and Changchun (7237). Improvement at the level ofhalf mm was noticed for stations Haleakala (7119),
Grasse (7845), and Zimmerwald (7810). The deterioration of average values of station coordinate repeatability
has been noticed for stations Monument Peak (7110)
and Yarragadee (7090). For station Shanghai (7821), we
observe an improvement exceeding 1 mm. For stations
Changchun (7237) and Simosato (7838), the improvement is at the level of 2 mm even for solutions with ZTD
derived from NWM w.r.t. the standard M–P solution.
The improvement of horizontal components is mainly
related to considering horizontal gradients.
3.4 Earth rotation parametersDrożdżewski et al. (2019) noticed that neglecting horizontal gradients leads to the deterioration of the consistency between SLR and IERS-C04-14 Earth Rotation
Parameters (ERPs). Indeed, Table 6 shows that the horizontal gradients reduce the mean offsets for the X andY pole coordinate components at the level of 20 µas
between the SLR solutions with and without gradientsand the IERS-C04 series. The horizontal gradients canbe taken from a simplified WGM parametrized modelof horizontal gradients (M–P model) or NWM (PMF or
VMF3o). In the case in which the ZTD model is usedfrom NWM without estimating ZTD corrections, theX and Y pole coordinates deteriorate in terms of offset
between the IERS-C04-14 combined solution and mean
sigma (see Table 5). When the ZTD from NWM is used,
the troposphere delay correction has to be estimated;otherwise, the ZTD derived only from NWM introduces

deterioration of the X and Y ERP components at the levels of 22 and 8 µas, respectively (see Table 5). In solutions,
VMF3o ZTD TRP and PMF ZTD TRP, the length-of-day(LOD) parameter deteriorates at the level of 18 and 20 µs/day for solutions PMF ZTD no est. and VMF3o ZTD
no est., respectively. Thus, the estimated tropospheredelay correction is not able to fully account for the large
sigma and offsets observed in pole coordinates when
the a priori ZTD values are biased. The aim of testing
the solution VMF3o ZWD TRP is to show the impact of
the separation mapping function on the crucial geodetic
parameters as well as to validate the SLR in situ humidity records. However, the solution VMF3o ZWD TRP is
consistent with the solution PMF TRP, where the separation of the mapping function is neglected. The mostconsistent solution with the IERS-C04-14 product is thesolution PMF TRP. Table 5 does not indicate one solution
with superior performance with respect to the other solutions. However, the best solutions are obtained when (1)horizontal gradients are considered, (2) ZHD is based onin situ meteorological data, and (3) tropospheric correction is estimated. The selection of the mapping function
and the separation between wet and hydrostatic delays
play a minor role. The set of best-performing models
includes the VMF3o TRP, VMF3o ZWD TRP, PMF TRP,
and M–P model TRP. Therefore, it is sufficient to consider tropospheric biases and gradients from the WGM
to reduce the mean pole offsets by approximately 20 µas.
3.5 Geocenter coordinates

We compare the solutions with horizontal gradients,
the mapping function derived from NWM (PMF or
VMF3o) and the ZTD correction with the M–P standard ILRS solution for the geocenter coordinates. Next,Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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Fig. 7 Comparison of temperature from in situ measurements at SLR stations and VMF3o interpolated for the SLR sites in years 2008–2012
in Celsius degrees as a function of the day of the year (color scale: winter-blue, spring-green, summer-red, fall-yellow in the Northern hemisphere)

Table 4 Description of analyzed solutions
Solution

ZTD bias
estimation

Horizontal
gradients

Separation of mapping
function: wet & dry fromNWM

ZHD NWMZWD NWM

ZHD M–PZWD M–P

Temp.
from
NWM

Mendes–Pavlis (M–P)

–

–

–

–

–

x

x

–

M–P TRP

x

–

–

–

–

x

x

–

VMF3o ZWD TRP

x

x

x

–

x

x

–

x

VMF3o ZTD TRP

x

x

x

x

x

–

–

x

PMF ZTD TRP

x

x

–

x

x

–

–

x

PMF TRP

x

x

–

–

–

x

x

x

M–P model TRP

x

x

–

–

–

x

x

–

PMF ZTD no est

–

x

–

x

x

–

–

x

VMF3o ZTD no est.

–

x

x

x

x

–

–

x

TRP denotes that the trosphere bias is estimated. M–P model denotes using the a prori WGM for horizontal gradients. Temp. denotes temperatureDrożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

Page 12 of 16

Fig. 8 Comparison of water vapor pressure (WVP) from in situ measurements at SLR stations and VMF3o interpolated for the SLR sites in 2008–2012
in hPa as a function of the day of the year (color scale: winter-blue, spring-green, summer-red, fall-yellow in the Northern Hemisphere)

the solutions with ZTD derived from NWM with theestimation of the troposphere delay correction are compared to the standard M–P solution and to the mixed
solutions: PMF TRP and VMF3o TRP. Figure 12 shows
the differences in the estimated geocenter coordinatesfrom the standard M–P solutions. Figure 12 (top) reveals
large differences and fluctuations of 1 mm at the end or
beginning of the year, which is probably related to thelower number of observations and poor observationgeometry that becomes sensitive to individual SLR stations contributing to the solution. However, the Z component of the geocenter coordinates reflects substantial
differences at the end of the year at the level of 5 mm.
Figure 12 (bottom) shows the Fast Fourier Transformation (FFT) results of the geocenter differences, confirming the moderate consistency of the annual signal forall of the analyzed solutions at the submillimeter level.
Figure 13 shows the comparisons of differences in geocenter coordinates for solutions in which the ZTDs arederived from NWM with respect to the standard M–P
solution. Although the troposphere delay correction isestimated in solutions PMF TRP and VMF3o TRP, highfrequency noise can be noticed (Fig. 13, bottom), andartificial signals occur. The annual signal of geocenter
coordinates is consistent at the level of 0.3 mm withother analyzed solutions (see Table 6). SLR observations
provide stable information about the origin of the reference frame that coincides with the long-term mean valueof the geocenter. Table 6 shows, however, that the geocenter coordinates are mostly sensitive to the estimationof tropospheric corrections. The offset of the Z component changes between − 1.7 and − 2.4 mm in the solutions with different handling of tropospheric biases, i.e.,with and without estimating TRP biases. Thus, the estimation of the tropospheric biases plays a main role in therecovery of the geocenter motion, whereas a priori ZTD
values, horizontal gradients and mapping functions playa minor role without any substantial offsets.

Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

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Fig. 9 Box plots of troposphere delay correction derived from solutions using different approaches to troposphere delay modeling based on in situ
and NWM data, according to Table 4. The black box describes the 25th and 75th percentiles, the magenta central line describes the median value,
and the whiskers describe the most extreme data points without outliers. Stations are sorted by the number of observations

4 Conclusions
The comparison of meteorological parameters (pressure, temperature, humidity) derived from in situ measurements and VMF3o pressure records together withtroposphere delay corrections as an indicator of biases
demonstrates that ZTD corrections should be estimated to properly handle biases in meteorological data.
The estimation of troposphere delay corrections slightly
increases the offsets between the ERPs derived from
SLR and the combined IERS-C04 solution. In the caseFig. 10 3D RMS of station coordinate repeatability in the period
2010.0–2019.0

Table 5 Difference in pole coordinates with respect to the IERS-C04-14 series
Solution

X-poleOffsetY-poleMean σµas

OffsetLOD
Mean σµas

OffsetNDAY
Mean σµs

M—P—standard solution

56

8.1

56

8.2

− 42

5.5

470

M–P TRP

60

8.0

53

7.7

− 43

5.5

470

VMF3o TRP

41

8.0

30

7.7

− 43

5.5

470

VMF3o ZWD TRP

42

8.0

31

7.7

− 43

5.5

470

VMF3o ZTD TRP

42

8.2

30

7.7

− 42

6.7

470

PMF ZTD TRP

40

8.1

31

7.8

− 45

6.7

470

PMF TRP

41

8.0

30

7.7

− 42

5.5

470

M–P model TRP

41

8.0

30

7.8

− 43

5.5

470

PMF ZTD no est.

62

10.1

38

11.1

− 63

8.8

470

VMF3o ZTD no est.

63

10.1

38

11.1

− 63

8.8

470

Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

Page 14 of 16

Table 6 Mean values of geocenter coordinates with respect to the SLRF2014 and the amplitude of the annual signal
Solution

X (mm)Y (mm)Z (mm)

OffsetAnnual ampl.

OffsetAnnual ampl.

OffsetAnnual ampl.

M–P

0.7

2.9

1.4

2.0

− 1.7

3.7

M–P model

0.6

2.9

1.3

2.0

− 1.7

4.0

VMF3o ZWD

0.6

3.0

1.4

2.1

− 1.7

4.0

PMF ZTD no est.

0.6

2.9

1.3

2.0

− 1.7

3.7

M–P TRP

0.6

3.0

1.0

2.1

− 2.4

3.4

VMF3o ZTD TRP

0.7

3.1

1.0

2.2

− 2.1

3.7

PMF ZTD TRP

0.6

2.8

0.9

2.0

− 2.3

3.8

Fig. 11 Differences in station coordinate repeatability for the north,east, and up components of station coordinates in the 2010.0–2019.0
period. The positive values denote the deterioration of station
coordinate repeatability with respect to the M–P solution, whereas
the negative values denote improvementsof solutions with the troposphere delay model based on
NWM, the mean offsets are absorbed by tropospheredelay correction, and the ZTD does not deteriorate theconsistency between SLR results and the IERS-C04
series. The ZTD model derived from the NWM is not
the most suitable approach for SLR data processing whenused as an a priori model without estimating corrections. However, the mixed approach of mapping function
based on NWM and zenith delays based on in situ data
may mitigate some malfunctioning of in situ records.
The troposphere delay correction reduces the deterioration of station coordinate estimates and the mean offsets
in pole coordinates to some extent in the case of solutions in which the ZTD is based only on the NWM. The

Fig. 12 Time series of differences in geocenter coordinates
for solutions with ZTD based only on the standard ILRS solution (top)w.r.t. the reference solution (M–P). and periodogram of geocenter
coordinates (bottom) for the X, Y and Z components, respectively.
The solid line represents the Savitzky–Golay filter with 11 weeks
of windowingDrożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

Page 15 of 16

simplified model of horizontal gradients, WGM, or PMF
and VMF3o models are highly recommended for future
ITRF realizations to account for the horizontal asymmetry of the troposphere. Moreover, further investigation
related to the detection of pressure issues should involve
the whole time series of SLR measurements contributing to the ITRF realization to avoid any systematic issues
with the origin, scale, and orientation of the reference
frame.
Abbreviations
ECMWF	European Centre for Medium-Range Weather Forecasts
ERA5	Is the fifth generation ECMWF reanalysis for the global climate
and weather
ERA Interim	is the fourth generation ECMWF reanalysis for the global climate and weather
FFT	Fast Furier Transform
GNSS	Global Navigational Satelltie Systems
IERS	International Earth Rotation and Reference Systems Service
C04	Combined solution for Earth Orientation Parameters
ILRS	International Laser Ranging Service
InSAR	Synthetic Aperture Radar Interferometry
ITRF	International Terrestrial Reference Frame
LAGEOS	Laser Geodynamics Satellite
NWM	Numerical weather model
PMF	Potsdam mapping function
SLR	Satellite laser ranging
VMF3o	Vienna Mapping Function 3 for optical frequencies
VLBI	Very Long Baseline Interferometry
WGM	Wroclaw Gradient Model
ZHD	Zenith hydrostatic delay
ZTD	Zenith total delay
ZWD	Zenith wet delay
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for
their valuable remarks. Finally we wuld like to thank you Janina Boisits andJohannes Boehm for delivering the VMF3o products and Kyriakos Balidakis
and Florian Zus for PMF mapping function. The APC is financed by Wrocław
University of Environmental and Life Sciences.

Fig. 13 Time series of differences in geocenter coordinates
for solutions with different approaches of ZTD modeling (Top)w.r.t. the M–P solution. The solid line represents the Savitzky–Golay
filter with 11 weeks of windowing. The bottom figure describes
the periodogram of geocenter coordinates for the X, Y, and Z
components

solution utilizing ZWD from NWM (VMF3o ZWD TRP)
is able to detect the malfunctioning of in situ humidity
records, such as in the case of stations Zimmerwald or
Mt Stromlo, where the IQR of troposphere delay correction is reduced with respect to the standard solution,
as well as the range of the whiskers of tropospheric bias
estimates. For the majority of analyzed stations, the PMF
TRP and VMF3o TRP solutions are consistent with eachother despite employing different assumptions for thenumber of parameters to describe horizontal gradientsand atmospheric asymmetry and the separation between
wet and hydrostatic tropospheric delays. Finally, theAuthor contributions
MD & KS implemented the VMF3o station-wise and PMF ERA 5 new release
into BERNESE GNSS software. MD introduced the parametrized troposphericmodel of horizontal gradients KS helped to implement them into BERNESE
GNSS software. MD conducted the experiments and performed the data
analysis. MD and KS wrote the article. All authors read and approved the final
manuscript.
Funding
Mateusz Drożdżewski has been funded by the Polish National Agency for
Academic Exchange, grant no. PPI/APM/2018/1/00013/U/001 “Interdisciplinaryinternational cooperation as the key to excellence in science and education”(INCREaSE). Krzysztof Sośnica has been funded by the National Science Centre,
Poland, grant no. 2019/35/B/ST10/00515. The APC is financed by Wrocław
University of Environmental and Life Sciences.
Availability of data and materials
The PMF ERA 5 model is publicly available and may be downloaded from:ftp://​ftp.​gfz-​potsd​am.​de/​pub/​home/​kg/​zusflo/​TRO/​SLR_​compl​ete/ (Zus et al.
2015), the VMF3o troposphere delay model is publicly available and may be
downloaded from: https://​vmf.​geo.​tuwien.​ac.​at/​trop_​produ​cts/​SLR/ (Boisits
et al. 2020). The SLR observations data used within this article has been
downloaded from Crustal Dynamics Data Information System (Noll et al. 2019).
The weekly solution based on observations to passive geodetic satellites werecomputed utilizing the Bernese GNSS Software version 5.3 http://​www.​berne​Drożdżewski and Sośnica ﻿Progress in Earth and Planetary Science(2024) 11:12

se.​unibe.​ch/ (Dach et al. 2015). Data sharing not applicable to this article as nodatasets were generated or analysed during the current study. Please contact
author for data requests.

Declarations
Competing interests
The authors declare that they have no competing interest.
Received: 28 November 2023 Accepted: 17 February 2024

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