Impact of different NWMderived mapping functions on VLBI and GPS analysis
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Abstract
Keywords
Troposphere modeling Mapping function VLBI GPS VMF1 UNBVMF1Abbreviations
 CMC
Canadian Meteorological Centre
 ECMWF
European Centre for MediumRange Weather Forecasts
 GAPS
Global Navigation Satellite System Analysis and Positioning software
 GDPS
Global Deterministic Prediction System
 GMF
Global Mapping Function
 GNSS
Global Navigation Satellite Systems
 GPS
Global Positioning System
 IERS
International Earth Rotation and Reference Systems Service
 IGS
International GNSS Service
 IMF
Isobaric Mapping Functions
 NCEP
National Center for Environmental Prediction
 NMF
Niell Mapping Function
 NOAA
National Oceanic and Atmospheric Administration
 PPP
Precise Point Positioning
 UNB
University of New Brunswick
 VieVS@GFZ
GFZ version of the Vienna VLBI Software
 VLBI
Very Long Baseline Interferometry
 VMF1
Vienna Mapping Functions 1
 TUW
Technische Universität Wien
Introduction and background
In precise space geodetic observation analysis, the propagation of signals through the Earth’s neutral atmosphere is one of the main sources of error. Retardation and bending of the electromagnetic signals toward their way down to Earth’s surface, a combined effect referred to as neutralatmospheric delay, mainly affects the vertical coordinate component of an observing station, if not modeled appropriately.
In the processing/analysis of microwavebased observations, the tropospheric delay is approximated by separation into a hydrostatic (Davis 1986) and a nonhydrostatic (Mendes 1999; wet^{1}) part, based on the properties of the main constituent gases: mixed humid air in hydrostatic equilibrium and water vapor, respectively.
Among the several evaluations of mapping functions that have been created in an attempt to achieve precision and accuracy, mapping functions derived employing numerical weather model (NWM) have dominated for at least the last decade. Ray tracing directly in an NWM has proven to be the most effective uptodate technique to derive mapping functions, such as the Vienna Mapping Functions 1 (VMF1) (Boehm et al. 2006b). Compared to the use of (a) spherical harmonics based on NWM, employed by the Global Mapping Function (GMF; Boehm et al. 2006a), (b) standard atmosphere profiles, used by the Niell Mapping Function (NMF; Niell 1996), or (c) the use of specific parameters such as the 200 hPa geopotential height, employed by Isobaric Mapping Functions (IMF; Niell 2001), discrete ray tracing has proven to yield the most accurate results.
Tesmer et al. (2007) compared four different mapping functions (NMF, GMF, IMF and VMF1) and demonstrated that VMF1 are the most precise ones with respect to station height repeatability. Two years later, they were recommended for all precise geophysical applications (Boehm and VanDam 2009). As of now, VMF1 are recommended by the latest International Earth Rotation and Reference Systems Service (IERS) Conventions (Petit and Luzum 2010) for all precise geophysical applications.

To improve the availability of the VMF1 corrections so as to maintain the undisrupted production thereof.

To be compatible with other numerical weather model products (for example, atmospheric pressure loading parameters, generated using NCEP ReAnalysis 1).

To increase robustness of the International GNSS Service (IGS)/IERS combined products as it utilizes different NWMs and an independent raytracing algorithm (Nievinski and Santos 2010) (considering that IGS would recommend VMF1 for operational analysis).
The spatial resolution of the NWM itself directly impacts the ability to model atmospheric conditions effectively. Moreover, since shortterm forecasts are able to capture the rapid changes of the surface temperature, especially at areas of complex terrain or along the coast, where rapid changes of orography occur, it is expected that tropospheric corrections would benefit from an NWM of higher spatial and temporal resolution. Thus, this research is focused on the impact of the NWM’s resolution when alternating the two operational mapping functions: VMF1 and UNBVMF1.
UNBVMF1 products have been compared with the VMF1 ones in the past by McAdam (2013). In the position domain, assessing the gridded products for a subset of (32) IGS stations over 11 years, he concluded that the global bias and RMSE of the two mapping functions agree well with each other at the submm level. A latitudedependent bias and a small trend at equatorial stations were also confirmed by the Precise Point Positioning (PPP) analysis. This study is distinguished from the previous work as it uses sitespecific products as opposed to grid based (interpolated) used in the past. The difference in zenith delays between the two product types has proven to reach cm level for VMF1 (Kouba 2008). Moreover, the CMC NWM is employed for the production of the UNBVMF1 products which has shown advantages over the NCEP model (McAdam 2013).
The motivation for this study originates from the fact that typically, higherresolution NWM allows for smallerscale weather patterns to be described and benefit from the more detailed orography employed. Since a lot of significant weather phenomena are related to the local orography and convective processes (Erfani et al. 2005), we investigate the potential benefit of using the CMC NWM for geophysical applications, which has nearly four times the resolution of the ECMWF. This study uses sitespecific products of both realizations of the VMF1 concept and validates their accuracy in GPS and VLBI processing by testing the station coordinate repeatability.
VMF1 development
VMF1 employ the ECMWF NWM, which has a temporal resolution of 6 h and uses a downscaled spatial resolution of the model of 2.5° × 2° (Boehm and Schuh 2004). To retrieve a coefficient, ray tracing at an initial elevation angle of 3.3° is performed which produces the elevation angle used in Eq. (2) (geometric elevation angle ~ 3°). Using the zenith and slant path delays and the predefined b and c coefficients and simply inverting the continued fraction form (Eq. 2), the value for the a coefficient is obtained. This procedure is done for the hydrostatic and wet components separately.
UNBVMF1 development
While it follows the same concept to produce the atmospheric parameters, UNBVMF1 utilize a different data source for the ray tracing, an independent raytracing algorithm and Gaussian Earth radius of curvature. The latter eliminates the hydrostatic mapping function bias that VMF1 shows as a possible result of using constant Earth radius (Urquhart 2010); both VMF1 and UNBVMF1 are based on a “normal sphere” whereby the center of the sphere is located along the ellipsoidal normal direction.
UNBVMF1 utilize two different NWMs: (1) NOAA NCEP ReAnalysis I with a global 2.5° × 2° analysis resolution, initialized every 6 h and (2) the CMC Global Deterministic Prediction System (GDPS) (Côté et al. 1997) with a temporal step of 3 h and a global resolution of approximately 0.6°. The latter, as a modern operational model, contains the latest application of atmospheric physics and parameterizations and the spatially based systematic effects are minimized. Therefore, for this study, we will focus on the CMC GDPS.
VMF1 and UNBVMF1 products usage
In spite of the improvement in NWM’s quality over the last years, its use alone can be ambiguous due to modeling and forecast errors. Thus, for highest accuracy, when processing microwave observations, a residual zenith delay (usually selected to be the wet delay) is estimated, along with the modeled delay attained utilizing the raytraced products. While the specific a coefficient (hydrostatic or wet) is used to evaluate the mapping function at any specified elevation angle, the hydrostatic zenith delay is utilized as an a priori value for the adjustment to follow, i.e., the a priori zenith wet delay is set to zero. Both hydrostatic and wet zenith delays can also be used to calibrate the tropospheric model before the adjustment of the observations as well as to access the quality of the results.
Data, products and assessment strategy
The chosen time span was 1 year: 2014. We acquired the site products of the hydrostatic and wet a coefficient and zenith delays for the VMF1 from the online repository^{3} of TUW for the GPS and VLBI stations. For the respective UNBVMF1 products, we followed the procedure described in section “VMF1 development” and raytraced through the CMC GDPS NWM. The primary data/products consisted of the zenith delays and mapping function coefficients with a temporal resolution of 6 h. The secondary products consisted of the daily series of the coordinates of the stations as a result of employing the mapping factors and zenith delays in the GPS and VLBI analyses.
To assess the performance of the two mapping functions on a site basis, the comparison took place in both the delay and the position domain. For the delay domain, 411 globally distributed GPS stations of the IGS network (Dow et al. 2009) were chosen and 61 VLBI stations, participating in the nonintensive IVSR1 and IVSR4 VLBI, 24hlong, sessions. Since the VLBI stations are colocated by GPS stations, only the results of the latter are presented in detail. For the position domain, 18 globally spread GPS stations and 29 VLBI stations were chosen for their location characteristics. Both analyses data/products span the entire year 2014.
Delay domain
Zenith delays and equivalent height error
Zenith delays are inversely correlated with the estimated station heights (and clock offset); the correlation is absolute at 90° elevation angle and drops to about half for low elevation angles, i.e., 5°. For that reason, the comparison of the zenith delays was considered imperative. The VMF1 and UNBVMF1 products were compared directly. Specifically, the respective zenith hydrostatic and wet delays were differenced to reveal possible systematic effects and biases. The equivalent height error was computed, according to the rule of thumb by Niell (2001) and refined by Boehm and Schuh (2004). The latter predicts that the delay path error at an elevation angle of 5° will map to the station height at a ratio of 1/5. Estimating this error provides an approximate indication of the impact of the difference of the mapping functions on position without any real application of the mapping function on geodetic observations.
Mapping factors and nominal slant delays
Mapping factors, defined as the ratio between the slant and zenith delay, contain possible errors of the Marini continued fraction form coefficients: a, b and c. Since VMF1 use constant/empirical values for the b and c coefficients, any mismatch in the mapping factors will reflect differences in the a coefficient induced by the ray tracing. Assuming azimuthal symmetry of the neutral atmosphere, the mapping function and zenith delay errors are inversely correlated in geodetic analysis. Therefore, an error in the mapping factor will not only propagate to the station position but also to the estimated delay (and clock offset). Moreover, opposite to the hydrostatic/wet mapping separation errors caused by inaccurate zenith hydrostatic delays, mapping factor errors cannot be compensated in the adjustment (Tregoning and Herring 2006). To assess this error, mapping factors were produced for both VMF1 and UNBVMF1 at eight elevation angles: 3°, 5°, 10°, 15°, 20°, 30°, 45° and 60°. In order to compare the mapping functions to metric units, the mapping factors were scaled by nominal delays at the elevation angle that showed the maximum differences. The produced nominal slant delays were the product of the evaluated mapping function with the nominal zenith delay: 2300 mm for the hydrostatic and 250 mm for the wet component, respectively.
Position domain
Although a common way to benchmark the solution would be comparing against the IGS station coordinate solution, most of the analysis centers contributing to the combined solution utilize VMF1. Since this choice directly affects the station coordinates, it would make such a comparison unfair. To acquire an objective representation of the accuracy of the two solutions (utilizing UNBVMF1 and VMF1 products), we calculated the weighted rootmeansquare error (wrms) of the height component for the 18 globally distributed stations. The daily station position solution was fitted using a linear polynomial in the local geodetic reference frame. Such fitting is necessary to remove possible displacement due to secular deformations primarily longterm crustal motion.
GPS Precise Point Positioning analysis
Employing the raytracing products (zenith delays and mapping factors) in the UNB’s Global Navigation Satellite System (GNSS) Analysis and Positioning software—GAPS—(Urquhart et al. 2014a, b), in Precise Point Positioning (PPP) (Zumberge et al. 1997) mode, we estimated the position of the station along with the residual zenith (wet) delay, station clock offset and ambiguities. The default modeling of the observations according to the GAPS processing strategy^{4} was used with cutoff elevation angle of 7°, tropospheric process noise of 5 mm/√h and elevation angledependent observation weighting (sine of elevation, correlations ignored). The raytraced zenith delays and the coefficients were computed using NWM data. Underlying errors in the NWM would certainly propagate into the height estimation during the PPP process. Thus, to unravel potential systematic errors, the two PPP solutions were processed using identical parametrization alternating only the mapping functions (either VMF1 or UNBVMF1).
Geodetic coordinates of the GPS stations used in the position domain analysis
Station  Latitude (°)  Longitude (°)  Ellipsoidal height (m) 

ABPO  − 19.02  47.23  1553.0 
ALRT  82.49  − 62.34  78.1 
BAKE  64.32  − 96.00  4.4 
BOGT  4.64  − 74.08  2576.8 
CAS1  − 66.28  110.52  22.6 
CRO1  17.76  − 64.58  − 31.5 
HOB2  − 42.80  147.44  41.1 
KOKB  22.13  − 159.66  1167.5 
MAC1  − 54.50  158.94  − 6.7 
MAT1  40.65  16.70  534.5 
MCM4  − 77.84  166.67  98.0 
MIZU  39.14  141.13  117.0 
MKEA  19.80  − 155.46  3754.7 
ROAP  36.46  − 6.21  73.7 
SCOR  70.49  − 21.95  128.5 
URUM  43.59  87.63  856.1 
WTZR  49.14  12.88  666.0 
ZIMM  46.88  7.47  956.7 
VLBI analysis
Summary of the parameters compared in the analyses and their characteristics
Compared parameters  Temporal resolution  Units  No. GPS stations  No. VLBI stations 

Delay domain analysis  
Zenith hydrostatic delay  6 h  mm  411  61 
Equivalent height error  
Zenith wet delay  
Hydrostatic mapping factors  –  
Wet mapping factors  
Position domain analysis  
Weighted root mean square error  Daily  mm  18  – 
Daily by session  –  29 
Geodetic coordinates of the VLBI position domain analysis stations and number of sessions participated in
Station  Latitude (°)  Longitude (°)  Ellipsoidal height (m)  No. of sessions 

AIRA  130.5998624  31.8237944  322.4  11 
BADARY  102.2339159  51.7702618  821.6  42 
FDVLBA  − 103.9448205  30.6350297  1606.4  5 
FORTLEZA  − 38.4258585  − 3.8778591  23.1  97 
HART15M  27.6842679  − 25.8897363  1409.6  96 
HARTRAO  27.6853927  − 25.8897518  1415.7  12 
HOBART12  147.4381401  − 42.8055739  41.0  134 
HOBART26  147.4405178  − 42.8035860  65.1  19 
KATH12M  132.1543178  − 14.3766210  189.0  121 
KOKEE  − 159.6650977  22.1266380  1176.6  80 
LAVLBA  − 106.2455957  35.7751235  1962.4  5 
MATERA  16.7040159  40.6495239  543.4  45 
MEDICINA  11.6469330  44.5204925  67.2  5 
NYALES20  11.8696917  78.9291103  87.3  104 
ONSALA60  11.9263544  57.3958363  59.3  30 
PIETOWN  − 108.1191894  34.3010175  2364.7  6 
SCVLBA  − 64.5836330  17.7565811  − 15.0  5 
SEJONG  127.3033611  36.5227208  194.6  5 
SESHAN25  121.1996589  31.0991628  29.4  7 
SVETLOE  29.7819372  60.5323443  86.0  10 
SYOWA  39.5862862  − 69.0063246  51.0  6 
TIGOCONC  − 73.0251485  − 36.8427183  171.0  27 
TSUKUB32  140.0887367  36.1031462  84.7  57 
WARK12M  174.6632547  − 36.4348089  127.9  66 
WESTFORD  − 71.4937938  42.6129481  86.8  34 
WETTZELL  12.8774503  49.1450079  669.1  113 
YARRA12M  115.3456213  − 29.0471648  250.5  119 
YEBES40M  − 3.0868621  40.5246653  989.1  38 
ZELENCHK  41.5651625  43.7878094  1175.0  38 
Results
Delay domain analysis (DDA)^{5},^{6}
VLBI zenith delay bias, standard deviation and equivalent height error
The zenith hydrostatic delay agreement between UNBVMF1 and VMF1 is 2.5 mm or less for 98% of the stations and for 40 of them (71%) less than 2 mm. Their standard deviation varies around 1–2 mm for 91% of the stations and exceeding this number only for 5 stations to maximum 3 mm.
The equivalent height error was computed, according to the refined rule of thumb by Boehm and Schuh (2004). Applying the rule, we observe that all the stations have a mean predicted vertical error of less than 2 mm and the standard deviation is zero for all the stations (maximum value of 0.7 mm). In other words, the rule predicts that, regardless of the mapping function employed, the maximum resulting height error will not exceed 2 mm.
For the zenith wet delay, the absolute bias varies more, up to 19 mm, due to the less predictable value of water vapor. The standard deviation fluctuates around 10–20 mm for 85% of the stations. The equivalent height error of the wet mapping factor is one order of magnitude less than the hydrostatic one (Boehm et al. 2006b) and is expected to be negligible if one was to employ it in VLBI processing or GPS PPP.
GPS zenith delays bias, standard deviation and equivalent height error
Zenith hydrostatic delays
Equivalent height error
Zenith wet delay
GPS mapping factors at different elevation angles and nominal slant delays
In sections (a) and (b), the differences in the mapping factors are presented in a relative (and absolute) manner; following that, in section (c), they are converted to metric system using nominal zenith delays.
Hydrostatic mapping factors
Wet mapping factors
Nominal slant delays
In order to compare the mapping factors in a metric system, a scale was applied to them using nominal delays. Specifically, the hydrostatic mapping factors were scaled using a nominal delay of 2300 and 250 mm for the wet, at the elevation angle that showed the maximum differences.
Position domain analysis (PDA)
GPS stations
GPS PDA stations height component weighted mean square error in (mm)
Station  wrms (mm)  

VMF1  UNBVMF1  
ABPO  5.19  5.18 
ALRT  7.91  7.97 
BAKE  9.20  9.40 
BOGT  9.08  9.09 
CAS1  6.10  6.10 
CRO1  7.79  7.80 
HOB2  12.32  12.35 
KOKB  5.84  5.83 
MAC1  7.53  7.45 
MAT1  4.18  4.18 
MCM4  6.14  6.25 
MIZU  7.49  7.43 
MKEA  5.50  5.51 
ROAP  11.82  11.90 
SCOR  9.75  9.76 
URUM  8.07  7.98 
WTZR  4.51  4.49 
ZIMM  6.92  4.20 
VLBI stations
VLBI PDA stations height component weighted mean square error in (mm)
Station  wrms (mm)  

VMF1  UNBVMF1  
AIRA  12.66  12.99 
BADARY  10.32  10.32 
FDVLBA  5.19  5.22 
FORTLEZA  16.14  16.50 
HART15M  8.71  8.77 
HARTRAO  6.69  8.42 
HOBART12  15.81  14.99 
HOBART26  16.07  24.57 
KATH12M  7.74  7.73 
KOKEE  9.48  8.74 
LAVLBA  6.22  6.11 
MATERA  8.91  8.77 
MEDICINA  8.70  10.38 
NYALES20  8.18  8.42 
ONSALA60  6.77  6.68 
PIETOWN  7.45  7.25 
SCVLBA  7.78  7.83 
SEJONG  27.14  9.65 
SESHAN25  10.41  9.24 
SVETLOE  12.20  11.58 
SYOWA  23.97  25.06 
TIGOCONC  22.93  22.83 
TSUKUB32  8.98  8.86 
WARK12M  9.55  9.50 
WESTFORD  8.69  9.70 
WETTZELL  7.13  7.46 
YARRA12M  6.93  7.06 
YEBES40M  7.56  7.71 
ZELENCHK  12.58  11.16 
Discussion
Conclusion
The choice of mapping function in space geodetic techniques affects the slant delays and thus the vertical position, in particular at elevation angles below 20°. In this study, the two stateoftheart mapping functions, namely the VMF1 and UNBVMF1, have been considered. Assessment took place in both delay and position domains and on a sitebysite basis.
This study has shown that the two current operational mapping functions, VMF1 and UNBVMF1, were consistent in terms of zenith hydrostatic delay computation with differences below 3.5 mm for the vast majority of the stations examined.
The zenith wet delay differences had a larger spread compared to the hydrostatic ones due to the unpredictable nature of the water vapor content. However, these did not exceed a few centimeters with the maximum values observed at the equator, where the largest water vapor signals reside.
The equivalent height error due to the hydrostatic mapping function did not exceed 2 mm globally for any of the GPS stations analyzed herein. However, both the delay components and the height error showed dependency on latitude as a result of different representations of the Earth’s shape in the two raytracing procedures.
Comparing the hydrostatic mapping functions, differences were invariable at the tenths; differences at the 0.01 level (unitless) occurred at 3° elevation angle for 87% of the stations. The respective wet mapping function, however, reached twice that value—differences at 0.02 level—for 14% of the stations at the same elevation angle. Yet, owing to the ZWD being at least 10 times smaller than ZHD, the slant delay differences did not exceed 20 mm for 61% of the stations.
In the position domain, results from GPS PPP processing using the GAPS software showed that the stations examined yield almost identical results, indicating that both mapping functions perform at the same level. The wrms for the height component did not exceed 12 mm for a yearly analysis, while the differences were at submm level.
In the VLBI data analysis, larger deviations were found, reaching up to 25 mm for the height wrms of the individual solutions. About half (48%) of the VLBI stations showed a reduction in the height wrms when using the VMF1, ranging from less than 1 to 26% maximum with an average reduction of 2.6%. On the other hand, the use of UNBVMF1 showed a reduction for 41% of the stations reaching up to 11%, while the average reduction was 1.7%. Moreover, the two mapping functions performed equally for a subset of stations with a large number of solutions available, and for the majority of the rest of the stations, the difference was at the submm level.
Although the wrms discrepancies between the two mapping functions were slightly larger for the VLBI stations, considering the elevationdependent weighting scheme of the GPS observations, makes the results more comparable. That being said and based on the statistics from both the GPS and VLBI analyses, the two mapping functions can be considered equal for geodetic analysis.
However, as the wrms is an index of the internal consistency and not an external validator, further assessment using an absolute indicator would be useful. Also, a lengthier VLBI analysis, i.e., spanning several years, would help to obtain more realistic wrms values and eliminate the outliers more efficiently. Still, the use of the weights in the formula corrects for the quality of observations.
For the case of GPS observations, processing using an alternative technique and/or software, perhaps more sensitive to the selection of the mapping function or modifying the weighting scheme of the observations, may reveal results more similar to the VLBI processing.
For the future, further analysis in the position domain including more sites could be beneficial to reach to locationbased conclusions regarding the equality of the two NWMs, particularly for stations affected by highly variable weather conditions and/or located at challenging topography, i.e., coastal areas and/or high altitudes. Additionally, expansion of the time span to several years could aid the assessment.
It should be added that in the recent months, the ray tracer employed by VMF1 has been improved and a new set of mapping functions has been developed, called the VMF3 (Landskron and Böhm 2017). Therefore, future work will necessarily involve these new developments.
Footnotes
 1.
We shall call the nonhydrostatic complement delay “wet”, following Davis (1986) hereafter for simplicity.
 2.
Hereafter we use “(UNB)VMF1” interchangeably for the mapping functions or the service. The specific meaning can be easily inferred from the context.
 3.
 4.
 5.
Note: all the limits described in the statistics are inclusive on the left side but not inclusive on the right side i.e., [a, b).
 6.
Outliers were excluded because they are not representative of the sample. Upon further investigation, we have confirmed that some stations had near zero heights even in mountain regions (Boehm, personal communication). Exact details about the discrepancies are unknown as the 2004 height values were not made available.
Notes
Authors’ contributions
TN performed the ray tracing to derive the UNBVMF1 products, carried out the GPS processing and analysis and wrote the main article. KB carried out the VLBI processing and contributed to the postprocessing analysis. FN contributed in the design of the delay domain analysis and assisted in issues concerning the raytracing process. MS helped to write the article and supervised the whole project. KB, FN, MS and HS helped to improve the manuscript and provided feedback throughout the project. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank: the International VLBI Service for Geodesy and Astrometry (IVS, Nothnagel et al. 2015; Schuh and Behrend 2012) for observing, correlating and providing the VLBI data used in this study; the International GNSS Service (IGS) (Dow et al. 2009) for the GPS data used in this study; the Technische Universität Wien (TUW) for providing the VMF1 tropospheric parameters; the European Centre for MediumRange Weather Forecasts and the Canadian Meteorological Centre for providing the data necessary for the creation of the mapping factors; last but not least, the editor and the two unknown reviewers for their constructive comments and helping to improve the manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article and the following links: http://ggosatm.hg.tuwien.ac.at/DELAY/, https://weather.gc.ca/grib/grib2_glb_66km_e.html and https://cddis.nasa.gov/Data_and_Derived_Products/GNSS/atmospheric_products.html.
Ethics approval and consent to participate
Not applicable.
Funding
The corresponding author is supported by an offer of financial assistance by University of New Brunswick.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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