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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[1] One major part in the error budget of GPS measurements is the imperfect modeling of the tropospheric delay. By processing a global network of 195 stations we have compared two different mapping techniques: (1) the commonly used Niell hydrostatic mapping function (NMF) and (2) the isobaric hydrostatic mapping function (IMF) based on numerical weather fields. The two solutions reveal significant differences in the derived zenith total delay (ZTD) parameters and site positions. The largest differences occur in Antarctica, where the annual mean heights differ by up to 15 mm. We infer that the significant differences are related to model deficiencies in NMF since a) IMF improves the repeatability in station heights in high southern latitudes significantly, and b) using IMF reduces the dependence of the solution on the elevation cut-off angle by about 20%. In conclusion, the use of mapping function (MF) parameters based on meteorological data is strongly recommended for global GPS analyses.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[2] A major part in the error budget of GPS measurements is the inaccurate modeling of the elevation dependence of the tropospheric path delay. In general the tropospheric delay at a given elevation angle ε, also referred to as slant total delay STD, is related to the zenith delay (ZD) by the mapping function m:

  • equation image

[3] The subscripts h and w stand for the separation of the delay into a hydrostatic and a wet part. The hydrostatic delay accounts for about 90% of the total delay. Therefore, the estimated height and zenith delay parameters are much more sensitive to modeling uncertainties of the hydrostatic than of the wet mapping function.

[4] Marini [1972] showed that for a spherically symmetric atmosphere the mapping function can be approximated by a continued fraction of the form

  • equation image

[5] In order to map the zenith delay to an elevation angle of 3°, three coefficients a, b and c are sufficient [Niell, 1996]. These coefficients have been determined by various authors using different strategies [e.g., Davis et al., 1985; Herring, 1992; Niell, 1996]. The Niell mapping function is widely used because of its global validity and its independence of surface meteorological observations. NMF uses 15° latitude grid tables and models the seasonal amplitudes for each of the three coefficients of equation (2).

[6] In recent time, several studies have been conducted to make use of numerical weather fields. Rocken et al. [2001] derived mapping function parameters from ray traces in numerical weather fields at ten different elevation angles. They showed that dry mapping based on ray tracing fits 2–3 times better to radiosonde data than mapping with NMF.

[7] Niell [2000] proposed a hydrostatic mapping function which uses only one meteorological input parameter, the height of the 200 hPa isobar. Hence, it is referred as isobaric mapping function (IMF). The Vienna mapping function (VMF) calculates ray tracing for only one initial elevation angle of 3.3° in order to optimize the computational effort. The VMF coefficients are determined by inverting for the continued fraction parameter a in equation 2, assuming b and c as given in IMF [Boehm and Schuh, 2004]. As a result of a VLBI analyses, the VMF yields very similar results compared to the IMF [Schuh and Boehm, 2004]. In the following we investigate the influence of the IMF compared to NMF in a global GPS network analysis. We concentrate on the hydrostatic component as it dominates for the tropospheric delay.

2. Model Differences Between NMF and IMF

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[8] The parametrization of NMF is based on ray traces of radiosonde profiles spanning the latitudes 43°S to 75°N. By assuming longitudinal homogeneity and symmetry between the southern and the northern hemisphere, this mapping function was extended globally [Niell, 1996]. Seasonal and spatial variations are parameterized in the NMF with a resolution of one day in time and 15° in latitude. The equatorial region from 15°S to 15°N is defined by the estimates of the 15°N latitude. The polar regions with latitudes higher than 75° on the northern and southern hemisphere are both approximated by the 75°N latitude values.

[9] IMF is directly derived from numerical weather fields. Numerical weather prediction (NWP) models provide a highly accurate description of the atmospheric state. IMF is estimated from the height of the 200 hPa surface [Niell, 2000]. The temporal and spatial resolution and the accuracy of IMF depend on the underlaying weather model. The global NWP models of the European Centre for Medium-Range Weather Forecasts (ECMWF) and the National Centers for Environmental Prediction (NCEP) provide values every 6 hours with a spatial resolution of 2.5° in longitude and 2° to 2.5° in latitude. Both models have their largest uncertainties in the polar regions. We validated the height of the 200 hPa surface (z200) from these models with radiosonde (RS) data at four Antarctic stations over a period of 1 year.

[10] The comparisons of NCEP data with RS show offsets of, in general, less than 20 m and standard deviations of the same order. The agreement between ECMWF data and RS measurements is better with offsets below 5 m and standard deviations of about 15 m. Test calculations were carried out using NCEP and ECMWF: a maximum difference of 40 m in the 200 hPa geopotential height between NCEP and ECMWF data in the Antarctic region results in a difference of 0.4 mm in the estimated station height. For most of the stations the effect on the station height due to the differences in the NWP models is less than 0.1 mm.

[11] Furthermore, we compare the NMF and IMF mapping function values by evaluating their effects on a hydrostatic delay of 2.3 m when mapping it from the zenith to an elevation angle of 5°. The IMF values are based on the NCEP model. As an example, NMF and IMF are compared for the station Davis, Antarctica over a period of 6 years in Figure 1. The differences in the mapped delay show an offset of about 90 mm, a seasonal signal, and short term variations.

image

Figure 1. (left) Comparison of hydrostatic slant delays (SD) created by the mapping of a zenith delay of 2.3 m to an elevation angle of 5° with NMF (dotted line) and with IMF (solid line) at station Davis, Antarctica. IMF values were derived from NCEP data with a 6-hour temporal resolution. (right) Corresponding difference between IMF and NMF mapped delay, represented as 30-day moving average.

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[12] The global distribution of the mean difference for the year 2002 is shown in Figure 2. For most of the globe there is clear evidence of a latitude dependence with larger differences at high latitudes. The differences are positive meaning that the slant delay derived from IMF is larger than the delay derived from NMF. Exceptions are the equatorial region and northern Europe. There the differences are negative, but small.

image

Figure 2. Global differences between the slant delay values of IMF and NMF at 5° elevation (mean over the year 2002).

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3. GPS Data Analysis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[13] We analyzed a set of nominally 195 globally distributed stations (Figure 3) with a modified version of the Bernese GPS Software Version 5.0 [Hugentobler et al., 2005]. One modification is the implementation of the IMF. Detailed information on the processing strategy is given by P. Steigenberger et al. (Reprocessing of a global GPS network, submitted to Journal of Geophysical Research, 2005). Here only the most important facts related to the atmospheric modeling are addressed. We estimated as continuous piecewise-linear the zenith wet delay in 2-hour intervals in conjunction with daily horizontal gradients in east-west and in north-south direction. The wet mapping function from NMF was used as the partial derivative for adjusting the zenith delay. The elevation cut-off angle was set to 3°. An elevation-dependent weighting (w = cos2z, with the zenith angle z) was applied to the observations. Absolute corrections for antenna phase center variations were applied both for receivers and for satellites. For this specific investigation all data of the year 2002 were processed twice. In the first run we used the NMF, and for the second we applied the IMF based on model data from NCEP.

image

Figure 3. Mean differences in GPS station height estimates between the solutions from IMF and NMF for the year 2002. Most of the stations show an increase in station height estimates when using IMF (black arrows) and only few a decrease (red arrows).

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4. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References

4.1. Influence of IMF on Station Height and ZTD Estimates

[14] Mean station height differences between the global solutions using NMF and IMF are presented in Figure 3. As expected from the model comparisons, the largest offsets occur in the polar regions and the smallest in the equatorial zone. Station heights in Antarctica show mean offsets in the range of 2 to 11 mm. In the equatorial region the mean differences are below 1 mm. For the majority of the stations the height increases when applying IMF.

[15] As examples, time series of the differences between the two solutions are presented for station Davis, Antarctica, and for Metsahovi, Finland, in Figure 4. It can be seen that the choice of the mapping function directly affects the height estimates and that differences between the mapping functions are negatively correlated with the ZTD estimates. Besides an offset, the differences in the station height and ZTD series reveal temporal variations. For station Davis the variation in the height component shows an annual signal with an amplitude of approximately 7 mm (Figure 4c). Even in the equatorial region the differences in the station height vary for many stations by up to 5 mm in the course of the year.

image

Figure 4. Comparison of time series derived using IMF and NMF mapping functions (left) for station Davis, Antarctica and (right) for station Metsahovi, Finland. (top) Difference in the slant delay between IMF and NMF, created by mapping of a zenith delay of 2.3 m to an elevation angle of 5° (see also Figure 1), (middle) difference in the zenith total delay between the two solutions (2-hour estimates), and (bottom) station height differences between IMF and NMF solution (daily estimates); gray line: 30-day moving average in all subplots.

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[16] As tropospheric delay and station height are highly correlated, ZTD estimates become smaller with the use of IMF for all the stations, while the station heights increase. At station Davis this effect results in a mean decrease in ZTD of 5 mm. The influence of mapping functions on the ZTD estimates is also significant for the derivation of integrated water vapor (IWV). This is especially true for Antarctica, where the largest differences in the MF parameters occur. Since the water vapor content of the atmosphere is very low in Antarctica [Vey et al., 2004], this consequently produces large relative errors in the IWV estimates over this region. For example, differences in the ZTD at station Davis cause an offset of 1 kg/m2 in IWV (1 mm in precipitable water), which is about 30% of the mean water vapor content. Furthermore, the seasonal signal in the derived integrated water vapor is significantly influenced by the choice of the mapping function. At station Davis the amplitude of the seasonal signal is reduced by 2 kg/m2 when applying IMF instead of NMF, corresponding to about 50% of the seasonal variation.

4.2. Influence on Scale, Origin and Geometry of the Terrestrial Reference Frame

[17] We conducted a 7-parameter transformation between the NMF and IMF solutions. The IMF solution shows a larger scale than the NMF solution by 0.15 ppb, which is not surprising as larger heights result for most of the stations when applying IMF. When using IMF instead of NMF the estimated origin shifts by less than 1 mm in each component. However, as shown in Figure 5, both scale and origin vary with time. The variations over the year reach peak-to-peak values of 0.3 ± 0.1 ppb for the scale and 3 ± 1 mm for the origin.

image

Figure 5. Difference in scale and origin between the global solutions with IMF and NMF (30-day averages).

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[18] The influence of MF differences is most pronounced for the height component. However, there are also effects on the horizontal station positions due to the correlation between the estimated parameters, especially the horizontal station positions on the one hand and the tropospheric gradients on the other hand. The residuals of the transformation reveal a change of horizontal station positions for most of the stations by up to 1 mm in almost northern direction (Figure 6).

image

Figure 6. Residuals in the horizontal coordinates were obtained by subtracting the NMF solutions from the transformed IMF coordinate sets (mean over the year 2002).

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4.3. Height Repeatability and Cut-Off Angle Sensitivity

[19] As demonstrated in Section 2, NCEP and ECMWF are in close agreement for the IMF input parameter (the 200 hPa geopotential height), the resulting effect on IMF is small. Assuming, IMF models the true atmospheric conditions better than NMF, the position repeatability and the dependence on the cut-off elevation angle should decrease in the IMF solution. Therefore, daily repeatabilities of station heights between the solutions using IMF and NMF were compared for the year 2002. IMF improves the repeatability of the height estimates for about two thirds of the stations. The largest improvements occur in the high southern latitudes. Here the repeatability reduces by up to 3 mm corresponding to an improvement of 20%.

[20] The better the MF is modeling the atmospheric state the less the solution depends on the cut-off angle. We conducted test calculations with 15°, 10°, 5° and 3° cut-off angles for the year 2002 using NMF and IMF. In general, the global solutions derived for different cut-off angles agree better with each other in the case of applying IMF instead of NMF. The differences in daily station heights estimated with 15° and 3° cut-off angles are reduced by about 1 mm for the IMF solution compared to NMF. However, also with IMF the results are not completely independent of the cut-off angle. As an example the cut-off angle sensitivity of the ZTD parameter is shown in Figure 7 for the stations Davis in Antarctica and Metsahovi in Finland. Globally, the reduction of the elevation mask from 15° to 3° causes ZTD differences larger than 3 mm for about 18% of all stations. IMF shows nearly 20% smaller offsets compared to NMF for all conducted cut-off angle changes.

image

Figure 7. Comparison of the elevation cut-off dependence of the ZTD when applying NMF and IMF at (top) station Davis, Antarctica and (bottom) Metsahovi, Finland for 22 November 2002 (day of year 236).

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[21] Mapping functions based on meteorological data, such as IMF, provide a high temporal and spatial resolution. They capture the real atmospheric conditions much better than NMF which is a parametrized mapping function based on many approximations. The choice of the mapping function significantly influences estimated parameters like station positions and tropospheric delays and, to a smaller extent, the origin and scale of the global network. Applying NMF introduces systematic offsets for the derived parameters, which in addition vary with time.

[22] The largest impact of IMF on estimated GPS heights and zenith delays was detected in the polar regions. In high southern latitudes the daily repeatability of the station heights is improved in the IMF solution by up to 3 mm, which is equivalent to 20% relative improvement. For most stations of the network the results are less dependent on elevation cut-off angle for the IMF than for the NMF solution.

[23] When determining long GPS time series on a global scale we strongly recommend the use of site and time-dependent mapping functions based on meteorological data rather than parametrized functions, especially as nowadays GPS observations at very low elevation angles are included in the analyses. IMF has proven to be computer-resource friendly for global network analyses.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References

[24] Sincere thanks go to A. E. Niell for making the IMF algorithm available to us. We acknowledge the cooperation with the CODE Analysis Center team concerning the setup of the GPS processing scheme. We gratefully acknowledge the constructive comments of the reviewers A. Niell and J. Kouba, which helped to improve the manuscript. We thank the IGS, NCEP and ECMWF for providing data via their Web sites and J. Boehm for making radiosonde data available to us. The research is funded by the German Research Foundation (DFG). One of the authors (S.V.) is supported by the ‘Studienstiftung des Deutschen Volkes’.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model Differences Between NMF and IMF
  5. 3. GPS Data Analysis
  6. 4. Results
  7. 5. Conclusions
  8. Acknowledgments
  9. References
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  • Herring, T. A. (1992), Modelling atmospheric delays in the analysis of space geodetic data, in Symposium on Refraction of Transatmospheric Signals in Geodesy, Neth. Geod. Comm. Ser., vol. 36, edited by J. C. De Munk, and T. A. Spoelstra, pp. 157164, Neth. Comm. Geod., Delft, Netherlands.
  • Hugentobler, U., R. Dach, P. Fridez, and M. Meindel (Eds.) (2005), Bernese GPS software 5.0, Astron. Inst., Univ. of Bern, Bern, Switzerland.
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  • Niell, A. E. (1996), Global mapping functions for the atmosphere delay at radio wavelengths, J. Geophys. Res., 101(B2), 32273246.
  • Niell, A. E. (2000), Improved atmospheric mapping functions for VLBI and GPS, Earth Planets Space, 52, 699702.
  • Rocken, C., S. Sokolovskiy, J. M. Johnson, and D. Hunt (2001), Improved mapping of tropospheric delays, J. Atmos. Oceanic Technol., 18(7), 12051213.
  • Schuh, H., and J. Boehm (2004), VMF and IMF mapping functions based on data from ECMWF, in Celebrating a Decade of the International GPS Service Workshop and Symposium, 1–5 March 2004, Bern, Switzerland [CD-ROM], edited by M. Meindel, Astron. Inst., Univ. of Bern, Bern, Switzerland.
  • Vey, S., R. Dietrich, K. P. Johnsen, J. Miao, and G. Heygster (2004), Comparison of tropospheric water vapour over Antarctica derived from AMSU-B data, ground-based GPS data and the NCEP/NCAR reanalysis, J. Meteorol. Soc. Jpn., 82(1B), 259267.