# Development of an improved empirical model for slant delays in the troposphere (GPT2w)

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## Abstract

Global pressure and temperature 2 wet (GPT2w) is an empirical troposphere delay model providing the mean values plus annual and semiannual amplitudes of pressure, temperature and its lapse rate, water vapor pressure and its decrease factor, weighted mean temperature, as well as hydrostatic and wet mapping function coefficients of the Vienna mapping function 1. All climatological parameters have been derived consistently from monthly mean pressure level data of ERA-Interim fields (European Centre for Medium-Range Weather Forecasts Re-Analysis) with a horizontal resolution of 1°, and the model is suitable to calculate slant hydrostatic and wet delays down to 3° elevation at sites in the vicinity of the earth surface using the date and approximate station coordinates as input. The wet delay estimation builds upon gridded values of the water vapor pressure, the weighted mean temperature, and the water vapor decrease factor, with the latter being tuned to ray-traced zenith wet delays. Comparisons with zenith delays at 341 globally distributed global navigation satellite systems stations show that the mean bias over all stations is below 1 mm and the mean standard deviation is about 3.6 cm. The GPT2w model with the gridded input file is provided at http://ggosatm.hg.tuwien.ac.at/DELAY/SOURCE/GPT2w/.

## Keywords

Tropospheric delay Zenith delay Slant delay Mapping function## Introduction

The total slant delay Δ*L* at an elevation angle *ε* is the sum of a hydrostatic and a wet portion, and each of them can be expressed as the product of a zenith delay and the corresponding mapping function. Hydrostatic zenith delay values Δ*L* _{h} ^{z} are determined with sufficient accuracy from the local instantaneous pressure and the approximate station coordinates following Davis et al. (1985). In contrast, characteristic “wet” surface parameters, such as the water vapor pressure, represent largely inadequate measures of the non-hydrostatic vertical refractivity as required in evaluating zenith wet delays Δ*L* _{w} ^{z} . These values are therefore usually estimated in the analysis of space geodetic observations as unknown parameters.

However, several positioning and navigation tasks like real-time applications do not have the benefit of post-processing analyses, necessitating the availability of accurate a priori estimates for Δ*L* _{w} ^{z} . Widely adopted utilities including proxies for the zenith wet delays are RTCA-MOPS (1999), originally called UNB3 (Collins et al. 1996), the European Space Agency Galileo User Receiver model (ESA GALTROPO) (Krueger et al. 2004, 2005; Martellucci 2012) and the concurrently developed TropGrid model (Krueger et al. 2004) with a recent update called TropGrid2 by Schüler (2014). The ESA model and both TropGrid versions account for annual and diurnal variations of the underlying parameters. Here, we present a new empirical zenith wet delay model, which, in combination with a corresponding utility for the pressure and fully consistent mapping functions, provides troposphere delays down to elevation angles of 3° after being fed with positional information and the time of observation.

A wide range of mapping functions has been used in the past. For example, the new mapping functions (NMF; Niell 1996) and the global mapping functions (GMF; Böhm et al. 2006a) are empirical or the so-called blind mapping functions which only need the day of year and approximate station coordinates as input. Alongside such climatological approaches, the isobaric mapping functions (IMF; Niell 2001) and the Vienna mapping functions 1 (VMF1; Böhm et al. 2006b) account for the actual refractivity being derived from the operational analysis fields of numerical weather models at the epoch of the observations. Consequently, VMF1 positively exceeds NMF and GMF in terms of accuracy, but it is also prone to being inapplicable if the underlying mapping functions have not been updated.

Pressure values as required for the determination of zenith hydrostatic delays should be preferably taken from local measurements close to the antenna or from the gridded output of numerical weather models as, e.g., provided with the VMF1 at http://ggosatm.hg.tuwien.ac.at/. In case these streams are unavailable, analysts need to resort to blind models by Berg (1948) and Hopfield (1969), to UNB3m (Leandro et al. 2006), or to the global pressure and temperature model (GPT; Böhm et al. 2007) for approximate surface pressure information.

*A*

_{0}) as well as annual (

*A*

_{1},

*B*

_{1}) and semiannual amplitudes (

*A*

_{2},

*B*

_{2}) for selected parameters

*r*on a regular 5° grid at mean ETOPO5 (earth topography) heights following

*a*

_{h}and

*a*

_{w}, pressure

*p*, temperature

*T*and its lapse rate d

*T*, and the water vapor pressure

*e*. The underlying routine evaluates (2) at the four grid points surrounding the target location before extrapolating the parameters vertically to the desired height and interpolating the data from those base points to the observational site in horizontal direction. It should be noted here that the extrapolation of

*a*

_{h}(strictly speaking of the hydrostatic mapping function) follows Niell (1996), whereas

*a*

_{w}is assumed to be constant in the vicinity of the earth surface. The extrapolation of the pressure relies on an exponential trend coefficient related to the inverse of the virtual temperature (see Böhm et al. 2013, Eq. 23–27,

*ibid*.), and the linear extrapolation of the temperature utilizes the GPT2 inherent temperature lapse rate d

*T*. Surface grids for specific humidity within that model have been derived from linear interpolation between pressure levels in the vicinity of earth’s surface. These parameters could be possibly used to determine values of zenith wet delays, e.g., by using the expressions of Saastamoinen (1972), though this approach is not optimal and in the following we are introducing GPT2w as an extension to GPT2 with an improved capability to determine zenith wet delays in blind mode. The next describes the development of this new, auxiliary model, which is then validated against zenith delays from GNSS observations. An outlook addresses possible applications of GPT2w and concludes the study.

## Development of GPT2w

*ibid.*) represent the starting point of our derivations,

*k*

_{2}

*′*and

*k*

_{3}are empirically determined coefficients,

*R*

_{d}denotes the specific gas constant for the dry constituents,

*g*

_{m}is the gravity acceleration at the mass center of the vertical column of the atmosphere, and

*e*

_{s}stands for the water vapor pressure at the site. The weighted mean temperature at height

*H*along the local vertical d

*z*can be formulated as

*λ*is defined implicitly via the expression

*e*

_{s}plus the total surface pressure

*p*

_{ s }) and those defined on any target level (

*e*,

*p*). In essence, to use (3) for the determination of zenith wet delays, we need the water vapor pressure at the site, the weighted mean temperature

*T*

_{ m }, and the water vapor decrease factor

*λ*. While

*T*

_{ m }is readily accessible via numerical integration through the pressure level data, the derivation of

*λ*evades a straightforward treatment because it can be calculated for any pair of pressure levels, always leading to significantly different results which is due to the variability of water vapor with height. On the other hand, we are able to determine the zenith wet delays by numerical integration of the wet refractivity along the site vertical (Nilsson et al. 2013), allowing us to simply invert (3) toward a global

*λ*grid, which will be very much representative of the decrease factor behavior through the entire troposphere and fully consistent with the zenith wet delay.

*λ*, its annual and semiannual amplitudes, and the standard deviations of the residuals as estimated by least-squares adjustment of the “observed” water vapor decrease factor field obtained by inverting (3). Gauged to 10 years of monthly mean pressure level data from ERA-Interim (1° horizontal resolution) via Δ

*L*

_{w}

^{z}and

*T*

_{ m }, the fitted

*λ*grids are fully consistent with GPT2. Both the global mean field (Fig. 1a) and the annual cycle (Fig. 2a) show typical atmospheric circulation patterns, e.g., the intertropical convergence zone standing out as a belt of small vertical gradients in the wake of upwelling moist air (Wallace and Hobbs 1977). On the contrary, horse latitudes (at about 30° north and south) are characterized by a large-scale subsidence of dry air, which in combination with strong regional evaporation over oceanic surfaces (e.g., Arabic and Mediterranean Sea, west coasts of Africa and America) implicates steep gradients of

*λ*. Overall, Fig. 1 clearly demonstrates that it is not sufficient to apply constant decrease factors for the water vapor pressure, neither in space nor in time.

Analogous plots of mean values, annual and semiannual amplitudes, as well as post-fit standard deviations are depicted in Fig. 2 for the weighted mean temperature *T* _{ m } of the water vapor pressure (4). The distribution of the mean values is mostly latitude-dependent, while a dominant annual variation of 10–15 K prevails over north-east Asia and the eastern part of Canada. Semiannual amplitudes of *T* _{ m } are strongest at latitudes higher than 60° and in subtropical regions experiencing bimodal (two-peak per year) patterns in precipitation and temperature attached to two distinct wet seasons; see, e.g., northern India, the Arabic Sea, and the western Sahel zone.

*λ*act to generate distinct and sharp annual peaks in the zenith wet delay of about 10 cm.

*T*

_{ m }and

*λ*for zenith wet delays, we add these two prime parameters to our set of variables determined from 10 years of ERA-Interim data. The horizontal resolution of 1° in latitude and longitude is not reduced toward coarser spacing as in the case of GPT2 because the wet part in the atmosphere has more small-scale structures, in particular at coastal areas (Fig. 1). With GPT2w, the vertical extrapolation of

*e*

_{ s }adheres to (5), utilizing the GPT2w-inherent values of the water vapor decrease factor

*λ*. Pressure reductions conform to GPT2’s use of exponential trend coefficients calculated from grid point-wise virtual temperature information. Such isothermal scale heights may be alternatively adjusted for adiabatic effects, but the benefit of this approach on the side of delay predictions is not entirely conclusive (Schüler, 2014). Table 1 summarizes the main features of the newly suggested model, which can be compared to Tab. 1 by Lagler et al. (2013).

Main features of GPT2w (compare with Tab. 1 by Lagler et al. 2013)

Numerical weather model (NWM) data | Monthly mean profiles from ERA-Interim (37 levels): 2001–2010 |
---|---|

Representation | 1° grid at mean ETOPO5-based heights |

Temporal variability | Mean, annual, and semiannual terms, with fitted phases |

Temperature reduction | Mean, annual, and semiannual terms of temperature lapse rate estimated at every grid point, with fitted phases |

Water vapor reduction | Mean, annual, and semiannual terms of water vapor decrease factor estimated at every grid point, with fitted phases |

Pressure reduction | Exponential approach with scale heights based on grid point-wise virtual temperature values |

Output parameters | Pressure, temperature and its lapse rate, water vapor pressure and its decrease factor, hydrostatic and wet VMF1 mapping function coefficients |

## Validation with zenith total delays from GNSS

Since 2011, the United States Naval Observatory (USNO) archives and distributes final tropospheric estimates from the observation data of all GNSS sites of the International GNSS Service (IGS) in the Solution Independent EXchange format (SINEX) with a temporal resolution of 5 min and with a latency of about 4 weeks (Hackman and Byram 2012). The accuracy of the tropospheric delays in zenith direction is specified with 4 mm by the IGS Central Bureau at http://igs.org/components/prods.html. For the purpose of validating GPT2w, all available SINEX files in 2012 (about 110,000) were downloaded and subsequently cleansed. A visual inspection showed that in some cases, no realistic tropospheric estimates could be derived from the GNSS observations and we consequently excluded these specific records from the comparison in order to avoid falsification of the validation outcome. Furthermore, only zenith total delays (i.e., the sum of hydrostatic and wet zenith delays) with a formal error smaller than 18 mm were retained, which is a reasonable threshold yielding a total of 341 GNSS sites with at least 110 days of tropospheric delays. In total, we removed about 3.4 % of all entries of zenith total delays in the SINEX files.

Global statistics of the differences between zenith total delays provided by IGS and four blind models calculated as mean values over 341 sites for the year 2012: RTCA-MOPS, ESA blind model, GPT2 (using the approximate equation by Saastamoinen (1972) for the zenith wet delay), and GPT2w

Mean bias (cm) | Mean RMS (cm) | |
---|---|---|

RTCA-MOPS | −2.50 | 4.55 |

ESA | 0.83 | 3.82 |

GPT2 | −0.28 | 3.79 |

GPT2w | −0.02 | 3.61 |

Of course, a 1-year analysis comparing predicted zenith delays against those from the IGS is rather tentative and should be extended in future. Schüler (2014; Fig. 1, *ibid.*) provides statistics for several years illustrating that the RMS values change by about 2 mm over the years depending on the distribution of IGS antennas available. Considering this distribution of RMS values, our results agree reasonably well with those for TropGrid2 by Schüler (2014). The RMS values in his study, however, contain systematic contributions from the biases between the time series.

An extended validation of GPT2w with IGS data and ray-traced delays as well as a comparison with state-of-the-art tropospheric models like RTCA-MOPS or the ESA tropospheric blind model can be found in Möller et al. (2014). The bias and RMS statistics in that research were, however, computed from the globally merged sample of zenith total delay differences instead of determining them as mean values over all stations as above. Furthermore, the grid files of water vapor pressure have been slightly modified in the course of the present study.

## Summary and outlook

With GPT2w, we have introduced a new blind tropospheric delay model which is based on gridded values of water vapor pressure, water vapor decrease factor, and weighted mean temperature. In terms of zenith total delays, the globally averaged bias is below 1 mm and the RMS difference is about 3.6 cm as when compared to zenith total delays from GNSS at 341 globally distributed sites. Since GPT2w is also equipped with fully consistent hydrostatic and wet VMF1 coefficients, it may not only be used for positioning and navigation purposes but also for high-precision applications, like geophysical studies, where the wet mapping functions are essential to estimate residual zenith wet delays. In the future, the addition of diurnal and semidiurnal amplitudes will be considered for selected parameters.

GPT2w also contains the mean values as well as annual and semiannual amplitudes of the weighted mean temperature. This is an important quantity for the determination of the integrated water vapor or precipitable water as required in GNSS meteorology (Bevis et al. 1992). However, detailed studies on that application still have to be carried out.

Another promising future application of GPT2w is the combination with local meteorological observations at the sites. It will be investigated whether, e.g., a local measurement of water vapor pressure can be reasonably connected to climatological values of weighted mean temperature and water vapor decrease factor in order to determine improved zenith wet delays or whether additional corrections will have to be applied to fully exploit local measurements.

The MATLAB source code of GPT2w is available at: http://ggosatm.hg.tuwien.ac.at/DELAY/SOURCE/GPT2w/.

## Notes

### Acknowledgments

This work has been supported by project TROPSY which is funded by ESA as part of the Technology Research Program. J. Böhm would also like to acknowledge the Austrian Science Fund (FWF) for financial support within project RADIATE VLBI (P25320). We would like to thank Prof. Leick and two anonymous reviewers for their valuable comments.

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