## 1. Introduction

[2] Modeling the path delays due to the neutral atmosphere for microwave signals emitted by satellites or radio sources is one of the major error sources in the analyses of Global Positioning System (GPS) and very long baseline interferometry (VLBI) observations. The concept is based on the separation of the path delays, ΔL, into a hydrostatic and a wet part [e.g., *Davis et al.*, 1985]:

In equation (1), the total delays ΔL(e) at an elevation angle e are made up of a hydrostatic (index h) and a wet (index w) part, and each of these terms is the product of the zenith delay (ΔL_{h}^{z} or ΔL_{w}^{z}) and the corresponding mapping function mf_{h} or mf_{w}. These mapping functions, which are independent of the azimuth of the observation, have been determined for the hydrostatic and the wet part separately by fitting the coefficients a, b, and c of a continued fraction form [*Marini*, 1972] (equation (2)) to standard atmospheres [e.g., *Chao*, 1974], to radiosonde data [*Niell*, 1996], or recently to numerical weather models (NWMs) [e.g., *Niell*, 2000; *Boehm and Schuh*, 2004]:

Whereas the hydrostatic zenith delays, ΔL_{h}^{z} (m), which can be determined from the total pressure p in hPa and the station coordinates (latitude ϕ and height h in m) at a site [*Saastamoinen*, 1973] (equation (3)), and the hydrostatic and wet mapping functions are assumed to be known, the wet zenith delays, ΔL_{w}^{z}, are estimated within the least squares adjustment of the GPS or VLBI analyses:

However, there might be errors in the hydrostatic zenith delays or the mapping functions, and their influence on station heights is well described with a rule of thumb by *Niell et al.* [2001, p. 839]: “The error in the station height is approximately one third of the delay error at the lowest elevation angle included in the analysis.” Following a refinement of this rule of thumb by *Boehm* [2004], the factor is rather 1/5 than 1/3 for a minimum elevation angle of 5°, which is also close to the value 0.22 found by *MacMillan and Ma* [1994]. The following two examples illustrate this rule of thumb, which holds for both GPS and VLBI, but which depends on the actual distribution of elevations and on whether elevation-dependent weighting is used: The hydrostatic and wet zenith delays are taken to be 2000 mm and 200 mm, respectively, the minimum elevation angle is 5°, and the corresponding values for the hydrostatic and wet mapping functions are 10.15 (mf_{h}(5°)) and 10.75 (mf_{w}(5°)). (1) We assume an error in the total pressure measured at the station of 10 hPa: 10 hPa correspond to ∼20 mm hydrostatic zenith delay (compare equation (3)), which is then mapped with the wrong mapping function (factor 0.6 = 10.75 − 10.15). At 5° elevation the mapping function error is 12 mm, and one fifth of it, i.e., 2.4 mm, would be the resulting station height error. (2) We consider an error in the wet mapping function of 0.01 (mf_{w}(5°) = 10.76 instead of 10.75) or in the hydrostatic mapping function of 0.001 (mf_{h}(5°) = 10.151 instead of 10.15). The error at 5° elevation in both cases is 20 mm; that is, the error in the station height would be approximately 4 mm.

[3] The Vienna mapping functions (VMF) introduced by *Boehm and Schuh* [2004] depend only on elevation angle and not on azimuth; that is, they assume that the troposphere is symmetric around the stations. For the b and c coefficients (see equation (2)) the best values available at that time were taken from the isobaric mapping functions (IMF) [*Niell and Petrov*, 2003] for the hydrostatic part and from the Niell mapping functions (NMF) at 45° latitude [*Niell*, 1996] for the wet part. (See Figure 8 in section 4.3 for the hydrostatic mapping function from NMF and VMF for the station Algonquin Park in 2002 and 2003.) In section 2, an updated version for the VMF [*Boehm and Schuh*, 2004] is developed, which is based on new b and c coefficients for the hydrostatic mapping functions and which will be called VMF1 hereinafter. For VMF1, the c coefficients from ray tracing are fit to a function of latitude and day of year to remove systematic errors. This is important for geophysical applications of geodesy, for instance, to determine the correct seasonal and latitude dependence of hydrology. An alternative approach to the traditional separation into wet and hydrostatic mapping functions is the introduction of the “total Vienna mapping function” (VMF1-T) for mapping the total delays, which uses the total refractivity instead of its hydrostatic and wet components. In section 3 different procedures are described for calculating a priori zenith delays that can be used for GPS and VLBI analyses, including their determination from the operational analysis pressure level data set of the European Centre for Medium-Range Weather Forecasts (ECMWF). In section 4, the impact of the different mapping functions and of the different a priori zenith delays on geodetic results is investigated using all IVS-R1 and IVS-R4 24-hour sessions of 2002 and 2003, including CONT02. CONT02 was a 2-week continuous VLBI campaign in the second half of October 2002 [*Thomas and MacMillan*, 2003].