Mapping Functions

As mentioned above, the total signal propagation delay is divided into a hydrostatic and a wet delay, and both components consist of a zenith delay correction and a corresponding elevation-dependent mapping function. This model is described by Eq. (3.25) under the assumption of azimuthal sym- metry of the neutral atmosphere around a station. In other words, this means that for a constant elevation angle the propagation delay is independent from the azimuth angle of the observation. Since the bending effect in Eq. (3.18) is accounted for by the hydrostatic mapping function, the (geometric) elevation angle in vacuum has to be used as input for the mapping functions instead of the refracted elevation angle.

According to Niell (2000), the mapping functions are defined as the ratio between the electrical path length L through the atmosphere at elevation ε and the electrical path length in zenith direction. Comparing the mapping functions for the hydrostatic and wet component, respectively, the hydrostatic mapping function is smaller than the wet mapping function, except for observations at very low elevation angles, where the geometric bending effect attributed to the hydrostatic mapping function is increasing considerably (Böhm 2004). The reason is the smaller scale height H for the wet part of the atmosphere (H ≈ 2 km) compared to the hydrostatic component (H ≈ 8 km).

30 3. Modeling the Atmosphere

Thus, the mapping functions can be interpreted as a measure for the thickness of the Earth’s atmosphere compared to the Earth’s radius (Niell 2000). If the thickness of the atmosphere decreases, the atmosphere appears to be flatter. Assuming the atmosphere to be planar and evenly stratified, the mapping function can be formulated as

mf (ε) = 1

sin(ε). (3.46)

This simple approach was used by Saastamoinen (1972), which, however, is only sufficient for observations with high elevation (ε > 20◦_{). Marini (1972) found that the continued fraction form}

mf (ε) = 1

sin(ε) + a

sin(ε)+_sin(ε)+cb

(3.47)

could be used to consider corrections accounting for the Earth’s curvature, where a, b and c are constants. During the last decades, many approaches were developed, which are generally slightly modified versions of the continued fraction form using constants from analytic fits to ray-tracing either for a defined standard atmosphere, for observed atmospheric profiles based on radiosonde measurements, or numerical weather models. In the following, a short overview of the different mapping function is given.

The first mapping function for space-geodetic applications with different coefficients for both map- ping functions was published by Chao (1971) who truncated the continued fraction form to a representation with two coefficients a and b and replaced the second sin(ε) by tan(ε) in order to force mf (ε) = 1 at the zenith. Davis et al. (1985) developed the mapping function CfA-2.2 for the hydrostatic delay down to 5◦ elevation. The three constants a, b, and c of the continued frac- tion form were derived from a ray-tracing analysis through idealized model atmospheres. Herring (1992) introduced the mapping function MTT, for which radiosonde data were used instead of standard atmospheres to fit the coefficients of the slightly modified continued fraction form

mf (ε) = 1 + a 1+_1+cb sin(ε) + a sin(ε)+_sin(ε)+cb . (3.48)

The coefficients in the mapping functions depend on the latitude and height of the site and the surface temperature, and were determined by a least squares fitting performed separately for the hydrostatic and wet component. The new mapping functions developed by Niell (1996) (often called Niell mapping functions, NMF) are unique in that global weather variations are represented analytically as a function of station latitude and height as well as the day of year instead of me- teorological parameters at the sites. The coefficients in the mapping functions were derived from profiles of standard atmosphere data down to 3◦ elevation angle using the continued fraction form in Eq. (3.48). Further, sine functions are introduced to describe the temporal variation of the co- efficients, and a height correction describing the increase of the mapping function with increasing height was used for the hydrostatic mapping function. The first mapping functions based on numer- ical weather models were the isobaric mapping functions (IMF) developed by Niell (2000). For the coefficients b and c empirical functions are used, whereas the coefficient a is determined from re-analysis data of the Data Assimilation Office (DAO) of the Goddard Space Flight Center. At present the most accurate mapping functions that are available globally are the Vienna mapping functions 1 (VMF1, Böhm et al. 2006b) which are based on a direct ray-tracing through the nu- merical weather model to make use of the entire model information provided instead of calculating

3.3. Definition of the Propagation Delay in the Neutral Atmosphere 31

intermediate parameters as necessary in the IMF. The coefficients a and b for both the hydrostatic and wet VMF1 are determined from empirical equations depending on the day of year and station latitude, whereas the a coefficients for both components are obtained based on different pressure level data sets from the ECMWF. The VMF1 is realized as discrete time series with 6 h resolution either on a global grid or for specific VLBI sites. Further, Böhm et al. (2006b) introduced the alternative approach of the so-called total Vienna Mapping Function 1 (VMF1-T) for a mapping of the total delays instead of separating the delays into a hydrostatic and a wet component. The principle idea was to introduce a mapping function which is not affected by poor a priori hydrostatic delays, because errors in the a priori model would not be fully compensated by the estimated wet part (cf. discussion in Sec. 3.3.3). However, this concept is not recommended since variations in the wet delay are faster than they could be described by the coefficients with a 6 h resolution (Nilsson et al. 2013). Böhm et al. (2006a) proposed the Global Mapping Function (GMF) to create, on the one hand, an easy-to-handle mapping function depending only on the day of year and station latitude, longitude and height, which, on the other hand, is consistent with VMF1 (Tesmer et al. 2007). The GMF is based on monthly mean profiles of meteorological data from the ECMWF 40-years reanalysis data (ERA-40).

For a more detailed description on mapping functions, the reader is referred to Mendes (1999), Nothnagel (2000), Böhm (2004), Tesmer et al. (2007), Böhm et al. (2007a) or Nilsson et al. (2013).