## 1. Introduction

[2] All electromagnetic signals propagating through the (neutral) atmosphere are delayed (and damped) since the refractivity index of the gases in the media is greater than one. Due to the complex structure of the atmosphere and its highly variable water vapor fields, it is very difficult to correct for such delays without a detailed knowledge of the current meteorologic field. Space geodetic applications such as the Global Positioning System (GPS) or Very Long Baseline Interferometry (VLBI) avoid an a priori correction of the measurements; however, they rely on the estimation of the troposphere delays using simple model assumptions that enable slant observations to be related to zenith delays, which are estimated as unknowns along with other geodetic parameters [e.g., *Hofmann-Wellenhof et al.*, 2001]. Numerical weather models have drawn the interest of the space geodetic community for over a decade, as they proved to be highly useful for the determination of the mapping functions [e.g., *Niell*, 1996, 2001; *Boehm et al.*, 2006a, 2006b] that describe the growth of the atmospheric path delay with increasing zenith distance. However, it was necessary to carry out ray-tracing through meteorologic fields in order to compute these numerical expressions. The obtained path delays were only used to obtain parameters of the mapping functions, while the atmospheric delays were not considered in space geodetic data processing.

[3] However, in recent years, such models have been improved with regard to their spatial and temporal resolution. Further, an increasingly large number of small-scale phenomena are now being considered in the model runs. The enhanced accuracy and precision has made it feasible to utilize ray-traced atmospheric delays directly for the analysis of space geodetic applications.

[4] In order to realize the targeted accuracy and provide ray-traced delays in real time, it is necessary to apply a dedicated processing scheme, while simultaneously taking into consideration the differences between the meteorologic and geodetic reference systems. Therefore in the following sections, we will discuss how numerical weather models can be transformed into geodetic coordinate systems, which facilitate a fast and efficient solution to the ray-tracing task.

### 1.1. Ray-Path and Atmospheric Delay

[5] The propagation path of electromagnetic rays can be deduced from the 3-D Eikonal equation [e.g., *Paris and Hurd*, 1969]

where *n*() is the index of refractivity at position . is referred to as the Eikonal, which enables the computation of the ray direction by computing ∇ϕ(). Therefore the following relation holds in a ray-based coordinate system of length *s*

This equation can be easily split into two coupled first-order differential equations and solved by standard methods such as Runge-Kutta and general linear methods. Thus when the ray-path is known, the atmospheric delay Δ*τ*_{a}, which is expressed by the following equation, can be computed:

The first integral in equation (3) is evaluated along the path of the ray from the transmitter, through the atmosphere, until it reaches the receiver and yields the electromagnetic delay Δ*τ*_{e}. The second term denotes the geometric excess resulting from the variation in the path of the ray as it passes through the atmosphere as compared to its path when propagating through vacuum. Due to ray-bending, the outgoing elevation angle, which is the elevation angle at the upper boundary of the atmosphere, will always be smaller than the initial elevation angle at the station where ray-tracing begins. This implies that a few iterations are required (the number of iterations depends on the elevation angle) until the outgoing elevation angle agrees with the vacuum elevation angle, which can be computed a priori from the observing geometry (refer to the discussion in section 4.4). In the case of satellite geodesy, orbit information can be used to compute the vacuum geometry. Further, in case of VLBI experiments, source catalogs provide the information required to compute the azimuth and elevation angles for the selected stations. Thus if the four-dimensional refractivity field can be deduced from the numerical weather models, it is possible to compute atmospheric path delays for any given station located within the model boundaries.

### 1.2. Obtaining Refractivity From Meteorologic Data Sets

[6] As explained by *Smith and Weintraub* [1953] and *Boudouris* [1963], atmospheric refractivity *N* (or the index of refractivity *n*) can be computed from the following equation

where *p*_{d} and *p*_{v} denote the partial pressures of dry air and water vapor (in hPa), respectively, and *T* represents the absolute temperature. The physical constants for our studies were set in accordance with *Bevis et al.* [1994]; therefore, *k*_{1} = 77.604 (K hPa^{−1}), *k*_{2} = 70.4 (K hPa^{−1}), and *k*_{3} = 373,900 (K^{2} hPa^{−1}). Since numerical weather models provide only the values of the total pressure *p*, the water vapor pressure should be computed first; subsequently, the pressure of dry air should be derived by applying the relationship *p*_{d} = *p* − *p*_{v}. Therefore as an intermediate step, the saturation vapor pressure *p*_{w}, which depends only on the temperature *T*, is computed as follows:

which denotes the well-known equation from *Goff* [1957], incorporating the corrections proposed by the *World Meteorological Organization* [2000]. Finally, *p*_{v} can be obtained from

where *RH* is the relative humidity (expressed as a percentage), which is also provided by numerical weather models.