Radio signals transmitted by GPS (Global Positioning System) satellites, traverse the atmosphere before they are recorded by ground-based receivers. Refraction in the atmosphere changes the phase and amplitude of the signals. From GPS dual-frequency phase-observations the signal travel-time delay induced by the neutral atmosphere, in this study referred to as the STD (Slant Total Delay), can be retrieved. Conversely, provided that the refractivity field is known, e.g. given by a NWM (Numerical Weather Model) analysis, STDs can be computed. A method to compute STDs for a given refractivity field is required in both, meteorological and geodetic applications. Potential applications in GPS meteorology [Bevis et al., 1992] include variational data assimilation and least-travel time tomography [Järvinen et al., 2007; Bender et al., 2010]. A typical application in geodesy is the determination of mapping functions [Rocken et al., 2001; Boehm et al., 2006].
 The computation of the propagation of radio signals in a refractivity field is based on Fermat's principle: the path taken by a ray between the satellite and the ground-based receiver is the path that can be traversed in the least time. From calculus of variations the ray-trajectory equation is derived and solved by a numerical algorithm. To date a number of different algorithms exist [Mendes, 1999; Pany, 2002; Nievinski, 2009]. Fast and accurate algorithms were recently summarized and analyzed by Hobiger et al. . In essence, all of them have in common that the ray-trajectory equation is solved as an initial value problem. In this study we present an alternative algorithm; the ray-trajectory equation is solved directly as a boundary value problem. The algorithm yields a similar performance, regarding the accuracy and the computational speed. In addition, we estimate the uncertainty of STDs due to simplifying assumptions in the algorithm and we estimate the uncertainty of STDs due to the uncertainty of the refractivity field. The algorithm is particularly suited to compute STDs for a large and continuously operating network of ground-based receivers. In a first application, we compare STDs retrieved from GPS phase-observations with STDs derived from the ECMWF (European Center for Medium-Range Weather Forecasts) analysis.
 This paper is structured as follows. In section 2 we provide a technical description of the algorithm. In section 3 we study the accuracy and the computational speed of the algorithm. In section 4the algorithm is used to compare STDs retrieved from the GPS phase-observations with STDs derived from meteorological analyses.Section 5 summarizes the main results.