Taking advantage of numerical integration, we solve the direct and inverse geodetic problems on the ellipsoid. In general, the solutions are composed of a strict solution for the sphere plus a correction to the ellipsoid determined by numerical integration. Primarily the solutions are integrals along the geodesic with respect to the reduced latitude or azimuth, but these techniques either have problems when the integral passes a vertex (i.e., point with maximum/minimum latitude of the arc) or a singularity at the equator. These problems are eliminated when using Bessel's idea of integration along the geocentric angle of the great circle of an auxiliary sphere. Hence, this is the preferred method. The solutions are validated by some numerical comparisons to Vincenty's iterative formulas, showing agreements to within 2 × 10 À10 of geodesic length (or 3.1 mm) and 4 × 10 À5 as seconds of azimuth and position for baselines in the range of 19,000 km.

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Abstract:In GPS positioning, usually the satellite ephemeris are fixed in the observation equations using broadcast or published values. Therefore, to have a realistic covariance matrix for the observations one must incorporate a well-defined covariance matrix of the satellite ephemeris into the observations covariance matrix. Contributions so far have discussed only the variance and covariance of the observations. Precise Point Positioning (PPP) is a technique aimed at processing of measurements from a single (stand-alone) GPS receiver to compute high-accurate position. In this paper, the covariance matrix of the satellite ephemeris and its impact on the position estimates through the PPP are discussed. Keywords

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The international GNSS service (IGS) started publishing the precise ephemeris files in the form of the standard products #3, version C (the sp3c files) in which the GPS satellite orbits and clocks and their uncertainties were available since 2004. Incorporating these uncertainties into the GPS observation equations results in a better stochastic model of the processing system. The reality of these uncertainties is questioned and studied in this paper. Precise point positioning (PPP) model, statistical tests and variance component estimation (VCE) techniques are employed for this study. The results confirm the efficiency of the proposed method in the assessment of reality of the published ephemeris uncertainties.