# Broadcast Ephemeris with Centimetric Accuracy: Test Results for GPS, Galileo, Beidou and Glonass
## 1 Introduction  
Is the limited accuracy of the broadcast ephemeris caused by inherent limitations of the model to compute the spacecraft coordinates and clock offset, or by the model coefficients in the broadcast message being insufficiently accurate? In this paper we provide arguments based first on detailed test cases and then on volume computations that the broadcast ephemeris can in fact provide spacecraft coordinates and clock offsets as accurate as the corresponding precise ephemeris provided that the coefficients of the broadcast ephemeris are appropriately tuned. For each GNSS, the Interface Control Document (ICD) and the documentation of the RINEX format of the Navigation Message provide full description of the parameters which enable the calculation of the Earth Centered Earth Fixed (ECEF) coordinates of a Space Vehicle (SV), and its clock offset relative to the common time scale of a specific GNSS. Most GNSSs adopt a model which approximates the orbit of the SV with an arc of a Keplerian ellipse with periodic and secular perturbations in the Keplerian orbital elements [1]. The length of the arc, or validity time of the set of these orbital parameters, is typically two hours, so that a new ephemeris message is normally issued every two hours, with a reference epoch called Toe (Time of ephemeris) and a corresponding orbital phase angle denoted by \(M_{0}\). The clock offset is instead described by a quadratic polynomial with a reference epoch called Toe (Time of clock). There are 15 parameters in the orbital model and 3 parameters of the clock polynomial. Often Toe and Toe coincide. GNSSs falling into this mathematical scheme of the orbit are GPS (USA), Galileo (Europe), Beidou (China), NAVIC/IRNSS (India) and QZSS (Japan). Each GNSS computes the broadcast message using different control stations and reference clocks, which may result in small but significant differences in the realization of the Terrestrial Reference Frame and System Time [2; 3; 4; 5; 6]. Regarding Glonass, the instantaneous position of each SV is computed by numerical integration of the equations of motion. The force field consists of a term approximating the gravitational field of the Earth truncated to the \(J_{2}\) zonal. Because the numerical integration is carried out in a rotating frame, the force includes also the centrifugal and Coriolis terms. Third body perturbations, typically caused by the gravity of the Moon and the Sun, are modeled by accelerations kept constant during the validity time of the message, which, for Glonass, is 30 min. Thus, there are in each Ephemeris Message three ECEF positions and three velocities at epoch Toe, and three constant Lunisolar accelerations, for a total of nine parameters. The clock offset is again modeled by a quadratic polynomial with a time Toe as reference. An important feature of the Glonass approach is that the Reference Time scale tracks the Leap Second, so that the computation epoch must be offset by the number of Leap Seconds corresponding to the epoch [7]. Further details can be found in the pertinent ICDs (Interface Control Documents). Noteworthy is the use of different values of the product of the gravitational constant by the mass of the Earth. Galileo, Glonass and Beidou, for example, use \(\mu=3.986004418\times 10^{14}\) m\({}^{3}\) s\({}^{-2}\), whereas GPS and QZSS use \(\mu=3.9860050\times 10^{14}\) m\({}^{3}\) s\({}^{-2}\). Merged files of Broadcast Ephemeris (BRDM) in the standard RINEX 3.04 format can be downloaded for example from the BKG server (_Bundesamt fuer Kartographie und Geodaseie_). As mentioned earlier, these messages inherit systematic differences due to the realization of the Reference Frame, the relative alignment of the GNSS-specific System Time and other peculiarities such as the gravity constant or the nominal Earth rotation rate. All these interoperability issues must be considered by a user who wants to compute its ECEF coordinates by simultaneously processing data from several GNSSs [8].