Orbit parameters errors

Ephemeris data, broadcast in the satellite navigation message, give the satellite positions as a function of time. They are predicted starting from satellite po-

Figure 3.5: Comparison between dual frequencies ionospheric delays affected by receiver biases (red) and corresponding ionospheric delays from Klobuchar model data (green).The period is 2010/01/02-2010/28/02. Data are from the station of Massa (Tuscany).

sitions regularly verified at the ground control stations. Typically, overlapping intervals of 4 hours of GPS data are used by the operational control system to update satellite orbital elements for a following period of 1 hour.

Ephemeris errors can be of 2 m to 5 m, and can reach up to 50 m under selective availability. The ephemeris error is usually decomposed into components along three orthogonal directions defined for the satellite orbit: radial, along- track and cross-track.

Figure 3.6: Ephemeris error components (cf [29]).

The radial component in the estimation and prediction of satellite position is smaller than the other by one order of magnitude. These are good news as we know that geometric errors impacts on atmospheric sounding only on their along the receiver-satellite (line-of-sight) vector direction, and this projection depends primarily on the radial component and secondarily on the cross-track and along-track components.

Due to the fact that this errors originate from a prediction, they increase with the time interval between observations and updating of the orbital parameters.

The increasing need of precise ephemeris in quasi-real time for an increasing number of applications has concentrated the efforts of the user community to develop products more and more precise with respect to ephemeris broadcast in

the navigation message. Several institutions, e.g., the International GPS Ser- vice for Geodynamics (IGS), the U.S. National Geodetic Survey (N GS), and Geomatics Canada, have developed postmission precise orbital services. Pre- cise ephemeris data is based on GPS data collected at a global GPS network coordinated by the Internation GNSS Service (IGS). Precise ephemeris data contain very accurate parameters for the correction of space vehicle clock offset and drift and for retrieving satellite position at transmission time. At present, precise ephemeris data are also near real-time available through the Ultra-Rapid ephemeris service4_{, with an accuracy of 5 cm. The best accuracy can be gained}

with the delayed service, after 12 ÷ 18 days, that is 2.5 cm. There are other two services with intermediate delivery-time, and consequent intermediate parameter accuracies.

3.4.5.1 Retrieval of satellite positions

For computing the user position it is more convenient to refer to an Earth rotating reference frame, the Earth Centred Earth Fixed (ECEF) Cartesian coordinate system. This simplifies the computation of geographical coordinates (latitude, longitude, height) for the receiver position. The ECEF reference system has the origin in the Earth centre of mass, the x-y plane coincident with the equatorial plane, the x axis oriented along 0◦ longitude, and the z axis normal to the x-y plane and oriented towards the north pole.

Satellite position is univocally determined from the knowledge of instant of time (epoch) we want to refer to, by means of orbital parameters (ephemeris) contained into the navigation message of the satellite signal, that are system- atically updated about every two hours. This parameters will be described in § 3.4.5.2. Once decoded, ephemeris must be processed through a (large) set of established equations. The steps start from the computation of the correct in- stant of the signal transmission and the resolution of the Kepler equation for the satellite orbit, in a loop fashion up to obtain the convergence to the satellite

position coordinates. A detailed explanations of such steps is not relevant for this work, however an exhaustive analysis of the process can be found in (cf. [3] and [11]).

In order to completely describe the observation geometry and its effect on signals, it is necessary to compute the Earth rotation during the signal travel time: this is known as the Sagnac effect and will be described in § 3.4.6.

3.4.5.2 Ephemeris

Orbital parameters are “written” into the navigation message. They are referred to a determined time (epoch). Their meaning is clear only knowing the orbital equation to be solved for retrieving satellite positions. However these parameters will be listed to give an idea of the number of features that need to be taken into account, according to the Interface Control Document (cf. [21]):

toc: clock reference time (in seconds) used for clock offset computation;

a0 , a1, a2: polynomial coefficient for clock offset computation;

tGD: (Group Delay), instrumental delay delay differential;

toe: reference time (in seconds) since the GPS week start (at the Saturday/Sunday

transition) of ephemeris values;

M0: mean anomaly;

∆n: mean motion difference from computed value;

e: satellite orbit eccentricity; √

a: square root of orbital semi-major axis

Ω0: latitude of ascending node of the orbit plane at the weekly epoch;

i0: inclination angle of the orbit plane with respect to the equatorial plane;

˙

Ω0 rate of right ascension;

˙i rate of inclination angle;

Crc: amplitude of the cosine harmonic correction term to the orbit radius

Crs: amplitude of the sine harmonic correction term to the orbit radius

Cuc: amplitude of the cosine harmonic correction term to the argument of

latitude

Cus: amplitude of the sine harmonic correction term to the argument of latitude

Cic: amplitude of the cosine harmonic correction term to the angle of inclination

Cis: amplitude of the sine harmonic correction term to the angle of inclination