Data Processing for Positioning - A Software-Defined GPS and Galileo Receiver: A Single-Frequency Approach - page 114

However, GPS satellites do not follow the presented normal orbit theory. We have
to use time-dependent, more accurate orbit values. They come to us as the socalled
broadcast ephemerides; see Section 8.2.2. We insert those values in a procedure
given below and finally we get a set of variables to be inserted into (8.8).
Obviously, the vector is time-dependent, and one speaks about the ephemeris
(plural: ephemerides , emphasis on “phem”) of the satellite. These are the param-
eter values at a specific time. Each satellite transmits its unique ephemeris data.
The parameters chosen for description of the actual orbit of a GPS satellite
and its perturbations are similar to the Keplerian orbital elements. The broadcast
ephemerides are calculated using the immediate previous part of the orbit and they
predict the following part of the orbit. The broadcast ephemerides are accurate to
1-2 m. For geodetic applications, better accuracy is needed. One possibility is to
obtain post-processed precise ephemerides , which are accurate at the dm-level.
An ephemeris is intended for use from the epoch t oe of reference counted in
seconds of the GPS week. It is nominally at the center of the interval over which
the ephemeris is useful. The broadcast ephemerides are intended for use during
this period. However, they describe the orbit to within the specified accuracy for
2 hours afterward. The broadcast ephemerides include the parameters in Table 8.2.
The coefficients C ω , C r ,and C i correct argument of perigee, orbit radius, and
orbit inclination due to inevitable perturbations of the theoretical orbit caused by
variations in the Earth's gravity field, albedo and sun pressure, and attraction from
sun and moon.
Given the transmit time t (in GPS time), the following procedure gives the
necessary variables to use in (8.8):
Time elapsed since t oe t j = t − t oe

GM

n t j Mean anomaly at time t j µ j = µ 0 + / a 3 +

Iterative solution for E j E j = µ j + e sin E j

arctan √ 1

e 2 sin E j
cos E j −

−

True anomaly f j =

e

= 0 + ( ˙ Longitude for ascending node j − ω e ) t j − ω e t oe

10 − 5 rad ω e = 7 . 2921151467 · / s

Argument of perigee ω j = ω + f j + C ω c cos 2 (ω + f j ) + C ω s sin 2 (ω + f j )

Radial distance r j = a ( 1 − e cos E j ) + C rc cos 2 (ω + f j )

+ C rs sin 2 (ω + f j )

i 0 + it j + Inclination i j = C ic cos 2 (ω + f j ) + C is sin 2 (ω + f j ).

The mean Earth rotation is denoted ω e . This algorithm is coded as the M -file

satpos . The function calculates the position of any GPS satellite at any time. It is
fundamental to every position calculation.