Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 308

There are a number of approaches for correcting for the Sagnac effect. One com-
mon approach is to avoid the Sagnac effect entirely by working within an ECI coor-
dinate system for satellite and user position computations. An ECI frame can be
conveniently obtained by freezing an ECEF frame at the instant of time when
pseudorange measurements are made to the set of visible satellites. The Sagnac effect
does not arise in an ECI frame. Importantly, the satellite positions that are used in
the standard GPS user position solution (Section 2.4) must correspond to the times
of transmission, which are generally not the same. The time of transmission for each
satellite, T S , is a natural measurement of a GPS receiver, as discussed in Section
5.7.1. Users of commercial equipment can access time of transmission for each satel-
lite by simply subtracting the pseudorange measurement (after applying the clock
corrections discussed in Section 7.2.1) divided by the speed of light from the
receiver's time-tag for the measurement. Next, each satellite position can be com-
puted in terms of its ECEF coordinates ( x S , y S , z S ) at its time of transmission using the
broadcast ephemeris data described in Table 2.3. Then, each satellite position can be
transformed into the common ECI frame using the rotation:

&

( )

&

( )

x
y
z

cos Ω TT − sin Ω TT − 0 x

y
z

eci u s u s s

&

( )

Ω

( )

= − sin Ω TT − cos TT − 0

eci u s u s s

0 0 1

eci s

In this formulation, the time of reception, T u , is initially unknown prior to the
position/time estimate. It may be initially approximated as the average time of trans-
mission among visible satellites plus 75 ms for an Earth-based user. Once the posi-
tion solution is generated using the least-squares technique described in Section 2.4,
the user clock correction can be applied to obtain a much better estimate of T u , and
the process can be iterated. The user's position coordinates are the same in both the
ECEF and ECI frames at the signal reception time, since by definition these two
frames were fixed at that instant. A number of alternative Earth rotation correction
formulations, along with numerical examples, are provided in the excellent refer-
ence [12].
Finally, the GPS signal experiences space-time curvature due to the gravitational
field of the Earth. The magnitude of this relativistic effect can range from 0.001 ppm
in relative positioning to about 18.7 mm for point positioning [13].

7.2.4 Atmospheric Effects
The propagation speed of a wave in a medium can be expressed in terms of the index
of refraction for the medium. The index of refraction is defined as the ratio of the
wave's propagation speed in free space to that in the medium by the formula

c
v

n = (7.5)

where c is the speed of light equal to 299,792,458 m/s as defined within the WGS-84
system. The medium is dispersive if the propagation speed (or, equivalently, the
index of refraction) is a function of the wave's frequency. In a dispersive medium,