Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 310

dv
d

p

vv =−λ (7.7)

g P

λ

which implies that the difference between the group velocity and phase velocity
depends on both the wavelength and the rate of change of phase velocity with wave-
length.
The corresponding indices of refraction are related by [13]

dn
df

p

nnf =+ (7.8)

g p

where the indices of refraction are defined by

c
v

c
v

n = n = (7.9)

p g

p g

and f denotes the signal frequency. In a nondispersive medium, wave propagation is
independent of frequency, and the signal phase and signal information propagate at
the same speed with v g = v p and n g = n p .

7.2.4.1 Ionospheric Effects
The ionosphere is a dispersive medium located primarily in the region of the atmo-
sphere between about 70 km and 1,000 km above the Earth's surface. Within this
region, ultraviolet rays from the sun ionize a portion of gas molecules and release
free electrons. These free electrons influence electromagnetic wave propagation,
including the GPS satellite signal broadcasts.
The following is based on a similar development in [13]. The index of refraction
for the phase propagation in the ionosphere can be approximated as

c
f

c
f

c
f

n =+ 1

2
2

+

3
3

+

4
4

(7.10)

L

p

where the coefficients c 2 , c 3 , and c 4 are frequency independent but are a function of
the number of electrons (i.e., electron density) along the satellite-to-user signal prop-
agation path. The electron density is denoted as n e . A similar expression for n g can be
obtained by differentiating (7.10) with respect to frequency and substituting the
result along with (7.10) into (7.8). This results in the following:

c
f

23 c

f

c
f

2
2

3
3

4
4

n =− 1 − − L

g

Neglecting higher-order terms, the following approximations are obtained:

c
f

c
f

n =+ 1

2

n =− 1

2
2

(7.11)

p g

2