Global Positioning System Reference
In-Depth Information
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
The products of phase index of refraction n ϕ and velocity c ϕ , and group index of
refraction n g and velocity c g equal the speed of light in vacuum, i.e.,
n ϕ c ϕ
=
c
(6.75)
n g c g =
c
(6.76)
Us ing the various relationships identified above, the group index of refraction can be
ex pressed as
f dn ϕ
df
n g
=
n ϕ
+
(6.77)
If the group index of refraction depends on the frequency f , i.e., the derivative dn ϕ /
df in (6.77) is not zero, then n ϕ
[21
n g and we call the medium dispersive. It follows
that the phase velocity and the group velocity are not the same in a dispersive medium
whereas in a nondispersive medium we have c ϕ
=
=
c g and the wave envelope moves
Lin
3.9
——
No
PgE
with the same velocity as the wave.
Expression (6.77) is applicable in quantifying the impact of the ionosphere on the
GPS signals. Substituting the phase index of refraction (6.63) into (6.77) and carrying
out the differentiation gives the expression for the group index of refraction,
f dn ϕ
df
n g =
n ϕ +
=
1
+
N I
(6.78)
Neglecting terms of the order N I squared and higher, the expressions for the phase
and group velocities become
[21
c
n ϕ =
c
c ϕ =
N I =
c ( 1
+
N I )
(6.79)
1
c
n g =
c
c g =
N I =
c ( 1
N I )
(6.80)
1
+
Since N I is a positive number, the phase velocity is larger than vacuum speed and the
group velocity is smaller than vacuum speed by the same amount
c ; i.e.,
40 . 30 c
f 2
c
=
cN I
=
N e
(6.81)
The time of a code delay or the phase advancement that is registered at the receiver
is directly related to the velocity difference
c and its variations along the path.
Integrating (6.81) over time and realizing that ds
=
cdt gives the ionospheric delay
in units of distance
f 2 N e ds
40 . 30
I k,f,P
I f,P
=
(6.82)
 
Search WWH ::




Custom Search