Global Positioning System Reference
In-Depth Information
frequencies (> 100 MHz), the refractive index mainly depends on the electron density, the
strength and direction of the geomagnetic field in relation to the ray path. Thus, the spatial
distribution of the electron density along the ray path and corresponding geomagnetic field
relationships determine the ionospheric impact on the electromagnetic wave.
2.1 Ionospheric refractive index
For high frequency (HF) radio waves with frequencies
f
> 100 MHz the phase refractive
index
n
can be derived from the Appleton - Hartree formula as (Appleton, 1932; Bassiri &
Hajj, 1993)
2
2
2
2
⎡
⎤
f
f
f
cos
Θ
f
f
(
)
p
p
g
p
p
2
2
n
=− ±
1
−
⎢
+
f
1
+
cos
Θ
⎥
(1)
g
2
3
4
2
2
f
2
f
4
f
⎢
⎥
⎣
⎦
in which
(
)
2
2
2
f
=
ne
/4
πε
m
p
e
0
(
)
f
=
eB
/2
π
g
where
f
p
is the plasma frequency,
f
g
is the gyro frequency,
ε
0
is the free space permittivity,
B
is the geomagnetic induction, Θ is the angle between the wave propagation direction and
the geomagnetic field vector
B
, and
e
,
n
e
,
m
are the electron charge, density and mass,
respectively. The wave with the upper (+) sign in Eq. (1) is called the ordinary wave and is
left-hand circularly polarized, whereas the wave with the lower (-) sign is called the
extraordinary wave and is right-hand circularly polarized (Hartmann & Leitinger, 1984).
The GPS signals are transmitted in right-hand circular polarization (Parkinson & Gilbert,
1983).
Equation (1) indicates that the phase refractive index is less than the unity resulting in a
phase velocity that is greater than the speed of light in vacuum (i.e., phase advance).
Therefore, the integration of
n
along a signal path gives a measure of the range / travel time
between a receiver and a satellite that is smaller than the geometric distance / travel time in
the vacuum.
To compute group delay measurements, the group refractive index
n
gr
should be
considered. The expression for
n
gr
can be determined by the relationship
n
gr
=
n
+
f
(
dn
/
df
).
2
2
2
⎡
2
⎤
f
f
f
cos
Θ
3
f
f
(
)
p
p
g
p
p
2
2
⎢
⎥
n
=+
1
∓
+
+
f
1
+
cos
Θ
(2)
gr
g
2
3
4
2
f
f
4
f
2
⎢
⎥
⎣
⎦
Equation (2) indicates that the group refractive index is greater than the unity resulting in a
group velocity that is less than the speed of light. Therefore, the integration of
n
gr
along a
signal path gives a measure of the range / travel time that is greater than the geometric
distance / travel time in the vacuum. Therefore, when GNSS signals propagate through the
ionosphere, the carrier-phase experiences an advance and the code experiences a group
delay. The carrier-phase pseudoranges are measured too short and the code pseudoranges
are measured too long compared to the geometric range between a satellite and a receiver.
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