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where
c
is the speed of light. In order to solve Eq.
36
, information of the magnetic
field
B
0
and the angle
θ
along the ray path have to be known. Since this is difficult
to accomplish, Brunner and Gu (
1991
) assumed t
hat
B
0
co
s
θ
does not vary greatly
along the ray path, so that onemay take the average
B
0
cos
θ
in front of the integration:
7527
c
2
f
3
ion
2
ph
Δρ
=−
B
0
cos
θ
N
e
ds
0
.
(37)
An alternative way was proposed by Bassiri and Hajj (
1993
) who assumed the
Earth's magnetic field as a co-centric tilted magnetic dipole and approximated the
ionospheric layer as a thin shell at the height of 400km. Thus, the magnetic field
vector
B
0
can be written as:
R
E
R
E
+
B
0
=
B
g
[
sin
θ
m
·
Y
m
−
2 cos
θ
m
·
Z
m
]
,
(38)
H
B
g
represents the magnitude of the magnetic field near the equator at surface height
(
B
g
10
−
5
T).
R
E
is the Earth's radius (
R
E
370 km).
H
denotes the
height of the ionospheric thin shell above the Earth's surface (
H
≈
3
.
12
×
≈
6
,
400 km).
Y
m
and
Z
m
are the
Y
and
Z
unit vectors in the geomagnetic coordinate system, and
=
θ
m
is the
angle between the ambient magnetic field vector and wave vector in the geomagnetic
coordinate system (see Sect.
4.3
). The scalar product of the magnitude field vector
B
0
and the signal propagation unit vector
k
is:
B
0
·
k
=
B
0
|
k
|
cos
θ
=
B
0
cos
θ.
(39)
Combining Eqs.
36
,
38
, and
39
, an expression similar to Eq.
37
can be derived
k
7527
c
2
f
3
ion
2
ph
Δρ
=−
B
0
·
N
e
ds
0
.
(40)
Equation
40
is sufficient to approximate the effect of the second order term to better
than 90% on the average (Fritsche et al.
2005
).
Third Order Delay
According to Eq.
31
and evaluating the constant
C
X
, the third order ionospheric phase
delay is expressed as
812
.
4
ion
3
ph
N
e
Δρ
=−
ds
0
.
(41)
f
4
Brunner and Gu (
1991
) applied the shape parameter
η
in such a way that the integral
in Eq.
41
can be approximated by