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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
 
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