Global Positioning System Reference
In-Depth Information
Considering ionospheric refraction the geometric distance (Euclidean line) or true range
ρ
between a transmitting satellite
S
and a receiver
R
can be written in units of length as
R
∫
len
I
(3)
ρ=+ − −
L
(1
n ds
)
d
S
R
=
∫
where the optical distance
L s
is the line integral of the refractive index between the
S
R
satellite and the receiver along the ray path,
(
)
∫
1
−
nds
is the ionospheric group delay and
S
len
I
d
is the excess path length of the signal in addition to the geometric path length caused
by the ray path bending and defined by
R
len
I
d
=
∫
ds
ρ
−
(4)
S
R
where
∫
ds
is the curved path length in the vacuum. The travel time of the signal can be
S
computed dividing the expression of
ρ
(Eq. 3) simply by the speed of light.
2.2 Group delay and phase advance
Assuming a right hand circularly polarized signal, the ionospheric group delay
d
Igr
and
carrier phase advance
d
I
can be written in units of length as (using Eqs. (1) and (2))
R
pq
u
(
)
() () ()
1
2
3
d
=++= − =++
d
d
d
∫
n ds
1
(5)
Igr
Igr
Igr
Igr
gr
2
3
4
f
f
f
S
R
p
q
u
() () ()
1
2
3
(
)
∫
dd d d
=++=− =+ +
1
n s
f
(6)
I
I
I
I
2
3
4
2
f
3
f
S
(
)
p
=
K
∫
n ds
= ⋅
K TEC
=
K TEC
+
Δ
TEC
(7)
e
LoS
bend
12
q
=
2.2566
×
10
∫
nB
cos
Θ
⋅
ds
(8)
e
2
22
2
2
u
=
2437
∫
n ds
+
4.74
×
10
∫
n B
(1
+
cos
Θ
)
ds
(9)
e
e
where
K
=
e
2
/(8π
2
ε
0
m
) = 40.3 m
3
s
-2
, the integration of
n
e
along signal paths
nd
∫
is called
the total electron content TEC and measured in TEC units (1 TECU = 10
16
electrons/m
2
). The
terms
()
()
()
()
()
()
3
I
d
in Eq. (5) / (6) are the first, second and third
order ionospheric group delays / phase advances, respectively. Due to the dispersive nature
of the ionosphere, satellite signals transmitted on different frequencies travel along different
1
Igr
1
2
Igr
2
3
Igr
d
/
d
,
d
/
d
and
d
/
I
I
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