Geoscience Reference
In-Depth Information
where:
ξ
4
=
f
L
1
/
f
L
2
≈−
1
−
0
.
647 factor (GPS) for relating the ionospheric refraction
on
L
4
to
L
1
,
B
4
=
λ
L
1
B
(
f
L
1
)
−
λ
L
2
B
(
f
L
2
)
ambiguity parameter with undefined wavelength,
thus defined in length units,
b
S
b
S
,
1
b
S
,
2
Δ
=
−
differential inter-frequency hardware delay of the
satellite
S
in time units,
Δ
b
R
=
b
R
,
1
−
b
R
,
2
differential inter-frequency hardware delay of the
receiver
R
in time units.
The ionospheric refraction
I
in Eqs.
87
and
88
can be related to the VTEC as a
function of the geomagnetic latitude and the sun-fixed longitude in the following
way:
I
=
ξ
E
STEC
(β,
s
)
=
ξ
E
F
(
z
)
VTEC
(β,
s
),
(89)
with:
F
(
z
)
mapping function evaluated at zenith distance
z
,
β
geomagnetic latitude,
s
sun-fixed longitude,
C
x
2
f
−
2
1
ξ
E
=
≈
0
.
162 m/TECU (GPS).
By substituting Eq.
89
in Eqs.
87
and
88
the ionospheric observable for code and
phase measurements reads
c
b
R
b
S
P
4
≈+
ξ
4
ξ
E
F
(
z
)
VTEC
(β,
s
)
+
Δ
−
Δ
,
(90)
and
L
4
≈−
ξ
4
ξ
E
F
(
z
)
VTEC
(β,
s
)
+
B
4
.
(91)
In Eqs.
90
and
91
, the equation sign '
=
' has been replaced by the approximate equa-
tion sign '
' because of including the simplified single layer assumption. Depending
on the study and whether we want to estimate VTEC on a local, regional or global
basis,
VTEC
≈
is represented with an appropriate base-function. As an example
Taylor series expansion can be used for local representation of TEC; B-splines are
very suitable for studying TEC in regional applications, and for global representa-
tion of TEC, spherical harmonics expansion is most commonly used. Here we briefly
discuss the spherical harmonics expansion approach:
(β,
s
)
Global TEC Representation Using Spherical Harmonics Expansion
In order to develop a global ionosphere model, the vertical TEC has to be represented
as a function of longitude, latitude and time, or according to the definition of the
adopted coordinate system given in Sect.
4.3
—as a function of the geomagnetic
latitude
β
and sun-fixed longitude
s
(Schaer
1999
):
