Geoscience Reference
In-Depth Information
n
max
n
0
P
nm
(
VTEC
(β,
s
)
=
sin
β)(
a
nm
cos
(
ms
)
+
b
nm
sin
(
ms
)) ,
(92)
n
=
0
m
=
where:
VTEC
(β,
s
)
vertical TEC in TECU,
P
nm
=
N
nm
P
nm
normalized Legendre function from degree
n
and order
m
,
N
nm
normalizing function,
P
nm
classical Legendre function,
a
nm
and
b
nm
unknown coefficients of the spherical harmonics expansion,
with the normalizing function written as:
(
n
−
m
)
!
(
2
n
+
1
)(
2
−
δ
0
m
)
N
nm
=
,
(93)
(
n
+
m
)
!
where
δ
0
m
denotes the Kronecker delta. The number of unknown coefficients of
spherical harmonics expansion Eq.
92
is given by:
2
=
(
n
max
+
)
,
u
1
(94)
and the spatial resolution of a truncated spherical harmonics expansion is given by:
π
n
max
,
2
π
m
max
,
2
Δβ
=
Δ
s
=
(95)
where
Δβ
is the resolution in latitude, and
Δ
s
is the resolution in sun-fixed longitude and local time, respectively.
It is shown that the mean VTEC (
VTEC
) of the global TEC distribution expressed
by Eq.
92
is generally represented by the zero-degree spherical harmonics coefficient
C
00
(Schaer
1999
):
+
2
2
π
1
4
N
00
C
00
=
C
00
.
VTEC
=
E
v
(β,
s
)
cos
β
d
β
ds
=
(96)
π
−
2
0
Parametrization and Estimation of VTEC
To estimate a global VTEC model, GNSS observations from a set of globally distrib-
uted GNSS stations are collected. The computation is carried out on a daily basis,
using observations with sampling rate of 30 s and elevation cut-off angle 10
◦
.For
all of the observations the ionospheric observable is calculated using Eqs.
90
or
91
.
This observable forms the observation equation. The observation equations are then
