Geoscience Reference
In-Depth Information
4.1 Modeling TEC Using Physical and Empirical Models
4.1.1 Klobuchar Model
In the mid-80s, a simple algorithmwas developed for the GPS single-frequency users
to correct about 50% of the ionospheric range error. This correction method was
established because the GPS satellite message had space for only eight coefficients
to describe the worldwide behavior of the Earth's ionosphere. Furthermore, these
coefficients could not be updated more often than once per day, and generally not
even that often. Finally, simple equations had to be used to implement the algorithm
to avoid causing excessive computational stress on the GPS users. The algorithm
was developed by Klobuchar (
1986
) and led to the model that approximated the
entire ionospheric vertical refraction by modeling the vertical time delay for the
code pseudo-ranges.
The Klobuchar model does not directly compute the TEC. Instead, it models
time delay due to ionospheric effects. Equation
52
shows time delay in nanoseconds.
Multiplying this expression by the speed of light will result the vertical ionospheric
range delay. The obtained range delay, after applying the SLM function, can be
used to correct the ionospheric error in the measurements. Although the model is an
approximation, it is nevertheless of importance because it uses the ionospheric coef-
ficients broadcast within the fourth sub frame of the navigation message (Hofmann-
Wellenhof et al.
1993
). The time delay derived from the Klobuchar model follows
from
A
2
cos
2
π(
t
−
A
3
)
T
ion
ν
Δ
=
A
1
+
,
(52)
A
4
with
10
−
9
s
A
1
=
5
·
=
5ns
,
m
m
IP
2
m
IP
3
A
2
=
α
1
+
α
2
ϕ
IP
+
α
3
ϕ
+
α
4
ϕ
,
14
h
A
3
=
local time
,
m
m
IP
2
m
IP
3
A
4
=
β
1
+
β
2
ϕ
IP
+
β
3
ϕ
+
β
4
ϕ
.
4are
uploaded daily from the control segment to the satellites and broadcast to the users
through the broadcast ephemeris.
t
is the local time of the
Ionospheric Pierce Point
(
IPP
), and is derived from:
The values
A
1
and
A
3
are constant values, the coefficients
α
i
,β
i
,
i
=
1
, ...,
=
λ
IP
t
15
+
t
UT
,
(53)
where
λ
IP
is the longitude of
IPP
in degrees (positive to East) and
t
UT
is the obser-
vation epoch in Universal Time. Finally
m
ϕ
IP
in Eq.
52
is the geomagnetic latitude of
IPP and is calculated by Lilov
1972
:
