Digital Signal Processing Reference
In-Depth Information
Namely:
2
3
2
3
2
3
v
1
.B
1
;L
1
/
VTEC
v
1
.B
2
;L
2
/
VTEC
:
v
1
.B
m
;L
m
/
VTEC
a
1
a
2
:
a
n
Q .B
1
;L
1
;B
i1
;L
i1
/
Q .B
1
;L
1
;B
in
;L
in
/
4
5
4
5
4
5
:
Q .B
m
;L
m
;B
i1
;L
i1
/
Q .B
m
;L
m
;B
in
;L
in
/
:
D
(4.21)
The error equation is as follows:
v
D
Qa
v
VTEC
(4.22)
Through the Eq. (
4.22
) we can obtain the vector
a
by a weighted least squares
adjustment to all GPS observations in each epoch, namely
a
D
Q
T
P
v
Q
1
Q
T
P
v
v
(4.23)
where
P
v
is the weight matrix of the VTEC. And that, we can obtain the VTEC of
random grid sites, namely: VTEC
h
D
Q
h
t
a
, where Q
t
h
D
Q
h1
Q
h2
Q
hi
.
4.3.1.4
Spherical Harmonics Functions
For global VTEC expression, spherical harmonics functions are widely used (Schaer
1999
).
n
max
X
X
n
P
nm
.sin ˇ/.a
nm
cos ms
C
b
nm
sin ms/
VTEC .'; /
D
(4.24)
n
D
0
m
D
0
ˇ is the latitude of the ionosphere pierce point,
s
D
0
is the sun-fixed
longitude. ,
0
are the longitude of the ionosphere pierce point and the apparent
solar time.
a
nm
,
b
nm
are the ionosphere model's coefficients.
P
nm
are normalized
Legendre polynomials. Normalization function is as follow.
s
.n
m/Š .2n
C
1/.2
ı
om
/
.n
C
m/Š
ƒ
D
(4.25)
4.3.1.5
Triangular Grid Method
Another VTEC description is used in JPL's global VTEC computation. They use
triangular grid to express the global VTEC distribution and variation. For each grid
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