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tion angle
e
, because close to the site this angle determines the refraction. It holds
that
e
=
e
+
δ
e
(
e
),
(141)
where
δ
e
can be described with (Davis et al.
1993
)
10
−
6
N
S
cot
δ
e
≈
(
e
).
(142)
5
◦
we get
2
◦
.
N
S
is the refractivity at the Earth surface, and for
N
S
=
300,
e
=
δ
e
≈
0
.
e
)
Since
can be expanded into a series, and we get for the deviation
from the symmetric mapping function
δ
e
is small, cot
(
10
−
6
N
S
csc
2
δ
mf
(
a
,
e
)
=
mf
0
(
e
)
cot
(
e
)(
1
−
(
e
))(
Z
n
cos
(
a
)
+
Z
e
sin
(
a
)).
(143)
With the delay gradients (or just gradients)
G
L
z
G
=
Z
·
Δ
,
(144)
we get for the path delay
10
−
6
N
S
csc
2
)).
(145)
The equations above can be used to determine gradients by integrating over the
horizontal gradients of refractivity along the site vertical (see e.g. MacMillan and
Ma (
1997
), Böhm and Schuh (
2007
))
Δ
L
(
a
,
e
)
=
Δ
L
0
(
e
)
+
mf
0
(
e
)
cot
(
e
)(
1
−
(
e
))(
G
n
cos
(
a
)
+
G
e
sin
(
a
10
−
6
∞
0
G
a
=
ξ
a
z
d
z
,
(146)
where
a
denotes the azimuth direction (e.g.
e
or
n
). Figure
12
shows refractivity
gradients at the site vertical for the station Fortaleza.
Fig. 12
Weighted (with
height) refractivity gradients
(
dN
30
hydrostatic
wet
z
) towards east at
Fortaleza (Brazil) on 21
November 2011 at 0:00 UT.
The
black line
shows the
hydrostatic, the
red line
the
wet gradients
(
z
)
·
25
20
15
10
5
0
−0.05
0
0.05
0.1
ppm