Geoscience Reference
In-Depth Information
x
N
0
(z)
N
0
(z)
+
(z)
⋅
x
z
Atmosphere
e
N
S
Earth
Fig. 11
The refractivity in the vicinity of the vertical profile above the site can be determined with
the linear horizontal gradients of refractivity
ξ
4.2.2 Azimuthal Asymmetry: Gradients
Mapping functions as described above allow for the modeling of path delays under
the assumption of azimuthal symmetry of the neutral atmosphere around the station.
Consequently, vertical refractivity profiles above the sites are sufficient to determine
the path delays at arbitrary elevation angles or the mapping functions, respectively,
because the refractivity is always taken from the vertical profile as it is the case for
the VMF1. However, due to certain climatic and weather phenomena path delays will
not be constant at varying azimuths. For example, at sites at northern latitudes the
path delay towards south will be systematically larger than towards north, because
the height of the troposphere above the equator is larger than above the poles.
In the following the derivation of linear horizontal gradients is shown following
Davis et al. (
1993
). The Taylor series up to degree one for the refractivity at a station
is (Fig.
11
)
N
(
x
,
z
)
=
N
0
(
z
)
+
ξ
(
z
)
·
x
,
(128)
x
=
0
.
∂
N
(
x
,
z
)
ξ
i
(
z
)
=
(129)
∂
x
i
N
0
(
is the refractivity above the site,
x
is the horizontal position vector (origin is
placed at the site), and
z
)
is the linear horizontal gradient vector of refractivity at
height
z
. The index
i
refers to the i-th component of
x
: 1 towards north and 2 towards
east. The path delay (hydrostatic or wet) at an arbitrary direction can be found by
integration of Eq. (
128
) along the path
s
. If expressed with elevation angle
e
and
azimuth angle
a
, we get
ξ
(
z
)
10
−
6
∞
0
10
−
6
∞
0
10
−
6
∞
0
Δ
L
(
a
,
e
)
=
N
(
s
)
d
s
=
N
0
(
z
)
d
s
+
ξ
(
z
)
·
x
d
s
,
(130)
10
−
6
∞
0
Δ
L
(
a
,
e
)
=
Δ
L
0
(
e
)
+
ξ
(
z
)
·
x
d
s
,
(131)
