Geoscience Reference
In-Depth Information
Δ
where
L
0
is the delay without gradients. In Eq. (
131
) the difference between the
paths with and without gradients has been neglected. Davis et al. (
1993
) state that
this difference is as large as 1 mm for
N
20
◦
.
If the concept of an azimuth-dependent mapping function is used, Eq. (
131
) can be
written as
1km
−
1
, and
e
=
300,
∂
N
/∂
x
i
=
=
L
z
Δ
L
(
a
,
e
)
=
Δ
·
mf
(
a
,
e
),
(132)
with
10
−
6
∞
0
mf
(
a
,
e
)
=
mf
0
(
e
)
+
δ
mf
(
a
,
e
)
=
mf
0
(
e
)
+
ζ
(
z
)
·
x
d
s
,
(133)
where
ξ
(
z
)
ζ
(
z
)
=
L
z
,
(134)
Δ
L
z
are the mapping function and the path delay in zenith direction
for the symmetric case. Thus, the gradients cause a change in the mapping function
which can be described by an additional term
and
mf
0
and
Δ
δ
mf
. With
e
)(
(
,
)
≈
·
(
(
)
ˆ
+
(
)
ˆ
),
x
a
e
z
cot
cos
a
n
sin
a
e
(135)
d
s
≈
d
z
·
mf
0
(
e
),
(136)
and normalized gradients of refractivity
ζ
(
z
)
=
ζ
n
· ˆ
n
+
ζ
e
· ˆ
e
,
(137)
e
refer to the unit vectors in north and east direction and
e
is the
refracted elevation angle (which only differs from the geometric elevation angle at
low elevations), we get
and when
n
and
ˆ
ˆ
cos
(
a
)
z
·
ζ
e
(
z
)
·
d
z
∞
∞
δ
mf
(
a
,
e
)
≈
10
−
6
mf
0
(
e
)
cot
(
e
)
z
·
ζ
n
(
z
)
·
d
z
+
sin
(
a
)
,
(138)
0
0
and
e
)(
δ
mf
(
a
,
e
)
=
mf
0
(
e
)
cot
(
Z
n
cos
(
a
)
+
Z
e
sin
(
a
)),
(139)
when
10
−
6
∞
0
Z
=
z
·
ζ
(
z
)
d
z
.
(140)
Equation (
138
) shows that the elevation-dependence of the azimuth-dependent map-
ping function
δ
mf
(
a
,
e
)
consists of two parts: the dependence on
mf
0
and on the
e
)
factor cot
. As already mentioned the mapping functions are dependent on the
geometric elevation angle
e
, whereas the cotangent depends on the refracted eleva-
(
