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for larger droplets the expression becomes more complicated (Solheim et al.
1999
).
However, normally the liquid water contribution to the refractivity (
k
4
(ν)ρ
lw
)is
neglected since it is small, especially outside of clouds. The variables
Z
d
and
Z
w
are compressibility factors for dry air and water vapor, respectively. These describe
the deviation of the atmospheric constituents from an ideal gas. The compressibility
factor for the
i
th constituent of air is given by
pM
i
ρ
i
RT
,
Z
i
=
(12)
where
M
i
is the molar mass and
R
is the universal gas constant. For an ideal gas we
have
Z
1. Owens (
1967
) obtained expressions for
Z
−
1
d
and
Z
−
w
by a least squares
fitting to thermodynamic data. These expressions are (for
p
d
and
p
w
in hPa and
T
in
K)
=
p
d
57
10
−
8
1
0
.
52
T
10
−
4
T
−
273
.
15
Z
−
1
d
=
1
+
.
97
·
+
−
9
.
4611
·
,
(13)
T
2
1650
p
w
Z
−
1
=
+
−
.
(
−
.
)
1
T
3
[1
0
01317
T
273
15
w
3
10
−
4
2
10
−
6
+
1
.
75
·
(
T
−
273
.
15
)
+
1
.
44
·
(
T
−
273
.
15
)
.
(14)
2.1 Microwaves
Figure
1
shows the total refractivity for frequencies between 0 and 100GHz for the
case when the total pressure is 1013hPa, the temperature is 300K, and the relative
humidity is 100 % (and for three different values for the concentration of liquid
water). The refractivitywas calculated using theMillimeter-wave PropagationModel
(MPM) (Liebe
1985
,
1989
; Liebe et al.
1993
). As can be seen, the variations in the
refractivity as function of frequency are relatively small. The biggest variations are
in the range 50-70GHz, a region where several strong absorption lines exist for
oxygen. Below 40GHz the refractivity is more or less constant. There are small
variations around the 22.235GHz water vapor absorption line, however these can
typically be neglected. Since all space geodetic techniques that use microwaves
operate at frequencies well below 40GHz, we can consider the refractivity to be
frequency independent for microwaves. Thus the phase (
c
p
=
c
0
/
n
) and group
velocities (
c
g
=
) in the troposphere will be equal.
In Fig.
1
three different cases are shown corresponding to different concentrations
of liquid water: 0 g/m
3
, 0.05 g/m
3
(e.g. fog), and 1 g/m
3
(e.g. inside a cloud). The
impact of liquid water on the refractivity is typically neglected since it is relatively
small, although in order to achieve highest accuracy in the presence of dense clouds
the effect should be considered. The difference between the case with 1 g/m
3
liquid
water and the case with no liquid water is about 1.44 mm/km for the frequencies
below 10 GHz, and then it decreases slightly with frequency to about 1.35 mm/km
c
0
/(
n
+
f
∂
n
/∂
f
)
