Geoscience Reference
In-Depth Information
The imaginary part of the refractivity, N (ν)
, causes absorption and is related to
α
the absorption coefficient
N (ν)
c
10 6 4
πν
α(ν) =
.
(8)
The power W of a signal received after propagating along the path S through the
atmosphere will be lower than it would have been ( W 0 ) in vacuum (i.e. no absorption)
W 0 e S α( s ,ν) d s
W 0 e τ ( ,ν) ,
W
=
=
(9)
where
is called the opacity.
Since the observables of space geodetic techniques (e.g. GNSS, VLBI, and SLR)
typically aremeasurements of the travel time of the signals, the absorption is typically
not important since it does not affect the propagation delay. Of course, absorption
will affect the delay measurements by increasing the noise; higher attenuation will
cause the signal-to-noise-ratio to be lower, and thus the accuracy of the measured
delay will be worse (in the worst case the signal cannot be detected). However, there
is typically no need for modeling this effect in the space geodetic data analysis.
Thus, in the following we will concentrate on the real part of the refractive index
and the effects caused by it. We will come back to the absorption in Sect. 4.4 , where
measurements of the absorption by microwave radiometry are used to estimate the
atmospheric path delay.
The (real part of) refractivity can be expressed as a function of the densities of
the different atmospheric gases and the temperature T (Debye 1929 )
τ ( , ν)
A i (ν)ρ i +
B i (ν) ρ i
T
N
=
,
(10)
i
ρ i is the density of the i th gas, and A i and B i are constants. The B i ρ T term is
caused by the permanent dipole moment of the molecules. Since water vapor is the
only major atmospheric gas having a permanent dipole moment, we can ignore this
term for all other gases. The relative concentrations of the dry atmospheric gases are
approximately constant (except carbon dioxide, see Sect. 2.1 ). Thus we can assume
that
where
ρ d is the density of dry air. This makes
it possible to express the refractivity as a function of pressure, temperature, and
humidity (Essen and Froome 1951 )
ρ i
=
x i ρ d , where x i is constant and
B w (ν) ρ w
N
=
A i (ν)
x i ρ d +
A w (ν)ρ w +
T +
A lw (ν)ρ lw
i
p d
T
p w
T
p w
Z 1
d
Z 1
T 2 Z 1
=
k 1 (ν)
+
k 2 (ν)
+
k 3 (ν)
+
k 4 (ν)ρ lw ,
(11)
w
w
where
ρ lw is the density of liquid water. It is here assumed here that the liquid water
droplets are small compared to the wavelength (
<
1 mm for microwave techniques),
 
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