Geoscience Reference
In-Depth Information
Emardson et al. (
1999
) and Nothnagel et al. (
2007
). The results indicate that the
accuracy of the VLBI estimates can be improved if WVRs are used to calibrate the
tropospheric delay compared to the normal method of estimating the tropospheric
delay as function in the data analysis (see Sect.
4.2
), although the results are incon-
clusive. It should be noted that the WVR calibration was only applied at a few sites
since most VLBI stations are not equipped with radiometers. One problem with
radiometers is that they cannot be used during rain for several reasons (liquid water
on radiometer antenna, saturation problems, water droplets may not be “small”, …).
Furthermore, most radiometer antennas have relatively large beam-widths (several
degrees), and thus the observations are limited to high elevation angles (
15-20
◦
)
in order to avoid picking up radiation from the ground. Hence, either VLBI observa-
tions made at low elevation angles have to be excluded, or the WVR measurements
need to be extrapolated to low elevation angles, which is a process that can introduce
errors.
>
5 Atmospheric Turbulence
The normal modeling of atmospheric delays in space geodesy, i.e. using mapping
functions and horizontal gradients assumes that the spatial variations in the refrac-
tivity are linear, and that the temporal variations can be described by e.g. piece-wise
linear functions. For the large-scale variations this is an adequate approximation,
however at small scales there are non-linear variations caused by atmospheric turbu-
lence. Although it is normally impossible to correct for these random fluctuations, it
can be important to model them in order to minimize their effect on the results.
Atmospheric turbulence occurs when energy from e.g. wind shears and temper-
ature gradients creates turbulent eddies. These eddies then break down into smaller
eddies until at very small scales the energy of the eddies are dissipated into heat.
Inside each eddy the air is mixed, and thus large-scale variations in any atmospheric
quantity, e.g. refractivity, will be mixed to create random small-scale variations.
A turbulent eddy with a size
R
will have a characteristic wind velocity
v
.Kol-
mogorov (
1941a
,
b
) assumed that the rate of which kinetic energy (per unit mass)
of an eddy is transferred to smaller eddies is only dependent on
R
and
v
. By dimen-
sional analysis it is clear that this rate must be proportional to
v
3
R
. For stationary
turbulence the kinetic energy for the eddies of a specific size will be constant, i.e.
the kinetic energy received from larger eddies must be equal to the energy lost to
smaller scale eddies (assuming no dissipation into heat at larger scale). At small
scales the kinetic energy is dissipated into heat with a dissipation rate
/
ε
, which thus
3
R
1
/
3
,or
equivalently that the structure function for the velocity fluctuations between
r
and
r
1
/
must be equal to the kinetic energy rate of all larger eddies. Thus
v
∝
ε
+
R
is given by
[
v
]
2
C
v
ε
2
/
3
2
/
3
D
v
(
R
)
=
(
r
)
−
v
(
r
+
R
)
=
R
,
(182)
