Geoscience Reference
In-Depth Information
Δ
L
and
N
are vectors containing the wet delays and the refractivities, respectively,
and
D
is a matrix containing the
D
ij
values. By inverting the system, the voxel
refractivities are estimated.
There are however a few problems with this method. One is how to estimate the
slant wet delays along the GNSS signal rays. One way is to estimate the zenith wet
delay and gradients in a normal GNSS analysis, then use these to calculate the slant
wet delays. This is for example done by Champollion et al. (
2005
). However, this
method will limit the accuracy since it assumes that all horizontal variations in the
refractivity above a station are linear, somethingwhich is not always realistic. In order
to improve the slant wet delays, it is often assumed that the post-fit residuals of the
GNSS analysis will contain the unmodeled parts of the slant wet delays, and thus
adding these to the calculated slant wet delays will give the true delays (Alber et al.
2000
; Troller et al.
2006
). This is not true, the residuals will contain also other errors
of the GNSS measurements (e.g. multi-path). Another approach is to model the slant
delays by using Eq. (
199
) in the GNSS data analysis instead of zenith delays and
gradients. First results using this approach are presented by Nilsson and Gradinarsky
(
2006
) and Nilsson et al. (
2007
).
Another problem is the normally weak geometry since tomography ideally
requires that there are rays crossing the investigated volume in all possible directions.
In GNSS tomography, however, all rays are going between the top of the troposphere
to the stations on the surface of the Earth, while there are no rays entering and/or
leaving the voxel grid at the sides. This makes the sensitivity to the vertical refractiv-
ity profile very low and as a result the equation system (
200
) will be ill-conditioned.
This problem can be solved by either constraining the refractivity field to some a pri-
ori field obtained either by models or external measurements like radiosondes. The
problem is not as big if the GNSS stations are placed at very different altitudes (e.g.
if there are differences of several kilometers between highest and lowest stations)
(Nilsson and Gradinarsky
2006
).
Furthermore, since the satellite geometry will change during the day, some vox-
els may at times have no or only a few rays passing through it. Thus, in order to
avoid singularity problems, constraints need to be applied. Simple constraints are
for example inter-voxel constraints which constrain the refractivity of a voxel to the
mean refractivity of the neighboring voxels (Flores et al.
2000
). A more advanced
approach is to use a Kalman filter with a covariance matrix for the voxel refractivity
calculated from turbulence theory using Eq. (
186
) (Gradinarsky and Jarlemark
2004
;
Nilsson and Gradinarsky
2006
).
Figure
17
shows the results of two simulations demonstrating the strengths and
weaknesses of GNSS tomography. In the upper plot the case where the refractivity is
20 mm/km in the layer between 3 and 4 km altitude and zero elsewhere is simulated.
As seen the tomographic reconstruction is not working well. This is because of the
weak geometry in the vertical direction, resulting in a very lowsensitivity to the height
of the layer with non-zero refractivity. Thus the refractivity is spread out over all
layers in the tomographic reconstruction. Most refractivity is put in the lower layers
simply because the tomographic software is set up to allow for a higher variability in
the refractivity at lower altitudes than at higher altitudes. The estimated refractivity
