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Resolution
This section analyses the resolution of the tomographic model system
outlined in the previous section both by singular value
decomposition and model tests.
In paper I, I tried to apply the standard method of calculating the 3D
resolution for ALIS by singular value decomposition (SVD). Due to the
curse of dimensionality the number of voxels had to be reduced from
to and the number of pixels per image had to be
reduced from to thus restricting the possible
conclusions about the threedimensional resolution.
Now with the twodimensional tomographic model system with imaging
stations at 50 km north, 0, 50 and 100 km south, it is possible to
apply the SVD analysis with a vertical slice with 128 voxels per side
and onedimensional images with 256 pixels. With the
cameras directed towards the centre of the voxel slice, the
transfer matrices from the stations have been
calculated. Concatenating the station transfer matrices gives a full
system transfer matrix:

(3.14) 
With SVD analysis the resolution in voxel space can
be calculated. Given the transfer matrix
from the voxel space down to all images
with a singular value decomposition
, the
maximum spatial resolution is given by the resolution matrix
( Menke, 1989).
In order to suppress noise in the reconstruction, only the singular values
larger than 1 in the singular value decomposition of
above are kept. To illustrate the result of
the SVD analysis, we calculate the point spread of three input points
at altitudes of 120, 152 and 184 km central in the slice. By doing this
we get the variation of the point spread function with altitude.
Figure 3.5:
Retrieval of point sources at altitudes of 120, 152 and 184 km.

As can be seen in Figure 3.5, the point is accurately
retrieved in position but there are starlike traces in the directions
away from and towards the stations. To quantify the results of the
point spread calculations we take vertical cuts through the centres of
the point spread functions. This gives the vertical resolution and its
variation with altitude.
Figure 3.6:
Altitude variation of the
point spread function of a groundbased imaging system.

As can be seen, the full width at half
maximum is 4 km at 120 km and growing to 10 km at 210 km indicating good
altitude resolution at all
altitudes. The maximum of the point spread function is, however, only
10 % of the input function and the 10 % width in the zenith
direction is smeared out at all altitudes.
This implies that the individual peaks in the emission distribution are
well retrieved, while internal structures at lower intensity levels
will be smeared out. To show whether it is possibile to retrieve internal
structures in the aurora with a groundbased system it is enlightening to
consider a simple volume distribution and set one or more voxels to zero
and then calculate the ``hole retrieval'' in the same way as the point
spread was calculated above. In the vertical retrieval of this test, where
the simple model is the first column of that corresponds to the
eigenvector with the largest eigenvalue, holes with vertical sizes of 2, 4,
8 and 16 km are used; the result is presented in
Figure 3.7. As can be seen, the hole is not retrieved
even for the largest hole size.
Figure 3.7:
Thin curves are input profiles with a hole centred at
120 km altitude. From left the size of the holes are 2, 4, 8 and
16 km respectively. The thick curves show the retrieved altitude
variations of the input distributions.

Next: Error sensitivity
Up: Tomography
Previous: Introduction
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copyright Björn Gustavsson 20001024
