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Introduction
One of the main purposes of ALIS is to determine the full 3D
spatial structure of the aurora. This cannot be done with
measurements from a single station and this is the main reason for
developing ALIS as a multistation system, with as many stations as
possible to optimise the coverage of northern Scandinavia. This system
gives simultaneous images of the aurora seen from different perspectives.
This chapter presents a short introduction to threedimensional
tomographic methods. Further it investigates the
tomographic resolution of a groundbased system and the error
sensitivity of such a system, and describes a general stopping
criterion that gives a sufficient condition for when to interrupt
iterative solutions for tomography.
Figure 3.1:
Cartoon of the
groundbased auroral tomographic problem. Here the images are
displayed in polar representation from the respective stations.

Tomography is the method to determine the internal structure of an
object starting from a set of images that in some way carry
information of the integral intensity along the lineofsight
through the object. For xray tomography a known
source radiates light from one side of the studied object and the
image intensity acquired on the other side depends on the total
absorption through the object along the ray path. For emission
tomography, like auroral tomography, the images are made up of the
total emission along the linesofsight. In Figure 3.1 a typical setup
of a groundbased auroral tomographic problem is presented. Here there
are three observation sites which record onedimensional (1D) images,
and an unknown source to be determined.
It is not evident that such inverse problems have solutions but for
twodimensional (2D) tomography, Radon (1917) showed that it was
possible to determine exactly the source function . Here
is given in a vertical plane, the measured intensities
, corresponding to
our images, are given by line integrals

(3.1) 
where and determine the position of the detector and
its lineofsight , as defined in Figure 3.2
and is a length element along .
Figure:
Definition of the parameters in equation (3.1) for
a 2D tomographic inversion. Here defines the
direction perpendicular to the imaging, defines the loci of
the image pixels, is the lineofsight, and
is the unknown source function.

It is possible to obtain an exact and unique solution of the source
function if and only if
is known for all
rays that cut the region where . That is for an
angle we have to move the detector along so that
measurements are made for all rays that go through the object. The
solution can be written as ( Gordon and Herman, 1974):

(3.2) 
For the threedimensional tomographic problem exact solutions can be
obtained if there is an observation point from which our image
projections are made on every plane
that cuts the region where the function
( Kudo and Saito, 1990; Smith, 1990). There exist a number of analytical
algorithms that determine from a set of image projections which
closely satisfy the above condition ( Jacobsson, 1996; Grangeat, 1990).
However, in most cases these tomographic problems are illposed,
meaning that they are sensitive to noise in the measurements or have
either an infinite number of exact solutions or no exact
solutions. With some regularization schemes, i.e. physical or
mathematical constraints, the problem can, in a restricted sense, be
transformed into a wellposed problem ( Ghosh Roy, 1991). Auroral and
airglow tomography from groundbased multistation measurements is
illposed because only a few projections/images of the
aurora are available and that the image data are all taken from below
the object and thus are close to being linearly dependent.
The tomographic problem of ALIS is threedimensional, and all performance,
resolution and error analysis on the actual geometry suffers badly from
the curse of dimensionality. That is, the total size of the
tomographic problem is proportional to the total number of pixels in
the images times the total number of volume elements (voxel = volume
element, small cube, c.f. pixel = picture element) in the retrieved
volume distribution. For a twodimensional tomographic problem with
voxels per side, the number of unknowns is projected onto
images with pixels per onedimensional image; the size of
the problem is
. For a threedimensional tomographic system
the total number of pixels is
and the number of voxels
rises to , giving a total size of the problem of
. For systems studying the ionosphere the
typical number of stations appears to be approximately 5 and typical
number of pixels is 1000 per image dimension. For the retrieved
volume distribution the number of voxels per side is of the
order of 100. When we go from a twodimensional to a threedimensional
tomographic problem the total size of the problem
increases by a factor
.
To simplify the analysis
in this and the following two chapters, which describe the geometrical
calibrations and the forward model, we will study a twodimensional problem where emission distribution is to be
determined in a vertical twodimensional slice from onedimensional
images. This tomographic model system makes onedimensional
image measurements with the imaging characteristics of the ALIS
cameras from four groundbased stations separated by 50 km from each
other. The principles of tomographic inversion and all the
characteristics of groundbased auroral tomography remain in this
2D problem but the size is small enough to allow use of the most
rigorous analysis of its resolution and retrieval characteristics.
Since ALIS currently only has six groundbased stations, and thus is
very far from fulfilling the necessary requirements for obtaining
exact reconstructions, we turn to approximate methods in order to
obtain the best solution possible.
In order to be able to solve the tomographic problem it is necessary to
have good knowledge about the forward problem or forward model. That
is, we must know what fraction of the photons emitted in a volume
element will be detected and where in the image the intensity will
appear. In this model case we have to know how the auroral distribution
in a slice is projected down onto the images, as depicted in
Figure 3.3.
Figure 3.3:
The imaging process or
the forward model from the 2D model
aurora to two 1D imagers at 50 km north and 50 km south.

The mathematical notation for this is:

(3.3) 
where is the image from stations , is
the twodimensional source function and is the forward
model for station . Here we will not go into the details of the
forward model; it is sufficient to note that it is important to know
the pixel linesofsight and the sensitivities of the cameras. Further
it is necessary to note that the cameras used by ALIS have a linear
response to intensity and that the emissions currently
measured by ALIS are all optically thin which implies that the forward
model in equation (3.3) can be simplified to a matrix
multiplication:

(3.4) 
where is the image from station and
is the transfer matrix from the voxel space to the image.
If we solve the tomographic problem analytically, we will encouner
the problem that the forward model is not a
deterministic process, but rather a random process

(3.5) 
where a component of is a sample from one
random variable with a
probability distribution function that depends on several processes
from the emission of photons in voxel to the detection in
the image. Neglecting the randomness in all processes from the camera
to the voxel, the random direction of photon emission remains. Since
the distribution of emission can safely be assumed to be isotropic
and the photon emissions are independent from each other, the
probability distribution for the number of photons emitted towards the
camera is a binomially distributed random variable:

(3.6) 
where is the probability of getting photons,
is the total number of photons emitted in voxel
and

(3.7) 
is the probability that a photon is emitted towards the front lens of
camera ,
is the area of the front lens as
seen from voxel . With respect to work on aurora for which the
number of photons emitted is large and the probability is small, the
binomial distribution in equation (3.6) is well
approximated by a normal distribution:

(3.8) 
Thus we see that the imaging process is a stochastic one for which the
probability of getting a set of measured images from a
source distribution is:

(3.9) 
where
is the conditional probability of
getting the measured images
given the source
distribution . For auroral imaging with ALIS the image
intensity has a probability distribution that is approximately:

(3.10) 
The solution to the inverse problem is then to find the
source distribution
that is most probable given
the measurements
and considering
equation (3.9) and Bayes' rule the resulting problem is

(3.11) 
This is the maximum likelihood solution of the inverse problem. For
this case in which the random processes have normal distributions the
maximum likelihood solution is the wellknown least square solution.
For the threedimensional inverse problem the system transfer matrix
is too large to invert directly, so we use
iterative methods to solve the set of linear equations.
The simple and robust iterative methods we have chosen are the
algebraic reconstruction technique (ART) and the simultaneous
iterative reconstruction technique (SIRT) ( Gordon and Herman, 1974). The
working principle of ART is then to take an initial guess
and project that guess down to an image by

(3.12) 
Then the current solution is updated voxel by voxel according to:

(3.13) 
This is a modified multiplicative ART update where the update is a
weighted average of the ratios of the measured image and
for the pixels whose linesofsight intersect the voxel at
.
Figure 3.4:
a) Initial guess for the ART reconstruction and the
projection to the southern station in red; blue is the measured
image. b) In the southern station the ratio (measured
image)/(projection of the start guess) is calculated. This ratio
is projected up/out into the start guess. The updated guess is
then projected down to the next imaging station (red). c) From
that station the ratio (measured image)/(projection of the current
guess) is projected into the current guess.

The initial progress of this iterative reconstruction scheme is
shown in Figure 3.4. First a constant initial guess is
projected down onto one image. Then the ratio between the measured
image and the projection of the start guess is calculated. This ratio
is then projected out into the solution. In this way the voxel values
along a pixel lineofsight is increased if the measure image is
larger than the projection from the start guess and viceverse. This
iteration is then proceeded over all stations in random order until a
stopping criteria is encountered.
ART is a reconstruction scheme that converges fast but is
sensitive to noise in the images ( Aso et al., 1993). A modified version
is SIRT for which the current guess/solution is projected down
to all stations at once and the voxel intensity in the reconstruction
is updated with the average ratio of pixels from all stations.
It is possible to use a priori knowledge about the aurora, such as
fieldalignedness of the auroral structures
( Semeter, 1997; Aso et al., 1990), in the reconstruction process in
order to bias the reconstruction and to avoid noise amplification
thereby stabilising the solution. Several more general constraints can
also be used to reduce the effect of noise in the images, e.g. spatial
low pass filtering in 3D as outlined in paper I, horizontal median
filtering, horizontal smoothing of field aligned intensity variation
Aso et al. (1993).
Several more advanced iterative schemes exist and the
technically interested reader is directed to Veklerov and Llacer (1989b).
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copyright Björn Gustavsson 20001024
