Dave,

The "blurb" on tomography by "J. Steele et Al" is given below. We
realize that it is longer than you wanted. We submit it anyway for your
team to edit.

1.5.1  Description of Experimental Methods

The NITEOWL experiment is going to measure ozone concentrations using
stellar occultation. The descent of the rocket through the atmosphere
and its rotation allow various stars to be viewed. Because the methods
applied do not depend on tracking any particular star, or on the
position of the sun or moon, measurements may be made at any local time
during the night. The data from the ISG is acquired at a rate of one
frame per second. In each frame several stars will be visible, however
the S/N ratio required to do accurate inversion will only be achieved by
stars of magnitude 3.0 or brighter. Each star provides a set of lines of
sight through the atmosphere. The combination of all the lines of sight
through the region to be reconstructed provides the raw data on which
tomographic analysis will be performed.

The initial measurement of each star viewed when the line of sight
remains above the stratosphere will be used as a benchmark for all
subsequent measurements of that star. This allows self-calibration of
the ISG. The ratio of the initial intensity, which will be effectively
unattenuated, to the intensity measured viewing the same star along a
different line of sight through the ozone will permit calculation of the
ozone column density. 

As starlight passes through an absorbing species, the resulting
occultation is an exponential decay with the amount of ozone along the
light path determining the attenuation ratio. Working with the log of
the ratio of light intensities transforms the governing equation to a
linear one, permitting a tomographic solution process.

The tomographic reconstruction of the ozone distribution presents a
number of challenges. These arise from the slenderness and curvature of
the domain, the dominance of the vertical component of the ozone density
gradients, the tiny absorption characteristic in the Chapuis bands
necessitating large column lengths, and the requirement to reconstruct
geographic variations using a small range of source elevation angles.
These factors conspire to make the problem ill-posed.

Because each potential source star must be viewed at least once above
the stratosphere, the instrument's 80 km apogee will realistically limit
data gathering to stars which are no lower than 5 degrees below the
horizon. (Note: horizon = the set of all points 90 degrees from the
local zenith.) Assuming such a star is present and bright enough and
observed by the instrument, the lowest of its rays that clear the
aerosol contaminated region below 10 km altitude will provide data on
ozone no further away than 1200 km from the instrument and then only at
40 km altitude. Above the horizon, elevation angles greater than 6
degrees pass through too little atmosphere for attenuation to be clearly
different from noise.

If ozone varied only with altitude, the tomographical problem would be
governed by Abel's integral equation [Vest 1979] which is only
moderately ill-posed. Our simulations, however, show that if geographic
ozone variation is at all significant, a purely vertical inversion
yields poor results. To incorporate the geographical variation in ozone,
a technique other than Abel inversion must be used.

To provide estimates of the reconstruction error, we developed a
preliminary code to generate synthetic noisy tomographic data using the
vertical ozone density profile taken from the GOMOS model [Kyrola 1993]
with a gentle geographic variation superimposed. We then attempted to
reconstruct the ozone density distribution within the domain from the
noisy data. 

A classical tomographical approach was used; i.e., we tracked starlight
paths through the domain generating line integrals, discretized the
field spatially thus reducing the governing integral transform to a
system of linear equations, and added random noise to the data
corresponding to an RMS S/N ratio of 100:1 before applying solution
techniques. Simplifying assumptions included no earth rotation during
the experiment and no payload drift though this will clearly not be the
case. 

The domain reconstructed lies between 10 and 40 km altitude and between
a hypothetical vertical line of instrument trajectory and another
locally vertical line 1200 km away. These boundaries define an annular
sector. The domain is defined by sweeping this sector between two
planes, each of constant azimuth, about the instrument trajectory line.
These planes are close enough together that ozone does not vary
significantly between them, but far enough apart to include two stars
near the horizon at +1 and -3 degrees elevation angles. From an analysis
of the star horizons available at Churchill and current estimates of
geographic ozone variations, 5 degrees is a reasonable width. 

(A picture here would be good.  We will supply one.)

Two approaches to solving the linear systems yielded very encouraging
results; both used regularization [Craig and Brown, 1986; Dubovikova and
Dubovikov, 1987 ] to stabilize the systems. One approach modelled the
gentle geographic distribution using an assumed form with one degree of
freedom. The second approach used a full domain discretization.

Our simulations give us confidence that for a S/N ratio of 100:1, the
average vertical ozone density profile may be determined to within 15%
of true values with a resolution of 3 km. 

The ultimate accuracy in reconstructing the geographic distribution will
depend on many factors including the nature of the actual geographic
distribution (for which little is known a priori), the type of
regularization used, and the model upon which inversion is based.
Alternatives to regularization, such as the maximum probability method [
McDade and Llewellyn, 1993], are available as well. The determination of
the optimum model and form of inversion, taking expected payload drift
and star rising and setting into account, will be conducted during Phase
A.



References

Craig, I. J. D. and Brown, J. C., Inverse Problems in Astronomy: a Guide
to Inversion Strategies for Remotely Sensed Data , Adam Hilger, Bristol,
England, 1986.

Dubovikova, E. A. and Dubovikov, M. S., "Regularization, Experimental
Errors, and Accuracy Estimation in Tomography and Interferometry", J.
Optical Society of America A, Vol. 4, No. 11, 1987, pp.2033-2038.

McDade, I.C., and Llewellyn, E.J., "Satellite Airglow Limb Tomography:
Methods for Recovering Structured Emission Rates in the Mesospheric
Airglow Layer", Canadian Journal of Physics, Vol. 71, No. 11-12, 1993,
pp. 552-563.