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Manuscript Title: ABEL: stable, high accuracy program for the inversion of Abel's integral equation.
Authors: I. Beniaminy, M. Deutsch
Program title: ABEL
Catalogue identifier: AAOK_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 27(1982)415
Programming language: Fortran.
Computer: IBM 370/168.
Operating system: OS/MVS-3.8.
RAM: 45K words
Word size: 32
Keywords: General purpose, Fit, Plasma physics, Data interpretation, Abel equation, Abel inversion, Plasma diagnostics.
Classification: 4.9, 19.4.

Nature of problem:
Abel's integral equation occurs in many fields of research. In particular, in flame and plasma diagnostics it relates the emission coefficient distribution function of optically thin cylindrically symmetric extended radiation source to the line-of-sight radiance measured in the laboratory. Thus, in order to obtain the physical information, which is the emission coefficient distribution function, one has to Abel-invert the measured radiance data.

Solution method:
Although two explicit analytic inverses are known for Abel's integral equation, their straightforward application greatly amplifies the experimental noise inherent in the radiance data. This is due to the fact that these formulae employ the first derivative of the radiance function. A third, analytic inverse was recently obtained by the present authors which does not require differentiation of the radiance data and thus avoids unnecessary error amplification. Based on these three analytic inverses, three formulae were developed for computing the inverse numerically. These were obtained by representing the radiance function by a piece-wise cubic spline function, least squares fitted to be measured data. The inverse of Abel's integral equation for a spline function is calculated analytically, so that once the parameters of the spline are determined by the fitting procedure, the inverse can readily be calculated using either one of the three formulae. The excellent smoothing properties of the spline function insure that both error amplification and over- smoothing is avoided while the piece-wise nature of the spline function keeps the termination errors small. A comparative study of our method with some of the best methods currently in use indicates that for highly error-free, very noisy or sparse radiance data, our method outperforms those methods in all cases studied. The difference was, in some cases, as much as two orders of magnitude in our favour.

Only 250 radiance values can be inverted by the present version in a single run.