New inverses of the attenuated Abel integral equation

Abel's integral equation relates the line-of-sight radiance to the emission coefficient distribution function of an extended, cylindrically symmetric transparent radiation source. In the presence of absorption this equation is modified. The author presents two new inversion formulae for this attenuated Abel equation for the case of constant absorption throughout the source. As one of them is completely derivative-free and the other requires differentiation of a weighted integral over the radiance rather than a differentiation of the radiance function itself, they are particularly well suited for use with measured radiance data, where differentiation invariably results in a very large amplification of the random experimental error inherent in the data. An illustrative example is also given.