Various methods for computing the inverse of Abel's integral equation, the underlying problem in the retrieval of temperature and concentration profiles from absorption measurements for axisymmetric spatially in homogeneous flows, have been reviewed. A nonlinear inversion method based on functional representation by cubic splines has been developed with a view to minimize the number of measurements required. Factors affecting this number, such as the choice of absorption frequencies and the phenomenon of error magnification inherent in the inversion process, have been studied. The efficacy of this inversion method has been demonstrated using absorption data synthetically generated from (previously reported) typical experimentally observed temperature and concentration profiles. The effect of pseudo-measurement error on the retrieval accuracy has been studied and the inversion method has been refined to limit error magnification in the solution process. The associated problem of low signal to noise ratio in regions with small absorption levels andpossible measures to overcome it have also been studied.