Estimation of confidence intervals for radial emissivity and optimization of data treatment techniques in Abel inversion

A novel method is described for finding confidence intervals for radially resolved emissivity derived from Abel inversion. The essence of the method is that the measured lateral emission profile (here a profile is defined as a line-of-sight spatial map) is split into two components, namely (a) an estimated noise-free lateral emission profile that mimics the lateral emission profile in the absence of measurement noise and (b) a noise profile that estimates the magnitude of noise along the lateral profile. Through random-number generation, noise with a magnitude defined by the noise profile is then artificially added to the estimated noise-free lateral profile. If this noise-addition and subsequent Abel inversion is iterated for a sufficient number of cycles, a statistical distribution of the Abel-inverted radial profiles can be obtained and confidence intervals can thus be estimated. It was verified that an effective means for the estimation of the noise-free lateral profile involves approximation of the lateral emission profile through polynomial fitting. A noise profile can then be obtained by assuming that the residual between the experimental lateral profile and the fitted profile is due solely to noise. The validity of this methodology was verified with seven lateral test profiles. In addition, the parameters that are commonly used in data treatment by Abel inversion are optimized. It was found that 100–300 data points are sufficient for accurate Abel inversion using the Nestor-Olsen inversion algorithm; more data points provide only minimal improvement in the inversion. Also, it was found that most commonly observed lateral emission profiles can be satisfactorily approximated by polynomials with a degree of fifteen. Further, polynomial fitting can be used to partially filter out noise in the lateral profile. Depending on the shape of the lateral profile and the magnitude of the noise, the optimal polynomial order, in general, ranges from ten to fifteen. For higher-degree polynomials, the effectiveness of a polynomial as a noise filter decreases and severely degrades the quality of the Abel-inverted radial profile.