Estimating the probability density of the scattering cross section from Rayleigh scattering experiments

An important parameter in the experimental study of dynamics of combustion is the probability distribution of the effective Rayleigh scattering cross section. This cross section cannot be observed directly. Instead, pairs of measurements of laser intensities and Rayleigh scattering counts are observed. Our aim is to provide estimators for the probability density function of the scattering cross section from such measurements. The probability distribution is derived first for the number of recorded photons in the Rayleigh scattering experiment. In this approach the laser intensity measurements are treated as known covariates. This departs from the usual practice of normalizing the Rayleigh scattering counts by the laser intensities. For distributions supported on finite intervals two estimators of the probability density are given: one based on expansion of the density in orthonormal polynomials and the other based on the maximum likelihood. The latter procedure is applied to Rayleigh scattering data from an open V-shaped premixed methane–air flame.