Resolving Asymmetric Spectral Bands

This paper introduces the basics of asymmetrical spectral band fitting and discusses the resolution of overlapping Gaussian shapes. We study how to fit overlapping bands with asymmetric Gaussian shapes. First, we derive an equation for an Asymmetric Gaussian shape. We then use this equation to derive a resolvability basis for the resolution of two nearby Gaussian bands. The so called Master Equation is then used to fit these two overlapping bands. We identify regions of the fitting space where the Asymmetric Gaussian fit is likely to be Optimal, Sub Optimal and Not Optimal. We then demonstrate the use of the Asymmetric Gaussian to fit four overlapping bands, and show how this is relevant to the olivine spectral complex at 1 $\mu$m. The limitations of the asymmetric band fitting method and a critical assessment of three commonly used numerical minimization and fitting methods are also provided.