This paper discusses the mathematical aspects of band fitting and
introduces the mathematics of the Asymmetric Gaussian shape and its
tangent space for the first time. First, we derive an equation for an
Asymmetric Gaussian shape. We then use this equation to derive a rule
for the resolution of two Gaussian shaped bands. We then use the
Asymmetrical Gaussian equation to derive a Master Equation to fit two
overlapping bands. We identify regions of the fitting space where the
Asymmetric Gaussian fit is optimal, sub optimal and not optimal. We then
demonstrate the use of the Asymmetric Gaussian to fit four overlapping
Gaussian bands, and show how this is relevant to the olivine spectral
complex at 1 ?m. We develop a new model of the olivine family spectrum
based on previous work by Runciman and Burns. The limitations of the
asymmetric band fitting method and a critical assessment of three
commonly used numerical minimization methods are also provided.