Although we use a spectrometer to measure the spectrum of a source,
what the spectrometer obtains is not the actual spectrum, but rather
photon counts () within specific instrument channels, (). This
observed spectrum is related to the actual spectrum of the source
Where is the instrumental response and is proportional to the probability that an incoming photon of energy will be detected in channel . Ideally, then, we would like to determine the actual spectrum of a source, , by inverting this equation, thus deriving for a given set of . Regrettably, this is not possible in general, as such inversions tend to be non-unique and unstable to small changes in . (For examples of attempts to circumvent these problems see Blissett & Cruise (1979); Kahn & Blissett (1980); Loredo & Epstein (1989)).
The usual alternative is to choose a model spectrum, , that can be described in terms of a few parameters (i.e., ), and match, or “fit” it to the data obtained by the spectrometer. For each , a predicted count spectrum () is calculated and compared to the observed data (). Then a “fit statistic” is computed from the comparison and used to judge whether the model spectrum “fits” the data obtained by the spectrometer.
The model parameters then are varied to find the parameter values that give the most desirable fit statistic. These values are referred to as the best-fit parameters. The model spectrum, , made up of the best-fit parameters is considered to be the best-fit model.
The most common fit statistic in use for determining the “best-fit” model is , defined as follows:
where is the (generally unknown) error for channel (e.g., if are counts then is usually estimated by ; see e.g. Wheaton et al. (1995) for other possibilities).
Once a “best-fit” model is obtained, one must ask two questions: