A method for merging the results of separate least-squares fits and testing for systematic errors

Abstract

A method is presented for merging the results of separate least-squares fits to obtain the most precise, single values for each of the molecular constants of a spectroscopic system. The output molecular constants and accompanying variance-covariance matrices from each of the separate fits are taken together as input to a correlated least-squares fit, which then yields the desired merged single values. The method is applied to a 1Σ- 1Σ diatomic model band system using well-defined synthetic data as an example. Other currently used, multistate methods of obtaining molecular constants for a spectroscopic system are special cases of the method described here. For the nonlinear least-squares fits required for multiplet states, the merge method is particularly advantageous. The properties of the merged values of the molecular constants and their standard errors are examined with regard to forming confidence limits for these estimates and it is shown that, in most spectroscopic applications, the confidence limits have their usual meaning. Furthermore, it is similarly shown that the confidence limits for the variance of the merged fit provide a simple and convenient test for the presence of relative systematic errors between the separate least-squares results being merged. Although the method is illustrated here by merged separate band-by-band results, it also can be applied to any type of spectroscopic results.