Separation of continuous lines mutually overlapping and smoothed by the instrumental function

Abstract

Direct and inverse spectroscopy problems concerning superposition of overlapping continuous spectral lines, as well as smoothing of the total spectrum by the instrumental function, are considered. The direct problem is formulated in two stages: initially, the total spectrum is formed by lines with a given intensity distribution, and, then, a smoothed by the spectrometer instrumental function and noisy spectrum is obtained. The inverse problem is also formulated in two stages: initially, the Fredholm integral equation of the first kind is solved by the Tikhonov regularization method (an ill-posed problem), and, then, the problem of reconstruction of separate line-component shapes from the total spectrum is solved, which is the problem of line separation (division). The individual line components are modeled by Gaussians and Lorentzians. Numerical illustrations are shown. Gaussian and dispersion (Lorentz) instrumental functions are considered.