Simple Estimation of Errors and Multi-Pixel Signal-to-Noise Ratio for Gaussian,Lorentzian,and Pseudo-Voigt functions Fit to Noisy Spectral Data

Abstract

Quantification of band parameters in spectral data typically requires iteratively fitting the spectroscopic feature to Gaussian, Lorentzian, or pseudo-Voigt functions for the extraction of band height, width, position, and area values. However, there are currently no adequate methods of measuring their errors. Previously, Lenz and Ayres [“Errors Associated with Fitting Gaussian Profiles to Noisy Emission-Line Spectra”. Publications of the Astronomical Society of the Pacific. 1992. 104: 1104. DOI: 10.1086/133096] addressed this problem for Gaussian fits by empirically fitting a set of model data to derive equations for the calculation of Gaussian band parameter errors. However, the treatment of only Gaussian fitting functions greatly limits utility. In this study, we extend their error calculation methods to include Lorentzian and pseudo-Voigt fits for error analysis in a wide range analytical techniques that utilize spectral fitting. We do this empirically by fitting a model dataset of noisy spectral signals to Gaussian, Lorentzian, and pseudo-Voigt functions to calculate the accuracy and precision of the fits. We derive a set of equations for the simple calculation of band fitting errors. In addition, the simple calculation of band height and area errors allows for easy calculation of multi-pixel signal-to-noise ratios that were previously shown to significantly improve instrument limit-of-detection [Jakubek et al. “Improving Spectroscopic Detection Limits with Multi-Pixel Signal-to-Noise Ratio Calculations: Application to the SHERLOC Instrument Aboard the Perseverance Rover”. Analytica Chimica Acta. 2025. 1357: 344072. DOI: 10.1016/j.aca.2025.344072]. The results of this work are broadly applicable to most analytical techniques that produce spectral data.

Graphical abstract

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