Precision in high resolution absorption line modelling, analytic Voigt derivatives, and optimisation methods

TL;DR

This paper improves absorption line modelling by deriving analytical Voigt derivatives, introducing a new optimization method, and addressing numerical issues for high-precision spectral analysis in future astronomical observations.

Contribution

It presents analytical derivatives of the Voigt function, a novel optimization approach combining Gauss-Newton and Levenberg-Marquardt principles, and practical solutions for ill-conditioning in spectral modelling.

Findings

Analytical Voigt derivatives enhance precision over finite differences.

A new optimization method improves convergence efficiency.

Practical fixes address ill-conditioning in spectral fitting.

Abstract

This paper describes the optimisation theory on which VPFIT, a non-linear least-squares program for modelling absorption spectra, is based. Particular attention is paid to precision. Voigt function derivatives have previously been calculated using numerical finite difference approximations. We show how these can instead be computed analytically using Taylor series expansions and look-up tables. We introduce a new optimisation method for an efficient descent path to the best-fit, combining the principles used in both the Gauss-Newton and Levenberg-Marquardt algorithms. A simple practical fix for ill-conditioning is described, a common problem when modelling quasar absorption systems. We also summarise how unbiased modelling depends on using an appropriate information criterion to guard against over- or under-fitting. The methods and the new implementations introduced in this paper are aimed at optimal usage of future data from facilities such as ESPRESSO/VLT and HIRES/ELT, particularly for the most demanding applications such as searches for spacetime variations in fundamental constants and attempts to detect cosmological redshift drift.