A constrained linear model for continuum normalization of stellar spectra

TL;DR

The paper tackles the challenge of continuum normalization in stellar spectroscopy by introducing a constrained linear forward model that simultaneously fits stellar absorption, telluric transmission, and the joint continuum-instrument response. It constructs the stellar component via non-negative matrix factorization of a grid of theoretical spectra in the form $-\log\mathbf{S} \approx \mathbf{W}_\star^{\mathsf{T}}\mathbf{F}$ and expresses both telluric and continuum terms as linear combinations, yielding the additive representation $\mathbf{Y} = -\mathbf{F}^{\mathsf{T}}\boldsymbol{\alpha} - \mathbf{G}^{\mathsf{T}}\boldsymbol{\beta} + \mathbf{H}\boldsymbol{\gamma}$ in $\log$-flux space. The inference is convex, fast, and requires no initial guess, with the model trained once on the BOSZ grid and then applied to ESO/HARPS spectra; it achieves sub-percent continuum normalization consistency across varying $S/N$, including $0.2\%$ for $S/N \approx 100$ and $<0.5\%$ for $S/N \approx 30$. This approach offers a practical, scalable path for automatic continuum normalization in large spectral surveys, and is readily extendable to include radial velocity and rotational broadening while remaining robust to model imperfections.

Abstract

Inferring stellar parameters and chemical abundances by forward modeling stellar spectra usually requires a spectral synthesis code, or an emulator constructed from a curated training set. In these situations continuum normalization is often implemented as a pre-processing step that is independent of stellar parameters. This leads to results that are biased, or inconsistent across signal-to-noise ratios. A more justified approach is to forward model spectra with all nuisances simultaneously, but in practice this can be an expensive or non-convex optimization procedure. Here we describe a constrained linear model that can fit stellar absorption, telluric transmission, the joint continuum-instrument response. Stellar absorption and telluric transmission are each modeled by factorizing a grid of rectified theoretical spectra into two non-negative matrices with a chosen number of basis components. This model characterizes all possible spectra in many fewer parameters than comparable data-driven models. The non-negativity constraint ensures basis vectors are strictly additive, which limits rectified flux to less than or equal to unity, such that we can distinguish normalized spectra from the joint instrument-continuum response. The model requires no initial guess, and the linearity ensures that inference is convex, stable, and fast. This model allows us to reliably fit nuisances (e.g., tellurics, continuum), and is readily extensible to radial velocity and rotational broadening, without any prior knowledge about the fundamental stellar properties. We demonstrate our method by fitting ESO/HARPS high-resolution echelle spectra of BAFGKM-type stars. With repeat observations of $α$-Centauri A we present results that are best in class: consistent across time to 0.2% at S/N ~ 100, and to better than 0.5% at S/N ~ 30.

Figures (4)

• Figure 1: A schematic showing a small wavelength range of the basis vectors computed by non-negative matrix factorisation from a grid of theoretical stellar spectra.
• Figure 2: An example fit to a single ESO/HARPS spectrum of HD 222595 (black), showing the continuum fits per echelle order , the stellar absorption model (blue) and the telluric transmission model (orange), all fit simultaneously with a constrained linear absorption model. Near the strong Ca H and K lines (see inset) it is clear that there are no 'continuum pixels' in those echelle orders, yet the model provides an excellent estimate of the true continuum. The telluric model used is imperfect (see inset), but sufficient to capture most telluric absorption.
• Figure 3: Continuum-rectified HARPS spectra of example spectral types from M to B. The model used here includes 16 basis vectors for stellar absorption, and 4 basis vectors for telluric transmission.
• Figure 4: Distribution (shaded) of rectified flux values of $\alpha$-Centauri A as a function of S/N ratio, with every spectrum fit independently. The shaded region line represents distribution of flux values for each spectrum (e.g., each vertical distribution represents all $\sim3\times10^6$ pixels in a single spectrum). The blue line indicates the 95th percentile, which has a 1-$\sigma$ spread of 0.46% for spectra with S/N $>$ 30, and 0.22% for everything with S/N $>$ 100.