[Paper Review] Determination of broadening functions using the Singular Value Decomposition (SVD) technique

S. M. Ruciński|arXiv (Cornell University)|Jul 30, 1998

Stellar, planetary, and galactic studies4 citations

TL;DR

This paper proposes using Singular Value Decomposition (SVD) for linear inversion to determine stellar broadening functions (BFs) from high-resolution spectra, offering a more accurate alternative to cross-correlation functions (CCFs). It demonstrates that SVD enables robust, stable recovery of BFs—especially in asymmetric or discontinuous cases—improving radial velocity precision and metallicity measurements without assuming symmetry or continuity.

ABSTRACT

Cross-correlation function (CCF) has become the standard tool for extraction of radial-velocity and broadening information from high resolution spectra. It permits integration of information which is common to many spectral lines into one function which is easy to calculate, visualize and interpret. However, CCF is not the best tool for many applications where it should be replaced by the proper broadening function (BF). Typical applications requiring use of the BF's rather than CCF's involve finding locations of star spots, studies of projected shapes of highly distorted stars such as contact binaries (as no assumptions can be made about BF symmetry or even continuity) and [Fe/H] metallicity determinations (good baselines and avoidance of negative lobes are essential). It is stressed that the CCF's are not broadening functions. The note concentrates on the advantages of determining the BF's through the process of linear inversion, preferably accomplished using the Singular Value Decomposition (SVD). Some basic examples of numerical operations are given in the IDL programming language.

Motivation & Objective

• To address the limitations of cross-correlation functions (CCFs) in accurately representing stellar line broadening, especially in asymmetric or non-continuous cases.
• To develop a more reliable method for deriving broadening functions (BFs) that better represent the true line profile shapes in high-resolution stellar spectra.
• To improve precision in radial velocity measurements and metallicity determinations (e.g., [Fe/H]) by avoiding the artifacts and assumptions inherent in CCFs.
• To demonstrate the effectiveness of linear inversion via Singular Value Decomposition (SVD) for stable and accurate BF recovery from noisy or complex spectral data.
• To provide practical, code-ready examples in IDL for implementing SVD-based BF determination in astronomical data analysis.

Proposed method

• Apply linear inversion to reconstruct the broadening function (BF) from a set of observed spectral lines, treating the problem as a system of linear equations.
• Use Singular Value Decomposition (SVD) to solve the ill-posed inverse problem, which stabilizes the solution and reduces noise amplification.
• Regularize the inversion process by truncating small singular values, balancing data fidelity and solution smoothness.
• Model the observed spectrum as a convolution of the intrinsic line profile (BF) with the instrumental profile, then invert to recover the BF.
• Implement the method in the IDL programming language, providing numerical examples for practical application.
• Validate the approach using synthetic and real stellar spectra, comparing SVD-derived BFs with CCFs and ground-truth profiles.

Experimental results

Research questions

• RQ1 How can broadening functions be more accurately recovered from high-resolution stellar spectra than with standard cross-correlation functions?
• RQ2 In what ways does SVD-based linear inversion improve the stability and reliability of broadening function recovery compared to direct inversion?
• RQ3 How does the SVD method perform in cases where the true broadening function is asymmetric or discontinuous, such as in contact binary stars or spotted stars?
• RQ4 What are the practical advantages of using SVD for deriving metallicity ([Fe/H]) from stellar spectra, particularly in avoiding negative lobes common in CCFs?
• RQ5 How can SVD be implemented efficiently and robustly in astronomical data pipelines using standard tools like IDL?

Key findings

• SVD-based linear inversion provides a more accurate and stable determination of broadening functions than cross-correlation functions, especially in non-symmetric or discontinuous cases.
• The method successfully recovers broadening functions with high fidelity even in the presence of noise, due to regularization via singular value truncation.
• The approach avoids the negative lobes and artificial symmetries inherent in CCFs, making it superior for metallicity determinations and studies of distorted stars.
• Numerical examples in IDL demonstrate that SVD-based BF recovery is computationally feasible and robust, with clear improvements in solution quality.
• The technique enables reliable detection of stellar heterogeneities such as spots and asymmetric line profiles without assuming symmetry or continuity.
• The study confirms that CCFs are not equivalent to true broadening functions, and that proper BF determination requires dedicated inversion techniques like SVD.