Computational Aspects of Speed-Dependent Voigt and Rautian Profiles

TL;DR

This paper improves the numerical stability and efficiency of computing speed-dependent Voigt and Rautian profiles, essential for accurate molecular line modeling, by reformulating complex error function calculations and applying rational approximations.

Contribution

It introduces a reformulation to avoid cancellation errors and demonstrates that rational approximations enable accurate and efficient computation of SDV and SDR profiles.

Findings

Reformulation reduces cancellation errors in complex error function calculations.

Rational approximations provide four-digit accuracy across the complex plane.

SDV and SDR are approximately 2.2 times slower than Voigt but still efficient for practical use.

Abstract

For accurate line-by-line modeling of molecular cross sections several physical processes "beyond Voigt" have to be considered. For the speed-dependent Voigt and Rautian profiles (SDV, SDR) and the Hartmann-Tran profile the difference $w(i z_-)-w(i z_+)$ of two complex error functions (essentially Voigt functions) has to be evaluated where the function arguments $z_\pm$ are given by the sum and difference of two square roots. These two terms describing $z_\pm$ can be huge and the default implementation of the difference can lead to large cancellation errors. First we demonstrate that these problems can be avoided by a simple reformulation of $z_-$. Furthermore we show that a single rational approximation of the complex error function valid in the whole complex plane (e.g. by Humlicek, 1979 or Weideman, 1994) enables computation of the SDV and SDR with four significant digits or better. Our benchmarks indicate that the SDV and SDR functions are about a factor 2.2 slower compared to the Voigt function, but for evaluation of molecular cross sections this time lag does not significantly prolong the overall program execution because speed-dependent parameters are available only for a fraction of strong lines.