Lbl4IR – Linebyline for InfraRed
(Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 112, pp. 10101025, 2011)
Abstract
Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued functions.
For the complex error function w(x+iy), whose real part is the Voigt function K(x,y), code optimizations of rational approximations are investigated.
An assessment of requirements for atmospheric radiative transfer modeling indicates a y range over many orders of magnitude and accuracy better than 10^{4}.
Following a brief survey of complex error function algorithms in general and rational function approximations in particular the problems associated with subdivisions of the x, y plane (i.e., conditional branches in the code) are discussed and practical aspects of Fortran and Python implementations are considered.
Benchmark tests of a variety of algorithms demonstrate that programming language, compiler choice, and implementation details influence computational speed and there is no unique ranking of algorithms.
A new implementation, based on subdivision of the upper halfplane in only two regions, combining Weideman's rational approximation for small x+y < 15 and Humlicek's rational approximation otherwise is shown to be efficient and accurate for all x, y.
(See the weideman.py module for Python implementations of various HumlicekWeideman combinations.)
(Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 187, pp. 4453, 2017)
Abstract
The increasing quality of atmospheric spectroscopy observations has indicated the limitations of the Voigt profile routinely used for linebyline modeling, and physical processes beyond pressure and Doppler broadening have to be considered. The speeddependent Voigt (SDV) profile can be readily computed as the difference of the real part of two complex error functions (i.e. Voigt functions). Using a highly accurate code as a reference, various implementations of the SDV function based on Humlicek's rational approximations are examined for typical speed dependences of pressure broadening and the range of wavenumber distances and Lorentz to Doppler width ratios encountered in infrared applications. Neither of these implementations appears to be optimal, and a new algorithm based on a combination of the Humlicek (1982) and Weideman (1994) rational approximations is suggested.
Supplement Material
 demo.py (called mmc1.txt in the JQSRT supplement)
 A (I)Python interactive session demonstrating the use of the humlicek, weideman, and sdv modules and how to generate the contour plots.
 sdv.py (called mmc4.txt in the JQSRT supplement)
 The 'main functions' of this module, sdvProfile and voigtProfile, evaluate the corresponding profiles (normalized to one) for given pressure & temperature, and line position & widths.
Further functions defined here provide implementations of the SDV function based on various codes (and combinations therof) for the complex error function.
 humlicek.py (called mmc2.txt in the JQSRT supplement)
 Various implementations of Humlicek's cpf12 and w4 rational approximations
 weideman.py (called mmc5.txt in the JQSRT supplement)
 The rational approximations with N=24 and N=32 terms and the HumlicekWeideman combinations.
 ir.py (called mmc3.txt in the JQSRT supplement)
 Mathematical and physical constants
Journal of Quantitative Spectroscopy and Radiative Transfer, 213, 1316, 2018;
Preprint available at arXiv:1806.10338
Abstract
The Voigt profile is important for spectroscopy, astrophysics, and many other fields of physics, but is notoriously difficult to compute.
McLean et al. [J. Electron Spectrosc. & Relat. Phenom., 1994] have proposed an approximation using a sum of Lorentzians. Our assessment indicates that this algorithm has significant errors for small arguments. After a brief survey of the requirements for spectroscopy we give a short list of both efficient and accurate codes and recommend implementations based on rational approximations.
Supplement Material and Further Links
 mcLean.py
 The Python script defining the Voigt function approximations and generating the figures.
 astropy
 The Astropy Collaboration, T.P. Robitaille, E.J. Tollerud, et al.
Astropy: A community Python package for astronomy
Astronomy & Astrophysics, Vol. 558, Article A33, October 2013
 spectraPlot.com
 C.S. Goldenstein, V.A. Miller, R.M. Spearrin, C.L. Strand:
SpectraPlot.com: Integrated spectroscopic modeling of atomic and molecular gases
J. Quant. Spectrosc. & Radiat. Transfer, 200, pp. 249257, 2017
(Monthly Notices of the Royal Astronomical Society, 479(3), 30683075, 2018;
Preprint available at arXiv:1806.11560
Abstract
Accurate yet efficient computation of the Voigt and complex error function is a challenge since decades in astrophysics and other areas of physics.
Rational approximations have attracted considerable attention and are used in many codes, often in combination with other techniques.
The 12term code "cpf12" of Humlicek (1979) achieves an accuracy of five to six significant digits throughout the entire complex plane.
Here we generalize this algorithm to a larger (even) number of terms.
The n = 16 approximation has a relative accuracy better than 10^{5} for almost the entire complex plane except for very small imaginary values of the argument even without the correction term required for the cpf12 algorithm.
With 20 terms the accuracy is better than 10^{6}.
In addition to the accuracy assessment we discuss methods for optimization and propose a combination of the 16term approximation with the asymptotic approximation of Humlicek (1982) for high efficiency.
Supplement Material
 cpfX.py  Various implementations of the Humlicek (1979) algorithm:

 Python/NumPy implementations of the original Fortran 77 source code (with two versions to combine the two regions);
 A generalization of the region I rational approximation to arbitrary (even) number of terms;
 An optimized implementation of the region I approximation for 16 terms using a single fraction;
 The combination Eq. (12) with the asymptotic approximation of Humlicek (1982) for NumPy.
 fig124.py
 Execution of this script inside the IPython interpreter should produce the Figures 1, 2, and 4 of the manuscript.
 weideman.py (originally provided as supplementary material of the JQSRT 2017 paper)
 Weideman's rational approximations with N=24 and N=32 terms and the HumlicekWeideman combinations.
Py4CAtS – Python for Computational ATmospheric Spectroscopy
Abstract
Py4CAtS is a Python reimplementation of the Fortran infrared radiative transfer code GARLIC, where computeintensive code sections utilize the Numeric/Scientific Python modules (Numpy/Scipy) for highly optimized arrayprocessing.
The individual steps of an infrared or microwave radiative transfer computation are implemented in separate scripts to extract lines of relevant molecules in the spectral range of interest, to compute linebyline cross sections for given pressure(s) and temperature(s), to combine cross sections to absorption coefficients and optical depths, and to integrate along the lineofsight to transmission and radiance/intensity.
Documentation
 The Voigt and complex error function: Humliceks rational approximation generalized MNRAS, 479(3), 30683075, 2018
(arxiv: physics.compph/1806.11560)
 Computational Aspects of Speeddependent Voigt Profiles JQSRT, Vol. 187, pp. 4453, 2017
 Optimized Implementations of Rational Approximations for the Voigt and Complex Error Function
JQSRT, Vol. 112(6), pp. 110125, 2011
 The Voigt and complex error function: A comparison of computational methods
JQSRT, Vol. 48, pp. 743762, 1992
 GARLIC  A General Purpose Atmospheric Radiative Transfer LinebyLine InfraredMicrowave Code: Implementation and Evaluation; JQSRT, Vol. 127, pp. 2950, 2014
 Py4CAtS  Python for Computational ATmospheric Spectroscopy; Atmosphere, Vol. 10(5), pp. 262, 2019
 An assessment of some closedform expressions for the Voigt function: III: Combinations of the Lorentz and Gauss functions JQSRT, 226, 8791, 2019
(arxiv: physics.compph/1901.08366)
 Comments on the Voigt function implementation in the Astropy and SpectraPlot.com packages JQSRT, 213, 1316, 2018
(arxiv: physics.compph/1806.10338)
 An assessment of some closedform expressions for the Voigt function: II: Utilizing rational approximations for the Gauss function JQSRT, 202, 8189, 2017
 An assessment of some closedform expressions for the Voigt function JQSRT, 176, 15, 2016
 J.Humlicek: An efficient method for evaluation of the complex probability function: the Voigt function and its derivatives; JQSRT, Vol. 21, 309313, 1979
 J.Humlicek: Optimized computation of the Voigt and complex probability function; JQSRT, Vol. 27, 437444, 1982
 J.A.C. Weideman: Computation of the Complex Error Function; SIAMJ. Num. Anal., Vol. 31, 14971518, 1992
What's new?
 October 2016
 Initial page setup
 April 2018
 JQSRT 2018 paper added
 June 2018
 MNRAS 2018 paper added
 August 2019
 "Assessment of some closedform expressions ..." paper series added
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