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Applied Optics

Applied Optics


  • Editor: James C. Wyant
  • Vol. 47, Iss. 2 — Jan. 10, 2008
  • pp: 153–156

Comment on “Improved ray tracing air mass numbers model”

Siebren Y. van der Werf  »View Author Affiliations

Applied Optics, Vol. 47, Issue 2, pp. 153-156 (2008)

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Air mass numbers have traditionally been obtained by techniques that use height as the integration variable. This introduces an inherent singularity at the horizon, and ad hoc solutions have been invented to cope with it. A survey of the possible options including integration by height, zenith angle, and horizontal distance or path length is presented. Ray tracing by path length is shown to avoid singularities both at the horizon and in the zenith. A fourth-order Runge–Kutta numerical integration scheme is presented, which treats refraction and air mass as path integrals. The latter may optionally be split out into separate contributions of the atmosphere's constituents.

© 2008 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.4030) Atmospheric and oceanic optics : Mirages and refraction

ToC Category:
Atmospheric and Oceanic Optics

Original Manuscript: August 10, 2007
Manuscript Accepted: September 5, 2007
Published: January 7, 2008

Siebren Y. van der Werf, "Comment on "Improved ray tracing air mass numbers model"," Appl. Opt. 47, 153-156 (2008)

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  1. F. Kasten, "A new table and approximation formula for the relative optical air mass," Arch. Meteorol. Geophys. Bioklimatol. Ser. B 14, 206-223 (1965). [CrossRef]
  2. F. Link and L. Neuzil, Tables of Light Trajectories in the Terrestrial Atmosphere (Hermann, 1969).
  3. F. Kasten and A. T. Young, "Revised optical air mass tables and approximation formula," Appl. Opt. 28, 4735-4738 (1989). [CrossRef] [PubMed]
  4. S. N. Kivalov, "Improved ray tracing air mass numbers model," Appl. Opt. 46, 7091-7098 (2007). [CrossRef] [PubMed]
  5. C. Runge, "Ueber die numerische Auflösung von Differentialgleichungen," Math. Ann. 46, 167-178 (1895). [CrossRef]
  6. W. M. Smart, Textbook on Spherical Astronomy, 6th ed., revised by R. M. Green (Cambridge U. Press, 1977).
  7. L. H. Auer and E. M. Standish, "Astronomical refraction: computational method for all zenith angles," Astron. J. 119, 2472-2477 (2000). [CrossRef]
  8. C. Y. Hohenkerk and A. T. Sinclair, "The computation of angular atmospheric refraction at large zenith angles," NAO Technical Note No. 63 (HM Nautical Almanac Office, 1985).
  9. P. K. Seidelmann, ed., Explanatory Supplement to the Astronomical Almanac (University Science Books, 1992).
  10. A. T. Young, "Sunset science. IV. Low-altitude refraction," Astron. J. 127, 3622-3637 (2004). [CrossRef]
  11. W. H. Lehn, "A simple parabolic model for the optics of the atmospheric surface layer," Appl. Math. Model. 9, 447-453 (1985). [CrossRef]
  12. D. Bruton, "Optical determination of atmospheric temperature profiles," Ph.D. thesis (Texas A&M University, 1996).
  13. S. Y. van der Werf, "Ray tracing and refraction in the modified US1976 atmosphere," Appl. Opt. 42, 354-366 (2003). [CrossRef] [PubMed]
  14. S. Y. van der Werf, G. P. Können, and W. H. Lehn, "Novaya Zemlya effect and sunsets," Appl. Opt. 42, 367-378 (2003). [CrossRef] [PubMed]
  15. D. Gutierrez, F. J. Seron, A. Munoz, and O. Anson, "Simulation of atmospheric phenomena," Comput. Graph. 30, 994-1010 (2006). [CrossRef]

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