OSA's Digital Library

Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 6 — Jun. 13, 2007

Extension of Chandrasekhar's formula to a nonhomogeneous Lambertian surface and comparison with the 6S formulation

Alain Sei  »View Author Affiliations


Applied Optics, Vol. 46, Issue 13, pp. 2471-2480 (2007)
http://dx.doi.org/10.1364/AO.46.002471


View Full Text Article

Enhanced HTML    Acrobat PDF (115 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The classical Chandrasekhar's formula, which relates the surface albedo to the top of the atmosphere radiance, rigorously applies to a homogeneous Lambertian surface. For a nonhomogeneous Lambertian surface in a plane-parallel atmosphere, an extension of this formula was proposed in the 1980s and has been implemented recently in the 6S algorithm. To analyze this extension, the rigorous formula of the top of the atmosphere signal in a plane-parallel atmosphere bounded by a nonhomogeneous Lambertian surface is derived. Then the 6S algorithm extension is compared to the exact formula and approximations and their validity is examined. The derivation of the exact formula is based on the separation of the radiation fields into direct and diffuse components, on the introduction of the Green's function of the problem, and on integrations of boundary values of the radiation fields with Green's function.

© 2007 Optical Society of America

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1310) Atmospheric and oceanic optics : Atmospheric scattering
(010.1320) Atmospheric and oceanic optics : Atmospheric transmittance
(280.1310) Remote sensing and sensors : Atmospheric scattering
(290.4210) Scattering : Multiple scattering

ToC Category:
Atmospheric and ocean optics

History
Original Manuscript: October 26, 2006
Revised Manuscript: December 18, 2006
Manuscript Accepted: December 19, 2006
Published: April 9, 2007

Virtual Issues
Vol. 2, Iss. 6 Virtual Journal for Biomedical Optics

Citation
Alain Sei, "Extension of Chandrasekhar's formula to a nonhomogeneous Lambertian surface and comparison with the 6S formulation," Appl. Opt. 46, 2471-2480 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ao-46-13-2471


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground-I. Theory," J. Quant. Spectrosc. Radiat. Transfer 31, 97-125 (1984). [CrossRef]
  2. D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground-II. Computational results," J. Quant. Spectrosc. Radiat. Transfer 31, 279-304 (1984). [CrossRef]
  3. K. F. Evans, "The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998). [CrossRef]
  4. A. Lyapustin and T. Z. Muldashev, "Method of spherical harmonics in the radiative transfer problem with non-Lambertian surface," J. Quant. Spectrosc. Radiat. Transfer 61, 545-555 (1999). [CrossRef]
  5. A. Lyapustin and Y. Knyazikhin, "Green's function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface," Appl. Opt. 41, 5600-5606 (2002). [CrossRef] [PubMed]
  6. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, "Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media," Appl. Opt. 27, 2502-2509 (1988). [CrossRef] [PubMed]
  7. A. Lyapustin, Radiative transfer code SHARM-3D for radiance simulations over a non-Lambertian nonhomogeneous surface: intercomparison study," Appl. Opt. 41, 5607-5615 (2002). [CrossRef] [PubMed]
  8. D. Tanré, M. Herman, P. Y. Deschamps, and A. de Leffe, "Atmospheric modeling for space measurements of ground reflectances, including bidirectional properties," Appl. Opt. 18, 3587-3594 (1979). [CrossRef] [PubMed]
  9. D. Tanré, M. Herman, and P. Y. Deschamps, "Influence of the background contribution upon space measurements of ground reflectance," Appl. Opt. 20, 3676-3684 (1981). [CrossRef] [PubMed]
  10. E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcette, "Second simulation of the satellite signal in the solar spectrum, 6S: An overview," IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997). [CrossRef]
  11. E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.
  12. A. Sei, "Extension of Chandrasekhar's formula to a homogeneous non-Lambertian surface and comparison with the 6S formulation," Appl. Opt. 45, 1010-1022 (2006). [CrossRef] [PubMed]
  13. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  14. G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, 1970).
  15. A. A. Ioltukhovski, "Radiative transfer over the surface with an arbitrary reflection: Green's functions method," Transp. Theory Stat. Phys. 28, 349-368 (1999). [CrossRef]
  16. Y. Qin, M. A. Box, and P. Douriaguine, "Computation of Green's function for radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 84, 159-168 (2004). [CrossRef]
  17. J. Lenoble, Atmospheric Radiative Transfer (Deepak, 1993).
  18. K. N. Liou, Introduction to Atmospheric Radiation (Academic, 2002).
  19. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  20. H. C. van de Hulst, "Scattering in a planetary atmosphere," Astrophys. J. 107, 220-246 (1948). [CrossRef]
  21. A. Sei, "Analysis of adjacency effects for two Lambertian half-spaces," Int. J. Remote Sens. (to be published).
  22. P. N. Reinersman and K. Carder, "Monte Carlo simulation of the atmospheric point-spread function with an application to correction for the adjacency effect," Appl. Opt. 34, 4453-4471 (1995). [CrossRef] [PubMed]
  23. P. R. Garabedian, Partial Differential Equations (American Mathematical Society, 1998).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited