OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 38, Iss. 9 — Mar. 20, 1999
  • pp: 1530–1542

Radiative transfer in the atmosphere–ocean system: the finite-element method

Barbara Bulgarelli, Viatcheslav B. Kisselev, and Laura Roberti  »View Author Affiliations

Applied Optics, Vol. 38, Issue 9, pp. 1530-1542 (1999)

View Full Text Article

Enhanced HTML    Acrobat PDF (289 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The finite-element method has been applied to solving the radiative-transfer equation in a layered medium with a change in the refractive index, such as the atmosphere–ocean system. The physical processes that are included in the algorithm are multiple scattering, bottom-boundary bidirectional reflectivity, and refraction and reflection at the interface between the media with different refractive properties. The incident radiation is a parallel flux on the top boundary that is characteristic of illumination of the atmosphere by the Sun in the UV, visible, and near-IR regions of the electromagnetic spectrum. The necessary changes, compared with the case of a uniformly refracting layered medium, are described. An energy-conservation test has been performed on the model. The algorithm has also been validated through comparison with an equivalent backward Monte Carlo code and with data taken from the literature, and optimal agreement was shown. The results show that the model allows energy conservation independently of the adopted phase function, the number of grid points, and the relative refractive index. The radiative-transfer model can be applied to any other layered system with a change in the refractive index. The fortran code for this algorithm is documented and is available for applications.

© 1999 Optical Society of America

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(010.7340) Atmospheric and oceanic optics : Water
(290.4210) Scattering : Multiple scattering

Original Manuscript: May 12, 1998
Revised Manuscript: November 18, 1998
Published: March 20, 1999

Barbara Bulgarelli, Viatcheslav B. Kisselev, and Laura Roberti, "Radiative transfer in the atmosphere–ocean system: the finite-element method," Appl. Opt. 38, 1530-1542 (1999)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. V. Kisselev, L. Roberti, G. Perona, “Finite-element algorithm for radiative transfer in a vertically inhomogeneous medium: numerical scheme and applications,” Appl. Opt. 34, 8460–8471 (1995). [CrossRef] [PubMed]
  2. R. F. Harringtion, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.
  3. O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).
  4. K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981). [CrossRef]
  5. K. Stamnes, H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: Intensity computations,” J. Atmos. Sci. 38, 2969–2706 (1981).
  6. K. Stamnes, S. Tsay, W. Wiscombe, 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. K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975). [CrossRef]
  8. B. Bulgarelli, “Radiative transfer in atmosphere and ocean,” Ph.D. dissertation (Politecnico di Torino, Torino, Italy, 1998).
  9. V. B. Kisselev, L. Roberti, G. Perona, “An application of the finite element method to the solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 51, 603–614 (1994). [CrossRef]
  10. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980).
  11. C. Devaux, Y. Fouquart, M. Herman, J. Lenoble, “Comparaisons de diverses methodes de resolution de l’equation de transfert du rayonnement dans un milieu diffusant,” J. Quant. Spectrosc. Radiant. Transfer 13, 1421–1431 (1973). [CrossRef]
  12. V. V. Sobolev, Scattering of Light in Planetary Atmosphere (Pergamon, New York, 1975).
  13. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974). [CrossRef]
  14. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993). [CrossRef] [PubMed]
  15. L. Roberti, “Monte Carlo radiative transfer in the microwave and in the visible: biasing techniques,” Appl. Opt. 36, 7929–7938 (1997). [CrossRef]
  16. T. J. Petzold, “Volume scattering functions for selected natural waters,” (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif., 1972).
  17. O. I. Smotky, Modeling of Radiation Fields in the Problems of Space Spectrophotometry (in Russian) (Nauka, Moscow, 1986), pp. 352–370.
  18. W. J. Wiscombe, “The Delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase func-tions,” J. Atmos. Sci. 34, 1408–1422 (1977). [CrossRef]
  19. Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994). [CrossRef] [PubMed]
  20. H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997). [CrossRef] [PubMed]

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.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited