OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 40, Iss. 21 — Jul. 20, 2001
  • pp: 3495–3501

Green’s function method for the radiative transfer problem. I. Homogeneous non-Lambertian surface

Alexei Lyapustin and Yuri Knyazikhin  »View Author Affiliations

Applied Optics, Vol. 40, Issue 21, pp. 3495-3501 (2001)

View Full Text Article

Enhanced HTML    Acrobat PDF (121 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



An application of the Green’s function method to the one-dimensional radiative transfer problem with a non-Lambertian surface is described. This method separates atmospheric radiative transport from the lower boundary condition and allows expressing a solution analytically for an arbitrary surface reflectance. In the physical sense, the Green’s function represents bidirectional atmospheric transmission for the unitary radiance source located at the bottom of the atmosphere. The boundary-value problem for the Green’s function is adjoint to the problem for atmospheric path radiance, and therefore it can be solved by use of existing numerical methods by reversal of the direction of light propagation. From an analysis of an exact operator solution and extensive numerical study, we found two accelerating parameterizations for computing the surface-reflected radiance. The first one is a maximum-eigenvalue method that is comparable in accuracy with rigorous radiative transfer codes in calculations with realistic land-cover types. It requires a total of the first three orders of the surface-reflected radiance. The second one is based on the Lambertian approximation of multiple reflections. Designed for operational applications, it is much faster: Already the first-order reflected radiance ensures an average accuracy of better than 1%.

© 2001 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
(290.4210) Scattering : Multiple scattering

Original Manuscript: January 2, 2001
Revised Manuscript: March 22, 2001
Published: July 20, 2001

Alexei Lyapustin and Yuri Knyazikhin, "Green’s function method for the radiative transfer problem. I. Homogeneous non-Lambertian surface," Appl. Opt. 40, 3495-3501 (2001)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. E. Hansen, J. W. Hovenier, “The doubling method applied to multiple scattering of polarized light,” J. Quant. Spectrosc. Radiat. Transfer 11, 809–812 (1971). [CrossRef]
  2. J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation,” J. Atmos. Sci. 32, 790–798 (1975). [CrossRef]
  3. K. Stamnes, S. C. 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]
  4. P. Koepke, K. T. Kriebel, “Influence of measured reflection properties of vegetated surfaces on atmospheric radiance and its polarization,” Appl. Opt. 17, 260–264 (1978). [CrossRef] [PubMed]
  5. E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997). [CrossRef]
  6. D. Tanre, M. Herman, P. Y. Deschamps, A. de Leffe, “Atmospheric modeling for space measurement of ground reflectances, including bidirectional properties,” Appl. Opt. 18, 3587–3594 (1979). [CrossRef] [PubMed]
  7. A. I. Lyapustin, T. Z. Muldashev, “Generalization of Marshak boundary condition for non-Lambert reflection,” J. Quant. Spectrosc. Radiat. Transfer 67, 457–464 (2000). [CrossRef]
  8. A. I. Lyapustin, “Atmospheric and geometrical effects on land surface albedo,” J. Geophys. Res. 104, 4123–4143 (1999).
  9. A. I. Lyapustin, J. L. Privette, “A new algorithm for retrieving surface BRDF from ground measurements: atmospheric sensitivity study,” J. Geophys. Res. 104, 6257–6268 (1999). [CrossRef]
  10. G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).
  11. T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, Russia, 1990).
  12. S. A. W. Gerstl, “Application of the adjoint method in atmospheric radiative transfer calculations,” in Atmospheric Aerosols: Their Formation, Optical Properties and Effects, A. Deepak, ed. (Spectrum, Hampton, Va., 1982), pp. 241–254.
  13. S. Twomey, “Green’s function formulae for the internal intensity in radiative transfer computations by matrix–vector methods,” J. Quant. Spectrosc. Radiat. Transfer 33, 575–579 (1985). [CrossRef]
  14. G. I. Marchuk, V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport, (Harwood Academic, New York, 1986).
  15. Y. Knyazikhin, “On the solvability of plane-parallel problems in the theory of radiation transport,” USSR Comput. Math. Math. Phys. 30, 557–569 (1990).
  16. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  17. L. Elterman, “UV, visible and IR attenuation for altitudes to 50 km,” (U.S. Air Force Cambridge Research Laboratories, Bedford, Mass., 1968).
  18. H. Rahman, B. Pinty, M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data,” J. Geophys. Res. 98, 20791–20801 (1993). [CrossRef]
  19. R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993). [CrossRef]
  20. D. J. Diner, J. V. Martonchik, “Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground—II. Computational considerations and results,” J. Quant. Spectrosc. Radiat. Transfer 32, 279–304 (1984). [CrossRef]
  21. F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).
  22. V. S. Vladimirov, “Mathematical problems in the one-velocity theory of particle transport,” (Atomic Energy of Canada Ltd., Chalk River, Ontario, 1963).
  23. R. K. Kaufmann, L. Zhou, Y. Knyazikhin, N. V. Shabanov, R. B. Myneni, C. J. Tucker, “Effect of orbital drift and sensor changes on the time series of AVHRR vegetation index data,” IEEE Trans. Geosci. Remote Sens. 38, 2584–2597 (2000). [CrossRef]
  24. O. Engelsen, B. Pinty, M. M. Verstraete, J. V. Martonchik, “Parametric bidirectional reflectance factor models: evaluation, improvements and applications,” (Space Applications Institute, Ispra, Italy, 1996).
  25. D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” (NASA Jet Propulsion Laboratory, Pasadena, Calif., 1999).

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

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited