Abstract

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

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References

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1997 (2)

1995 (1)

1994 (2)

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]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

1993 (1)

1988 (1)

1981 (2)

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]

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).

1977 (1)

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]

1975 (1)

K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975).
[CrossRef]

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1973 (1)

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]

Boynton, G. C.

Bulgarelli, B.

B. Bulgarelli, “Radiative transfer in atmosphere and ocean,” Ph.D. dissertation (Politecnico di Torino, Torino, Italy, 1998).

Dale, H.

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).

Devaux, C.

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]

Fouquart, Y.

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]

Gentili, B.

Gordon, H. R.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Harringtion, R. F.

R. F. Harringtion, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.

Herman, M.

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]

Jayaweera, K.

Jin, Z.

Kattawar, G. W.

Kisselev, V.

Kisselev, V. B.

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]

Lenoble, J.

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]

Liou, K. N.

K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975).
[CrossRef]

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980).

Mobley, C. D.

Morel, A.

Perona, G.

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]

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]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected natural waters,” (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif., 1972).

Reinersman, P.

Roberti, L.

Smotky, O. I.

O. I. Smotky, Modeling of Radiation Fields in the Problems of Space Spectrophotometry (in Russian) (Nauka, Moscow, 1986), pp. 352–370.

Sobolev, V. V.

V. V. Sobolev, Scattering of Light in Planetary Atmosphere (Pergamon, New York, 1975).

Stamnes, K.

Stavn, R. H.

Swanson, R. A.

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]

Taylor, R. L.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Tsay, S.

Wiscombe, W.

Wiscombe, W. J.

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]

Zienkiewicz, O. C.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).

Appl. Opt. (6)

J. Atmos. Sci. (3)

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]

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).

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]

J. Geophys. Res. (1)

K. N. Liou, “Application of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres,” J. Geophys. Res. 80, 3434–3444 (1975).
[CrossRef]

J. Quant. Spectrosc. Radiant. Transfer (1)

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]

J. Quant. Spectrosc. Radiat. Transfer (1)

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]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Other (7)

R. F. Harringtion, Field Computation by Moment Methods (Macmillan, London, 1968), Chap. 1, pp. 1–9.

O. C. Zienkiewicz, R. L. Taylor, The Finite Element Method (McGraw-Hill, London, 1989).

V. V. Sobolev, Scattering of Light in Planetary Atmosphere (Pergamon, New York, 1975).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, Toronto, 1980).

B. Bulgarelli, “Radiative transfer in atmosphere and ocean,” Ph.D. dissertation (Politecnico di Torino, Torino, Italy, 1998).

T. J. Petzold, “Volume scattering functions for selected natural waters,” (Scripps Institution of Oceanography, Visibility Laboratory, San Diego, Calif., 1972).

O. I. Smotky, Modeling of Radiation Fields in the Problems of Space Spectrophotometry (in Russian) (Nauka, Moscow, 1986), pp. 352–370.

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