Abstract

Asymptotic expressions for the reflected, transmitted, and internal scattered radiation field in optically thick, vertically homogeneous, plane-parallel media are derived from first principles by using the discrete ordinates method of radiative transfer. Compact matrix equations are derived for computing the escape function, diffusion pattern, diffusion exponent, and the reflection function of a semi-infinite atmosphere in terms of the matrices, eigenvectors, and eigenvalues that occur in the discrete ordinates method. These matrix equations are suitable for numerical computations and are valid throughout the full range of single scattering albedos. The present formulations are validated by comparing them with established methods of radiative transfer.

© 1992 Optical Society of America

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  1. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), Chap. 2, pp. 180–188.
  2. T. A. Germogenova, “The nature of the solution of the transfer equation for a plane layer,” Zh. Vych. Mat. Mat. Fiz. 1, 1001–1019 (1961) [USSR Comput. Math. Math. Phys. 1, 1168–1186(1962)].
  3. H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).
  4. V. V. Sobolev, “Radiation diffusion in a medium of large optical thickness with nonisotropic scattering,” Dokl. Akad. Nauk SSSR 179, 41–44 (1968) [Sov. Phys. Dokl. 13, 180–182 (1968)].
  5. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975), Chaps. 2 and 3, pp. 24–29, 60–65.
  6. H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, Chaps. 4–6, pp. 37, 49, 70–87, 94–97.
  7. J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985), Chap. 1, pp. 75–82.
  8. M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
    [CrossRef]
  9. M. D. King. “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
    [CrossRef]
  10. T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
    [CrossRef]
  11. M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
    [CrossRef]
  12. T. A. Germogenova, N. V. Konovalov, “Asymptotic characteristics of the solution of the transport equation in the inhomogeneous layer problem,” Zh. Vych. Mat. Mat. Fiz. 14, 928–946 (1974) [USSR Comp. Math. Math. Phys. 14, 107–125 (1974).
  13. N. V. Konovalov, “Range of validity of asymptotic expressions for calculation of monochromatic radiation in an inhomogeneous optically thick plane-parallel layer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 11, 1263–1271 (1975) [Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 790–795 (1976)].
  14. P. Gabriel, S. Lovejoy, A. Davis, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer, 2, renormalization approach for homogeneous and fractal clouds,” J. Geophys. Res. 95, 11,717–11,728 (1990).
    [CrossRef]
  15. H. C. van de Hulst, “Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers,” J. Comput. Phys. 3, 291–306 (1968).
    [CrossRef]
  16. T. Duracz, N. J. McCormick, “Equations for estimating the similarity parameters from radiation measurements within weakly absorbing optically thick clouds,” J. Atmos. Sci. 43, 486–492 (1986).
    [CrossRef]
  17. H. C. Yi, N. J. McCormick, R. Sanchez, “Cloud optical thickness estimation from irradiance measurements,” J. Atmos. Sci. 47, 2567–2579 (1990).
    [CrossRef]
  18. M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
    [CrossRef]
  19. 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]
  20. T. Nakajima, M. Tanaka, “Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13–21 (1986).
    [CrossRef]
  21. K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. 24, 299–309 (1986).
    [CrossRef]
  22. 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]
  23. T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
    [CrossRef]
  24. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, pp. 20–23.
  25. J. V. Dave, B. H. Armstrong, “Computations of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
    [CrossRef]
  26. M. D. King, “Number of terms required in the Fourier expansion of the reflection function of optically thick atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 30, 143–161 (1983).
    [CrossRef]
  27. K. Stamnes, S. C. Tsay, T. Nakajima, “Computation of eigenvalues and eigenvectors for the discrete ordinate and matrix operator methods in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 39, 415–419 (1988).
    [CrossRef]
  28. G. Yamamoto, M. Tanaka, S. Asano, “Radiative heat transfer in water clouds by infrared radiation,” J. Quant. Spectrosc. Radiat. Transfer 11, 697–708 (1971).
    [CrossRef]
  29. R. D. M. Garcia, C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 42, 385–394 (1989).
    [CrossRef]
  30. K. Stamnes, P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 31, 273–282 (1984).
    [CrossRef]
  31. G. N. Plass, G. W. Kattawar, F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314–329 (1973).
    [CrossRef] [PubMed]
  32. 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, 2696–2706 (1981).
    [CrossRef]
  33. W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
    [CrossRef]
  34. W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637–658 (1976).
    [CrossRef]

1990

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

P. Gabriel, S. Lovejoy, A. Davis, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer, 2, renormalization approach for homogeneous and fractal clouds,” J. Geophys. Res. 95, 11,717–11,728 (1990).
[CrossRef]

H. C. Yi, N. J. McCormick, R. Sanchez, “Cloud optical thickness estimation from irradiance measurements,” J. Atmos. Sci. 47, 2567–2579 (1990).
[CrossRef]

1989

R. D. M. Garcia, C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 42, 385–394 (1989).
[CrossRef]

1988

K. Stamnes, S. C. Tsay, T. Nakajima, “Computation of eigenvalues and eigenvectors for the discrete ordinate and matrix operator methods in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 39, 415–419 (1988).
[CrossRef]

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]

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

1987

M. D. King. “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

1986

T. Nakajima, M. Tanaka, “Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13–21 (1986).
[CrossRef]

K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. 24, 299–309 (1986).
[CrossRef]

M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

T. Duracz, N. J. McCormick, “Equations for estimating the similarity parameters from radiation measurements within weakly absorbing optically thick clouds,” J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

1984

K. Stamnes, P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 31, 273–282 (1984).
[CrossRef]

1983

M. D. King, “Number of terms required in the Fourier expansion of the reflection function of optically thick atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 30, 143–161 (1983).
[CrossRef]

1981

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, 2696–2706 (1981).
[CrossRef]

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]

M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

1977

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

1976

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637–658 (1976).
[CrossRef]

1975

N. V. Konovalov, “Range of validity of asymptotic expressions for calculation of monochromatic radiation in an inhomogeneous optically thick plane-parallel layer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 11, 1263–1271 (1975) [Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 790–795 (1976)].

1974

T. A. Germogenova, N. V. Konovalov, “Asymptotic characteristics of the solution of the transport equation in the inhomogeneous layer problem,” Zh. Vych. Mat. Mat. Fiz. 14, 928–946 (1974) [USSR Comp. Math. Math. Phys. 14, 107–125 (1974).

1973

1971

G. Yamamoto, M. Tanaka, S. Asano, “Radiative heat transfer in water clouds by infrared radiation,” J. Quant. Spectrosc. Radiat. Transfer 11, 697–708 (1971).
[CrossRef]

1970

J. V. Dave, B. H. Armstrong, “Computations of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
[CrossRef]

1968

H. C. van de Hulst, “Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers,” J. Comput. Phys. 3, 291–306 (1968).
[CrossRef]

H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).

V. V. Sobolev, “Radiation diffusion in a medium of large optical thickness with nonisotropic scattering,” Dokl. Akad. Nauk SSSR 179, 41–44 (1968) [Sov. Phys. Dokl. 13, 180–182 (1968)].

1961

T. A. Germogenova, “The nature of the solution of the transfer equation for a plane layer,” Zh. Vych. Mat. Mat. Fiz. 1, 1001–1019 (1961) [USSR Comput. Math. Math. Phys. 1, 1168–1186(1962)].

Armstrong, B. H.

J. V. Dave, B. H. Armstrong, “Computations of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
[CrossRef]

Asano, S.

G. Yamamoto, M. Tanaka, S. Asano, “Radiative heat transfer in water clouds by infrared radiation,” J. Quant. Spectrosc. Radiat. Transfer 11, 697–708 (1971).
[CrossRef]

Austin, G. L.

P. Gabriel, S. Lovejoy, A. Davis, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer, 2, renormalization approach for homogeneous and fractal clouds,” J. Geophys. Res. 95, 11,717–11,728 (1990).
[CrossRef]

Catchings, F. E.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 1, pp. 20–23.

Conklin, P.

K. Stamnes, P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 31, 273–282 (1984).
[CrossRef]

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, 2696–2706 (1981).
[CrossRef]

Dave, J. V.

J. V. Dave, B. H. Armstrong, “Computations of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
[CrossRef]

Davis, A.

P. Gabriel, S. Lovejoy, A. Davis, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer, 2, renormalization approach for homogeneous and fractal clouds,” J. Geophys. Res. 95, 11,717–11,728 (1990).
[CrossRef]

Duracz, T.

T. Duracz, N. J. McCormick, “Equations for estimating the similarity parameters from radiation measurements within weakly absorbing optically thick clouds,” J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), Chap. 2, pp. 180–188.

Gabriel, P.

P. Gabriel, S. Lovejoy, A. Davis, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer, 2, renormalization approach for homogeneous and fractal clouds,” J. Geophys. Res. 95, 11,717–11,728 (1990).
[CrossRef]

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 42, 385–394 (1989).
[CrossRef]

Germogenova, T. A.

T. A. Germogenova, N. V. Konovalov, “Asymptotic characteristics of the solution of the transport equation in the inhomogeneous layer problem,” Zh. Vych. Mat. Mat. Fiz. 14, 928–946 (1974) [USSR Comp. Math. Math. Phys. 14, 107–125 (1974).

T. A. Germogenova, “The nature of the solution of the transfer equation for a plane layer,” Zh. Vych. Mat. Mat. Fiz. 1, 1001–1019 (1961) [USSR Comput. Math. Math. Phys. 1, 1168–1186(1962)].

Hobbs, P. V.

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

Jayaweera, K.

Kattawar, G. W.

King, M. D.

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

M. D. King. “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

M. D. King, “Number of terms required in the Fourier expansion of the reflection function of optically thick atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 30, 143–161 (1983).
[CrossRef]

M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

King Harshvardhan, M. D.

M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

Konovalov, N. V.

N. V. Konovalov, “Range of validity of asymptotic expressions for calculation of monochromatic radiation in an inhomogeneous optically thick plane-parallel layer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 11, 1263–1271 (1975) [Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 790–795 (1976)].

T. A. Germogenova, N. V. Konovalov, “Asymptotic characteristics of the solution of the transport equation in the inhomogeneous layer problem,” Zh. Vych. Mat. Mat. Fiz. 14, 928–946 (1974) [USSR Comp. Math. Math. Phys. 14, 107–125 (1974).

Lovejoy, S.

P. Gabriel, S. Lovejoy, A. Davis, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer, 2, renormalization approach for homogeneous and fractal clouds,” J. Geophys. Res. 95, 11,717–11,728 (1990).
[CrossRef]

McCormick, N. J.

H. C. Yi, N. J. McCormick, R. Sanchez, “Cloud optical thickness estimation from irradiance measurements,” J. Atmos. Sci. 47, 2567–2579 (1990).
[CrossRef]

T. Duracz, N. J. McCormick, “Equations for estimating the similarity parameters from radiation measurements within weakly absorbing optically thick clouds,” J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), Chap. 2, pp. 180–188.

Nakajima, T.

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

K. Stamnes, S. C. Tsay, T. Nakajima, “Computation of eigenvalues and eigenvectors for the discrete ordinate and matrix operator methods in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 39, 415–419 (1988).
[CrossRef]

T. Nakajima, M. Tanaka, “Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13–21 (1986).
[CrossRef]

Plass, G. N.

Radke, L. F.

M. D. King, L. F. Radke, P. V. Hobbs, “Determination of the spectral absorption of solar radiation by marine stratocumulus clouds from airborne measurements within clouds,” J. Atmos. Sci. 47, 894–907 (1990).
[CrossRef]

Sanchez, R.

H. C. Yi, N. J. McCormick, R. Sanchez, “Cloud optical thickness estimation from irradiance measurements,” J. Atmos. Sci. 47, 2567–2579 (1990).
[CrossRef]

Schertzer, D.

P. Gabriel, S. Lovejoy, A. Davis, D. Schertzer, G. L. Austin, “Discrete angle radiative transfer, 2, renormalization approach for homogeneous and fractal clouds,” J. Geophys. Res. 95, 11,717–11,728 (1990).
[CrossRef]

Siewert, C. E.

R. D. M. Garcia, C. E. Siewert, “On discrete spectrum calculations in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 42, 385–394 (1989).
[CrossRef]

Sobolev, V. V.

V. V. Sobolev, “Radiation diffusion in a medium of large optical thickness with nonisotropic scattering,” Dokl. Akad. Nauk SSSR 179, 41–44 (1968) [Sov. Phys. Dokl. 13, 180–182 (1968)].

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975), Chaps. 2 and 3, pp. 24–29, 60–65.

Stamnes, K.

K. Stamnes, S. C. Tsay, T. Nakajima, “Computation of eigenvalues and eigenvectors for the discrete ordinate and matrix operator methods in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 39, 415–419 (1988).
[CrossRef]

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]

K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. 24, 299–309 (1986).
[CrossRef]

K. Stamnes, P. Conklin, “A new multi-layer discrete ordinate approach to radiative transfer in vertically inhomogeneous atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 31, 273–282 (1984).
[CrossRef]

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, 2696–2706 (1981).
[CrossRef]

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]

Tanaka, M.

T. Nakajima, M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

T. Nakajima, M. Tanaka, “Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 35, 13–21 (1986).
[CrossRef]

G. Yamamoto, M. Tanaka, S. Asano, “Radiative heat transfer in water clouds by infrared radiation,” J. Quant. Spectrosc. Radiat. Transfer 11, 697–708 (1971).
[CrossRef]

Tsay, S. C.

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]

K. Stamnes, S. C. Tsay, T. Nakajima, “Computation of eigenvalues and eigenvectors for the discrete ordinate and matrix operator methods in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 39, 415–419 (1988).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).

H. C. van de Hulst, “Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers,” J. Comput. Phys. 3, 291–306 (1968).
[CrossRef]

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vol. 1, Chaps. 4–6, pp. 37, 49, 70–87, 94–97.

Wiscombe, W.

Wiscombe, W. J.

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637–658 (1976).
[CrossRef]

Yamamoto, G.

G. Yamamoto, M. Tanaka, S. Asano, “Radiative heat transfer in water clouds by infrared radiation,” J. Quant. Spectrosc. Radiat. Transfer 11, 697–708 (1971).
[CrossRef]

Yi, H. C.

H. C. Yi, N. J. McCormick, R. Sanchez, “Cloud optical thickness estimation from irradiance measurements,” J. Atmos. Sci. 47, 2567–2579 (1990).
[CrossRef]

Appl. Opt.

Bull. Astron. Inst. Neth.

H. C. van de Hulst, “Radiative transfer in thick atmospheres with an arbitrary scattering function,” Bull. Astron. Inst. Neth. 20, 77–86 (1968).

Dokl. Akad. Nauk SSSR

V. V. Sobolev, “Radiation diffusion in a medium of large optical thickness with nonisotropic scattering,” Dokl. Akad. Nauk SSSR 179, 41–44 (1968) [Sov. Phys. Dokl. 13, 180–182 (1968)].

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana

N. V. Konovalov, “Range of validity of asymptotic expressions for calculation of monochromatic radiation in an inhomogeneous optically thick plane-parallel layer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 11, 1263–1271 (1975) [Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 790–795 (1976)].

J. Atmos Sci.

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]

J. Atmos. Sci.

T. Duracz, N. J. McCormick, “Equations for estimating the similarity parameters from radiation measurements within weakly absorbing optically thick clouds,” J. Atmos. Sci. 43, 486–492 (1986).
[CrossRef]

H. C. Yi, N. J. McCormick, R. Sanchez, “Cloud optical thickness estimation from irradiance measurements,” J. Atmos. Sci. 47, 2567–2579 (1990).
[CrossRef]

M. D. King Harshvardhan, “Comparative accuracy of selected multiple scattering approximations,” J. Atmos. Sci. 43, 784–801 (1986).
[CrossRef]

M. D. King, “A method for determining the single scattering albedo of clouds through observation of the internal scattered radiation field,” J. Atmos. Sci. 38, 2031–2044 (1981).
[CrossRef]

M. D. King. “Determination of the scaled optical thickness of clouds from reflected solar radiation measurements,” J. Atmos. Sci. 44, 1734–1751 (1987).
[CrossRef]

T. Nakajima, M. D. King, “Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: theory,” J. Atmos. Sci. 47, 1878–1893 (1990).
[CrossRef]

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Tables (7)

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Table 1 Minimum Eigenvalues for Several Fourier Frequencies and for Various Values of the Single Scattering Albedo ω0a

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Table 2 Diffusion Exponent k Derived by Several Different Methodsa

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Table 3 Asymptotic Constants I, m and n Derived by Several Different Methodsa

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Table 4 Escape Function K(μ) Derived by Several Different Methodsa

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Table 5 Diffusion Pattern P(μ) Derived by Several Different Methodsa

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Table 6 Plane Albedo of a Semi-infinite Atmosphere Derived by Several Different Methodsa

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Table 7 Reflection Function of a Semi-Infinite Atmosphere with Normal Incidence (μ0 = 1) Derived by Several Different Methodsa

Equations (127)

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u ( 0 ; - μ , μ 0 , ϕ ) = u ( - μ , μ 0 , ϕ ) - m l exp ( - 2 k τ c ) 1 - l 2 exp ( - 2 k τ c ) × K ( μ ) K ( μ 0 ) μ 0 F 0 π ,
u ( τ c ; + μ , μ 0 , ϕ ) = m exp ( - k τ c ) 1 - l 2 exp ( - 2 k τ c ) × K ( μ ) K ( μ 0 ) μ 0 F 0 π .
u ( τ ; ± μ , μ 0 , ϕ ) = exp ( - k τ ) 1 - l 2 exp ( - 2 k τ c ) × { P ( ± μ ) - l exp [ - 2 k ( τ c - τ ) P ( μ ) ] } K ( μ 0 ) μ 0 F 0 π ,
μ d u ( τ ; μ , μ 0 , ϕ ) d τ = - u ( τ ; μ , μ 0 , ϕ ) + ω 0 4 π - 1 1 0 2 π Φ ( μ , ϕ ; μ , ϕ ) u ( τ ; μ , μ 0 , ϕ ) d ϕ d μ + ω 0 4 π Φ ( μ , ϕ ; μ 0 , ϕ 0 ) F 0 exp ( - τ / μ 0 ) ,
1 4 π - 1 1 0 2 π Φ ( μ , ϕ ; μ , ϕ ) d ϕ d μ = 1.
ω 0 Φ ( cos Θ ) = l = 0 L ω l P l ( cos Θ ) ,
ω 0 Φ ( μ , ϕ ; μ , ϕ ) = h 0 ( μ , μ ) + 2 m = 1 L h m ( μ , μ ) cos m ( ϕ - ϕ ) ,
h m ( μ , μ ) = l = m L ω l Y l m ( μ ) Y l m ( μ ) ,
Y l m ( μ ) = [ ( l - m ) ! ( l + m ) ! ] 1 / 2 P l m ( μ ) .
u ( τ ; μ , μ 0 , ϕ ) = u 0 ( τ ; μ , μ 0 ) + 2 m = 1 L u m ( τ ; μ , μ 0 ) cos m ( ϕ - ϕ 0 ) ,
μ d u m ( τ ; μ , μ 0 ) d τ = - u m ( τ ; μ , μ 0 ) + ½ - 1 1 h m ( μ , μ ) μ m ( τ ; μ , μ 0 ) d μ + h m ( μ , μ 0 ) 4 π F 0 exp ( - τ / μ 0 ) .
± μ i d u m ( τ ; ± μ i , μ 0 j ) d τ = - u m ( τ ; ± μ i , μ 0 j ) + h m ( ± μ i , μ 0 j ) 4 π F 0 exp ( - τ / μ 0 j ) + ½ n = 1 N [ h m ( ± μ i , μ n ) u m ( τ ; μ n , μ 0 j ) + h m ( ± μ i , - μ n ) u m ( τ ; - μ n , μ 0 j ) ] w n
± M d u ± ( τ ) d τ = - u ± ( τ ) + h ± Wu ± ( τ ) + h Wu - ( τ ) + S ± E 0 ( τ ) ,
u ± ( τ ) = [ u m ( τ ; ± μ i , μ 0 j ) ] , i = 1 , , N , j = 1 , , M ; h ± = [ ½ h m ( ± μ i , μ j ) ] , i , j = 1 , , N ; S ± = [ F 0 4 π h m ( ± μ i , μ i ) ] , i = 1 , , N , j = 1 , , M ; M = [ μ i δ i j ] , i , j = 1 , , N ; W = [ w i δ i j ] , i , j = 1 , , N ; E 0 ( τ ) = [ exp ( - τ / μ 0 j ) δ i j ] , i , j = 1 , , M .
u ^ ± ( τ ) = W + u ± ( τ ) ,
W + = WM ,             W - = W M - 1 .
± d u ^ ± ( τ ) d τ = - M - 1 u ^ ± ( τ ) + h ^ ± u ^ ± ( τ ) + h ^ u ^ - ( τ ) + S ^ ± E 0 ( τ ) ,
h ^ ± = W - h ± W - ,             S ^ ± = W - S ± .
d ψ ^ ( τ ) d τ = - X ± ψ ^ ± ( τ ) + σ ^ ± E 0 ( τ ) ,
X ± = M - 1 - W - ( h + ± h - ) W - ,
σ ^ ± = W - ( S + ± S - ) .
d 2 ψ ^ + ( τ ) d τ 2 = G ψ ^ + ( τ ) + gE 0 ( τ ) ,
G = X - X + ,
g = - X - σ ^ + - σ ^ - M 0 - 1 ,
M 0 = [ μ 0 j δ i j ] ,             i , j = 1 , , M .
G = QL 2 Q - 1 ,
L = [ λ i δ i j ] ,             i , j = 1 , , N ,
C ( τ ) = ( ½ { exp [ - λ i ( τ c - τ ) ] + exp ( - λ i τ ) } δ i j ) ,             i , j = 1 , , N ,
S ( τ ) = ( ½ { exp [ - λ i ( τ c - τ ) ] - exp ( - λ i τ ) } δ i j ) ,             i , j = 1 , , N ,
γ = [ ( Q - 1 g ) i j 1 μ 0 j 2 - λ i 2 ] ,             i = 1 , , N ; j = 1 , , M .
u ^ ± ( τ ) = A ± ( τ ) α + B ± ( τ ) β + V ± E 0 ( τ ) ,
A ± ( τ ) = QC ( τ ) Q ˜ LS ( τ ) ,
B ± ( τ ) = QL - 1 S ( τ ) Q ˜ C ( τ ) ,
V ± = ½ [ Q γ ± Q ˜ γ M 0 - 1 ± ( X - ) - 1 σ ^ - ] ,
( α β ) = 1 2 [ ( A - ) - 1 ( A - ) - 1 - ( B - ) - 1 ( B - ) - 1 ] [ u ^ + ( 0 ) - V + u ^ - ( τ c ) - V - E 0 ( τ c ) ] ,
A ± = A ± ( τ c ) = A ( 0 ) ,
B ± = B ± ( τ c ) = - B ( 0 ) .
R ^ = ½ [ A + ( A - ) - 1 + B + ( B - ) - 1 ] ,
T ^ = ½ [ A + ( A - ) - 1 - B + ( B - ) - 1 ] ,
[ u ^ - ( 0 ) u ^ + ( τ c ) ] = ( R ^ T ^ T ^ R ^ ) [ u ^ + ( 0 ) - V + u ^ - ( τ c ) - V - E 0 ( τ c ) ] + ( V - V + E 0 ( τ c ) ) ,
A + ( A - ) - 1 = 2 QC ( τ c ) ( A - ) - 1 - I , = 2 [ I + Q ˜ S ( τ c ) LC ( τ c ) - 1 Q ˜ T ] - 1 - I , = ( A - Q ˜ a + Q ˜ T ) - 1 - I , = A - 1 ( I - Q ˜ a + Q ˜ T A - 1 ) - 1 - I , = A - 1 - I + A - 1 Q ˜ ( I - a + q ) - 1 a + Q ˜ T A - 1 ,
B + ( B - ) - 1 = A - 1 - I + A - 1 Q ˜ ( I - a - q ) - 1 a - Q ˜ T A - 1 ,
A = ½ ( I + Q ˜ L Q ˜ T ) ,
q = Q ˜ T A - 1 Q ˜
a + = ½ [ L - S ( τ c ) LC ( τ c ) - 1 ] ,
a - = ½ [ L - C ( τ c ) LS ( τ c ) - 1 ] .
( AB ) - 1 = B - 1 A - 1 ,             ( I - B ) - 1 = I + B + B 2 + ,
a i j ± { ± k exp ( - k τ c ) 1 ± exp ( - k τ c ) , if i = j = N 0 , otherwise .
[ ( I - a ± q ) - 1 a ± ] i j { ± k exp ( - k τ c ) 1 ± ( 1 - k q ) exp ( - k τ c ) , if i = j = N 0 , otherwise .
q = q N N .
R ^ = R ^ - k l exp ( - 2 k τ c ) 1 - l 2 exp ( - 2 k τ c ) K ^ * K ^ T ,
T ^ = k exp ( - k τ c ) 1 - l 2 exp ( - 2 k τ c ) K ^ * K ^ T ,
R ^ = A - 1 - I ,
l = 1 - k q ,
K ^ = A - 1 Q ˜ N ,
K ^ * K ^ T = { K ^ i K ^ j T } ,
R ^ = R ^ - T ^ ,
T ^ = 1 2 ( τ c + q ) K ^ * K ^ T .
u ^ + ( τ ) = ( I - R ^ a R ^ b ) - 1 T ^ a u ^ + ( 0 ) ,
u ^ - ( τ ) = R ^ b u ^ + ( τ ) ,
u ^ + ( τ ) = k exp ( - k τ ) 1 - l 2 exp ( - 2 k τ ) × ( I - R ^ a R ^ b ) - 1 K ^ * K ^ T u ^ + ( 0 ) .
v ( τ ) = ( I - R ^ a R ^ b ) - 1 K ^ , = [ ( 1 - c 1 + c 3 ) - c 2 R ^ ] ( I - R ^ 2 ) - 1 K ^ ,
c 1 = γ a K ^ T R ^ v ( τ ) ,
c 2 = γ b K ^ T v ( τ ) ,
c 3 = γ a γ b K ^ T K ^ * K ^ T v ( τ ) ,
γ a = k l exp ( - 2 k τ ) 1 - l 2 exp ( - 2 k τ ) ,
γ b = k l exp [ - 2 k ( τ c - τ ) ] 1 - l 2 exp [ - 2 k ( τ c - τ ) ] .
P ^ + = k ( I - R ^ 2 ) - 1 K ^ ,
P ^ - = R ^ P ^ + ,
P ^ + = k A ( 2 A - I ) - 1 Q ˜ N , = k 2 [ ( Q ˜ L Q ˜ T ) - 1 + I ] Q ˜ N , = ½ ( Q N + k Q ˜ N ) ,
P ^ - = k ( I - I + R ^ ) ( I - R ^ 2 ) - 1 K ^ , = P ^ + - k ( I + R ^ ) - 1 K ^ , = P ^ + - k Q ˜ N .
K ^ T P ^ + = k Q ˜ N T ( 2 A - I ) - 1 Q ˜ N , = k Q ˜ N T ( Q ˜ L Q ˜ T ) - 1 Q ˜ N , = 1 ,
K ^ T P ^ - = K ^ T ( P ^ + - k Q ˜ N ) , = 1 - k q , = l .
( I - R ^ a R ^ b ) - 1 K ^ = 1 - l 2 exp ( - 2 k τ ) k - k l 2 exp ( - 2 k τ c ) × { P ^ + - l exp [ - 2 k ( τ c - τ ) ] P ^ - } .
u ^ ± ( τ ) = exp ( - k τ ) 1 - l 2 exp ( - 2 k τ c ) × { P ^ ± - l exp [ - 2 k ( τ c - τ ) ] P ^ ± } * K ^ T u ^ + ( 0 ) .
u ( 0 ; - μ , μ 0 , ϕ ) = 1 π 0 2 π 0 1 R ( τ c ; μ , ϕ ; μ , ϕ ) × u ( 0 ; μ , μ 0 , ϕ ) μ d μ d ϕ ,
u m ( 0 ; - μ , μ 0 ) = 2 0 1 R m ( τ c ; μ , μ ) u m ( 0 ; μ , μ 0 ) μ d μ ,
u - ( 0 ) = 2 RWM u + ( 0 ) .
R ^ = 2 W + R W + ,
R = ½ ( W + ) - 1 R ^ ( W + ) - 1 .
u + ( 0 ) = [ F 0 2 π w i δ i j ] ,             i = 1 , , N ; j = 1 , , M ,
R = ½ ( W + ) - 1 R ^ ( W + ) - 1 ,
K = k 2 m ( W + ) - 1 K ^ ,
P ± = m 2 k ( W + ) - 1 P ^ ± .
1 2 - 1 1 P ( μ ) d μ = 1 2 m 2 k 1 T W ( W + ) - 1 ( P ^ + - P ^ - ) , = 1 2 m 2 k 1 T W - Q N , = 1 ,
2 0 1 K ( μ ) P ( μ ) μ d μ = 1 ,
2 0 1 K ( μ ) P ( - μ ) μ d μ = l ,
2 - 1 1 [ P ( μ ) ] 2 μ d μ = m .
2 0 1 K ( μ ) μ d μ = n .
u ^ + ( τ c ) = - T ^ V + - R ^ V - E 0 ( τ c ) + V + E 0 ( τ c ) ,
u ^ - ( 0 ) = - R ^ V + - T ^ V - E 0 ( τ c ) + V - .
K 0 = - 2 π F 0 k 2 m M 0 - 1 ( V + ) T K ^ ,
P ( μ ) = l = 0 L + 1 ( 2 l + 1 ) g l P l ( μ ) ,
( l + 1 ) g l + 1 - ( 2 l + 1 - ω l ) k - 1 g l + l g l - 1 = 0 ,
R ( μ , μ 0 , ϕ ) = π u ( - μ , μ 0 , ϕ ) μ 0 F 0 .
u - ( 0 ) = lim τ c M - 1 0 τ c E ( τ ) J - ( τ ) d τ ,
J - ( τ ) = h - Wu + ( τ ) + h + Wu - ( τ ) + S - E 0 ( τ ) .
J - ( τ ) = [ HC ( τ ) + H ˜ LS ( τ ) ] α + [ HL - 1 S ( τ ) + H ˜ C ( τ ) ] β + J 0 - E 0 ( τ ) ,
H = ( h + + h - ) W - Q ,
H ˜ = ( h + - h - ) W - Q ˜ ,
J 0 - = h - W - V + + h + W - V - + S - .
u - ( 0 ) = ½ ( - H + H ˜ L ) ( M , L ) Q ˜ T A - 1 V + + J 0 - ( M , M 0 ) M 0 ,
H ( M , L ) = [ H i j μ i + λ j ] ,             i , j = 1 , , N ,
J 0 - ( M , M 0 ) = [ J 0 i j - μ i + μ 0 j ] , i = 1 , , N ; j = 1 , , M .
ω l * = ω l - ω 0 f ( 2 l + 1 ) 1 - f ,             l = 0 , , L ,
f = ω 2 N ω 0 ( 4 N + 1 ) .
k = ( 1 - ω 0 f ) k truncated .
q = q truncated / ( 1 - ω 0 f ) .
1 - ω 0 = k 2 a 1 - 4 k 2 a 2 - 9 k 2 a 3 - ,
a l = ( 2 l + 1 - ω l ) .
r ( μ 0 ) = 1 μ 0 F 0 0 2 π 0 1 u ( - μ , μ 0 , ϕ ) μ d μ d ϕ , = 2 0 1 R 0 ( μ , μ 0 ) μ d μ .
v ( τ ) = 1 k [ ( 1 - c 1 + c 3 ) P ^ + - c 2 P ^ - ] .
c 1 = γ a k K ^ T R ^ [ ( 1 - c 1 + c 3 ) P ^ + - c 2 P ^ - ] ,
c 2 = γ b k K ^ T [ ( 1 - c 1 + c 3 ) P ^ + - c 2 P ^ - ] ,
c 3 = γ a γ b k K ^ T K ^ * K ^ T [ ( 1 - c 1 + c 3 ) P ^ + - c 2 P ^ - ] .
K ^ T R ^ P ^ + = K ^ T P ^ - = l ,
K ^ T R ^ P ^ - = K ^ T R ^ 2 P ^ + , = K ^ T [ P ^ + - ( I - R ^ 2 ) P ^ + ] = 1 - k K ^ T K ^ .
c 1 = γ a k [ ( 1 - c 1 + c 3 ) l - c 2 ( 1 - k K ^ T K ^ ) ] ,
c 2 = γ b k ( 1 - c 1 + c 3 - c 2 l ) ,
c 3 = γ a c 2 K ^ T K ^ ,
c 2 = ( 1 - c 1 + c 3 ) l exp [ - 2 k ( τ c - τ ) ] ,
1 - c 1 + c 3 = 1 - l 2 exp ( - 2 k τ ) 1 - l 2 exp ( - 2 k τ c ) .
v ( τ ) = 1 - l 2 exp ( - 2 k τ ) k - k l 2 exp ( - 2 k τ c ) × { P ^ + - l exp [ - 2 k ( τ c - τ ) ] P ^ - } .
u ^ + ( τ ) = exp ( - k τ ) 1 - l 2 exp ( - 2 k τ c ) × { P ^ + - l exp [ - 2 k ( τ c - τ ) ] P ^ - } * K ^ T u ^ + ( 0 ) .
R ^ b { P ^ + - l exp [ - 2 k ( τ c - τ ) ] P ^ - } = ( R ^ - γ b K ^ * K ^ T ) × { P ^ + - l exp [ - 2 k ( τ c - τ ) ] P ^ - } , = P ^ - - l exp [ - 2 k ( τ c - τ ) ] ( P ^ + - k K ^ ) - γ b K ^ * K ^ T { P ^ + - l exp [ - 2 k ( τ c - τ ) ] P ^ - } ,
γ b K ^ * K ^ T { P ^ + - l exp [ - 2 k ( τ c - τ ) ] P ^ - } = k l exp [ - 2 k ( τ c - τ ) ] K ^ ,
u ^ - ( τ ) = R ^ b u ^ + ( τ ) , = exp ( - k τ ) 1 - l 2 exp ( - 2 k τ c ) × { P ^ - - l exp [ - 2 k ( τ c - τ ) ] P ^ + } * K ^ T u ^ + ( 0 ) .

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