Abstract

Advantages and disadvantages of modern discrete-ordinates finite-element methods for the solution of radiative transfer problems in meteorology, climatology, and remote sensing applications are evaluated. After the common basis of the formulation of radiative transfer problems in the fields of neutron transport and atmospheric optics is established, the essential features of the discrete-ordinates finite-element method are described including the limitations of the method and their remedies. Numerical results are presented for 1-D and 2-D atmospheric radiative transfer problems where integral as well as angular dependent quantities are compared with published results from other calculations and with measured data. These comparisons provide a verification of the discrete-ordinates results for a wide spectrum of cases with varying degrees of absorption, scattering, and anisotropic phase functions. Accuracy and computational speed are also discussed. Since practically all discrete-ordinates codes offer a builtin adjoint capability, the general concept of the adjoint method is described and illustrated by sample problems. Our general conclusion is that the strengths of the discrete-ordinates finite-element method outweight its weaknesses. We demonstrate that existing general-purpose discrete-ordinates codes can provide a powerful tool to analyze radiative transfer problems through the atmosphere, especially when 2-D geometries must be considered.

© 1985 Optical Society of America

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  2. E. P. Shettle, J. A. Weinman, “The Transfer of Solar Irradiance Through Inhomogeneous Turbid Atmospheres Evaluated by Eddington's Approximation,” J. Atmos. Sci. 27, 1048 (1970).
    [CrossRef]
  3. R. E. Samuelson, “Non-Local Thermodynamic Equilibrium in Cloudy Planetary Atmospheres,” J. Atmos. Sci. 27, 711 (1970).
    [CrossRef]
  4. K. N. Liou, “Numerical Experiments with Chandrasekhar's Discrete-Ordinate Method,” J. Atmos. Sci. 30, 1303 (1973).
    [CrossRef]
  5. K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, New York, 1980).
  6. S. H. Schneider, R. E. Dickinson, “Climate Modelling,” Rev. Geophys. Space Phys. 12, 447 (1974).
    [CrossRef]
  7. V. Ramanathan, J. A. Coakley, “Title,” Rev. Geophys. Space Phys. 16, 465 (1978).
    [CrossRef]
  8. J. V. Dave, “A Direct Solution of the Spherical-Harmonics Approximation to the Transfer Equation for a Plane-Parallel, Nonhomogeneous Atmosphere,” Lawrence Livermore Laboratory Report UCRL-51581 (1974).
  9. J. V. Dave, “Extensive Datasets of the Diffuse Radiation in Realistic Atmospheric Models with Aerosols and Common Absorbing Gases,” IBM Palo Alto Scientific Center Report G320-3366 (Mar.1978).
  10. W. J. Wiscombe, “On Initialization Errors and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637 (1976).
    [CrossRef]
  11. J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
    [CrossRef]
  12. G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).
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  16. A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Application of the 2-D Discrete-Ordinates Method to Multiple Scattering of Laser Radiation,” Appl. Opt. 22, 1346 (1983).
    [CrossRef] [PubMed]
  17. S. A. W. Gerstl, A. Zardecki, “Coupled Atmosphere/Canopy Model for Remote Sensing of Plant Reflectance Features,” Appl. Opt. 23, 0000 (1984), same issue.
  18. A. Zardecki, S. A. W. Gerstl, “Calculations of Solar Irradiances in Clear and Polluted Atmospheres and Potential Effects on Plant Life,” Los Alamos National Laboratory Report LA-9010-MS (Oct.1981).
  19. B. G. Carlson, K. D. Lathrop, “Transport Theory, The Method of Discrete Ordinates,” in Computing Methods of Reactor Physics, H. Greenspan, C. N. Kelber, D. Okrent, Eds. (Gordon & Breach, New York, 1968), Chap. 3, pp. 171–261.
  20. K. D. Lathrop, “Discrete-Ordinates Methods for the Numerical Solution of the Transport Equation,” Reactor Technol. 15, 107 (1972).
  21. K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45 (3), 255 (1971).
  22. A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Multiple Scattering Effects in Spatial Frequency Filtering,” Appl. Opt. 23, 4124 (1984).
    [CrossRef] [PubMed]
  23. D. R. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for ONEDANT: A Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).
  24. T. R. Hill, “onetran: A Discrete-Ordinates Finite-Element Code for the Solution of the One-Dimensional Multigroup Transport Equation,” Los Alamos National Laboratory Report LA-5990-MS (June1975).
  25. R. E. Alcouffe, F. W. Brinkley, D. R. Marr, R. D. O'Dell, “User's Guide for twodant: A Code Package for Two-Dimensional Diffusion-Accelerated Neutral Particle Transport,” Los Alamos National Laboratory Report LA-10049-M (Mar.1984).
  26. K. D. Lathrop, F. W. Brinkley, “twotran-II: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).
  27. W. W. Engle, “A User's Manual for anisn, a One-Dimensional Discrete-Ordinates Transport Code with Anisotropic Scattering,” Oak Ridge National Laboratory, Union Carbide Corp. Report K-1693 (Mar.1967).
  28. W. A. Rhoades, F. R. Mynatt, “The DOT-III Two-Dimensional Discrete-Ordinates Transport Code,” Oak Ridge National Laboratory Report ORNL-TM-4280 (Sept.1973).
  29. “RSIC Computer Code and Data Collection: A Capsule Review of the Computer Code Collection (CCC), Peripheral Shielding Routines (RSR), and Data Library Collection (DLC) Packaged by the Radiation Shielding Information Center,” Informal Report, revisedNov.1983;available from RSIC, P.O. Box X, Oak Ridge, Tenn. 37831.
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  31. E. P. Shettle, A. E. S. Green, “Multiple Scattering Calculation of the Middle Ultraviolet Reaching the Ground,” Appl. Opt. 13, 1567 (1974).
    [CrossRef] [PubMed]
  32. H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980), Tables 35 of Vol. 2.
  33. L. T. Chan, “Results of Telephotometer Measurements,” Los Alamos Scientific Laboratory memorandum to A. Zardecki (20Nov., 1980).
  34. R. W. Bergstrom, B. L. Bobson, T. P. Ackerman, “Calculation of Multiple Scattered Radiation in Clean Atmospheres,” in Symposium on Plumes and Visibility, Grand Canyon, Colo. (Nov. 1980).
  35. T. Mekler, Y. J. Kaufman, “The Effect of the Atmosphere on Contrast Reduction for a Non-Uniform Surface Albedo and “Two-Halves” Field,” J. Geophys. Res. 85, 4067 (1980).
    [CrossRef]
  36. J. Otterman, R. S. Fraser, “Adjacency Effects on Imaging by Surface Reflection and Atmospheric Scattering Cross Radiance to Zenith,” Appl. Opt. 18, 2852 (1979).
    [CrossRef] [PubMed]
  37. Y. J. Kaufman, “Atmospheric Effects on Remote Sensing of Surface Reflectance,” in Proceedings, SPIE Conference 475 on Critical Reviews of Remote Sensing, Washington, D.C., 1–2 May, 1984.
  38. W. A. Pierce, “A Study of the Effects of the Atmosphere on Thematic Mapper Observations,” EG&G Report 004-77, Washington Analysis Service Center, Riverdale, Md. (1977).
  39. E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).
  40. R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program Part I: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memorandum 86078, Goddard Space Flight Center, Greenbelt, Md. 20771 (Mar.1984).
  41. 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 Press, Hampton, Va., 1982), pp. 241–254;also Los Alamos National Laboratory Report LA-UR-80-17 (1980).
  42. S. A. W. Gerstl, A. Zardecki, E. P. Shettle, “Solar Irradiance Calculations in the UV and Visible Using the Adjoint Discrete Ordinates Method,” in Proceedings, Fourth Conference on Atmospheric Radiation, Toronto, Canada, 16–18 June, 1981 (American Meteorological Society, Boston, 1981), pp. 67–80.
  43. J. Lewins, Importance, The Adjoint Function (Pergamon, Oxford, 1965).
  44. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).
  45. S. A. W. Gerstl, W. M. Stacey, “A Class of Second-Order Approximate Formulations of Deep-Penetration Radiation Transport Problems,” Nucl. Sci. Eng. 51, 339 (1973).
  46. W. M. Stacey, Variational Methods in Nuclear Reactor Physics (Academic, New York, 1974).

1984 (2)

S. A. W. Gerstl, A. Zardecki, “Coupled Atmosphere/Canopy Model for Remote Sensing of Plant Reflectance Features,” Appl. Opt. 23, 0000 (1984), same issue.

A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Multiple Scattering Effects in Spatial Frequency Filtering,” Appl. Opt. 23, 4124 (1984).
[CrossRef] [PubMed]

1983 (1)

1980 (2)

L. T. Chan, “Results of Telephotometer Measurements,” Los Alamos Scientific Laboratory memorandum to A. Zardecki (20Nov., 1980).

T. Mekler, Y. J. Kaufman, “The Effect of the Atmosphere on Contrast Reduction for a Non-Uniform Surface Albedo and “Two-Halves” Field,” J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

1979 (1)

1978 (1)

V. Ramanathan, J. A. Coakley, “Title,” Rev. Geophys. Space Phys. 16, 465 (1978).
[CrossRef]

1976 (1)

W. J. Wiscombe, “On Initialization Errors and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637 (1976).
[CrossRef]

1974 (3)

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[CrossRef]

S. H. Schneider, R. E. Dickinson, “Climate Modelling,” Rev. Geophys. Space Phys. 12, 447 (1974).
[CrossRef]

E. P. Shettle, A. E. S. Green, “Multiple Scattering Calculation of the Middle Ultraviolet Reaching the Ground,” Appl. Opt. 13, 1567 (1974).
[CrossRef] [PubMed]

1973 (2)

S. A. W. Gerstl, W. M. Stacey, “A Class of Second-Order Approximate Formulations of Deep-Penetration Radiation Transport Problems,” Nucl. Sci. Eng. 51, 339 (1973).

K. N. Liou, “Numerical Experiments with Chandrasekhar's Discrete-Ordinate Method,” J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

1972 (1)

K. D. Lathrop, “Discrete-Ordinates Methods for the Numerical Solution of the Transport Equation,” Reactor Technol. 15, 107 (1972).

1971 (1)

K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45 (3), 255 (1971).

1970 (2)

E. P. Shettle, J. A. Weinman, “The Transfer of Solar Irradiance Through Inhomogeneous Turbid Atmospheres Evaluated by Eddington's Approximation,” J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

R. E. Samuelson, “Non-Local Thermodynamic Equilibrium in Cloudy Planetary Atmospheres,” J. Atmos. Sci. 27, 711 (1970).
[CrossRef]

Ackerman, T. P.

R. W. Bergstrom, B. L. Bobson, T. P. Ackerman, “Calculation of Multiple Scattered Radiation in Clean Atmospheres,” in Symposium on Plumes and Visibility, Grand Canyon, Colo. (Nov. 1980).

Alcouffe, R. E.

R. E. Alcouffe, F. W. Brinkley, D. R. Marr, R. D. O'Dell, “User's Guide for twodant: A Code Package for Two-Dimensional Diffusion-Accelerated Neutral Particle Transport,” Los Alamos National Laboratory Report LA-10049-M (Mar.1984).

Bell, G. I.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

Bergstrom, R. W.

R. W. Bergstrom, B. L. Bobson, T. P. Ackerman, “Calculation of Multiple Scattered Radiation in Clean Atmospheres,” in Symposium on Plumes and Visibility, Grand Canyon, Colo. (Nov. 1980).

Bobson, B. L.

R. W. Bergstrom, B. L. Bobson, T. P. Ackerman, “Calculation of Multiple Scattered Radiation in Clean Atmospheres,” in Symposium on Plumes and Visibility, Grand Canyon, Colo. (Nov. 1980).

Brinkley, F. W.

K. D. Lathrop, F. W. Brinkley, “twotran-II: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).

R. E. Alcouffe, F. W. Brinkley, D. R. Marr, R. D. O'Dell, “User's Guide for twodant: A Code Package for Two-Dimensional Diffusion-Accelerated Neutral Particle Transport,” Los Alamos National Laboratory Report LA-10049-M (Mar.1984).

D. R. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for ONEDANT: A Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

Carlson, B. G.

B. G. Carlson, K. D. Lathrop, “Transport Theory, The Method of Discrete Ordinates,” in Computing Methods of Reactor Physics, H. Greenspan, C. N. Kelber, D. Okrent, Eds. (Gordon & Breach, New York, 1968), Chap. 3, pp. 171–261.

Chan, L. T.

L. T. Chan, “Results of Telephotometer Measurements,” Los Alamos Scientific Laboratory memorandum to A. Zardecki (20Nov., 1980).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950).

Coakley, J. A.

V. Ramanathan, J. A. Coakley, “Title,” Rev. Geophys. Space Phys. 16, 465 (1978).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

Dave, J. V.

J. V. Dave, “A Direct Solution of the Spherical-Harmonics Approximation to the Transfer Equation for a Plane-Parallel, Nonhomogeneous Atmosphere,” Lawrence Livermore Laboratory Report UCRL-51581 (1974).

J. V. Dave, “Extensive Datasets of the Diffuse Radiation in Realistic Atmospheric Models with Aerosols and Common Absorbing Gases,” IBM Palo Alto Scientific Center Report G320-3366 (Mar.1978).

Deering, D. W.

R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program Part I: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memorandum 86078, Goddard Space Flight Center, Greenbelt, Md. 20771 (Mar.1984).

Dickinson, R. E.

S. H. Schneider, R. E. Dickinson, “Climate Modelling,” Rev. Geophys. Space Phys. 12, 447 (1974).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Embury, J. F.

Engle, W. W.

W. W. Engle, “A User's Manual for anisn, a One-Dimensional Discrete-Ordinates Transport Code with Anisotropic Scattering,” Oak Ridge National Laboratory, Union Carbide Corp. Report K-1693 (Mar.1967).

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).

Fraser, R. S.

Gerstl, S. A. W.

S. A. W. Gerstl, A. Zardecki, “Coupled Atmosphere/Canopy Model for Remote Sensing of Plant Reflectance Features,” Appl. Opt. 23, 0000 (1984), same issue.

A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Multiple Scattering Effects in Spatial Frequency Filtering,” Appl. Opt. 23, 4124 (1984).
[CrossRef] [PubMed]

A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Application of the 2-D Discrete-Ordinates Method to Multiple Scattering of Laser Radiation,” Appl. Opt. 22, 1346 (1983).
[CrossRef] [PubMed]

S. A. W. Gerstl, W. M. Stacey, “A Class of Second-Order Approximate Formulations of Deep-Penetration Radiation Transport Problems,” Nucl. Sci. Eng. 51, 339 (1973).

S. A. W. Gerstl, “Application of Modern Neutron Transport Methods to Atmospheric Radiative Transfer,” Los Alamos Scientific Laboratory Report LA-UR-80-1403;also, Volume of Extended Abstracts, Proceedings, International Radiation Symposium, Fort Collins, Colo., 11–16 Aug.(1980), pp. 500–502.

A. Zardecki, S. A. W. Gerstl, “Calculations of Solar Irradiances in Clear and Polluted Atmospheres and Potential Effects on Plant Life,” Los Alamos National Laboratory Report LA-9010-MS (Oct.1981).

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 Press, Hampton, Va., 1982), pp. 241–254;also Los Alamos National Laboratory Report LA-UR-80-17 (1980).

S. A. W. Gerstl, A. Zardecki, E. P. Shettle, “Solar Irradiance Calculations in the UV and Visible Using the Adjoint Discrete Ordinates Method,” in Proceedings, Fourth Conference on Atmospheric Radiation, Toronto, Canada, 16–18 June, 1981 (American Meteorological Society, Boston, 1981), pp. 67–80.

Glasstone, S.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

Green, A. E. S.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

Hill, T. R.

T. R. Hill, “onetran: A Discrete-Ordinates Finite-Element Code for the Solution of the One-Dimensional Multigroup Transport Equation,” Los Alamos National Laboratory Report LA-5990-MS (June1975).

Kaufman, Y. J.

T. Mekler, Y. J. Kaufman, “The Effect of the Atmosphere on Contrast Reduction for a Non-Uniform Surface Albedo and “Two-Halves” Field,” J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

Y. J. Kaufman, “Atmospheric Effects on Remote Sensing of Surface Reflectance,” in Proceedings, SPIE Conference 475 on Critical Reviews of Remote Sensing, Washington, D.C., 1–2 May, 1984.

Lathrop, K. D.

K. D. Lathrop, “Discrete-Ordinates Methods for the Numerical Solution of the Transport Equation,” Reactor Technol. 15, 107 (1972).

K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45 (3), 255 (1971).

B. G. Carlson, K. D. Lathrop, “Transport Theory, The Method of Discrete Ordinates,” in Computing Methods of Reactor Physics, H. Greenspan, C. N. Kelber, D. Okrent, Eds. (Gordon & Breach, New York, 1968), Chap. 3, pp. 171–261.

K. D. Lathrop, F. W. Brinkley, “twotran-II: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).

Lewins, J.

J. Lewins, Importance, The Adjoint Function (Pergamon, Oxford, 1965).

Liou, K. N.

K. N. Liou, “Numerical Experiments with Chandrasekhar's Discrete-Ordinate Method,” J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, New York, 1980).

Marr, D. R.

D. R. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for ONEDANT: A Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

R. E. Alcouffe, F. W. Brinkley, D. R. Marr, R. D. O'Dell, “User's Guide for twodant: A Code Package for Two-Dimensional Diffusion-Accelerated Neutral Particle Transport,” Los Alamos National Laboratory Report LA-10049-M (Mar.1984).

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Mekler, T.

T. Mekler, Y. J. Kaufman, “The Effect of the Atmosphere on Contrast Reduction for a Non-Uniform Surface Albedo and “Two-Halves” Field,” J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

Murphy, R. E.

R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program Part I: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memorandum 86078, Goddard Space Flight Center, Greenbelt, Md. 20771 (Mar.1984).

Mynatt, F. R.

W. A. Rhoades, F. R. Mynatt, “The DOT-III Two-Dimensional Discrete-Ordinates Transport Code,” Oak Ridge National Laboratory Report ORNL-TM-4280 (Sept.1973).

O'Dell, D. R.

D. R. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for ONEDANT: A Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

O'Dell, R. D.

R. E. Alcouffe, F. W. Brinkley, D. R. Marr, R. D. O'Dell, “User's Guide for twodant: A Code Package for Two-Dimensional Diffusion-Accelerated Neutral Particle Transport,” Los Alamos National Laboratory Report LA-10049-M (Mar.1984).

Otterman, J.

Pierce, W. A.

W. A. Pierce, “A Study of the Effects of the Atmosphere on Thematic Mapper Observations,” EG&G Report 004-77, Washington Analysis Service Center, Riverdale, Md. (1977).

Ramanathan, V.

V. Ramanathan, J. A. Coakley, “Title,” Rev. Geophys. Space Phys. 16, 465 (1978).
[CrossRef]

Rhoades, W. A.

W. A. Rhoades, F. R. Mynatt, “The DOT-III Two-Dimensional Discrete-Ordinates Transport Code,” Oak Ridge National Laboratory Report ORNL-TM-4280 (Sept.1973).

Samuelson, R. E.

R. E. Samuelson, “Non-Local Thermodynamic Equilibrium in Cloudy Planetary Atmospheres,” J. Atmos. Sci. 27, 711 (1970).
[CrossRef]

Schneider, S. H.

S. H. Schneider, R. E. Dickinson, “Climate Modelling,” Rev. Geophys. Space Phys. 12, 447 (1974).
[CrossRef]

Shettle, E. P.

E. P. Shettle, A. E. S. Green, “Multiple Scattering Calculation of the Middle Ultraviolet Reaching the Ground,” Appl. Opt. 13, 1567 (1974).
[CrossRef] [PubMed]

E. P. Shettle, J. A. Weinman, “The Transfer of Solar Irradiance Through Inhomogeneous Turbid Atmospheres Evaluated by Eddington's Approximation,” J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).

S. A. W. Gerstl, A. Zardecki, E. P. Shettle, “Solar Irradiance Calculations in the UV and Visible Using the Adjoint Discrete Ordinates Method,” in Proceedings, Fourth Conference on Atmospheric Radiation, Toronto, Canada, 16–18 June, 1981 (American Meteorological Society, Boston, 1981), pp. 67–80.

Stacey, W. M.

S. A. W. Gerstl, W. M. Stacey, “A Class of Second-Order Approximate Formulations of Deep-Penetration Radiation Transport Problems,” Nucl. Sci. Eng. 51, 339 (1973).

W. M. Stacey, Variational Methods in Nuclear Reactor Physics (Academic, New York, 1974).

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980), Tables 35 of Vol. 2.

Weinman, J. A.

E. P. Shettle, J. A. Weinman, “The Transfer of Solar Irradiance Through Inhomogeneous Turbid Atmospheres Evaluated by Eddington's Approximation,” J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, “On Initialization Errors and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637 (1976).
[CrossRef]

Zardecki, A.

S. A. W. Gerstl, A. Zardecki, “Coupled Atmosphere/Canopy Model for Remote Sensing of Plant Reflectance Features,” Appl. Opt. 23, 0000 (1984), same issue.

A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Multiple Scattering Effects in Spatial Frequency Filtering,” Appl. Opt. 23, 4124 (1984).
[CrossRef] [PubMed]

A. Zardecki, S. A. W. Gerstl, J. F. Embury, “Application of the 2-D Discrete-Ordinates Method to Multiple Scattering of Laser Radiation,” Appl. Opt. 22, 1346 (1983).
[CrossRef] [PubMed]

A. Zardecki, S. A. W. Gerstl, “Calculations of Solar Irradiances in Clear and Polluted Atmospheres and Potential Effects on Plant Life,” Los Alamos National Laboratory Report LA-9010-MS (Oct.1981).

S. A. W. Gerstl, A. Zardecki, E. P. Shettle, “Solar Irradiance Calculations in the UV and Visible Using the Adjoint Discrete Ordinates Method,” in Proceedings, Fourth Conference on Atmospheric Radiation, Toronto, Canada, 16–18 June, 1981 (American Meteorological Society, Boston, 1981), pp. 67–80.

Appl. Opt. (5)

J. Geophys. Res. (1)

T. Mekler, Y. J. Kaufman, “The Effect of the Atmosphere on Contrast Reduction for a Non-Uniform Surface Albedo and “Two-Halves” Field,” J. Geophys. Res. 85, 4067 (1980).
[CrossRef]

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

W. J. Wiscombe, “On Initialization Errors and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 16, 637 (1976).
[CrossRef]

J. Atmos. Sci. (1)

R. E. Samuelson, “Non-Local Thermodynamic Equilibrium in Cloudy Planetary Atmospheres,” J. Atmos. Sci. 27, 711 (1970).
[CrossRef]

J. Atmos. Sci. (2)

K. N. Liou, “Numerical Experiments with Chandrasekhar's Discrete-Ordinate Method,” J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

E. P. Shettle, J. A. Weinman, “The Transfer of Solar Irradiance Through Inhomogeneous Turbid Atmospheres Evaluated by Eddington's Approximation,” J. Atmos. Sci. 27, 1048 (1970).
[CrossRef]

Los Alamos Scientific Laboratory memorandum to A. Zardecki (1)

L. T. Chan, “Results of Telephotometer Measurements,” Los Alamos Scientific Laboratory memorandum to A. Zardecki (20Nov., 1980).

Nucl. Sci. Eng. (2)

S. A. W. Gerstl, W. M. Stacey, “A Class of Second-Order Approximate Formulations of Deep-Penetration Radiation Transport Problems,” Nucl. Sci. Eng. 51, 339 (1973).

K. D. Lathrop, “Remedies for Ray Effects,” Nucl. Sci. Eng. 45 (3), 255 (1971).

Reactor Technol. (1)

K. D. Lathrop, “Discrete-Ordinates Methods for the Numerical Solution of the Transport Equation,” Reactor Technol. 15, 107 (1972).

Rev. Geophys. Space Phys. (2)

S. H. Schneider, R. E. Dickinson, “Climate Modelling,” Rev. Geophys. Space Phys. 12, 447 (1974).
[CrossRef]

V. Ramanathan, J. A. Coakley, “Title,” Rev. Geophys. Space Phys. 16, 465 (1978).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Other (29)

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

R. W. Roussin et al., Eds., Proceedings, Fifth International Conference on Radiation Shielding (Science Press, Princeton, N.J., 1977).Also, Proceedings, Sixth International Conference, Radiation Shielding, 16–20 May, 1983, Tokyo, Japan, in press.

S. A. W. Gerstl, “Application of Modern Neutron Transport Methods to Atmospheric Radiative Transfer,” Los Alamos Scientific Laboratory Report LA-UR-80-1403;also, Volume of Extended Abstracts, Proceedings, International Radiation Symposium, Fort Collins, Colo., 11–16 Aug.(1980), pp. 500–502.

D. R. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for ONEDANT: A Code Package for One-Dimensional, Diffusion-Accelerated, Neutral-Particle Transport,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

T. R. Hill, “onetran: A Discrete-Ordinates Finite-Element Code for the Solution of the One-Dimensional Multigroup Transport Equation,” Los Alamos National Laboratory Report LA-5990-MS (June1975).

R. E. Alcouffe, F. W. Brinkley, D. R. Marr, R. D. O'Dell, “User's Guide for twodant: A Code Package for Two-Dimensional Diffusion-Accelerated Neutral Particle Transport,” Los Alamos National Laboratory Report LA-10049-M (Mar.1984).

K. D. Lathrop, F. W. Brinkley, “twotran-II: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).

W. W. Engle, “A User's Manual for anisn, a One-Dimensional Discrete-Ordinates Transport Code with Anisotropic Scattering,” Oak Ridge National Laboratory, Union Carbide Corp. Report K-1693 (Mar.1967).

W. A. Rhoades, F. R. Mynatt, “The DOT-III Two-Dimensional Discrete-Ordinates Transport Code,” Oak Ridge National Laboratory Report ORNL-TM-4280 (Sept.1973).

“RSIC Computer Code and Data Collection: A Capsule Review of the Computer Code Collection (CCC), Peripheral Shielding Routines (RSR), and Data Library Collection (DLC) Packaged by the Radiation Shielding Information Center,” Informal Report, revisedNov.1983;available from RSIC, P.O. Box X, Oak Ridge, Tenn. 37831.

“Listing of Software Titles and Computer Codes from the National Energy Software Center (NESC),” Informal Report, updated monthly;available from NESC, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Ill. 60439.

J. V. Dave, “A Direct Solution of the Spherical-Harmonics Approximation to the Transfer Equation for a Plane-Parallel, Nonhomogeneous Atmosphere,” Lawrence Livermore Laboratory Report UCRL-51581 (1974).

J. V. Dave, “Extensive Datasets of the Diffuse Radiation in Realistic Atmospheric Models with Aerosols and Common Absorbing Gases,” IBM Palo Alto Scientific Center Report G320-3366 (Mar.1978).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1950).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, New York, 1980).

H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980), Tables 35 of Vol. 2.

A. Zardecki, S. A. W. Gerstl, “Calculations of Solar Irradiances in Clear and Polluted Atmospheres and Potential Effects on Plant Life,” Los Alamos National Laboratory Report LA-9010-MS (Oct.1981).

B. G. Carlson, K. D. Lathrop, “Transport Theory, The Method of Discrete Ordinates,” in Computing Methods of Reactor Physics, H. Greenspan, C. N. Kelber, D. Okrent, Eds. (Gordon & Breach, New York, 1968), Chap. 3, pp. 171–261.

W. M. Stacey, Variational Methods in Nuclear Reactor Physics (Academic, New York, 1974).

R. W. Bergstrom, B. L. Bobson, T. P. Ackerman, “Calculation of Multiple Scattered Radiation in Clean Atmospheres,” in Symposium on Plumes and Visibility, Grand Canyon, Colo. (Nov. 1980).

Y. J. Kaufman, “Atmospheric Effects on Remote Sensing of Surface Reflectance,” in Proceedings, SPIE Conference 475 on Critical Reviews of Remote Sensing, Washington, D.C., 1–2 May, 1984.

W. A. Pierce, “A Study of the Effects of the Atmosphere on Thematic Mapper Observations,” EG&G Report 004-77, Washington Analysis Service Center, Riverdale, Md. (1977).

E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).

R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program Part I: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memorandum 86078, Goddard Space Flight Center, Greenbelt, Md. 20771 (Mar.1984).

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 Press, Hampton, Va., 1982), pp. 241–254;also Los Alamos National Laboratory Report LA-UR-80-17 (1980).

S. A. W. Gerstl, A. Zardecki, E. P. Shettle, “Solar Irradiance Calculations in the UV and Visible Using the Adjoint Discrete Ordinates Method,” in Proceedings, Fourth Conference on Atmospheric Radiation, Toronto, Canada, 16–18 June, 1981 (American Meteorological Society, Boston, 1981), pp. 67–80.

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Figures (12)

Fig. 1
Fig. 1

Discrete-ordinates finite-element mesh in 1-D slab geometry.

Fig. 2
Fig. 2

Comparison of discrete-ordinates transmissivity calculations with Liou's results4 and the doubling method.

Fig. 3
Fig. 3

Comparison of discrete-ordinates reflectivity calculations with Liou's results4 and the doubling method.

Fig. 4
Fig. 4

Transmission of solar UV radiation through clear atmosphere. Comparison with results of Shettle and Green.31

Fig. 5
Fig. 5

Transmission of solar UV radiation through turbid atmosphere. Comparison with results of Shettle and Green.31

Fig. 6
Fig. 6

Comparison of angular-dependent transmission with results of van de Hulst.32

Fig. 7
Fig. 7

Comparison of angular-dependent reflection with results of van de Hulst.32

Fig. 8
Fig. 8

Comparison of predicted and measured solar radiance as a function of azimuth angle.

Fig. 9
Fig. 9

Upwelling solar radiance above surface discontinuity. Solar zenith angle 24°, rural aerosols of optical depth 0.312.

Fig. 10
Fig. 10

Scattered radiance at λ = 0.55 μm for three off-nadir observation directions above surface discontinuity. Solar zenith angle 24°, rural aerosols of optical depth 0.117.

Fig. 11
Fig. 11

Scattered radiance above two-halves field for model atmosphere of total optical depth 0.45 and ω = 0.4. Solar zenith angle 24°, specular surface reflectance.

Fig. 12
Fig. 12

Concept of an adjoint transfer calculation compared with the regular (forward) method.

Equations (31)

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Ω · I ( r , Ω ) + σ t ( r ) I ( r , Ω ) = σ s ( r ; Ω , Ω ) I ( r , Ω ) d Ω .
μ d I ( τ , μ , ϕ ) d τ = I ( τ , μ ϕ ) ω 4 π ( 4 π ) P ( τ , Ω · Ω ) I ( τ , μ , ϕ ) d Ω ,
μ d I ( z , μ , ϕ ) d z + σ t ( z ) I ( z , μ , ϕ ) = ( 4 π ) σ s ( z , Ω · Ω ) I ( z , μ , ϕ ) d Ω ,
τ = z 0 σ t ( z ) d z ,
σ s ( z , Ω · Ω ) = σ 0 s 1 4 π P ( τ , Ω · Ω ) .
F ( τ ) = 0 2 π 0 1 μ · I ( τ , μ , ϕ ) d μ d ϕ ,
F ( τ ) = 0 2 π 1 0 μ · I ( τ , μ , ϕ ) d μ d ϕ .
μ d I 0 ( τ , μ ) d τ = I 0 ( τ , μ ) ω 2 1 + 1 P 0 ( τ , μ , μ ) I 0 ( τ , μ ) d μ ,
μ I ( x , z , μ , ϕ ) z + 1 μ 2 cos ϕ I ( x , z , μ , ϕ ) x + σ t ( x , z ) I ( x , z , μ , ϕ ) = RHS ( 3 ) ,
μ m I i + 1 / 2 , m I i 1 / 2 , m Δ z i + σ i t I i , m = m = 1 M M σ i , m , m s I i , m Δ Ω m .
I i m = ½ ( I i + 1 / 2 , m + I i 1 / 2 , m ) ,
σ s ( z , Ω · Ω ) = l = 0 L 2 l + 1 4 π σ l s ( z ) P l ( Ω · Ω ) ,
σ l s ( z ) = 1 + 1 σ s ( z , Ω · Ω ) P l ( Ω · Ω ) d ( Ω · Ω ) .
RHS ( 10 ) = i = 0 L ( 2 l + 1 ) σ i , l s [ P l ( μ m ) m = 1 M M w m P l ( μ m ) I i , m + 2 r = 1 l ( l r ) ! ( l + r ) ! P l r ( μ m ) m = 1 M M w m P l r ( μ m ) × cos r ( ϕ m ϕ m ) I i , m ] .
m = 1 M M w m = 1 ,
m = 1 M M Δ Ω m = 4 π .
μ m 2 + η m 2 + ξ m 2 = 1
m w m μ m = 0 , m w m η m = 0 , m w m ξ m = 0 .
P ( z , μ ̂ ) = 1 g 2 ( 1 + g 2 2 g μ ̂ ) 3 / 2 ,
σ l s = σ 0 s · g 1 for all l .
T = F ( τ 0 ) / π μ 0 F 0 ,
R = F ( 0 ) / π μ 0 F 0 ,
d = H · tan θ · cos ϕ ,
L I = Q ,
Q ( z , μ , ϕ ) = μ 0 F 0 δ ( z z 0 ) δ ( μ μ 0 ) δ ( ϕ ϕ 0 ) ,
Response = 0 1 + 1 0 2 π R ( z , μ , ϕ ) I ( z , μ , ϕ ) dzd μ d ϕ R , I .
I + , L I = I , L + I + .
L + I + = R ,
I + , Q = I , R .
Response = I + , Q = 0 1 + 1 0 2 π I + ( z , μ , ϕ ) Q ( z , μ , ϕ ) dzd μ d ϕ .
F ( 0 ) = μ 0 F 0 I + ( z 0 , μ 0 , ϕ 0 ) .

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