Abstract

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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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).

2000 (2)

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]

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]

1999 (2)

A. I. Lyapustin, “Atmospheric and geometrical effects on land surface albedo,” J. Geophys. Res. 104, 4123–4143 (1999).

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]

1997 (1)

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]

1993 (2)

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]

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

1990 (1)

Y. Knyazikhin, “On the solvability of plane-parallel problems in the theory of radiation transport,” USSR Comput. Math. Math. Phys. 30, 557–569 (1990).

1988 (1)

1985 (1)

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]

1984 (1)

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]

1979 (1)

1978 (1)

1975 (1)

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]

1971 (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]

Asrar, G.

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

Bell, G. I.

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

Borel, C.

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).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Dave, J. V.

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]

de Leffe, A.

Deepak, A.

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.

Deschamps, P. Y.

Deuze, J. L.

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]

Diner, D. J.

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]

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).

Elterman, L.

L. Elterman, “UV, visible and IR attenuation for altitudes to 50 km,” (U.S. Air Force Cambridge Research Laboratories, Bedford, Mass., 1968).

Engelsen, O.

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).

Gerstl, S. A. W.

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).

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.

Glasstone, S.

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

Gordon, H. R.

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).

Hansen, J. E.

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]

Herman, M.

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]

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]

Hovenier, J. W.

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]

Ioltuhovskii, A. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, Russia, 1990).

Jayaweera, K.

Kaufmann, R. K.

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]

Knyazikhin, Y.

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]

Y. Knyazikhin, “On the solvability of plane-parallel problems in the theory of radiation transport,” USSR Comput. Math. Math. Phys. 30, 557–569 (1990).

Koepke, P.

Kriebel, K. T.

Lebedev, V. I.

G. I. Marchuk, V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport, (Harwood Academic, New York, 1986).

Lyapustin, A. I.

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]

A. I. Lyapustin, “Atmospheric and geometrical effects on land surface albedo,” J. Geophys. Res. 104, 4123–4143 (1999).

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]

Marchuk, G. I.

G. I. Marchuk, V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport, (Harwood Academic, New York, 1986).

Martonchik, J. V.

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]

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).

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).

Mocrette, J.-J.

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]

Muldashev, T. Z.

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]

Myneni, R.

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).

Myneni, R. B.

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]

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

Pinty, B.

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]

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).

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).

Privette, J. L.

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]

Rahman, H.

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]

Riesz, F.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).

Shabanov, N. V.

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]

Stamnes, K.

Strelkov, S. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, Russia, 1990).

Sushkevich, T. A.

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, Russia, 1990).

Sz.-Nagy, B.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).

Tanre, D.

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]

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]

Tsay, S. C.

Tucker, C. J.

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]

Twomey, S.

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]

Vermote, E. F.

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]

Verstraete, M. M.

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]

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).

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).

Vladimirov, V. S.

V. S. Vladimirov, “Mathematical problems in the one-velocity theory of particle transport,” (Atomic Energy of Canada Ltd., Chalk River, Ontario, 1963).

Wiscombe, W.

Zhou, L.

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]

Appl. Opt. (3)

IEEE Trans. Geosci. Remote Sens. (2)

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]

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]

J. Atmos. Sci. (1)

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]

J. Geophys. Res. (3)

A. I. Lyapustin, “Atmospheric and geometrical effects on land surface albedo,” J. Geophys. Res. 104, 4123–4143 (1999).

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]

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]

J. Quant. Spectrosc. Radiat. Transfer (5)

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

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]

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]

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]

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]

USSR Comput. Math. Math. Phys. (1)

Y. Knyazikhin, “On the solvability of plane-parallel problems in the theory of radiation transport,” USSR Comput. Math. Math. Phys. 30, 557–569 (1990).

Other (10)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

L. Elterman, “UV, visible and IR attenuation for altitudes to 50 km,” (U.S. Air Force Cambridge Research Laboratories, Bedford, Mass., 1968).

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).

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).

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

T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, Moscow, Russia, 1990).

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.

G. I. Marchuk, V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport, (Harwood Academic, New York, 1986).

F. Riesz, B. Sz.-Nagy, Functional Analysis (Dover, New York, 1990).

V. S. Vladimirov, “Mathematical problems in the one-velocity theory of particle transport,” (Atomic Energy of Canada Ltd., Chalk River, Ontario, 1963).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Different orders of the surface-reflected radiance as functions of the view zenith angle (VZA) at ϕ = 180°. Parameters of calculations: solar zenith angle, 60°; wavelength, 0.75 µm; aerosol optical thickness, 0.5; single-scattering albedo, 0.98; scattering function of Elterman17 at wavelength 0.75 µm. The surface BRDF was modeled by the Rahman–Pinty–Verstraete function18 fitted to the near-IR BRDF of grasses.19

Fig. 2
Fig. 2

Error of the total surface-reflected radiance calculated with the maximum-eigenvalue method [Eq. (26), dotted curve) and Lambertian approximations [relation (27), solid curve; relation 30, dashed curve]. Circles, squares, and triangles correspond to solar zenith angles of 77.4°, 53.2°, and 3.1°, respectively. The error is calculated with respect to the numerical MSH solution (1-I/MSH) 100%. The atmospheric parameters are the same as those of Fig. 1 except for the aerosol optical thickness (τ a = 0.8). The surface BRDF corresponds to irrigated wheat24 in the near-IR spectral range.

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

μ Iτ; sτ+Iτ; s=ωτ4πΩ χτ, γIτ; sds+ωτ4 χτ, γ0Sλ exp-τ/μ0,
I0; s=0, μ>0,
Iτ0; s=Sλμ0 exp-τ0/μ0ρs0, s+1πΩ+ Iτ0; sρs, sμds, μ<0.
Sˆ=ωτ/4πΩdsχτ, γ
Rˆ=1/πΩ+dsρs, sμ.
Iτ; s0, s=Dτ; s0, s+Jτ; s0, s.
Lˆ1D=SˆD+Sλωτ4 χτ, γ0exp-τ/μ0,
D+0=0, D-τ0=0;
Lˆ1J=SˆJ,
J+0=0, J-τ0=RˆI+0τ0+RˆJ+τ0.
I+0τ0I0τ0, s =πSλ exp-τ0/μ0δs-s0+Dτ0; s0, s, μ>0.
Jτ; s=k1 Jkτ, s.
Lˆ1Jk=SˆJk,
J+k0=0, J-kτ0=RˆJ+k-1τ0.
Jkτ; s=Ω- Gτ; s1, sJkτ0, s1ds1.
Lˆ1G=SˆG,
G+0=0, G-τ0=δs-s1, μ<0.
Gτ; s1, s=Gdτ; s1, s, μ>0,
Gτ; s1, s=exp-τ0-τ/|μ1|δs-s1+Gdτ; s1, s, μ<0,
Lˆ1Gd=SˆGd+ωτ4π χτ; γ1exp-τ0-τ/|μ1|,
G+d0=0, G+dτ0=0,
Γˆτ,s=Ω-ds1Gτ; s1, s.
J-kτ0=RˆΓˆτ0+J-k-1τ0.
J-τ0=k1 J-kτ0=k0RˆΓˆτ0+kRˆJ+0τ0=Iˆ-RˆΓˆτ0+-1RˆJ+0τ0.
Jτ; s=Γˆτ,sJ-τ0=Γˆτ,sIˆ-RˆΓˆτ0+-1RˆJ+0τ0.
Iτ=0; s0, s=D0; s0, s+Γˆτ0-Iˆ-RˆΓˆτ0+-1×RˆJ+0τ0, μ<0.
RˆJ+0τ0=qE0μ0,
E0μ0=Sλμ0 exp-τ0/μ0+1/π Ω+ Dτ0; s0, sμds
RˆΓˆτ0+J-kτ0=qJ-kτ01πΩ+ μds×Ω-Gdτ0; s1, sds1=qc0J-kτ0,
c0=1/πΩ+ μds Ω- Gdτ0; s1, sds1
J-τ0=k1 J-kτ0=k1qc0k-1RˆJ+0τ0=qE0μ01-qc0.
Γˆτ0-=Ω- G0; s1, sds1=exp-τ0/|μ|+Ω- Gd0; s1, sds1=Tμ.
Is0, s=Ds0, s+exp-τ0/|μ|Jτ0; s+Ω- Gd0; s1, sJτ0, s1ds1, μ<0,
J1τ0; s=Sλμ0 exp-τ0/μ0ρs0, s+1πΩ+ Dτ0; s0, sρs, sμds,
Jkτ0; s=1πΩ- Hτ0; s1, sJk-1τ0, s1ds1,
Hτ0; s1, s=Ω+ Gdτ0, s1, sμρs, sds.
Iτ0; s0, s=πSλ exp-τ0/μ0δs0-s+Dτ0; s0, s+Ω- Gdτ0; s1, sJτ0, s1ds1, μ>0.
Jk+1τ0; s/Jkτ0; sconst
Tkus/usηk,
J-k+1τ0=RˆΓˆτ0,s+J-kτ0ηJ-kτ0,
Jτ0; s=J1τ0; s+J2τ0; s1-η.
Jτ0; sJ1τ0; s1-qθ0c0.
Īrτ0; s=RˆI+δ+D+τ0+RˆΓˆτ0+RˆI+δ+D+τ01-η,
RˆΓˆτ0+RˆD+τ0ηRˆD+τ0.
RˆΓˆτ0+RˆI+δ=Sλμ0 exp-τ0/μ0RˆΓˆτ0+ρs0, s =Sλμ0 exp-τ0/μ01πΩ+ ρs, s×Ω- Gdτ0; s1, sρs0, s1ds1μds.
RˆΓˆτ0+RˆI+δSλμ0 exp-τ0/μ01πΩ+ μdsΩ-×Gdτ0; s1, sds1Ω+ds Ω-ds1×Ω+ ρs, sds Ω- ρs0, s1ds1 =Sλμ0 exp-τ0/μ0c0ρav1sρav2s0,
ρav1s=Ω+ ρs, sdsΩ+ds =12πΩ+ ρs, sds, ρav2s0=12πΩ- ρs0, sds.
Īrτ0; sSλμ0 exp-τ0/μ0×ρs0, s+c0ρav1sρav2s01-qθ0c0+1π1-qθ0c0Ω+×Dτ0; s0, sρs, sμds.

Metrics