Abstract

The code SHARM-3D, developed for fast and accurate simulations of the monochromatic radiance at the top of the atmosphere over spatially variable surfaces with Lambertian or anisotropic reflectance, is described. The atmosphere is assumed to be laterally uniform across the image and to consist of two layers with aerosols contained in the bottom layer. The SHARM-3D code performs simultaneous calculations for all specified incidence-view geometries and multiple wavelengths in one run. The numerical efficiency of the current version of code is close to its potential limit and is achieved by means of two innovations. The first is the development of a comprehensive precomputed lookup table of the three-dimensional atmospheric optical transfer function for various atmospheric conditions. The second is the use of a linear kernel model of the land surface bidirectional reflectance factor (BRF) in our algorithm that has led to a fully parameterized solution in terms of the surface BRF parameters. The code is also able to model inland lakes and rivers. The water pixels are described with the Nakajima–Tanaka BRF model of wind-roughened water surface with a Lambertian offset, which is designed to model approximately the reflectance of suspended matter and of a shallow lake or river bottom.

© 2005 Optical Society of America

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  1. T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
    [CrossRef]
  2. A. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface,” Appl. Opt. 41, 5600–5606 (2002).
    [CrossRef] [PubMed]
  3. A. Lyapustin, “Radiative transfer code SHARM-3D for radiance simulations over a non-Lambertian nonhomogeneous surface: intercomparison study,” Appl. Opt. 41, 5607–5615 (2002).
    [CrossRef] [PubMed]
  4. K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
    [CrossRef]
  5. W. Lucht, C. B. Schaaf, A. H. Strahler, “An algorithm for the retrieval of albedo from space using semiempirical BRDF models,” IEEE Trans. Geosci. Remote Sens. 38, 977–998 (2000).
    [CrossRef]
  6. C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
    [CrossRef]
  7. A. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt. 40, 3495–3501 (2001).
    [CrossRef]
  8. J.-L. Roujean, M. Leroy, P. Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of the remote sensing data,” J. Geophys. Res. 97, 20,455–20,468 (1992).
    [CrossRef]
  9. A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).
  10. A. I. Lyapustin, “Radiative transfer code SHARM for atmospheric and terrestrial applications,” Appl. Opt. (to be published).
  11. S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).
  12. T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, 1990; in Russian).
  13. A. A. Ioltukhovskii, “Radiative transfer over the surface with an arbitrary reflection: Green’s functions method,” Transp. Theory Stat. Phys. 28, 349–368 (1999).
    [CrossRef]
  14. A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
    [CrossRef]
  15. M. Wang, H. R. Gordon, “Radiance reflected from the ocean–atmosphere system: synthesis from individual components of the aerosol size distribution,” Appl. Opt. 33, 7088–7095 (1994).
    [CrossRef] [PubMed]
  16. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).
  17. T. Nakajima, M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
    [CrossRef]
  18. A. Lyapustin, SHARM Manual (NASA Goddard Space Flight Center), available at ftp://tpftp.gsfc.nasa.gov/projects/asrvn/.

2003 (1)

T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
[CrossRef]

2002 (3)

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

A. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface,” Appl. Opt. 41, 5600–5606 (2002).
[CrossRef] [PubMed]

A. Lyapustin, “Radiative transfer code SHARM-3D for radiance simulations over a non-Lambertian nonhomogeneous surface: intercomparison study,” Appl. Opt. 41, 5607–5615 (2002).
[CrossRef] [PubMed]

2001 (2)

A. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt. 40, 3495–3501 (2001).
[CrossRef]

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

2000 (1)

W. Lucht, C. B. Schaaf, A. H. Strahler, “An algorithm for the retrieval of albedo from space using semiempirical BRDF models,” IEEE Trans. Geosci. Remote Sens. 38, 977–998 (2000).
[CrossRef]

1999 (1)

A. A. Ioltukhovskii, “Radiative transfer over the surface with an arbitrary reflection: Green’s functions method,” Transp. Theory Stat. Phys. 28, 349–368 (1999).
[CrossRef]

1998 (1)

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

1994 (1)

1992 (1)

J.-L. Roujean, M. Leroy, P. Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of the remote sensing data,” J. Geophys. Res. 97, 20,455–20,468 (1992).
[CrossRef]

1983 (1)

T. Nakajima, M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
[CrossRef]

1958 (1)

S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).

Anderson, T. L.

T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
[CrossRef]

Barnsley, M.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).

Charlson, R. J.

T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
[CrossRef]

d’Entremont, R. P.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Deschamps, P. Y.

J.-L. Roujean, M. Leroy, P. Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of the remote sensing data,” J. Geophys. Res. 97, 20,455–20,468 (1992).
[CrossRef]

Disney, M.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Doll, C.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Dunderdale, M.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Evans, K. F.

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Gao, F.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Gordon, H. R.

Hobson, P.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Holben, B.

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

Holmen, K.

T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
[CrossRef]

Hu, B.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Ioltuhovskii, A. A.

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

Ioltukhovskii, A. A.

A. A. Ioltukhovskii, “Radiative transfer over the surface with an arbitrary reflection: Green’s functions method,” Transp. Theory Stat. Phys. 28, 349–368 (1999).
[CrossRef]

Jin, Y.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Knyazikhin, Yu.

Leroy, M.

J.-L. Roujean, M. Leroy, P. Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of the remote sensing data,” J. Geophys. Res. 97, 20,455–20,468 (1992).
[CrossRef]

Lewis, P.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Li, X.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Liang, S.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Lucht, W.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

W. Lucht, C. B. Schaaf, A. H. Strahler, “An algorithm for the retrieval of albedo from space using semiempirical BRDF models,” IEEE Trans. Geosci. Remote Sens. 38, 977–998 (2000).
[CrossRef]

Lyapustin, A.

Lyapustin, A. I.

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

A. I. Lyapustin, “Radiative transfer code SHARM for atmospheric and terrestrial applications,” Appl. Opt. (to be published).

Martonchik, J.

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

Muldashev, T. Z.

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

Muller, J.-P.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Nakajima, T.

T. Nakajima, M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
[CrossRef]

Ogren, J. A.

T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Privette, J.

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

Privette, J. L.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Roberts, G.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Roujean, J.-L.

J.-L. Roujean, M. Leroy, P. Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of the remote sensing data,” J. Geophys. Res. 97, 20,455–20,468 (1992).
[CrossRef]

Roy, D.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Schaaf, C. B.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

W. Lucht, C. B. Schaaf, A. H. Strahler, “An algorithm for the retrieval of albedo from space using semiempirical BRDF models,” IEEE Trans. Geosci. Remote Sens. 38, 977–998 (2000).
[CrossRef]

Sinyuk, A.

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

Slutsker, I.

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

Smirnov, A.

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

Strahler, A. H.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

W. Lucht, C. B. Schaaf, A. H. Strahler, “An algorithm for the retrieval of albedo from space using semiempirical BRDF models,” IEEE Trans. Geosci. Remote Sens. 38, 977–998 (2000).
[CrossRef]

Strelkov, S. A.

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

Strugnell, N. C.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Sushkevich, T. A.

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

Tanaka, M.

T. Nakajima, M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Tsang, T.

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Wang, M.

Wang, Y.

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

Winker, D. M.

T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
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C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
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Appl. Opt. (4)

IEEE Trans. Geosci. Remote Sens. (1)

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T. L. Anderson, R. J. Charlson, D. M. Winker, J. A. Ogren, K. Holmen, “Mesoscale variations of tropospheric aerosols,” J. Atmos. Sci. 60, 119–136 (2003).
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A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
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T. Nakajima, M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transfer 29, 521–537 (1983).
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S. Chandrasekhar, “On the diffuse reflection of a pencil of radiation by a plane-parallel atmosphere,” Proc. Natl. Acad. Sci. USA 44, 933–940 (1958).

Remote Sens. Environ. (1)

C. B. Schaaf, F. Gao, A. H. Strahler, W. Lucht, X. Li, T. Tsang, N. C. Strugnell, X. Zhang, Y. Jin, J.-P. Muller, P. Lewis, M. Barnsley, P. Hobson, M. Disney, G. Roberts, M. Dunderdale, C. Doll, R. P. d’Entremont, B. Hu, S. Liang, J. L. Privette, D. Roy, “First operational BRDF, albedo nadir reflectance products from MODIS,” Remote Sens. Environ. 83, 135–148 (2002).
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T. A. Sushkevich, S. A. Strelkov, A. A. Ioltuhovskii, Method of Path Integration in the Problems of Atmospheric Optics (Nauka, 1990; in Russian).

A. Lyapustin, Y. Wang, J. Martonchik, J. Privette, B. Holben, I. Slutsker, A. Sinyuk, A. Smirnov, “Local analysis of MISR surface BRF and albedo over GSFC and Mongu AERONET sites,” IEEE Trans. Geosci. Remote Sens. (to be published).

A. I. Lyapustin, “Radiative transfer code SHARM for atmospheric and terrestrial applications,” Appl. Opt. (to be published).

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Tables (1)

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Table 1 Relative Efficiency of SHARM-3D Code in Three-Dimensional and Independent Pixel Approximation Modesa

Equations (51)

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L ( r r s ; s 0 , s ) = D ( s 0 , s ) + exp ( τ / | μ | ) L s ( r ; s 0 , s ) + L ¯ s d ( s 0 , s ) + L s d ( r ; s 0 , s ) .
L s ( r ; s 0 , s ) S λ μ 0 exp ( τ / μ 0 ) [ ρ ( r ; s 0 , s ) + α c 0 ρ 1 ( r ; μ ) ρ ¯ 2 ( μ 0 ) ] + α π Ω + D s ( s 0 , s ) ρ ( r ; s , s ) μ d s ,
ρ 1 ( μ ) = 1 2 π Ω + ρ ( s , s ) d s ; ρ 2 ( μ 0 ) = 1 2 π Ω ρ ( s 0 , s ) d s ;
L ¯ s d ( s 0 , s ) = Ω G d ( s 1 , s ) L ¯ s ( s 0 , s 1 ) d s 1 .
q ( r ; μ 0 ) = F Up ( r ; μ 0 ) / F ¯ Down ( μ 0 ) ,
F ¯ Down ( μ 0 ) = π S λ μ 0 exp ( r / μ 0 ) + Ω + D s ( s 0 , s ) μ d s = F s Dir ( μ 0 ) + F s Dif ( μ 0 ) ,
F Up ( r ; μ 0 ) = π S λ μ 0 exp ( τ / μ 0 ) q 2 ( r ; μ 0 ) + Ω + μ q 2 ( r ; μ ) D s ( s 0 , s ) d s , q 2 ( r ; μ 0 ) = 1 π Ω ρ ( r ; s 0 , s ) μ d s .
L s d ( r r s ; μ 0 ; s ) α E 0 ( μ 0 ) ( 2 π ) 2 + q ( μ 0 ; p ) A ( p ; s ) 1 q ¯ ( μ 0 ) c ( p ) × exp { i [ p ( r r s ) Φ ( p ; s ) ] } d p .
ρ ( s , s ) = k L + k go f go ( s , s ) + k υ f υ ( s , s ) .
h ( φ φ ) = m 0 ( 2 δ 0 , m ) h ( m ) cos m ( φ φ ) , h ( m ) = 1 2 π 0 2 π h ( φ ) cos m φ d φ ,
f k 1 ( μ ) = 1 2 π 0 1 d μ 0 2 π f k ( μ , μ , φ φ ) d φ = 0 1 f k ( 0 ) ( μ , μ ) d μ ,
f k 2 ( μ 0 ) = 1 2 π 1 0 d μ 1 0 2 π f k ( μ 0 , μ 1 , φ 1 φ 0 ) d φ 1 = 1 0 f k ( 0 ) ( μ 0 , μ 1 ) d μ 1 ,
f k 3 ( μ ) = 1 π 1 0 μ d μ 0 2 π f k ( μ , μ , φ φ ) d φ = 2 1 0 f k ( 0 ) ( μ , μ ) μ d μ .
ρ 1 ( r ; μ ) = k L ( r ) + k go ( r ) f go 1 ( μ ) + k υ ( r ) f υ 1 ( μ ) ,
ρ ¯ 2 ( μ 0 ) = k ¯ L + k ¯ go f go 2 ( μ 0 ) + k ¯ υ f υ 2 ( μ 0 ) .
q ( r ; μ 0 ) = q Dir ( r ; μ 0 ) + q Dif ( r ; μ 0 ) ,
q Dir ( r ; μ 0 ) = F s Dir ( μ 0 ) π E 0 ( μ 0 ) [ k L ( r ) + k go ( r ) f go 3 ( μ 0 ) + k υ ( r ) f υ 3 ( μ 0 ) ] ,
q Dif ( r ; μ 0 ) = E 0 1 ( μ 0 ) [ k L ( r ) F s Dif ( μ 0 ) π + k go ( r ) D go 3 ( μ 0 ) + k υ ( r ) D υ 3 ( μ 0 ) ] ,
D k 3 ( μ 0 ) = 2 0 1 μ f k 3 ( μ ) D s ( 0 ) ( μ 0 , μ ) d μ .
D k 1 ( s 0 , s ) = 1 π 0 1 μ d μ 0 2 π d φ D s ( μ 0 , μ , φ φ 0 ) f k ( μ , μ , φ φ ) = m 0 ( 2 δ 0 , m ) D k 1 ( m ) ( μ 0 , μ ) × cos m ( φ φ 0 ) ,
D k 1 ( m ) ( μ 0 , μ ) = 2 0 1 D ( m ) ( μ 0 , μ ) f k ( m ) ( μ , μ ) μ d μ .
G av ( μ ) = 1 0 d μ 1 0 2 π G d ( μ 1 , μ , φ φ 1 ) d φ 1 = 2 π 1 0 G d ( 0 ) ( μ 1 , μ ) d μ 1 ,
G k 11 ( μ ) = 1 0 f k 1 ( μ 1 ) d μ 1 × 0 2 π G d ( μ 1 , μ , φ φ 1 ) d φ 1 = 2 π 1 0 f k 1 ( μ 1 ) G d ( 0 ) ( μ 1 , μ ) d μ 1 ,
G k 1 ( s 0 , s ) = 1 0 d μ 1 0 2 π G d ( μ 1 , μ , φ φ 1 ) × f k ( μ 0 , μ 1 , φ 1 φ 0 ) d φ 1 = m 0 ( 2 δ 0 , m ) G k 1 ( m ) ( μ 0 , μ ) × cos m ( φ φ 0 ) ,
G k 1 ( m ) ( μ 0 , μ ) = 2 π 0 1 G ( m ) ( μ 1 , μ ) × f k ( m ) ( μ 0 , μ 1 ) d μ 1 ,
H k 1 ( s 0 , s ) = 1 0 d μ 1 0 2 π G d ( μ 1 , μ , φ φ 1 ) × D k 1 ( μ 0 , μ 1 , φ 1 φ 0 ) d φ 1 = m 0 ( 2 δ 0 , m ) H k ( m ) ( μ 0 , μ ) × cos m ( φ φ 0 ) ,
H k 1 ( m ) ( μ 0 , μ ) = 2 π 0 1 G ( m ) ( μ 1 , μ ) D k 1 ( m ) ( μ 0 , μ 1 ) d μ 1 .
L s ( r ; s 0 , s ) S λ μ 0 exp [ τ ( 1 μ 0 + 1 | μ | ) ] [ ρ ( r ; s 0 , s ) + α c 0 ρ 1 ( r ; μ ) ρ ¯ 2 ( μ 0 ) ] + exp ( τ / | μ | ) α [ k L ( r ) E 0 d ( μ 0 ) + k go ( r ) D go 1 ( s 0 , s ) + k υ ( r ) D υ 1 ( s 0 , s ) ] ,
L ¯ s d ( s 0 , s ) = S λ μ 0 exp ( τ / μ 0 ) { [ k ¯ L G av ( μ ) + k ¯ go G go 1 ( s 0 , s ) + k ¯ υ G υ 1 ( s 0 , s ) ] + α c 0 [ k ¯ L G av ( μ ) + k ¯ go G go 11 ( μ ) + k ¯ υ G υ 11 ( μ ) ] ρ ¯ 2 ( μ 0 ) } + α [ k ¯ L E 0 d ( μ 0 ) G av ( μ ) + k ¯ go H go 1 ( s 0 , s ) + k ¯ υ H υ 1 ( s 0 , s ) ] .
μ Ψ ( z ; p ; s 0 , s ) z + [ k ( z ) i ( p , υ ) ] Ψ ( z ; p ; s 0 , s ) = σ ( z ) 4 π Ω χ ( z ; s , s ) Ψ ( z ; p ; s 0 , s ) d s ;
Ψ ( 0 ; p ; s 0 ; s ) = 0 , μ > 0 ; Ψ ( H ; p ; s 0 ; s ) = δ ( s s 0 ) , μ < 0 .
Ψ ( z ; p ; s 0 , s ) = exp [ i ( p , υ ) ( H z ) | μ | ] { exp [ τ 0 τ ( z ) | μ | ] × δ ( s s 0 ) + A ( z ; p ; s 0 , s ) × exp [ i Φ ( z ; p ; s 0 , s ) ] } .
Ψ = Ψ ( 0 ) + Ψ ( 1 ) + Ψ ( m ) .
μ Ψ ( 0 ) ( z ) z + [ k ( z ) i ( p , υ ) ] Ψ ( 0 ) ( z ) = 0 ,
Ψ ( 0 ) ( 0 ) = 0 , μ > 0 ; Ψ ( 0 ) ( H ) = δ ( s s 0 ) , μ < 0
Ψ ( 0 ) ( z ; p ; s 0 , s ) = { exp [ τ 0 τ ( z ) i ( p , υ ) ( H z ) | μ | ] δ ( s s 0 ) , μ < 0 0 , μ > 0 .
μ Ψ ( 1 ) ( z ) z + [ k ( z ) i ( p , υ ) ] Ψ ( 1 ) ( z ) = σ ( z ) 4 π exp [ τ 0 τ ( z ) i ( p , υ 0 ) ( H z ) | μ 0 | ] × χ ( z ; s 0 , s ) ;
Ψ ( 1 ) ( 0 ) = 0 , μ > 0 ; Ψ ( 1 ) ( H ) = 0 , μ < 0 .
Z = A exp ( i Φ ) = τ 0 ω χ ( s 0 , s ) 4 π | μ | { exp ( τ 0 | μ | ) exp [ τ 0 | μ 0 | + i ( ɛ 0 ɛ ) H ] } / [ τ 0 ( | μ 0 | 1 | μ | 1 ) + i ( ɛ ɛ 0 ) H ] ,
A ( s = s 0 ) = τ 0 ω χ ( 0 ) 4 π | μ 0 | exp ( τ 0 | μ 0 | ) , Φ ( s = s 0 ) = 0 .
Z ( 0 ; H ) = exp [ τ ( z 1 ) | μ | ] Z ( z 1 ; H ) + Z ( 0 ; z 1 ) .
Ψ L ( 1 ) ( s ) = Ω Ψ ( 1 ) ( s 1 , s ) d s 1 .
A ( p ; ω ) = β ( p ss ) A ( p ; ω Ref ) , β ( p ss ) = ω ω Ref , p p ss .
β ( 0 ) = T d ( ω ; μ ) A ( p = 0 ; ω Ref ; μ ) .
A ( p ; ω ) = [ β ( 0 ) β ( 0 ) β ( p ss ) p ss p ] A ( p ; ω Ref ) , 0 p < p ss .
Ψ L ( μ ; p x , p y , φ ) = Ψ L ( μ ; p 1 , 0 ; φ 1 ) ,
p 1 = p x 2 + p y 2 , φ 1 = φ arctan ( p y / p x ) .
A ( μ ; p 1 ; φ 1 ) = A ( μ ; p 1 ; φ 1 ) = A ( μ ; p 1 ; π φ 1 ) , Φ ( μ ; p 1 ; φ 1 ) = Φ ( μ ; p 1 ; φ 1 ) = Φ ( μ ; p 1 ; π φ 1 ) ,
τ = τ a + τ m , ω = f a ω a + f m ω m , χ ( γ ) = [ f a ω a χ a ( γ ) + f m ω m χ m ( γ ) ] / ω .
D ( τ ) = f a D a ( τ ) + f m D m ( τ ) ,
Z ( z 1 , H ) = f a Z a ( z 1 , H ) + f m Z m ( z 1 , H ) .

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