Abstract

Radiation-vegetation canopy interaction is analyzed using a semi-analytic discrete ordinates characteristics solution. The plant canopy is considered as a single layer containing a set of leave ensembles of bi-Lambertian surfaces and the radiation-plant canopy interaction is described by a radiative transfer equation in which the plant canopy radiative properties depend on the incident radiation direction. An analytical expression for anisotropic plant canopy radiance is derived. Radiance and hemispherical reflectance/transmittance for different boundary conditions and the plant canopy in the visible and near infrared are predicted. Results show that the discrete ordinates characteristics solution using the moderate number of double-Gauss ordinates direction compete well with the FN or “facile” and analytical discrete ordinates methods, which are two other literature quasi-analytical methods for phonon transport in the anisotropic plant canopy. Results also reveal that the proposed method matches exact Chandrasekhar benchmark, except for view cosine direction µ≥0.9, where the accuracy on transmitted radiance is less than 0.3% for low angular discretization. Comparisons between numerical predictions and soybean reflectance experiments indicate that the leaf area index and soil reflectance effects are significant in the near infrared plateau.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Coupled atmosphere/canopy model for remote sensing of plant reflectance features

Siegfried A. W. Gerstl and Andrew Zardecki
Appl. Opt. 24(1) 94-103 (1985)

References

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  1. J. Ross, The radiation regime and architecture of plant stands (The Hague, 1981)
  2. R. B. Myneni and J. Ross, Photon-vegetation interactions: applications in optical remote sensing and plant ecology (Springer-Verlag, 1991).
  3. S. A. W. Gerstl and A. Zardecki, “Coupled atmosphere/canopy model for remote sensing pf plant reflectance features,” Appl. Opt. 24(1), 94 (1985).
    [Crossref]
  4. S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
    [Crossref]
  5. H. T. Kamdem Tagne and D. Baillis, “Reduced models for radiative heat transfer analysis through anisotropic fibrous medium,” J. Heat Transfer 132(7), 072703 (2010).
    [Crossref]
  6. C. Zhang, A. Kribus, and R. Ben-Zvi, “Effective Radiative Properties of a Cylinder Array,” J. Heat Transfer 124(1), 198–200 (2002).
    [Crossref]
  7. A. L. Marshak, “The effect of the hot spot on the transport equation in plant canopies,” J. Quant. Spectrosc. Radiat. Transfer 42(6), 615–630 (1989).
    [Crossref]
  8. B. D. Ganapol, “Radiative transfer in dense plant canopies with azimuthal symmetry,” Transp. Theory Stat. Phys. 18(5-6), 475–491 (1989).
    [Crossref]
  9. A. A. Kokhanovsky, Aerosol Optics:Light Absorption and Scattering by Particles in the Atmosphere (Praxis Publishing Ltd, 2008).
  10. W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology (Cambridge University Press, 2007).
  11. L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).
  12. M.F. Modest, Radiative Heat Transfer, 3rd edition, Acad. Press, New York, 2013.
  13. J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer. 6th ed. (Taylor & Francis, CRC Press, 2015).
  14. M. Lazard, S. Andre, and D. Maillet, “Transient coupled radiative-conductive heat transfer in a gray planar medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 69(1), 23–33 (2001).
    [Crossref]
  15. F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
    [Crossref]
  16. F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
    [Crossref]
  17. Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
    [Crossref]
  18. F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
    [Crossref]
  19. R. Tapimo, H. T. T. Kamdem, and D. Yemele, “Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere,” J. Opt. Soc. Am. A 35(7), 1081 (2018).
    [Crossref]
  20. V. S. Antyufeev and A. L. Marshak, “Monte Carlo Method and Transport in Plant Canopies,” Remote Sens. Environ. 31(3), 183–191 (1990).
    [Crossref]
  21. K. Cooper, J. A. Smith, and D. Pitts, “Reflectance of a vegetation canopy using the adding method,” Appl. Opt. 21(22), 4112 (1982).
    [Crossref]
  22. R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Reflectance of a soybean canopy using the method of successive orders of scattering (SOSA),” Agricultural and Forest Meteorology 40(1), 71–87 (1987).
    [Crossref]
  23. J. K. Shultis and R. B. Myneni, “Radiative transfer in vegetation canopies with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 39(2), 115–129 (1988).
    [Crossref]
  24. T. Nilson and A. Kuusk, “A Reflectance Model for the Homogeneous Plant Canopy and Its Inversion,” Remote Sens. Environ. 27(2), 157–167 (1989).
    [Crossref]
  25. F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
    [Crossref]
  26. H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
    [Crossref]
  27. N. S. Goel, “Models of vegetation canopy reflectance and their use in estimation of biophysical parameters from reflectance data,” Remote Sensing Reviews 4(1), 1–212 (1988).
    [Crossref]
  28. S. Liang and A. H. Strahler, “An analytical BRDF model of canopy radiative transfer and its inversion,” IEEE Transactions on Geoscience and remote sensing 31(5), 1081–1092 (1993).
    [Crossref]
  29. B. D. Ganapol, “The determination of the reflected intensity for a single leaf angle plant canopy via Chandrasekhar’s method,” Transp. Theory Stat. Phys. 19(3-5), 251–271 (1990).
    [Crossref]
  30. B. D. Ganapol and R. B. Myneni, “The FN method for the one-angle radiative transfer transfer equation applied to plant canopies,” Remote Sens. Environ. 39(3), 213–231 (1992).
    [Crossref]
  31. R. Furfaro and B. D. Ganapol, “Spectral Theory for Photon Transport in Dense Vegetation Media: Caseology for the Canopy Equation,” Transp. Theory Stat. Phys. 36(1-3), 107–135 (2007).
    [Crossref]
  32. P. Picca, R. Furfaro, and B. D. Ganopol, “On radiative transfer in dense vegetation canopies,” Transp. Theory Stat. Phys. 41(3-4), 223–244 (2012).
    [Crossref]
  33. P. Picca and R. Furfaro, “Analytical discrete ordinate method for radiative transfer in dense vegetation canopies,” J. Quant. Spectrosc. Radiat. Transfer 118, 60–69 (2013).
    [Crossref]
  34. H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The Discrete Ordinates Characteristics Solution to the One-Dimensional Radiative Transfer Equation,” J. Comput. Theor. Transp. 46(5), 346–365 (2017).
    [Crossref]
  35. L. G. Ymeli and T. T. H. Kamdem, “Hyperbolic conduction–radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” ASME J. Heat Transf. 139(4), 042703 (2017).
    [Crossref]
  36. S. Chandrasekhar, Radiative Transfer (Dover Publication, Inc., 1960).
  37. K. J. Ranson, L. L. Biehl, and C. S. T. Daughtry, “Soybean canopy reflectance modeling data sets,” LARS Tech. Report 071584, Purdue Univ., W. Lafayette, IN (1984).
  38. N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997).
    [Crossref]
  39. R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988).
    [Crossref]
  40. W.-M. Wang, Z.-L. Li, and H.-B. Su, “Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage,” Agric. For. Meteorol. 143(1-2), 106–122 (2007).
    [Crossref]
  41. S. Otto and T. Trautmann, “A note on G-functions within the scope of radiative transfer in turbid vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 109(17-18), 2813–2819 (2008).
    [Crossref]
  42. B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
    [Crossref]
  43. R. B. Myneni, J. Ross, and G. Asrar, “A review on the theory of photon transport in leaf canopies,” Agric. For. Meteorol. 45(1-2), 1–153 (1989).
    [Crossref]

2019 (1)

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

2018 (2)

F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
[Crossref]

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere,” J. Opt. Soc. Am. A 35(7), 1081 (2018).
[Crossref]

2017 (5)

F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
[Crossref]

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The Discrete Ordinates Characteristics Solution to the One-Dimensional Radiative Transfer Equation,” J. Comput. Theor. Transp. 46(5), 346–365 (2017).
[Crossref]

L. G. Ymeli and T. T. H. Kamdem, “Hyperbolic conduction–radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” ASME J. Heat Transf. 139(4), 042703 (2017).
[Crossref]

2013 (2)

P. Picca and R. Furfaro, “Analytical discrete ordinate method for radiative transfer in dense vegetation canopies,” J. Quant. Spectrosc. Radiat. Transfer 118, 60–69 (2013).
[Crossref]

F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
[Crossref]

2012 (1)

P. Picca, R. Furfaro, and B. D. Ganopol, “On radiative transfer in dense vegetation canopies,” Transp. Theory Stat. Phys. 41(3-4), 223–244 (2012).
[Crossref]

2010 (1)

H. T. Kamdem Tagne and D. Baillis, “Reduced models for radiative heat transfer analysis through anisotropic fibrous medium,” J. Heat Transfer 132(7), 072703 (2010).
[Crossref]

2009 (1)

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

2008 (1)

S. Otto and T. Trautmann, “A note on G-functions within the scope of radiative transfer in turbid vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 109(17-18), 2813–2819 (2008).
[Crossref]

2007 (2)

W.-M. Wang, Z.-L. Li, and H.-B. Su, “Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage,” Agric. For. Meteorol. 143(1-2), 106–122 (2007).
[Crossref]

R. Furfaro and B. D. Ganapol, “Spectral Theory for Photon Transport in Dense Vegetation Media: Caseology for the Canopy Equation,” Transp. Theory Stat. Phys. 36(1-3), 107–135 (2007).
[Crossref]

2002 (1)

C. Zhang, A. Kribus, and R. Ben-Zvi, “Effective Radiative Properties of a Cylinder Array,” J. Heat Transfer 124(1), 198–200 (2002).
[Crossref]

2001 (1)

M. Lazard, S. Andre, and D. Maillet, “Transient coupled radiative-conductive heat transfer in a gray planar medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 69(1), 23–33 (2001).
[Crossref]

1999 (1)

B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
[Crossref]

1997 (1)

N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997).
[Crossref]

1993 (1)

S. Liang and A. H. Strahler, “An analytical BRDF model of canopy radiative transfer and its inversion,” IEEE Transactions on Geoscience and remote sensing 31(5), 1081–1092 (1993).
[Crossref]

1992 (1)

B. D. Ganapol and R. B. Myneni, “The FN method for the one-angle radiative transfer transfer equation applied to plant canopies,” Remote Sens. Environ. 39(3), 213–231 (1992).
[Crossref]

1990 (2)

B. D. Ganapol, “The determination of the reflected intensity for a single leaf angle plant canopy via Chandrasekhar’s method,” Transp. Theory Stat. Phys. 19(3-5), 251–271 (1990).
[Crossref]

V. S. Antyufeev and A. L. Marshak, “Monte Carlo Method and Transport in Plant Canopies,” Remote Sens. Environ. 31(3), 183–191 (1990).
[Crossref]

1989 (4)

T. Nilson and A. Kuusk, “A Reflectance Model for the Homogeneous Plant Canopy and Its Inversion,” Remote Sens. Environ. 27(2), 157–167 (1989).
[Crossref]

A. L. Marshak, “The effect of the hot spot on the transport equation in plant canopies,” J. Quant. Spectrosc. Radiat. Transfer 42(6), 615–630 (1989).
[Crossref]

B. D. Ganapol, “Radiative transfer in dense plant canopies with azimuthal symmetry,” Transp. Theory Stat. Phys. 18(5-6), 475–491 (1989).
[Crossref]

R. B. Myneni, J. Ross, and G. Asrar, “A review on the theory of photon transport in leaf canopies,” Agric. For. Meteorol. 45(1-2), 1–153 (1989).
[Crossref]

1988 (3)

R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988).
[Crossref]

N. S. Goel, “Models of vegetation canopy reflectance and their use in estimation of biophysical parameters from reflectance data,” Remote Sensing Reviews 4(1), 1–212 (1988).
[Crossref]

J. K. Shultis and R. B. Myneni, “Radiative transfer in vegetation canopies with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 39(2), 115–129 (1988).
[Crossref]

1987 (1)

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Reflectance of a soybean canopy using the method of successive orders of scattering (SOSA),” Agricultural and Forest Meteorology 40(1), 71–87 (1987).
[Crossref]

1985 (1)

1982 (1)

Andre, S.

M. Lazard, S. Andre, and D. Maillet, “Transient coupled radiative-conductive heat transfer in a gray planar medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 69(1), 23–33 (2001).
[Crossref]

Antyufeev, V. S.

V. S. Antyufeev and A. L. Marshak, “Monte Carlo Method and Transport in Plant Canopies,” Remote Sens. Environ. 31(3), 183–191 (1990).
[Crossref]

Asner, G. P.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Asrar, G.

R. B. Myneni, J. Ross, and G. Asrar, “A review on the theory of photon transport in leaf canopies,” Agric. For. Meteorol. 45(1-2), 1–153 (1989).
[Crossref]

R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988).
[Crossref]

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Reflectance of a soybean canopy using the method of successive orders of scattering (SOSA),” Agricultural and Forest Meteorology 40(1), 71–87 (1987).
[Crossref]

Bacour, C.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Baillis, D.

H. T. Kamdem Tagne and D. Baillis, “Reduced models for radiative heat transfer analysis through anisotropic fibrous medium,” J. Heat Transfer 132(7), 072703 (2010).
[Crossref]

L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

Baret, F.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Ben-Zvi, R.

C. Zhang, A. Kribus, and R. Ben-Zvi, “Effective Radiative Properties of a Cylinder Array,” J. Heat Transfer 124(1), 198–200 (2002).
[Crossref]

Biehl, L. L.

K. J. Ranson, L. L. Biehl, and C. S. T. Daughtry, “Soybean canopy reflectance modeling data sets,” LARS Tech. Report 071584, Purdue Univ., W. Lafayette, IN (1984).

Bond, B.

B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
[Crossref]

Bott, A.

W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology (Cambridge University Press, 2007).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publication, Inc., 1960).

Cooper, K.

Dai, Q.

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

Dai, Y.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

Daughtry, C. S. T.

K. J. Ranson, L. L. Biehl, and C. S. T. Daughtry, “Soybean canopy reflectance modeling data sets,” LARS Tech. Report 071584, Purdue Univ., W. Lafayette, IN (1984).

Dickinson, R. E.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

Dombrovsky, L. A.

L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

François, C.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Furfaro, R.

P. Picca and R. Furfaro, “Analytical discrete ordinate method for radiative transfer in dense vegetation canopies,” J. Quant. Spectrosc. Radiat. Transfer 118, 60–69 (2013).
[Crossref]

P. Picca, R. Furfaro, and B. D. Ganopol, “On radiative transfer in dense vegetation canopies,” Transp. Theory Stat. Phys. 41(3-4), 223–244 (2012).
[Crossref]

R. Furfaro and B. D. Ganapol, “Spectral Theory for Photon Transport in Dense Vegetation Media: Caseology for the Canopy Equation,” Transp. Theory Stat. Phys. 36(1-3), 107–135 (2007).
[Crossref]

Ganapol, B. D.

R. Furfaro and B. D. Ganapol, “Spectral Theory for Photon Transport in Dense Vegetation Media: Caseology for the Canopy Equation,” Transp. Theory Stat. Phys. 36(1-3), 107–135 (2007).
[Crossref]

B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
[Crossref]

B. D. Ganapol and R. B. Myneni, “The FN method for the one-angle radiative transfer transfer equation applied to plant canopies,” Remote Sens. Environ. 39(3), 213–231 (1992).
[Crossref]

B. D. Ganapol, “The determination of the reflected intensity for a single leaf angle plant canopy via Chandrasekhar’s method,” Transp. Theory Stat. Phys. 19(3-5), 251–271 (1990).
[Crossref]

B. D. Ganapol, “Radiative transfer in dense plant canopies with azimuthal symmetry,” Transp. Theory Stat. Phys. 18(5-6), 475–491 (1989).
[Crossref]

Ganopol, B. D.

P. Picca, R. Furfaro, and B. D. Ganopol, “On radiative transfer in dense vegetation canopies,” Transp. Theory Stat. Phys. 41(3-4), 223–244 (2012).
[Crossref]

Gerstl, S. A. W.

Gobron, N.

N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997).
[Crossref]

Goel, N. S.

N. S. Goel, “Models of vegetation canopy reflectance and their use in estimation of biophysical parameters from reflectance data,” Remote Sensing Reviews 4(1), 1–212 (1988).
[Crossref]

Govaerts, Y.

N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997).
[Crossref]

Gutschick, V.P.

R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988).
[Crossref]

Hlavka, C. A.

B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
[Crossref]

Howell, J. R.

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer. 6th ed. (Taylor & Francis, CRC Press, 2015).

Hu, S.

F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
[Crossref]

Jacquemoud, S.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Jing, X.

F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
[Crossref]

Johnson, L. F.

B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
[Crossref]

Kamdem, H. T. T.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere,” J. Opt. Soc. Am. A 35(7), 1081 (2018).
[Crossref]

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The Discrete Ordinates Characteristics Solution to the One-Dimensional Radiative Transfer Equation,” J. Comput. Theor. Transp. 46(5), 346–365 (2017).
[Crossref]

Kamdem, T. T. H.

L. G. Ymeli and T. T. H. Kamdem, “Hyperbolic conduction–radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” ASME J. Heat Transf. 139(4), 042703 (2017).
[Crossref]

Kamdem Tagne, H. T.

H. T. Kamdem Tagne and D. Baillis, “Reduced models for radiative heat transfer analysis through anisotropic fibrous medium,” J. Heat Transfer 132(7), 072703 (2010).
[Crossref]

Kanemasu, E. T.

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Reflectance of a soybean canopy using the method of successive orders of scattering (SOSA),” Agricultural and Forest Meteorology 40(1), 71–87 (1987).
[Crossref]

Kanemasu, E.T.

R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988).
[Crossref]

Kokhanovsky, A. A.

A. A. Kokhanovsky, Aerosol Optics:Light Absorption and Scattering by Particles in the Atmosphere (Praxis Publishing Ltd, 2008).

Kribus, A.

C. Zhang, A. Kribus, and R. Ben-Zvi, “Effective Radiative Properties of a Cylinder Array,” J. Heat Transfer 124(1), 198–200 (2002).
[Crossref]

Kuusk, A.

T. Nilson and A. Kuusk, “A Reflectance Model for the Homogeneous Plant Canopy and Its Inversion,” Remote Sens. Environ. 27(2), 157–167 (1989).
[Crossref]

Lazard, M.

M. Lazard, S. Andre, and D. Maillet, “Transient coupled radiative-conductive heat transfer in a gray planar medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 69(1), 23–33 (2001).
[Crossref]

Lei, Y.

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

Li, J.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
[Crossref]

F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
[Crossref]

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
[Crossref]

Li, Z.-L.

W.-M. Wang, Z.-L. Li, and H.-B. Su, “Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage,” Agric. For. Meteorol. 143(1-2), 106–122 (2007).
[Crossref]

Liang, S.

S. Liang and A. H. Strahler, “An analytical BRDF model of canopy radiative transfer and its inversion,” IEEE Transactions on Geoscience and remote sensing 31(5), 1081–1092 (1993).
[Crossref]

Liu, P.

F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
[Crossref]

Ma, L.

F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
[Crossref]

Maillet, D.

M. Lazard, S. Andre, and D. Maillet, “Transient coupled radiative-conductive heat transfer in a gray planar medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 69(1), 23–33 (2001).
[Crossref]

Marshak, A. L.

V. S. Antyufeev and A. L. Marshak, “Monte Carlo Method and Transport in Plant Canopies,” Remote Sens. Environ. 31(3), 183–191 (1990).
[Crossref]

A. L. Marshak, “The effect of the hot spot on the transport equation in plant canopies,” J. Quant. Spectrosc. Radiat. Transfer 42(6), 615–630 (1989).
[Crossref]

Mengüç, M. P.

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer. 6th ed. (Taylor & Francis, CRC Press, 2015).

Modest, M.F.

M.F. Modest, Radiative Heat Transfer, 3rd edition, Acad. Press, New York, 2013.

Myneni, R. B.

B. D. Ganapol and R. B. Myneni, “The FN method for the one-angle radiative transfer transfer equation applied to plant canopies,” Remote Sens. Environ. 39(3), 213–231 (1992).
[Crossref]

R. B. Myneni, J. Ross, and G. Asrar, “A review on the theory of photon transport in leaf canopies,” Agric. For. Meteorol. 45(1-2), 1–153 (1989).
[Crossref]

J. K. Shultis and R. B. Myneni, “Radiative transfer in vegetation canopies with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 39(2), 115–129 (1988).
[Crossref]

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Reflectance of a soybean canopy using the method of successive orders of scattering (SOSA),” Agricultural and Forest Meteorology 40(1), 71–87 (1987).
[Crossref]

R. B. Myneni and J. Ross, Photon-vegetation interactions: applications in optical remote sensing and plant ecology (Springer-Verlag, 1991).

Myneni, R.B.

R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988).
[Crossref]

Nilson, T.

T. Nilson and A. Kuusk, “A Reflectance Model for the Homogeneous Plant Canopy and Its Inversion,” Remote Sens. Environ. 27(2), 157–167 (1989).
[Crossref]

Otto, S.

S. Otto and T. Trautmann, “A note on G-functions within the scope of radiative transfer in turbid vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 109(17-18), 2813–2819 (2008).
[Crossref]

Peng, Y.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

Peterson, D. L.

B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
[Crossref]

Picca, P.

P. Picca and R. Furfaro, “Analytical discrete ordinate method for radiative transfer in dense vegetation canopies,” J. Quant. Spectrosc. Radiat. Transfer 118, 60–69 (2013).
[Crossref]

P. Picca, R. Furfaro, and B. D. Ganopol, “On radiative transfer in dense vegetation canopies,” Transp. Theory Stat. Phys. 41(3-4), 223–244 (2012).
[Crossref]

Pinty, B.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997).
[Crossref]

Pitts, D.

Ranson, K. J.

K. J. Ranson, L. L. Biehl, and C. S. T. Daughtry, “Soybean canopy reflectance modeling data sets,” LARS Tech. Report 071584, Purdue Univ., W. Lafayette, IN (1984).

Ross, J.

R. B. Myneni, J. Ross, and G. Asrar, “A review on the theory of photon transport in leaf canopies,” Agric. For. Meteorol. 45(1-2), 1–153 (1989).
[Crossref]

R. B. Myneni and J. Ross, Photon-vegetation interactions: applications in optical remote sensing and plant ecology (Springer-Verlag, 1991).

J. Ross, The radiation regime and architecture of plant stands (The Hague, 1981)

Shangguan, W.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

Shen, Z.

F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
[Crossref]

Shi, Y.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

Shultis, J. K.

J. K. Shultis and R. B. Myneni, “Radiative transfer in vegetation canopies with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 39(2), 115–129 (1988).
[Crossref]

Siegel, R.

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer. 6th ed. (Taylor & Francis, CRC Press, 2015).

Smith, J. A.

Strahler, A. H.

S. Liang and A. H. Strahler, “An analytical BRDF model of canopy radiative transfer and its inversion,” IEEE Transactions on Geoscience and remote sensing 31(5), 1081–1092 (1993).
[Crossref]

Su, H.-B.

W.-M. Wang, Z.-L. Li, and H.-B. Su, “Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage,” Agric. For. Meteorol. 143(1-2), 106–122 (2007).
[Crossref]

Tapimo, R.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere,” J. Opt. Soc. Am. A 35(7), 1081 (2018).
[Crossref]

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The Discrete Ordinates Characteristics Solution to the One-Dimensional Radiative Transfer Equation,” J. Comput. Theor. Transp. 46(5), 346–365 (2017).
[Crossref]

Trautmann, T.

S. Otto and T. Trautmann, “A note on G-functions within the scope of radiative transfer in turbid vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 109(17-18), 2813–2819 (2008).
[Crossref]

W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology (Cambridge University Press, 2007).

Ustin, S. L.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Verhoef, W.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Verstraete, M. M.

N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997).
[Crossref]

Wang, L.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

Wang, W.-M.

W.-M. Wang, Z.-L. Li, and H.-B. Su, “Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage,” Agric. For. Meteorol. 143(1-2), 106–122 (2007).
[Crossref]

Wang, Z.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

Wu, K.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
[Crossref]

F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
[Crossref]

Yan, J.-R.

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

Yang, Q.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

Yemele, D.

Ymeli, G. L.

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The Discrete Ordinates Characteristics Solution to the One-Dimensional Radiative Transfer Equation,” J. Comput. Theor. Transp. 46(5), 346–365 (2017).
[Crossref]

Ymeli, L. G.

L. G. Ymeli and T. T. H. Kamdem, “Hyperbolic conduction–radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” ASME J. Heat Transf. 139(4), 042703 (2017).
[Crossref]

Yuan, H.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

Zarco-Tejada, P. J.

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

Zardecki, A.

Zdunkowski, W.

W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology (Cambridge University Press, 2007).

Zhang, C.

C. Zhang, A. Kribus, and R. Ben-Zvi, “Effective Radiative Properties of a Cylinder Array,” J. Heat Transfer 124(1), 198–200 (2002).
[Crossref]

Zhang, F.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
[Crossref]

F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
[Crossref]

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
[Crossref]

Zhang, H.

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
[Crossref]

Zhang, S.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

Zhao, J.-Q.

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

Zhou, X.

F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
[Crossref]

Zhu, S.

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

Adv. Atmos. Sci. (1)

F. Zhang, Y. Lei, J.-R. Yan, J.-Q. Zhao, J. Li, and Q. Dai, “A new parameterization of canopy radiative transfer for land surface radiative models,” Adv. Atmos. Sci. 34(5), 613–622 (2017).
[Crossref]

Agric. For. Meteorol. (3)

R.B. Myneni, V.P. Gutschick, G. Asrar, and E.T. Kanemasu, “Photon transport in vegetation canopies with anisotropic scattering part I. Scattering phase functions in one angle,” Agric. For. Meteorol. 42(1), 1–16 (1988).
[Crossref]

W.-M. Wang, Z.-L. Li, and H.-B. Su, “Comparison of leaf angle distribution functions: Effects on extinction coefficient and fraction of sunlit foliage,” Agric. For. Meteorol. 143(1-2), 106–122 (2007).
[Crossref]

R. B. Myneni, J. Ross, and G. Asrar, “A review on the theory of photon transport in leaf canopies,” Agric. For. Meteorol. 45(1-2), 1–153 (1989).
[Crossref]

Agricultural and Forest Meteorology (1)

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Reflectance of a soybean canopy using the method of successive orders of scattering (SOSA),” Agricultural and Forest Meteorology 40(1), 71–87 (1987).
[Crossref]

Appl. Opt. (2)

ASME J. Heat Transf. (1)

L. G. Ymeli and T. T. H. Kamdem, “Hyperbolic conduction–radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” ASME J. Heat Transf. 139(4), 042703 (2017).
[Crossref]

IEEE Transactions on Geoscience and remote sensing (1)

S. Liang and A. H. Strahler, “An analytical BRDF model of canopy radiative transfer and its inversion,” IEEE Transactions on Geoscience and remote sensing 31(5), 1081–1092 (1993).
[Crossref]

J. Adv. Model. Earth Syst. (1)

H. Yuan, Y. Dai, R. E. Dickinson, B. Pinty, W. Shangguan, S. Zhang, L. Wang, and S. Zhu, “Reexamination and further development of two-stream canopy radiative transfer models for global land modeling,” J. Adv. Model. Earth Syst. 9(1), 113–129 (2017).
[Crossref]

J. Atmos. Sci. (1)

F. Zhang, Z. Shen, J. Li, X. Zhou, and L. Ma, “Analytical delta-four-stream doubling-adding method for radiative transfer parameterizations,” J. Atmos. Sci. 70(3), 794–808 (2013).
[Crossref]

J. Comput. Theor. Transp. (1)

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The Discrete Ordinates Characteristics Solution to the One-Dimensional Radiative Transfer Equation,” J. Comput. Theor. Transp. 46(5), 346–365 (2017).
[Crossref]

J. Geophys. Res.: Atmos. (1)

N. Gobron, B. Pinty, M. M. Verstraete, and Y. Govaerts, “A semidiscrete model for the scattering of light by vegetation,” J. Geophys. Res.: Atmos. 102(D8), 9431–9446 (1997).
[Crossref]

J. Heat Transfer (2)

H. T. Kamdem Tagne and D. Baillis, “Reduced models for radiative heat transfer analysis through anisotropic fibrous medium,” J. Heat Transfer 132(7), 072703 (2010).
[Crossref]

C. Zhang, A. Kribus, and R. Ben-Zvi, “Effective Radiative Properties of a Cylinder Array,” J. Heat Transfer 124(1), 198–200 (2002).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (8)

M. Lazard, S. Andre, and D. Maillet, “Transient coupled radiative-conductive heat transfer in a gray planar medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 69(1), 23–33 (2001).
[Crossref]

J. K. Shultis and R. B. Myneni, “Radiative transfer in vegetation canopies with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 39(2), 115–129 (1988).
[Crossref]

P. Picca and R. Furfaro, “Analytical discrete ordinate method for radiative transfer in dense vegetation canopies,” J. Quant. Spectrosc. Radiat. Transfer 118, 60–69 (2013).
[Crossref]

A. L. Marshak, “The effect of the hot spot on the transport equation in plant canopies,” J. Quant. Spectrosc. Radiat. Transfer 42(6), 615–630 (1989).
[Crossref]

F. Zhang, K. Wu, P. Liu, X. Jing, and J. Li, “Accounting for Gaussian quadrature in four-stream radiative transfer algorithms,” J. Quant. Spectrosc. Radiat. Transfer 192, 1–13 (2017).
[Crossref]

Q. Yang, F. Zhang, H. Zhang, Z. Wang, J. Li, K. Wu, Y. Shi, and Y. Peng, “Assessment of Two Two-stream Approximations in a Climate Model,” J. Quant. Spectrosc. Radiat. Transfer 225, 25–34 (2019).
[Crossref]

F. Zhang, K. Wu, J. Li, H. Zhang, and S. Hu, “Radiative transfer in the region with solar and infrared spectra overlap,” J. Quant. Spectrosc. Radiat. Transfer 219, 366–378 (2018).
[Crossref]

S. Otto and T. Trautmann, “A note on G-functions within the scope of radiative transfer in turbid vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 109(17-18), 2813–2819 (2008).
[Crossref]

Remote Sens. Environ. (5)

B. D. Ganapol, L. F. Johnson, C. A. Hlavka, D. L. Peterson, and B. Bond, “LCM2: A coupled leaf/cnopy radiative transfer model,” Remote Sens. Environ. 70(2), 153–166 (1999).
[Crossref]

S. Jacquemoud, W. Verhoef, F. Baret, C. Bacour, P. J. Zarco-Tejada, G. P. Asner, C. François, and S. L. Ustin, “PROSPECT + SAIL models: A review of use for vegetation characterization,” Remote Sens. Environ. 113, S56–S66 (2009).
[Crossref]

B. D. Ganapol and R. B. Myneni, “The FN method for the one-angle radiative transfer transfer equation applied to plant canopies,” Remote Sens. Environ. 39(3), 213–231 (1992).
[Crossref]

T. Nilson and A. Kuusk, “A Reflectance Model for the Homogeneous Plant Canopy and Its Inversion,” Remote Sens. Environ. 27(2), 157–167 (1989).
[Crossref]

V. S. Antyufeev and A. L. Marshak, “Monte Carlo Method and Transport in Plant Canopies,” Remote Sens. Environ. 31(3), 183–191 (1990).
[Crossref]

Remote Sensing Reviews (1)

N. S. Goel, “Models of vegetation canopy reflectance and their use in estimation of biophysical parameters from reflectance data,” Remote Sensing Reviews 4(1), 1–212 (1988).
[Crossref]

Transp. Theory Stat. Phys. (4)

R. Furfaro and B. D. Ganapol, “Spectral Theory for Photon Transport in Dense Vegetation Media: Caseology for the Canopy Equation,” Transp. Theory Stat. Phys. 36(1-3), 107–135 (2007).
[Crossref]

P. Picca, R. Furfaro, and B. D. Ganopol, “On radiative transfer in dense vegetation canopies,” Transp. Theory Stat. Phys. 41(3-4), 223–244 (2012).
[Crossref]

B. D. Ganapol, “The determination of the reflected intensity for a single leaf angle plant canopy via Chandrasekhar’s method,” Transp. Theory Stat. Phys. 19(3-5), 251–271 (1990).
[Crossref]

B. D. Ganapol, “Radiative transfer in dense plant canopies with azimuthal symmetry,” Transp. Theory Stat. Phys. 18(5-6), 475–491 (1989).
[Crossref]

Other (9)

A. A. Kokhanovsky, Aerosol Optics:Light Absorption and Scattering by Particles in the Atmosphere (Praxis Publishing Ltd, 2008).

W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the Atmosphere: A Course in Theoretical Meteorology (Cambridge University Press, 2007).

L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

M.F. Modest, Radiative Heat Transfer, 3rd edition, Acad. Press, New York, 2013.

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer. 6th ed. (Taylor & Francis, CRC Press, 2015).

J. Ross, The radiation regime and architecture of plant stands (The Hague, 1981)

R. B. Myneni and J. Ross, Photon-vegetation interactions: applications in optical remote sensing and plant ecology (Springer-Verlag, 1991).

S. Chandrasekhar, Radiative Transfer (Dover Publication, Inc., 1960).

K. J. Ranson, L. L. Biehl, and C. S. T. Daughtry, “Soybean canopy reflectance modeling data sets,” LARS Tech. Report 071584, Purdue Univ., W. Lafayette, IN (1984).

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Figures (9)

Fig. 1.
Fig. 1. Plant canopy schematic considered as a single layer containing leaves ensemble of bi-Lambertian surfaces
Fig. 2.
Fig. 2. View cosine direction and number of ordinates direction dependences of a finite single leaf plant canopy discrete ordinates characteristics solutions (DOCS) radiance angular discretization error in the visible and near infrared: : ${\mu _0} = 1$, ${r_L} = {t_L}$ and ${\theta _L} = 60^\circ $.
Fig. 3.
Fig. 3. View zenith angle dependence on the reflected and transmitted single canopy plant radiance in the near infrared: ${\tau _L} = 1$, ${\omega _L} = 0.9$, ${r_L} = 0.25$, ${r_s} = 0.2$, ${\mu _0} = 1$ and ${\theta _L} = 60^\circ $. The discrete ordinates characteristics solution (DOCS) is compared to FN [30] and analytical discrete ordinates (ADO) [33] results.
Fig. 4.
Fig. 4. View zenith angle dependence on the reflected and transmitted single plant canopy radiance in the visible predicted using discrete ordinates characteristics solution: ${\tau _L} = 2$, ${\mu _0} = 1$, ${r_s} = 0.1$, ${\omega _L} = 0.1$ and ${r_L} = 0.07$.
Fig. 5.
Fig. 5. View zenith angle dependence for the reflected and transmitted plant canopy radiance in the near infrared for five representative leaves angle distribution.
Fig. 6.
Fig. 6. Angular discretization dependences of discrete ordinates characteristics solutions (DOCS) radiance computational time for different canopies types: ${\mu _0} = 1$, ${\omega _L} = 0.9$, ${r_L} = {t_L}$ and ${\theta _L} = 60^\circ $.
Fig. 7.
Fig. 7. View cosine direction and number of ordinates direction dependences of an infinite single leaf plant canopy discrete ordinates characteristics solutions (DOCS) reflected radiance angular discretization error: ${\mu _0} = 1$, ${\omega _L} = 0.1$, ${r_L} = {t_L}$ and ${\theta _L} = 60^\circ $
Fig. 8.
Fig. 8. Spectral dependences of a single soybeans leaves plant canopy: sensibility of leaf-area index (LAI) standard deviation error on DOCS predictions comparative with Ranson et al. [37] measurements.
Fig. 9.
Fig. 9. Leaf angle distribution and soil reflectance effects on discrete ordinates characteristics solutions soybeans spectral reflectance factor comparative with Ranson et al. [37] measurements.

Tables (6)

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Table 1. Discrete ordinates characteristics solution reflected and transmitted radiances in the visible and near infrared for a finite single leaf plant canopy:τL=1, μ0=1, and θL=60

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Table 2. Discrete ordinates characteristics solution reflected and transmitted radiances in the near infrared of a finite single leaf canopy: μ0=1, τL=1, ωL=0.9, rL=0.25, θL=60, rs=0.2

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Table 3. Discrete ordinates characteristics solutions (DOCS) hemispherical reflectance and transmittance in the visible for finite plant canopy considering four leaf orientations compared to FN [30] and analytical discrete ordinates (ADO) [33] predictions : τL=2, ω=0.1, rL=0.07 rs=0.1 and μ0=1.

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Table 4. Discrete ordinates characteristics solutions (DOCS) hemispherical reflectance and transmittance predictions for different canopies compared with FN [30] and analytical discrete ordinates (ADO) [33] predictions: τL=1, μ0=1, rs=0.

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Table 5. Discrete ordinates characteristics solutions (DOCS) reflected radiances, 102×I(τ,μ), for a semi-infinite canopy in the visible compared with benchmark Chandrasekhar, and FN [30] and analytical discrete ordinates (ADO) [33] semi-analytical results: μ0=1, ωL=0.1, rL=0.05 and θL=60

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Table 6. Common canopy leaf angle distribution function and orientation characteristics [1,40,41]

Equations (83)

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μIτ+G(Ω)I(τ,Ω)=ωL4π4πG(Ω)P(ΩΩ)I(τ,Ω)dΩ
I(τ0=0,Ω)=I0δ(ΩΩ0), μ>0
I(τL,Ω)=rsπ2π+ I(τL,Ω)μdΩ , μ<0
I(τ,Ω)=Id(τ,Ω)+I0exp(η0τ)δ(ΩΩ0)
μIτ+G(μ)I(τ,μ)=J(τ,μ)
I(τ0=0,μ)=F(0), μ>0
I(τL,μ)=F(τL)+2ρd01I(τL,μ)μdμ ,μ<0
J(τ,μ)=ωL211G(μ)P(μ,μ)I(τ,μ)dμ+Q0(τ,μ)
F(τ)=2ρdμ0I0exp(η0τ)τ/τL
Q0(τ,μ)=I02ωLG(μ0)P(μ0,μ)exp(η0τ)
dIdτ+PI=S0exp(η0τ)
R0I(0)=1F(τ0=0)
RτLI(τL)=1F(τL)
R0=[I,R0], RτL=[RτL,I]
R{0,τL},=2ρ{0,τL}dwμ
P,=(Gδ,ωL2wGP,)/μ
S0,=ωL0G0P0,/4πμ
P,=4Γ,/ωLG
G=1NqwL,igL(μL,i)ψ(μ,μL,i)
Γ,=i=1NqwL,igL(μL,i)Ψ(μ,μ,μL,i)
I(τ)=VL(τ)
ddτ[L+L]+[Λ+00Λ][L+L]=S0
S0=[S0+,S0]T=V1S0
L±(τ)=exp(λ±ξ)C±+exp~ξ[(λ±η0)ξ]S0±
exp(λ±ξ)=diag{exp(λ±ξ)}
exp~ξ(λ±ξ)=exp(η0τ){diag{ξ}δ(λ±η0)+[1δ(λ±η0)]exp~(λ±ξ)}
exp~(λ±ξ)=diag{exp~(λ±ξ)}withexp~(λ±ξ)=[1exp(λ±ξ)]/λ±
[R0Vexp(ξ0)RLVexp(ξL)][C+C]=[1F(0)R0Vexp~ξ(ξ0)S0+1F(τL)RLVexp~ξ(ξL)S0]
exp(ξ)=[exp(λ+ξ),exp(λξ)]T,exp~ξ(ξ)=[exp~ξ(λ+ξ),exp~ξ(λξ)]T
Thm=[2π01I(τ,μ)μdμ+qcol]/qInc
Rhm=2π01I(τ,μ)μdμ/qInc
q(τ)=Thm+Rhm
Thm=[μ+W+I+(τ)+qcol(τ)]/qInc
Rhm=μ+W+I(τ)/qInc
I(τ,+μ)=η0τJ(t,+μ)exp[η(τt)]dt+I(0,+μ)exp(ητ)
I(τ,0)=J(τ,0)
I(τ,μ)=ηττLJ(t,μ)exp[η(tτ)]dt+I(τL,μ)exp[η(τLτ)]
J(τ,μ)=ωL2P(μ)WI(τ)+Q0(τ,μ)
I(τ,+μ)=η[ωL2P(μ)WI^(τ,μ)+Q0+(τ,+μ)]
I(τ,μ)=μ+W+I+(τL)exp[η(τLτ)]+η[ωL2P(μ)WI^(τ,μ)+Q0(τ,μ)]
L^±(τ,μ)=EXP±^(μ)C+EXP±(μ)S0
Q0+(τ,μ)=Q0(τ,μ){τδ(η0η)+[1δ(η0η)]exp~[(ηη0)τ]}
Q0(τ,μ)=Q0exp~[(η0+η)(τLτ)]
EXP+^(μ)=exp(λξ){τδ(λη)+[1δ(λη)]exp~[(λη)τ]}
EXP^(μ)=exp(λξ)exp~[(λ+η)(τLτ)]
EXP±(μ)=exp(η0τ){diag{Δξ±}δ(λη0)+[1δ(λη0)]diag{Δ±}}
Δξ+=τ(ξ+τ2)δ(η0η)+[1δ(η0η)]ζτ+ζ0+exp[(η0η)τ]η0η
Δξ=1η0+η{ζτ+ζ0exp[(η0+η)(τLτ)]}
ζτ±=ξ1/(η0η)
Δ+=(Δ+,1+Δ+,2)/(λη0)
Δ+,1=τδ(η0η)+[1δ(η0η)]exp~[(ηη0)τ]
Δ+,2={τexp[(ηη0)ξ]λ=ηexp[(λη0)ξ]exp~[(λη)τ]else
Δ=exp[(λη0)ξ]exp~[(λ+η)(τLτ)]+exp~[(η0+η)(τLτ)]
uL=AL/ϑL
τ=0zuL(z)dz
12π2πgL(z,ΩL)dΩL=1
0π/2gL(z,θL)sinθLdθL=1
χL=±120π/2|sinθLgL(z,θL)|dθL
G(z,Ω)=2πgL(z,Ω)|ΩLΩ|dΩL
14π4πG(z,Ω)dΩ=1
G(z,θ)=0π/2gL(z,θL)ψ(θ,θL)sinθLdθL
ψ={|cosθcosθf|,|ctnθctnθL|>1cosθcosθf[2cos1(ctnθctnθL)1]+2πsinθsinθfsinϕ(θ),else
G(z,θ)=ψ(θ,θL)
p(z,Ω)={1+G(z,Ω)z/cosθexp[G(z,Ω)z/cosθ][1+G(z,Ω)z/cosθ]1
σ{e,s}(Ω)=uL{G(Ω),ω(Ω)G(Ω)}
ω(Ω)=σs(Ω)/σe(Ω)
4πf(ΩΩ;ΩL)dΩ=ωL(Ω)
ωL(Ω)=rL(Ω)+tL(Ω)
{rL,tL}=2π±f(ΩΩ;ΩL)dΩ
f(ΩΩ;ΩL)=rL|cosζ|π,cosζcosζ<0
f(ΩΩ;ΩL)=tL|cosζ|π,cosζcosζ<0
ω=ωL=rL+tL
14πΦ(ΩΩ)=uL(z)πΓ(ΩΩ)
Γ(Ω,Ω)=122π|ΩΩL|gL(ΩL)f(Ω,Ω;ΩL)dΩL
1π4πΓ(ΩΩ)dΩ=ωL(Ω)G(Ω)
P(ΩΩ)=Φ(ΩΩ)/σs(Ω)
14π4πP(ΩΩ)dΩ=1
P(ΩΩ)=4Γ(ΩΩ)/ωLG(Ω)
P(μμ)=4Γ(μμ)/ωLG(μ)
Γ(μμ)=01gL(μ)Ψ(μ,μ)dμL
Ψ(μ,μ)=[tLΨ+(μ,μ)+rLΨ(μ,μ)]Ψ±(μ,μ)=H(μ)H(±μ)+H(μ)H(μ)H(μ)={cosfθ,ctnθctnθL>10,ctnθctnθL<1[cosfθϕθ+sinfθsinϕθ]/π,else
Γ(μμ)=Γ(μμ)=Γ(μμ)
G(μ)=G(μ)

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