Abstract

Solar radiative transfer through a coupled system of atmosphere and plant canopy is modeled as a multiple-scattering problem through a layered medium of random scatterers. The radiative transfer equation is solved by the discrete-ordinates finite-element method. Analytic expressions are derived that allow the calculation of scattering and absorption cross sections for any plant canopy layer from measurable biophysical parameters such as the leaf area index, leaf angle distribution, and individual leaf reflectance and transmittance data. An expression for a canopy scattering phase function is also given. Computational results are in good agreement with spectral reflectance measurements directly above a soybean canopy, and the concept of greenness- and brightness-transforms of Landsat MSS data is reconfirmed with our computed results. A sensitivity analysis with the coupled atmosphere/canopy model quantifies how satellite-sensed spectral radiances are affected by increased atmospheric aerosols, by varying leaf area index, by anisotropic leaf scattering, and by non-Lambertian soil boundary conditions. Possible extensions to a 2-D model are also discussed.

© 1985 Optical Society of America

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References

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  1. N. M. Short, The Landsat Tutorial Workbook: Basics of Satellite Remote Sensing, NASA Publ. 1078 (1982).Also, Landsat Sensor Design and Operation, Proceedings Landsat Short Course, 16–19 Aug. 1983, U. California, Santa Barbara.
  2. R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program, Part 1: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memo. 86078, Goddard Space Flight Center (Mar.1984).
  3. Quarterly Technical Interchange: Assessing Key Vegetation Characteristics from Remote Sensing, NASA Report JSC-18894, Lyndon B. Johnson Space Center (Oct.1983).
  4. R. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. 1 (Los Alamos Scientific Laboratory, June1953).
  5. J. A. Weinman, P. J. Guetter, “Penetration of Solar Irradiances Through the Atmosphere and Plant Canopies,”J. Appl. Meteorol. 11, 136 (1972).
    [CrossRef]
  6. R. S. Fraser, “Computed Atmospheric Corrections for Satellite Data,” Proc. Soc. Photo-Opt: Instrum. Eng. 51, 64 (1974).
  7. Y. J. Kaufman, “Atmospheric Effects on Remote Sensing of Surface Reflectance,” Proc. SPIE Conference 475 on Critical Reviews of Remote Sensing, Washington, D.C., 1–2 May 1984.
  8. G. H. Suits, “The Calculation of the Directional Reflectance of a Vegetative Canopy,” Remote Sensing Environ. 2, 117 (1972).
    [CrossRef]
  9. J. A. Smith, “Role of Scene Radiation Models in Remote Sensing,” in Proceedings, 1982 Machine Processing of Remotely Sensed Data Symposium (1982), pp. 546–549.
  10. S. A. W. Gerstl, A. Zardecki, “Discrete-Ordinates Finite-Element Method for Atmospheric Radiative Transfer and Remote Sensing,” Appl. Opt. 23, 94 (1984) (this issue).
  11. R. D. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for onedant,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).
  12. B. G. Carlson, K. D. Lathrop, “Transport Theory—The Method of Discrete Ordinates,” in Computing Methods of Reactor Physics, H. Greenspan et al., Eds. (Gordon & Breach, New York, 1968), Chap. 3, pp. 171–261.
  13. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).
  14. H. C. van de Hulst, Multiple Light Scattering Tables, Formulas, and Applications (Academic, New York, 1980).
  15. E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).
  16. J. Ross, T. Nilson, in Photosynthesis of Productive Systems, A. A. Nichiporovich, Ed. (Israel Program for Science Translation, Jerusalem, 1967), pp. 75–99.
  17. J. Ross, The Radiation Regime and Architecture of Plant Stands (Dr. W. Junk Publ., The Hague, 1981).
    [CrossRef]
  18. D. M. Gates, H. J. Keegan, J. C. Schleter, V. R. Weidner, “Spectral Properties of Plants,” Appl. Opt. 4, 11 (1965).
    [CrossRef]
  19. N. J. J. Bunnik, “The Multispectral Reflectance of Shortwave Radiation by Agricultural Crops in Relation with their Morphological and Optical Properties,” Thesis, Agricultural University Wageningen, The Netherlands (1978).
  20. H. T. Breece, R. A. Holmes, “Bidirectional Scattering Characteristics of Healthy Green Soybean and Corn Leaves in vivo,” Appl. Opt. 10, 119 (1971).
    [CrossRef]
  21. M. H. Randolph, J. A. Smith, Colorado State University; private communication (Jan.1984).
  22. K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, M. E. Bauer, “Soybean Canopy Reflectance as a Function of View and Illumination Geometry,” AGRISTARS Tech. Rep. SR-P2-04278 (NAS9-15466) (Purdue U, Laboratory for Applications of Remote Sensing, Apr.1982).
  23. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical Considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. U.S. Monogr. 160, Institute for Basic Standards, Washington, D.C. (1977).
  24. R. J. Kauth, G. S. Thomas, “The Tasselled Cap-A Graphic Description of the Spectral-Temporal Development of Agricultural Crops as Seen by Landsat,” in Proceedings, Symposium on Machine Processing of Remotely Sensed Data (Purdue U., West Lafayette, Ind., 1976).
  25. W. R. Johnson, M. L. Sestak, “An Analysis of Haze Effects on Landsat Multispectral Scanner Data,” NASA Report JSC-17127 (SR-L1-04071), Lyndon B. Johnson Space Center (Mar.1981).
  26. F. Hall, NASA Johnson Space Center, Earth Sciences Branch; private communication (Feb.1984).

1984 (1)

1982 (2)

J. A. Smith, “Role of Scene Radiation Models in Remote Sensing,” in Proceedings, 1982 Machine Processing of Remotely Sensed Data Symposium (1982), pp. 546–549.

N. M. Short, The Landsat Tutorial Workbook: Basics of Satellite Remote Sensing, NASA Publ. 1078 (1982).Also, Landsat Sensor Design and Operation, Proceedings Landsat Short Course, 16–19 Aug. 1983, U. California, Santa Barbara.

1974 (1)

R. S. Fraser, “Computed Atmospheric Corrections for Satellite Data,” Proc. Soc. Photo-Opt: Instrum. Eng. 51, 64 (1974).

1972 (2)

G. H. Suits, “The Calculation of the Directional Reflectance of a Vegetative Canopy,” Remote Sensing Environ. 2, 117 (1972).
[CrossRef]

J. A. Weinman, P. J. Guetter, “Penetration of Solar Irradiances Through the Atmosphere and Plant Canopies,”J. Appl. Meteorol. 11, 136 (1972).
[CrossRef]

1971 (1)

1965 (1)

Bauer, M. E.

K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, M. E. Bauer, “Soybean Canopy Reflectance as a Function of View and Illumination Geometry,” AGRISTARS Tech. Rep. SR-P2-04278 (NAS9-15466) (Purdue U, Laboratory for Applications of Remote Sensing, Apr.1982).

Biehl, L. L.

K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, M. E. Bauer, “Soybean Canopy Reflectance as a Function of View and Illumination Geometry,” AGRISTARS Tech. Rep. SR-P2-04278 (NAS9-15466) (Purdue U, Laboratory for Applications of Remote Sensing, Apr.1982).

Breece, H. T.

Brinkley, F. W.

R. D. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for onedant,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

Bunnik, N. J. J.

N. J. J. Bunnik, “The Multispectral Reflectance of Shortwave Radiation by Agricultural Crops in Relation with their Morphological and Optical Properties,” Thesis, Agricultural University Wageningen, The Netherlands (1978).

Carlson, B. G.

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

Case, R. M.

R. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. 1 (Los Alamos Scientific Laboratory, June1953).

de Hoffmann, F.

R. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. 1 (Los Alamos Scientific Laboratory, June1953).

Deering, D. W.

R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program, Part 1: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memo. 86078, Goddard Space Flight Center (Mar.1984).

Fenn, R. W.

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

Fraser, R. S.

R. S. Fraser, “Computed Atmospheric Corrections for Satellite Data,” Proc. Soc. Photo-Opt: Instrum. Eng. 51, 64 (1974).

Gates, D. M.

Gerstl, S. A. W.

Guetter, P. J.

J. A. Weinman, P. J. Guetter, “Penetration of Solar Irradiances Through the Atmosphere and Plant Canopies,”J. Appl. Meteorol. 11, 136 (1972).
[CrossRef]

Hall, F.

F. Hall, NASA Johnson Space Center, Earth Sciences Branch; private communication (Feb.1984).

Holmes, R. A.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical Considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. U.S. Monogr. 160, Institute for Basic Standards, Washington, D.C. (1977).

Johnson, W. R.

W. R. Johnson, M. L. Sestak, “An Analysis of Haze Effects on Landsat Multispectral Scanner Data,” NASA Report JSC-17127 (SR-L1-04071), Lyndon B. Johnson Space Center (Mar.1981).

Kaufman, Y. J.

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

Kauth, R. J.

R. J. Kauth, G. S. Thomas, “The Tasselled Cap-A Graphic Description of the Spectral-Temporal Development of Agricultural Crops as Seen by Landsat,” in Proceedings, Symposium on Machine Processing of Remotely Sensed Data (Purdue U., West Lafayette, Ind., 1976).

Keegan, H. J.

Lathrop, K. D.

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

Liou, K. N.

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

Marr, D. R.

R. D. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for onedant,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

Murphy, R. E.

R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program, Part 1: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memo. 86078, Goddard Space Flight Center (Mar.1984).

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical Considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. U.S. Monogr. 160, Institute for Basic Standards, Washington, D.C. (1977).

Nilson, T.

J. Ross, T. Nilson, in Photosynthesis of Productive Systems, A. A. Nichiporovich, Ed. (Israel Program for Science Translation, Jerusalem, 1967), pp. 75–99.

O'Dell, R. D.

R. D. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for onedant,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

Placzek, G.

R. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. 1 (Los Alamos Scientific Laboratory, June1953).

Randolph, M. H.

M. H. Randolph, J. A. Smith, Colorado State University; private communication (Jan.1984).

Ranson, K. J.

K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, M. E. Bauer, “Soybean Canopy Reflectance as a Function of View and Illumination Geometry,” AGRISTARS Tech. Rep. SR-P2-04278 (NAS9-15466) (Purdue U, Laboratory for Applications of Remote Sensing, Apr.1982).

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical Considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. U.S. Monogr. 160, Institute for Basic Standards, Washington, D.C. (1977).

Robinson, B. F.

K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, M. E. Bauer, “Soybean Canopy Reflectance as a Function of View and Illumination Geometry,” AGRISTARS Tech. Rep. SR-P2-04278 (NAS9-15466) (Purdue U, Laboratory for Applications of Remote Sensing, Apr.1982).

Ross, J.

J. Ross, The Radiation Regime and Architecture of Plant Stands (Dr. W. Junk Publ., The Hague, 1981).
[CrossRef]

J. Ross, T. Nilson, in Photosynthesis of Productive Systems, A. A. Nichiporovich, Ed. (Israel Program for Science Translation, Jerusalem, 1967), pp. 75–99.

Schleter, J. C.

Sestak, M. L.

W. R. Johnson, M. L. Sestak, “An Analysis of Haze Effects on Landsat Multispectral Scanner Data,” NASA Report JSC-17127 (SR-L1-04071), Lyndon B. Johnson Space Center (Mar.1981).

Shettle, E. P.

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

Short, N. M.

N. M. Short, The Landsat Tutorial Workbook: Basics of Satellite Remote Sensing, NASA Publ. 1078 (1982).Also, Landsat Sensor Design and Operation, Proceedings Landsat Short Course, 16–19 Aug. 1983, U. California, Santa Barbara.

Smith, J. A.

J. A. Smith, “Role of Scene Radiation Models in Remote Sensing,” in Proceedings, 1982 Machine Processing of Remotely Sensed Data Symposium (1982), pp. 546–549.

M. H. Randolph, J. A. Smith, Colorado State University; private communication (Jan.1984).

Suits, G. H.

G. H. Suits, “The Calculation of the Directional Reflectance of a Vegetative Canopy,” Remote Sensing Environ. 2, 117 (1972).
[CrossRef]

Thomas, G. S.

R. J. Kauth, G. S. Thomas, “The Tasselled Cap-A Graphic Description of the Spectral-Temporal Development of Agricultural Crops as Seen by Landsat,” in Proceedings, Symposium on Machine Processing of Remotely Sensed Data (Purdue U., West Lafayette, Ind., 1976).

van de Hulst, H. C.

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

Vanderbilt, V. C.

K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, M. E. Bauer, “Soybean Canopy Reflectance as a Function of View and Illumination Geometry,” AGRISTARS Tech. Rep. SR-P2-04278 (NAS9-15466) (Purdue U, Laboratory for Applications of Remote Sensing, Apr.1982).

Weidner, V. R.

Weinman, J. A.

J. A. Weinman, P. J. Guetter, “Penetration of Solar Irradiances Through the Atmosphere and Plant Canopies,”J. Appl. Meteorol. 11, 136 (1972).
[CrossRef]

Zardecki, A.

Appl. Opt. (3)

J. Appl. Meteorol. (1)

J. A. Weinman, P. J. Guetter, “Penetration of Solar Irradiances Through the Atmosphere and Plant Canopies,”J. Appl. Meteorol. 11, 136 (1972).
[CrossRef]

Proc. Soc. Photo-Opt: Instrum. Eng. (1)

R. S. Fraser, “Computed Atmospheric Corrections for Satellite Data,” Proc. Soc. Photo-Opt: Instrum. Eng. 51, 64 (1974).

Proceedings, 1982 Machine Processing of Remotely Sensed Data Symposium (1)

J. A. Smith, “Role of Scene Radiation Models in Remote Sensing,” in Proceedings, 1982 Machine Processing of Remotely Sensed Data Symposium (1982), pp. 546–549.

Remote Sensing Environ. (1)

G. H. Suits, “The Calculation of the Directional Reflectance of a Vegetative Canopy,” Remote Sensing Environ. 2, 117 (1972).
[CrossRef]

The Landsat Tutorial Workbook: Basics of Satellite Remote Sensing (1)

N. M. Short, The Landsat Tutorial Workbook: Basics of Satellite Remote Sensing, NASA Publ. 1078 (1982).Also, Landsat Sensor Design and Operation, Proceedings Landsat Short Course, 16–19 Aug. 1983, U. California, Santa Barbara.

Other (18)

R. E. Murphy, D. W. Deering, “Fundamental Remote Sensing Science Research Program, Part 1: Status Report of the Scene Radiation and Atmospheric Effects Characterization Project,” NASA Tech. Memo. 86078, Goddard Space Flight Center (Mar.1984).

Quarterly Technical Interchange: Assessing Key Vegetation Characteristics from Remote Sensing, NASA Report JSC-18894, Lyndon B. Johnson Space Center (Oct.1983).

R. M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. 1 (Los Alamos Scientific Laboratory, June1953).

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

R. D. O'Dell, F. W. Brinkley, D. R. Marr, “User's Manual for onedant,” Los Alamos National Laboratory Report LA-9184-M (Feb.1982).

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

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

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

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

J. Ross, T. Nilson, in Photosynthesis of Productive Systems, A. A. Nichiporovich, Ed. (Israel Program for Science Translation, Jerusalem, 1967), pp. 75–99.

J. Ross, The Radiation Regime and Architecture of Plant Stands (Dr. W. Junk Publ., The Hague, 1981).
[CrossRef]

N. J. J. Bunnik, “The Multispectral Reflectance of Shortwave Radiation by Agricultural Crops in Relation with their Morphological and Optical Properties,” Thesis, Agricultural University Wageningen, The Netherlands (1978).

M. H. Randolph, J. A. Smith, Colorado State University; private communication (Jan.1984).

K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, M. E. Bauer, “Soybean Canopy Reflectance as a Function of View and Illumination Geometry,” AGRISTARS Tech. Rep. SR-P2-04278 (NAS9-15466) (Purdue U, Laboratory for Applications of Remote Sensing, Apr.1982).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, “Geometrical Considerations and Nomenclature for Reflectance,” Natl. Bur. Stand. U.S. Monogr. 160, Institute for Basic Standards, Washington, D.C. (1977).

R. J. Kauth, G. S. Thomas, “The Tasselled Cap-A Graphic Description of the Spectral-Temporal Development of Agricultural Crops as Seen by Landsat,” in Proceedings, Symposium on Machine Processing of Remotely Sensed Data (Purdue U., West Lafayette, Ind., 1976).

W. R. Johnson, M. L. Sestak, “An Analysis of Haze Effects on Landsat Multispectral Scanner Data,” NASA Report JSC-17127 (SR-L1-04071), Lyndon B. Johnson Space Center (Mar.1981).

F. Hall, NASA Johnson Space Center, Earth Sciences Branch; private communication (Feb.1984).

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Figures (12)

Fig. 1
Fig. 1

Concepts of modeling solar radiative transfer for remote sensing of plant canopies, B.C. = boundary condition.

Fig. 2
Fig. 2

Horizontally and vertically projected leaf area with leaf inclination angle θL.

Fig. 3
Fig. 3

Measured vs calculated canopy reflectance at the top of the canopy. Clear rural atmosphere with low aerosol component of optical depth 0.1 at λ = 0.55 μm.

Fig. 4
Fig. 4

Measured vs calculated canopy reflectance at the top of the canopy. Polluted rural atmosphere with high aerosol component of optical depth 0.6 at λ = 0.55 μm.

Fig. 5
Fig. 5

Angular distributions of soybean canopy reflectance, measured and calculated in visible (RF1: 0.5 ≤ λ ≤ 0.6 μm) and near-IR (RF3: 0.7 ≤ λ ≤ 0.8 μm).

Fig. 6
Fig. 6

Spectral radiance directly above the plant canopy and above the atmosphere for varying aerosol levels indicated by the surface visual range V0. Solar zenith angle θs =34°, near nadir view direction, θυ = 15°.

Fig. 7
Fig. 7

Spectral radiance directly above the plant canopy and above the atmosphere for three different canopy phase function asymmetry parameters: ●, g = 0, Δ, g = −0.25; ×, g = −0.5. Solar zenith angle θs = 34°, near nadir view direction, θυ = 15°.

Fig. 8
Fig. 8

Effects of varying leaf area index on reflected radiance above canopy and above atmosphere. Lambertian soil boundary condition with reflectance 0.2, moderate aerosol load with surface visual range V0 = 23 km.

Fig. 9
Fig. 9

Effects of different soil BRDF boundary conditions on reflected radiance above canopy and atmosphere for near nadir observation. Light aerosol load with surface visual range V0 = 50 km; LAI = 1.5, integral surface reflectance 0.2.

Fig. 10
Fig. 10

Effects of different soil BRDF boundary conditions on reflected radiance for off-nadir observation. V0 = 50 km, LAI = 1.5.

Fig. 11
Fig. 11

Computed Landsat MSS greenness and brightness for varying leaf area index and fixed atmosphere.

Fig. 12
Fig. 12

Computed Landsat MSS greenness and brightness for varying total atmospheric optical depth above plant canopy.

Equations (24)

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

μ d I ( z , μ , ϕ ) d z + σ t ( z ) I ( z , μ , ϕ ) = σ s ( z , Ω Ω ) I ( z , μ , ϕ ) d Ω + μ 0 F 0 δ ( z z 0 ) δ ( μ μ 0 ) δ ( ϕ ϕ 0 ) ,
τ ATM = 0 z σ t ( z ) d z ,
σ s ( z , Ω Ω ) = σ 0 s ( z ) 1 4 π P ( z , Ω Ω ) , = ω σ t 4 π P ( z , Ω Ω ) ,
ω = σ 0 s / σ t = σ 0 s / ( σ a + σ 0 s ) ,
P ( z , Ω Ω ) = 1 g 2 ( z ) [ 1 + g 2 ( z ) 2 g ( z ) ( Ω Ω ) ] 3 / 2 ,
1 2 π g ( θ L , ϕ L )
1 2 π g ( Ω L ) d Ω L = 1
1 2 π 0 π / 2 0 2 π g ( θ L , ϕ L )
σ 0 s ( z , λ ) = [ ρ ( λ ) + τ ( λ ) ] G ( z ) LAI ( z ) Δ ,
σ a ( z , λ ) = [ 1 ρ ( λ ) τ ( λ ) ] G ( z ) LAI ( z ) Δ ,
σ t ( z , λ ) = 1 G ( z ) LAI ( z ) Δ .
G ( z ) = ( 2 π ) ( 4 π ) | Ω Ω L | 1 2 π g ( Ω L ) 1 4 π h ( Ω ) d Ω L d Ω = 1 2 π 0 π / 2 sin θ L d θ L 0 2 π d ϕ L 1 4 π π / 2 π / 2 sin θ d θ × 0 2 π d ϕ | Ω Ω L | g ( θ L , ϕ L ) h ( θ , ϕ ) .
G 0 ( z ) = 1 2 π 0 π / 2 0 2 π | Ω Ω L | g ( θ L , ϕ L ) sin θ L d θ L d ϕ L ,
G s ( z ) = 1 4 π ( 4 π ) d Ω 1 2 π ( 2 π ) d Ω L | Ω Ω L | g ( Ω L ) ,
G s ( z ) = 0 1 | cos Ω ̂ | d ( cos Ω ̂ ) = ½ .
H = 0 π / 2 f ( θ L ) cos θ L d θ L ,
V = 0 π / 2 f ( θ L ) sin θ L d θ L .
g ( θ L , ϕ L ) = H sin θ L δ ( θ L 0 ) + V δ ( θ L π 2 ) .
G 0 ( z ) = H | cos θ 0 | + 2 π V sin θ 0 ,
f ( θ L ) = 2 π ( 1 cos 4 θ L ) .
P ( z , Ω , Ω ) = 1 2 π 0 2 π 0 π / 2 γ ( Ω , Ω ) | Ω Ω L | | Ω Ω L | × g ( θ L , ϕ L ) sin θ L d θ L d ϕ L .
R c ( λ ) = I λ ( z V , θ V , ϕ V ) I λ L ( z V , θ V , ϕ V ) ,
G = 0.48 X 1 0.65 X 2 + 0.09 X 3 + 0.59 X 4 ,
B = 0.33 X 1 + 0.40 X 2 + 0.59 X 3 + 0.62 X 4 ,

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