Abstract

The classical Chandrasekhar's formula relating the surface reflectance to the top of the atmosphere radiance rigorously applies to a Lambertian surface. For a homogeneous non-Lambertian surface in a plane-parallel atmosphere, an extension of this formula was proposed in the 1980s and has been recently implemented in the second simulation of the satellite signal in the solar spectrum (6S) algorithm. To analyze this extension, the rigorous formula of the top of the atmosphere signal is derived in a plane-parallel atmosphere bounded by a homogeneous non-Lambertian surface. Then the 6S algorithm extension is compared with the exact formula and approximations and their validity are pointed out. The methods used for the derivation of the exact formula are classical. They are based on the separation of direct and diffuse components of the radiation fields, on the introduction of the Green's function of the problem, and on integrations of boundary values of the radiation fields with the Green's function.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, 1985).
  2. D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory," J. Quant. Spectrosc. Radiat. Transfer 31, 97-125 (1984).
    [CrossRef]
  3. D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational results," J. Quant. Spectrosc. Radiat. Transfer 31, 279-304 (1984).
    [CrossRef]
  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. A. Lyapustin and T. Z. Muldashev, "Method of spherical harmonics in the radiative transfer problem with non-Lambertian surface," J. Quant. Spectrosc. Radiat. Transfer 61, 545-555 (1999).
    [CrossRef]
  6. A. Lyapustin and Y. Knyazikhin, "Green's function method for radiative transfer problem. I. Homogeneous non-Lambertian surface," Appl. Opt. 40, 3495-3501 (2001).
  7. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, "Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media," Appl. Opt. 27, 2502-2509 (1988).
  8. D. Tanré, M. Herman, P. Y. Deschamps, and A. de Leffe, "Atmospheric modeling for space measurements of ground reflectances, including bidirectional properties," Appl. Opt. 18, 3587-3594 (1979).
  9. D. Tanré, M. Herman, and P. Y. Deschamps, "Influence of the atmosphere on space measurements of directional properties," Appl. Opt. 22, 733-741 (1983).
  10. E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcette, "Second simulation of the satellite signal in the solar spectrum, 6S: an overview," IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
    [CrossRef]
  11. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  12. H. C. van de Hulst, "Scattering in a planetary atmosphere," Astrophys. J. 107, 220-246 (1948).
    [CrossRef]
  13. E. F. Vermote and A. Vermeulen, "Atmospheric correction algorithm: spectral reflectances (MOD09). Algorithm Technical Background Document (1999)," available on-line at http://mod/is/data.gsfc.nasa.gov/modis/atbd/atbdlowbarmod08.pdf.
  14. K. M. Case, "Elementary solutions of the transport equation and their applications," Ann. Phys. (N.Y.) 9, 1-23 (1960).
    [CrossRef]
  15. G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Rheinholt, 1970).
  16. G. I. Marchuk and V. I. Lebedev, Numerical Methods in the Theory of Neutron Transport (Harwood Academic, 1986).
  17. A. A. Ioltukhovski, "Radiative transfer over the surface with an arbitrary reflection: Green's functions method," Transp. Theory Stat. Phys. 28, 349-368 (1999).
  18. Y. Knyazikhin and A. Marshak, "Mathematical aspects of BRDF modeling: adjoint problem and Green's function," Remote Sens. Rev. 18, 1-18 (2000).
  19. Y. Qin, M. A. Box, and P. Douriaguine, "Computation of Green's function for radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 84, 159-168 (2004).
    [CrossRef]
  20. J. Lenoble, Atmospheric Radiative Transfer (Deepak, 1993).
  21. K. N. Liou, Introduction to Atmospheric Radiation (Academic, 2002).

2004

Y. Qin, M. A. Box, and P. Douriaguine, "Computation of Green's function for radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 84, 159-168 (2004).
[CrossRef]

2001

2000

Y. Knyazikhin and A. Marshak, "Mathematical aspects of BRDF modeling: adjoint problem and Green's function," Remote Sens. Rev. 18, 1-18 (2000).

1999

A. A. Ioltukhovski, "Radiative transfer over the surface with an arbitrary reflection: Green's functions method," Transp. Theory Stat. Phys. 28, 349-368 (1999).

A. Lyapustin and T. Z. Muldashev, "Method of spherical harmonics in the radiative transfer problem with non-Lambertian surface," J. Quant. Spectrosc. Radiat. Transfer 61, 545-555 (1999).
[CrossRef]

1998

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

1997

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcette, "Second simulation of the satellite signal in the solar spectrum, 6S: an overview," IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

1988

1984

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory," J. Quant. Spectrosc. Radiat. Transfer 31, 97-125 (1984).
[CrossRef]

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational results," J. Quant. Spectrosc. Radiat. Transfer 31, 279-304 (1984).
[CrossRef]

1983

1979

1960

K. M. Case, "Elementary solutions of the transport equation and their applications," Ann. Phys. (N.Y.) 9, 1-23 (1960).
[CrossRef]

1948

H. C. van de Hulst, "Scattering in a planetary atmosphere," Astrophys. J. 107, 220-246 (1948).
[CrossRef]

Bell, G. I.

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Rheinholt, 1970).

Box, M. A.

Y. Qin, M. A. Box, and P. Douriaguine, "Computation of Green's function for radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 84, 159-168 (2004).
[CrossRef]

Case, K. M.

K. M. Case, "Elementary solutions of the transport equation and their applications," Ann. Phys. (N.Y.) 9, 1-23 (1960).
[CrossRef]

Chandrasekhar, S.

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

de Leffe, A.

Deschamps, P. Y.

Deuzé, J. L.

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcette, "Second simulation of the satellite signal in the solar spectrum, 6S: an overview," IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

Diner, D. J.

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory," J. Quant. Spectrosc. Radiat. Transfer 31, 97-125 (1984).
[CrossRef]

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational results," J. Quant. Spectrosc. Radiat. Transfer 31, 279-304 (1984).
[CrossRef]

Douriaguine, P.

Y. Qin, M. A. Box, and P. Douriaguine, "Computation of Green's function for radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 84, 159-168 (2004).
[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]

Glasstone, S.

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Rheinholt, 1970).

Herman, M.

Ioltukhovski, A. A.

A. A. Ioltukhovski, "Radiative transfer over the surface with an arbitrary reflection: Green's functions method," Transp. Theory Stat. Phys. 28, 349-368 (1999).

Jayaweera, K.

Knyazikhin, Y.

A. Lyapustin and Y. Knyazikhin, "Green's function method for radiative transfer problem. I. Homogeneous non-Lambertian surface," Appl. Opt. 40, 3495-3501 (2001).

Y. Knyazikhin and A. Marshak, "Mathematical aspects of BRDF modeling: adjoint problem and Green's function," Remote Sens. Rev. 18, 1-18 (2000).

Lebedev, V. I.

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

Lenoble, J.

J. Lenoble, Atmospheric Radiative Transfer (Deepak, 1993).

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, 1985).

Liou, K. N.

K. N. Liou, Introduction to Atmospheric Radiation (Academic, 2002).

Lyapustin, A.

A. Lyapustin and Y. Knyazikhin, "Green's function method for radiative transfer problem. I. Homogeneous non-Lambertian surface," Appl. Opt. 40, 3495-3501 (2001).

A. Lyapustin and T. Z. Muldashev, "Method of spherical harmonics in the radiative transfer problem with non-Lambertian surface," J. Quant. Spectrosc. Radiat. Transfer 61, 545-555 (1999).
[CrossRef]

Marchuk, G. I.

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

Marshak, A.

Y. Knyazikhin and A. Marshak, "Mathematical aspects of BRDF modeling: adjoint problem and Green's function," Remote Sens. Rev. 18, 1-18 (2000).

Martonchik, J. V.

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational results," J. Quant. Spectrosc. Radiat. Transfer 31, 279-304 (1984).
[CrossRef]

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory," J. Quant. Spectrosc. Radiat. Transfer 31, 97-125 (1984).
[CrossRef]

Morcette, J. J.

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcette, "Second simulation of the satellite signal in the solar spectrum, 6S: an overview," IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

Muldashev, T. Z.

A. Lyapustin and T. Z. Muldashev, "Method of spherical harmonics in the radiative transfer problem with non-Lambertian surface," J. Quant. Spectrosc. Radiat. Transfer 61, 545-555 (1999).
[CrossRef]

Qin, Y.

Y. Qin, M. A. Box, and P. Douriaguine, "Computation of Green's function for radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 84, 159-168 (2004).
[CrossRef]

Stamnes, K.

Tanré, D.

Tsay, S. C.

van de Hulst, H. C.

H. C. van de Hulst, "Scattering in a planetary atmosphere," Astrophys. J. 107, 220-246 (1948).
[CrossRef]

Vermeulen, A.

E. F. Vermote and A. Vermeulen, "Atmospheric correction algorithm: spectral reflectances (MOD09). Algorithm Technical Background Document (1999)," available on-line at http://mod/is/data.gsfc.nasa.gov/modis/atbd/atbdlowbarmod08.pdf.

Vermote, E. F.

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcette, "Second simulation of the satellite signal in the solar spectrum, 6S: an overview," IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

E. F. Vermote and A. Vermeulen, "Atmospheric correction algorithm: spectral reflectances (MOD09). Algorithm Technical Background Document (1999)," available on-line at http://mod/is/data.gsfc.nasa.gov/modis/atbd/atbdlowbarmod08.pdf.

Wiscombe, W.

Ann. Phys.

K. M. Case, "Elementary solutions of the transport equation and their applications," Ann. Phys. (N.Y.) 9, 1-23 (1960).
[CrossRef]

Appl. Opt.

Astrophys. J.

H. C. van de Hulst, "Scattering in a planetary atmosphere," Astrophys. J. 107, 220-246 (1948).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcette, "Second simulation of the satellite signal in the solar spectrum, 6S: an overview," IEEE Trans. Geosci. Remote Sens. 35, 675-686 (1997).
[CrossRef]

J. Atmos. Sci.

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

J. Quant. Spectrosc. Radiat. Transfer

A. Lyapustin and T. Z. Muldashev, "Method of spherical harmonics in the radiative transfer problem with non-Lambertian surface," J. Quant. Spectrosc. Radiat. Transfer 61, 545-555 (1999).
[CrossRef]

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. I. Theory," J. Quant. Spectrosc. Radiat. Transfer 31, 97-125 (1984).
[CrossRef]

D. J. Diner and J. V. Martonchik, "Atmospheric transfer of radiation above an inhomogeneous non-Lambertian reflective ground. II. Computational results," J. Quant. Spectrosc. Radiat. Transfer 31, 279-304 (1984).
[CrossRef]

Y. Qin, M. A. Box, and P. Douriaguine, "Computation of Green's function for radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 84, 159-168 (2004).
[CrossRef]

Remote Sens. Rev.

Y. Knyazikhin and A. Marshak, "Mathematical aspects of BRDF modeling: adjoint problem and Green's function," Remote Sens. Rev. 18, 1-18 (2000).

Transp. Theory Stat. Phys.

A. A. Ioltukhovski, "Radiative transfer over the surface with an arbitrary reflection: Green's functions method," Transp. Theory Stat. Phys. 28, 349-368 (1999).

Other

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, 1985).

J. Lenoble, Atmospheric Radiative Transfer (Deepak, 1993).

K. N. Liou, Introduction to Atmospheric Radiation (Academic, 2002).

E. F. Vermote and A. Vermeulen, "Atmospheric correction algorithm: spectral reflectances (MOD09). Algorithm Technical Background Document (1999)," available on-line at http://mod/is/data.gsfc.nasa.gov/modis/atbd/atbdlowbarmod08.pdf.

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Rheinholt, 1970).

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

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

Cited By

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

Alert me when this article is cited.


Equations (63)

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

{ μ I τ ( τ , μ , ϕ ) + I ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π 1 1 p ( τ , μ , ϕ , μ , ϕ ) I ( τ , μ , ϕ ) d μ d ϕ I ( 0 , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) μ > 0 I ( τ o , μ , ϕ ) = a π 0 2 π 0 1 I ( τ o , μ , ϕ ) μ d μ d ϕ μ < 0 ,
{ μ I d τ ( τ , μ , ϕ ) + I d ( τ , μ , ϕ ) = 0 I d ( 0 , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) μ > 0 .
I d ( τ , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) exp ( τ / μ )
{ μ I s τ ( τ , μ , ϕ ) + I s ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π 1 1 p ( τ , μ , ϕ , μ , ϕ ) I s ( τ , μ , ϕ ) d μ d ϕ + ω ( τ ) F o 4 π p ( τ , μ , ϕ , μ o , ϕ o ) exp ( τ / μ o ) I s ( 0 , μ , ϕ ) = 0 μ > 0 I s ( τ o , μ , ϕ ) = a [ F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 I s ( τ o , μ , ϕ ) μ d μ d ϕ ] μ < 0 .
I s ( τ , μ , ϕ ) = I atm ( τ , μ , ϕ ) + I sur ( τ , μ , ϕ ) ,
{ μ I atm τ ( τ , μ , ϕ ) + I atm ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π 1 1 p ( τ , μ , ϕ , μ , ϕ ) I atm ( τ , μ , ϕ ) d μ d ϕ + ω ( τ ) F o 4 π p ( τ , μ , ϕ , μ o , ϕ o ) exp ( τ / μ o ) I atm ( 0 , μ , ϕ ) = 0 μ > 0 I atm ( τ o , μ , ϕ ) = 0 μ < 0 ,
{ μ I sur τ ( τ , μ , ϕ ) + I sur ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π - 1 1 p ( τ , μ , ϕ , μ , ϕ , ) I sur ( τ , μ , ϕ ) I sur ( 0 , μ , ϕ ) = 0                   μ > 0 I sur τ o , μ , ϕ ) = a [ F o μ o π exp ( - τ o / μ o ) + 1 π 0 2 π 0 1 I atm ( τ o , μ , ϕ ) μ + 1 π 0 2 π 0 1 I sur ( τ o , μ , ϕ ) μ ]       μ < 0
{ μ G τ ( τ , μ , ϕ , μ 1 , ϕ 1 ) + G ( τ , μ , ϕ , μ 1 , ϕ 1 ) = ω ( τ ) 4 π 0 2 π 1 1 p ( τ , μ , ϕ , μ , ϕ ) G ( τ , μ , ϕ , μ 1 , ϕ 1 ) d μ d ϕ G ( 0 , μ , ϕ , μ 1 , ϕ 1 ) = 0 μ > 0 G ( τ o , μ , ϕ , μ 1 , ϕ 1 ) = δ ( μ μ 1 ) δ ( ϕ ϕ 1 ) μ < 0 .
I sur ( τ , μ , ϕ ) = 0 2 π - 1 0 I sur ( τ 0 , μ , ϕ ) G ( τ , μ , ϕ , μ , ϕ ) d μ d ϕ .
{ μ G d τ ( τ , μ , ϕ , μ 1 , ϕ 1 ) + G d ( τ , μ , ϕ , μ 1 , ϕ 1 ) = 0 G d ( τ o , μ , ϕ , μ 1 , ϕ 1 ) = δ ( μ μ 1 ) δ ( ϕ ϕ 1 ) μ < 0 ,
G d ( τ , μ , ϕ , μ 1 , ϕ 1 ) = exp [ ( τ o τ ) / | μ | ] δ ( μ μ 1 ) δ ( ϕ ϕ 1 )
{ μ G s τ ( τ , μ , ϕ , μ 1 , ϕ 1 ) + G s ( τ , μ , ϕ , μ 1 , ϕ 1 ) = ω ( τ ) 4 π 0 2 π 1 1 p ( τ , μ , ϕ , μ , ϕ ) G s ( τ , μ , ϕ , μ 1 , ϕ 1 ) d μ d ϕ + ω ( τ ) 4 π p ( τ , μ , ϕ , μ 1 , ϕ 1 ) exp [ ( τ o τ ) / | μ 1 | ) ] G s ( 0 , μ , ϕ , μ 1 , ϕ 1 ) = 0 μ > 0 G s ( τ o , μ , ϕ , μ 1 , ϕ 1 ) = 0 μ < 0 .
I sur ( τ , μ , ϕ ) = I sur ( τ o , μ , ϕ ) exp [ ( τ o τ ) / | μ | ] + 0 2 π - 1 0 I sur ( τ o , μ , ϕ ) G s ( τ , μ , ϕ , μ , ϕ ) d μ d ϕ .
I sur ( τ o , μ , ϕ ) = a [ F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 I atm ( τ o , μ , ϕ ) μ d μ d ϕ + 1 π 0 2 π 0 1 I sur ( τ o , μ , ϕ ) μ d μ d ϕ ] μ < 0.
I sur ( τ o ) = a [ F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 I atm ( τ o , μ , ϕ ) μ d μ d ϕ + I sur ( τ o ) π 0 2 π 0 1 0 2 π - 1 0 G ( τ o , μ , ϕ , μʺ , ϕʺ ) μ dμʺdϕʺd μ d ϕ ] μ < 0.
S = 1 π 0 2 π 0 1 0 2 π - 1 0 G ( τ o , μ , ϕ , μʺ , ϕʺ ) μ dμʺdϕʺd μ d ϕ ,
I sur ( τ o ) = a [ F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 I atm ( τ o , μ , ϕ ) μ d μ d ϕ + I sur ( τ o ) S ] ,
I sur ( τ o ) = a 1 a S [ F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 I atm ( τ o , μ , ϕ ) μ d μ d ϕ ] .
I sur ( 0 , μ , ϕ ) = I sur ( τ o ) [ exp ( τ o / | μ | ) + 0 2 π 0 1 G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ ] .
I s ( 0 , μ , ϕ ) = I atm ( 0 , μ , ϕ ) + [ exp ( τ o / | μ | ) + 0 2 π - 1 0 G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ ] a 1 a S [ F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 I atm ( τ o , μ , ϕ ) μ d μ d ϕ ] .
ρ s ( 0 , μ , ϕ ) = ρ atm ( 0 , μ , ϕ ) + [ exp ( τ o / | μ | ) + 0 2 π - 1 0 G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ ] a 1 a S [ exp ( τ o / μ o ) + 1 π 0 2 π 0 1 ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ ] .
{ t ( μ , ϕ ) = 0 2 π - 1 0 G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ t ( μ o , ϕ o ) = 1 π 0 2 π 0 1 ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ ,
{ T ( μ , ϕ ) = exp ( τ o / | μ | ) + t ( μ , ϕ ) T ( μ o , ϕ o ) = exp ( τ o / μ o ) + t ( μ o , ϕ o ) ,
ρ s ( 0 , μ , ϕ ) = ρ atm ( 0 , μ , ϕ ) + T ( μ , ϕ ) T ( μ o , ϕ o ) a 1 a S .
{ I 0 sur ( τ o ) = a [ F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 I atm ( τ o , μ , ϕ ) μ d μ ] I n + 1 sur ( τ o ) = I 0 sur ( τ o ) + 1 π 0 2 π 0 1 a I n sur ( τ o , μ , ϕ ) μ d μ .
L ( I n sur ) = 1 π 0 2 π 0 1 a I n sur ( τ o , μ , ϕ ) μ d μ d ϕ = I n sur ( τ o ) a π 0 2 π 0 1 0 2 π - 1 0 G ( τ o , μ , ϕ , μʺ , ϕʺ ) μ dμʺdϕʺd μ d ϕ = I n sur ( τ o ) a S .
I n + 1 sur ( τ o ) = I 0 sur ( τ o ) + I n sur ( τ o ) a S ,
I n + 1 sur ( τ o ) = I 0 sur ( τ o ) k = 0 n + 1 ( a S ) k ,
I sur ( τ o ) = I 0 sur ( τ o ) 1 - a S ,
{ μ I τ ( τ , μ , ϕ ) + I ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π - 1 1 p ( τ , μ , ϕ , μ , ϕ ) I ( τ , μ , ϕ ) d μ d ϕ I ( 0 , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) μ > 0 I ( τ o , μ , ϕ ) = 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) I ( τ o , μ , ϕ ) μ d μ d ϕ μ < 0 .
{ μ I s τ ( τ , μ , ϕ ) + I s ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π - 1 1 p ( τ , μ , ϕ , μ , ϕ ) I s ( τ , μ , ϕ ) d μ d ϕ + ω ( τ ) F o 4 π p ( τ , μ , ϕ , μ o , ϕ o ) exp ( τ / μ o ) I s ( 0 , μ , ϕ ) = 0 μ > 0 I s ( τ o , μ , ϕ ) = a ( μ , ϕ , μ o , ϕ o ) F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) I s ( τ o , μ , ϕ ) μ d μ d ϕ μ < 0 ,
{ μ I sur τ ( τ , μ , ϕ ) + I sur ( τ , μ , ϕ ) = ω ( τ ) 4 π 0 2 π - 1 1 p ( τ , μ , ϕ , μ , ϕ ) I sur ( τ , μ , ϕ ) d μ d ϕ I sur ( 0 , μ , ϕ ) = 0 μ > 0 I sur ( τ o , μ , ϕ ) = a ( μ , ϕ , μ o , ϕ o ) F o μ o π exp ( τ o / μ o ) + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) I atm ( τ o , μ , ϕ ) μ d μ d ϕ + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) I sur ( τ o , μ , ϕ ) μ μ < 0 .
I sur ( 0 , μ , ϕ ) = I sur ( τ o , μ , ϕ ) exp ( τ o / | μ | ) + 0 2 π - 1 0 I sur ( τ o , μ , ϕ ) G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ ,
ρ sur ( 0 , μ , ϕ ) = ρ sur ( τ o , μ , ϕ ) exp ( τ o / | μ | ) + 0 2 π - 1 0 ρ sur ( τ o , μ , ϕ ) G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ .
ρ sur ( τ o , μ , ϕ ) = a ( μ , ϕ , μ o , ϕ o ) exp ( τ o / μ o ) + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ sur ( τ o , μ , ϕ ) μ d μ d ϕ μ < 0.
ρ sur ( 0 , μ , ϕ ) = a ( μ , ϕ , μ 0 , ϕ o ) exp ( τ o / μ o ) exp ( τ o / | μ | ) + exp ( τ o / | μ | ) π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ + exp ( τ o / μ o ) 0 2 π - 1 0 a ( μ , ϕ , μ o , ϕ o ) G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ + 1 π 0 2 π 0 1 0 2 π - 1 0 ρ atm ( τ o , μʺ , ϕʺ ) a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) μʺdμʺdϕʺd μ d ϕ + exp ( τ o / | μ | ) π 0 2 π 0 1 ρ sur ( τ o , μ , ϕ ) a ( μ , ϕ , μ , ϕ ) μ d μ d ϕ + 1 π 0 2 π 0 1 0 2 π - 1 0 ρ sur ( τ o , μʺ , ϕʺ ) a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) μʺdμʺdϕʺd μ d ϕ .
a ¯ ( μ , ϕ , μ o , ϕ o ) = 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ t ( μ o , ϕ o ) ,
a ̄ ( μ , ϕ , μ o , ϕ o ) = 0 2 π - 1 0 a ( μ , ϕ , μ o , ϕ o ) G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ t ( μ , ϕ ) ,
a ̿ ( μ , ϕ , μ o , ϕ o ) = 1 π 0 2 π 0 1 0 2 π - 1 0 ρ atm ( τ o , μʺ , ϕʺ ) a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) μʺdμʺdϕʺd μ d ϕ t ( μ , ϕ ) t ( μ o , ϕ o ) .
ρ sur ( 0 , μ , ϕ ) = exp ( τ o / μ o ) a ( μ , ϕ , μ o , ϕ o ) exp ( τ o / | μ | ) + t ( μ o , ϕ o ) a ¯ ( μ , ϕ , μ o , ϕ o ) exp ( τ o / | μ | ) + exp ( τ o / μ o ) a ¯ ( μ , ϕ , μ o , ϕ o ) t ( μ , ϕ ) + t ( μ o , ϕ o ) a ̿ ( μ , ϕ , μ o , ϕ o ) t ( μ , ϕ ) + exp ( τ o / | μ | ) π 0 2 π 0 1 ρ sur ( τ o , μ , ϕ ) a ( μ , ϕ , μ , ϕ ) μ d μ d ϕ + 1 π 0 2 π 0 1 0 2 π - 1 0 ρ sur ( τ o , μʺ , ϕʺ ) a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) μʺdμʺdϕʺd μ d ϕ .
{ ρ 0 sur ( τ o , μ , ϕ ) = a ( μ , ϕ , μ o , ϕ o ) exp ( τ o / μ o ) + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ ρ n + 1 sur ( τ o , μ , ϕ ) = ρ 0 sur ( τ o , μ , ϕ ) + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ n sur ( τ o , μ , ϕ ) μ d μ d ϕ .
L ( ρ n sur ) ( τ o , μ , ϕ ) = 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ n sur ( τ o , μ , ϕ ) μ d μ d ϕ .
L ( ρ n sur ) ( τ o , μ , ϕ ) = a ̿ ( μ , ϕ , μ o , ϕ o ) ρ n sur ( τ o , μ , ϕ ) 1 π 0 2 π 0 1 0 2 π - 1 0 a ( μ , ϕ , μ , ϕ ) a ̿ ( μ , ϕ , μ o , ϕ o ) ρ n sur ( τ o , μʺ , ϕʺ ) ρ n sur ( τ o , μ , ϕ ) G ( τ o , μ , ϕ , μʺ , ϕʺ ) μ dμʺdϕʺd μ d ϕ .
L ( ρ n sur ) ( τ o , μ , ϕ ) = a ̿ ( μ , ϕ , μ o , ϕ o ) ρ n sur ( τ o , μ , ϕ ) S n ( μ , ϕ , μ o , ϕ o ) ,
ρ 6 S sur ( 0 , μ , ϕ ) = exp ( τ o / μ o ) a ( μ , ϕ , μ o , ϕ o ) exp ( τ o / | μ | ) + t ( μ o , ϕ o ) a ¯ 6 S ( μ , ϕ , μ o , ϕ o ) exp ( τ o / | μ | ) + exp ( τ o / μ o ) a ¯ 6 S ( μ , ϕ , μ o , ϕ o ) t ( μ , ϕ ) + t ( μ o , ϕ o ) a ̿ 6 S ( μ , ϕ , μ o , ϕ o ) t ( μ , ϕ ) + a ̿ 6 S 2 T ( μ o , ϕ o ) T ( μ , ϕ ) S 1 - S a ̿ 6 S .
a ̄ 6 S ( μ , ϕ , μ o , ϕ o ) = a ¯ 6 S ( μ o , ϕ o , μ , ϕ ) .
a ̿ 6 S = 0 2 π 0 1 0 2 π - 1 0 a ( μ , ϕ , μʺ , ϕʺ ) μ μʺdμʺdϕʺd μ d ϕ 0 2 π 0 1 0 2 π - 1 0 μ μʺdμʺdϕʺd μ d ϕ .
a ̿ Tanré ( μ , ϕ , μ o , ϕ o ) = 1 π 0 2 π 0 1 0 2 π - 1 0 ρ atm ( τ o , μʺ , ϕʺ , μ o , ϕ o ) a ( μ , ϕ , μʺ , ϕʺ ) ρ atm ( 0 , μ , ϕ , μ , ϕ ) μʺdμʺdϕʺ μ d μ d ϕ t ( μ , ϕ ) 0 2 π - 1 0 ρ atm ( 0 , μ , ϕ ) μ d μ d ϕ ,
L ( ρ n sur ) ( μ , ϕ ) = 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) ρ n sur ( τ o , μ , ϕ ) μ d μ d ϕ
= a ̿ ( μ , ϕ , μ o , ϕ o ) ρ n sur ( τ o , μ , ϕ ) S n ( τ o , μ , ϕ , μ o , ϕ o )
ρ ˜ n + 1 sur ( τ o , μ , ϕ ) = ρ 0 sur ( τ o , μ , ϕ ) + a ̿ ( μ , ϕ , μ o , ϕ o ) S ρ ˜ n sur ( τ o , μ , ϕ ) ,
ρ ˜ n + 1 sur ( τ o , μ , ϕ ) = ρ 0 sur ( τ o , μ , ϕ ) k = 0 n + 1 [ a ̿ ( μ , ϕ , μ o , ϕ o ) S ] k ,
ρ ˜ sur ( τ o , μ , ϕ ) = ρ 0 sur ( τ o , μ , ϕ ) 1 a ̿ ( μ , ϕ , μ o , ϕ o ) S .
ρ 0 sur ( τ o , μ , ϕ ) = a ̿ ( μ , ϕ , μ o , ϕ o ) [ a ( μ , ϕ , μ o , ϕ o ) a ̿ ( μ , ϕ , μ o , ϕ o ) exp ( τ o / μ o ) + 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) a ̿ ( μ , ϕ , μ o , ϕ o ) ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ ] .
t ˜ ( μ , ϕ , μ o , ϕ o ) = 1 π 0 2 π 0 1 a ( μ , ϕ , μ , ϕ ) a ̿ ( μ , ϕ , μ o , ϕ o ) ρ atm ( τ o , μ , ϕ ) μ d μ d ϕ .
ρ 0 sur ( τ o , μ , ϕ ) = a ̿ ( μ , ϕ , μ o , ϕ o ) [ a ( μ , ϕ , μ o , ϕ o ) a ̿ ( μ , ϕ , μ o , ϕ o ) exp ( τ o / μ o ) + t ˜ ( μ , ϕ , μ o , ϕ o ) ] = a ̿ ( μ , ϕ , μ o , ϕ o ) T ˜ ( μ , ϕ , μ o , ϕ o ) .
ρ ˜ sur ( τ o , μ , ϕ ) = a ̿ ( μ , ϕ , μ o , ϕ o ) T ˜ ( μ , ϕ , μ o , ϕ o ) 1 - a ̿ ( μ , ϕ , μ o , ϕ o ) S ,
1 π 0 2 π 0 1 ρ sur ( τ o , μ , ϕ ) a ( μ , ϕ , μ , ϕ ) μ d μ d ϕ = L ( ρ sur ) ( μ , ϕ ) ρ ˜ sur ( τ o , μ , ϕ ) a ̿ ( μ , ϕ , μ o , ϕ o ) S ,
1 π 0 2 π 0 1 0 2 π - 1 0 ρ sur ( τ o , μʺ , ϕʺ ) a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) μʺdμʺdϕʺd μ d ϕ = 1 π 0 2 π 0 1 0 2 π - 1 0 ρ sur ( τ o , μ‴ , ϕ‴ ) G ( τ o , μʺ , ϕʺ , μ‴ , ϕ‴ ) μʺdμ‴dϕ‴ × 0 2 π - 1 0 a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) dμʺdϕʺd μ d ϕ .
1 π 0 2 π 0 1 0 2 π - 1 0 ρ sur ( τ o , μ‴ , ϕ‴ ) G ( τ o , μʺ , ϕʺ , μ‴ , ϕ‴ ) μʺdμ‴dϕ‴dμʺdϕʺ ρ ˜ sur ( τ o , μ , ϕ ) 1 π 0 2 π 0 1 0 2 π - 1 0 G ( τ o , μʺ , ϕʺ , μ‴ , ϕ‴ ) μʺdμ‴dϕ‴dμʺdϕʺ = ρ ˜ sur ( τ o , μ , ϕ ) S ,
0 2 π - 1 0 a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ = a ̿ ( μ , ϕ , μ o , ϕ o ) 0 2 π - 1 0 a ( μ , ϕ , μʺ , ϕʺ ) a ̿ ( μ , ϕ , μ o , ϕ o ) G s ( 0 , μ , ϕ , μ , ϕ ) d μ d ϕ a ̿ ( μ , ϕ , μ o , ϕ o ) t ˜ ( μ , ϕ ) .
exp ( τ o / | μ | ) π    0 2 π 0 1 ρ sur ( τ o , μ , ϕ ) a ( μ , ϕ , μ , ϕ ) μ d μ d ϕ + 1 π 0 2 π 0 1 0 2 π - 1 0 ρ sur ( τ o , μʺ , ϕʺ ) a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) μʺd μʺdϕʺd μ d ϕ ρ ˜ sur ( τ o , μ , ϕ ) a ̿ ( μ , ϕ , μ o , ϕ o ) [ exp ( τ o / | μ | ) + t ˜ ( μ , ϕ ) ] S = ρ ˜ sur ( τ o , μ , ϕ ) a ̿ ( μ , ϕ , μ o , ϕ o ) T ˜ ( μ , ϕ ) S .
exp ( τ o / | μ | ) π    0 2 π 0 1 ρ sur ( τ o , μ , ϕ ) a ( μ , ϕ , μ , ϕ ) μ d μ d ϕ + 1 π 0 2 π 0 1 0 2 π - 1 0 ρ sur ( τ o , μʺ , ϕʺ ) a ( μ , ϕ , μʺ , ϕʺ ) G s ( 0 , μ , ϕ , μ , ϕ ) μʺd μʺdϕʺd μ d ϕ a ̿ 2 ( μ , ϕ , μ o , ϕ o ) T ˜ ( μ , ϕ ) T ˜ ( μ , ϕ , μ o , ϕ o ) S 1 a ̿ ( μ , ϕ , μ o , ϕ o ) S .

Metrics