Abstract

The classical Chandrasekhar's formula, which relates the surface albedo to the top of the atmosphere radiance, rigorously applies to a homogeneous Lambertian surface. For a nonhomogeneous Lambertian surface in a plane-parallel atmosphere, an extension of this formula was proposed in the 1980s and has been implemented recently in the 6S algorithm. To analyze this extension, the rigorous formula of the top of the atmosphere signal in a plane-parallel atmosphere bounded by a nonhomogeneous Lambertian surface is derived. Then the 6S algorithm extension is compared to the exact formula and approximations and their validity is examined. The derivation of the exact formula is based on the separation of the radiation fields into direct and diffuse components, on the introduction of the Green's function of the problem, and on integrations of boundary values of the radiation fields with Green's function.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. 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]
  3. K. F. Evans, "The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998).
    [CrossRef]
  4. 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]
  5. A. Lyapustin and Y. Knyazikhin, "Green's function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface," Appl. Opt. 41, 5600-5606 (2002).
    [CrossRef] [PubMed]
  6. 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).
    [CrossRef] [PubMed]
  7. A. Lyapustin, Radiative transfer code SHARM-3D for radiance simulations over a non-Lambertian nonhomogeneous surface: intercomparison study," Appl. Opt. 41, 5607-5615 (2002).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  9. D. Tanré, M. Herman, and P. Y. Deschamps, "Influence of the background contribution upon space measurements of ground reflectance," Appl. Opt. 20, 3676-3684 (1981).
    [CrossRef] [PubMed]
  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. E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.
  12. A. Sei, "Extension of Chandrasekhar's formula to a homogeneous non-Lambertian surface and comparison with the 6S formulation," Appl. Opt. 45, 1010-1022 (2006).
    [CrossRef] [PubMed]
  13. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  14. G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, 1970).
  15. A. A. Ioltukhovski, "Radiative transfer over the surface with an arbitrary reflection: Green's functions method," Transp. Theory Stat. Phys. 28, 349-368 (1999).
    [CrossRef]
  16. 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]
  17. J. Lenoble, Atmospheric Radiative Transfer (Deepak, 1993).
  18. K. N. Liou, Introduction to Atmospheric Radiation (Academic, 2002).
  19. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  20. H. C. van de Hulst, "Scattering in a planetary atmosphere," Astrophys. J. 107, 220-246 (1948).
    [CrossRef]
  21. A. Sei, "Analysis of adjacency effects for two Lambertian half-spaces," Int. J. Remote Sens. (to be published).
  22. P. N. Reinersman and K. Carder, "Monte Carlo simulation of the atmospheric point-spread function with an application to correction for the adjacency effect," Appl. Opt. 34, 4453-4471 (1995).
    [CrossRef] [PubMed]
  23. P. R. Garabedian, Partial Differential Equations (American Mathematical Society, 1998).

2006 (1)

2004 (1)

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]

2002 (2)

1999 (2)

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]

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

1998 (1)

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

1997 (1)

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]

1995 (1)

1988 (1)

1984 (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]

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]

1981 (1)

1979 (1)

1948 (1)

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 Reinhold, 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]

Carder, K.

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

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]

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.

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]

Garabedian, P. R.

P. R. Garabedian, Partial Differential Equations (American Mathematical Society, 1998).

Glasstone, S.

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

Herman, M.

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]

D. Tanré, M. Herman, and P. Y. Deschamps, "Influence of the background contribution upon space measurements of ground reflectance," Appl. Opt. 20, 3676-3684 (1981).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.

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).
[CrossRef]

Jayaweera, K.

Knyazikhin, Y.

Lenoble, J.

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

Liou, K. N.

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

Lyapustin, A.

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]

Morcrette, J. J.

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.

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]

Reinersman, P. N.

Sei, A.

Stamnes, K.

Tanré, D.

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]

D. Tanré, M. Herman, and P. Y. Deschamps, "Influence of the background contribution upon space measurements of ground reflectance," Appl. Opt. 20, 3676-3684 (1981).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.

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]

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, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.

Wiscombe, W.

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt. (7)

Astrophys. J. (1)

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

IEEE Trans. Geosci. Remote Sens. (1)

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. (1)

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 (4)

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]

Transp. Theory Stat. Phys. (1)

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

Other (8)

P. R. Garabedian, Partial Differential Equations (American Mathematical Society, 1998).

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

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

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

A. Sei, "Analysis of adjacency effects for two Lambertian half-spaces," Int. J. Remote Sens. (to be published).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

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

E. F. Vermote, D. Tanré, J. L. Deuzé, M. Herman, and J. J. Morcrette, "6S User Guide Version 2," (1997), available at ftp://kratmos.gsfc.nasa.gov/pub/6S/.

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 (68)

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

( 1 μ 2 cos ( ϕ ) x + 1 μ 2 sin ( ϕ ) y + μ   z ) I ( x , y , z , μ , ϕ ) = κ ( z ) I ( x , y , z , μ , ϕ ) + κ ( z ) ω ( z ) 4 π 0 2 π 1 1 p ( z , μ , ϕ , μ , ϕ ) I ( x , y , z , μ , ϕ ) d ϕ , I ( x , y , 0 , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) , μ > 0 , I ( x , y , z o , μ , ϕ ) = a ( x , y ) π 0 2 π 0 1 I ( x , y , z o , μ , ϕ ) μ d ϕ μ < 0 ,
1 4 π 0 2 π 1 1 p ( z , μ , ϕ , μ , ϕ ) d μ d ϕ = 1 .
τ ( z ) = 0 z κ ( ξ ) d ξ , d τ = κ ( z ) d z .
κ ˜ ( τ ) = κ ˜ ( τ ( z ) ) = κ ( z ) ,
( 1 μ 2 κ ˜ ( τ ) cos ( ϕ ) x + 1 μ 2 κ ˜ ( τ ) sin ( ϕ ) y + μ   τ ) I ( x , y , τ , μ , ϕ ) = I ( x , y , τ , μ , ϕ ) + ω ˜ ( τ ) 4 π 0 2 π 1 1 p ˜ ( τ , μ , ϕ , μ , ϕ ) I ( x , y , τ , μ , ϕ ) d ϕ I ( x , y , 0 , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) , μ > 0 , I ( x , y , τ o , μ , ϕ ) = a ( x , y ) π 0 2 π 0 1 I ( x , y , τ o , μ , ϕ ) μ d ϕ μ < 0.
μ   I d i r τ ( τ , μ , ϕ ) + I d i r ( τ , μ , ϕ ) = 0 , I d i r ( 0 , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) , μ > 0.
I d i r ( τ , μ , ϕ ) = F o δ ( μ μ o ) δ ( ϕ ϕ o ) e τ / μ f o r   μ > 0.
( 1 μ 2 κ ˜ ( τ ) cos ( ϕ ) x + 1 μ 2 κ ˜ ( τ ) sin ( ϕ ) y + μ   τ ) I d i f ( x , y , τ , μ , ϕ ) = I d i f ( x , y , τ , μ , ϕ ) + ω ˜ ( τ ) 4 π 0 2 π 1 1 p ˜ ( τ , μ , ϕ , μ , ϕ ) I d i f ( x , y , τ , μ , ϕ ) d ϕ + ω ˜ ( τ ) F o 4 π p ˜ ( τ , μ , ϕ , μ o , ϕ o ) e τ / μ o , I d i f ( x , y , 0 , μ , ϕ ) = 0 , μ > 0 , I d i f ( x , y , τ o , μ , ϕ ) = a ( x , y ) π { F o μ o e τ o / μ o + 0 2 π 0 1 I d i f ( x , y , τ o , μ , ϕ ) μ d ϕ } , μ < 0.
I d i f ( x , y , τ , μ , ϕ ) = I a t m ( τ , μ , ϕ ) + I s u r ( x , y , τ , μ , ϕ ) ,
μ   I a t m τ ( τ , μ , ϕ ) + I a t m ( τ , μ , ϕ ) = ω ˜ ( τ ) 4 π 0 2 π 1 1 p ˜ ( τ , μ , ϕ , μ , ϕ ) I a t m ( τ , μ , ϕ ) d ϕ     + ω ˜ ( τ ) F o 4 π p ˜ ( τ , μ , ϕ , μ o , ϕ o ) e τ / μ o , I a t m ( 0 , μ , ϕ ) = 0 μ > 0 , I a t m ( τ o , μ , ϕ ) = 0 μ < 0,
( 1 μ 2 κ ˜ ( τ ) cos ( ϕ ) x + 1 μ 2 κ ˜ ( τ ) sin ( ϕ ) y + μ   τ ) I s u r ( x , y , τ , μ , ϕ ) =   I s u r ( x , y , τ , μ , ϕ ) + ω ˜ ( τ ) 4 π 0 2 π 1 1 p ˜ ( τ , μ , ϕ , μ , ϕ ) I s u r ( x , y , τ , μ , ϕ ) d ϕ ,   I s u r ( x , y , 0 , μ , ϕ ) = 0 , μ > 0 , I s u r ( x , y , τ o , μ , ϕ ) = a ( x , y ) π { F o μ o e τ o / μ o + 0 2 π 0 1 I a t m ( τ o , μ , ϕ ) μ d ϕ   + 0 2 π 0 1 I s u r ( x , y , τ o , μ , ϕ ) μ d ϕ } , μ < 0.
( 1 μ 2 κ ˜ ( τ ) cos ( ϕ ) x + 1 μ 2 κ ˜ ( τ ) sin ( ϕ ) y + μ   τ ) G ( x , y , τ , μ , ϕ ) = G ( x , y , τ , μ , ϕ ) + ω ˜ ( τ ) 4 π 0 2 π 1 1 p ˜ ( τ , μ , ϕ , μ , ϕ ) G ( x , y , τ , μ , ϕ ) d ϕ , G ( x , y , 0 , μ , ϕ ) = 0 , μ > 0 , G ( x , y , τ o , μ , ϕ ) = δ ( x ) δ ( y ) , μ < 0.
I s u r ( x , y , τ , μ , ϕ ) = + + I s u r ( x , y , τ o ) × G ( x x , y y , τ , μ , ϕ ) d x d y .
( 1 μ 2 κ ˜ ( τ ) cos ( ϕ ) x + 1 μ 2 κ ˜ ( τ ) sin ( ϕ ) y + μ τ ) G d i r ( x , y , τ , μ , ϕ ) = G d i r ( x , y , τ , μ , ϕ ) ,
G d i r ( x , y , τ o , μ , ϕ ) = δ ( x ) δ ( y ) , μ < 0.
G d i r ( x , y , τ , μ , ϕ ) = δ ( x + 1 μ 2 μ cos ( ϕ ) τ τ o d ξ κ ˜ ( ξ ) ) × δ ( y + 1 μ 2 μ sin ( ϕ ) τ τ o d ξ κ ˜ ( ξ ) ) × e ( τ o τ ) / | μ | , μ < 0 .
α ( τ ) = τ τ o d ξ κ ˜ ( ξ ) ,
G d i r ( x , y , τ , μ , ϕ ) = δ ( x + tan ( θ ) cos ( ϕ ) α ( τ ) ) δ ( y + tan ( θ ) × sin ( ϕ ) α ( τ ) ) e ( τ o τ ) / | μ | .
α ( τ ) = τ τ o d ξ κ ˜ ( ξ ) = z z o κ ( z ) d z κ ( z ) = z z o d z = ( z o z ) so   α o = z o ,
( 1 μ 2 κ ˜ ( τ ) cos ( ϕ ) x + 1 μ 2 κ ˜ ( τ ) sin ( ϕ ) y + μ   τ ) G d i f ( x , y , τ , μ , ϕ ) = G d i f ( x , y , τ , μ , ϕ ) + ω ˜ ( τ ) 4 π 0 2 π 1 1 p ˜ ( τ , μ , ϕ , μ , ϕ ) G d i f ( x , y , τ , μ , ϕ ) d ϕ + ω ˜ ( τ ) 4 π 0 2 π 1 1 p ˜ ( τ , μ , ϕ , μ , ϕ ) G d i r ( x , y , τ , μ , ϕ ) d ϕ , G d i f ( x , y , 0 , μ , ϕ ) = 0 , μ > 0 , G d i f ( x , y , τ o , μ , ϕ ) = 0 , μ < 0.
I s u r ( x , y , τ , μ , ϕ ) = e ( τ o τ ) / | μ | I s u r ( x + tan ( θ ) cos ( ϕ ) × α ( τ ) , y + tan ( θ ) sin ( ϕ ) α ( τ ) , τ 0 ) + + + I s u r ( x , y , τ o ) × G d i f ( x x , y y , τ , μ , ϕ ) d x d y .
I s u r ( x , y , τ o ) = a ( x , y ) π { F o μ o e τ o / μ o + 0 2 π 0 1 I a t m ( τ o , μ , ϕ ) × μ d ϕ + 0 2 π 0 1 I s u r ( x , y , τ o , μ , ϕ ) × μ d ϕ } , μ < 0.
I s u r ( x , y , τ o , μ , ϕ ) = + + I s u r ( x , y , τ o ) × G ( x x , y y , τ o ,  μ , ϕ ) d x d y ,
0 2 π 0 1 I s u r ( x , y , τ o , μ , ϕ ) μ d ϕ = 0 2 π 0 1 + + I s u r ( x , y , τ o ) G × ( x x , y y , τ o , μ , ϕ ) d x d y μ d ϕ = + + I s u r ( x , y , τ o ) 0 2 π 0 1 × G ( x x , y y , τ o , μ , ϕ ) μ d ϕ d x d y .
G ˜ ( x , y , τ 0 ) = 0 2 π 0 1 G ( x , y , τ o , μ , ϕ ) μ d ϕ ,
0 2 π 0 1 I s u r ( x , y , τ o , μ , ϕ ) μ d ϕ = + + I s u r ( x , y , τ o ) G ˜ ( x x , y y , τ o ) d x d y .
I s u r ( x , y , τ o ) = a ( x , y ) π { F o μ o e τ o / μ o + 0 2 π 0 1 I a t m ( τ o , μ , ϕ ) × μ d ϕ + + + I s u r ( x , y , τ o ) × G ˜ ( x x , y y , τ o ) d x d y } .
I 0 s u r ( x , y , τ o ) = a ( x , y ) ( F o μ o π e τ o / μ o + 1 π 0 2 π 0 1 I a t m ( τ o , μ , ϕ ) μ d ϕ ) = a ( x , y ) F o μ o π T ( μ o , ϕ o ) , I n + 1 s u r ( x , y , τ o ) = I 0 s u r ( x , y , τ o ) + a ( x , y ) π + + I n s u r ( x , y , τ o ) G ˜ ( x x , y y , τ o ) d x d y .
G ˜ , I ( x , y ) = 1 π + + I ( x , y , τ o ) × G ˜ ( x x , y y , τ o ) d x d y ,
G ˜ , I 1 ( x , y ) = a ( x , y ) G ˜ , I ( x , y ) ,
G ˜ , I k + 1 ( x , y ) = G ˜ , G ˜ , I k 1 ( x , y ) .
I n + 1 s u r ( x , y , τ o ) = I 0 s u r ( x , y , τ o ) + k = 1 n G ˜ , I 0 s u r k ( x , y ) .
I n + 1 s u r ( x , y , τ o ) = F o μ o π T ( μ o , ϕ o ) ( a ( x , y ) + k = 1 n G ˜ , a k ( x , y ) ) .
I s u r ( x , y , τ o ) = F o μ o π T ( μ o , ϕ o ) ( a ( x , y ) + k = 1 + G ˜ , a k ( x , y ) ) .
G ˜ , a 1 = a ( x , y ) + + a ( x , y ) × G ˜ ( x x , y y , τ o ) d x d y .
G ˜ , a 2 = a ( x , y ) + + G ˜ ( x x , y y , τ o ) a ( x , y ) × + + G ˜ ( x x , y y , τ o ) × a ( x , y ) d x d y d x d y .
G ˜ , a k = a k + 1 S k where S = G ˜ , 1 .
I s u r ( x , y , τ o ) = F o μ o π T ( μ o , ϕ o ) ( a + k = 1 + a k + 1 S k ) = F o μ o π a T ( μ o , ϕ o ) 1 a S ,
G ˜ , a k a a ¯ k S k ,
I s u r ( x , y , τ o ) F o μ o π a T ( μ o , ϕ o ) 1 a ¯ S .
I s u r ( x , y , 0 , μ , ϕ ) = I s u r ( x + x 1 , y + y 1 , τ o ) e τ o / | μ | + + + I s u r ( x , y , τ o ) × G d i f ( x x , y y , 0 , μ , ϕ ) d x d y .
I d i f ( x , y , 0 , μ , ϕ ) = I a t m ( 0 , μ , ϕ ) + I s u r ( x + x 1 , y + y 1 , τ o ) e τ o / | μ | + + + I s u r ( x , y , τ o ) × G d i f ( x x , y y , 0 , μ , ϕ ) d x d y .
ρ d i f ( x , y , 0 , μ , ϕ ) = ρ a t m ( 0 , μ , ϕ ) + ρ s u r ( x + x 1 , y + y 1 , τ o ) e τ o / | μ | + + + ρ s u r ( x , y , τ o ) × G d i f ( x x , y y , 0 , μ , ϕ ) d x d y .
ρ s u r ( x , y , τ o ) = T ( μ o , ϕ o ) ( a ( x , y ) + k = 1 + G ˜ , a k ( x , y ) ) .
ρ d i f ( x , y , 0 , μ , ϕ ) = ρ a t m ( 0 , μ , ϕ ) + + + ρ s u r ( x , y , τ o ) × G ( x x , y y , 0 , μ , ϕ ) d x d y .
t ( μ , ϕ ) = + + G d i f ( x x , y y , 0 , μ , ϕ ) d x d y , t ( μ o , ϕ o ) = 1 π 0 2 π 0 1 ρ a t m ( τ o , μ , ϕ ) μ d ϕ ,
T ( μ , ϕ ) = e τ o / | μ | + t ( μ , ϕ ) , T ( μ o , ϕ o ) = e τ o / μ o + t ( μ o , ϕ o ) ,
ρ s u r ( x , y , τ o ) = a T ( μ o , ϕ o ) 1 a S ,
ρ d i f ( 0 , μ , ϕ ) = ρ a t m ( 0 , μ , ϕ ) + a T ( μ o , ϕ o ) 1 a S × ( e τ o / | μ | + + + × G d i f ( x x , y y , 0 , μ , ϕ ) d x d y ) ,
ρ d i f ( 0 , μ , ϕ ) = ρ a t m ( 0 , μ , ϕ ) + a T ( μ o , ϕ o ) T ( μ , ϕ ) 1 a S .
ρ 6 S d i f ( x , y , 0 , μ , ϕ ) = ρ a t m ( 0 , μ , ϕ ) + T ( μ o , ϕ o ) 1 S a ( x , y ) × ( a ( x , y ) e τ o / | μ | + a ( x , y ) t ( μ , ϕ ) ) ,
a ( x , y ) = + + a ( x , y ) m ( x x , y y ) d x d y .
m ( x , y ) = f ( x 2 + y 2 ) 2 π x 2 + y 2 ,
f ( r ) = i = 1 4 β i exp ( α i r ) with i = 1 4 β i α i = 1 ,
ρ d i f ( x , y , 0 , μ , ϕ ) ρ 6 S d i f ( x , y , 0 , μ , ϕ ) = T ( μ o , ϕ o ) × { e τ o / | μ | ( ρ s u r ( x + x 1 , y + y 1 , τ o ) T ( μ o , ϕ o ) a ( x , y ) 1 a ( x , y ) S ) + + + ( ρ s u r ( x , y , τ o ) T ( μ o , ϕ o ) a ( x , y ) 1 a ( x , y ) S ) × G d i f ( x x , y y , 0 , μ , ϕ ) d x d y } .
ρ s u r ( x , y , τ o ) T ( μ o , ϕ o ) a 1 a ¯ S .
( a ( τ ) x + b ( τ ) y + c τ ) G d i r ( x , y , τ , μ , ϕ ) = G d i r ( x , y , τ , μ , ϕ ) , G d i r ( x , y , τ o , μ , ϕ ) = δ ( x ) δ ( y ) , μ < 0,
a ( τ ) = 1 μ 2 κ ˜ ( τ ) cos ( ϕ ), b ( τ ) = 1 μ 2 κ ˜ ( τ ) sin ( ϕ ), c = μ .
d x ( t ) d t = a ( τ ( t ) ) , x ( 0 ) = x o , d y ( t ) d t = b ( τ ( t ) ) , x ( 0 ) = y o , d τ ( t ) d t = c , τ ( 0 ) = τ o .
x ( t ) = x o + 0 t a ( τ ( ξ ) ) d ξ , y ( t ) = y o + 0 t b ( τ ( ξ ) ) d ξ , τ ( t ) = τ o + c t .
d G ˜ d t = a ( τ ) G d i r x + b ( τ ) G d i r y + c G d i r τ ,
d G ˜ d t = G ˜ , G ˜ ( 0 ) = δ ( x o ) δ ( y o ) ,
G ˜ ( t ) = δ ( x o ) δ ( y o ) e t .
G d i r ( x , y , τ ) = δ ( x + 1 c τ τ o a ( ξ ) d ξ ) δ ( y + 1 c τ τ o b ( ξ ) d ξ ) × e ( τ τ o ) / c .
G d i r ( x , y , τ ) = δ ( x + 1 μ 2 μ cos ( ϕ ) τ τ o d ξ κ ˜ ( ξ ) ) × δ ( y + 1 μ 2 μ sin ( ϕ ) τ τ 0 d ξ κ ˜ ( ξ ) ) e ( τ τ o ) / c ,
I n + 1 s u r = I 0 s u r + G ˜ , I n s u r 1 .
I n s u r = I 0 s u r + k = 1 n 1 G ˜ , I 0 s u r k ,
I n + 1 s u r = I 0 s u r + G ˜ , I 0 s u r + k = 1 n 1 G ˜ , I 0 s u r k 1 , = I 0 s u r + G ˜ , I 0 s u r 1 + k = 1 n 1 G ˜ , G ˜ , I 0 s u r k 1 , = I 0 s u r + G ˜ , I 0 s u r 1 + k = 1 n 1 G ˜ , I 0 s u r k + 1 , = I 0 s u r + k = 1 n G ˜ , I 0 s u r k ,

Metrics