Abstract

An approximate solution to the radiative transfer equation for bistatic scattering from a rough surface covered by a tenuous distribution of particulate (scattering and absorbing) media is derived by means of a series expansion in the scattering coefficient κs of the covering layer up to the first order. The formulation of the successive orders of a scattering series is reviewed, and an analytic solution to the first-order interaction contribution is given by means of a series expansion of the azimuthally averaged product of the bidirectional reflectance distribution function of the surface and the scattering phase function of the covering layer.

© 2016 Optical Society of America

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References

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  1. R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
    [Crossref]
  2. O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996).
    [Crossref]
  3. J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006).
    [Crossref]
  4. H. Lievens and N. Verhoest, “On the retrieval of soil moisture in wheat fields from L-band SAR based on water cloud modeling, the IEM, and effective roughness parameters,” IEEE Geosci. Remote Sens. Lett. 8, 740–744 (2011).
    [Crossref]
  5. W. T. Crow, W. Wagner, and V. Naeimi, “The impact of radar incidence angle on soil-moisture-retrieval skill,” IEEE Geosci. Remote Sens. Lett. 7, 501–505 (2010).
    [Crossref]
  6. A. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).
  7. F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.
  8. E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
    [Crossref]
  9. S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).
  10. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).
  11. P. Liang, L. E. Pierce, and M. Moghaddam, “Radiative transfer model for microwave bistatic scattering from forest canopies,” IEEE Trans. Geosci. Remote Sens. 43, 2470–2483 (2005).
    [Crossref]
  12. F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).
  13. E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear approximation of reflectance functions,” in Proceedings of Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (ACM/Addison-Wesley, 1997), pp. 117–126.
  14. G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).
  15. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [Crossref]
  16. Q. Liu and F. Weng, “Combined Henyey–Greenstein and Rayleigh phase function,” Appl. Opt. 45, 7475–7479 (2006).
    [Crossref]
  17. M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983).
    [Crossref]
  18. Q. Yin and S. Luo, “An approximation frame of the particle scattering phase function with a delta function and a Legendre polynomial series,” in Light, Energy and the Environment (Optical Society of America, 2015), paper EW3A.3.
  19. B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975).
    [Crossref]

2011 (1)

H. Lievens and N. Verhoest, “On the retrieval of soil moisture in wheat fields from L-band SAR based on water cloud modeling, the IEM, and effective roughness parameters,” IEEE Geosci. Remote Sens. Lett. 8, 740–744 (2011).
[Crossref]

2010 (1)

W. T. Crow, W. Wagner, and V. Naeimi, “The impact of radar incidence angle on soil-moisture-retrieval skill,” IEEE Geosci. Remote Sens. Lett. 7, 501–505 (2010).
[Crossref]

2006 (2)

J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006).
[Crossref]

Q. Liu and F. Weng, “Combined Henyey–Greenstein and Rayleigh phase function,” Appl. Opt. 45, 7475–7479 (2006).
[Crossref]

2005 (1)

P. Liang, L. E. Pierce, and M. Moghaddam, “Radiative transfer model for microwave bistatic scattering from forest canopies,” IEEE Trans. Geosci. Remote Sens. 43, 2470–2483 (2005).
[Crossref]

2001 (1)

R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
[Crossref]

1996 (1)

O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996).
[Crossref]

1983 (1)

M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983).
[Crossref]

1978 (1)

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

1975 (1)

B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975).
[Crossref]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Alvarez-Mozos, J.

J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006).
[Crossref]

Arfken, G.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

Attema, E. P. W.

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

Barros, A. P.

R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
[Crossref]

Bindlish, R.

R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
[Crossref]

Boisvert, R. F.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Box, M. A.

M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983).
[Crossref]

Casali, J.

J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

Clark, C. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Crow, W. T.

W. T. Crow, W. Wagner, and V. Naeimi, “The impact of radar incidence angle on soil-moisture-retrieval skill,” IEEE Geosci. Remote Sens. Lett. 7, 501–505 (2010).
[Crossref]

Emblanch, C.

O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996).
[Crossref]

Foo, S.

E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear approximation of reflectance functions,” in Proceedings of Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (ACM/Addison-Wesley, 1997), pp. 117–126.

Fung, A.

A. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).

Gonzalez-Audicana, M.

J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006).
[Crossref]

Greenberg, D.

E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear approximation of reflectance functions,” in Proceedings of Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (ACM/Addison-Wesley, 1997), pp. 117–126.

Greenstein, J. L.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Harris, F.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

Henyey, L. G.

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).

Lafortune, E.

E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear approximation of reflectance functions,” in Proceedings of Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (ACM/Addison-Wesley, 1997), pp. 117–126.

Liang, P.

P. Liang, L. E. Pierce, and M. Moghaddam, “Radiative transfer model for microwave bistatic scattering from forest canopies,” IEEE Trans. Geosci. Remote Sens. 43, 2470–2483 (2005).
[Crossref]

Lievens, H.

H. Lievens and N. Verhoest, “On the retrieval of soil moisture in wheat fields from L-band SAR based on water cloud modeling, the IEM, and effective roughness parameters,” IEEE Geosci. Remote Sens. Lett. 8, 740–744 (2011).
[Crossref]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).

Liu, Q.

Lozier, D. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Luo, S.

Q. Yin and S. Luo, “An approximation frame of the particle scattering phase function with a delta function and a Legendre polynomial series,” in Light, Energy and the Environment (Optical Society of America, 2015), paper EW3A.3.

Moghaddam, M.

P. Liang, L. E. Pierce, and M. Moghaddam, “Radiative transfer model for microwave bistatic scattering from forest canopies,” IEEE Trans. Geosci. Remote Sens. 43, 2470–2483 (2005).
[Crossref]

Moore, R.

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

Naeimi, V.

W. T. Crow, W. Wagner, and V. Naeimi, “The impact of radar incidence angle on soil-moisture-retrieval skill,” IEEE Geosci. Remote Sens. Lett. 7, 501–505 (2010).
[Crossref]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).

Normand, M.

O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996).
[Crossref]

Olver, F. W.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

Phong, B. T.

B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975).
[Crossref]

Pierce, L. E.

P. Liang, L. E. Pierce, and M. Moghaddam, “Radiative transfer model for microwave bistatic scattering from forest canopies,” IEEE Trans. Geosci. Remote Sens. 43, 2470–2483 (2005).
[Crossref]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).

Taconet, O.

O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996).
[Crossref]

Torrance, K.

E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear approximation of reflectance functions,” in Proceedings of Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (ACM/Addison-Wesley, 1997), pp. 117–126.

Ulaby, F.

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

Ulaby, F. T.

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

Verhoest, N.

H. Lievens and N. Verhoest, “On the retrieval of soil moisture in wheat fields from L-band SAR based on water cloud modeling, the IEM, and effective roughness parameters,” IEEE Geosci. Remote Sens. Lett. 8, 740–744 (2011).
[Crossref]

J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006).
[Crossref]

Vidal-Madjar, D.

O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996).
[Crossref]

Wagner, W.

W. T. Crow, W. Wagner, and V. Naeimi, “The impact of radar incidence angle on soil-moisture-retrieval skill,” IEEE Geosci. Remote Sens. Lett. 7, 501–505 (2010).
[Crossref]

Weber, H.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

Weng, F.

Yin, Q.

Q. Yin and S. Luo, “An approximation frame of the particle scattering phase function with a delta function and a Legendre polynomial series,” in Light, Energy and the Environment (Optical Society of America, 2015), paper EW3A.3.

ACM Commun. (1)

B. T. Phong, “Illumination for computer generated pictures,” ACM Commun. 18, 311–317 (1975).
[Crossref]

Appl. Opt. (1)

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Aust. J. Phys. (1)

M. A. Box, “Power series expansion of the Mie scattering phase function,” Aust. J. Phys. 36, 701–706 (1983).
[Crossref]

IEEE Geosci. Remote Sens. Lett. (2)

H. Lievens and N. Verhoest, “On the retrieval of soil moisture in wheat fields from L-band SAR based on water cloud modeling, the IEM, and effective roughness parameters,” IEEE Geosci. Remote Sens. Lett. 8, 740–744 (2011).
[Crossref]

W. T. Crow, W. Wagner, and V. Naeimi, “The impact of radar incidence angle on soil-moisture-retrieval skill,” IEEE Geosci. Remote Sens. Lett. 7, 501–505 (2010).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (2)

P. Liang, L. E. Pierce, and M. Moghaddam, “Radiative transfer model for microwave bistatic scattering from forest canopies,” IEEE Trans. Geosci. Remote Sens. 43, 2470–2483 (2005).
[Crossref]

J. Alvarez-Mozos, J. Casali, M. Gonzalez-Audicana, and N. Verhoest, “Assessment of the operational applicability of RADARSAT-1 data for surface soil moisture estimation,” IEEE Trans. Geosci. Remote Sens. 44, 913–924 (2006).
[Crossref]

Radio Sci. (1)

E. P. W. Attema and F. T. Ulaby, “Vegetation modeled as a water cloud,” Radio Sci. 13, 357–364 (1978).
[Crossref]

Remote Sens. Environ. (2)

R. Bindlish and A. P. Barros, “Parameterization of vegetation backscatter in radar-based, soil moisture estimation,” Remote Sens. Environ. 76, 130–137 (2001).
[Crossref]

O. Taconet, D. Vidal-Madjar, C. Emblanch, and M. Normand, “Taking into account vegetation effects to estimate soil moisture from C-band radar measurements,” Remote Sens. Environ. 56, 52–56 (1996).
[Crossref]

Other (8)

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, Geometric Considerations and Nomenclature for Reflectance, (National Bureau of Standards, 1977).

A. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, 1994).

F. Ulaby, R. Moore, and A. Fung, Microwave Remote Sensing (Artech House, 1986), Vol. 3.

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. (Cambridge University, 2010).

E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, “Non-linear approximation of reflectance functions,” in Proceedings of Conference on Computer Graphics and Interactive Techniques (SIGGRAPH) (ACM/Addison-Wesley, 1997), pp. 117–126.

G. Arfken, H. Weber, and F. Harris, Mathematical Methods for Physicists, 7th ed. (Elsevier, 2013).

Q. Yin and S. Luo, “An approximation frame of the particle scattering phase function with a delta function and a Legendre polynomial series,” in Light, Energy and the Environment (Optical Society of America, 2015), paper EW3A.3.

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Figures (8)

Fig. 1.
Fig. 1. Schematic illustration of the model geometry and the individual contributions considered in the calculated intensity.

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Fig. 2.
Fig. 2. Angles introduced in the separation of the RTE.

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Fig. 3.
Fig. 3. Illustration of the problem geometry.

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Fig. 4.
Fig. 4. Illustration of the scattering distributions in Eqs. (A12), (A13), and (A16) as used in their following examples.

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Fig. 5.
Fig. 5. Visualization of the resulting contributions in Eqs. (17)–(19) to the scattered intensity in linear scale as a function of the outgoing direction ( θ ex , φ ex ) using the phase functions of Fig. 4’s Example 1 with θ 0 = 45 ° , τ = 0.7 , and ω = 0.3 .

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Fig. 6.
Fig. 6. Visualization of the resulting contributions in Eqs. (17)–(19) to the scattered intensity in linear scale as a function of the outgoing direction ( θ ex , φ ex ) using the phase functions of Fig. 4’s Example 2 with θ 0 = 45 ° , τ = 0.7 , and ω = 0.3 .

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Fig. 7.
Fig. 7. Illustration of the backscattering contributions [Eqs. (17)–(19) with θ ex = θ 0 and φ ex = π ] in decibel scale using the phase functions of Fig. 4’s Example 1 with τ = 0.7 and ω = 0.3 .

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Fig. 8.
Fig. 8. Illustration of the backscattering contributions [Eqs. (17)–(19) with θ ex = θ 0 and φ ex = π ] in decibel scale using the phase functions of Fig. 4’s Example 2 with τ = 0.7 and ω = 0.3 .

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Equations (67)

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

cos ( θ ) ∂ I f ( r , Ω ) ∂ r = − κ ex I f ( r , Ω ) + κ s ∬ Ω ′ = 4 π I f ( r , Ω ′ ) p ^ ( Ω ′ → Ω ) d Ω ′ ,
∫ 0 2 π ∫ 0 π p ^ ( Ω i → Ω ′ ) sin ( θ ′ ) d θ ′ d φ ′ = 1 .
∫ θ ′ = 0 π I f ( θ ′ ) p ^ ( θ ′ → θ ) sin ( θ ′ ) d θ ′ = ∫ θ ′ = 0 π / 2 [ I f ( θ ′ ) p ^ ( θ ′ → θ ) sin ( θ ′ ) + I f ( π − θ ′ ) p ^ ( π − θ ′ → θ ) sin ( θ ′ ) ] d θ ′ ,
upwelling   gradiation : If       θ ∈ [ 0 , π / 2 ] θ u ≔ θ and I + ( θ u ) ≔ I f ( θ u ) ,
downwelling   radiation : If    θ ∈ [ π / 2 , π ] θ d ≔ π − θ and I − ( θ d ) ≔ I f ( π − θ d ) .
θ ∈ [ 0 , π / 2 ] : μ u ∂ I + ( μ u ) ∂ r + κ ex I + ( μ u ) = F + ( μ u ) ,
θ ∈ [ π / 2 , π ] : − μ d ∂ I − ( μ d ) ∂ r + κ ex I − ( μ d ) = F − ( μ d ) ,
F ± ( μ ) = κ s ∫ 0 2 π ∫ 0 1 [ I + ( μ ′ ) p ^ ( μ ′ → ± μ ) + I − ( μ ′ ) p ^ ( − μ ′ → ± μ ) ] d μ ′ d φ ′ .
I − ( z = 0 , μ d ) = I inc δ ( μ d − μ 0 ) δ ( φ d − φ i ) ,
I + ( z = − d , μ u ) = ∫ 0 2 π ∫ 0 1 I − ( z = − d , μ ′ ) ⁢ BRDF ( − μ ′ → μ u ) μ ′ d μ ′ d φ .
∫ 0 2 π ∫ 0 π / 2 BRDF ( θ i → θ ′ ) cos ( θ ′ ) ⁢ sin ( θ ′ ) d θ ′ d φ ′ = ρ ( θ i , φ i ) ≤ 1 ,
BRDF ( θ i → θ s ) = BRDF ( θ s → θ i ) ,
p ^ ( θ i → θ s ) = p ^ ( θ s → θ i ) .
I + ( z , μ u ) = I + ( − d , μ u ) exp [ − κ ex μ u ( z + d ) ] + ∫ − d z 1 μ u   exp [ − κ ex μ u ( z − z ′ ) ] F + ( z ′ , μ u ) d z ′ ,
I − ( z , μ d ) = I − ( 0 , μ d ) exp [ κ ex μ d z ] + ∫ z 0 1 μ d   exp [ κ ex μ d ( z − z ′ ) ] F − ( z ′ , μ d ) d z ′ .
I + ( z , μ u ) = I surf + + ( I vol + + I int + + I svs + ) ⏟ terms    proportional   to   κ s + O ( κ s 2 ) .
I surf + ( μ 0 , μ ex ) = I inc   exp [ − τ μ 0 − τ μ ex ] μ 0 BRDF ( − μ 0 → μ ex ) ,
I vol + ( μ 0 , μ ex ) = I inc ω μ 0 μ 0 + μ ex ( 1 − exp [ − τ μ 0 − τ μ ex ] ) p ^ ( − μ 0 → μ ex ) ,
I int + ( μ 0 , μ ex ) = I inc μ 0 ω { exp [ − τ μ ex ] F int ( μ 0 , μ ex ) + exp [ − τ μ 0 ] F int ( μ ex , μ 0 ) } ,
F int ( μ 0 , μ ex ) = ∫ 0 2 π ∫ 0 1 μ ′ μ 0 − μ ′ ( exp [ − τ μ 0 ] − exp [ − τ μ ′ ] ) × p ^ ( μ 0 → μ ′ ) BRDF ( − μ ′ → μ ex ) d μ ′ d φ ′ ,
Single   Scattering   Albedo :    ω = κ s κ ex ,
Optical Depth :    τ = κ ex d .
∫ 0 2 π p ^ ( μ 0 → μ s ) BRDF ( − μ s → μ ex ) d φ s = ∑ n = 0 ∞ f n ( μ 0 , μ ex ) μ s n .
F int ( μ 0 , μ ex ) = ∑ n = 0 ∞ f n ( μ 0 , μ ex ) [ exp ( − τ / μ 0 ) A ( n + 1 ) − B ( n + 1 ) ] ,
A ( N ) = ∫ 0 1 ( μ ′ ) N μ 0 − μ ′ d μ ′ ,
B ( N ) = ∫ 0 1 ( μ ′ ) N μ 0 − μ ′   exp ( − τ / μ ′ ) d μ ′ .
P ∫ a b f ( x ) d x = lim ε → 0 ( ∫ a x 0 − ε f ( x ) d x + ∫ x 0 + ε b f ( x ) d x ) .
( μ ′ ) N μ 0 − μ ′ = μ 0 N μ 0 − μ ′ − ∑ k = 1 N μ 0 N − k ( μ ′ ) k − 1 .
A ( N ) = ∫ 0 1 μ 0 N μ 0 − μ ′ d μ ′ − ∑ k = 1 N ∫ 0 1 μ 0 N − k ( μ ′ ) k − 1 d μ ′ ,
B ( N ) = ∫ 0 1 μ 0 N μ 0 − μ ′   exp ( − τ / μ ′ ) d μ ′ − ∑ k = 1 N ∫ 0 1 μ 0 N − k ( μ ′ ) k − 1   exp ( − τ / μ ′ ) d μ ′ .
P ∫ 0 1 μ 0 N μ 0 − μ ′ d μ ′ = μ 0 N   ln ( μ 0 1 − μ 0 ) ,
∫ 0 1 μ 0 N − k ( μ ′ ) k − 1 d μ ′ = μ 0 N − k k ,
P ∫ 0 1 μ 0 N μ 0 − μ ′   exp ( − τ / μ ′ ) d μ ′ = μ 0 N [ E i ( − τ ) − exp ( − τ / μ 0 ) × E i ( τ / μ 0 − τ ) ] ,
P ∫ 0 1 μ 0 N − k ( μ ′ ) k − 1   exp ( − τ / μ ′ ) d μ ′ = μ 0 N − k E k + 1 ( τ ) ,
A ( N ) = μ 0 N [ ln ( μ 0 1 − μ 0 ) − ∑ k = 1 N μ 0 − k k ] ,
B ( N ) = μ 0 N [ E i ( − τ ) − exp ( − τ / μ 0 ) E i ( τ / μ 0 − τ ) − ∑ k = 1 N E k + 1 ( τ ) μ 0 k ] .
F int ( μ 0 , μ ex ) = ∑ n = 0 ∞ f n ( μ 0 , μ ex ) μ 0 n + 1 { exp ( − τ / μ 0 ) ln ( μ 0 1 − μ 0 ) − E i ( − τ ) + exp ( − τ / μ 0 ) E i ( τ / μ 0 − τ ) + ∑ k = 1 n + 1 μ 0 − k ( E k + 1 ( τ ) − exp ( − τ / μ 0 ) k ) } .
i ^ = ( sin ( θ i ) cos ( φ i ) sin ( θ i ) sin ( φ i ) cos ( θ i ) ) s ^ = ( sin ( θ s ) cos ( φ s ) sin ( θ s ) sin ( φ s ) cos ( θ s ) ) ,
cos ( Θ ˜ i ) = i ^ T M i · s ^ with M j = ( a i 0 0 0 b i 0 0 0 c i ) ,
cos ( Θ ˜ i ) = i ^ T M i · s ^ = a i   cos ( θ i ) cos ( θ s ) + sin ( θ i ) sin ( θ s ) [ b i   cos ( φ i ) cos ( φ s ) + c i   sin ( φ i ) sin ( φ s ) ] .
p ^ ∼ ∑ n = 0 ∞ p n   cos ( Θ ˜ 1 ) n BRDF ∼ ∑ n = 0 ∞ b n   cos ( Θ ˜ 2 ) n .
p ^ ∼ ∑ n = 0 ∞ { α n + β n [ b ˜ 1   cos ( φ s ) + c ˜ 1   sin ( φ s ) ] n } ,
BRDF ∼ ∑ n = 0 ∞ { γ n + η n [ b ˜ 2   cos ( φ s ) + c ˜ 2   sin ( φ s ) ] n } ,
α n = ∑ i [ α n ] i   cos ( θ s ) i β n = sin ( θ s ) n ∑ i [ β n ] i   cos ( θ s ) i ,
γ n = ∑ i [ γ n ] i   cos ( θ s ) i η n = sin ( θ s ) n ∑ i [ η n ] i   cos ( θ s ) i .
BRDF · p ^ = ∑ n = 0 ∞ ∑ k = 0 n { α n − k γ k + β n − k γ k [ b ˜ 1   cos ( φ s ) + c ˜ 1   sin ( φ s ) ] n − k + η k α n − k [ b ˜ 2   cos ( φ s ) + c ˜ 2   sin ( φ s ) ] k + η k β n − k [ b ˜ 1   cos ( φ s ) + c ˜ 1   sin ( φ s ) ] n − k ⁢ [ b ˜ 2   cos ( φ s ) + c ˜ 2   sin ( φ s ) ] k } .
∫ 0 2 π [ b ˜ 1   cos ( φ s ) + c ˜ 1   sin ( φ s ) ] n d φ s { ≠ 0 if n … even 0 if n … odd ,
∫ 0 2 π [ b ˜ 1   cos ( φ s ) + c ˜ 1   sin ( φ s ) ] k ⁢ [ b ˜ 2   cos ( φ s ) + c ˜ 2   sin ( φ s ) ] n − k d φ s { ≠ 0 if n … even 0 if n … odd .
∫ μ 0 N μ 0 − μ ′ d μ ′ = − μ 0 N   ln ( μ 0 − μ ′ ) + const .
P ∫ 0 1 μ 0 N μ 0 − μ ′ d μ ′ = μ 0 N lim ε → 0   [ ln ( μ 0 ) + ln ( − ε ) − ln ( μ 0 − 1 ) − ln ( ε ) ] .
P ∫ 0 1 μ 0 N μ 0 − μ ′ d μ ′ = μ 0 N lim ε → 0 [ ln ( − ε μ 0 ) − ln ( ε [ μ 0 − 1 ] ) ] = μ 0 N lim ε → 0 [ ln ( − ε μ 0 ε ( μ 0 − 1 ) ) ] = μ 0 N   ln ( μ 0 1 − μ 0 ) .
E i ( x ) = P ∫ − ∞ x exp ( t ) t d t with x > 0 .
1 μ 0 − μ ′ = 1 μ ′ + μ 0 μ ′ ( μ 0 − μ ′ ) .
P ∫ 0 1 μ 0 N μ 0 − μ ′   exp ( − τ / μ ′ ) d μ ′ = μ 0 N P ∫ 0 1 exp ( − τ / μ ′ ) μ ′ d μ ′ + μ 0 N P ∫ 0 1 μ 0   exp ( − τ / μ ′ ) μ ′ ( μ 0 − μ ′ ) .
P ∫ 0 1 exp ( − τ / μ ′ ) μ ′ d μ ′ = | t = − τ / μ ′ d μ ′ = τ / t 2 d t = P ∫ − ∞ − τ exp ( t ) t d t = E i ( − τ ) .
P ∫ 0 1   exp [ f ( x ) ] f ( x ) ′ f ( x ) d x = P ∫ f ( 0 ) f ( 1 ) e t t d t = E i [ f ( 1 ) ] if { f ( 0 ) = − ∞ f ( 1 ) ∈ R ,
f ( μ ′ ) = τ μ ′ − τ μ 0 ⇒ f ′ ( μ ′ ) f ( μ ′ ) = − μ 0 μ ′ ( μ 0 − μ ′ ) .
P ∫ 0 1 μ 0 μ ′ ( μ 0 − μ ′ )   exp ( − τ / μ ′ ) = − exp ( − τ μ 0 ) P ∫ 0 1 [ − μ 0 μ ′ ( μ 0 − μ ′ ) ] exp ( − τ μ ′ + τ μ 0 ) = − exp ( − τ μ 0 ) E i ( τ − τ μ 0 ) .
P ∫ 0 1 μ 0 N μ 0 − μ ′   exp ( − τ μ 0 ) d μ ′ = μ 0 N [ E i ( − τ ) − exp ( − τ μ 0 ) E i ( τ − τ μ 0 ) ] .
E n ( x ) = P ∫ 1 ∞ exp ( − x t ) t n d t .
P ∫ 0 1 μ 0 N − k ( μ ′ ) k − 1   exp ( − τ / μ ′ ) d μ ′ = | t = ( μ ′ ) − 1 d μ ′ = − d t / t 2 = μ 0 N − k P ∫ 1 ∞ exp ( − τ t ) t k + 1 d t = μ 0 N − k E k + 1 ( τ )
p ^ ( θ i , θ s ) = 3 16 π ( 1 + cos ( Θ ) 2 ) ,
BRDF ( θ i , θ s ) = Max [ cos ( Θ ′ ) 5 , 0 ] = ∑ n = 0 ∞ ( 2 n + 1 ) 15 π 16 Γ ( 7 − n 2 ) Γ ( 8 + n 2 ) P n ( cos ( Θ ′ ) )
cos ( Θ ) = cos ( θ i ) cos ( θ s ) + sin ( θ i ) sin ( θ s ) cos ( φ i − φ s ) ,
cos ( Θ ′ ) = − cos ( θ i ) cos ( θ s ) + sin ( θ i ) sin ( θ s ) cos ( φ i − φ s ) .
p ^ ( θ i , θ s ) = 1 4 π 1 − t 2 [ 1 + t 2 − 2 t   cos ( Θ ) ] 3 / 2
= 1 4 π ∑ n = 0 ∞ ( 2 n + 1 ) t n P n ( cos ( Θ ) ) .

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