Abstract

The auxiliary function method consists of taking full advantage of the expansion of the phase function on spherical harmonics in order to deduce an integral equation from the radiative transfer equation. In contrast to the discrete-ordinate method, it is free of the channel concept, the unknowns being a function only of the optical depth. After presenting the method, we show that it is very accurate and particularly well fitted when the scattering medium is continuously inhomogeneous in albedo and phase function and also for sublayers with different refractive index.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. K. Stamnes, S. Chee Tsay, W. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media,” Appl. Opt. 27, 2502–2510 (1988).
    [Crossref] [PubMed]
  3. L. Tsang, J. A. Jong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications, Wiley Series in Remote Sensing (Wiley, New York, 2000).
  4. A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
    [Crossref]
  5. M. Elias, G. Elias, “New and fast calculation for incoherent multiple scattering,” J. Opt. Soc. Am. A 19, 894–901 (2002).
    [Crossref]
  6. A. da Silva, M. Elias, C. Andraud, J. Lafait, “Comparison between the auxiliary function method and the discrete-ordinate method for solving the radiative transfer equation for light scattering,” J. Opt. Soc. Am. A 20, 2321–2329 (2003).
    [Crossref]
  7. L. Simonot, M. Elias, “Special visual effect of art glazes explained by the radiative transfer equation,” Appl. Opt. (to be published).
  8. R. G. Barrera, A. Garca-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach,” J. Opt. Soc. Am. A 20, 296–311 (2003).
    [Crossref]

2004 (1)

A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
[Crossref]

2003 (2)

2002 (1)

1988 (1)

Andraud, C.

A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
[Crossref]

A. da Silva, M. Elias, C. Andraud, J. Lafait, “Comparison between the auxiliary function method and the discrete-ordinate method for solving the radiative transfer equation for light scattering,” J. Opt. Soc. Am. A 20, 2321–2329 (2003).
[Crossref]

Barrera, R. G.

A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
[Crossref]

R. G. Barrera, A. Garca-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective-medium approach,” J. Opt. Soc. Am. A 20, 296–311 (2003).
[Crossref]

Chandrasekhar, S.

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

Chee Tsay, S.

da Silva, A.

A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
[Crossref]

A. da Silva, M. Elias, C. Andraud, J. Lafait, “Comparison between the auxiliary function method and the discrete-ordinate method for solving the radiative transfer equation for light scattering,” J. Opt. Soc. Am. A 20, 2321–2329 (2003).
[Crossref]

Ding, K.-H.

L. Tsang, J. A. Jong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications, Wiley Series in Remote Sensing (Wiley, New York, 2000).

Elias, G.

Elias, M.

Garca-Valenzuela, A.

Jayaweera, K.

Jong, J. A.

L. Tsang, J. A. Jong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications, Wiley Series in Remote Sensing (Wiley, New York, 2000).

Lafait, J.

Laffait, J.

A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
[Crossref]

Robin, T.

A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
[Crossref]

Simonot, L.

L. Simonot, M. Elias, “Special visual effect of art glazes explained by the radiative transfer equation,” Appl. Opt. (to be published).

Stamnes, K.

Tsang, L.

L. Tsang, J. A. Jong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications, Wiley Series in Remote Sensing (Wiley, New York, 2000).

Wiscombe, W.

Appl. Opt. (1)

J. Mod. Opt. (1)

A. da Silva, C. Andraud, J. Laffait, T. Robin, R. G. Barrera, “A model of the angular distribution of light scattered by multilayered media,” J. Mod. Opt. 51, 313–332 (2004).
[Crossref]

J. Opt. Soc. Am. A (3)

Other (3)

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

L. Tsang, J. A. Jong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications, Wiley Series in Remote Sensing (Wiley, New York, 2000).

L. Simonot, M. Elias, “Special visual effect of art glazes explained by the radiative transfer equation,” Appl. Opt. (to be published).

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.


Figures (8)

Fig. 1
Fig. 1

Definition of collimated and diffuse fluxes.

Fig. 2
Fig. 2

Medium with varying albedo ϖ(τ)=0.5+β(τ-0.5), n=1, θ0=0: overall diffuse flux as a function of the albedo slope β. Solid curve, reflected Frefdif; dotted curve, transmitted Ftransdif; dashed curve, total Ftotdif.

Fig. 3
Fig. 3

Geometry of the sublayers with different refractive indices. Reflection and transmission Fresnel coefficients.

Fig. 4
Fig. 4

Particular case of four layers for the numerical simulation: refractive indices and thickness.

Fig. 5
Fig. 5

Particular case of four layers with continuously varying phase function and a unit albedo: collimated flux as a function of the incidence angle. Solid curve, reflected Frefcol; dashed curve, transmitted Ftranscol.

Fig. 6
Fig. 6

Particular case of four layers with continuously varying phase function and a unit albedo: overall diffuse flux as a function of the incidence angle. Solid curve, reflected Frefdif; dashed curve, transmitted Ftransdif.

Fig. 7
Fig. 7

Particular case of four layers with a continuously varying phase function and a unit albedo: relative error =1-|Ftot| in the flux balance.

Fig. 8
Fig. 8

Particular case of four layers with a continuously varying phase function and a linearly varying albedo: overall diffuse flux as a function of the incidence angle. Solid curve, reflected Frefdif; dashed curve, transmitted Ftransdif.

Equations (89)

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

μdi(u, z)dz=-[k(z)+s(z)]i(u, z)+s(z)4πi(u1, z)p(u, u1, z)dΩ1+kIk(z)p(u, uk, z),
τ(z)=0z[k(y)+s(y)]dy,
ϖ(τ)=s(τ)/[k(τ)+s(τ)],
f(μ, ϕ, τ)=|μ|i(μ, ϕ, τ),
Fk(τ)=|μk|Ik(τ),
df(μ, ϕ, τ)dτ=-f(μ, ϕ, τ)μ+ϖ(τ)4πμ|μ|f(μ1, ϕ1, τ)|μ1|p(u, u1, τ)×dΩ1+kFk(τ)|μk|p(u, uk, τ).
dFk(τ)dτ=-Fk(τ)μk,
p(u, u1, τ)=l=0νmaxpl(τ)Pl(cos γ).
p(u, u1, τ)=m=0νmaxl=mνmaxα(l, m)×pl(τ)Plm(μ)Plm(μ1)cos m(ϕ-ϕ1),
α(l, m)=1form=0,
α(l, m)=2(l-m)!/(l+m)!form0.
f(μ, ϕ, τ)=m=0νmaxf(m)(μ, τ)cos mϕ
ηm=2form=0,ηm=1form0,
rl(m)(τ)=ϖ(τ)α(l, m)ηmpl(τ)/4,
df(m)(μ, τ)dτ=-f(m)(μ, τ)μ+l=mνmaxrl(m)(τ)Plm(μ)×-1+1f(m)(μ1, τ)|μ1|Plm(μ1)dμ1+kFk(τ)πηm|μk|Plm(μk).
Al(m)(τ)=-11f(m)(μ, τ)|μ|Plm(μ)dμ,
sl(m)(τ)=kFk(τ)Plm(μk)/(πηm|μk|),
tl(m)(τ)=Al(m)(τ)+sl(m)(τ).
Pl(m)(-x)=(-1)l+mPl(m)(x),
df+(m)(μ, τ)dτ=-f+(m)(μ, τ)μ+l=mνmaxrl(m)(τ)Plm(μ)tl(m)(τ),
df-(m)(-μ, τ)dτ=f-(m)(-μ, τ)μ-l=mνmax(-1)l+mrl(m)(τ)Plm(μ)tl(m)(τ).
f+(m)(μ, τ)=g+(m)(μ)exp(-τ/μ)+l=mνmaxPlm(μ)×0τrl(m)(y)tl(m)(y)exp[(y-τ)/μ]dy,
f-(m)(-μ, τ)=g-(m)(μ)exp((τ-τh)/μ)+l=mνmax(-1)l+mPlm(μ)ττhrl(m)(y)tl(m)(y)×exp[(τ-y)/μ]dy,
g+(m)(μ)=f+(m)(μ, 0),g-(m)(μ)=f-(m)(-μ, τh).
f+m(μ, τh)=g+(m)(μ)exp(-τh/μ)+Δ+(m)(μ),
Δ+(m)(μ)=lmPlm(μ)0τhrl(m)(y)tl(m)(y)×exp[-(τh-y)/μ]dy,
f-(m)(-μ, 0)=g-(m)(μ)exp(-τh/μ)+Δ-(m)(μ),
Δ-(m)(μ)=l=mνmax(-1)l+mPlm(μ)0τhrl(m)(y)tl(m)(y)×exp(-y/μ)dy.
Al(m)(τ)=01[f+(m)(μ, τ)+(-1)l+mf-(m)(μ, τ)]Plm(μ)μdμ,
Al(m)(τ)=l1=mνmax0τhH(m)(l, l1, τ, y)[Al1(m)(y)+sl1(m)(y)]dy+Bl(m)(τ),
H(m)(l, l1, τ, y)=rl1(m)(y)K(m)(l, l1, |τ-y|)  fory<τ,
H(m)(l, l1, τ, y)=(-1)l+l1rl1(m)(y)×K(m)(l, l1,|τ-y|)  fory>τ,
K(m)(l, l1, x)=01Plm(μ)Pl1m(μ)μexp(-x/μ)dμ,
Bl(m)(τ)=01[g+(m)(μ)exp(-τ/μ)+(-1)l+mg-(m)(μ)×exp((τ-τ1)/μ)]Plm(μ)μdμ.
Al(m)(τ)=l1=mνmax0τh[H(m)(l, l1, τ, y)+HBC(m)×(l, l1, τ, y)]Al(m)(y)dy+Ql(col)(m)(τ).
Ql(col)(m)(τ)=l1=mνmax0τhH(m)(l, l1, τ, y)sl1(m)(y)dy,
Al(m)(τ)=l1=mνmax0τhH(m)(l, l1, τ, y)×[Al1(m)(y)+sl1(m)(y)]dy.
I=01Plm(μ)Pl1m(μ)μ0Δτ/2exp(-x/μ)dµdx=01Plm(μ)Pl1m(μ)[1-exp[-Δτ/(2µ)]dμ,
ϖ(τ)=0.5+β(τ-0.5),
nj sin θj=nj+1 sin θj+1=x.
Fj+1+(μj+1, τj)=Tj,j+1(μj)Fj+(μj, τj)+Rj+1,j(μj+1)Fj+1-(-μj+1, τj),
Fj-(-μj, τj)=Rj,j+1(μj)Fj+(μj, τj)+Tj+1,j(μj+1)Fj+1-(-μj+1, τj).
fj+1+(μj+1, τj)dμj+1=Tj,j+1(μj)fj+(μj, τj)dμj+Rj+1,j(μj+1)fj+1-(-μj+1, τj)×dμj+1,
fj-(-μj, τj)dμj=Rj,j+1(μj)fj+(μj, τj)dμj+Tj+1,j(μj+1)fj+1-(-μj+1,τj)×dμj+1.
nj2(1-μj)2=nj+12(1-μj+1)2,
nj2μjdμj=nj+12μj+1dμj+1.
T˜j,j+1(μj)=Tj,j+1(μj)nj+12μj+1nj2μj,
fj+1+(μj+1, τj)=T˜j,j+1(μj)fj+(μj, τj)+Rj+1,j(μj+1)fj+1-(-μj+1, τj),
fj-(-μj, τj)=Rj,j+1(μj)fj+(μj, τj)+T˜j+1,j(μj+1)fj+1-(-μj+1, τj).
Rj,j+1(x)=Rj+1,j(x)=12nj2-x2-nj+12-x2nj2-x2+nj+12-x22×1+nj2-x2nj+12-x2-x2nj2-x2nj+12-x2+x22,
fornj2-x2>0,nj+12-x2>0,
Rj,j+1(x)=Rj+1,j(x)=0otherwise,
andR+T=1.
Fj+(μ0,j, τ)=Gj+(μ0,j)exp[-(τ-τj-1)/μ0,j],
Fj-(-μ0,j, τ)=Gj-(μ0,j)exp[-(τj-τ)/μ0,j],
Gj+1+(μ0,j+1)=Gj+(μ0,j)Tj,j+1(μ0,j)exp(-Δτj/μ0,j)+Gj+1-(μ0,j+1)Rj+1,j(μ0,j+1)×exp(-Δτj+1/μ0,j+1),
Gj-(μ0,j)=Gj+(μ0,j)Rj,j+1(μ0,j)exp(-Δτj/μ0,j)+Gj+1-(μ0,j+1)Tj+1,j(μ0,j+1)×exp(-Δτj+1/μ0,j+1).
G1+(μ0,1)=F0T0,1(μ0)+G1-(μ0,1)R1,0(μ0,1)exp(-Δτ1/μ0,1),
G0-(μ0)=F0R0,1(μ0)+G1-(μ0,1)T1,0(μ0,1)exp(-Δτ1/μ0,1),
Gj+1+(μ0,J+1)=GJ+(μ0,J)TJ,J+1(μ0,J)exp(-ΔτJ/μ0,J),
GJ-(μ0,J)=GJ+(μ0,J)RJ,J+1(μJi)exp(-ΔτJ/μJi).
Al(τ)=l1=mνmaxτj-1τjH(l, l1, τ, y)[Al1(τ)+sl1(τ)]dy+Bj,l(τ)
forτj-1<τ<τj,
Bj,l(τ)=01{gj+(μ)exp[-(τ-τj-1)/μ]+(-1)l+mgj-(μ)exp[-(τj-τ)/μ]}Plm(μ)μdμ.
fi+(μ, τi)=gi+(μ)exp(-Δτi/μ)+Δi+(μ),
Δi+(μ)=l=mνmaxPlm(μ)τi-1τirl(y)tl(y)×exp[-(y-τi)/μ]dy,
fi-(-μ,τi-1)=gi-(μ)exp(-Δτi/μ)+Δi-(μ),
Δi-(μ)=l=mνmax(-1)l+mPlm(μ)τi-1τirl(y)tl(y)exp(-y/μ)dy.
ni sin θj,i=nj sin θj.
gi+1+(μj,i+1)=T˜i,i+1(μj,i)[gi+(μj,i)exp(-Δτi/μj,i)+Δi+(μj,i)]+Ri+1,i(μj,i+1)×[gi+1-(μj,i+1)exp(-Δτi+1/μj,i+1)+Δi+1-(μj,i+1)],
gi-(μj,i)=Ri,i+1(μj,i)[gi+(μj,i)exp(-Δτi/μj,i)+Δi+(μj,i)]+T˜i+1,i(μi+1)×[gi+1-(μj,i+1)exp(-Δτi+1/μj,i+1)+Δi+1-(μj,i+1)].
g1+(μj,1)=R1,0(μj,1)[g1-(μj,1)exp(-Δτ1/μj,1)+Δ1-(μj,1)]
g0-(μj,1)=+T˜1,0(μj,1)[g1-(μj,1)exp(-Δτ1/μj,1)+Δ1-(μj,1)],
gJ+1+(μj,J+1)=T˜J,J+1(μj,J)[gJ+(μj,J)exp(-ΔτJ/μj,J)+ΔJ+(μj,J)],
gJ-(μj,J)=RJ,J+1(μj,J)[gJ+(μj,J)exp(-ΔτJ/μj,J)+ΔJ+(μj,J)].
gj+(μj)=i[Uj,i+(μj)Δi+(μj,i)+Uj,i-(μj)Δi-(μj,i)],
gj-(μj)=i[Vj,i+(μj)Δi+(μj,i)+Vj,i-(μj)Δi-(μj,i)],
Bj,l(τ)=i01{[Uj,i+Δi+(μj,i)+Uj,i-Δi-(μj,i)]×exp[-(τ-τj-1)/μj]+[Vj,i+Δi+(μj,i)+(-1)l+mVj,i-Δi-(μj,i)]×exp[(τ-τj)/μj]}Plm(μj)μjdμj.
Ej,i(l, l1, τ, s)=rl1(s)01Q(j, i, μj,i, μj)×Pl1m(μj,i)Plm(μj)μjdμj,
Q(j, i, μj,i, μj)=Uj,i+(μj)exp[-(τi-s)]/μj,i)exp[(τj-1-τ)/μj]+(-1)l1+mUj,i-(μj)exp[-(s-τi-1)/μj,i]×exp[(τj-1-τ)/μj]+(-1)l+mVj,i+(μj)×exp[-(τi-s))/μj,i]exp[(τ-τj)/μj]+(-1)l1+mVj,i-(μj)exp[-(s-τi-1)/μj,i]×exp[(τ-τj)/μj].
Bj,l(τ)=i,l1τi-1τiEj,i(l, l1, τ, y)tl(y)dy.
Al(τ)=l1=mνmaxτj-1τjH(l, l1, τ, y)[Al1(τ)+sl1(τ)]dy+i=1Jτi-1τiEj,i(l, l1, τ, y)[Al1(τ)+sl1(τ)]dy,
forτj-1<τ<τj.
Al(τ)=l1=mνmaxτ0τJ+1H(l, l1, τ, y)[Al1(τ)+sl1(τ)]dy
forτ0<τ<τJ+1,
p(γ, τ)=1-τ+τ(1+cos γ)31+τ.
p(γ, τ)=P0+2τ1+τ35P1+P2+25P3.
Ftot=Frefcol+Ftranscol+Frefdif+Ftransdif
ϖ(τ)=0.6+0.4τ.

Metrics