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The 26th International Conference on Transport Theory (ICTT-26)

Paris, France, September 23-27, 2019

The Albedo Problem in Nonexponential Radiative Transfer

Eugene d’Eon

Autodesk

Level 5, Building C - 11 Talavera Road - North Ryde NSW 2113 - Australia

ejdeon@gmail.com

We derive exact solutions of the generalized albedo problem of isotropic scattering in a half space in Rd

with smooth vacuum boundary under monodirectional and uniform diffuse illumination conditions. We

consider general dimension d≥1and nonclassical transport [1, 2] with a general, nonexponential free-path

distribution pc(s)between collisions and nonstochastic phase function Pand single-scattering albedo

0< c ≤1, where absorption is restricted to the collision sites. Photons entering the medium from the

boundary draw their initial free-path lengths from distribution pu(s). The two attenuation laws are

Xc(s)=1−Rs

0pc(s0)ds0between collisions and Xu(s)=1−Rs

0pu(s0)ds0from the boundary.

We derive the Green’s function for the half space and the general law of diffuse reﬂection (BRDF) and

diffuse albedo are also attained, provided the Fourier and inverse Laplace transforms of the Wiener-Hopf

kernel are known. In the talk, we present Monte Carlo validation of these results over a wide variety of

nonclassical media types in a variety of dimensions.

Integral equations: In Rdthe surface area Ωd(r)of the sphere of radius ris

Ωd(r) = dπd/2rd−1/(Γ (d/2 + 1)), and the isotropic scattering phase function is P(ωi→ωo)=1/Ωd(1).

The generalized Peierls integral equation for the scalar collision rate density C(x)is [2, 3]

C(x) = C0(x) + cZRd

C(x0)pc(||x−x0||)

Ωd(||x−x0||)dx0,(1)

where C0(x)is the scalar rate density of initial collisions in the system. Under generalized plane-parallel

symmetry in a half space, where Cis uniform in all but one axis x, we ﬁnd the Wiener-Hopf equation,

C(x) = C0(x) + cZ∞

0

C(x0)KC(x−x0)dx0(2)

where the collision-rate density kernel KCand its Fourier transform ˜

KC(t)are

KC(x) = 1

2Z1

0

pc(|x|/µ)1

µG(µ)dµ, ˜

KC(t)≡Z∞

−∞

KC(x)eixtdx (3)

using angular measure

G(µ) = 2(1 −µ2)d−3

2Γ(d

2)

√πΓd−1

2, d > 1(4)

with G(µ)=1in 3D. After determining C(x), the collided scalar ﬂux φc(x)follows from convolution of

c C(x)with kernel Kφgiven by Eq.(3) with pc(s)replaced by Xc(s).

Eugene d’Eon

Green’s function: We have, from Ivanov ([4], Eqs. (19) and (21)), that the double Laplace transform of

the reciprocal Green’s function is

¯

¯

G(s, s0) = Ls[Ls0[G(x, x0)]] = H(1/s)H(1/s0)

s+s0

(5)

in terms of the Hfunction for the given kernel KC.His given uniquely by [4]

H(z) = exp z

πZ∞

0

1

1 + z2t2log 1

1−c˜

KC(t)dt,Re z > 0(6)

with universal limits H(0) = 1 and H(∞) = (1 −c)−1/2. The Hfunction satisﬁes [4]

H(1/s) = 1 + H(1/s)cZ∞

0

H(1/s0)k(s0)

s+s0ds0,(7)

where k(s)is the inverse Laplace transform of the collision rate density kernel,

Z∞

0

e−s|x|k(s) = KC(x), k(s) = 1

2Z1

0L−1

s u [pc(x)]G(u)du. (8)

Albedo problem: In nonclassical random media, the density of initial collisions for a single photon

entering along direction µ0is not an exponential, but rather C0(x) = pu(x/µ0)/µ0, which creates a less

direct relationship between the Laplace-transformed Green’s function and the diffuse reﬂection law. Using

the inverse Laplace transforms of pu(s)and Xc(s), we ﬁnd the generalized law of diffuse reﬂection for the

half space in terms of a superposition of the transformed Green’s function,

I(0, µ;−µ0) = c

2Z∞

0Z∞

0L−1

s0(pu(x))L−1

s(Xc(x))H(µ/s)H(µ0/s0)

sµ0+s0µds ds0.(9)

Here, I(0, µ;−µ0)is an “azimuthally-integrated” radiance such that I(0, µ;−µ0)µG(µ)is the rate density

of energy ﬂowing in directions with cosine µthrough a surface patch of unit area. Reciprocity is achieved

if and only if the free-path distribution for entering the medium pu(s)is proportional to the extinction

function for leaving a collision Xc(s), in agreement with a previous derivation over single-scattering

paths [5]. Chandrasekhar’s classical result is included above in the case that pu(s) = Xc(s) = e−s, with

L−1

si(pu(x)) and L−1

si(Xc(x)) given by Dirac deltas. The diffuse albedo under unidirectional illumination

along cosine µ0is likewise generalized,

R(µ0) = Z1

0

I(0, µ;−µ0)µG(µ)dµ = 1 −√1−cZ∞

0L−1

si(pu(x))H(µ0/si)

si

dsi.(10)

Universal properties of nonclassical transport: We also prove a number of universal properties of half

space transport, which hold for any continuous free-path distribution pc(s)and any dimension d≥1,

provided scattering is isotropic. When illuminated by a one-sided isotropic plane source at the boundary,

the half space has universal albedo

R=2−c−2√1−c

c.(11)

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The Albedo Problem in Nonexponential Radiative Transfer

Similarly, for grazing illumination, the total reﬂectance is always R(0) = 1 −√1−c. For two adjacent

half spaces with differing single scattering albedos c1, c2and an isotropic source at their interface, we ﬁnd

the total collision rate in the system

Z∞

−∞

C(x)dx =1

√1−c1√1−c2

.(12)

Markovian Binary Mixtures: For multidimensional Levermore-Pomraning random media [1] with

pc(s) = w−r2

−e−r−s+w+r2

+e−r+s

hΣi, Xc(s) = w−r−e−r−s+w+r+e−r+s

hΣi,(13)

pu(s) = w−r−e−r−s+w+r+e−r+s, Xu(s) = w−e−r−s+w+e−r+s,(14)

and constants w−+w+= 1,0< r−< r+,hΣi>0, we ﬁnd

L−1

s(pu(x)) = w−r−δ(s−r−) + w+r+δ(s−r+)(15)

in terms of Dirac delta functions δ, from which, using Eq.(9), we ﬁnd the law of diffuse reﬂection,

I(0, µ;−µ0) = c

2

r+w+Hµ

r+

w+Hµ0

r+

u+µ0

+

w−r−Hµ0

r−

r−µ+r+µ0

+r−w−Hµ

r−

w−Hµ0

r−

µ+µ0

+

w+r+Hµ0

r+

r−µ0+r+µ

.(16)

Combining Eqs.(15) and (10), we ﬁnd the diffuse albedo under monodirectional illumination to be

R(µ0)=1−√1−cw−Hµ0

r−+w+Hµ0

r+.(17)

The Hfunctions for Markovian binary mixtures in Flatland are determined from

˜

KC(z) = 1

hΣi

r2

−w−

qr2

−+z2

+r2

+w+

qr2

++z2

,(18)

and in 3D from

˜

KC(z) = 1

hΣi

w−r2

−tan−1z

r−+w+r2

+tan−1z

r+

z.(19)

Gamma random ﬂights Gamma random ﬂights derive from an intercollision free-path distribution that

is the Laplace transform of the fractional derivative of a Dirac delta,

pc(s) = e−ssa−1

Γ(a)=Z∞

0

e−st δ(a−1)(t−1)

Γ(a)dt (20)

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Eugene d’Eon

with parameter a > 0, and include classical exponential media when a= 1. The Fourier transform of the

kernels and the Case eigenfunctions generalize, respectively, to

˜

KC(z) = 2F1a

2,a+ 1

2;d

2;−z2, φ(µ, ν) = c

2ν

ν−µa

.(21)

In 3D, we ﬁnd exact analytic solutions of the albedo problem for a∈ {2,3,4}. When a= 2, we ﬁnd C(x)

to exactly satisfy a diffusion equation. With the escape probability from the medium given by a simple

exponential of depth, we construct the perfectly zero-variance walk for escaping the medium. The diffuse

albedos for monodirectional and uniform diffuse illuminations are,

R(µ0) = c√1−cµ0+ 2

2√1−c+ 1√1−cµ0+ 12, R =c

√1−c+ 12.(22)

The diffuse reﬂection law for the “diffusion-transport” half space is then the algebraic expression

I(0, µ;−µ0) = c

4(µ+µ0)3µ(µ+µ0)H0(µ)µ0(µ+µ0)H0(µ0) + H(µ0)(2µ+µ0)

+H(µ)µ0(µ+µ0)(µ+ 2µ0)H0(µ0)+2H(µ0)µ2+ 3µµ0+µ2

0 (23)

with H(µ) = (1 + µ)/(1 + √1−cµ).

Power-law random ﬂights For power law random media [6] with pu(s) = a

a+sa+1, we ﬁnd, in 3D,

L−1

t[pc(s)] = aae−atta+1

Γ(a), k(s) = Z1

0

aae−asu(su)a+1

2Γ(a)du =Γ(a+ 2) −Γ(a+ 2, as)

2a2sΓ(a).(24)

In the general case, the complexity of the Fourier transform of KCpresented numerical difﬁculty, but we

were able to derive the closed form single-scattering BRDF in 3D [5],

f1(µi, µo) = c

4π

a2F11, a + 1; 2(a+ 1); 1 −µo

µi

(2a+ 1)µi

.(25)

REFERENCES

[1] S. Audic, H. Frisch, Monte-Carlo simulation of a radiative transfer problem in a random medium:

Application to a binary mixture, Journal of Quantitative Spectroscopy and Radiative Transfer,50

(2), 127–147 (1993).

[2] E. W. Larsen, R. Vasques, A generalized linear Boltzmann equation for non-classical particle

transport, Journal of Quantitative Spectroscopy and Radiative Transfer,112 (4) 619–631, (2011).

[3] C. Grosjean, The Exact Mathematical Theory of Multiple Scattering of Particles in an Inﬁnite

Medium. Memoirs Kon. Vl. Ac. Wetensch.,13 (36) , (1951).

[4] V. V. Ivanov, Resolvent method: exact solutions of half-space transport problems by elementary

means, Astronomy and Astrophysics,286, 328-337 (1994).

[5] E. d’Eon, A Reciprocal Formulation of Nonexponential Radiative Transfer. 1: Sketch and

Motivation, Journal of Computational and Theoretical Transport,47 (1-3) 84–115, (2018).

[6] A. B. Davis, In Computational Methods in Transport. Springer, 85–140. (2006).

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