The Albedo Problem in Nonexponential Radiative Transfer

Abstract

We derive exact solutions of the generalized albedo problem of isotropic scattering in a half space in R d with smooth vacuum boundary under monodirectional and uniform diffuse illumination conditions. We consider general dimension d ≥ 1 and nonclassical transport [1, 2] with a general, nonexponential free-path distribution p c (s) between collisions and nonstochastic phase function P and single-scattering albedo 0 < c ≤ 1, where absorption is restricted to the collision sites.

Full text

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)
√πΓd−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+Hµ
r+

w+Hµ0
r+
u+µ0
+
w−r−Hµ0
r−
r−µ+r+µ0

+r−w−Hµ
r−

w−Hµ0
r−
µ+µ0
+
w+r+Hµ0
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−cw−Hµ0
r−+w+Hµ0
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−1z
r−+w+r2
+tan−1z
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) = 2F1a
2,a+ 1
2;d
2;−z2, φ(µ, ν) = c
2ν
ν−µa
.(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+ 12, R =c
√1−c+ 12.(22)
The diffuse reflection law for the “diffusion-transport” half space is then the algebraic expression
I(0, µ;−µ0) = c
4(µ+µ0)3µ(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+sa+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
a2F11, a + 1; 2(a+ 1); 1 −µo
µi
(2a+ 1)µi
.(25)
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