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The Albedo Problem in Nonexponential Radiative Transfer



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.
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
Level 5, Building C - 11 Talavera Road - North Ryde NSW 2113 - Australia
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 d1and 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
0pc(s0)ds0between collisions and Xu(s)=1Rs
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 d(r)of the sphere of radius ris
d(r) = d/2rd1/(Γ (d/2 + 1)), and the isotropic scattering phase function is P(ωiωo)=1/d(1).
The generalized Peierls integral equation for the scalar collision rate density C(x)is [2, 3]
C(x) = C0(x) + cZRd
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
where the collision-rate density kernel KCand its Fourier transform ˜
KC(x) = 1
µG(µ)dµ, ˜
KC(x)eixtdx (3)
using angular measure
G(µ) = 2(1 µ2)d3
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)
in terms of the Hfunction for the given kernel KC.His given uniquely by [4]
H(z) = exp z
1 + z2t2log 1
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
where k(s)is the inverse Laplace transform of the collision rate density kernel,
es|x|k(s) = KC(x), k(s) = 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)0, 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
0+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) = es, with
si(pu(x)) and L1
si(Xc(x)) given by Dirac deltas. The diffuse albedo under unidirectional illumination
along cosine µ0is likewise generalized,
R(µ0) = Z1
I(0, µ;µ0)µG(µ)= 1 1cZ
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 d1,
provided scattering is isotropic. When illuminated by a one-sided isotropic plane source at the boundary,
the half space has universal albedo
The Albedo Problem in Nonexponential Radiative Transfer
Similarly, for grazing illumination, the total reflectance is always R(0) = 1 1c. 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
C(x)dx =1
Markovian Binary Mixtures: For multidimensional Levermore-Pomraning random media [1] with
pc(s) = wr2
hΣi, Xc(s) = wrers+w+r+er+s
pu(s) = wrers+w+r+er+s, Xu(s) = wers+w+er+s,(14)
and constants w+w+= 1,0< r< r+,hΣi>0, we find
s(pu(x)) = wrδ(sr) + w+r+δ(sr+)(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
Combining Eqs.(15) and (10), we find the diffuse albedo under monodirectional illumination to be
The Hfunctions for Markovian binary mixtures in Flatland are determined from
KC(z) = 1
and in 3D from
KC(z) = 1
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) = essa1
est δ(a1)(t1)
Γ(a)dt (20)
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;z2, φ(µ, ν) = c
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) = c10+ 2
21c+ 110+ 12, R =c
1c+ 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).
Power-law random flights For power law random media [6] with pu(s) = a
a+sa+1, we find, in 3D,
t[pc(s)] = aaeatta+1
Γ(a), k(s) = Z1
2Γ(a)du =Γ(a+ 2) Γ(a+ 2, as)
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
a2F11, a + 1; 2(a+ 1); 1 µo
(2a+ 1)µi
[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).
ResearchGate has not been able to resolve any citations for this publication.
Previous proposals to permit non-exponential free-path statistics in radiative transfer have not included support for volume and boundary sources that are spatially uncorrelated from the scattering events in the medium. Birth-collision free paths are treated identically to collision-collision free paths and application of this to general, bounded scenes with inclusions leads to non-reciprocal transport. Beginning with reciprocity as a desired property, we propose a new way to integrate non-exponential transport theory into general scenes. We distinguish between the free-path-length statistics between correlated medium particles and the free-path-length statistics beginning at locations not correlated to medium particles, such as boundary surfaces, inclusions and uncorrelated sources. Reciprocity requires that the uncorrelated free-path distributions are simply the normalized transmittance of the correlated free-path distributions. The combination leads to an equilibrium imbedding of a previously derived generalized transport equation into bounded domains. We compare predictions of this approach to Monte Carlo simulation of multiple scattering from negatively-correlated suspensions of monodispersive hard spheres in bounded two-dimensional domains and demonstrate improved performance relative to previous work. We also derive new, exact, reciprocal, single-scattering solutions for plane-parallel half-spaces over a variety of non-exponential media types.
This paper considers monochromatic radiative transfer in a diffusive three dimensional random binary mixture. The absorption coefficient, along any line-of-sight is a homogeneous Markov process, which is described by a three-dimensional Kubo-Anderson process. The transfer equation is solved numerically by Monte-Carlo simulations on a massively parallel computer (a Connection Machine) by attaching one or several photons to each processor. The implementation of the simulations on the machine is discussed in detail, in particular the association between photons and processors and the storage of the data concerning the photons and the realizations of the statistics. With a CM-2 having 8000 processors, it is possible, with an adequate strategy, to follow simultaneously millions of photons in hundreds of realizations and to reach optical thicknesses up to 100 with dispersions of order 10-2 for the reflection and transmission coeficients. The simultations are validated, in the case of the one-dimensional rod geometry, by comparison with the exact analytical solution, constructed by averaging the solution of the non-stochastic problem (diffusion in a rod of given optical thickness) over the probability density of the optical thickness. The latter obeys a stochastic Liouville equation which is solved by a Green's function method. The influence of the dimension of the Kubo-Anderson process is studied for the case of a slab and it is shown that a slab consisting in a pile of layers (1D process) is more transparent than one which consists in a stack of lumps (3D process). A strategy for improving the efficiency of Monte-Carlo simulations, based on the distribution of the lengths of the individuals steps of the photons, is presented and discussed.
A new version of the resolvent method for solving half-space transport problems is presented. We solve the integral transfer equation with the {LAMBDA}-operator. The approach is based on the use of a non-linear equation relating the surface Green function G_0_(τ) for the semi-infinite medium and the Green function Ginfinity_(τ) for the infinite medium. This Ginfinity_(τ) is assumed known. The equation is first used to derive, literally in one line, the famous sqrt(ɛ) law. Then, by a short series of elementary transformations, we obtain the Fock-Chandrasekhar explicit expression for the H-function. Assuming that the kernel function of the {LAMBDA}-operator is a Laplace-type integral, we derive the usual explicit expression for G_0_(τ) in the form of a similar integral. The integrand involves only the functions appearing in the explicit expression for G_0_(τ), and the H-function. And again, the derivation is elementary, i.e., without any use of the complex plane, and surprisingly short; the non-linear H-equation is completely by-passed. Alternative equations satisfied by the H-function are presented, among them a linear integral equation with a non-singular kernel. The standard non-linear H-equation is used to find the asymptotic behavior of the solution of the homogeneous transfer equation in deep layers of the semi-infinite atmosphere. In conclusion, a brief comment on the development of the resolvent method is given.
This paper presents a derivation and initial study of a new generalized linear Boltzmann equation (GLBE), which describes particle transport for random statistically homogeneous systems in which the distribution function for chord lengths between scattering centers is non-exponential. Such problems have recently been proposed for the description of photon transport in atmospheric clouds; this paper is a first attempt to develop a Boltzmann-like equation for these and other related applications.
The Exact Mathematical Theory of Multiple Scattering of Particles in an Infinite Medium
  • C Grosjean
C. Grosjean, The Exact Mathematical Theory of Multiple Scattering of Particles in an Infinite Medium. Memoirs Kon. Vl. Ac. Wetensch., 13 (36), (1951).
  • A B Davis
A. B. Davis, In Computational Methods in Transport. Springer, 85-140. (2006).