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Radiative transfer in half spaces of arbitrary dimension

Eugene d’Eona, Norman J. McCormickb

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

ejdeon@gmail.com

bDepartment of Mechanical Engineering - University of Washington - Seattle, WA

98195-2600, USA - mccor@uw.edu

Abstract

We solve the classic albedo and Milne problems of plane-parallel illumination of

an isotropically-scattering half-space when generalized to a Euclidean domain

Rdfor arbitrary d≥1. A continuous family of pseudo-problems and related H

functions arises and includes the classical 3D solutions, as well as 2D “Flatland”

and rod-model solutions, as special cases. The Case-style eigenmode method is

applied to the general problem and the internal scalar densities, emerging distri-

butions, and their respective moments are expressed in closed-form. Universal

properties invariant to dimension dare highlighted and we ﬁnd that a discrete

diﬀusion mode is not universal for d > 3 in absorbing media. We also ﬁnd unex-

pected correspondences between diﬀering dimensions and between anisotropic

3D scattering and isotropic scattering in high dimension.

Keywords: Albedo problem, Flatland, MacDonald kernel, Hypergeometric,

Case’s method, Wiener-Hopf

1. Introduction

Linear transport theory [1, 2, 3] has a long history of utility in many ﬁelds

for modeling the motion of particles and waves in random media and is most

often concerned with transport in a spatially three-dimensional (3D) domain

(even when symmetries in the solution reduce the equations to a single spa-

tial coordinate, prompting a “1D” label). It is, however, occasionally useful to

consider Euclidean domains apart from R3. For inﬁnite homogeneous media,

monoenergetic problems in Rdhave been mostly solved [4]. In this paper, we

solve the classic problem of diﬀuse reﬂection from an isotropically-scattering

half space in Rdfor general d≥1. The familiar Hfunctions, rigorous diﬀusion

lengths, extrapolation constants, and singular eigenfunctions of radiative trans-

fer and neutron transport are given new mathematical context as special cases

in a general family of solutions expressed in closed form using hypergeometric

functions. We also derive a number of new analytic results for the “Flatland”

(d= 2) domain, which is used in a wide number of applied settings.

Preprint May 21, 2019

arXiv:1905.07825v1 [physics.class-ph] 19 May 2019

1.1. Motivation and related work

There are both practical and theoretical reasons to consider transport prob-

lems in general dimension. The 1D rod has long been a useful domain for

transport research and education [5, 6, 7, 8]. The same can be said for the 2D

“Flatland” domain [9], where it is possible to visualize the entire lightﬁeld with

2D images [10].

Transport in both the rod and Flatland settings ﬁnds numerous applica-

tion in practice. The rod model is equivalent to the two-stream approxima-

tion in plane-parallel atmospheric scattering [11, 12], which is still a com-

mon method of solution for radiation budgets [13]. Transport in Flatland also

has many real-world applications, such as sea echo [14], seismology [15], an-

imal migration [16, 17], and wave propagation and diﬀraction in plates and

ice [18, 19, 20]. Also, planar waveguides comprised of dielectric plates with con-

trolled or random patterns of holes lead to 2D transport and have proven useful

for studying engineered disorder [21]. Similarly, bundles of aligned dielectric

ﬁbers, such as clumps of hair or fur, can also be treated with a Flatland ap-

proach [22, 23, 24, 25], where it is common to employ an approximate separable

product of 1D and 2D solutions [26]. Reactor design also makes use of such

2D/1D decompositions [27].

Going beyond 3D, higher-order dimensions occasionally ﬁnd application in

practice. Exact time-dependent solutions in 2D and 4D have been combined to

approximate the unknown 3D solution for the isotropic point source in inﬁnite

media [28], and later applied to a time-dependent searchlight problem using the

method of images [29]. In the study of cosmic microwave background radiation

it has even been considered to change dimension over the course of a single

random ﬂight [30].

The study of transport problem in the case of general dimension can reveal

how dimension dimpacts various aspects of the solution. New insights about

3D transport have been found by identifying correspondences between problems

of diﬀering conﬁgurations and dimensionalities [31, 32, 33], which often appear

unexpectedly. Most investigations of this nature have considered only inﬁnite

media. In this work, we identify some new exact correspondences regarding

anisotropic scattering in a 3D half space.

The literature on stochastic processes includes a number of fully general

studies in Rd(see [34, 4, 35, 36, 37, 38] and references therein) that reveal

the inﬂuence of dimension. Such studies often permit d≥1 to be a free real

parameter [39] by using the general surface area Ωdof the unit sphere Sd−1≡

{x∈Rd,||x|| = 1}in ddimensions,

Ωd=dπd/2

Γd

2+ 1,(1)

as expressed using the Gamma function [40], and taking the meaning of a non-

integral dimension ﬂight abstractly and “by deﬁnition” from the equations that

follow. Where possible, we consider the same freedom on din this paper.

2

For inﬁnite homogeneous media, Green’s functions for monoenergetic linear

transport with isotropic [28, 41, 42, 38, 43, 44] and anisotropic [4, 45, 46] scatter-

ing are known in domains apart from 3D. The unidirectional point source was

also considered in Flatland [47]. In these inﬁnite domains, the non-universal

role of diﬀusion as a “rigorous asymptote” of the full solution for general di-

mension with absorption was observed [48, 43]. We expand on these ﬁndings for

the isotropic scattering case, and ﬁnd a simple algebraic condition for diﬀusion

asymptotics to arise.

For the case of bounded domains, isotropic scattering in a Flatland half space

has been solved in a number of works [18, 19, 49, 25]. Slab geometry [23] and

layered problems [50] in Flatland have also been solved. The study of inverse

problems in plane-parallel domains of general dimension has been considered

in a number of works (see, for example, [51]). We expand on these solutions

by considering general dimension and by producing the singular eigenfunctions,

whose orthogonality properties allow derivation of the moments of the internal

scalar ﬂux and angular distributions. These moment derivations complement

the mean, variance and general moments previously produced for 3D [52, 53]

and are useful for forming approximate searchlight approximations [54] and for

guiding Monte Carlo estimators towards zero-variance [55].

1.2. Outline

Section 2 provides a discussion of general dimensional (dD) radiometry. The

integral and integrodiﬀerential transport equations are given in Section 3, to-

gether with the dispersion equation and related eigenvalues. The albedo problem

is solved in Section 4 using Chandrasekhar H-functions. This section also dis-

cusses evaluation strategies for the Hfunctions, escape probabilities from within

the half space, and a generalized enhancement factor for coherent backscatter-

ing. Case’s method is applied to the general problem in Section 5, and used to

derive equations for spatial and directional moments. Extrapolation distances

for the Milne problem for a half-space and two adjacent half spaces are derived

in Section 6. We derive some universal properties of half-space transport in Sec-

tion 7 before concluding in Section 8. A closed-form equation for the discrete

eigenvalues is given in Appendix A and additional closed-form equations for the

Hfunction are given in Appendix B. Monte Carlo sampling and connections

between anisotropic scattering and nonclassical transport are given in Appendix

C and Appendix D, respectively.

2. Radiometry in Rd

Previous studies of the transport equation for d-space [56, 57] implicitly

assumed the generalization of standard radiometric quantities, which we brieﬂy

review here, along with a derivation of the generalized Lambertian bidirectional

reﬂectance distribution function (BRDF).

3

2.1. Radiance

Let us consider time-independent mono-energetic speciﬁc intensity (radi-

ance) I(x, ω) at a position x∈Rdsuch that the rate of energy dE ﬂowing

across a surface element of area dσ in directions comprising a solid angle dω

about direction ω∈Sd−1is

dE =I(x, ω) cos θdσdω (2)

where θis the angle between ωand the outward surface normal of dσ. With

Ωd“steradians” in the unit sphere Sd−1, the integral of an angularly-uniform

(I(x, ω) = 1) ﬁeld of unit radiance is

ZSd−1

I(x, ω)dω = Ωd(3)

and the isotropic phase function is

P(ωi→ωo)=1/Ωd.(4)

2.2. Axial symmetry

In plane-parallel problems with axial symmetry the radiance is assumed to be

symmetric in all but one axis and rotationally invariant about that axis, allowing

us to express the transport equations over an integrated radiance that depends

on a single position, the optical depth x, and a single direction parameter µ=

cos θ, the cosine with respect to the depth axis. The parameter µindexes a pair

of directions in Flatland, a cone of directions in 3D, and higher-dimensional

hypercones for d > 3.

We choose a deﬁnition of integrated radiance I(x, µ) that is constant in µ∈

[−1,1] whenever I(x, ω) is constant in ω∈Sd−1by introducing the appropriate

angular measure G(µ)dµ. The total rate of radiant energy ﬂowing across dσ at

depth xconﬁned to directions with cosines in dµ about µis then

dE =I(x, µ)µ G(µ)dµ dσ. (5)

We normalize G(µ) such that

1

2Z1

−1

G(µ)dµ = 1,(6)

a uniform integral over the sphere Sd−1. The function G(µ) that satisﬁes these

conditions is ([4], Eq.(33))

G(µ) = 2(1 −µ2)d−3

2Γ(d

2)

√πΓd−1

2, d > 1.(7)

For integer dimensionalities d= 2 to 7, G(µ) is

2

πp1−µ2; 1; 4p1−µ2

π;3

2(1 −µ2); 16(1 −µ2)3/2

3π;15

81−µ22.

4

Scattering in a 1D rod is included in this deﬁnition by noting [28]

lim

d→1+ G(µ) = δ(µ−1) + δ(µ+ 1).(8)

The probability that a single photon leaves an isotropic collision into dµ about

µis (c/2)G(µ)dµ, where 0 < c ≤1 is the single-scattering albedo, and so the

integrated radiance about xarising from isotropic collisions happening at a unit

rate is I(x, µ) = c/2.

2.3. Uniform diﬀuse illumination

In additional to unidirectional illumination of the half space, we will also

consider the case of uniform diﬀuse illumination, which we deﬁne to be uniform

radiance in all directions arriving from the hemisphere to a given surface patch.

Such a source produces ﬂux across the patch with an intensity proportional

to I(µ)µ. To produce a unit ﬂux across a patch of unit area, we require a

normalization constant such that a µ-weighted integral over the hemisphere is

1, √πΓd+1

2

Γd

2Z1

0

µG(µ)dµ = 1.(9)

The uniform diﬀuse boundary source condition is thus

I(0, µ) = √πΓd+1

2

Γd

2,−1≤µ≤0 (10)

or I(0, ω)=1/πdwhere

πd≡Ωd

2Z1

0

µ G(µ)dµ =πd−1

2

Γd+1

2,(11)

π1= 1, π2= 2, π3=π, π4=4π

3, ...

2.4. Bidirectional reﬂectance distribution function

The BRDF gives the radiance fL(ωi, ωo) leaving surface area dσ in direction

ωodue to a unit radiance arriving at dσ from direction ωi. This form of ex-

pressing the diﬀuse reﬂection law is convenient for image synthesis [58] and for

comparing the behaviour to other known BRDFs. Of particular interest is the

Lambertian BRDF with total diﬀuse albedo 0 < kd≤1, whose generalization

to arbitrary dimension is

fL(ωi, ωo) = kd

πd

.(12)

5

3. Transport equations

We now review the plane-parallel transport equations for isotropic scattering

in a halfspace in Rd. Energy balance in an inﬁnitesimal slab in plane geometry

with axial symmetry yields a transport equation of one spatial and one angular

variable. If the intensity distribution arriving at the slab of thickness dx is given

by I(x, µ0), the ﬂux in direction µ0crossing dx is proportional to µ0G(µ0), and

the track lengths extending through the slab are dx/µ0, so the rate of photons

entering collisions within dx is

C(x)dx =Z1

−1

I(x, µ0)G(µ0)µ0dx

µ0dµ0.(13)

The inscattered contribution to I(x, µ) is thus

c

2C(x) = c

2Z1

−1

I(x, µ0)G(µ0)dµ0(14)

and the full integrodiﬀerential form of the transport equation is then

µ∂

∂x + 1I(x, µ) = c

2Z1

−1

I(x, µ0)G(µ0)dµ0,(15)

which reduces to the familiar 3D form with G(µ) = 1 and the Flatland equa-

tion [22, 19, 25, 49] with G(µ) = 2/(πp1−µ2).

Equation (15) is a “pseudo problem” of the form studied by Chandrasekhar

(Section 89 of [1]). In his notation,

µ∂

∂x + 1I(x, µ) = Z1

−1

Ψ(µ0)I(x, µ0)dµ0.(16)

Chandrasekhar considered pseudo problems in relation to anisotropic scattering

in a 3D half space. Multiple pseudo problems with polynomial characteristic

functions Ψi(µ) arise in each case, and their related Hfunctions appear in

the exact solution. No individual pseudo problem on its own corresponds to a

complete transport problem, hence the label. For isotropic scattering in d-space,

however, we see a single pseudo problem does describe the complete problem.

Comparing Eqs.(15) and (16), we ﬁnd the characteristic function for our problem

to be

Ψ(µ) = c

2G(µ).(17)

For d > 1, Ψ(µ) is an even, non-negative function satisfying

Z1

0

Ψ(µ)dµ =c

2≤1

2,(18)

provided 0 ≤c≤1. Further, for d > 1, Ψ(µ) is also regular on (−1,1). At the

boundaries,

Ψ(±1) =

∞,(1 ≤d < 3)

c/2,(d= 3)

0,(d > 3).

(19)

6

Busbridge studied a very general class of pseudo problems, relaxing the assump-

tion of polynomial characteristic function. With the above conditions satisﬁed,

we can apply the ﬁndings of Chapter 2 of Busbridge [59].

Before solving Eq.(15), we consider the related integral equation for the

collision-rate density C(x) at optical depth xin the half space. This can be

formed by integrating the total attenuated intensity at xarriving from collisions

at each depth x0inside the half space,

C(x) = C0(x) + cZ∞

0

K(x−x0)C(x0)dx0,(20)

where C0(x) is the forcing function (the collision-rate density of ﬁrst collisions

from any external source, in this case). The kernels Kare symmetric and

account for the total collision rate density at optical depth xarising from energy

that leaves a collision from a hyperplane at depth x0.

Eq.(20) is an integral equation of the Wiener-Hopf (W-H) kind, named after

the authors who ﬁrst solved it for isotropic scattering in 3D. That original W-H

equation was ﬁrst posed by Chwolson [60], who considered the Schwarz-schild-

Milne (exponential integral) displacement kernel

K(x) = 1

2E1(|x|) = 1

2Z∞

1

e−|x|t

tdt (21)

in his study of the translucent appearance of milk glass1.

For 1D and 2D, the kernels are also already known, and their W-H equations

have been studied. The Picard-Lalesco kernel

K(x) = 1

2e|x|(22)

describes exponential ﬂights in a rod. The MacDonald / Hankel kernel

K(x) = K0(|x|)

π(23)

describes isotropic scattering in Flatland [49], and has also appeared in studies of

wave diﬀraction problems [62, 63]. It is interesting that Fock, Case and Krein

each considered the three kernels (21), (22), and (23) in papers [18, 64, 65]

on general W-H methods, but did not explicitly identify them as pertaining

to isotropic scattering in 3D, 1D and 2D, respectively. To the best of our

knowledge, they have not previously been shown as members of the same uniﬁed

family. We show this now, expressing the general kernel in terms of the plane-

geometry measure for d-space,

K(x) = 1

2Z1

0

e−|x|/µ 1

µG(µ)dµ =1

2πΓd

2G3,0

1,3x2

4

d−1

2

0,0,1

2.(24)

1112 years later, the computer rendering of a glass of milk with multiple scattering was

one of the iconic images in a seminal paper [61] that sparked the subsurface revolution in ﬁlm

rendering and earned the authors an Academy award.

7

Here, G3,0

1,3is a Meijer Gfunction. Again, we have used the assumption of

isotropic scattering (which can be lifted, at considerable complexity, and won’t

be treated here) and also that the free-path distribution between collisions pc(x)

is an exponential, pc(|x|/µ) = e−|x|/µ, which can be easily generalized for the

case of complete-frequency redistribution in line formation [66] and non-classical

media with non-exponential free paths [44] (see Appendix D.2).

The kernels in Eq.(24) are positive symmetric normalized displacement/convolution

kernels Z∞

−∞

K(x)dx = 1 (25)

of the Laplace type [66], expressible as

K(x) = Z∞

0

h(s)e−|x|sds (26)

where

h(s) = 1

2Θ(s−1)G(1/s)/s, (27)

using the Heaviside theta function Θ(x). The kernels are singular at x= 0 for

d > 1,

lim

x→0K(x) = ∞.(28)

The Fourier transform ˜

K(t) of the kernels will play a central role in solving

the albedo problem and can be expressed for the general case d≥1 using the

hypergeometric function 2F1[40], by taking the Fourier transform of Eq.(24)

and exchanging the order of integration,

˜

K(t)≡Z∞

−∞

K(x)eixtdx =2F11

2,1; d

2;−t2.(29)

The common cases of d∈ {1,2,3}reduce to the familiar set

1

1 + t2,1

√1 + t2,tan−1t

t⊂˜

K(t),(30)

of Fourier transforms of the Picard, MacDonald and Schwarz-schild-Milne ker-

nels, respectively.

We see that a change in dimension damounts to a change of kernel K(x)

and related characteristic function Ψ(µ), and these two functions completely

characterize the problem. The solutions that follow will have much in common

with analogous variations of Kand Ψ that arise when considering general phase

function [1], reﬂectance conditions at the boundary [67], and line-formation and

other energy-dependent problems [68, 69, 70]. As such, we will rely on general

studies of W-H equations [18, 59, 65, 68, 71, 72, 70, 66].

8

3.1. Dispersion equation and eigenvalues

For inﬁnite, half space or slab geometry problems, the solutions all depend

on the eigenspectra of the transport kernel K(x), which can include both a

continuous and a discrete component. We review the eigenvalues now, and

their conditions for existence, before solving the general albedo problem.

The discrete eigenvalues, when they exist, are real zeros ν0of the dispersion

function Λ(z) [59], which is related to the Fourier transform of the kernel by

Λ(i/t)=1−cZ∞

−∞

K(x)eitxdx (31)

or, equivalently, from the characteristic function Ψ(µ),

Λ(z)=1−c z

2Z1

−1

G(µ)

z−µdµ. (32)

From Eq.(29) we have the generalized dispersion equation in terms of a hyper-

geometric function

Λ(z)=1−c2F11

2,1; d

2;1

z2,(33)

in agreement with previous derivations in inﬁnite spherical geometry [48, 28, 38].

By known properties of 2F1[40], Λ(z) satisﬁes the diﬀerential equation

Λ00(z)z2−z4+ Λ0(z)zz2(d−3) −2+ 2Λ(z)−2=0.(34)

Here we see d= 3 as a special case, the unique dimension where the z3Λ0(z)

term vanishes.

The discrete eigenvalues fall into three cases [59], based on dimension dand

absorption c. For the case of conservative scattering c= 1, double zeros at

inﬁnity arise for all d≥1. This follows immediately from the limit as z→ ∞,

Λ(∞)=1−c

2Z1

−1

G(µ)dµ

= 1 −c , (35)

and from Λ0(∞) = 0. For absorbing media, 0 <c<1, given the symmetry and

non-negativity of K, there will be either 0 or 1 pairs of real eigenvalues ±ν0,

satisfying Λ(±ν0) = 0 [65].

For d≤3, Ψ(1) 6= 0 and so Λ(z) always has a real root ν0>1. For d > 3,

discrete eigenvalues will only exist when Λ(1) <0 [59]. After observing that

2F11

2,1; d

2; 1=d−2

d−3, d > 3,(36)

we ﬁnd that Λ(z) admits a ﬁnite zero ν0>1 if and only if

(d−3)/(d−2) <c<1.(37)

9

This condition simpliﬁes several prior observations and conditions for diﬀusion

modes disappearing in dimensions d > 3 [48, 43]2and is new, to the best of our

knowledge. Closed form expressions are known for d∈ {1,2,4,6}[43],

ν0=±1/√1−c, (d= 1)

ν0=±1/p1−c2,(d= 2)

ν0=±1/2p(c−c2),(d= 4, c > 1/2)

ν0=±3

2q(9 −8c)c−pc(4c−3)3

,(d= 6, c > 3/4).

Mathematica is able to ﬁnd the roots in 8D and 10D in closed form but we omit

these bulky expressions for space reasons.

In 3D, the eigenvalues always exist, the dispersion equation being

0 = 1 −c ν0tanh−1(1/ν0).

In odd dimensionalities d≥5, the discrete eigenvalues are (like in 3D) also

solutions of transcendental equations of increasing complexity, such as in 5D,

6cν0ν0+ν2

0−coth−1(ν0)+ coth−1(ν0)= 4.(38)

In Appendix A we derive a general closed-form expression for ν0for any dimen-

sion d≥1.

A related function that plays an important role in the solution of half space

problems is

λ(ν)=1−cν

2PZ1

−1

G(µ)

ν−µdµ (39)

with Pindicating the Cauchy principal value of any integral over νor µmust

be taken. For ν∈[−1,1] the principal value integral can be expressed in closed

form,

λ(ν) = 1 −c+c2F11,1−d

2;1

2;ν2,(40)

which simpliﬁes to

1,1,1−1

2cν ln 1 + ν

1−ν,1−2cν2⊂λ(ν) (41)

for d∈ {1,2,3,4}, respectively. Equation (40) can also be written

λ(ν) = 1 + cd−2

d−3ν2−12F11,2−d

2;−1

2;ν2+

(d−6)ν2+ 12F11,2−d

2;1

2;ν2 (42)

2It was incorrectly reported in [43] that the number of discrete eigenvalues increases past

d= 4.

10

d=2

d=3

d=4

d=5

d=6

d=10

Λ(z)

λ(z)

c=0.63

(1-c)

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

z

Figure 1: The functions Λ(z) and λ(z) for real zand c= 0.63.

which shows how c= (d−3)/(d−2) is a special value. Again, using known

properties of 2F1, it is straightforward to ﬁnd the diﬀerential equation satisﬁed

by λ(z),

z2−1λ00(z)−(d−5)zλ0(z) + (4 −2d)λ(z) + 2(1 −c)(d−2) = 0,(43)

with conditions

λ(0) = 1, λ0(0) = 0.(44)

From the Plemelj-Sohotski formula[73] the boundary values of Λ(z) are

Λ±(ν) = λ(ν)±iπcνG(ν)/2, ν ∈[−1,1],(45)

from which it follows that

Λ+(ν)+Λ−(ν)=2λ(ν).(46)

Also, Λ(0) = Λ±(0) = 1. It is straightforward to verify that Eq.(45) satisﬁes

the diﬀerential equation in Eq.(34).

Both Λ(z) and λ(z) are shown in Figure 1 for a variety of dimensions d. For

the illustrated case of c= 0.63 we see that there are no discrete roots ν0>1 of

Λ(ν0) for d > 4 and also that λ(ν) admits either 0, 1, or 2 roots 0 < ν < 1.

4. The albedo problem

Let us now consider a unit current azimuthally-symmetric plane-parallel

illumination arriving along direction cosine µ`, given by

I(0, µ;−µ`) = δ(µ−µ`)/(µ`G(µ`)) (47)

11

for µand µ`∈[0,1]. The density of initial collisions inside the half space at

optical depth x≥0 is then,

C0(x) = (1/µ`) exp(−x/µ`),(48)

which is independent of dimension. Here, C0(x)dx is the rate of particles enter-

ing their ﬁrst collision within dx about x.

The inhomogeneous integral equation (20) with the forcing function C0(x) in

Eq.(48) deﬁnes the dD albedo problem with a unidirectional source. Its solution

can be found in a number of ways, and directly yields the internal distribution

and, indirectly, the emerging distribution.

Once the collision rate density C(x) is found, the scalar ﬂux of particles

in ﬂight inside the medium is known immediately because we are assuming

a classical medium with no correlation or memory between scattering events.

Given our parametrization of optical depth x, the scalar ﬂux is proportional to

C(x) by a unit constant, the mean-free path, which is simply a change of units.

The integrated radiance at the boundary is

I(0, µ;−µ`) = Z∞

0

c

2C(x) exp(−x/µ)1

µdx, (49)

and the rate of photons leaving the medium along directions in dµ about µ

is I(0, µ;−µ`)µG(µ)dµ. Eq.(49) is simply a reduced-intensity calculation of

radiance leaving collisions at depth x, using the exponential Beer-Lambert law.

The quantity (c/2)C(x)G(µ)dµ is the integrated radiance leaving collisions at

xinto dµ about µ, exp(−x/µ) is the Beer-Lambert calculation, and (1/µ)dx is

the source measure at depth x; i.e. the length of the line segment tilted by µ

inside the slab of thickness dx from which the source of collided photons arises.

Equation (49) gives the law of diﬀuse reﬂection for the half space and shows

that the emerging distribution is related to the internal distribution by a Laplace

transform. The Laplace transform of the internal distribution is given in terms

of H-functions, which we consider next.

4.1. dD Chandrasekhar H-functions

Given the characteristic function Ψ(µ) = (c/2)G(µ) for isotropic scattering

in dD (Section 3), the related Hfunctions satisfy the integral equation [59],

1

H(µ)= 1 −c µ

2Z1

0

H(µ0)

µ+µ0G(µ0)dµ0,(50)

for 0 ≤µ≤1 or, more generally

1

H(z)= 1 −c z

2Z1

0

H(µ0)

z+µ0G(µ0)dµ0, z /∈[−1,0] (51)

12

Regardless of dimension d≥1 or absorption 0 < c ≤1, the solution of Eq.(50)

is known in closed form by the Fock/Chandrasekhar equation [18, 59]

H(z) = exp z

2πi Zi∞

−i∞

1

t2−z2ln Λ(t)dt

= exp −z

πZ∞

0

1

1 + z2k2ln Λ(i/k)dk,Re z > 0.(52)

Figure 2 illustrates values of 1/H(z).

Let us pause for a moment to consider the signiﬁcance of the appearance of

Λ(i/k) in Eq.(52). By Eq.(31),

1

Λ(i/t)=1

1−c˜

K(t)= 1 + c˜

K(t) + c2˜

K(t)2+..., (53)

which is the Fourier-space Neumann-series Green’s function for the isotropic

plane source in inﬁnite geometry. So we see an exact inﬁnite-space solution

inside the Hfunction expression for the half space,

H(z) = exp z

πZ∞

0

1

1 + z2t2ln 1

1−c˜

K(t)dt.

Ivanov [66] noted a similar relationship for general displacement kernels K.

From his analysis we also have, for all d≥1,

H(0) = 1,(54)

H(∞) = (1 −c)−1/2.(55)

4.1.1. H-function moments

The moments of the Hfunctions can form an additional integral equation for

H, and also arise in later expressions for the internal and emerging distributions

of the albedo problem (and related extrapolation distances), and so we look at

these now.

If Eq.(51) is expanded around inﬁnity, one obtains

1

H(z)= 1 −c

2

∞

X

n=0

(−1)nαn

zn,|z|>1,(56)

where αnare moments of the H-function deﬁned by

αn=Z1

0

µnH(µ)G(µ)dµ . (57)

In her general study of pseudo problems, Busbridge [59, 74] found that the αn

moments of Eq.(57) satisfy the recurrence equations

α2n√1−c=g2n+c

4

2n−1

X

k=1

(−1)kα2n−kαk, n = 0,1,2, . . . (58)

13

d=2

d=3

d=4

d=5

d=6

1

2

1

H-(z)+1

H+(z)1

H(z)

c=0.63

(1-c)

d=3

d=4

d=5

d=6

d=2

-3-2-1 0 1 2 3

-1.0

-0.5

0.0

0.5

1.0

z

Figure 2: The H-functions of isotropic scattering in Rd. The continuous curves show 1/H(z)

for real zand c= 0.63. The arithmetic mean of the boundary values is plotted (dot-dashed)

for z∈(−1,0).

where moments related to the characteristic function are given by

g2n=Z1

0

µ2nG(µ)dµ. (59)

For our present study of isotropic scattering in d-space, we ﬁnd

g2n=Γd

2Γn+1

2

√πΓd

2+n,(60)

which, for n∈ {0,1,2,3}, is

1; 1/d;3

d2+ 2d;15

d3+ 6d2+ 8d.

In Flatland, g2nreduces to

g2n=(2n−1)!!

(2n)!! ,(61)

diﬀering from the familiar 3D case, g2n= (2n+ 1)−1.

For nonconservative scattering 0 < c < 1, the odd moments must be evalu-

ated numerically, and the even moments are given from the recurrence relations.

14

α1α2α3α4α5α6α7α8

d=2 1.26655 1.02971 0.889917 0.794985 0.725119 0.670931 0.627316 0.591231

d=3 1.02718 0.721955 0.557658 0.45458 0.383773 0.332099 0.292712 0.261689

d=4 0.886147 0.556488 0.392232 0.295955 0.233692 0.190647 0.159405 0.135873

d=5 0.790625 0.452909 0.295462 0.208513 0.15522 0.120126 0.0957642 0.078151

d=6 0.720483 0.381903 0.232999 0.155012 0.109429 0.0809076 0.0616274 0.0487154

d=7 0.666181 0.330172 0.189885 0.119836 0.0806849 0.057 0.041798 0.03158

Table 1: Hfunction moments of various orders αiand various dimensions d, for c= 0.99.

For example,

α0=2

c1−√1−c(62)

α2=1

√1−c1

d−c

4α2

1.(63)

For conservative (c= 1) scattering, it is the even moments which must be

numerically evaluated, and the odd moments deduced,

α0= 2 (64)

α1= 2/√d. (65)

Figure 3 and Table 1 illustrate a peculiar, but not exact, pairing of moments

for diﬀerent dimensions that we cannot explain and leave as an interesting area

for future investigation.

Equation (62) can be used to rewrite Eq.(50) as

1

H(µ)= (1 −c)1/2+c

2Z1

0

µ0H(µ0)

µ+µ0G(µ0)dµ0,(66)

which is the form that should be used if evaluating Hdirectly from an integral

equation [1].

4.1.2. H(z)calculation methods

Given their central role in the solutions that follow, we now consider im-

portant details regarding the uniqueness and evaluation strategies for the H

functions.

Equation (52) is the unique solution of Eq.(51) unless there is a ﬁnite discrete

eigenvalue ν0of the dispersion Eq.(33). In the latter case, there is one other

non-physical solution [59], which is not relevant to our problem. Equation (52)

is the unique solution of Eq.(66) in all cases [59].

Fox [75] considered the expression of Hwith general characteristic Ψ as the

solution to a Riemann-Hilbert problem involving the function

tan θ(t) = πtΨ(t)

λ(t)=c

2

πtG(t)

λ(t)(67)

15

α1(d=2)

α1(d=3)

α1(d=4)

α1(d=5)

α2(d=2)

α2(d=3)

α2(d=4)

α2(d=5)

α3(d=2)

α3(d=3)

α3(d=4)

α4(d=2)

α4(d=3)

α0

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

c

αi

(a) H-function moments αifor varying c.

α2(c,2)

α1(c,3)

α3(c,2)

α1(c,4)

α3(c,3)

α2(c,4)

0.5 0.6 0.7 0.8 0.9 1.0

1.0010

1.0015

1.0020

1.0025

1.0030

1.0035

1.0040

1.0045

c

ratio of moments

(b) Ratios of close pairs of moments αi(c, d).

Figure 3: The Hfunction moments appear to exhibit a peculiar pairing over various orders

and dimensions (a). Upon closer inspection (b), we see that the agreement is not exact.

16

where θ(0) = 0 and 0 ≤θ(t)≤π. When (d−3)/(d−2) < c < 1 and one real

eigenvalue ν0>1 exists, a useful extension of Fox’s approach is [71, 76]

H(z) = 1 + z

ν0+z

1

√1−cexp −1

πZ1

0

θ(t)

t+zdt.(68)

For c≤(d−3)/(d−2), with no eigenvalues ν0>1, the simpler result is [71]

H(z) = 1

√1−cexp −1

πZ1

0

θ(t)

t+zdt, z /∈[−1,0].(69)

We noted improved numerical stability over Eq.(52) for d > 4 when using these

last two forms of H(z). In practice, when λ(ν) has zeros, negative tan−1results

need to be detected manually and remapped to ensure 0 < θ(t)< π when

performing numerical evaluation of Eqs.(68) and (69).

Carlstedt and Mullikin [71] declare that when there are no discrete roots

ν0>1 of Λ(z) that there are then no roots of λ(ν) for ν∈[−1,1], which we

found to not hold in general (Figure 1 shows that λ(ν) will admit 0,1 or 2 roots,

depending on dand c), but we noted no issues in applying Eqs.(68) and (69),

provided θ(t) was strictly non-negative.

Additional closed-form expressions for computing H(z) are given in Ap-

pendix B.

4.2. Law of diﬀuse reﬂection

We can derive the law of diﬀuse reﬂection for the half space by solving for

the internal collision rate density C(x) due to external unidirectional illumina-

tion along cosine µ`, the solution of the W-H equation (20) with source term

(48). This collision rate can be found using the Green’s function for an internal

isotropic plane source at depth x0(with C0(x) = δ(x−x0)).

The Green’s function G(x, x0) is a source function such that G(x, x0)dx is

the rate of photons leaving collisions (or the source directly) from depths dx

about x. If we deﬁne the Laplace transform

Ls[f(x)] ≡Z∞

0

f(x)e−sxdx, (70)

then we have, from Ivanov ([66], Eqs. (19) and (21)), that the double Laplace

transform of the Green’s function is

¯

¯

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

s+s0

.(71)

Before considering the external source, we ﬁrst note a number of exact properties

that relate to the life of a photon in the half space. From G(x, x0), we can ﬁnd

the collision rate density due to isotropic emission (or leaving a collision) at

depth x0. We convert the source function G(x, x0) to a collision rate density by

17

removing the Dirac delta for direct emission (since this is not a collision), which

is

Ls[Ls0[δ(x−x0)]] = 1

s+s0

,(72)

and then apply a factor 1/c, to convert densities for leaving collisions into den-

sities for entering collisions,

¯

¯

C(s;s0) = 1

cH(1/s)H(1/s0)−1

s+s0.(73)

The double Laplace inversion of ¯

¯

C(s;s0) gives the collision rate density C(x;x0)

at any position x≥0 inside the half space, due to an isotropic plane source at

depth x0. We also easily have the mean number of collisions, by taking the

Laplace inversion of ¯

C(0; s0), which is, by Eq.(55),

¯

C(0; s0) = 1

c(1 −c)−1/2H(1/s0)−1

s0.(74)

Of those collisions, 1 −cwill absorb the photon, so the escape probability p(x0)

after isotropic emission (or leaving a collision) at x0is then

p(x0)=1−(1 −c)L−1

x0¯

C(0; s0).(75)

For unidirectional illumination along cosine µ`, the collision-rate density

inside the half space will be

C(x) = Z∞

0

G(x, x0)e−x0/µ`

µ`

dx0=1

µ`

¯

G(x, 1/µ`).(76)

After combination of Eqs.(49) and (76), we ﬁnd

I(0, µ;−µ`) = c

2

1

µµ`

¯

¯

G(1/µ, 1/µ`) = c

2

H(µ)H(µ`)

µ+µ`

,(77)

which is the generalized law of diﬀuse reﬂection for a half space of general

dimension. The probability that a photon arriving along cosine µ`escapes the

half space along a direction within dµ of µis µI(0, µ;−µ`)G(µ)dµ, and so the

total albedo of the half space is

R(µ`) = Z1

0

µI(0, µ;−µ`)G(µ)dµ = 1 −√1−cH (µ`),(78)

where we have used Eq.(66). Equations (77) and (78) are in agreement with

previous derivations for Flatland [49] and show how the familiar expressions for

3D are universal over dimension d≥1, with all variation due to the Hfunction

and the measure G(µ)dµ. The variation of the emergent distribution and albedo

with respect to dimension dis illustrated in Figures 4 and 5.

18

d=3

d=4

d=5

d=2

0.0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

1.4

μ

I(0,μ;-1/2)

Figure 4: Comparison of integrated emergent intensity I(0, µ;−µ`) for various dimensions,

c= 0.9, µ`= 1/2. A Lambertian exitance (constant) of matched total albedo R(µ`) is shown

(dashed) for reference.

4.2.1. Low-order scattering

The once- and twice-scattered portions of the reﬂection law and albedo shine

further light on the structure of the solutions and can also provide accurate

approximations for high absorption. These are found via Taylor expansions

about c= 0 of Equations (77) and (78). The once-scattered reﬂection law

reduces to

I(0, µ;−µ`|1) = c

2

1

µ+µ`

.(79)

In the Taylor expansion of the reﬂection law for the twice-scattered emergent

distribution, we encounter an integral that we observe is equal to the Laplace

transform of the kernel,

Z∞

0

µ1−Λi

t

c(π(1 + t2µ2)) dt =L1/µ [K(x)] .(80)

The twice-scattered component of the reﬂection is then,

I(0, µ;−µ`|2) = c2

2L1/µ`[K(x)] + L1/µ [K(x)]

µ`+µ.(81)

We found the general form of the Laplace transform to reduce to

L1/µ [K(x)] = 1

22F11

2,1; d

2;1

µ2−G(0)

µ(d−1) 2F11,1; 1

2+d

2;1

µ2.(82)

Special cases include the known results for Flatland [49]

L1/µ [K(x)] = µ

π

sech−1(µ)

p1−µ2,(83)

19

0.0 0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

2.5

3.0

μ

I[0,μ;-0.25]

Figure 6: Emergent distribution I(0, µ;−µ`) from a 4D half space with isotropic scattering,

c= 0.8, µ`= 0.25. Total emergent distribution (Eq.(77), thin), single-scattered component

(Eq.(79), dashed), and double-scattered (Eq.(81), thick). MC simulation shown as dots.

for 3D [77]

L1/µ [K(x)] = µcoth−1(1 + 2µ),(84)

and our new result for 4D,

L1/µ [K(x)] = µ

π2p1−µ2sech−1(µ) + πµ −2.(85)

Figure 6 shows Monte Carlo validation of the emergent distributions for a 4D

half space.

The escape probability after exactly one scattering also involves the Laplace

transform of the kernel,

R(µ`|1) = Z1

0

µI(0, µ;−µ`|1)G(µ)dµ (86)

=c1

2− L1/µ`[K(x)].(87)

An expression for R(µ`|1) in terms of the characteristic function is also known [72]

R(µ`|1) = c1

2−µ`

2Z1

0

G(µ)

µ`+µdµ.(88)

21

4.2.2. Uniform diﬀuse illumination

We now consider the case of uniform diﬀuse illumination of the half space,

with boundary condition given by Eq.(10). Integrating the diﬀuse reﬂection law

(77) against the source function

I(0, µ) = √πΓd+1

2

Γd

2Z1

0

µ`I(0, µ;−µ`)G(µ`)dµ`(89)

and applying Eq.(66), we ﬁnd the emergent distribution

I(0, µ) = √πΓd+1

2

Γd

21−√1−cH(µ)(90)

with albedo

R=Z1

0

µ I(0, µ)G(µ)dµ. (91)

The emergent distribution I(0, µ) is proportional to Eq.(78), as it must be, by

the optical reciprocity theorem. For the conservative case, c= 1, Rreduces to

Eq.(9), which is 1 by construction.

We ﬁnd a number of analytic results for the single- and double-scattered

components of the albedo for diﬀuse illumination, from a Taylor expansion about

c= 0 of Eq.(91). For the probability of escape after one collision, in Flatland

we ﬁnd

R1=c

16

G3,2

3,31−1

2,0,3

2

−1

2,0,0

√π+ 12 −3π

≈0.214601836602552c, (92)

for 3D[78]

R1=2

3c(1 −ln(2)),(93)

and for d≥4

R1=c

333F21,1,1−d

2;3

2,d

2+1

2; 1

−4(d−2)

d+ 1 3F22,2,2−d

2;5

2,d

2+3

2; 1.(94)

It appears that for even dimensionalities d≥4, the escape probabilities are

rational multiplies of c, including the remarkably simple single-scattering albedo

for 4D, R1=c/5. The double-scattering albedo for diﬀuse illumination in 4D

is also a rational factor of c2,R2= 4c2/35.

22

2 4 6 8 10 12 14 16

1.75

1.80

1.85

1.90

1.95

d

η

Maximal backscattering enhancement factor η

(c=1, μ=1)

Figure 7: Maximal coherent backscattering enhancement factor ηunder weak localization for

an isotropically-scattering halfspace in Rd(normal incidence and conservative scattering).

4.3. Coherent backscattering

In applications such as remote sensing it can be important to consider the

eﬀects of coherent backscattering. Under the conditions of weak localization,

the reﬂectance will be ampliﬁed in the backward direction by an enhancement

factor ηthat reaches a maximum for normal incidence and c= 1. The maximal

enhancement factors for 2D and 3D half spaces are [22, 24]

η= 2 −H(1)−2,(95)

which generalize to arbitrary dimension dusing Eq.(52), producing a monotoni-

cally increasing factor as dimension dincreases (Figure 7), beginning at η= 7/4

for a 1D rod and approaching η→2.

5. The albedo problem by Case’s method

5.1. Introduction

In this section we consider Case’s approach [79] to solving the dD albedo

problem to illustrate its relationship to the Wiener-Hopf and resolvent ap-

proaches. To solve Eq.(15) with the classic Case-style eigenmode method[79],

we separate variables with the ansatz

I(x, µ) = φ(ν, µ) exp(−x/ν),(96)

23

for eigenvalues in the spectrum ν∈σ, to obtain

(ν−µ)φ(ν, µ) = cν

2Z1

−1

φ(ν, µ)G(µ)dµ. (97)

If we impose the normalization condition for the eigenmodes to be

Z1

−1

φ(ν, µ)G(µ)dµ = 1 , ν ∈σ, (98)

then σ={ν∈[−1,1] ∪ ±ν0}is the eigenvalue spectrum and the eigenmodes

φ(ν, µ) satisfy the equation

(ν−µ)φ(ν, µ) = cν/2,(99)

identical to the form in a 3D domain (relative to our angular measure G(µ)).

From Eq.(99) the discrete eigenmodes, when they occur, satisfy

φ(±ν0, µ) = cν0

2(ν0∓µ)(100)

with roots ν0obtained from the dispersion function (33), which we have sum-

marized in Section 3.1. The eigenmodes for the continuum are

φ(ν, µ) = cν

2P1

ν−µ+λ(ν)

G(ν)δ(ν−µ), ν ∈[−1,1] (101)

From Eqs.(7) and (98) it follows that the continuum eigenmodes also satisfy

Z1

−1

φ(ν, µ)G(µ)dµ =c2F11

2,1; d

2;1

ν2, ν ∈σ. (102)

In this section, the collimated incident and diﬀuse illuminations for the

albedo problem are selected to be

I(0, µ;µ`) = δ(µ−µ`), µ, µ`∈[0,1] (103)

Ik(0, µ) = µk, µ ∈[0,1] . . . , k =−1,0,1 (104)

with I(x, µ;µ`) and Ik(x, µ) tending to 0 as x→ ∞. The special case k=−1

corresponds to a unit current illumination. The notation ψ(µ) will be used

henceforth to denote either surface illumination.

The constraint I(x, µ)→0 as x→ ∞ forces the eigenmode expansion for

the albedo problem to be written as

I(x, µ) = A(ν0)φ(ν0, µ) exp(−x/ν0) + Z1

0

A(ν)φ(ν, µ) exp(−x/ν)dν

≡Zσ+

A(ν)φ(ν, µ) exp(−x/ν)dν , µ ∈[−1,1],(105)

24

where σ+ = {ν∈[ 0,1] ∪ν0}deﬁnes the half-spectrum subset of the full

spectrum of eigenvalues σ. From Eqs.(103), (104), and (105) it follows that the

expansion coeﬃcients A(ν0) and A(ν), ν∈[0,1], are to be determined from

I(0, µ)≡ψ(µ) = Zσ+

A(ν)φ(ν, µ)dν, µ ∈[0,1].(106)

We follow the approach in [80] and [53]3to construct a Chandrasekhar H(µ)

function by forcing the eigenfunctions φ(ν, µ) to obey a half-range-in-µtransport

equation analogous to Eq.(97),

(ν−µ)φ(ν, µ) = cν

2Z1

0

φ(ν, µ)H(µ)G(µ)dµ , ν ∈σ+.(107)

Equation (97) then forces the constraint

Z1

0

φ(ν, µ)H(µ)G(µ)dµ = 1 , ν ∈σ+.(108)

With the substitution of φ(ν, µ) from Eqs.(100) and (101) into Eq.(108), the

following equations are valid,

cν0

2Z1

0

H(µ)G(µ)

ν0−µdµ = 1 (109)

and cν

2PZ1

0

H(µ)G(µ)

ν−µdµ +λ(ν)H(ν) = 1, ν ∈[0,1] .(110)

This suggests we construct H(µ) by considering in the complex plane the equa-

tion cz

2Z1

0

H(µ)G(µ)

z−µdµ + Λ(z)H(z) = 1 , z /∈[−1,1] (111)

and examining the analyticity properties of H(µ), 0 ≤µ≤1. The factors

Λ(z)H(z) and H(−z) are both continuous across (−1,0); similarly, Λ(z)H(−z)

and H(z) are continuous across (0,1) so Λ(z)H(z)H(−z) is analytic along

(−1,1). The points z=±1 can be included so Λ(z)H(z)H(−z) is analytic

in the entire plane and from Liouville’s theorem approaches a constant. With

H(0) = Λ(0) = 1 the Wiener-Hopf identity results,

H(z)H(−z)=1/Λ(z),(112)

3Tables 1, 3 and 4 of reference [53] have numerical errors. The values for the column

labeled jratio,1(µ0) in Table 1 are all incorrect. The correct values for Table 3 with c= 0.7

are <x2(µ0)>= 3.36647 and <x3(µ0)>= 12.45400 for µ0= 0.9 and

<x2(µ0)>= 3.83320 and <x3(µ0)>= 14.86410 for µ0= 1.0. The top right value of Table 4

should be 1.91257.

25

with H(z) satisfying Eq.(51) subject to the constraint

1/H(−ν0) = 0 imposed by Eq.(109). Thus, for H(µ), 0 ≤µ≤1, Eq.(50)

is recovered. Equation (66) is needed, for example, to show that multiplication

of Eq.(107) by G(µ)H(µ) and integration over µ∈[0,1] gives

Z1

0

φ(ν, µ)µ H(µ)G(µ)dµ =ν(1 −c)1/2, ν ∈σ+ (113)

after use of Eq.(62).

The results of Eqs.(108) and (113) can be generalized, following a partial

fraction analysis along with use of Eq.(57), to show that

Uk+1(ν) = Z1

0

µk+1φ(ν, µ)H(µ)G(µ)dµ =

νk+1(1 −c)1/2−c

2

k

X

j=1

νk+1−jαj, k ≥0, ν ∈σ+ (114)

if the convention P0

j=1 Xj≡0 is understood here and elsewhere. For k=−1,

Eq.(108) gives U0(ν) = 1, ν∈σ+.

5.2. Orthogonality relations

Multiply Eq.(107) by ν−1φ(ν0, µ) and, in a second equation for ν0, multiply

by ν0−1φ(ν, µ) and then integrate both results over µ∈[0,1] and subtract to

obtain

Z1

0

φ(ν, µ)φ(ν0, µ)µ H(µ)G(µ)dµ = 0, ν 6=ν0, ν, ν 0∈σ+.(115)

The corresponding normalization equations when (d−3)/(d−2) <c<1 are

Z1

0

φ2(ν0, µ)µ H(µ)G(µ)dµ =N(ν0)H(ν0) (116)

Z1

0

φ(ν, µ)φ(ν0, µ)µ H(µ)G(µ)dµ =N(ν)H(ν)δ(ν−ν0),(117)

for ν, ν0∈(0,1).

The discrete normalization (116) can be derived by ﬁrst diﬀerentiating Eq.(66),

written for −z, and using Eq.(112); then for (d−3)/(d−2) <c<1 and with

Eq.(100) and Λ(ν0) = 0, it follows that

N(ν0) = Z1

−1

φ2(ν0, µ)µ G(µ)dµ =cν2

0

2

dΛ(z)

dz

z=ν0

(118)

=c2

2F13

2,2; d

2+ 1; 1

ν2

0

d ν0

(119)

=cν0

ν2

0−11−(d−1) ν2

0

21−c2F1−1

2,1; d

2;1

ν2

0.(120)

26

For d∈ {1,2,3,4}we ﬁnd, respectively,

(√1−c, √1−c2

2c,cν0

2cν2

0

ν2

0−1−1,p(1 −c)c

4c−2)⊂N(ν0).

Another convenient expression for N(ν0), generalizing the derivation of Case

and Zwiefel [79] (p. 68), is found from writing the dispersion equation (33)

Λ(ν0)=1−cF (ν0) = 0,(121)

where

F(z) = 2F11

2,1; d

2;1

z2.(122)

Diﬀerentiating (121) with respect to c, we ﬁnd

c∂ν0

∂c F0(ν0) + F(ν0)=0.(123)

From Eq.(121), we also have F(ν0)=1/c. Combining with Eq.(118), we ﬁnd

1

2N(ν0)=1

ν2

0

∂ν0

∂c .(124)

The continuum normalization of

N(ν) = νΛ+(ν)Λ−(ν)/G(ν), ν ∈[−1,1] (125)

can be derived with the Poincar´e-Bertrand formula and Λ±(ν) from Eq.(45)

yielding

N(ν) = ν

G(ν)"λ(ν)2+1

2cπνG(ν)2#,(126)

where λ(ν) is given by Eq.(40). Use of that formula enables an interchange

of the order of integration from R1

0dµ Rσ+dν to Rσ+dν R1

0dµ when using the

orthogonality relations.

With the orthogonality relations we can determine the half-range expansion

coeﬃcients A(ν) in Eq.(106) from

A(ν) = 1

N(ν)H(ν)Z1

0

ψ(µ)φ(ν, µ)µ H(µ)G(µ)dµ, ν ∈σ+.(127)

Completeness of the eigenfunctions is assured by virtue of the closure relation

δ(µ−µ`) = µ`G(µ`)H(µ`)Zσ+

φ(ν, µ)φ(ν, µ`)

N(ν)H(ν)dν, (128)

for µ, µ`∈(0,1). This last equation can be derived as in [81, 82], again with the

Poincar´e-Bertrand formula, or easily conﬁrmed by solving Eq.(127) for ψ(µ) =

δ(µ−µ`) and substituting the result into Eq.(106) to verify closure.

27

Three identities directly follow from Eq.(128). For the ﬁrst, multiply by

1/µ`and integrate over µ`∈[0,1]; after interchanging orders of integration and

using Eq.(108) it follows that

Zσ+

φ(ν, µ)

N(ν)H(ν)dν =1

µ.(129a)

Similarly, a direct integration of Eq.(128) over µ`∈[0,1] gives, with the help of

Eq.(113),

Zσ+

νφ(ν, µ)

N(ν)H(ν)dν =1

(1 −c)1/2.(129b)

Multiplication of this last equation by G(µ)H(µ), integration over µ∈[0,1],

and Eq.(57) gives

Zσ+

ν

N(ν)H(ν)dν =α0

(1 −c)1/2.(129c)

Yet another identity can be derived if Eq.(128) is multiplied by µn

`dµ`and

G(µ)H(µ)dµ and the result integrated for both variables over [0,1] to obtain

Zσ+

Un+1(ν)

N(ν)H(ν)dν =αn, n ≥0.(130)

For n= 0 the last result is consistent with Eq.(113).

Other equations can be derived by rewriting Eq.(66) for −zand −z0and

subtracting the resulting equations to obtain

Z1

0

cz

2

1

z−µ0

cz0

2

1

z0−µ0µ0H(µ0)G(µ0)dµ0=

czz0

2(z−z0)1

H(−z)−1

H(−z0), z 6=z0(131)

before specializing zand z0to variables νand/or µ. After taking the appropriate

limits as zand z0approach eigenvalues, all the orthogonality results can be

condensed into the formula

Z1

0

φ(ν, µ)φ(ν0, µ)µG(µ)H(µ)dµ

= [1 −Ξ(ν)]N(ν)H(ν)δ(ν−ν0)−Ξ(ν)νφ(ν0, ν )

H(−ν)−Ξ(ν0)ν0φ(ν, ν0)

H(−ν0),(132)

where Ξ(ν) = 0 for 0 ≤ν≤1 and 1 otherwise. Thus, we ﬁnd

φ(ν, −µ) = µ−1H(µ)Z1

0

φ(ν, µ0)φ(−µ, µ0)µ0H(µ0)G(µ0)dµ0.(133)

Equation (133) leads to the reﬂection relation (or “albedo operator”) that

allows us to conveniently express the outgoing radiation from the surface in

28

terms of the ingoing radiation. To derive that equation, observe from Eq.(106)

that

I(0,−µ) = Zσ+

A(ν)φ(ν, −µ)dν, µ ∈[0,1] (134)

so insert Eq.(133) into Eq.(134), interchange the order of integrations, and use

Eq.(106) to obtain

I(0,−µ) = µ−1H(µ)Z1

0

ψ(µ0)φ(−µ, µ0)µ0H(µ0)G(µ0)dµ0.(135)

5.3. Albedo problem spatial moments

Substitution of Eqs.(103) and (104) into Eq.(127), followed by the use of

Eq.(105), yields general equations for the angular intensities for the collimated

and diﬀuse illuminations, respectively, as

I(x, µ;µ`) = µ`G(µ)H(µ`)Zσ+

φ(ν, µ)φ(ν, µ`)

N(ν)H(ν)exp(−x/ν)dν (136a)

and

Ik(x, µ) = Zσ+

exp(−x/ν)

N(ν)H(ν)dν Z1

0

µkφ(ν, µ)µH(µ)G(µ)dµ. (136b)

From Eq.(133), Eq.(136b) also can be written as

Ik(x, µ) = Zσ+

Uk+1(ν)

N(ν)H(ν)exp(−x/ν)dν. (136c)

The objective here is to compute the nth spatial moments of the ﬂux within

the half-space, as given for n= 0,1,2,etc., by

ρn(µ`) = Z∞

0

xndx Z1

−1

I(x, µ;µ`)G(µ)dµ , (137)

ρn,k =Z∞

0

xndx Z1

−1

Ik(x, µ)G(µ)dµ , k =−1,0,1. . . . (138)

Also of interest are the ratios of ﬂuxes that give the nth mean distances of travel

in dD before absorption or escape from the half space,

hxn(µ`)i=ρn(µ`)/ρ0(µ`) (139a)

hxnik=ρn,k/ρ0,k .(139b)

Substitution of Eq.(105) into either Eq.(137) or (138) gives a spatial moment

ρn=Zσ+

A(ν)dν Z1

−1

φ(ν, µ)G(µ)dµ Z∞

0

xnexp(−x/ν)dx. (140)

29

After substitution of A(ν) from Eq.(127), followed by integration over xand µ

and use of Eq.(98), the result is

ρn=n!Zσ+

νn+1

N(ν)H(ν)dν Z1

0

ψ(µ)φ(ν, µ)µH(µ)G(µ)dµ, n = 0,1,2. . . (141)

that is the dD equivalent to the 3D Eq.(I42)4that has no factor G(µ) in the

integral.

Use of the Dirac delta ψ(µ) of Eq.(103) in Eq.(141) gives

ρn(µ`) = n!µ`G(µ`)H(µ`)Mn+1(µ`) (142)

where

Mn+1(µ`) = Zσ+

νn+1φ(ν, µ`)

N(ν)H(ν)dν , µ`∈[ 0,1].(143)

Multiplication of closure relation (128) by µn+1H(µ)G(µ)dµ and integration

over µgives

µn

`=Zσ+

Un+1(ν)φ(ν, µ`)

N(ν)H(ν)dν. (144)

After substitution of Eq.(114) into (144) and a rearrangement of terms, it follows

from Eq.(142) that

ρn(µ`) = n!

(1 −c)1/2

µn+1

`G(µ`)H(µ`) + c

2

n

X

j=1

αj

(n−j)! ρn−j(µ`)

, n ≥0,

(145)

with the understanding that P0

j=1 Xj≡0.

Equations (139a) and (145) then give the recursion relation

hxn(µ`)i=n!

µn

`+c

2(1 −c)1/2

n

X

j=1

αj

(n−j)! xn−j(µ`)

, n ≥1 (146)

with x0(µ`)≡1.The ﬁrst two equations, for example, are

hx(µ`)i=µ`+cα1

2(1 −c)1/2(147)

x2(µ`)= 2µ2

`+c

(1 −c)1/2α1µ`+cα2

1

2(1 −c)1/2+α2.(148)

The results for x2(µ`)and hx(µ`)iare useful because they can be combined

to give the variance V(x(µ`)) of the spatial distribution,

V(x(µ`)) = (x(µ`)− hx(µ`)i)2=x2(µ`)− hx(µ`)i2.(149)

4Eq.(XX) from [53] will be cited as Eq.(IXX)

30

d=2

d=3

d=4

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.6

0.8

1.0

1.2

1.4

1.6

μl

〈x(μl)〉

(a) Fixed absorption c= 0.7

d=2

d=3

d=4

0.5 0.6 0.7 0.8 0.9 1.0

2

4

6

8

c

〈x(1)〉

(b) Normally-incident illumination

d=2

d=3

d=4

0.0 0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

μl

V(x(μl))

(c) Fixed absorption c= 0.7

d=2

d=3

d=4

0.5 0.6 0.7 0.8 0.9 1.0

2

5

10

20

50

c

V(x(1))

(d) Normally-incident illumination

Figure 8: Mean hx(µ`)iand variance V(x(µ`)) of the optical depth of a photon undergoing

isotropic scattering in a half space of dimension dhaving arrived along cosine µ`, (Eqs.(147)

and (149)). Monte Carlo reference solution shown as dots.

31

d=2

d=3

d=4

0.5 0.6 0.7 0.8 0.9 1.0

1

2

5

c

〈x〉k

(a) Mean optical depth

d=2

d=3

d=4

0.5 0.6 0.7 0.8 0.9 1.0

0.5

1

5

10

50

c

Vk

(b) Variance of optical depth

Figure 9: Mean hxikand variance Vkof the optical depth of a photon undergoing isotropic

scattering in a half space in dimension dfor diﬀuse illumination conditions, (Eqs.(152) and

(154)). Uniform diﬀuse illumination k= 0 is shown as continuous, with the Monte Carlo

reference solution shown as dots. Isotropic illumination k=−1 is shown as dashed.

For a diﬀuse illumination, the spatial moments of Eq.(141) for a diﬀuse

illumination can be derived for n≥1 with Eq.(145) used as a Green’s function

to obtain(I55)

ρn,k =n!

(1 −c)1/2

αn+k+1 +c

2

n

X

j=1

αj

(n−j)!ρn−j,k

, n ≥0,(150)

with the starting condition

ρ0,k =αk+1

(1 −c)1/2, k =−1,0,1. . . . (151)

Equations (139b) and (150) then lead to

hxik=αk+2

αk+1

+cα1

2(1 −c)1/2(152)

x2k=2αk+3

αk+1

+cα1αk+2

(1 −c)1/2αk+1

+c2α2

1

2(1 −c)+cα2

(1 −c)1/2(153)

for k=−1,0,1, etc. We denote the variance of the optical depth of the photon

under diﬀuse illumination by

Vk=x2k− hxi2

k.(154)

The dD half-space results in Eqs.(145) to (153) are the same as for the 3D

half-space except the integrals in H(µ) and αnhave the additional factor G(µ).

The equations for the mean and variance of the optical depth of a photon under

unidirectional and diﬀuse illuminations are illustrated and compared to Monte

Carlo simulation in Figures 8 and 9. Monte Carlo sampling methods for Rdare

described in Appendix C.

32

5.4. Albedo problem directional moments

Moments of the direction of photons emerging from the surface of the half-

space also can be derived, in analogy to the emerging directional moments for

a 3D half-space[53]. Equation (135) for the collimated incident beam boundary

condition of Eq.(103) immediately gives

I(0,−µ;µ`) = cµ`G(µ`)H(µ`)H(µ)

2(µ+µ`),(155)

in agreement with Eq.(77) once the diﬀerences between Eqs.(47) and (103) are

accounted for. We choose to weight I(0,−µ;µ`) with the powers µnG(µ), n=

0,1,etc., so for the outward and inward moments of a collimated incident

illumination we have

jout,n(µ`) = Z1

0

µn+1I(0,−µ;µ`)G(µ)dµ , n = 0,1,2. . .

=c µ`G(µ`)H(µ`)

2Z1

0

µn+1H(µ)G(µ)

µ+µ`

dµ (156)

jin,n(µ`) = Z1

0

µn+1I(0, µ;µ`)G(µ)dµ , n = 0,1,2. . .

=µn+1

`G(µ`).(157)

The ratios Rnfor n= 0,1,etc.,

Rn(µ`) = jout,n(µ`)/j in,0(µ`) (158a)

Rn,k =jout,n,k/j in,n,k ,(158b)

are the fractions of the incident current propagated in the nth outward direc-

tional moment, with R0the probability that an entering photon will escape the

half space.

With the partial fraction analysis of

µn+1

µ+µ`

= (−1)nµ µn

`

µ+µ`

+

n

X

j=1

(−1)n+jµjµn−j

`, n ≥0,(159)

and Eqs.(66) and (156), the directional moments for a collimated surface illu-

mination equal the 3D moments of Eq.(I37a) except for the extra factor G(µ`),

jout,n(µ`)=(−1)nG(µ`)µn+1

`

1−H(µ`)

(1 −c)1/2−c

2

n

X

j=1

(−1)jαjµ−j

`

,

(160)

33