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We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain R d for 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 distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension d are highlighted and we find that a discrete diffusion mode is not universal for d > 3 in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.
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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 d1. 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 d1. 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 d1 to be a free real
parameter [39] by using the general surface area Ωdof the unit sphere Sd1
{xRd,||x|| = 1}in ddimensions,
d=d/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 xRdsuch that the rate of energy dE flowing
across a surface element of area in directions comprising a solid angle
about direction ωSd1is
dE =I(x, ω) cos θdσ(2)
where θis the angle between ωand the outward surface normal of . With
d“steradians” in the unit sphere Sd1, the integral of an angularly-uniform
(I(x, ω) = 1) field of unit radiance is
ZSd1
I(x, ω)= Ωd(3)
and the isotropic phase function is
P(ωiωo)=1/d.(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 ωSd1by introducing the appropriate
angular measure G(µ). The total rate of radiant energy flowing across at
depth xconfined to directions with cosines in about µis then
dE =I(x, µ)µ G(µ)dµ dσ. (5)
We normalize G(µ) such that
1
2Z1
1
G(µ)= 1,(6)
a uniform integral over the sphere Sd1. The function G(µ) that satisfies these
conditions is ([4], Eq.(33))
G(µ) = 2(1 µ2)d3
2Γ(d
2)
πΓd1
2, d > 1.(7)
For integer dimensionalities d= 2 to 7, G(µ) is
2
πp1µ2; 1; 4p1µ2
π;3
2(1 µ2); 16(1 µ2)3/2
3π;15
81µ22.
4
Scattering in a 1D rod is included in this definition by noting [28]
lim
d1+ G(µ) = δ(µ1) + δ(µ+ 1).(8)
The probability that a single photon leaves an isotropic collision into about
µis (c/2)G(µ), 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)= 1.(9)
The uniform diffuse boundary source condition is thus
I(0, µ) = πΓd+1
2
Γd
2,1µ0 (10)
or I(0, ω)=1dwhere
πdd
2Z1
0
µ G(µ)=πd1
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 in direction
ωodue to a unit radiance arriving at 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 < kd1, 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
µ00.(13)
The inscattered contribution to I(x, µ) is thus
c
2C(x) = c
2Z1
1
I(x, µ0)G(µ0)0(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)0,(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/(πp1µ2).
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)0.(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)=c
21
2,(18)
provided 0 c1. 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(xx0)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|1
µG(µ)=1
2πΓd
2G3,0
1,3x2
4
d1
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|) = e−|x|, 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(s1)G(1/s)/s, (27)
using the Heaviside theta function Θ(x). The kernels are singular at x= 0 for
d > 1,
lim
x0K(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 d1 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,tan1t
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)=1cZ
−∞
K(x)eitxdx (31)
or, equivalently, from the characteristic function Ψ(µ),
Λ(z)=1c z
2Z1
1
G(µ)
zµdµ. (32)
From Eq.(29) we have the generalized dispersion equation in terms of a hyper-
geometric function
Λ(z)=1c2F11
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)z2z4+ Λ0(z)zz2(d3) 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 d1. This follows immediately from the limit as z→ ∞,
Λ()=1c
2Z1
1
G(µ)
= 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 d3, 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=d2
d3, d > 3,(36)
we find that Λ(z) admits a finite zero ν0>1 if and only if
(d3)/(d2) <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=±1/1c, (d= 1)
ν0=±1/p1c2,(d= 2)
ν0=±1/2p(cc2),(d= 4, c > 1/2)
ν0=±3
2q(9 8c)cpc(4c3)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 ν0tanh1(10).
In odd dimensionalities d5, the discrete eigenvalues are (like in 3D) also
solutions of transcendental equations of increasing complexity, such as in 5D,
60ν0+ν2
0coth1(ν0)+ coth1(ν0)= 4.(38)
In Appendix A we derive a general closed-form expression for ν0for any dimen-
sion d1.
A related function that plays an important role in the solution of half space
problems is
λ(ν)=1
2PZ1
1
G(µ)
νµ(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+c2F11,1d
2;1
2;ν2,(40)
which simplifies to
1,1,11
2ln 1 + ν
1ν,122λ(ν) (41)
for d∈ {1,2,3,4}, respectively. Equation (40) can also be written
λ(ν) = 1 + cd2
d3ν212F11,2d
2;1
2;ν2+
(d6)ν2+ 12F11,2d
2;1
2;ν2 (42)
2It was incorrectly reported in [43] that the number of discrete eigenvalues increases past
d= 4.
10
Figure 1: The functions Λ(z) and λ(z) for real zand c= 0.63.
which shows how c= (d3)/(d2) is a special value. Again, using known
properties of 2F1, it is straightforward to find the differential equation satisfied
by λ(z),
z21λ00(z)(d5)zλ0(z) + (4 2d)λ(z) + 2(1 c)(d2) = 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(µµ`)/(µ`G(µ`)) (47)
11
for µand µ`[0,1]. The density of initial collisions inside the half space at
optical depth x0 is then,
C0(x) = (1`) 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 about µ
is I(0, µ;µ`)µG(µ). 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)is the integrated radiance leaving collisions at
xinto about µ, exp(x/µ) is the Beer-Lambert calculation, and (1)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)0,(50)
for 0 µ1 or, more generally
1
H(z)= 1 c z
2Z1
0
H(µ0)
z+µ0G(µ0)0, z /[1,0] (51)
12
Regardless of dimension d1 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
t2z2ln 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
1c˜
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
1c˜
K(t)dt.
Ivanov [66] noted a similar relationship for general displacement kernels K.
From his analysis we also have, for all d1,
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
α2n1c=g2n+c
4
2n1
X
k=1
(1)kα2nkαk, 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=(2n1)!!
(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
c11c(62)
α2=1
1c1
dc
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)0,(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 (d3)/(d2) < 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
1cexp 1
πZ1
0
θ(t)
t+zdt.(68)
For c(d3)/(d2), with no eigenvalues ν0>1, the simpler result is [71]
H(z) = 1
1cexp 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 tan1results
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(xx0)).
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)esxdx, (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(xx0)]] = 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 x0 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)L1
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)ex0`
µ`
dx0=1
µ`
¯
G(x, 1`).(76)
After combination of Eqs.(49) and (76), we find
I(0, µ;µ`) = c
2
1
µµ`
¯
¯
G(1/µ, 1`) = 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 of µis µI(0, µ;µ`)G(µ), and so the
total albedo of the half space is
R(µ`) = Z1
0
µI(0, µ;µ`)G(µ)= 1 1cH (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 d1, with all variation due to the Hfunction
and the measure G(µ). 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
µ1Λi
t
c(π(1 + t2µ2)) dt =L1[K(x)] .(80)
The twice-scattered component of the reflection is then,
I(0, µ;µ`|2) = c2
2L1`[K(x)] + L1[K(x)]
µ`+µ.(81)
We found the general form of the Laplace transform to reduce to
L1[K(x)] = 1
22F11
2,1; d
2;1
µ2G(0)
µ(d1) 2F11,1; 1
2+d
2;1
µ2.(82)
Special cases include the known results for Flatland [49]
L1[K(x)] = µ
π
sech1(µ)
p1µ2,(83)
19
d=2
d=3
d=4
d=7
0.0 0.2 0.4 0.6 0.8 1.0
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
μ
R(μ)
(a) Fixed single-scattering albedo (c= 0.9)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
c
R(1)
(b) Normal incidence (µ`= 1)
Figure 5: Comparison of Eq.(78) for the total diffuse albedo of the half space for 2D, 3D, 4D,
and 7D.
20
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[K(x)] = µcoth1(1 + 2µ),(84)
and our new result for 4D,
L1[K(x)] = µ
π2p1µ2sech1(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)(86)
=c1
2− L1`[K(x)].(87)
An expression for R(µ`|1) in terms of the characteristic function is also known [72]
R(µ`|1) = c1
2µ`
2Z1
0
G(µ)
µ`+µ.(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`)`(89)
and applying Eq.(66), we find the emergent distribution
I(0, µ) = πΓd+1
2
Γd
211cH(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
3,311
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 d4
R1=c
333F21,1,1d
2;3
2,d
2+1
2; 1
4(d2)
d+ 1 3F22,2,2d
2;5
2,d
2+3
2; 1.(94)
It appears that for even dimensionalities d4, 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 η2.
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 νσ, to obtain
(νµ)φ(ν, µ) =
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)= 1 , ν σ, (98)
then σ={ν[1,1] ∪ ±ν0}is the eigenvalue spectrum and the eigenmodes
φ(ν, µ) satisfy the equation
(νµ)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) = 0
2(ν0µ)(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) =
2P1
νµ+λ(ν)
G(ν)δ(νµ), ν [1,1] (101)
From Eqs.(7) and (98) it follows that the continuum eigenmodes also satisfy
Z1
1
φ(ν, µ)G(µ)=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(µµ`), 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)
Zσ+
A(ν)φ(ν, µ) exp(x/ν) , 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),
(νµ)φ(ν, µ) =
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)= 1 , ν σ+.(108)
With the substitution of φ(ν, µ) from Eqs.(100) and (101) into Eq.(108), the
following equations are valid,
0
2Z1
0
H(µ)G(µ)
ν0µ= 1 (109)
and
2PZ1
0
H(µ)G(µ)
νµ+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)
zµ+ Λ(z)H(z) = 1 , z /[1,1] (111)
and examining the analyticity properties of H(µ), 0 µ1. 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 µ1, 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)=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)=
νk+1(1 c)1/2c
2
k
X
j=1
νk+1jαj, k 0, ν σ+ (114)
if the convention P0
j=1 Xj0 is understood here and elsewhere. For k=1,
Eq.(108) gives U0(ν) = 1, νσ+.
5.2. Orthogonality relations
Multiply Eq.(107) by ν1φ(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)= 0, ν 6=ν0, ν, ν 0σ+.(115)
The corresponding normalization equations when (d3)/(d2) <c<1 are
Z1
0
φ2(ν0, µ)µ H(µ)G(µ)=N(ν0)H(ν0) (116)
Z1
0
φ(ν, µ)φ(ν0, µ)µ H(µ)G(µ)=N(ν)H(ν)δ(νν0),(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 (d3)/(d2) <c<1 and with
Eq.(100) and Λ(ν0) = 0, it follows that
N(ν0) = Z1
1
φ2(ν0, µ)µ G(µ)=2
0
2
dΛ(z)
dz
z=ν0
(118)
=c2
2F13
2,2; d
2+ 1; 1
ν2
0
d ν0
(119)
=0
ν2
011(d1) ν2
0
21c2F11
2,1; d
2;1
ν2
0.(120)
26
For d∈ {1,2,3,4}we find, respectively,
(1c, 1c2
2c,0
22
0
ν2
011,p(1 c)c
4c2)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)=1cF (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(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
0Rσ+to Rσ+R1
0when 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(µµ`) = 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(µµ`) and substituting the result into Eq.(106) to verify closure.
27
Three identities directly follow from Eq.(128). For the first, multiply by
1`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)=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)=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)=α0
(1 c)1/2.(129c)
Yet another identity can be derived if Eq.(128) is multiplied by µn
``and
G(µ)H(µ)and the result integrated for both variables over [0,1] to obtain
Zσ+
Un+1(ν)
N(ν)H(ν)=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
zµ0
cz0
2
1
z0µ0µ0H(µ0)G(µ0)0=
czz0
2(zz0)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)
= [1 Ξ(ν)]N(ν)H(ν)δ(νν0)Ξ(ν)νφ(ν0, ν )
H(ν)Ξ(ν0)ν0φ(ν, ν0)
H(ν0),(132)
where Ξ(ν) = 0 for 0 ν1 and 1 otherwise. Thus, we find
φ(ν, µ) = µ1H(µ)Z1
0
φ(ν, µ0)φ(µ, µ0)µ0H(µ0)G(µ0)0.(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) = µ1H(µ)Z1
0
ψ(µ0)φ(µ, µ0)µ0H(µ0)G(µ0)0.(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)(136a)
and
Ik(x, µ) = Zσ+
exp(x/ν)
N(ν)H(ν)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`)0(µ`) (139a)
hxnik=ρn,k0,k .(139b)
Substitution of Eq.(105) into either Eq.(137) or (138) gives a spatial moment
ρn=Zσ+
A(ν)Z1
1
φ(ν, µ)G(µ)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)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)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
(nj)! ρnj(µ`)
, n 0,
(145)
with the understanding that P0
j=1 Xj0.
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
(nj)! xnj(µ`)
, n 1 (146)
with x0(µ`)1.The first two equations, for example, are
hx(µ`)i=µ`+1
2(1 c)1/2(147)
x2(µ`)= 2µ2
`+c
(1 c)1/2α1µ`+2
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
xk
(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 n1 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
(nj)!ρnj,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
+1
2(1 c)1/2(152)
x2k=2αk+3
αk+1
+1αk+2
(1 c)1/2αk+1
+c2α2
1
2(1 c)+2
(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`) = `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`
(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+jµjµnj
`, 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
`
1H(µ`)
(1 c)1/2c
2
n
X
j=1
(1)jαjµj
`
,
(160)
33