Comparative analysis of radiative transfer approaches for calculation of diffuse reflectance of plane-parallel light-scattering layers

Abstract

We present an analysis of a number of different approximations for the diffuse reflectance (spherical and plane albedo) of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from full absorption to full scattering and from fully forward to fully backward scattering) at collimated, diffuse, and combined illumination conditions were considered. The approximations were analyzed from the point of view of their physical limitations and compared to the numerical radiative transfer solutions, whenever it was possible. The main factors impacting the spherical and plane albedo were revealed for the known and unknown scattering phase functions. The main criterion for inclusion of the models in analysis was the possibility of practical use, i.e., approximations were well parameterized and only included easily measured or estimated parameters. We give a detailed analysis of errors for different models. An algorithm for recalculation of results under combined (direct and diffuse) illumination also has been developed.

Full text

Comparative analysis of radiative transfer approaches
for calculation of diffuse reflectance of
plane-parallel light-scattering layers
Leonid G. Sokoletsky,1,2,* Alexander A. Kokhanovsky,3,4 and Fang Shen5
1SPE “LAZMA”Ltd., Tvardovsky Street 8, Technopark “Strogino,”Moscow 125252, Russian Federation; currently at
2State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200062, China
3Institute of Remote Sensing, University of Bremen, O. Hahn Allee 1, D-28334 Bremen, Germany; currently at
4EUMETSAT, Eumetsat Allee 1, D-64295 Darmstadt, Germany
5State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200062, China
*Corresponding author: sokoletsky.leonid@gmail.com
Received 3 June 2013; revised 27 September 2013; accepted 8 October 2013;
posted 28 October 2013 (Doc. ID 191006); published 4 December 2013
We present an analysis of a number of different approximations for the diffuse reflectance (spherical and
plane albedo) of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The
maximally wide optical conditions (from full absorption to full scattering and from fully forward to fully
backward scattering) at collimated, diffuse, and combined illumination conditions were considered. The
approximations were analyzed from the point of view of their physical limitations and compared to the
numerical radiative transfer solutions, whenever it was possible. The main factors impacting the spheri-
cal and plane albedo were revealed for the known and unknown scattering phase functions. The main
criterion for inclusion of the models in analysis was the possibility of practical use, i.e., approximations
were well parameterized and only included easily measured or estimated parameters. We give a detailed
analysis of errors for different models. An algorithm for recalculation of results under combined (direct
and diffuse) illumination also has been developed. © 2013 Optical Society of America
OCIS codes: (030.5620) Radiative transfer; (120.5700) Reflection; (290.7050) Turbid media.
http://dx.doi.org/10.1364/AO.52.008471
1. Introduction
An estimation of the reflected and transmitted light
in turbid medium as a function of layer thickness,
illumination, and observation conditions, along with
the inherent optical properties (IOPs) of the medium,
is a main step to a solution of many different prob-
lems. Examples of such problems are an estimation
of the amounts of various substances dissolved and
suspended in natural waters, noninvasive determi-
nation of the optical absorption and scattering
properties of human tissue, evaluation of particle
size distribution, and refractive indices of pigments
and other particles in paint and varnish, calibration
of optical instruments, etc. However, in many cases
direct measurements of IOPs are difficult; for exam-
ple, in cases of medical diagnostics or ocean color
remote sensing. In such cases, measurements of ap-
parent optical properties such as reflectance and
transmittance remain the main sources of informa-
tion about the target under investigation. Therefore
the use of indirect methods and algorithms for the
conversion of measured reflectance and transmit-
tance into IOPs is still an utmost issue.
1559-128X/13/358471-13$15.00/0
© 2013 Optical Society of America
10 December 2013 / Vol. 52, No. 35 / APPLIED OPTICS 8471

Though a large number of such methods have been
developed over the last century in the radiative
transfer theory and such applicative fields as
astrophysics, ocean and atmospheric optics, biomedi-
cal optics, chemistry-technological optics etc., the se-
lection of simple and accurate methods remains a
nontrivial task for many investigators. One of the
reasons for this consists in using very different sys-
tems of approaches and nomenclatures in different
scientific fields. Our purpose was to express these dif-
ferent approaches and nomenclatures to the common
language. In the current work, we focus on an analy-
sis of the accuracy for different optical models and
their combinations, limiting ourselves to considera-
tion of reflected light in plane-parallel homogeneous
unbounded layers. Additionally, we considered only
models for collimate (at incidence angles smaller
than 45°) or diffuse incoming light and diffuse col-
lected light. Such conditions are fulfilled in the most
real situations, especially if we consider propagation
and reflection of light in the media with the refrac-
tive indices different from this of the air. The major-
ity of the models have been selected from numerous
literature sources, while several other models have
been developed for the first time.
We consider this study as a continuation of the
previous studies attempting to compare different
numerical and analytical radiative transfer reflec-
tance approximations [1–13] with the aim of seeking
simple but reliable solutions. We put forward a task
to yield a comprehensive review of the radiative
transfer approximations for diffuse reflectance, espe-
cially in their analytical form, which is convenient for
fast routine calculations. This purpose responds to
the utmost requirements of many various fields of
knowledge.
2. Main Definitions
We will consider diffuse reflectance (plane and
spherical albedo) in plane-parallel homogeneous un-
bounded layers illuminated by an external light
source. Thus we neglect the refractive index mis-
match between the plane-parallel layer and the sur-
rounding medium. However, readers interested in
how to account accurately for this mismatch are in-
vited to use algorithms described by numerous
authors beginning from Saunderson [14] and his fol-
lowers [15–17]. Both types of illumination (direct and
diffuse) with the diffusely collected reflected light
will be considered. Finally, we show how to deal with
the combined (direct and diffuse) illumination.
A. Plane Albedo
The plane albedo Rpis defined as the ratio of radia-
tion reflected diffusively from the layer to the incom-
ing direct radiation. This optical property is often
called by other names such as “directional-
hemispherical reflectance,”“hemispherical albedo,”
“hemispherical reflectance,”and “diffuse reflectance
of the surface, illuminated by the direct rays.”We use
the term “plane albedo,”which is more acceptable in
publications on radiative transfer. The plane albedo
for an infinite layer is defined as [9,18–20]
Rpμi1
πZ2π
0Z1
0
Rμi;μv;φμvdμvdφ;(1)
where μiis the cosine of the incidence angle θiin the
medium, μvis the cosine of the viewing angle θv
in the medium, and φis the azimuthal angle between
the incident and scattered beam directions. The re-
flectance factor Rμi;μv;φis defined [20,21] as the
ratio of the intensity of light reflected from a given
layer to the intensity of light reflected from the
Lambertian absolutely white surface. For the practi-
cal aims, it is suitable to use an azimuthally aver-
aged reflectance factor ¯
Rμi;μv[11] as follows:
Rpμi2Z1
0
¯
Rμi;μvμvdμv;
¯
Rμi;μv 1
2πZ2π
0
Rμi;μv;φdφ:(2)
Five physical limitations may be imposed on Rpμi
[22–24]: (1) Rpμi0at the single-scattering albedo
ω0b∕abb∕c0; (2) Rpμi1at ω01;
(3) 0<R
pμi<1for 0<ω0<1; (4) ∂Rpμi;ω0∕
∂ω0≥0; and (5) ∂Rpμi;B∕∂B≥0.
Here a,b, and care coefficients of absorption,
scattering, and attenuation, respectively. Table 1
contains these and several other IOPs used in this
study. Note also that although the Rpμi≤1,
Rμi;μv;φmay be >1at ω0close to 1 [23,24].
B. Spherical Albedo
The spherical albedo ris another type of reflectance
at which incoming and reflected light are diffused. In
the literature, other names of this physical quantity
like “global albedo,”“diffuse-diffuse reflectance,”and
“spherical reflectance”may be met. The spherical al-
bedo for an infinite layer is defined as [9,18,20,25]
r2Z1
0
Rpμiμidμi4Z1
0Z1
0
¯
Rμi;μvμiμvdμidμ:
(3)
From Eq. (3) it follows that rdoes not depend on the
angular conditions of illumination or observation;
thus it may be considered as an IOP of the medium.
Five physical limitations similar to this were im-
posed on the Rpμiand may also be applied for r:
(1) r0at the ω00; (2) r1at ω01;
(3) 0<r<1for 0<ω0<1; (4) ∂rω0∕∂ω0≥0;
and (5) ∂rμi;B∕∂B≥0.
3. Calculation Methods and Numerical Results
A. Scattering Phase Functions
Three different scattering phase functions pθ
(Fig. 1) have been used in our study for modeling
reflective and transmitted properties of the layer:
8472 APPLIED OPTICS / Vol. 52, No. 35 / 10 December 2013

We used two optical properties for characterization
of pθ; namely, scattering asymmetry parameter g
and backscattering probability Bas follows:
(1) g0.0019,B0.4986 (the pθwith such
parameters corresponding to the case of the balance
between the forward and back scattering);
(2) g0.5033,B0.1559 (process of forward
scattering is prevailing); (3) g0.9583,B0.0087
(process of backscattering is almost negligible com-
pared to forward scattering). The pθhave been cal-
culated using exact Mie theory for spherical particles
distributed in the medium according to the gamma
particle size distribution [9,24] for different values
of the effective radius reff and particles (relative to
medium) of refractive indices mn−iχ(at the
wavelength of 550 nm) as specified in Table 2. Only
models satisfying the above-mentioned physical
limitations were included in the final tables. For
comparison, we plot also Henyey–Greenstein pθ
[26] computed for the same values of Bas selected
pθ, but for gvalues ensuring a maximal closness
to the selected pθ.
Accuracy of the approximated models was
evaluated by computing the mean absolute percent-
age error (MAPE) and the normalized (to the stan-
dard deviation s) root-mean-square error (NRMSE):
MAPE%100% Pn
i1j~
xi−xi∕~
xij
n;(4)
Table 1. Compendium of IOPs Used in the Work and Their Mathematical Definitions
Name of Optical Property Symbol Definition
Absorption coefficient (m−1)a
Scattering coefficient (m−1)bbbfbb
Attenuation coefficient (m−1)ccab
Single-scattering albedo ω0ω0b
cb
ab
Backscattering coefficient (m−1)bbbb2πRπ
π∕2βθsin θdθ, where βθis the volume scattering
function at the light propagation angle θ
Forward scattering coefficient (m−1)bfbf2πRπ∕2
0βθsin θdθ
Backscattering probability BBbb
b1
2Rπ
π∕2pθsin θdθ, where pθis the scattering phase
function
Forward scattering probability FFbf
b1−B1
2Rπ∕2
0pθsin θdθ
Scattering asymmetry parameter gg1
2Rπ
0pθsin θcos θdθthis parameter may also be
expressed through the refractive indices of particles and the
medium, wavelength, and particle sizes (i.e., by Mie theory)
Transport (reduced) scattering
coefficient (m−1)
btr btr b1−g
Scattering phase function pθpθβθ
b
Gordon’s parameter GGbb
abbBω0
1−Fω0
Similarity (Hulst’s) parameter ss
1−ω0
1−gω0
q
1−ωtr
p
Transport (reduced) single-
scattering albedo
ωtr ωtr btr
abtr 1−gω0
1−gω0
Diffuse absorption coefficient (m−1)KDefined in the frame of the two-flux Gurevich–Kubelka–Munk
(GKM) theory
Diffuse scattering coefficient (m−1)SDefined in the frame of the GKM theory
Thennadil’s parameter C0C04.8446 0.472g−0.114g2
Hapke’s reflectance parameter r0r01−
1−ω0
p
1
1−ω0
p
Wu’s parameter WW−
ln ω0
1−g
Conversion absorption parameter ηηa∕K
Chandrasekhar–Klier parameter ξω
tr 2ξ
ln1ξ∕1−ξ
Conversion-scattering parameter χχbtr∕S
Haltrin’s parameter ΨΨ142
2
pbb∕a−1∕2
h1−G
132
2
pGi1∕2
Rozenberg’s parameter yy
1−ω0
1−g
q
10 December 2013 / Vol. 52, No. 35 / APPLIED OPTICS 8473

NRMSE%100% 
Pn
i1~
xi−xi2
n−1
q¯
x;(5)
where ~
xiand xiare the analytical (approximated) and
numerical (accepted as a reference) values of the op-
tical property under investigation, respectively; ¯
xis
the averaged value for all xivalues derived for a
given phase function pθ. The MAPE yields an aver-
aged absolute error, while the NRMSE indicates
whether the prediction is better than a simple mean
prediction. An NRMSE 0indicates predictions are
perfect, and an NRMSE 1indicates that the pre-
diction is no more accurate than taking the mean
of numerical results for given modeling parameters.
B. Plane Albedo Modeling
Seven various approximations for Rpderived by a
number of authors and three approximations derived
currently for the first time were examined for their
accuracy (Table 3, Figs. 2–5). We used for computa-
tions three different pθdescribed above. The values
of ω0for modeling were taken from the range of
0.1,0.2…0.9, 0.95, 0.99, 0.999, 0.9999, while values
of μiwere taken almost evenly (51 values) within
the range from 0.70 to 1. However, in a final analysis
we used only three different ranges for the ω0;
namely, 0.1≤ω0≤0.6,0.6≤ω0≤0.9, and 0.9≤
ω0≤1, and only one value of μicos 30.5°
0.862. A reason for the last choice is a rather flat
dependence Rpμi, excluding the range of θi>180°−
θrwhere θris the rainbow scattering angle. The rea-
sonable range for the θrvalues [27–29] is from 114°
to 124° at the selected values of refractive indices n
(Table 2). Therefore, θi30.5° is approximately the
middle of the zone free from the rainbow effect. This
θivalue corresponds to 42.8° for the angle of inci-
dence from the air to water (with refractive index
n1.34) and 49.6° for the incidence angle from the
air to glass (n1.5), which are typical values for
many optical applications. Such choice of parameters
allows obtaining some conclusions for the very differ-
ent scattering media.
All plane albedo approximation models have been
compared with the results derived by using the
invariant imbedding method (IIM) described in de-
tail by Mishchenko et al. (1999) [30] and Sokoletsky
et al. (2009) [12]. This method has been verified pre-
viously [12] for two very distinct scattering phase
functions (the Rayleigh and Fournier–Forand–
Mobley phase functions with g0.00 and g0.94,
respectively) by comparison of the plane albedo IIM
computations with two other numerical methods
(namely, different variants of the discrete ordinate
method). Divergence of the results was within
1.8% for any combination of geometrical and optical
parameters.
Below we give a derivation of new approximations
used for the plane albedo modeling.
1. Replacement Method
An idea of this method is a replacement of computa-
tion for the plane albedo Rpμiby the computation
for the spherical albedo r. Taking into account that
dependences of Rpμiare strictly monotonic, and the
“effective”angles θi;ef arccos μef at which Rpμef 
rlie generally in the range from 48° to 61° [10], a pro-
cedure of Rpμicalculation may be presented as
RpμirRpμi∕Rp1
Rpμef ∕Rp1;(6)
where an effective angle θef is a function of the
similarity parameter as follows:
θef 48 14.12s−22.77s219.24s3deg;(7)
while the Rpμi∕Rp1ratio approximated (with the
relative errors generally smaller than 10%) by the
following expression:
Rpμi
Rp1expf−3.599 ln1−s−0.550 ln21−s
0.0416 ln1−g×ln1−s1−μi2g(8)
at any values of gand μi>0.7. An expression for
calculation of rused in the form developed by Hulst
[18,31]:
r1−0.139s1−s
11.170s:(9)
The numerator of the fraction in Eq. (6) is the plane
albedo at the given illumination angle normalized by
the plane albedo at the vertical illumination, and the
Table 2. Parameters Used for the pθGeneration
gBr
eff μmnχ
0.0019 0.4986 0.006 1.2 0
0.5033 0.1559 0.116 1.25 0.001
0.9583 0.0087 5 1.2 0.01
Fig. 1. Scattering phase functions pθused for modeling. The
main selected pθis shown by solid lines, while corresponding
Henyey–Greenstein pθ(with the same Bvalues) is shown by
dash lines.
8474 APPLIED OPTICS / Vol. 52, No. 35 / 10 December 2013

denominator of the fraction is the plane albedo at
the effective angle normalized by the plane albedo
at the vertical illumination. Calculations carried
for five very different scattering phase functions
[10] show that the relative errors of Eq. (7) are
generally smaller than 2%.
Table 3. Accuracy of Selected Models for the Plane Albedo Rpθiat θi30.5°a
Model References MAPE (%) NRMSE (%)
Rpμiexp h−
412μis
3
p
7i[10,32] -; 2.7; 0.7
-; 29; 0.9
-; -; 2.4
49; 3.6; 1.0
-; 24; 0.9
-; -; 6.3
Rpμi0.5G
BPN
j0−1jxjPjμiQjμi,
pθPN
j0xjPjθ,
QjμiR1
0
Pjμvμvdμv
μiμv,
where Pjθ) are the Legendre
polynomials of order j, and xjare the expansion
coefficients for a given phase function pθ
[10,33,34]; abbreviated
as GKS
5.0; 15; -
9.6; 24; -
2.4; 11; 47
12; 21; -
20; 37; -
6.3; 30; -
RpμirRpμi∕Rp1
Rpμi;eff ∕Rp1,
rRpμi;eff 1−0.139s1−s
11.170s,
Rpμi
Rp1expf−3.599 ln1−s−0.550
×ln
21−s0.0416 ln1−g
×ln1−s1−μi2g
μi;eff cos θi;eff ;
θi;eff 48 14.12s−22.77s219.24s3deg
[18,31]; current
(replacement) method
44; 20; 4.4
36; 14; 1.5
27; 26; 4.1
43; 24; 3.3
36; 12; 1.3
38; 29; 0.8
Rpμi0.0001 0.3244G0.1425G20.1308G3∕μi[22,35]; abbreviated
as Gordon
13; 20; 26
2.7; 2.2; 15
28; 5.4; 20
22; 25; 34
3.2; 2.2; 29
7.1; 20; 33
Rpμi 1−
1−ω0
p
12μi
1−ω0
p[20]; abbreviated
as Hapke 1
3.5; 2.5; 1.0
-; -; 23
-; -; -
4.0; 3.0; 1.1
-; -; 13
-; -; -
Rpμi 1−s
12μis[10,18,20,31]; abbreviated
as HKS 1
3.3; 2.4; 0.9
46; 20; 3.6
-; -; 19
3.9; 2.9; 1.1
48; 19; 2.1
-; -; 4.2
RpμiΦζi1−s
12μis,ΦζiexpfA1ζiA2ζ2
is
A3ζiA4ζ2
is2g;
AjP3
k1αjkgk−1,ζiμi−0.5,
αjk 0
B
B
B
@
−0.991 3.139 −1.874
1.435 −4.294 2.089
0.719 −5.801 2.117
−0.509 0.418 3.360
1
C
C
C
A
[10,18,20,31]; abbreviated
as HKS 2
3.4; 0.7; 0.5
3.9; 2.0; 0.2
22; 19; 3.2
2.4; 0.9; 0.7
4.3; 1.5; 0.2
28; 19; 0.6
Rpμi0.51−s212μi
s12μis
×n11−sln12s
2s2μis−1
μis2μi−1ln μi−ln1μi
2μis−1o
Derived from [18,20,31];
abbreviated as HH
4.7; 4.2; 3.4
34; 12; 1.8
-; -; 15
4.9; 4.4; 3.8
34; 10; 2.6
-; -; 4.0
Rpμi0.5ω0npθ−11μiln μi
−ln1μi  HμiR1
0
Hμvμv
μiμvdμvo;
Hμf1−μω0r00.5−μr0ln11∕μg−1;
×ln11∕μg−1
Derived from [20,36];
abbreviated as Hapke 2
13; 9.8; 4.5
43; 33; 12
-; -; -
15; 12; 4.2
-; 28; 11
-; -; 43
Rpμi 1−¯μ2
1μi¯μ4−¯μ2,¯μ
12G−
G45G
p
1G
r[4] 14; 11; 2.7
23; 26; 16
23; 28; 21
16; 13; 2.0
34; 33; 10
33; 50; 18
aThe error values (MAPE and NRMSE) derived for the Rpθiare shown in the upper, middle, and bottom rows for the pθwith
g0.0019, 0.5033, and 0.9583, respectively, while the error values derived for Rpθicomputed for the ranges of 0.1≤ω0≤0.6,
0.6≤ω0≤0.9, and 0.9≤ω0≤1are shown in the left, middle, and right columns of the same columns. Errors more than 50% are
noted by “-”.
10 December 2013 / Vol. 52, No. 35 / APPLIED OPTICS 8475

2. Hapke and van de Hulst (HH) Method
This method was derived from the Hapke [20] ex-
pression obtained for the azimuthally averaged re-
flectance factor in the case of the isotropic scattering:
¯
Rμi;μv0.25ω0HμiHμv
μiμv
;(10)
where the Ambartsumian–Chandrasekhar function
Hμis expressed as
Hμ 12μ
12μ
1−ω0
p:(11)
To generalize Eqs. (10) and (11) for media with arbi-
trary phase functions, we apply Hulst’s[
18,31]
similarity rule using the replacement of the single-
scattering albedo ω0by the transport (reduced)
single-scattering albedo ωtr and then by the similar-
ity parameter sas follows:
ω0b
ab
→ωtr btr
abtr 1−s2:(12)
Then
¯
Rμi;μv0.251−s2HμiHμv
μiμv
(13)
and
Fig. 2. Plane albedo as a function of transport single-scattering albedo ωtr. (a) Computations performed by the numerical (IIM).
(b)–(d) Selected analytical methods at incidence angle θi30.5° shown for three different phase functions with g0.00 [(a), (b)];
0.50 [(a), (c)]; and 0.96 [(a), (d)].
Fig. 3. Errors of selected plane albedo approximations compared
to the IIM-derived values at θi30.5° for different phase func-
tions versus ωtr.
Fig. 4. Plane albedo as a function of Gordon’s parameter Gcom-
puted by numerical and selected analytical methods at θi30.5°
for different phase functions.
Fig. 5. Same as Fig. 3, but errors versus G.
8476 APPLIED OPTICS / Vol. 52, No. 35 / 10 December 2013

Hμ 12μ
12μs:(14)
Finally, an expression for the Rpμihas been derived
by substitution of Eqs. (13) and (14) into Eq. (2)and
taking the definite integral:
Rpμi0.51−s212μi
s12μis×11−sln12s
2s2μis−1
μis2μi−1ln μi−ln1μi
2μis−1:(15)
3. Hapke (Hapke 2) Method
This method uses the ¯
Rμi;μvin the form derived by
Hapke [20]:
¯
Rμi;μv0.25ω0pθHμiHμv−1
μiμv
(16)
and Hμin the form derived by Hapke [36] for aniso-
tropic scattering with any g:
Hμf1−μω0r00.5−μr0ln11∕μg−1;
r01−
1−ω0
p
1
1−ω0
p:(17)
Substitution of Eqs. (16) and (17) into Eq. (2) leads to
expression for the Rpμi:
Rpμi0.5ω0pθ−11μiln μi
−ln1μi  HμiZ1
0
Hμvμv
μiμv
dμv:(18)
An integral in Eq. (18) cannot be taken analytically;
instead, we estimated it by the standard numerical
method using a composite of Simpson’s rule with 250
subintervals.
4. Plane Albedo Modeling Results
Numerical results demonstrate a great variability
of different approximations for the plane albedo
(Table 3and Fig. 2), an accuracy of which mostly de-
pends on the selected model and values of μi,g,and
ω0. As shown for calculations carried out by different
models, the best result demonstrates the HKS 2
model at ωtr >0.1; however, at smaller values of
ωtr, the best result was obtained by the GKS model
(Fig. 3). The HKS 2 model has been first derived by
Kokhanovsky and Sokoletsky [10] as a modification
of Hapke [20] solution. This modification may be re-
duced to two subsequent steps: (1) the replacement
ω0by the ωtr according to Hulst’s[18,31] similarity
rule and (2) using the additional multiplicative
factor Φζito improve accuracy of the model. The
GKS model for the plane albedo was derived by
Kokhanovsky and Sokoletsky [10] from the Gordon
[33] and Golubitsky et al. [34] quasi-single-scattering
approximation obtained initially as a solution for the
reflectance factor.
Another finding is that the Rpμiis rather a func-
tion of the Gordon’s parameter G(Fig. 4) than of the
transport albedo ωtr [Fig. 2(a)]. This fact is confirmed
by the numerical and approximated computations.
Especially good results were obtained by applying
the Gordon, Haltrin, and GKS models. Note that
in a practice if, for example, the ωtr and ω0are
known, then it is easy to find an asymmetry param-
eter g(see Table 1) and then to estimate a backscat-
tering contribution B(for a given scattering phase
function) and G. To show better the impact of Gon
Rpμi, we used an alternative absciss axis with G
values (Figs. 4and 5) for demonstrating the absolute
values of Rpμi;Gand relative errors for several
models. Note that since the GKS and HKS 2 models
give values of Rpμialmost independent on selected
scattering phase functions, we have shown here the
values of Rpμiobtained only for one phase function;
namely, with g0.50 (Fig. 4).
Overall, among all considered approximations, the
best results show the GKS approximation at G<
0.05 and HKS 2 approximation at all other values
of G(see Fig. 5). Combined application of these
two models gives a relative error δ<16%.
5. Plane Albedo Modeling Under Lack of pθ
Information
In real practice, a scattering phase function is often
inaccessible. This is also means inaccessibility of
information about the g,ωtr,andsparameters. How-
ever, it is possible to estimate g(and, hence, ωtr and s)
from measured Bbb∕bvalues. Below we suggest
an algorithm for the pθand gestimation and test
for the impact of this inaccuracy in gon the accuracy
of estimated Rpμi.
For current modeling, we selected the widely ap-
plied Henyey–Greenstein pθ:
pθ 1−g2
1−2gcos θg21.5:(19)
Fig. 6. Errors of selected plane albedo approximations compared
to the IIM-derived values at θi30.5° for different phase func-
tions versus Gin situations when the scattering phase function
is unknown.
10 December 2013 / Vol. 52, No. 35 / APPLIED OPTICS 8477

Table 4. Accuracy of Selected Models for the Spherical Albedo ra
Model References MAPE (%) NRMSE (%)
r1−
1−G2
p
G[38–40]; abbreviated
as GKM
17; 14; 1.0
18; 14; 3.7
44; 44; 18
20; 16; 1.7
22; 15; 3.5
-; -; 11
rln1ξ−ξ
ln1−ξξ,ωtr 2ξ
ln1ξ∕1−ξ [41–43]; abbreviated
as CKS
17; 7.1; 0.5
15; 7.5; 0.9
-; 17; 3.8
16; 8.4; 1.0
18; 7.2; 0.9
-; 22; 1.3
rexp −4

3
py[32,44] -; 11; 0.5
-; 3.6; 0.4
-; -; 13
-; 13; 1.0
50; 3.5; 0.4
-; -; 3.6
r1K
S
−
K
SK
S2
r,K
S8
3
1−ωtr
ωtr GKM; [45,46]; abbreviated
as GKM MR
7.2; 4.1; 0.5
3.4; 3.0; 0.6
0.5; 1.8; 1.1
7.5; 4.9; 1.0
4.7; 3.2; 0.6
1.0; 4.1; 0.7
rG
BPN
j0R1
0−1jxjPjμiQiμiμidμi;
pθPN
j0xjPjθ;
QiμiR1
0
Piμvμvdμv
μiμv
,
where Pjθare the Legendre polynomials
of order j, and xjare the expansion
coefficients for a given phase function pθ.
[33,34,43]; abbreviated
as GSK
3.0; 12; -
1.0; 1.7; 22
0.8; 1.0; 9.1
7.9; 16; -
2.1; 1.9; 37
1.4; 1.2; 13
r1−0.139s1−s
11.170s[18,31]; abbreviated
as Hulst
2.9; 1.3; 0.5
1.5; 0.2; 0.0
5.8; 3.5; 0.4
2.8; 1.5; 1.0
1.2; 0.2; 0.0
6.8; 3.2; 0.1
r0.0003 0.3687G0.1802G20.0740G3[22] 4.3; 1.5; 35
36; 27; 27
47; -; 37
3.1; 2.2; 44
45; 29; 37
-; -; 43
r1K
S
−
K
SK
S2
r,
Ka∕η,Sbtr∕χ,
η1
224 132 −55ωtr 35
×
121−ωtr∕35 1211−ωtr 2∕49
p;
χ2ηωtr16η−3
15η2−1−ωtr16η−3
GKM; [47] with the
replacement: ω0→ωtr
10; 5.5; 0.5
27; 3.9; 0.8
-; 46; 1.4
8.7; 6.6; 1.0
13; 4.6; 0.8
-; 33; 1.0
r1−0.681s1−s
10.792s[1] 41; 24; 0.3
45; 27; 5.0
-; 48; 13
43; 29; 0.6
-; 27; 4.7
-; -; 5.7
r1−exp−
24a∕btr
p

24a∕btr
pexp−2a∕btr[48] -;-;1.1
-; 19; 5.2
-; -; -
-; -; 1.8
-; 19; 5.2
-; -; 37
r1K
S
−
K
SK
S2
q,
Ka∕η,Sbtr∕χ,η0.253−ωtr ,
χ98 −38ωtr ∕45
GKM; [49]; abbreviated
as GS
12; 6.2; 0.5
10; 5.8; 0.8
7.9; 8.7; 2.5
12; 7.4; 1.0
11; 5.8; 0.8
11; 13; 1.1
r1−ΨΨ−
1Ψ2
p2
1Ψ[50] 11; 1.1; 0.7
44; 29; 7.0
-; -; 24
7.5; 1.4; 1.3
-; 31; 6.5
-; -; 15
r1−s2
15
3
p∕3s2s2[51]; abbreviated
as Flock
17; 15; 0.2
13; 14; 5.4
9.4; 11; 7.0
21; 18; 0.3
18; 16; 5.2
14; 18; 5.7
r1−s
1sDerived from [52] with the
replacement: ω0→ωtr
17; 14; 1.0
25; 17; 4.4
33; 29; 8.4
20; 17; 1.7
29; 18; 4.1
43; 36; 4.6
r0.51−s2exp −
3
ps
×h1exp −4
3
p
3si
[53] 45; 33; 0.3
46; 34; 9.4
48; 47; 17
-; 40; 0.4
-; 38; 8.8
-; -; 9.9
(Table continued)
8478 APPLIED OPTICS / Vol. 52, No. 35 / 10 December 2013

The relationship between gand Bis easily deter-
mined by integrating within the equation for B
(see Table 1) leading to [37]
B1−g
2g1g

1g2
p
−1:(20)
Inverting this equation and fitting it by the 5° poly-
nomial leads to the following result:
g1−4.440B12.11B2−23.87B323.52B4
−9.317B5;(21)
Table 4. Continued
Model References MAPE (%) NRMSE (%)
r1−sh1
s
−
0.5ln12s
s2i[10,20]; abbreviated
as HKS 1
6.5; 4.1; 0.6
13; 6.6; 1.0
20; 16; 3.3
7.1; 5.0; 1.0
14; 6.6; 0.9
25; 19; 1.2
r
6
pf2
0.45 W
p−
W
p−
0.9W
pg[54] 23; 26; 3.6
29; 32; 10
-; -; 21
34; 32; 4.4
49; 35; 9.1
-; -; 11
r1−¯μ
1¯μ2;¯μ
12G−
G45G
p
1G
r[4] 17; 6.7; 0.5
48; 34; 9.1
-; -; -
15; 7.9; 1.0
-; 37; 8.6
-; -; 17
r13a∕btr−
9a∕btr26a∕btr 
p
1a∕btr [55] -; 39; 0.3
-; 45; 7.3
-; -; 22
-; 47; 0.6
-; 45; 7.0
-; -; 8.9
rexp h−
6.3740.3569btr∕a0.2879

31btr∕a
piDerived from [56] 45; -; 3.8
-; -; 22
-; -; 26
-; -; 5.8
-; -; 21
-; -; 23
rωtr
13−12ωtr [57], their Eq. (2) -; 42; 3.4
-; 43; 10
-; -; 21
-; 50; 5.5
-; 46; 9.7
-; -; 12
r1
1−13 ln ωtr [57], their Eq. (10) -; -; 1.7
-; -; 42
-; -; -
-; -; 2.7
-; -; 43
-; -; 48
rexp−2
2.68W
p[57], their Eq. (15) -; -; 0.8
-; -; 14
-; -; 31
-; -; 1.2
-; -; 13
-; -; 16
r1
2hexp −
3
psexp −7

3
psi [57], their Eq. (16) -; 5.2; 0.2
-; 16; 4.8
-; -; 24
-; 6.8; 0.4
-; 16; 4.8
-; -; 7.6
rexpf−
61−ωtr∕ωtr 
pgDerived from [58] with the
replacement: ω0→ωtr
-; 24; 0.3
-; 30; 3.8
-; -; 18
-; 28; 0.6
-; 29; 3.6
-; -; 5.7
r1K
S
−
K
SK
S2
r,K
SC2
0
6
1−ωtr
ωtr [59] 33; 25; 0.1
37; 30; 8.8
40; 40; 16
38; 31; 0.2
47; 34; 8.3
-; -; 10
r1K
S
−
K
SK
S2
r,Ka∕η,Sbtr∕χ,
ηϕ−11−ωtr
ϕ1ξ,ϕξln1−ξ
ξ−ln1ξ,χ−
0.5ωtrϕ−1∕ϕ
ξ,
ξ
47
52 31
49 ωtr −49
54 ω2
tr −17
27 ω3
tr
q
GKM; [60] -; 6.9; 0.7
-; 7.2; 1.5
-; -; 2.7
41; 8.2; 1.4
-; 7.0; 1.6
-; -; 1.3
r1K
S
−
K
SK
S2
r,Ka∕η,
Sbtr∕χ,
η1−0.6864ωtr −0.1727ω2
tr 0.6783ω3
tr −0.3196ω4
tr,
χ3.321 −3.495ωtr 1.777ω2
tr −0.2670ω3
tr
GKM with the new equations
for ηωtrand χωtr ;
abbreviated as GKM-new
17; 7.2; 0.5
16; 7.6; 0.8
19; 17; 3.7
16; 8.5; 0.9
18; 7.3; 0.8
25; 22; 1.3
aThe error values (in %) derived for rcomputed for pθwith g0.0019, 0.5033, and 0.9583 are shown in the upper, middle, and
bottom rows, respectively, in the columns labeled as MAPE (%) and NRMSE (%), while the error values derived for rcomputed for
the ranges of 0.1≤ω0≤0.6,0.6≤ω0≤0.9, and 0.9≤ω0≤1are shown in the left, middle, and right columns of the same columns.
Errors more than 50% are noted by “-”.
10 December 2013 / Vol. 52, No. 35 / APPLIED OPTICS 8479

with the NRMSE 1.2% and R20.99998 over the
whole possible ranges of parameters Band g:0≤B≤
1and −1≤g≤1.
The values of the expansion coefficients xjneeded
for computation Rpμiby the GKS model (see
Table 3) may be now easily obtained from the expan-
sion of pθinto the Legendre polynomial series [37]:
xj2j1gj:(22)
The selection of Henyey–Greenstein pθmay be ex-
plained by its convinience for computations along
with its closeness to the initial pθat given values
of B(see Fig. 2). The errors for the Rpμicomputed
for the same models that were used for plotting Fig. 6
at μi0.862 and three values of Btaken from the
Table 2are shown in Fig. 6as a function of parameter
G. As it follows from the computations, the inexact
values of pθand glead typically to small decreases
in the Rpμivalues (up to 12% and 10% for GKS and
HKS 2 models, respectively) comparative to the
Rpμivalues computed at exact (known) values of
pθ. However, an accuracy of the models under
consideration is still high. The HKS 2 model is
obviously better than the other models at any pθ
and G≥0.05, while at G<0.05 the best results yield
a GKS model. Combined application of these two
models (or only the HKS 2 model) gives a relative er-
ror δ<15% (Fig. 6) that is even better than an error
obtained for the case of known pθand g.
C. Spherical Albedo Modeling
Table 4and Fig. 7show various radiative transfer
approximations for the spherical albedo rderived
by a number of authors. The separate values of ω0,
their ranges, and scattering phase functions were se-
lected the same as were used for the Rpμicomputa-
tions. Again, all models were compared with the
numerical calculations (IIM). The results show that
the transport albedo ωtr (or, alternatively, Hulst’s
similarity parameter s) better related to rthan
Gordon’s parameter G. A logical explanation of this
fact may be found on p. 211 by van de Hulst [18].
Hulst also gave numerical confirmation of this fact
for the wide ranges of ω0(from 0.2 to 0.99) and g
(from 0 to 7/8). Our work confirms this fact again.
Numerical results contained in Table 4and shown
in Figs. 7and 8demonstrate an excellent accuracy
for Hulst’s and six other testing approximations. Five
of these models (Hulst, GKM MR, Flock, HKS 1,
and GSK) were considered earlier [7,9,10,43] and
also demonstrated encouraging results. Two new
models were analyzed now for a first time and also
show reasonable results; namely, the model by
Gemert and Star (“GS”) and the “GKM-new”models.
Both models are similar (as well as several other
models listed in Table 4) and present actual attempts
to express the parameters of absorption (K) and
scattering (S) of the two-flux GKM theory via
well-established IOPs: a,btr,andωtr.
The “GKM-new”model was developed by following
[7,41,42,60] findings. More specifically, for this model
Fig. 7. Spherical albedo as a function of ωtr computed by numeri-
cal and selected analytical methods for different phase functions.
Fig. 8. Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus ωtr.
8480 APPLIED OPTICS / Vol. 52, No. 35 / 10 December 2013

we have solved a system of the equations
r1K
S
−
K
SK
S2
s;rln1ξ−ξ
ln1−ξξ;
ωtr 2ξ
ln1ξ∕1−ξ;(23)
with the boundary conditions
ηa∕K1at ωtr 0;
η0.5at ωtr 1;and
χbtr∕S4∕3at ωtr 1:(24)
The first equation in Eq. (23) is a classical
GKM, while two others were derived from the
Chandrasekhar–Klier equations with Hulst’s
replacement: ω0→ωtr (CKS model) to the better ac-
counting of a scattering anisotropy. The boundary
conditions [Eq. (24)] were derived by Yudovsky and
Pilon [60], and they seem as reasonable and close
to conditions derived by other investigators (see
Table 4). The solution has been found in the form
of strictly decreasing ηωtrand χωtr polynomial
values that maximally satisfy Eqs. (23) and (24).
A full solution presented in the Table 4yields rval-
ues almost coinciding with the values following from
the CKS model and the Yudovsky and Pilon (2009)
model as well; however, their solutions failed at val-
ues ωtr <0.06 (or s>0.97). Contrarily, the GKM-new
model has a solution at any values of ωtr or s. Overall,
among all considered approximations, the GSK
approximation demonstrated the best results at
ωtr <0.36 (or s>0.80), and the Hulst approximation
is the best at the other values of ωtr or s(see Fig. 8).
Combined application of two these models yields a
relative error δ<3% at any values of parameters.
However, taking into account the relative complexity
of the GSK model, application of only Hulst’s model
(with δ<7%) also is reasonable. It is worth mention-
ing that the GSK approximation for the spherical al-
bedo was derived by the direct integration (Eq. 3)
from the GKS model for the plane albedo (Table 3)
by Sokoletsky and Kokhanovsky [43], while Hulst’s
model has been developed by van de Hulst [18].
1. Spherical Albedo Modeling Under Lack of pθ
Information
This is the case similar to that for the plane albedo
(Subsection 3.B.5). Again, we replaced the initial pθ
by the Henyey–Greenstein pθ[Eq. (19)] and the g
values by their calculated values [Eq. (21)] in the
cases when knowledge of pθand/or gwas necessary
for model computations. Results show that, similarly
to the Rpμicalculations, inaccurate gvalues lead
typically to a small decrease in the rvalues (up to
11%) comparative to the rvalues computed at known
(exact) values of g(Fig. 9). As before, Hulst’s model
demonstrates superior results in most cases with the
relative errors <7%. Thus we recommend to use this
spherical albedo model in the case, if the precise
values of pθis unknown.
D. Diffuse Reflectance Under Combined Illumination
Let us consider now a case of combined (direct and
diffuse) illumination with the diffuse irradiance
Ii;dif contribution dEinto the total (Ii) irradiance.
Then the reflectance may be easily determined as
follows:
RcμiIr
IiIi;dirμiRpμiIi;dif r
Ii
1−dERpμidEr: (25)
Fig. 9. Errors of selected spherical albedo approximations compared to the IIM-derived values for different phase functions versus Gin
situations when the scattering phase function is unknown. The values of backscattering ratio Bare (a) 0.4986, (b) 0.1559, and (c) 0.0087.
10 December 2013 / Vol. 52, No. 35 / APPLIED OPTICS 8481

4. Conclusions
A large number of different approximate analytic
models for plane and spherical albedo (specifically,
10 and 28 for Rpand r, respectively) of unbounded
plane-parallel turbid layers were considered to re-
veal the most theoretically grounded and accurate
models. For this aim, all models were checked for
their correspondence with the physical limitations
and/or compared when it was possible with the accu-
rate numerical results. As concluded elsewhere [7]
and confirmed again, among spherical albedo ap-
proximations, the Hulst and the GKM MR models
are the most accurate ones with the errors NRMSE
<6.8% and 7.5%, respectively, under any optical con-
ditions (see Table 4). The current study revealed a
number of other approximations, which may also
yield accurate solutions in definite situations. Such
an impressive result may be explained by the fact
that the ris safely governed by only one parameter,
namely, s(or ωtr ) (Fig. 7).
A much more difficult situation is with plane al-
bedo modeling. This property is governed by three
optical parameters: μi,g(or B), and ω0, though this
also may be expressed by only two parameters: μiand
ωtr (Fig. 2)orμiand G(Fig. 4). The best result here is
demonstrated in the HKS 2 and Gordon’s model with
the NRMSE <28% and 34%, respectively (Table 3).
However, using a replacement method [Eqs. (6)to(9)]
at which a calculation of the plane albedo Rpμire-
placed by the calculation of the spherical albedo r
also seems to be a perspective. The replacement
method yields good results only at strong scattering
(i.e., ω0close to 1) in its current form. Probably, the
more complicate (and accurate) approximation for
Rpμi∕Rp1and θi;eff may help in improvement of
the current solution.
The study also considers briefly the issue of the
layers illuminated by combined (collimated and
diffuse) light (Subsection 3.D).
The obtained results may be useful for the solution
to many problems relating to the light reflected from
turbid media—from very clear skies and oceanic
waters to extremely turbid inland waters, biological
tissues, and paint and varnishes. Better knowledge
of the relationships between the measured coeffi-
cients of reflectance and the modeled IOPs (beam
scattering, absorption, asymmetry parameters, etc.)
will allow a more accurate solution to such problems
as remote monitoring of water environments and de-
veloping multifunctional laser systems for noninva-
sive diagnostics.
The authors would like to acknowledge the valu-
able comments of Dr. Dmitrii A. Rogatkin (Moscow
Regional Research and Clinical Institute “MONIKI”)
on earlier versions of the manuscript. The research
leading to these results has received funding from
the European Community’s Seventh Framework
Programme (FP7-PEOPLE-2009-IAPP) under grant
agreement number no. 251531 (MEDI-LASE
project). During the final stage of the manuscript
preparation, it was also supported by the State
Key Laboratory of Estuarine and Coastal Research
(SKLEC) grant 2012KYYW02 and by the 111 project
(B08022).
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