Comparative analysis of radiative transfer approaches for calculation of plane transmittance and diffuse attenuation coefficient of plane-parallel light scattering layers

Abstract

We present an analysis of a number of different approximations for the plane transmittance T_p and diffuse attenuation coefficient K_d of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from the full absorption to the full scattering and from the fully forward to the fully backward scattering) were considered. The approximations were analyzed from the point of view of their physical limitations and closeness to the numerical solution of the radiative transfer equation for the plane transmittance. The main criterion for inclusion of the models for analysis was the possibility of practical use, i.e., approximations were well parameterized and included only easily measured or estimated parameters. A detailed analysis of errors for different T_p and K_d models showed that the two-stream radiative transfer Ben-David model yields the best results over all optical conditions and depths. However, the quasi-single-scattering and polynomial Gordon's approximations proved to be the best for the depths close to zero.

Full text

Comparative analysis of radiative transfer approaches
for calculation of plane transmittance and diffuse
attenuation coefficient of plane-parallel
light scattering layers
Leonid G. Sokoletsky,1,2,* Vladimir P. Budak,3Fang Shen,4and
Alexander A. Kokhanovsky5,6
1SPE “LAZMA”Ltd., Tvardovsky str. 8, Technopark “Strogino”, Moscow 125252, Russia
2Currently at State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200062, China
3National Research University “Moscow Power Engineering Institute”, Krasnokazarmennaya Str. 14,
Moscow 111250, Russia
4State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200062, China
5Institute of Remote Sensing, University of Bremen, O. Hahn Allee 1, Bremen D-28334, Germany
6Currently at EUMETSAT, Eumetsat Allee 1, Darmstadt D-64295, Germany
*Corresponding author: sokoletsky.leonid@gmail.com
Received 3 June 2013; revised 28 October 2013; accepted 29 November 2013;
posted 5 December 2013 (Doc. ID 191116); published 17 January 2014
We present an analysis of a number of different approximations for the plane transmittance Tpand dif-
fuse attenuation coefficient Kdof a semi-infinite, unbounded, plane-parallel, and optically homogeneous
layer. The maximally wide optical conditions (from the full absorption to the full scattering and from the
fully forward to the fully backward scattering) were considered. The approximations were analyzed from
the point of view of their physical limitations and closeness to the numerical solution of the radiative
transfer equation for the plane transmittance. The main criterion for inclusion of the models for analysis
was the possibility of practical use, i.e., approximations were well parameterized and included only easily
measured or estimated parameters. A detailed analysis of errors for different Tpand Kdmodels showed
that the two-stream radiative transfer Ben-David model yields the best results over all optical conditions
and depths. However, the quasi-single-scattering and polynomial Gordon’s approximations proved to be
the best for the depths close to zero. © 2014 Optical Society of America
OCIS codes: (030.5620) Radiative transfer; (120.7000) Transmission; (290.7050) Turbid media.
http://dx.doi.org/10.1364/AO.53.000459
1. Introduction
A numerical estimation of the amount of light passed
(transmitted) through the scattering and absorbing
media is a key for the solution of many practical
tasks applicable in various fields, from paint technol-
ogy and biomedicine to atmospheric and oceanic
optics. This task is closely related to the task of
numerical estimation of the amount of light reflected
from the layer’s surface [1–6]. A consideration and
comparison of different numerical and analytical
models for calculation of diffuse reflectance (plane
and spherical albedo) of a semi-infinite, unbounded,
plane-parallel, and optically homogeneous layer was
the main topic of our preceding publication [7].
The current study is devoted to modeling the
plane transmittance Tpin the plane-parallel layer.
1559-128X/14/030459-10$15.00/0
© 2014 Optical Society of America
20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS 459

Although the transmittance is directly related to re-
flectance, its modeling seems to be more complicated.
Moreover, an analysis of the current literature leads
to understanding serious difficulties in both the
numerical and analytical modeling of diffuse trans-
mittance. For example, using the numerical results
obtained by Dlugach and Yanovitskij [8] or analytical
expressions by King and Harshvardhan [9] based on
these results leads to the unusual pick in the depend-
ence of transmittance versus depth at some values of
parameters. The other researchers have encountered
similar problems in their numerical computations
(see, e.g., [10,11]).
This study is an attempt to fill this gap by using a
new powerful numerical method, the modified dis-
crete ordinates method [(MDOM) combining the
classical DOM with the small-angle modification of
the spherical harmonic method (MSH)], already
validated before for reflectances and transmittances
by comparing with the other numerical methods
[12–15] and empirically [15]. However, simple and
physically based approximate models remain widely
applicable for many practical tasks, including in-
verse optical problems, processing data in a real
time, or processing the multispectral and hyperspec-
tral images. Therefore, we present here a series of
different plane transmittance approximate models
and compare them with the MDOM. All selected
models used well-defined inherent optical properties
(IOPs), such as single-scattering albedo ω0and back-
scattering probability Balong with the cosine μiof
the incidence angle θiin the medium. A choice of
these model parameters was realized so as to reduce
their number to minimum.
We also investigated another optical property that
has an extremely wide application in ocean optics
and is closely related to the plane transmittance,
namely, diffuse attenuation coefficient, Kd.
2. Main Definitions
A. Plane and Spherical Transmittances
We will consider plane transmittance in plane-paral-
lel homogeneous unbounded layers illuminated by
the external light source. The plane transmittance
Tpis defined as the ratio of radiation transmitted dif-
fusively on the optical depth τcz (zis the layer
depth) to the incoming direct radiation. The Tp
may be mathematically expressed through the
geometrical consideration as [9,10,16]
Tpμi;τ1
πZ2π
0Z1
0
Tμi;μv;ϕ;τμvdμvdϕ
2Z1
0
¯
Tμi;μv;τμvdμv;¯
Tμi;μv;τ
1
2πZ2π
0
Tμi;μv;ϕ;τdϕ;(1)
where Tμi;μv;φ;τis the transmission function, de-
fined as the ratio of the direct radiation transmitted
on the layer depth zto the incoming direct radiation;
¯
Tμi;μv;τis the azimuthally averaged transmission
function; 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
other names for Tpused in the literature for the
plane transmittance are “diffuse transmittance of
the surface, illuminated by the direct rays”and “total
transmission.”
We input also another optical property related to
plane transmittance, namely, the spherical transmit-
tance (the other literature names are “diffuse trans-
mittance of the surface, illuminated by the diffuse
light”and “global transmittance”) defined as [10,16]
tτ2Z1
0
Tpμi;τμidμi
4Z1
0Z1
0
¯
Tμi;μvi;τμiμvdμidμv:(2)
It is obvious that transmittance is an optical prop-
erty more complicated for modeling than reflectance
because it contains one additional parameter, τ.An
additional difficulty is the necessity to consider
multi-flux scatterings, transmissions, and reflections
processes in the layer. Nevertheless, several physical
limitations may be established for Tpμi;τ:
(1) Tpμi;01; (2) 0<T
pμi;τ<1for 0<τ<∞;
(3) Tpμi;∞0excepting the case of ω0F1;
(4) ∂Tpμi;τ∕∂τ<0for 0<τ<∞; (5) Tpμi;τ1
for ω0F1and 0≤τ<∞; (6) Tpμi;τef 
exp−τef for ω00; (7) Tpμi;τef exp−1−ω0
τef at B0; and (8) ∂Tpμi;τ;F∕∂F≥0.
Here Fis the forward scattering probability equal
to the ratio of forward scattering coefficient bfto the
total scattering b;B1−Fis the backscattering
probability equal to the ratio of backscattering
coefficient bbto the b;ω0is the single-scattering
albedo equal to the ratio of bto the attenuation co-
efficient cab(ais the absorption coefficient);
τef in an effective optical depth, which may be pre-
sented as τef τ∕μifor collimated light with a good
accuracy [17].
These limitations undoubtedly help in transmit-
tance analytical modeling. The limitations (5)to
(8) appeared to be little less obvious than the first
four, but they have a clear meaning. Actually, they
mean that all obstacles for the successful light propa-
gation through the medium are related to two differ-
ent processes, namely, absorption in the medium and
backscattering [18,19]. Therefore, if light is not scat-
tered at all (ω00) or scattered only in the forward
direction (i.e., F1), then light’s propagation will be
weakened only due to absorption. This is in complete
correspondence with the classical Bouguer–Lam-
bert–Beer law for absorption, BLB-a (e.g., [20–22]).
On the other hand, if light is not absorbed at all
(ω01), then light’s propagation will be weakened
only due to backscattering that is in contradiction
to the Bouguer–Lambert–Beer law for attenuation,
BLB-c [23,24], in its general form.
460 APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014

To take into account backscattering, several
other approximate models were developed
[11,17,18,22,25–28]; however, one should be very
cautious about their applicability, especially at high
values of B. We consider this question below by com-
paring all considered approximations to the numeri-
cal results.
Note that the physical limitations similar to those
written for the Tpμi;τmay also be established for
tτ; however, the spherical transmittance will not
be considered in depth in this publication.
Some of the limitations noted above may be
derived from a scheme for multiple reflections and
transmissions in the slab (Fig. 1). A summation of
all reflected and transmitted irradiances in the slab
of thickness zgives the total irradiance
ItIif1−Rp1zgf1−r1zgTp1z1r2
1zt2
1z
r4
1zt4
1z…
Iif1−Rp1zgf1−r1zgTp1z
1−r2
1zt2
1z;(3)
where Iiis the irradiance incoming on the slab;
Tp1zand t1zare the plane transmittance and
spherical transmittance, respectively, after the first
light passing through the slab of thickness z; and,
similarly, Rp1zand r1zare the plane albedo and
spherical albedo, respectively, before the first light
passing through the slab of thickness z. Thus, a total
plane albedo will be
TpzItz
Ii
f1−Rp1zgf1−r1zgTp1z
1−r2
1zt2
1z:(4)
A detailed analysis of Eq. (4) leads to the abovemen-
tioned limitations. A similar consideration of multi-
ple reflections and transmissions in the slab in the
case of diffuse illumination leads to an equation
for the spherical transmittance:
tzf1−r1zg2t1z
1−r2
1zt2
1z:(5)
The same equation has been derived, for instance,
by Stokes [1], Tuckerman [2], and Bohren and
Huffman [6], their Eq. (2.76). The similar equations
were derived also by Gurevich [3] and Kubelka [5]in
the frame of their two-flux theory; however, their
equations deal with the reflectance from the infi-
nitely thick layer.
3. Calculation Methods and Numerical Results
A. Input Data for Modeling
Three different scattering phase functions pθ
(Fig. 2) have been used in our study for modeling
transmitted properties of the layer.
We used two optical properties for characterization
of pθ, namely, scattering asymmetry parameter
gand backscattering probability Bas follows:
(1) g0.0019,B0.4986 [the pθwith such param-
eters corresponds to the case of the balance between
the forward and backscattering]; (2) g0.5033,
Fig. 1. Schematic picture for the case of plane transmittance. The Tp1zand t1zare the plane transmittance and spherical transmit-
tance, respectively, after the first light passing through the slab of thickness z. Similarly, the Rp1zand r1zare the plane albedo and
spherical albedo, respectively, before the first light passing through the slab of thickness z.
Fig. 2. Scattering phasefunctions pθusedfor modeling. The main
selected pθshown by solid lines, while corresponding Henyey–
Greenstein pθ(with the same Bvalues) shown by dash lines.
20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS 461

B0.1559 (process of forward scattering is prevail-
ing); (3) g0.9583,B0.0087 (process of backscat-
tering is almost negligible compared to forward
scattering). The pθhave been calculated using exact
Mie theory for spherical particles distributed in
the medium according to the gamma particle size
distribution [29,30] for different values of the
effective radius reff and particles (relative to medium)
refractive indices mn−iχ(at the wavelength of
550 nm) as specified in Table 1. For comparison,
we plot also Henyey–Greenstein pθ[31] computed
for the same values of Bas selected pθ, but for
gvalues ensuring a maximal closeness to the selected
pθ.
The 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;(6)
NRMSE%100% 
Pn
i1
~
xi−xi2
n−1
q¯
x;(7)
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 0indicated predictions are
perfect and an NRMSE 1indicates that prediction
is no more accurate than taking the mean of numeri-
cal results for the given modeling parameters.
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;
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. Additionally, we
accepted θi30.5° to ensure compatibility with
the reflectance results [7] and 0≤τ≤10. So a wide
choice of optical parameters along with the very
different scattering phase functions allows us to
perform the best possible test of approximations
under study.
One numerical method (MDOM), two forms of the
classical BLB law (for absorption and attenuation
linear coefficients), and six simple analytical expres-
sions found in the literature were used for modeling
the plane transmittance Tpμi;τ. Below we give a
short description of used numerical method.
B. MDOM Method for the Plane Transmittance
The benchmark numerical method of radiative trans-
fer equation solution (called MDOM) used in the cur-
rent publication is based on the superposition of the
smooth regular part and the most anisotropic singu-
lar parts [12,32]. The MDOM algorithm computes
the radiation diffusively reflected from or transmit-
ted through a plane-parallel homogeneous absorbing
and scattering slab. The solution for the regular part
is found by a classical DOM (e.g., see [33]), while for
the singular part the small-angle MSH has been
developed [34] and exploited [12–14]. A solution
is written in the form of the matrix exponential func-
tions [14]. The calculation time for the one point was
0.01, 0.03, and 0.2 s for the scattering phase func-
tions with the asymmetry parameter g0.0019,
0.5033, and 0.9583, respectively, which allows for
performing a large number of calculations in reason-
able time.
C. Comparison Results
Figure 3demonstrates a transmission process
through the medium for different optical conditions
(expressed via ω0and B) in accordance with the
MDOM algorithm along with both forms of BLB
law. The following parameters were selected for plot-
ting this figure: θi30.5°; ω00.1, 0.5, 0.9999. An
important and well observed feature of the Tpτre-
lationships is the almost ideal exponential decreas-
ing for the most real conditions (excluding only
cases of ω00.9999,B0.16, and ω00.9999,
B0.50). Numerical calculations prove that the
BLB-a law does not take backscattering impact into
account and, therefore, significantly overestimates
Tpτat B≫0. Oppositely, the BLB-c law dealing
with total scattering (in addition to absorption)
and, therefore, significantly underestimates Tpτ
at ω0≫0.
The next step is a validation of different analytical
models found in the literature (they listed in Table 2)
Table 1. Parameters Used for 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. 3. Plane transmittance as a function of optical depth τ, single-
scattering albedo ω0, and backscattering ratio Bcomputed by the
MDOM method at θi30.5°. The dependencies calculated by
the BLB-a and BLB-c laws also shown for comparison.
462 APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014

by testing them from the physical point of view. We
have found that the MDOM, quasi-single-scattering
approximation (QSSA), Kirk, Cornet, and Ben-David
models satisfy all physical limitations established for
the Tpθi;τ(Subsection 2.A), while the Gordon and
Lee models show results slightly distinct from the
BLB-a results at ω00.
Figure 4shows variations of relative errors for
modeling values of Tpτwith the depths. All models
included in analysis yield the Tp01and, there-
fore, zero error at the layer’s top. However, in general
case, errors grow fast with a depth, and no model pro-
vides an acceptable accuracy at large depths and at
any values of optical parameters. This fact makes dif-
ficult an estimation these models at certain depths.
Instead of this, we compared all analytical models
with the MDOM algorithm at the same angle θi
30.5° and the first 10 optical depths (Table 2
and Fig. 5).
A comparison shows a high closeness between all
models at small ω0values and between most models
at small Bvalues (e.g., at Fand gclose to 1). For ex-
ample, at g0.96 and τ1, the BLB-a, QSSA,
Gordon, Kirk, Ben-David, and Lee models yielded
errors jδj<1.7%, 1.0%, 3.7%, 3.3%, 1.1%, and
1.7%, respectively, comparative to MDOM at any
values of ω0. This is not surprising, because most
of the models were developed just for small Bvalues.
For example, the QSSA model has been tested (by the
Matrix Operator method and the Monte Carlo simu-
lations) with the eight phase functions at values of B
from 0.0236 to 0.1462 [18]; similarly, Kirk [26] inves-
tigated different natural and artificial waters with
the gvalues varied from 0.660 to 0.947 (that corre-
sponded to a range of Bfrom 0.146 to 0.013);
the Lee [28] model has been constructed following
the numerical radiative transfer computations
(by Hydrolight simulations) for three pθwith the
values of B0.010, 0.018, and 0.040.
On the other hand, the Cornet’s model [22]was
developed by modification of the Schuster’s[
35] radi-
ative transfer model and, similar to the other histori-
cal approximations, like BLB-a and BLB-c, does not
include scattering asymmetry parameters, therefore
yielding generally the best results for the intermedi-
ate values of B. The least sensitivity of accuracy on
the scattering asymmetry revealed the Ben-David’s
model that was developed as a modification of the
two-stream van de Hulst’s radiative transfer model
[36, Chapter 14]. Results demonstrate the best ro-
bustness for the Ben-David model having errors
NRMSE <3% for any range of parameters and 0≤
τ≤10 (Table 2).
D. Diffuse Attenuation Coefficient Modeling
As was shown in our study and many others, a propa-
gation of light in a plane-parallel layer generally
does not obey the BLB law for both absorption and
attenuation. This situation was recognized as mini-
mum not later than in the 1970s in atmospheric
Table 2. Accuracy of Selected Models for the Plane Transmittance Tpθi;τat θi30.5°and 0≤τ≤10a
Model Source MAPE (%) NRMSE (%)
Tpμi;τef exp −az
μi
expf−1−ω0τef g
[20–22]; abbreviated as BLB-a -; -; - 8.4; 24; 48
67; -; 75 5.0; 17; 35
3.1; 5.4; 5.2 0.5; 2.8; 5.3
Tpμi;τef exp −cz
μiexp−τef [23,24]; abbreviated as BLB-c 14; 26; 31 4.7; 12; 20
19; 30; 34 8.1; 18; 29
24; 33; 34 12; 28; 47
Tpμi;τef Tp1μi;τef 
expf−1−ω0Fτef g
[18,25]; quasi-single-scattering
approximation (QSSA)
14; 10; 17 1.0; 1.3; 8.2
29; 41; 9.1 2.3; 5.1; 4.3
2.5; 4.0; 3.6 0.4; 1.9; 2.7
Tpμi;τef exp−τef P3
n1knτ1−ω0Fn,
k11.3197,k2−0.7559,k30.4655
[17]; abbreviated as Gordon 11; 9.0; 18 0.7; 1.6; 8.9
21; 24; 6.0 1.7; 2.8; 3.8
1.4; 2.7; 0.8 0.7; 1.7; 1.2
Tpμi;τef expf−τef 1−ω0
1g1μi−g2ω0∕1−ω0
pg,
g12.636∕g−2.447,g20.849∕g−0.739
[26]; abbreviated as Kirk 61; 60; 39 57; 59; 39
9.0; 11; 19 3.4; 4.6; 20
0.7; 2.2; 2.6 0.5; 1.9; 3.1
Tpμi;τ 4α
1α2expδ−1−α2exp−δ,
α
1−ω0
p,δατef
[22]; abbreviated as Cornet 24; 41; 11 1.5; 2.7; 2.3
1.9; 5.1; 11 2.2; 5.4; 9.6
16; 23; 24 6.3; 17; 32
Tpμi;τef  1
coshzτef x∕zsinhzτefx1−0.51gω0,
z
1−ω01−gω0
p
[11,27]; abbreviated as Ben-David 24; 42; 11 1.5; 2.7; 2.3
18; 26; 6.9 1.2; 2.9; 2.6
1.6; 2.3; 1.8 0.2; 0.9; 1.2
Tpμi;τexpf−10.005θi1−ω03.47Bω0τg,
θiin degrees
[28]; abbreviated as Lee 22; 33; 37; 9.1; 19; 27
3.3; 14; 23; 2.1; 6.9; 17
2.1; 1.9; 1.4 0.2; 0.6; 2.6
aThe errors values (MAPE and NRMSE) derived for the Tpθi;τ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 errors values derived for Tpθi;τ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 positions, respectively. Note that τef τ∕μi. Errors more than
100% noted by “-.”
20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS 463

and ocean optics [18,37,38]. To overcome this disad-
vantage of the BLB law, instead of aor c, a special
optical quantity, called diffuse attenuation coefficient
Kd[17,38–40], was exploited as a parameter. A
strong mathematical definition of the Kdat the
depth zis
Fig. 4. Relative errors for the Tpτcomputed by the (a) BLB-a, (b) BLB-c, (c) QSSA, (d) Gordon, (e) Kirk, (f) Cornet, (g) Ben-David, and
(h) Lee models relative to Tpτcomputed by the MDOM algorithm at θi30.5° as a function of optical depth τ, single-scattering albedo ω0,
and backscattering ratio B.
464 APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014

Kdz−
d
dz ln Itz
Ii
−
1
Itz
dItz
dz (8)
from which follows for the plane transmittance
Tpμi;zItμi;z
Iiμiexp−¯
Kd0→zz
exp −
¯
Kd0→z
cτ;(9)
where ¯
Kd0→zis the average diffuse attenuation
coefficient in the layer of thickness z:
¯
Kd0→z1
zZz
0
Kdz0dz0:(10)
There follows from Eq. (9) a simple equation
expressing a ratio between ¯
Kd0→zand cin terms
of plane transmittance and optical depth:
¯
Kd0→z
c−
ln Tpμi;z
τ:(11)
A plot for ¯
Kd0→z∕cas a function of τ,ω0, and Bfor
three selected phase functions is shown in Fig. 6.All
calculations here were carried out by the MDOM
method and Eq. (11). The important feature of
¯
Kd0→z∕c[and, hence, of ¯
Kd0→zand Kd] func-
tion is relatively weak variations with the layer’s
depth, as could be predicted from Fig. 3. The plot
shows that in a general case there are three different
vertical regions for ¯
Kd0→z∕c: (1) a stable light
regime in the top of the layer; (2) transitional light
regime; and (3) asymptotic light regime. However,
curves for different optical parameters (ω0and B)
revealed very different behavior for ¯
Kd0→z∕c;
for example, some of them have very different sizes
of the each light region or do not have them at all (see
also Fig. 3in [17]).
Further, we have computed the vertical profiles of
the ¯
Kd0→z∕cby different approximations de-
scribed above and compared results with the MDOM
(Table 3, Fig. 7). Again, as in the case of Tp, the best
results overall demonstrate Ben-David’s model;
Fig. 6. Average diffuse attenuation coefficient ¯
Kd0→znormal-
ized to the linear attenuation coefficient cas a function of optical
depth τ, single-scattering albedo ω0and backscattering ratio B
computed by the MDOM method at θi30.5°.
Fig. 5. Plane transmittance Tpas a function of ω0computed by the selected analytical methods at incidence angle θi30.5°, optical
depth τ1, and three different phase functions with (a) g0.00, (b) 0.50, and (c) 0.96.
20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS 465

however, the QSSA and Gordon’s polynomial models
prove to be superior for the top of the layer, yielding
errors in the range 20% at any values of optical
parameters (Fig. 7).
To show a connection between the QSSA and
numerical computations more clearly, we plotted
(Fig. 8) a dependence of ¯
Kd0→zμi∕cversus
1−ω0Fat the stable light regime in the top of the
layer (at τ10−6) and at the asymptotic light regime
(at τ1000). At the top of the layer, where the QSSA
works well, this dependence is very close to the 1∶1
line; however, at large depths this approximation is
held only for very strong forward scattering or at
large scattering contribution. Therefore, using more
Fig. 7. Same as Fig. 4, but for the ¯
Kd0→z.
466 APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014

precise numerical or analytical models for modeling
Tpat large depths would be the better solution.
4. Conclusions
One numerical method (MDOM), two forms of the
classical BLB law, and six simple analytical expres-
sions found in the literature were used for modeling
the plane transmittance Tpand diffuse attenuation
coefficient Kdof unbounded plane-parallel turbid
layers illuminated by direct beam radiation. For this
aim, three very different phase functions and a wide
range of the single-scattering albedo ω0were used for
modeling. All models were checked for their corre-
spondence with the physical limitations and com-
pared with the accurate numerical results.
Results show that the accuracy in Tpnormally de-
teriorates with the depth while an estimation of Kdis
much less depth dependent. Different models re-
vealed their applicability under different optical con-
ditions. More specifically, both versions of the BLB
law (for absorption and attenuation) work relatively
well at small ω0, while such optical models as QSSA,
Gordon, Kirk, and Lee, which were developed
primarily for ocean applications, have shown good re-
sults at typical conditions characterizing natural
waters, namely, small Band wide range of ω0. Among
these models, the QSSA and Gordon’s models demon-
strated the best results, however, the Gordon’s model
(and, similarly, the Lee’s model) has small deviations
from the BLB-a law at ω00. By contrast, the Cor-
net and Ben-David models were developed from the
more general radiative transfer assumptions and
have shown good results at different values of B.
Overall, for all values of 0≤τ≤10,0.01 ≤B≤0.50
and 0.1≤ω0≤0.9999, the Ben-David model has
demonstrated the best results (with the NRMSE 
2.1% and 7.4% for Tpand Kd, respectively) among
analytical models; however, the QSSA and Gordon’s
models were superior for the Kd0estimation (2.5%
and 3.8%, respectively, versus 5.3% for the Ben-
David model). However, it should be noted that we
could not find from the literature or be able to de-
velop ourselves a model that could provide an accept-
able accuracy of the plane transmittance at large
depths and at any values of optical parameters.
Thus, we see a development of such model as a future
task. Additionally, it is worth noting that the authors
considered in details only plane transmittance but
did not investigate the spherical one. Thus, this is
also a possible task for future studies.
Nevertheless, the obtained results may be useful
for solution of many problems relating to the light
propagation through turbid media—from very clear
skies and clear oceanic waters to extremely turbid
inland waters, biological tissues, and paint and
varnishes systems.
Table 3. Accuracy of Selected Models for the Diffuse
Attenuation Coefficient Kdθi;τat θi30.5°and 0≤τ≤10a
Model MAPE (%) NRMSE (%)
BLB-a 24; 62; 93 26; 61; 95
14; 44; 88 15; 43; 86
1.3; 6.2; 11 1.5; 5.8; 29
BLB-c 19; 62; - 21; 61; -
39; -; - 39; -; -
63; -; - 58; -; -
QSSA 2.5; 2.8; 21 3.9; 4.9; 24
5.8; 14; 17 7.6; 17; 19
0.8; 2.9; 4.2 1.1; 3.9; 13
Gordon 2.3; 4.7; 28 3.2; 5.8; 27
3.8; 7.1; 6.7 5.9; 11; 9.3
3.0; 13; 16 2.8; 11; 26
Kirk -; -; - -; -; -
17; 33; 50 17; 34; 53
2.3; 13; 22.4 2.2; 10; 44
Cornet 3.5; 6.3; 3.5 5.6; 10; 6.3
11; 44; - 12; 44; -
29; -; - 27; -; -
Ben-David 3.5; 6.4; 3.5 5.6; 10; 6.3
4.2; 12; 19 5.4; 13; 22
0.7; 3.7; 6.6 0.8; 3.2; 20
Lee 40; -; -; 43; -; -
10; 46; -; 2.1; 6.9; 17
0.9; 4.5; 8.9 0.2; 0.6; 2.6
aThe errors values (MAPE and NRMSE) derived for
the Kdθi;τ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
Kdθi;τ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 positions, respectively. Errors more
than 100% noted by “-.”
Fig. 8. Dependence of ¯
Kd0→zμi∕con 1−Fω0computed by the MDOM for the stable light regime in the top of the layer (a) and at the
asymptotic light regime (b). Note that a 1∶1line corresponds to the QSSA approximation.
20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS 467

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 n°251531 (MEDI-LASE project).
During a 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).
References
1. G. G. Stokes, “On the intensity of the light reflected from or
transmitted through a pile of plates,”Math. Phys. Papers 4,
145–156 (1862).
2. L. B. Tuckerman, “On the intensity of the light reflected from
or transmitted through a pile of plates,”J. Opt. Soc. Am. 37,
818–825 (1947).
3. M. Gurevich, “Über eine rationelle klassifikation der lichten-
streuenden medien,”Physikalische Zeitschrift,”31, 753–763
(1930).
4. P. Kubelka and F. Munk, “Ein beitrag zur optik der farban-
striche,”Z. Tech. Phys. 12, 593–601 (1931).
5. P. Kubelka, “New contributions to the optics of intensely light-
scattering material. Part I,”J. Opt. Soc. Am. 38, 448–457
(1948).
6. C. F. Bohren and D. R. Huffman, Absorption and Scattering of
Light by Small Particles (Wiley-VCH, 2004).
7. L. G. Sokoletsky, A. A. Kokhanovsky, and F. Shen, “Compar-
ative analysis of radiative transfer approaches for calculation
of diffuse reflectance of plane-parallel light scattering layers,”
Appl. Opt. 52, 8471–8483 (2013).
8. J. M. Dlugach and E. G. Yanovitskij, “The optical properties of
Venus and the Jovian planets. II. Methods and results of cal-
culations of the intensity of radiation diffusely reflected from
semi-infinite homogeneous atmospheres,”Icarus 22,66–81
(1974).
9. M. King and Harshvardhan, “Comparative accuracy of
selected multiple scattering approximations,”J. Atmos. Sci.
43, 784–801 (1986).
10. A. A. Kokhanovsky, “Physical interpretation and accuracy of
the Kubelka-Munk theory,”J. Phys. D 40, 2210–2216 (2007).
11. A. Ben-David, “Multiple-scattering effects on differential ab-
sorption for the transmission of a plane-parallel beam in a
homogeneous medium,”Appl. Opt. 36, 1386–1398 (1997).
12. L. G. Sokoletsky, O. V. Nikolaeva, V. P. Budak, L. P. Bass, R. S.
Lunetta, V. S. Kuznetsov, and A. A. Kokhanovsky, “A compari-
son of numerical and analytical radiative-transfer solutions
for plane albedo of natural waters,”J. Quant. Spectr. Rad.
Transfer 110, 1132–1146 (2009).
13. A. A. Kokhanovsky, V. P. Budak, C. Cornet, M.Duan, C. Emde,
I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B.
Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Ro-
zanov, T. Yokota, and E. P. Zege, “Benchmark results in vector
atmospheric radiative transfer,”J. Quant. Spectrosc. Radiat.
Transfer 111, 1931–1946 (2010).
14. V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Complete
matrix solution of radiative transfer equation for PILE of hori-
zontally homogeneous slabs,”J. Quant. Spectrosc. Radiat.
Trans. 112, 1141–1148 (2011).
15. V. P. Afanas’ev, D. S. Efremenko, and A. V. Lubenchenko, “On
the application of the invariant embedding method and the
radiative transfer equation codes for surface state analysis,”
in Light Scattering Reviews 8: Radiative Transfer and
Light Scattering, A. A. Kokhanovsky, ed. (Springer, 2013),
pp. 363–423.
16. H. C. van de Hulst, “Asymptotic fitting, a method for solving
anisotropic transfer problems in thick layers,”J. Comput.
Phys. 3, 291–306 (1968).
17. H. G. Gordon, “Can the Lambert-Beer law be applied to the
diffuse attenuation coefficient of ocean water?”Limnol.
Oceanogr. 34, 1389–1409 (1989).
18. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed re-
lationships between the inherent and apparent optical proper-
ties of a flat homogeneous ocean,”Appl. Opt. 14, 417–427
(1975).
19. S. Sathyendranath and T. Platt, “The spectral irradiance field
at the surface and in the interior of the ocean: a model for ap-
plications in oceanography and remote sensing,”J.
Geophys. Res. 93, 9270–9280 (1988).
20. R. H. Stavn, “Light attenuation in natural waters: Gershun’s
law, Lambert-Beer law, and the mean light path,”Appl. Opt.
20, 2326–2327 (1981).
21. R. H. Stavn, “Lambert-Beer law in ocean waters: optical prop-
erties of water and of dissolved/suspended material, optical
energy budgets,”Appl. Opt. 27, 222–231 (1988).
22. J. F. Cornet, C. G. Dussap, and G. Dubertret, “A structured
model for simulation of cultures of the cyanobacterium
Spirulina platensis in photobioreactors: I. Coupling between
light transfer and growth kinetics,”Biotechnol. Bioeng. 40,
817–825 (1992).
23. B. A. Bodhaine, N. B. Wood, E. G. Dutton, and R. Slusser, “On
Rayleigh optical depth calculations,”J. Atmos. Ocean. Tech-
nol. 16, 1854–1861 (1999).
24. A. B. Kostinski, “On the extinction of radiation by a homo-
geneous but spatially correlated random medium,”J. Opt.
Soc. Am. 18, 1929–1933 (2001).
25. H. R. Gordon and O. B. Brown, “Influence of bottom depth and
albedo on the diffuse reflectance of a flat homogeneous ocean,”
Appl. Opt. 13, 2153–2159 (1974).
26. J. T. O. Kirk, “Volume scattering function, average cosines,
and the underwater light field,”Limnol. Oceanogr. 36,
455–467 (1991).
27. A. Ben-David, “Multiple-scattering transmission and an effec-
tive average photon path length of a plane-parallel beam in a
homogeneous medium,”Appl. Opt. 34, 2802–2810 (1995).
28. Z. P. Lee, K. P. Du, and R. Arnone, “A model for the diffuse
attenuation coefficient of downwelling irradiance,”J.
Geophys. Res. 110, C02016 (2005).
29. A. A. Kokhanovsky, Light Scattering Media Optics (Springer-
Praxis, 2004).
30. A. A. Kokhanovsky and L. G. Sokoletsky, “Reflection of light
from semi-infinite absorbing turbid media. Part 1: spherical
albedo,”Color Res. Appl. 31, 491–497 (2006).
31. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the
galaxy,”Astrophys. J. 93,70–83 (1941).
32. V. P. Budak and S. V. Korkin, “On the solution of a vectorial
radiative transfer equation in an arbitrary three-dimensional
turbid medium with anisotropic scattering,”J. Quant. Spectr.
Rad. Transfer. 109, 220–234 (2008).
33. C. E. Siewert, “A discrete-ordinates solution for radiative-
transfer models that include polarization effects,”J. Quant.
Spectrosc. Radiat. Transfer 64, 227–254 (2000).
34. V. P. Budak and S. E. Sarmin, “Solution of the radiation
transfer equation by the method of spherical harmonics in
the small-angle modification,”Atmos. Opt. 3, 898–903
(1990).
35. A. Schuster, “Radiation through a foggy atmosphere,”Astro-
phys. J. 21,1–22 (1905).
36. H. C. van de Hulst, Multiple Light Scattering, Vol. 2
(Academic, 1980).
37. H. C. van de Hulst, “The spherical albedo of a planet covered
with a homogeneous cloud layer,”Astron. Astrophys. 35,
209–214 (1974).
38. K.-N. Liou, An Introduction to Atmospheric Radiation
(Academic, 1980).
39. D. A. Siegel and T. D. Dickey, “Observations of the vertical
structure of the diffuse attenuation coefficient spectrum,”
Deep Sea Res. 34, 547–563 (1987).
40. D. R. Mishra, S. Narumalanu, D. Rundquist, and M. Lawson,
“Characterizing the vertical diffuse attenuation coefficient for
downwelling irradiance in coastal waters: implications for
water penetration by high resolution satellite data,”ISPRS
J. Photogrammetry Rem. Sens. 60,48–64 (2005).
468 APPLIED OPTICS / Vol. 53, No. 3 / 20 January 2014