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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 τcz (zis the layer

depth) to the incoming direct radiation. The Tp

may be mathematically expressed through the

geometrical consideration as [9,10,16]

Tpμi;τ1

πZ2π

0Z1

0

Tμi;μv;ϕ;τμvdμvdϕ

2Z1

0

¯

Tμi;μv;τμvdμv;¯

Tμi;μv;τ

1

2πZ2π

0

Tμi;μv;ϕ;τdϕ;(1)

where Tμi;μv;φ;τis the transmission function, de-

fined as the ratio of the direct radiation transmitted

on the layer depth zto the incoming direct radiation;

¯

Tμi;μv;τis 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]

tτ2Z1

0

Tpμi;τμidμi

4Z1

0Z1

0

¯

Tμi;μvi;τμiμvdμidμv:(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 Tpμi;τ:

(1) Tpμi;01; (2) 0<T

pμi;τ<1for 0<τ<∞;

(3) Tpμi;∞0excepting the case of ω0F1;

(4) ∂Tpμi;τ∕∂τ<0for 0<τ<∞; (5) Tpμi;τ1

for ω0F1and 0≤τ<∞; (6) Tpμi;τef

exp−τef for ω00; (7) Tpμi;τef exp−1−ω0

τef at B0; and (8) ∂Tpμi;τ;F∕∂F≥0.

Here Fis the forward scattering probability equal

to the ratio of forward scattering coefficient bfto the

total scattering b;B1−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 cab(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 (ω00) or scattered only in the forward

direction (i.e., F1), 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

(ω01), 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 Tpμi;τmay also be established for

tτ; 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

ItIif1−Rp1zgf1−r1zgTp1z1r2

1zt2

1z

r4

1zt4

1z…

Iif1−Rp1zgf1−r1zgTp1z

1−r2

1zt2

1z;(3)

where Iiis the irradiance incoming on the slab;

Tp1zand t1zare the plane transmittance and

spherical transmittance, respectively, after the first

light passing through the slab of thickness z; and,

similarly, Rp1zand r1zare 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

TpzItz

Ii

f1−Rp1zgf1−r1zgTp1z

1−r2

1zt2

1z:(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:

tzf1−r1zg2t1z

1−r2

1zt2

1z:(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 pθ

(Fig. 2) have been used in our study for modeling

transmitted properties of the layer.

We used two optical properties for characterization

of pθ, namely, scattering asymmetry parameter

gand backscattering probability Bas follows:

(1) g0.0019,B0.4986 [the pθwith such param-

eters corresponds to the case of the balance between

the forward and backscattering]; (2) g0.5033,

Fig. 1. Schematic picture for the case of plane transmittance. The Tp1zand t1zare the plane transmittance and spherical transmit-

tance, respectively, after the first light passing through the slab of thickness z. Similarly, the Rp1zand r1zare the plane albedo and

spherical albedo, respectively, before the first light passing through the slab of thickness z.

Fig. 2. Scattering phasefunctions pθusedfor modeling. The main

selected pθshown by solid lines, while corresponding Henyey–

Greenstein pθ(with the same Bvalues) shown by dash lines.

20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS 461

B0.1559 (process of forward scattering is prevail-

ing); (3) g0.9583,B0.0087 (process of backscat-

tering is almost negligible compared to forward

scattering). The pθhave 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 mn−iχ(at the wavelength of

550 nm) as specified in Table 1. For comparison,

we plot also Henyey–Greenstein pθ[31] computed

for the same values of Bas selected pθ, but for

gvalues ensuring a maximal closeness to the selected

pθ.

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

i1j~

xi−xi∕~

xij

n;(6)

NRMSE%100%

Pn

i1

~

xi−xi2

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 pθ. 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 θi30.5° 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 Tpμi;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 g0.0019,

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: θi30.5°; ω00.1, 0.5, 0.9999. An

important and well observed feature of the Tpτre-

lationships is the almost ideal exponential decreas-

ing for the most real conditions (excluding only

cases of ω00.9999,B0.16, and ω00.9999,

B0.50). Numerical calculations prove that the

BLB-a law does not take backscattering impact into

account and, therefore, significantly overestimates

Tpτat B≫0. Oppositely, the BLB-c law dealing

with total scattering (in addition to absorption)

and, therefore, significantly underestimates Tpτ

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 pθGeneration

gBr

eff μmnχ

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 θi30.5°. 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 Tpθi;τ(Subsection 2.A), while the Gordon and

Lee models show results slightly distinct from the

BLB-a results at ω00.

Figure 4shows variations of relative errors for

modeling values of Tpτwith the depths. All models

included in analysis yield the Tp01and, 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 g0.96 and τ1, 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 pθwith the

values of B0.010, 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 Tpθi;τat θi30.5°and 0≤τ≤10a

Model Source MAPE (%) NRMSE (%)

Tpμi;τef exp −az

μi

expf−1−ω0τef 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

Tpμi;τef exp −cz

μiexp−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

Tpμi;τef Tp1μi;τef

expf−1−ω0Fτef 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

Tpμi;τef exp−τef P3

n1knτ1−ω0Fn,

k11.3197,k2−0.7559,k30.4655

[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

Tpμi;τef expf−τef 1−ω0

1g1μi−g2ω0∕1−ω0

pg,

g12.636∕g−2.447,g20.849∕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

Tpμi;τ 4α

1α2expδ−1−α2exp−δ,

α

1−ω0

p,δατef

[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

Tpμi;τef 1

coshzτef x∕zsinhzτefx1−0.51gω0,

z

1−ω01−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

Tpμi;τexpf−10.005θi1−ω03.47Bω0τg,

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 Tpθi;τare shown in the upper, middle, and bottom rows for the pθwith

g0.0019, 0.5033, and 0.9583, respectively, while the errors values derived for Tpθi;τcomputed 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 Tpτcomputed 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 Tpτcomputed by the MDOM algorithm at θi30.5° 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

Kdz−

d

dz ln Itz

Ii

−

1

Itz

dItz

dz (8)

from which follows for the plane transmittance

Tpμi;zItμi;z

Iiμiexp−¯

Kd0→zz

exp −

¯

Kd0→z

cτ;(9)

where ¯

Kd0→zis the average diffuse attenuation

coefficient in the layer of thickness z:

¯

Kd0→z1

zZz

0

Kdz0dz0:(10)

There follows from Eq. (9) a simple equation

expressing a ratio between ¯

Kd0→zand cin terms

of plane transmittance and optical depth:

¯

Kd0→z

c−

ln Tpμi;z

τ:(11)

A plot for ¯

Kd0→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

¯

Kd0→z∕c[and, hence, of ¯

Kd0→zand 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 ¯

Kd0→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 ¯

Kd0→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 ¯

Kd0→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 ¯

Kd0→znormal-

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 θi30.5°.

Fig. 5. Plane transmittance Tpas a function of ω0computed by the selected analytical methods at incidence angle θi30.5°, optical

depth τ1, and three different phase functions with (a) g0.00, (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 ¯

Kd0→zμi∕cversus

1−ω0Fat the stable light regime in the top of the

layer (at τ10−6) and at the asymptotic light regime

(at τ1000). 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 ¯

Kd0→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 ω00. 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 Kd0estimation (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 Kdθi;τat θi30.5°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 Kdθi;τare shown in the upper, middle, and

bottom rows for the pθwith g0.0019, 0.5033, and

0.9583, respectively, while the error values derived for

Kdθi;τcomputed 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 ¯

Kd0→zμi∕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).

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