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Comparative analysis of selected
radiative transfer approaches for aquatic environments
Leonid Sokoletsky
A comparative analysis is presented of simple approaches to radiative transfer in plane-parallel layers,
such as the self-consistent Haltrin approach, the Chandrasekhar–Klier exact solution for isotropic
scatters, an extended version of two-flux radiative Kubelka–Munk theory, the neutron-diffuse Gate–
Brinkworth theory, and different versions of the ␦-Eddington theory. It is shown that the Haltrin
approach is preferable to others and can be used for the solution of an inverse optical problem of the
estimation of absorption and backscattering coefficients of aquatic environments from measured appar-
ent optical properties. Two different methods of transformation from measured irradiance reflectance at
combined illumination to irradiance reflectance induced by diffuse illumination only are developed. An
analysis of the use of the different models for estimation of the effect of the bottom albedo is also
presented. © 2005 Optical Society of America
OCIS codes: 010.4450, 120.5700, 260.2160, 290.1350, 290.7050, 350.5610.
1. Introduction
Inherent optical properties (IOP’s), such as the aver-
age cosine of scattering ¯s, the volume coefficients of
absorption a, scattering b, and backscattering bb, and
the volume beam-attenuation coefficient c, are poten-
tially expedient variables for the characterization of
aquatic environments. The independence of IOP’s on
Sun position and weaker spatial (horizontal and ver-
tical) variability of IOP’s in comparison with appar-
ent optical properties (AOP’s), such as the average
cosine of the underwater light field ¯, the diffuse
attenuation coefficient k, and the reflectance coeffi-
cient R, suggest the use of these properties as the
output of inverse optical models, whereas measured
AOP’s serve as the input of such models.
The majority of existing practical methods for the
estimation of IOP’s in natural waters is based on
semianalytic inverse optical models.
1–11
In turn,
these models are derived from the radiative-transfer
consideration with the application of such tools as the
Snell, the Fresnel, and the Gershun laws, the Mie
and the Hulst theories, Monte Carlo simulations, al-
gebraic nonlinear optimization, principal component
analysis, and the neural network approach.
12
In ad-
dition to the above-mentioned simulations, Haltrin
and colleagues
13–18
developed in recent years a sim-
ple two-flux approach to the solution of the light
transfer problem for irradiance in waters with arbi-
trary turbidity, depth, and surface illumination. The
basis of this approach is the presentation of the un-
derwater light field with irradiance traveling in two
directions: downward and upward. The distinctive
features of such an approach are as follows: (1) the
system of equations used is equivalent to the original
radiative transfer equation, yielding the same values
of irradiances; (2) the scattering phase function is
chosen to obtain an analytical solution that relates
IOP’s to AOP’s; and (3) the diminution of the loss of
accuracy is achieved by use of empirical relations
between the average cosine for downward irradiance
¯dand the total average cosine ¯.
Verification of the self-consistent Haltrin approach
for the reflectance coefficient in a semi-infinite me-
dium illuminated by diffuse light was carried out by
comparison with some other approaches and demon-
strated excellent agreement with the experimental
results of Timofeeva
19
and the semianalytical model
of Gordon et al.
20
It is clear, however, that, in real
aquatic environments, the reflectance coefficient de-
pends on two constituents of incoming irradiance:
direct solar and diffuse sky. Thus, for the solution of
the radiative transfer problem under natural condi-
tions, at least two additional parameters should be
taken into consideration, namely, the parameter rep-
Israel Oceanographic and Limnological Research, Yigal Allon
Kinneret Limnological Laboratory, P.O. Box 447, Migdal 14950,
Israel.
Received 20 April 2004; accepted 27 July 2004.
0003-6935/05/010136-13$15.00/0
© 2005 Optical Society of America
136 APPLIED OPTICS 兾Vol. 44, No. 1 兾1 January 2005
resenting the relation between direct and diffuse in-
coming radiation and the parameter describing the
Sun’s position.
We consider two different approaches for the solu-
tion of the problem of estimation of the reflectance
coefficient of a plane-parallel semi-infinite layer illu-
minated by diffuse light R
dif. This coefficient is de-
rived from the easily measured reflectance coefficient
Rof such a layer when it is illuminated by natural
(direct and diffuse) light, the solar zenith angle 0,
and the direct-to-diffuse incoming irradiance ratio s.
Two other problems are connected with the first prob-
lem and to each other; they also are considered in this
paper: (2) an estimation of the diffuse reflectance
coefficient Rdif at any distance to the bottom Z
ZBZ(where ZBis the bottom depth and Zis the
current depth) from R
dif, the diffuse attenuation co-
efficient in the asymptotic light regime kand the
bottom albedo AB; and (3) an estimation of the ab-
sorption and the scattering properties [the diffuse
absorption coefficient Kand the diffuse scattering
coefficient Sof the Kubelka–Munk (KM) theory, and
the volume absorption coefficient aand the volume
backscattering coefficient bbof other theories and ap-
proaches] in finite and semi-infinite layers from Z,
R
dif, and k.
The main tool for the solution of the above-
mentioned problems is the self-consistent Haltrin
approach.
13–18
In addition, this study includes an
analysis of other theories and approaches, such as
Monte Carlo simulations by Kirk
21
and by Morel and
Gentili,
22
the exact solution of Chandrasekhar’s ra-
diative transfer equation
23
obtained by Klier
24
for
isotropic scattering, the extended version of the two-
flux radiative KM theory,
25–33
the neutron diffuse
Gate–Brinkworth theory,
34,35
different versions of
the ␦-Eddington theory,
36
and some other ap-
proaches.
2. Reflectance Models
A. Haltrin Model
We represent the total irradiance reflectance within
any plane-parallel layer REu兾Ed(where Euis the
total upward irradiance and Edthe total downward
irradiance) as a superposition of the diffuse reflec-
tance coefficient owing to direct illumination Rdir and
the diffuse reflectance coefficient owing to diffuse il-
lumination Rdif, as follows
17
:
REu
dir Eu
dif
Ed
dir Ed
dif Rdir Ed
dir Rdif Ed
dif
Ed
dir Ed
dif
sRdir Rdif
1s,sEd
dir
Ed
dif , (1)
where Ed
dir and Ed
dif are the direct and the diffuse
parts of the downward irradiance, respectively, and
Eu
dir and Eu
dif are parts of the diffuse upward irradi-
ance that are caused by direct and diffuse illumina-
tion, respectively.
In accord with the Haltrin model, the values of
reflectance constituents and hence the reflectance co-
efficient itself, are assumed to be the same through-
out the water column far away from the colored
bottom layer. For an optically semi-infinite plane-
parallel layer with a scattering isotropic phase func-
tion in the backward hemisphere, the reflectance
coefficient of the surface illuminated by direct sun-
light R
dir expressed as a function of the cosine of the
refracted solar zenith angle just below the surface w
and the average cosine of the asymptotic light field
(i.e., when optical properties are assumed to be in-
variant with depth) ¯is as follows:
R
dir (1 ¯)2
1
w¯(4 ¯
2). (2)
The reflectance coefficient of an optically semi-
infinite plane-parallel layer that is illuminated by
diffuse skylight R
dif with the transport approxima-
tion to the scattering phase function and far away
from the bottom was related with ¯by the following
simple formula.
13,15,17,18
R
dif
冉
1¯
1¯
冊
2
. (3)
The closeness of both reflectance constituents R
dir
and R
dif to one another at any parameter value fol-
lows from Eqs. (2) and (3) (see Fig. 1) and was man-
ifested also by Gordon and Brown
37
and by Brown
38
for oceanic waters, and by other investigators consid-
ered more generally any plane-parallel homogeneous
layers.
32,39,40
Moreover, in sufficiently thick layers,
for example, in most natural waters, the incoming
global (direct plus diffuse) radiation becomes mixed
during light propagation into deeper layers, under-
going two opposite processes: direct light becomes
partly diffuse,
26,28
whereas diffuse light becomes
Fig. 1. Direct R
dir and diffuse R
dif constituents of the total irra-
diance reflectance coefficient calculated within the framework of
the Haltrin approach plotted as functions of the cosine of the
refracted solar zenith angle just below the surface wand the
average cosine of the asymptotic light field ¯. The values of ware
represented by numbers in the inset of the legend.
1 January 2005 兾Vol. 44, No. 1 兾APPLIED OPTICS 137
partly collimated, regardless of the angle of solar in-
cidence.
27
Thus, we can assume that at sufficiently
deep layers the approximated equalities Rdir共Z兲
⬇Rdif共Z兲⬇R共Z兲hold. Note that the equality of Rdir to
Rdif for case 1 waters (see Table 1) {with
R僆关0.001, 0.1兴, see, e.g., Refs. 3 and 12} is achieved
at lower solar zenith angles than for case 2 waters
{for which the range R僆关0.01, 0.3兴was assumed}. A
more rigorous analysis of Eqs. (2) and (3) shows that
the relations 0 R
dir R
dif R1兾9 holds at
values of w, satisfying the following conditions:
2兾3
w关3R2共R1兲1兾2兴兾共9R1兲1兾共2
¯兲1. In a more general case for any
w僆关0.666, 1兴,R
dir R
dif R, and it is possible to
estimate R
dir and R
dif from measured wand Rby
approximated formulas. Below, a formula con-
structed for R
dif is presented that is more suitable
than R
dir for the purposes of the present study:
R
dif R
兵
12R关w(1 R)兴3
其
. (4)
The accuracy of Eq. (4) for the model’s complete pa-
rameter set of w僆关0.666, 1兴,¯僆关0, 1兴, and
s僆关0, 兲was estimated; further, ¯was calculated
by analytical inversion of Eq. (3) and then also esti-
mated. Calculations show that, for the total natural
range of variability of parameter s(s僆关1, 7兴, see Ref.
41) the error, expressed in this paper by means of the
normalized root-mean-square error, (NRMSE), for
R
dif and ¯did not exceed 8.6% and 2.0%, respec-
tively (see Table 1).
Equation (2) was derived under the assumption of
an isotropic phase function in the backward hemi-
sphere. However, it may be not valid in real, espe-
cially turbid, aquatic environments for either case 1
or 2 waters. Besides, experimental data from differ-
ent authors contradict this formula and demonstrate
a stronger dependence of Rdir on the solar zenith an-
gle. A more detailed consideration of this issue and an
alternative approach to the estimation of R
dif from
Rare proposed in Section 4 of this paper.
Now let us consider the reflectance Rdif from a
homogeneous shallow layer with a bottom depth ZB
and a Lambertian bottom albedo AB. In accord with
the Haltrin model, Rdif at depth Zcan be expressed
as
17,18
Rdif(Z)R
dif (Z)
1R0dif(Z), (5)
where
R0dif 2¯
2¯
R
dif, (6)
(Z)ABR
dif
1R0difAB
exp
冋
k(ZBZ)(7 2¯
2¯
4
3¯
2
册
,
(7)
where kis the diffuse attenuation coefficient in the
asymptotic light regime.
Equations (3) and (5)–(7) permit the prediction of
vertical variations in Rdif共Z兲if R
dif and k共ZBZ兲are
known or estimated. The physical sense of the last
product follows from the Lambert–Beer law for irra-
diance transmittance Tbof a layer with a perfectly
black bottom (i.e., at AB0) from a current depth Z
to the bottom depth ZBin the asymptotic regime with
a radiance distribution symmetrical about the verti-
cal axis
42
:
TbE0(ZB)
E0(Z)exp
冋
a
¯
(ZBZ)
册
exp关k(ZBZ)兴, (8)
where E0共Z兲and E0共ZB兲are scalar irradiances at
depths of Zand ZB, respectively.
It is important to note that, in real aquatic envi-
ronments, a quality kcan be replaced with high
accuracy by the attenuation coefficient for downward
irradiance in the asymptotic light regime.
38,431–47
Following the authors of Refs. 45, 48, and 49, we call
a quality kZ(Z is the thickness of any plane-
parallel layer, in particular, ZZBZ) an optical
depth for the diffuse attenuation coefficient, denoted
by k.
Numerical simulations of the vertical variability of
Rdif共Z兲with knear the bottom were carried out for
values of R
dif (2%, 10%, and 20%) typical for natural
waters
12
and AB(0%, 20%, and 60%) typical for dif-
Table 1. Estimated Accuracy (NRMSE for the Estimation of R
dif and ¯by Use of Eqs. (4) and (3), Respectively
Estimated
Parameter
Water-Type
Case s0s1s2s3s4s5s6s7
R
dif 1 5.2 3.9 5.9 7.0 7.7 8.1 8.4 8.6
R
dif 2 13.7 4.9 2.2 1.5 1.9 2.3 2.6 2.9
R
dif 1 and 2 16.7 6.0 2.7 2.0 4.0 2.9 3.2 3.6
¯1 0.4 0.5 0.7 0.8 0.8 0.9 0.9 0.9
¯2 5.4 2.0 1.1 1.1 1.4 1.5 1.7 1.8
¯1 and 2 3.0 1.2 0.8 1.0 1.1 1.2 1.3 1.3
In percent.
138 APPLIED OPTICS 兾Vol. 44, No. 1 兾1 January 2005
ferent bottom substrates
49
[Fig. 2(a)]. The results
clearly demonstrate that, at any value of R
dif and AB,
Rdif共Z兲approaches R
dif and becomes invariant to the
impact of the bottom when k2–3.
B. Kubelka–Munk Model
The basis of another popular two-flux radiative the-
ory for turbid plane-parallel layers with isotropic
scattering [the Kubelka–Munk (KM) theory] was for-
mulated in the beginning of the twentieth
century.
25–29
This theory has essentially been ex-
panded and further developed throughout the follow-
ing years.
30–33
For the reflectance coefficient of a
layer illuminated by diffuse light Rdif the extended
KM theory yields
Rdif 2ARbABRbAB
RbAB1, (9)
where the reflectance coefficient for the layer with a
perfectly black bottom Rbis determined as
32,33
Rb1K兾S关K兾S(K兾S2) Tb
2兴,Tb
exp(k). (10)
In Eq. (10) parameter Aand the ratio of the diffuse
absorption coefficient Kto the diffuse scattering co-
efficient S(the so-called remission function) are the
functions of R
dif as follows:
Fig. 2. Dependence of the reflectance coefficient for a plane-parallel layer illuminated by diffuse light Rdif on the optical depth for the
diffuse attenuation coefficient kat selected values of the diffuse reflectance coefficient for an infinite layer R
dif (solid curves, squares, and
triangles for 2%, 10%, and 20%, respectively) and the bottom albedo AB(solid, long-dashed, and short-dashed curves for 0%, 20%, and 60%,
respectively): (a) Haltrin model, (b) KM model, (c) CK model, (d) Lyzenda model, and (e) Albert—Mobley model.
1 January 2005 兾Vol. 44, No. 1 兾APPLIED OPTICS 139
A0.5(R
dif 1兾R
dif) (11)
K兾SA10.5(R
dif 1)2兾R
dif. (12)
Note that Eq. (9) can also be expressed in the equiv-
alent form
Rdif 1AB关ABcoth(BSZ)兴
ABcoth(BSZ)AB
, (13)
where
BSZarcoth
冉
1ARb
BRb
冊
, (14)
B(A21)1兾20.5(R
dif 1兾R
dif). (15)
The vertical variations of Rdif共k兲were plotted for the
same parameters of R
dif and ABas for the Haltrin
model [Fig. 2(b)] and demonstrated a similar behav-
ior but with one small difference: The optical depth
value, for which Rdif共k兲becomes indistinguishable
from R
dif, is greater than that for the Haltrin model
(beginning from k⬇3–4).
C. Chandrasekhar–Klier Model
In 1960, Chandrasekhar formulated the radiative
transfer equation,
23
and, in 1972, Klier
24
developed
its exact solution for isotropic distribution of scat-
tered photons. Below, we compare the solution of
Klier for the reflectance Rdif with the less exact KM
approximation. Taking into account that the param-
eter of Klier can be expressed in the form
1兾R
dif and changing designations in the
Chandrasekhar–Klier (CK) model to designations
that are more adequate for the present study, we get
Rdif 1AB(ABcothk)
ABcothkAB
, (16)
where the parameters Aand Bare determined by
Eqs. (11) and (15).
The above solution, in its general form, is numer-
ically different from the KM solution [Eq. (13)], al-
though Klier in his paper states (without a proof) that
both solutions are formally identical. I believe that
the identities of both approaches exist only in the case
of infinite layers, and I prove this statement in Sub-
section 3.A, below. Now I wish only to note that the
difference between the KM and the CK solutions in-
crease with the increase of the layer reflectance and
with the decrease of the optical depth. For aquatic
environments with a reflectance of R
dif 20% the
difference seems quite insignificant [compare Figs.
2(b) and 2(c) for both solutions]. Numerical analysis
of Eqs. (13) and (16) carried out for k僆关0, 4兴and the
model values of parameters R
dif : 2%, 10%, 20% and
AB: 0%, 20%, 60% yields a NRMSE difference of ap-
proximately 1.4% for the KM solution in comparison
with the CK solution.
D. Lyzenda Model
The following model was proposed by Lyzenda
50
for
irradiance reflectance:
Rdif R
dif关1exp(2k)兴ABexp(2k). (17)
It is the simplest approach and yields results close to
those of the KM and the CK models [Fig. 2(d)]. How-
ever, they were slightly different from those of the
Haltrin model.
E. Albert–Mobley Model
The recent Albert–Mobley model
51
represents an im-
proved Lyzenda model with the following differences:
the new model distinguishes between the downward
and the upward attenuation coefficients and between
radiation reflected in the water column and from that
reflected from the bottom. In addition, the authors
introduce two coefficients to fit their model to results
derived from the radiative-transfer program Hydroli-
ght Version 3.1.
52
However, the results of their sim-
ulations
51
demonstrate that, at concentrations of
total suspended matter greater than 3 mg/l, a back-
scatter probability of bb兾b0.019, and at some other
assumptions, characteristic for case 2 waters, all
three attenuation coefficients are close to one an-
other. Taking these simplifications into account, one
obtains from their model
Rdif R
dif关11.0546
exp(2k)兴0.9755ABexp(2k), (18)
i.e., results that are close to the above-considered
models [see Fig. 2(e)] and that show maximal, al-
though insignificant, divergence from Haltrin model.
3. Inverse Optical Models
In this section, we consider an inverse optical prob-
lem of the estimation of scattering properties (such as
parameters Sand bb) and absorption (such as param-
eters Kand a) from the measured reflectance coeffi-
cient. The solution of the inverse optical problem is
only the first step to the elucidation of an inverse
physical problem, such as an estimation of the con-
centration of a disperse phase in a two-phase colloidal
system or of phytoplankton cells in an aquatic envi-
ronment, particle size or cell distribution, individual
optical and structural characteristics of pigmented
and other particles, error predictions for IOP retriev-
als, and so forth. However, optical properties ob-
tained as a result of the solution of an inverse optical
problem cannot be considered to be true, as they de-
pend on the approach selected. Therefore, this study
refers not only to various optical models but also to
the formulas connecting optical characteristics de-
rived by different approaches. For simplicity a diffuse
scattering coefficient Sand a diffuse absorption coef-
ficient Kof the KM theory are chosen as the basic
optical properties.
140 APPLIED OPTICS 兾Vol. 44, No. 1 兾1 January 2005
A. Kubelka–Munk Model
The principal distinction of the extended KM theory
from any other theory that considers nonasymptotic
optical fields consists of two assumptions: (1) the pa-
rameters Kand Svary along the vertical coordinate,
and (2) the ratio K兾Sis considered to be a constant
that is independent of the vertical location for any
given medium.
30,31
According to this theory Svaries
with Z, as was described above by Eq. (14).
From Eqs. (10), (11), (14), and (15) it follows that
the product pS共Z兲S(usually referred as a scat-
tering power, see, for example, Refs. 31 and 53) is a
function of R
dif and k. A plot of pSversus kat various
values of R
dif (Fig. 3) demonstrates approximately
linear behavior at small R
dif values, characteristic
for aquatic environments. With the increase of R
dif,
and at small kvalues, the total relation between pS
and kis nonlinear; however, with the increase of k
the relation tends to be linear. After achieving values
of k⬇10 (at R
dif 0.3) or k⬇100 (at R
dif 0.9),
Sbecomes constant. Analysis of Eqs. (10) and (14)
yields the following expressions for Rb,pS, and Sat
large values of k:
Rb,R
dif(1 Tb
2)R
dif关1exp(2k)兴, (19)
pS,
k兾B,Sk兾B. (20)
Now, if we compare Eqs. (13) and (16) and take into
account Eq. (20), it can be proved that the KM and the
CK models are identical for the case of infinite layers.
It should also be noted that, in the calculation of pS,
one encounters serious difficulties at large values of
kbecause of numerical errors arising from the cal-
culation of Rbby Eq. (10). Therefore, the calculation of
pSat large k(at approximately 10–20, depending on
R
dif) could be carried out from the preliminarily es-
tablished linear regression between pSand k. Note
that, at R
dif 21兾21⬇0.414, parameter B1;
hence, pS→kand S→kat k→.
Now, taking into account Eqs. (12), (15), and (20),
we get an equation for asymptotic values of Kat large
k:
Kk(1 R
dif)兾(1 R
dif). (21)
By analogy with a scattering power, a term, the
absorption power pK共Z兲K, is coined. The depen-
dence of pKversus k(Fig. 4) is similar to the depen-
dence of pSversus k, with, however, an oppositely
oriented dependence on R
dif. It is also interesting to
note that, for a completely nonscattering medium
when S0 and R
dif 0, from Eq. (21) follows an
expected equivalency between Kand k; hence,
equivalency exists between pKand k.
B. Haltrin Model
The relation between the true or volume absorption
coefficient aand the input parameters kand R
dif
can be established from Eq. (3) and the Gershun
law
54
ka兾¯, (22)
as follows:
ak
1(R
dif)1兾2
1(R
dif)1兾2. (23)
The relation between the ratio a兾bband R
dif,
a
bb
关1(R
dif)1兾2兴2关14(R
dif)1兾2R
dif兴
4R
dif , (24)
was derived by Haltrin
13,17
within the framework of
his self-consistent approach and exhibited a high de-
gree of closeness
14,17
to the well-known equation of
Gordon et al.
20
Solving Eqs. (23) and (24) together, we
obtain a relation between the backscattering coeffi-
cients bb,k, and R
dif:
bb4k
(1 兾R
dif 1)关14(R
dif)1兾2R
dif兴. (25)
Plots of the relations between ¯a兾k,
a兾bb,bb兾k,R
dif, and the input R
dif are shown in Fig.
5.
Fig. 3. Dependence of the scattering power pS共Z兲Son the
optical depth for the diffuse attenuation coefficient kkZ and
R
dif computed according to the extended KM theory. The curves
correspond to values of R
dif of 0.1, 0.2,...,0.9(from left to right).
Fig. 4. Dependence of the absorption power pK共Z兲Kon the
optical depth for the diffuse attenuation coefficient kkZ and
R
dif computed according to the extended KM theory. The curves
correspond to values of R
dif of 0.01, 0.1, 0.2,...,0.9(from right to
left).
1 January 2005 兾Vol. 44, No. 1 兾APPLIED OPTICS 141
C. Comparison of Different Approaches with the
Kubelka–Munk Model
In this subsection we compare the backscattering
coefficient bband the absorption coefficient aob-
tained within the framework of the Haltrin model
with the corresponding parameters Sand Kcalcu-
lated within the framework of the KM model for any
plane-parallel layer. Then we compare the same pa-
rameters within the framework of other models for
the case of a plane-parallel semi-infinite layer. In
both cases, we neglect the influence of the bottom
albedo (in other words, we assume that ABR
dif and
k→).
Remembering that k共Z兲kand taking into ac-
count Eqs. (14), (15), and (25), one can derive an
expression relating the ratio S兾bbwith the other
optical parameters:
(H)arcoth关(1 ARb)兾BRb兴
2k
14(R
dif)1兾2R
dif
1R
dif ,
(26)
where the index H notes the Haltrin model. Simi-
larly, from Eqs. (12), (14), (15), and (25), we derive an
expression for K兾a:
(H) arcoth关(1 ARb)兾BRb兴
k
12(R
dif)1兾2R
dif
1R
dif .
(27)
Thus, Eqs. (10), (11), (15), (26), and (27) allow one to
express the ratios (Fig. 6) and (Fig. 7) as functions
of the input parameters kand R
dif. The figure plots
demonstrate a relatively weak dependence of and
(and hence, of Sand K, respectively) on the optical
depth at R
dif 0.3; however, at increasing values of
R
dif this dependence is strengthened. At k→, tak-
ing into account Eqs. (20) and (21), we get the ratios
共H兲S兾bband
共H兲K兾aas functions of R
dif (Fig.
8, below):
(H) 14(R
dif)1兾2R
dif
2(1 R
dif), (28)
(H) 12(R
dif)1兾2R
dif
1R
dif , (29)
respectively. The limits of the coefficients
共H兲and
共H兲
for small absorptions 共R
dif →0兲and for large absorp-
tions 共R
dif →1兲are
Within the framework of the CK model the follow-
ing expressions (in transformed form) for and
were derived by Klier
24
for isotropic scattering:
(CK) 4
0(1 兾R
dif R
dif), (31)
(CK) (1 R
dif)
(1
0)(1 R
dif), (32)
where the index (CK) denotes the CK model. These
lim
(H)0.5
R
dif →0,lim
(H)1
R
dif →0,lim
(H)1.5
R
dif →1,lim
(H)2
R
dif →1. (30)
Fig. 5. Relations between ¯a兾k,a兾bb,bb兾kb, and R
dif calcu-
lated within the framework of the self-consistent Haltrin approach.
Fig. 6. Dependence of the ratio S兾bbon the parameters kand
R
dif.Swas computed within the framework of the KM theory,
whereas bbwas computed based on the Haltrin self-consistent
approach.
142 APPLIED OPTICS 兾Vol. 44, No. 1 兾1 January 2005
relations are also shown in Fig. 8. Here the eigenvalue
of the radiative-transfer equation k兾cand the
single-scattering albedo 0b兾care related with the
reflectance R
dif by means of the following equations:
R
dif ln(1 )
ln(1 ), (33)
02
ln关(1 )兾(1 )兴. (34)
Equations (31)–(34) do not permit one to obtain an-
alytical expressions for ,0,
共CK兲, and
共CK兲relative to
the input variable R
dif; however, these optical prop-
erties can be outstandingly approximated by the fol-
lowing polynomials:
10.1104R
dif 4.705(R
dif)26.114(R
dif)3
2.533(R
dif)4, (35)
05.949R
dif 21.59(R
dif)251.99(R
dif)3
73.64(R
dif)454.23(R
dif)515.94(R
dif)6,
(36)
(CK) 0.6137 3.846R
dif 7.107(R
dif)2
6.400(R
dif)32.259(R
dif)4, (37)
(CK) 14.651R
dif 9.324(R
dif)29.019(R
dif)3
3.360(R
dif)4, (38)
for R
dif 僆共0, 1兲. The limits of the coefficients
共CK兲
and
共CK兲for R
dif →0 and for R
dif →1 are
Several other approaches that permit the predic-
tion of and were also derived within the frame-
work of the neutron diffuse theory by Gate
34
and
Brinkworth
35
and of the ␦-Eddington approximation
of the second-order and the fourth-order by Meador
and Weaver.
36
For isotropic scattering these ap-
proaches have the following forms:
lim
(CK) 2(1 ln 2) ⬇0.6137
R
dif →0,lim
(CK) 1
R
dif →0,lim
(CK) 1.5
R
dif →1,lim
(CK) 2
R
dif →1. (39)
(GB) 704
20
,
(GB) 2, (40)
(MW2) 401
20
,
(MW2) 2; (41)
(MW4) 15 (1
0)
(MW4)(16 3
(MW4))
0(16 3
(MW4)),
(MW4) 224
132 55035关12(1
0)兾35 121(1
0)2兾49兴1兾2, (42)
Fig. 7. Dependence of the ratio K兾aon the parameters kand
R
dif.Kwas computed within the framework of the KM theory,
whereas awas computed based on the Haltrin self-consistent ap-
proach.
1 January 2005 兾Vol. 44, No. 1 兾APPLIED OPTICS 143
where the indices (GB), (MW-2), and (MW-4) respec-
tively denote the Gate–Brinkworth, the second-order
␦-Eddington, and the fourth-order ␦-Eddington mod-
els. To express the models through the parameter
R
dif, one combines them all with Eqs. (33) and (34).
The most important conclusion to be drawn from this
consideration is a strong variation of the ratio with
R
dif for all models (Fig. 8), especially in the range for
which R
dif 0.3. An analogous conclusion for the
ratio holds for only several models (Haltrin, CK,
and MW-4); these models show a monotonic increase
of from 1, 1, and 4/3, respectively, to 2, with an
increase of R
dif from 0 to 1. For the GB and the MW-2
models an equality, 2, is assumed. Monotonic
behavior for is yielded by only two models: the
Haltrin and the CK. It is also interesting to note that
linear relations between and are close for some
sets of models:
(CK) 再
(CK) (0.449 0.027), R
dif 僆(0, 0.3]
(CK) (0.464 0.031), R
dif 僆(0, 1)
(43)
(MV-4) 再
(CK) (0.500 0.018), R
dif 僆(0, 0.3]
(CK) (0.499 0.015), R
dif 僆(0, 1)
(44)
(H)
(H) 0.5, for any values of R
dif. (45)
The Haltrin model appears to be more attractive
than other models owing to the fact that it deals with
assumptions that are more adequate for aquatic en-
vironments, such as the conspicuously anisotropic
phase-scattering function. It is also important to note
that, from a comparison of the Haltrin and the CK
models (Fig. 8), it is clear that the scattering anisot-
ropy affects the measured reflectance coefficient. This
influence is minimal at extreme values of the 兾
ratio (i.e, at extreme values of R
dif), and it increases
toward intermediate values. Therefore, the effect of
selected scattering phase functions would be greater
in the case of turbid, highly productive waters (when
兾ratios vary from approximately 1.37 to approx-
imately 1.72) than in the case of clear oceanic waters
(when 兾ratios vary from approximately 1.47 to
approximately 2.00). This conclusion also corre-
sponds to the findings of other experimental and the-
oretical investigations.
55,56
4. Alternative Approach to the Estimation of R
dif from
Rfor Turbid, Highly Productive Waters
The dependence of the irradiance reflectance from the
surface when illuminated by direct Sun rays Rdir on
the solar zenith angle is stronger than is predicted by
the Haltrin model, Eq. (2).
21,22,57
For example, Morel
and Gentili
22
proposed a linear Morel–Gentili model
(MGM) for the calculation of Rdir in semi-infinite oce-
anic layers at solar zenith angles of 070°obtained
by Monte Carlo simulation. Toward this aim, they
used a forward-peaked, strongly asymmetric scatter-
ing phase function for particles adopted from a well-
known Petzold function, with a constant
backscattering particle probability (it was accepted
as 0.0190). As a result, their model has the form of
R
dir [(0.6279 0.2227b0.0513b
2)
(0.2465b0.3119)0](bb兾a), (46)
where 0cos 0and bis the ratio of the molecular
water backscattering bbw to the total (water plus par-
ticles) backscattering bb. The result of the calcula-
tions of R
dir共0兲, normalized to R
dir for a zenith Sun,
at selected values of the parameter bis represented
in Fig. 9(a). The corresponding values of bb兾a, esti-
mated from the Morel bio-optical model for case 1
waters
22,58
are also shown in Fig. 9.
Another model was proposed by Kirk
21
that was
based on Monte Carlo simulations and on extensive
experimental material obtained for a range of water
types from clear oceanic to turbid harbor waters. Sim-
ulations were carried out for different phase func-
tions (i.e., with varied bb兾bratios) but at a constant
ratio of b兾a2. Taking Eqs. (20) and (24) of Kirk’s
paper (Ref. 21) into account and generalizing formu-
las presented in his Table 3 (and using his data from
Table 1 for water types 2 to 6), I derived the following
dependence:
R
dir 关0.31 (2.181¯s1.654)(011)兴(bb兾a),
(47)
where ¯sis the average cosine of scattering. Further,
taking into account that ¯sis closely connected with
bb兾bby exponential dependence (as was derived by
Sokoletsky et al.
59
based on the data of Kirk
21
), and
Fig. 8. Dependence of the ratios S兾bband K兾aon R
dif
for different approximations. Sand Kwere computed within the
framework of the KM theory, whereas aand bbwere computed
based on the approaches of Haltrin (H), Chandrasekhar and Klier
(CK), Gate and Brinkwort (GB), and Meador and Weaver (MW-2
and MW-4).
144 APPLIED OPTICS 兾Vol. 44, No. 1 兾1 January 2005
thus with bb兾a, we obtain the following dependence of
R
dir on 0and bb兾a:
R
dir {0.31 [2.126 exp(1.297bb兾a)
1.654]01)}(bb兾a). (48)
A plot of this dependence (constructed again with a
normalization of R
dir for a zenith Sun) is shown in
Fig. 9(b) for the same values of bb兾aas in Fig. 9(a) and
demonstrates the similarity between the models, es-
pecially at small zenith angles (NRMSE 1.4% at
070°): the normalized values of R
dir increase al-
most linearly with the decrease in 0and with the
decrease in the bb兾aratio. The latter relation corre-
sponds to the increase in water turbidity and the
decrease in the parameter b. At solar zenith angles
070°(i.e., at 00.34) the KM, similarly to the
MGM, shows an increasingly flat dependence of R
dir
on 0.
Another important conclusion to be made from the
comparison of these models is the existence of an
upper limit of dependence of R
dir on 0for turbid,
highly productive aquatic environments. It seems
that the upper enveloped curve can be derived from
Eq. (46) at btending to 0 (as was also noted by
Højerslev
60
) as follows:
R
dir (0.6279 0.31190)(bb兾a). (49)
This conclusion is also valid on a qualitative level for
the self-consistent Haltrin model [Fig. 9(c)]; however,
there is a much flatter R
dir dependence than for the
models considered above in this section. Calculations
for the Haltrin model were carried out for selected
values of bb兾a(including values used in the MGM and
the KM), based on Eq. (2) and the inversion of Eqs.
(22)–(24) [or, identically, from Eq. (37) of Ref. 17]:
再a兾bb
3a兾bb关94(a兾bb)兴1兾2冎1兾2
, (50)¯
Taking into account that the Haltrin model is based
on an idealized phase function with isotropic behav-
ior in the backward direction, whereas the MGM and
the KM (and also a model by Gordon
57
that is similar
to both models) are based more on natural aquatic-
based sources, we propose for the calculation of R
dir
versus 0and bb兾afor highly turbid aquatic environ-
ments the use of Eq. (49) as an alternative to Eqs. (2)
and (50).
Let us now show how the above findings can be
used as an alternative to Eq. (4) in the estimation of
R
dif from Rin highly turbid waters. Fitting the re-
lation between bb兾aand R
dif, Eq. (24), for 0.001
R
dif 0.3 by a quadratic polynomial (with
NRMSE 2.4%)
bb
a0.01439 2.279R
dif 10.54(R
dif)2(51)
and then introducing Eqs. (49) and (51) to Eq. (1), we
obtain a quadratic equation relative to R
dif:
10.54q(R
dif)2(2.279q1)R
dif
关0.01439q(1 s)R兴0, q(0.6279
0.31190)s, (52)
a positive root of which is the required solution. The
direct-to-diffuse incoming irradiance ratio scan be
estimated from the regional meteorological observa-
tions or with the help of different algorithms if the
Fig. 9. Dependence of the normalized reflectance coefficient of a
plane-parallel infinite layer illuminated by direct light Rdir on the
cos 0of the incoming radiation angle (in air): (a) Morel and Gentili
model (MGM). The numbers in parentheses in the legend are
values of bb兾a(numbers to the left) and b(numbers to the right);
(b) KM model. The numbers in parentheses in the legend are also
values of bb兾a: (c) Haltrin model (HM). Numbers in parentheses in
the legend are values of bb兾a.
1 January 2005 兾Vol. 44, No. 1 兾APPLIED OPTICS 145
solar position, the wavelength of the incoming irra-
diation, the sky condition, and geographical coordi-
nates (see, e.g., References 60– 63) are taken into
account; however, this issue is not be considered here
in detail.
5. Conclusion
Selected radiative transfer approaches have been
compared and evaluated from the point of view of
their applicability to natural aquatic environments.
Primary attention has been paid to simple two-flux
approaches: an extended KM theory and the self-
consistent Haltrin approach. The KM theory is
widely used in different technological applications,
medical physics, and atmospheric optics owing to the
easy evaluation from optical measurements of the
absorption and the scattering properties of the me-
dium. However, an application of the KM theory to
aquatic environments is rather rare,
64,65
apparently
owing to the fact that scattering processes in such
environments are anisotropic (although isotropic
scattering is the basic assumption for the KM theory).
From this point of view, the Haltrin model is more
applicable to natural waters of different types be-
cause it does use the phase function as a sum of the
isotropic and the anisotropic parts. Besides, the Hal-
trin model is completely equivalent to the original
radiative-transfer equation.
The KM and the Haltrin approaches have been
compared with each other and with additional ap-
proaches for the solution of optical problems, either
directly (the estimation of the irradiance reflectance
coefficient R) or inversely (the estimation of the vol-
ume absorption aand the backscattering bbcoeffi-
cients). The common feature of the models under
consideration has underlined the effect of the bottom
albedo ABon R. As has been shown, this influence is
significant up to values of the optical depth for a
diffuse attenuation coefficient kless than 2 to 4,
depending on the optical parameters of the medium
and the selected model. This study has also shown
that the KM model does not allow the transformation
of the parameters of absorption and scattering to the
IOP’s aand bbwithout the addition of the parameters
R
dif and kfor shallow and R
dif solely for optically
deep waters. However, taking into account that
within the frame of the Haltrin model R
dif is a func-
tion of bb兾aonly [Eqs. (24) or (51)], i.e., an IOP, one
can state that the KM model’s remission function
K兾S, calculated from R
dif solely is also an IOP, inde-
pendent of sun position or depth. That fact alone
might explain the success of using the K兾Sratio for
the solution of different bio-optical problems in sev-
eral studies.
64,65
Consistent with the KM theory, all
the other radiative transfer approaches considered in
the paper (the self-consistent Haltrin model, the CK
radiative transfer solution, the neutron diffuse Gate–
Brinkworth model, and different versions of the
␦-Eddington theory) permit one’s obtaining bb兾a
solely from R
dif. On the other hand, the above-
mentioned theories not permit one’s obtaining aor bb
separately from only R
dif, requiring additional pa-
rameters such as kor 0for such an assignment.
Nevertheless, the self-consistent Haltrin model
seems to be more preferable than other approaches,
primarily owing to the use of the more realistic scat-
tering phase function.
A transformation from a measured total irradiance
reflectance of the semi-infinite layer Rto the part of
the reflectance that depends on only the diffuse illu-
mination R
dif is, therefore, an important step of the
solution of various optical and bio-optical tasks. Two
different methods of such a transformation have been
developed, and their analysis has been carried out.
Several other problems, such as the estimation of the
direct-to-diffuse incoming irradiance ratio sand the
transformation of the measured above-water irradi-
ance reflectance to the measured subsurface irradi-
ance reflectance are out of the scope of the current
study and need special consideration.
This study was facilitated by a postdoctoral schol-
arship from the Admiral Yohai Ben-Nun Foundation
for Marine and Freshwater Research. The author
thanks Y. Z. Yacobi from the Kinneret Limnological
Laboratory, I. and A. A. Kokhanovsky from the
Stepanov Institute of Physics, Belarus, for useful
comments, critique, and discussion about earlier ver-
sions of the manuscript. Special thanks to A. A. Gi-
telson, Center for Advanced Land Management and
Information Technologies, University of Nebraska–
Lincoln, owing to whom this study was undertaken.
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