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Comparative analysis of selected radiative transfer approaches for aquatic environments

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Applied Optics
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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 δ-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 apparent 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.
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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 Vol. 44, No. 1 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 REuEd(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
1¯
1¯
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 Vol. 44, No. 1 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 RdirZ
RdifZ兲⬇RZhold. Note that the equality of Rdir to
Rdif for case 1 waters (see Table 1) {with
R0.001, 0.1, see, e.g., Refs. 3 and 12} is achieved
at lower solar zenith angles than for case 2 waters
{for which the range R0.01, 0.3was assumed}. A
more rigorous analysis of Eqs. (2) and (3) shows that
the relations 0 R
dir R
dif R19 holds at
values of w, satisfying the following conditions:
23
w3R2R112兴兾共9R11兾共2
¯1. In a more general case for any
w0.666, 1,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
12Rw(1 R)3
. (4)
The accuracy of Eq. (4) for the model’s complete pa-
rameter set of w0.666, 1,¯0, 1, and
s0, 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(s1, 7, 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
k(ZBZ)(7 2¯
2¯
4
3¯
2
,
(7)
where kis the diffuse attenuation coefficient in the
asymptotic light regime.
Equations (3) and (5)–(7) permit the prediction of
vertical variations in RdifZif R
dif and kZBZare
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
a
¯
(ZBZ)
expk(ZBZ), (8)
where E0Zand E0ZBare 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
RdifZwith 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 Vol. 44, No. 1 1 January 2005
ferent bottom substrates
49
[Fig. 2(a)]. The results
clearly demonstrate that, at any value of R
dif and AB,
RdifZapproaches R
dif and becomes invariant to the
impact of the bottom when k23.
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
Rb1KSKS(KS2) Tb
2,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 Vol. 44, No. 1 APPLIED OPTICS 139
A0.5(R
dif 1R
dif) (11)
KSA10.5(R
dif 1)2R
dif. (12)
Note that Eq. (9) can also be expressed in the equiv-
alent form
Rdif 1ABABcoth(BSZ)
ABcoth(BSZ)AB
, (13)
where
BSZarcoth
1ARb
BRb
, (14)
B(A21)120.5(R
dif 1R
dif). (15)
The vertical variations of Rdifkwere 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 Rdifkbecomes indistinguishable
from R
dif, is greater than that for the Haltrin model
(beginning from k34).
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
1R
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 k0, 4and 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
dif1exp(2k)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 bbb0.019, 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
dif11.0546
exp(2k)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 Vol. 44, No. 1 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 KSis 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 pSZS(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 k10 (at R
dif 0.3) or k100 (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
dif1exp(2k), (19)
pS,
kB,SkB. (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 21210.414, parameter B1;
hence, pSkand Skat 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)(1 R
dif). (21)
By analogy with a scattering power, a term, the
absorption power pKZK, 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¯, (22)
as follows:
ak
1(R
dif)12
1(R
dif)12. (23)
The relation between the ratio abband R
dif,
a
bb
1(R
dif)12214(R
dif)12R
dif
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 R
dif 1)14(R
dif)12R
dif. (25)
Plots of the relations between ¯ak,
abb,bbk,R
dif, and the input R
dif are shown in Fig.
5.
Fig. 3. Dependence of the scattering power pSZSon 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 pKZKon 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 Vol. 44, No. 1 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 kZkand taking into ac-
count Eqs. (14), (15), and (25), one can derive an
expression relating the ratio Sbbwith the other
optical parameters:
(H)arcoth(1 ARb)BRb
2k
14(R
dif)12R
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 Ka:
(H) arcoth(1 ARb)BRb
k
12(R
dif)12R
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
HSbband
HKaas functions of R
dif (Fig.
8, below):
(H) 14(R
dif)12R
dif
2(1 R
dif), (28)
(H) 12(R
dif)12R
dif
1R
dif , (29)
respectively. The limits of the coefficients
Hand
H
for small absorptions R
dif 0and for large absorp-
tions R
dif 1are
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 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 ¯ak,abb,bbkb, and R
dif calcu-
lated within the framework of the self-consistent Haltrin approach.
Fig. 6. Dependence of the ratio Sbbon 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 Vol. 44, No. 1 1 January 2005
relations are also shown in Fig. 8. Here the eigenvalue
of the radiative-transfer equation kcand the
single-scattering albedo 0bcare related with the
reflectance R
dif by means of the following equations:
R
dif ln(1 )
ln(1 ), (33)
02
ln(1 )(1 ). (34)
Equations (31)–(34) do not permit one to obtain an-
alytical expressions for ,0,
CK, and
CKrelative 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 0, 1. The limits of the coefficients
CK
and
CKfor 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 5503512(1 
0)35 121(1 
0)24912, (42)
Fig. 7. Dependence of the ratio Kaon 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 Vol. 44, No. 1 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)
(CK) (0.449 0.027), R
dif (0, 0.3]
(CK) (0.464 0.031), R
dif (0, 1)
(43)
(MV-4)
(CK) (0.500 0.018), R
dif (0, 0.3]
(CK) (0.499 0.015), R
dif (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
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
ratios vary from approximately 1.37 to approx-
imately 1.72) than in the case of clear oceanic waters
(when 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](bba), (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
dir0, normalized to R
dir for a zenith Sun,
at selected values of the parameter bis represented
in Fig. 9(a). The corresponding values of bba, 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 bbbratios) but at a constant
ratio of ba2. 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 0.31 (2.181¯s1.654)(011)(bba),
(47)
where ¯sis the average cosine of scattering. Further,
taking into account that ¯sis closely connected with
bbbby exponential dependence (as was derived by
Sokoletsky et al.
59
based on the data of Kirk
21
), and
Fig. 8. Dependence of the ratios Sbband Kaon 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 Vol. 44, No. 1 1 January 2005
thus with bba, we obtain the following dependence of
R
dir on 0and bba:
R
dir {0.31 [2.126 exp(1.297bba)
1.654]01)}(bba). (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 bbaas 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 bbaratio. 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)(bba). (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 bba(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]:
abb
3abb94(abb)1212
, (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 bbafor 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 bbaand 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
0.01439q(1 s)R0, 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 bba(numbers to the left) and b(numbers to the right);
(b) KM model. The numbers in parentheses in the legend are also
values of bba: (c) Haltrin model (HM). Numbers in parentheses in
the legend are values of bba.
1 January 2005 Vol. 44, No. 1 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 bbaonly [Eqs. (24) or (51)], i.e., an IOP, one
can state that the KM model’s remission function
KS, 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 KSratio 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 bba
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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148 APPLIED OPTICS Vol. 44, No. 1 1 January 2005
... Note that equations for transmittance such as Eqs. (14) and (16) may be found in numerous publications, e.g., by Stokes (1862), Ambartsumian (1942), Tuckerman (1947, Sobolev (1957), Ueno (1961), Hébert and Machizaud (2012), Sokoletsky et al. (2014a), and Sokoletsky andBudak (2016a, b, 2017). However, to the best of our knowledge, a presentation of ef in the form of ...
... was considered in only few publications, e.g., by Ambartsumian (1942), Sobolev (1957), Ueno (1961), Carder et al. (1999), Hébert and Machizaud (2012), Sokoletsky et al. (2014a), and Sokoletsky andBudak (2016a, b, 2017). In the other publications, instead of ...
... Overall, at all values of g, the King and Harshvardhan (1986) model has an accuracy better than 9% at g ≥ 0.3 ( Fig. A.3a), which is the best result among all models tested (not shown here), and the Sokoletsky et al. (2013) model is superior at g ≤ 0.3 (Fig. A.3b) with a relative error less than 7%. Thus, the Kattawar and Plass (1976) and Sokoletsky et al. (2013) models were ultimately used for the quasi-Raleigh p() #1, while the Sobolev (1975) and King and Harshvardhan (1986) in the form similar to that was successfully used by Berwald et al. (1995), Sokoletsky et al. (2003), and Sokoletsky andBudak (2016b, 2017) : ...
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The transmission of light is one of the key optical processes in the terrestrial environment (the atmosphere and underlying surfaces). The dependence of light transmittance on the illumination/observation conditions and optical properties of the atmosphere–underlying system can be studied using the integro-differential radiative transfer equation. However, for numerous applications a set of analytical equations is needed to describe the transmitted light intensity and flux. In this paper, we describe various analytical techniques to study light transmittance through light scattering and absorbing media. A physical significance and improved mathematical accuracy of approximations are provided using the analytical models for the diffusion exponent, average cosine of the light field, spherical and plane albedos. The accuracy of various approximations is studied using exact radiative transfer calculations with various scattering phase functions, single-scattering albedos, observational conditions, and optical depths.
... This permits the expression of reflectance via IOPs and angular characteristics. Some examples of the expressions for R using PA and SA can be found in the literature (Gordon, 1976;Morel and Gentili, 1991;Haltrin, 1998a;Højerslev, 2004;Sokoletsky, 2004Sokoletsky, , 2005Sokoletsky and Gallegos, 2010). The expression for r rs using RF and SA will be derived in the "Modeling remote-sensing reflectance" sub-section of the current study. ...
... Thus, minimal values of optical depths, calculated as ZK d , have been estimated as equal to 4.0 for the NRE. This allowed us to consider the NRE water layers as semi-infiniteeffectively neglecting the influence of bottom reflectance on upwelling radiance and resultant surface reflectance (Gordon and Brown, 1974;Sokoletsky, 2005;Sokoletsky et al., 2009). ...
... Note, that irradiance reflectance may be similarly presented as a superposition of PA and SA (Haltrin, 1998a;Sokoletsky, 2004Sokoletsky, , 2005Sokoletsky and Gallegos, 2010). An expression by Højerslev (2001) ...
Article
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Bio-geo-optical data collected in the Neuse River Estuary, North Carolina, USA were used to develop a semi-empirical optical algorithm for assessing inherent optical properties associated with water quality components (WQCs). Three wavelengths (560, 665 and 709 nm) were explored for algorithm development. WQCs included chlorophyll a (Chl), volatile suspended solids (VSS), fixed suspended solids (FSS), total suspended solids (TSS) and absorption of chromophoric dissolved organic matter (aCDOM). The relationships between the measured remote-sensing reflectance and the WQCs were derived based on the radiative transfer consideration. We simulated and analyzed impact of the CDOM absorption in the red and near infrared spectral domains, multiple scattering, and scattering phase function on accuracy of WQCs prediction. The algorithm was validated by comparing experimental Chl dynamics with predicted values and a numerical comparison between measured and modeled Chl values. The numerical comparison yielded the highest correlation between predicted and measured WQCs for Chl (R2 = 0.88) and the lowest for FSS (R2 = 0.00), while the best and worst mean-normalized root-mean-squares errors were obtained for aCDOM(412.5) and FSS (35% and 59%, respectively). WQCs retrieval accuracy was typically significantly better at values of aTSS, red > 0.5 m-1.
... This permits the expression of reflectance via IOPs and angular characteristics. Some examples of the expressions for R using PA and SA can be found in the literature (Gordon, 1976;Morel and Gentili, 1991;Haltrin, 1998a;Højerslev, 2004;Sokoletsky, 2004Sokoletsky, , 2005Sokoletsky and Gallegos, 2010). The expression for r rs using RF and SA will be derived in the "Modeling remote-sensing reflectance" sub-section of the current study. ...
... Thus, minimal values of optical depths, calculated as ZK d , have been estimated as equal to 4.0 for the NRE. This allowed us to consider the NRE water layers as semi-infiniteeffectively neglecting the influence of bottom reflectance on upwelling radiance and resultant surface reflectance (Gordon and Brown, 1974;Sokoletsky, 2005;Sokoletsky et al., 2009). ...
... Note, that irradiance reflectance may be similarly presented as a superposition of PA and SA (Haltrin, 1998a;Sokoletsky, 2004Sokoletsky, , 2005Sokoletsky and Gallegos, 2010). An expression by Højerslev (2001) ...
Data
Bio-geo-optical data collected in the Neuse River Estuary, North Carolina, USA were used to develop a semi-empirical optical algorithm for assessing inherent optical properties associated with water quality components (WQCs). Three wavelengths (560, 665 and 709 nm) were explored for algorithm development. WQCs included chlorophyll a (Chl), volatile suspended solids (VSS), fixed suspended solids (FSS), total suspended solids (TSS) and absorption of chromophoric dissolved organic matter (aCDOM). The relationships between the measured remote-sensing reflectance and the WQCs were derived based on the radiative transfer consideration. We simulated and analyzed impact of the CDOM absorption in the red and near infrared spectral domains, multiple scattering, and scattering phase function on accuracy of WQCs prediction. The algorithm was validated by comparing experimental Chl dynamics with predicted values and a numerical comparison between measured and modeled Chl values. The numerical comparison yielded the highest correlation between predicted and measured WQCs for Chl (R2 = 0.88) and the lowest for FSS (R2 = 0.00), while the best and worst mean-normalized root-mean-squares errors were obtained for a_CDOM(412.5) and FSS (35% and 59%, respectively). WQCs retrieval accuracy was typically significantly better at values of a_TSS, red > 0.5 m^(-1).
... We consider this study as a continuation of the previous studies attempting to compare different numerical and analytical radiative transfer reflectance approximations [1][2][3][4][5][6][7][8][9][10][11][12][13] with the aim of seeking simple but reliable solutions. We put forward a task to yield a comprehensive review of the radiative transfer approximations for diffuse reflectance, especially in their analytical form, which is convenient for fast routine calculations. ...
... However, using a replacement method [Eqs. (6) to (9)] at which a calculation of the plane albedo R p μ i replaced by the calculation of the spherical albedo r also seems to be a perspective. The replacement method yields good results only at strong scattering (i.e., ω 0 close to 1) in its current form. ...
Article
Full-text available
We present an analysis of a number of different approximations for the diffuse reflectance (spherical and plane albedo) of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from full absorption to full scattering and from fully forward to fully backward scattering) at collimated, diffuse, and combined illumination conditions were considered. The approximations were analyzed from the point of view of their physical limitations and compared to the numerical radiative transfer solutions, whenever it was possible. The main factors impacting the spherical and plane albedo were revealed for the known and unknown scattering phase functions. The main criterion for inclusion of the models in analysis was the possibility of practical use, i.e., approximations were well parameterized and only included easily measured or estimated parameters. We give a detailed analysis of errors for different models. An algorithm for recalculation of results under combined (direct and diffuse) illumination also has been developed.
... In order to study the upwelling irradiance in aquatic environments the global irradiance (composed by direct and diffuse irradiance beams), applied to a stratified medium, can be used to study the water radiative transfer (Sokoletsky 2005). We analyze the homogeneous single layer medium for the PAR irradiance, then define I 0,PAR as the PAR band irradiance just below the surface water; r PAR as the reflectance of the bottom of the layer; and Δh as the thickness of the layer. ...
... The proposed simple model represents the relationship between Secchi disk depth and attenuation coefficient expressed in terms of an hyperbolic function and losing part of absorption and scattering. The debate on the use of a linear (a perfect hyperbolic function), a non linear relationships and with the beam attenuation coefficient (e.g., K d,PAR × Z d γ = CONST, with γ ≠ 1 or c + K d = 8.69 × Z d -1 ) between Secchi disk depth and attenuation coefficients is still open (Montes-Hugo & Alvarez-Borrego 2003, 2005Padial & Thomaz 2008), but from our results and model interpretation the linear approach appears to be useful to estimate PAR upwelling irradiance. ...
Article
Full-text available
A simple model for upwelling irradiance has been developed. The model represents the relationship between Photosynthetically Active Radiation diffuse attenuation coefficients and Secchi disk depth described with a physical-mathematical expression. This physical mathematical expression allows the evaluation of the sub surface upwelling irradiance that was generated by the interaction between downwelling irradiance and the water column. The validation of the relation was performed using experimental data collected from five different aquatic ecosystems at different latitudes, solar elevations and irradiance levels. We found a good linear, positive correlation between the theoretical and measured upwelling irradiance (R2 = 0.96). The residues were well distributed, around the null value, according a Gaussian curve (R2 = 0.92). The results confirm the importance and the versatility of the Secchi
... The R and r rs may be defined in terms of physical (optical) quantities as follows (Haltrin, 1998; Sokoletsky, 2005; Sokoletsky et al., 2009a): ...
Article
Radiative transfer modeling is used to compare the performance of two spectral reflectance ratios (omega-R and omega-r), and these against of the Gordon spectral ratio (omega-g)'s performance. All of these ratios are used to extract water quality parameters (WQP) from the ocean. Inputs to the model are: five different scattering phase functions with particulate backscattering probability B-p (from 0.009 to 0.156), fourteen values of solar zenith angle in the water theta-i (from 0 degrees to 46 degrees), and seven values of delta-e (from 0.125 to 0.5). The results indicate that omega-R is more accurate than omega-r because it's less sensitive to different solar and atmospheric parameters. The Gordon's spectral ratio, omega-g is more accurate than any of the reflectance ratios omega-R and omega-r for any natural waters because it is totally independent on solar and atmospheric conditions. It is recommended that it always be used to extract WQP in extremely clear or turbid natural waters.
... 20 The derivation of Eq. (8) and more discussions on the applicability of this and other similar approximations can be found elsewhere. 5,11,16,17,21,22 We note that it follows from Eqs. (8) and (9) s ϭ ͱ 3␥ 3␥ ϩ 8 ...
Article
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The purpose of this article is to study the accuracy of a number of simple approximate equations for the spherical albedo r (or the hemispherical reflectance under diffuse illumination conditions) of semi‐infinite light scattering absorbing media, using numerical solutions of the nonlinear integral equation for the reflection function of a semi‐infinite turbid medium, as formulated by V. A. Ambartsumian. We find that the van de Hulst approximation provides the most accurate approximation for the diffuse reflectance r under diffuse illumination conditions. The value of r depends almost exclusively on the value of the similarity parameter s=(1ω0)(1gω0)1s=\sqrt{(1-{\omega}_{0})(1-g{\omega}_{0})^{-1}} , where ω 0 is the single scattering albedo and g is the asymmetry parameter. A simple approach to derive the normalized absorption spectra of particulate matter from reflectance measurements is proposed. © 2006 Wiley Periodicals, Inc. Col Res Appl, 31, 491–497, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.20262
Conference Paper
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We compare the approximations of reflective features (reflection function, plane albedo, and spherical albedo) with the exact radiative transfer calculations for semi-infinite light-scattering and absorbing media. We show that the level of complexity of the models and their relative error increase simultaneously with the transition from the completely directionally independent reflective properties (i.e., the spherical albedo) via directionally dependent illumination (i.e., the plane albedo) to the completely directionally dependent reflective properties (i.e., the reflection function). Moreover, we show that Gordon's quasi-single-scattering approximation (QSSA) yields acceptable results compared with the exact results and with other approximations for all reflective properties, if one is limited by optically clear natural waters. However, when diffuse reflectance (i.e. plane or spherical albedo) of turbid waters is considered, the more simple and exact approximations can be used by aquatic researchers. For example, from our simulations, it follows that at an asymmetry factor g ≈ 0.96 and the single-scattering albedo w0 > 0.9, an error for the plane albedo, calculated in the framework of QSSA, can reach 20% and more, whereas polynomial approximation, derived by us, yields a maximum of 15% error. Similarly, QSSA is more accurate for spherical albedo of modeled clear waters, but it worse than other simple approximations, for example, those by van de Hulst and by Madgett-Richards, if we consider turbid waters with w0 > 0.9. Thus, the simple approximations presented here are more accurate for the bio-optical modeling of very turbid natural waters than using the QSSA approach.
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We present a new analytical approach for retrieving the apparent optical properties (such as the attenuation coefficient for downwelling irradiance and the average attenuation coefficient for upwelling irradiance within the depth interval) and the inherent optical properties (such as the backscattering and absorption coefficients); in situ or remote sensing vertically weighted chlorophyll a concentration ( sat); and the water depth, which is "seen" by a satellite sensor. This approach is based on improved sub-algorithms developed by us that take into account optical processes occurring at the air-water interface and within the water column. The latter is ensured, in particular, by very simple and accurate approximations that apply to the plane and spherical albedos (for details see the paper by Sokoletsky and Kokhanovsky in this proceedings book). The stability and accuracy of the corresponding algorithms are assumed suitable for conditions where the highly stratified water column, with sat spans from 5 to 290 mg m-3 (an example is Lake Kinneret). We assume that the presented approach to bio-optical modeling is instrumental for developing optimal locally adapted algorithms for in situ and remote sensing of Chl a and other water components primarily for turbid natural waters. However, we expect that this approach can also be applied to much clearer waters, if the necessary changes in the sub-algorithms will be made.
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We demonstrate that, under ideal circumstances, passive optical measurements can yield surface water depth estimates with an accuracy of a few centimeters. Our target area is the Salar de Uyuni, in Bolivia. It is a large, active salt flat or playa, which is maintained as an almost perfectly level and highly reflective surface by annual flooding, to a mean depth of 30–50 cm. We use MISR data to estimate spatial and temporal variations in water depth during the waning portion of the 2001 flooding cycle. We use a single ICESat laser altimetry profile to calibrate our water depth model. Though the salt surface is probably the smoothest surface of its size on Earth, with less that 30 cm RMS height variations over an area of nearly 104 km2, it is not completely featureless. Topography there includes a peripheral depression, or moat, around the edge of the salt, and several sets of prominent parallel ridges, with 5 km wavelength and 30 cm amplitude. The process by which these features form is still not well characterized.
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Chapter
Systematic studies were achieved for correlating the global radiation with the relative sunshine duration and for determining the variations of their coefficients on a geographical scale as function of the radiation climatic area. The sum A + B of the coefficients of the well — know Angström regression equation can be regarded as a parameter characterizing the average monthly or annual conditions of the transparency of the atmosphere at a given station. This index called “atmospheric transparency index” (ATI) is a linear function of the latitude and of the Linke turbidity factor. A recapitulative table summarizes the different indices allowing to characterize a site. By an appropriate choice of them; it is possible to estimate with a reasonable accuracy the daily global irradiation of a site.
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The extent to which the relationship between the apparent optical properties of water bodies K d , , R(0) (vertical attenuation coefficient, reflectance) and inherent optical properties of the water ( a, b, b b ,—absorption, scattering, backscattering coefficients) varies with the shape of the volume scattering function has been investigated via Monte Carlo modeling of the underwater light field for 12 different optical water types. The previously derived equation for the vertical attenuation coefficient of downward irradiance urn:x-wiley:00243590:media:lno19913630455:lno19913630455-math-0001 has been shown to be of general applicability, but the coefficient G ( µ o ) is a function not only of the cosine of the refracted incident photons ( µ o ) but also of a shape factor of the volume scattering function in accordance with urn:x-wiley:00243590:media:lno19913630455:lno19913630455-math-0002 where µ ̄, is the average cosine of scattering derived from the volume scattering function in question, and c t , c 2 , c 3 , and c 4 are numerical constants estimated from the modeling data. The well‐known relationship urn:x-wiley:00243590:media:lno19913630455:lno19913630455-math-0003 between the subsurface irradiance reflectance under zenith sun and the ratio of backscattering coefficient to absorption coefficient is shown to apply satisfactorily to all the water types for b b /a values up to ∼0.25, but is less satisfactory at higher values of this ratio. As the incident beam departs from the vertical, so the reflectance increases to an extent that depends on the shape of the volume scattering function.
Article
A nonlinear statistical method for the inversion of ocean color spectra is used to determine three inherent optical properties (IOPs), the absorption coefficients for phytoplankton and dissolved and detrital materials, and the backscattering coefficient due to particulates. The inherent optical property inversion model assumes that (1) the relationship between remote- sensing reflectance and backscattering and absorption is well known, (2) the optical coefficients for pure water are known, and (3) the spectral shapes of the specific absorption coefficients for phytoplankton and dissolved and detrital materials and the specific backscattering coefficient for particulates are known. This leaves the magnitudes for the three unknown coefficients to be determined. A sensitivity analysis is conducted to determine the best IOP model configuration for the Sargasso Sea using existing bio-optical models. The optical and biogeochemical measurements used were collected as part of the Bermuda Bio-Optics Project and the U.S. Joint Global Ocean Flux Study Bermuda Atlantic Time Series (BATS). The results demonstrate that the absorption by dissolved and detrital materials. The retrieved chlorophyll a estimates show excellent correspondence to chlorophyll a determinations (r 2 = 81%), similar to estimates from standard band ratio pigment algorithms, while providing two additional retrievals simultaneously. The temporal signal of retrieved estimates of absorption by colored dissolved and detrital materials is mirrored in ratios of Kd(410) to Kd(488), a qualitative indicator for nonalgal light attenuation coefficients. The backscatter coefficient for particles is nearly constant in time and shows no correspondence with the temporal signal observed for chlorophyll a concentrations. Last, the IOP model is evaluated using only those wavelengths which closely match the Sea Viewing Wide Field of View Sensor wave bands. This results in only a 1 to 6% decrease in hindcast skill with the BATS biogeochemical data set. This is encouraging for the long-range goal of applying the IOP model to data from upcoming ocean color satellite missions.