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548 J. Opt. Soc. Am. A/Vol. 24, No. 2 /February 2007 Per
Edström
Examination of the revised Kubelka–Munk theory:considerations of
modeling strategies
Per Edström
Department of Engineering, Physics and Mathematics, Mid Sweden
University, SE-87188 Härnösand, Sweden
Received April 5, 2006; revised July 3, 2006; accepted July 18,
2006;posted September 11, 2006 (Doc. ID 70185); published January
10, 2007
The revised Kubelka–Munk theory is examined theoretically and
experimentally. Systems of dyed paper sheetsare simulated, and the
results are compared with other models. The results show that the
revised Kubelka–Munk model yields significant errors in predicted
dye-paper mixture reflectances, and is not self-consistent.The
absorption is noticeably overestimated. Theoretical arguments show
that properties in the revisedKubelka–Munk theory are inadequately
derived. The main conclusion is that the revised Kubelka–Munktheory
is wrong in the inclusion of the so-called
scattering-induced-path-variation factor. Consequently, thetheory
should not be used for light scattering calculations. Instead, the
original Kubelka–Munk theory shouldbe used where its accuracy is
sufficient, and a radiative transfer tool of higher resolution
should be used wherehigher accuracy is needed. © 2007 Optical
Society of America
OCIS codes: 000.3860, 290.4210, 290.7050.
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. INTRODUCTIONropagation of light in scattering and absorbing
media isescribed by general radiative transfer theory.
Solutionethods for radiative transfer problems have been stud-
ed throughout the last century. One of the earliest solu-ion
methods was developed by Kubelka and Munk1 andubelka2,3 (hereafter
referred to as KM). Later, achieve-ents in radiative transfer
theory4–8 have brought about
efined solution methods, used in areas with higher de-ands on
accuracy, such as neutron diffusion, stellar at-ospheres, optical
tomography, and atmospheric re-
earch. The coarsest resolution of these methods gives thearlier
so-called two-flux methods, of which KM is an ex-mple.Several
limitations for the KM model have been re-
orted, for example concerning dependencies between thecattering
and absorption coefficients s and k for translu-ent or strongly
absorbing media,9–13 and attempts haveeen made to attribute some of
this behavior to intrinsicrrors of the KM model14–18 or to
phenomena not includedn it. Despite these limitations, the KM model
is in wide-pread use for multiple-scattering calculations in
paper,aper coatings, printed paper, paint, plastic and
textile,robably due to its explicit form and ease of use. The
KModel has been modified and extended for different pur-
oses in a variety of ways19; most suggestions are, how-ver, of
limited generality, although they yield somewhatmproved results for
certain purposes.
In a recent series of papers,20–23 Yang and co-workersresented
their revised KM theory (hereafter referred tos Rev KM) as a way to
explain and overcome the prob-ems with strongly absorbing media
reported for KMheory. They argue that there was an oversight in
theerivation of the original KM theory that failed to takento
account the scattered path of individual photons,hus
underestimating the traveled path length. To corrector this, they
introduce what they call the scattering-
1084-7529/07/020548-9/$15.00 © 2
nduced-path-variation (SIPV) factor.20 This is then usedo derive
new relations23 between the KM scattering andbsorption coefficients
s and k, and the physically objec-ive scattering and absorption
parameters (in this paperenoted as and �s and �a) of the medium.The
purpose of this paper is to examine the suggested
ev KM theory, and thereby comment on the validity ofifferent
modeling strategies and their combinations.ore specifically, the
point is to inspect the inclusion of
he SIPV factor in the end results. (The purpose of thisaper is
not to explain or resolve the reported limitationsf KM theory.
However, a detailed analysis of that issueas been performed and
will be reported elsewhere.) Inections 2 and 3, some theoretical
reasoning is applied,nd in Sections 4 and 5 simulation results from
Rev KMre compared with KM, two discrete ordinate radiativeransfer
models and a Monte Carlo model. The results areiscussed in Section
6.
. THEORETICAL REASONING:ACKGROUNDhe KM theory is applicable in
plane-parallel geometryith infinite horizontal extension, meaning
that there areo boundary effects at the sides. The boundary
conditions,
ncluding illumination, are assumed to be time and
spacendependent at the top and bottom boundary surfaces.he medium
is assumed to be random and homogenousnd the radiation
monochromatic, to make scattering andbsorption constant. The
scattering is assumed to be iso-ropic and to take place without a
change in the frequencyetween incoming and outgoing radiation. The
medium isreated as a continuum of scattering and absorption sites.M
theory is limited to diffuse light distribution, consid-ring only
the averaged directions up and down.
The KM equations can be written
007 Optical Society of America
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Per Edström Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. A
549
− di = − �s + k�idx + sjdx,
dj = − �s + k�jdx + sidx �1�
or a thin layer dx, where i�x� is the intensity in the down-ard
direction, and j�x� is the intensity in the upward di-
ection, s and k are the light scattering and absorption
co-fficients, and x is the distance measured from theackground and
upward. This is a differential equationhat is easily integrated to
give the well-known relationsetween s and k and various reflectance
quantities.The KM coefficients s and k have no direct
physicaleaning on their own, but should only be interpretedithin
the KM model; they do not represent anythinghysically objective
outside the KM model. This is con-rary to the general formulation
of the radiative transferroblem, where the scattering and
absorption coefficientsre related to the mean free path in a
medium, and arehus model and geometry independent. They can
there-ore be given a physically objective interpretation, whichs a
desirable feature for any model.
Approximate relations between the KM coefficients andhysically
objective parameters have been suggested,uch as
s = �s,
k = 2�a, �2�
ttributed to original KM theory, and
s = 3�s/4,
k = 2�a, �3�
y Mudgett and Richards.5,6 These relations are approxi-ate,
since dependencies between s and k have been
eported,9–13 while �s and �a are considered to be inde-endent.
Other relations have been suggested in differentelds of application
to explain the apparent dependenceetween s and k. These relations
must all be approxi-ate, however, since KM is incommensurable
with
igher-order models; KM is fundamentally simpler and aranslation
to higher-order models could never be com-lete. Indeed, the
existence of a complete translationould imply that the higher-order
model was equivalent
o the simpler KM model, which would be a contradictionn terms.
Instead, relations such as these should be re-arded as the first
term of some series expansion.
A recent contribution in this matter is from Yang
ando-workers,20–23 who in their Rev KM theory propose a
re-erivation to correct an oversight of the original KMheory. The
setting is identical to the one given for KMbove, except that the
continuity assumption is invali-ated. They argue that KM did not
take into account thenfluence of internal scattering on the total
path length.sing a statistical line of reasoning, they obtain a
number
f relations used in statistical physics. The main result ishat
they call the SIPV factor � which they define with
espect to Fig. 1 as the ratio of averages of the true pathength
between B and C and the corresponding straight-ine
displacement.23
They also derive the explicit expression
� = ��sD, �4�
here � is a factor dependent on the angular distributionf light
intensity in the medium, and D is the averageepth of turning
points23; see Fig. 1. For optically thickedia, this is simplified
to
� = ��s2/��a
2 + �a�s��1/4. �5�
he traveled path length through a given layer is arguedo be on
the average � times longer than the straight lineetween the points
of entrance and exit; see Fig. 2. Theylaim that this effect was
ignored in KM theory, andence derive the relations20
s = ���s/2,
k = ���a. �6�
or perfectly diffuse light distribution throughout the me-ium,
�=2 and relations (6) become relations (2) with thextra factor �.
Considering expression (5) for �, the rela-
ig. 1. (Color online) Scattered photon path used in Rev KM
tobtain the SIPV factor.
ig. 2. (Color online) Longer path through a layer according toev
KM.
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550 J. Opt. Soc. Am. A/Vol. 24, No. 2 /February 2007 Per
Edström
ions (6) for s and k are strongly nonlinear with respect tooth
�s and �a.
. THEORETICAL REASONING:XAMINATION. Limiting Process Omittedang
and co-workers derive their expressions for �, s, andfor a layer of
finite thickness. But they inadequately
ombine this with KM theory—which is a differentialquation—and
thereby implicitly use infinitesimal layers.o get adequate results,
the limiting process should be ex-licitly carried out to obtain
expressions for �, s, and k fornfinitesimal layers. However, this
cannot actually be per-ormed because of the incompletely described
averagingrocesses, as discussed briefly below. Since the
limitingrocess is omitted in the derivation of Rev KM, this prob-em
is overlooked. Unfortunately, this is what causes therror in Rev
KM.
. Geometrical Exampleiven that the limiting process for � is not
easily per-
ormed, it can be enlightening to study a geometrical ex-mple.
One cannot have curves in an infinitesimal layer,uch as in Fig. 2.
It is easy to compare with the calcula-ion of the arc length of a
curve, where small line seg-ents are approximated with straight
lines between the
nd points of the segments. As the line segments areade smaller,
the straight lines get closer to the curve,
nd in the limit, the quotient between a true segmentength and
its straight-line approximation tends to unitysee Fig. 3).
Therefore, this must also happen to the SIPVactor � in the limit,
whereby original KM theory is re-ained. If one also corrects for a
known curvature by in-roducing a factor for the line segments, the
resulting arcength in the limit would be too large by precisely
theame factor when summing up the segments (and if oneere not to go
to the limit, the resulting arc length wouldlso depend on the
partitioning of the curve, since finerartitioning makes the
straight-line approximationsloser to the curve). That would be to
introduce somethinghat vanishes in the limiting process. In an
infinitesimalayer, only the direction matters, and thereby the
angularistribution of the intensity as a function of depth is
suf-cient in the light-scattering case. The differential equa-ion
of radiative transfer4 treats this exactly, but of coursehe
accuracy of a given radiative transfer tool depends onts
resolution. This means that KM is as exact as it can beithin the
two-flux approximation. It is unreasonable to
hange the model parameters, as Rev KM suggests, just
ig. 3. (Color online) In the calculation of the arc length of
aurve, small line segments are approximated with straight lines.s
the line segments are made smaller, the straight lines get
loser to the curve, and in the limit the quotient between a
trueegment length and its straight-line approximation tends tonity
(which is what must also happen to the SIPV factor �).
ecause the resolution is not sufficient. Instead, theroper thing
to do is to use a model with higher resolu-ion.
. Explicit Erroro be very explicit, the derivation of Rev KM
uses finiteayers in the reasoning concerning Figs. 1 and 2 to
obtainxpressions for the SIPV factor and for s and k, and
evenxplicitly insists that the layer be thick enough to
containsufficient number of scatterers. On the other hand, s
nd k—now ascribed properties of finite layers—are thensed in the
well-known KM relations for reflectance, rela-ions that are
explicitly derived using infinitesimal layers.
. Incompletely Described Averaging Processess mentioned above,
there are some problems with theveraging processes in the
derivation of Rev KM. There isn explicit averaging over different
directions of thetraight line B–C in Fig. 1, weighted with the
light distri-ution. But there is no averaging over incident light
di-ections, different turning points B, different exit points, or
number of scatterings N, weighted with the respec-
ive probabilities. Furthermore, establishing these unem-loyed
probabilities is nontrivial.
. Unknown Angular Distribution of Intensitynother problem in Rev
KM is the angular distribution of
he intensity, which is explicitly included through the fac-or �.
There is no way of determining it within Rev KM,o the assumption is
made that the light is perfectly dif-use throughout the medium. The
problem here is two-old. First, the light distribution is never
constanthroughout a medium; second, it is never perfectly
diffuseven if the single-scattering process is isotropic, not
evenor theoretically idealized media. Any radiative transferool
with sufficient resolution will show this, and there isn abundance
of examples within for instance tomogra-hy or astrophysics. While
the deviation from constantiffuse light distribution may not be so
great in many cir-umstances, it can be very large indeed in samples
withigh absorption24; since this is a case where Rev KM isupposed
to give better results, the assumption of con-tant diffuse light
distribution is not adequate.
. Modeling with Finite and Infinitesimal Layershe Rev KM
argument for finite layers is that in a realedium, e.g. paper, an
infinitesimal layer would containo physical particles; therefore a
finite layer is needed inrder to contain anything, and then these
phenomena ap-ear. But paper is not unique in this respect. In the
end,ll real media are discrete, be they particles, molecules,
ortoms. The infinitesimal layer is not real, but forms a partf the
mathematical description; it is a mathematical tool.he validity of
working with infinitesimal layers and dif-
erential equations for real media—apart from being com-on use in
any natural science or technology
pplication—has been thoroughly discussed byoedecke.25 A real
physical medium with finite thicknessnd macroscopic parameters can
always be modeled as andealized medium with average parameters.
According tooedecke, general radiative transfer theory, of which
KM
s a subset, assumes that the medium is random, homog-
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Per Edström Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. A
551
nous, and continuous. While the conditions of random-ess and
homogeneity most often are fulfilled, Goedeckehows that the
condition of continuity might not be. Forhose cases, he proposes a
difference equation instead ofhe traditional differential equation
of radiative transfer.his has the practical drawback that
difference equationsre in general much harder to solve. However,
Goedeckelso shows that for most media of practical interest
theraditional differential equation will suffice. For close-acked
media, it might be necessary to replace the phaseunction with one
appropriately describing near-field—aspposed to ordinary
far-field—scattering. Only fortrongly absorbing close-packed media
would the differ-nce equation be necessary. Thus, working with
infinitesi-al layers and differential equations is nearly always
ap-
ropriate, especially in paper applications, but in no cases it
valid to combine finite layers with differential equa-ions, as is
done in Rev KM.
. Where the Error Isven though the error in the derivation of
the Rev KM
heory might be theoretically fundamental, it is howeverot easily
identified in the outline of the papers. This isecause the error is
done implicitly in a part that was notncluded in the papers, the
limiting process for �. How-ver, when viewing it all from a more
general perspective,n this case from general radiative transfer
theory, it isasier to analyze the reasoning as a special case
thanhen working exclusively within it.
. SIMULATIONS: BACKGROUNDISORT (Ref. 7) and DORT2002 (Ref. 8)
are both modern dis-rete ordinate radiative transfer (hereafter
referred to asORT) solution methods. They are fast and accurate
toolsor solving radiative transfer problems in vertically
inho-ogeneous turbid media. DORT2002 is adapted to light-
cattering simulations in paper and print, while DISORT isostly
applied to atmospheric research. However, apart
rom being designed for much more challenging tasks,oth fully
include the KM situation as a simple specialase. As they also can
achieve any desired angular reso-ution (both polar and azimuthal),
they are well suited foromparison with KM and Rev KM.
GRACE26 is a modern Monte Carlo simulation tool foright
scattering in paper. It does not consider computa-ional layers at
all, finite or infinitesimal, and is not basedn either differential
or difference equations. Instead, itses a Monte Carlo approach with
probability distribu-ions for all constituents of the medium, and
collection oftatistics from a large number of incident photons
whosenteraction with the medium is governed by fundamentalhysical
laws.
. SIMULATIONS: EXAMINATION. Quantitative Experimental Setups
pointed out by Yang and Miklavcic,23 the exact amountf dye in a
dyed paper sheet is in practice not known sinceome of the dye
remains in the drain water. This preventsxact quantitative
comparison between simulations andeal measurements. To obtain a
relevant quantitative
omparison despite this practical problem, a Monte Carloxperiment
was designed and performed. The purpose ofhe theoretical experiment
was to simulate exactly thoserocesses that Rev KM aims to treat.
The Monte Carloodel GRACE was thus used to simulate diffuse
illumina-
ion of a homogenous, noncontinuous medium of a givenrammage,
with randomly distributed scattering and ab-orption sites of given
average densities. The scatteringas isotropic; i.e., for each
scattering event, every direc-
ion is of equal probability, and there were no surface
re-ections. This makes the simulated photons move in ex-ctly the
way the derivation of Rev KM assumes. As aheoretical experiment,
this has the advantage over realeasurements that the results are
not contaminated with
ny effects of other processes that are not modeled. Fur-hermore,
the amount of dye is known exactly, and theheoretical dye only
affects the light absorption. Hence,his Monte Carlo simulation is
ideally suited as a refer-nce in this examination, and is even
better than realeasurements. The experiment, as outlined below,
com-
ared results from Rev KM, original KM, the two DORTodels
DORT2002 and DISORT, and the Monte Carlo model
RACE.
. Real Input Datahe spectral data used as input were real
reflectance fac-
or measurements for the paper, and real s and k valuesor the dye
(originally obtained from reflectance factoreasurements). The s and
k values were then trans-
ormed to equivalent reflectance factor values via KMheory. It
should be pointed out that these real valuesere used for two
reasons: because they are relevant inractice, and because they are
identical to those Yang andiklavcic used,23 which facilitates
comparison. The theo-
etical experiment could, however, start with any reason-ble
spectral properties for the paper and dye, not neces-ary measured
values at all.
. Verification of Data and Procedurehe experimental and
computational procedure describedy Yang and co-workers20,23,27 was
followed closely. As aerification of the data and procedure, all
their spectralesults [their Figs. 2–6 and 7(a) (Ref. 23)] were
repro-uced with Rev KM and were found to be identical. Sincehe
measurements were made in accordance with ISO469,28 all
simulations, when applicable, were adapted tohe d /0° instrument
geometry specified therein.
. First Part of the Experimentn the first part of the
experiment, the reflectances for pa-er and dye were used as the
input for all models in ordero calculate scattering and absorption
parameters of theaper and dye (all models can do this, either by
them-elves or with a suitable optimization routine). The mod-ls
were then used to predict reflectances for dye–paperixtures with
different amounts of dye. It was assumed
hat the commonly used additivity principle is applicable,hich
essentially says that the parameters of a mixturere the mass
averages of the constituents’ parameters.he Monte Carlo model was,
as argued above, used as aeference.
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Frf4�pTatn
552 J. Opt. Soc. Am. A/Vol. 24, No. 2 /February 2007 Per
Edström
. Revised Kubelka–Munk Not Accuratehe accuracy of the models was
then evaluated by com-aring the predicted values with the Monte
Carlo refer-nce values [compare Figs. 4(b)–4(d) with Fig. 4(a)].
Anccurate model should obviously produce predictions closeo the
reference values. The two DORT models gave iden-ical results, and
their results were nearly identical to theonte Carlo model. The KM
model performed almost asell, but with slight deviations in the
absorptive band of
he dye. However, the Rev KM model gave good resultsnly for the
undyed sample, i.e. pure paper, and yieldedignificant errors for
all other samples. The absorptionas clearly overestimated.
. Second Part of the Experimenthe second part of the experiment
consisted of using RevM, KM, and the two DORT models to once again
calculate
ig. 5. (Color online) Rev KM s and k dye–paper mixture
pa-ameters (a) as predicted from additivity, and (b) as
calculatedrom dye–paper mixture reflectances. The paper grammage
was0 g/m2, and the dye grammages
were0,0.005,0.01,0.02,0.05,0.1,0.2� g/m2. Note the parameter
de-endencies (decrease in s with increased k) for predicted
values.he model is clearly not self-consistent, as (a) and (b) are
not atll similar neither in s nor in k. The statistical noise
inherent inhe Monte Carlo process is visible in the last pane, but
that doesot affect the conclusion.
ig. 4. (Color online) Dye–paper mixture reflectances for (a)
theonte Carlo reference values, (b) Rev KM, (c) original KM,
and
d) the DORT models. The paper grammage was 40 g/m2, and theye
grammages were �0,0.005,0.01,0.02,0.05,0.1,0.2� g/m2.he DORT models
give results nearly identical to the reference,M almost as well
except for slight deviations in the absorptionand of the dye, while
Rev KM yields significant errors with
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Per Edström Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. A
553
he scattering and absorption parameters of the dye-aper mixtures
(again, all models can do this, either byhemselves or with a
suitable optimization routine). How-ver, this time the Monte Carlo
reference reflectance val-es of the mixtures just calculated were
the startingoint.
. Revised Kubelka–Munk Not Self-Consistenthe consistency of the
models was then evaluated by com-aring these mixture parameters,
for the respectiveodel, with the ones obtained earlier from
additivity
compare the first pane with the last in the respectiveigs. 5–8).
The statistical noise inherent in the Montearlo process is visible
in the last pane of these figures,ut does not affect the
conclusions. A self-consistentodel should obviously give similar
values. Again, the
wo DORT models gave identical results, and they wereound to be
self-consistent. The KM model performed al-ost as well again, but
the deviations in the absorptive
and of the dye were somewhat larger. However, the RevM model
once again gave good results only for the un-
ig. 6. (Color online) Rev KM intrinsic �s and �a dye–paperixture
parameters (a) as predicted from additivity, and (b) as
alculated from dye–paper mixture reflectances. The model
islearly not self-consistent, as (a) and (b) are not at all similar
ina. (See the caption for Fig. 5 for grammages and comments
onoise.)
yed sample and was clearly not self-consistent in thether cases.
In the absorptive band of the dye, the devia-ion was more than a
factor of 10.
Two additional items can be compared for the Rev KModel. Since
it uses the same objective scattering and ab-
orption parameters as the DORT models, their respectives and �a
predictions should be similar. Furthermore,ince Rev KM uses the KM
parameters as well, their re-pective s and k predictions should be
similar too. It wasound that the parameter values of Rev KM were
notimilar to the ones of the DORT (compare Figs. 6 and 8)nd KM
(compare Figs. 5 and 7) models, respectively,hich would be expected
from an accurate model.
. Erroneous Parameter Dependencies in Revisedubelka–Munk
t was also noted that the s and k predicted from additiv-ty by
Rev KM in Fig. 5(a), as specifically pointed out byang and
Miklavcic,22,23 indeed show a decrease in s for
ig. 7. (Color online) Original KM s and k dye–paper
mixturearameters (a) as predicted from additivity, and (b) as
calculatedrom dye–paper mixture reflectances. The model is fairly
self-onsistent, as (a) and (b) are rather similar, but there are
someeviations in the absorption band of the dye. Note that no
param-ter dependencies are present. (See the caption for Fig. 5
forrammages and comments on noise.)
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554 J. Opt. Soc. Am. A/Vol. 24, No. 2 /February 2007 Per
Edström
ncreased k. This is in contrast with the parameters ob-ained
from dye-paper mixture reflectances from any ofhe tested models,
including Rev KM itself (although theast pane of the figures is
somewhat blurred by the statis-ical noise inherent in the Monte
Carlo process, it is clearhat they do not show this decrease in s).
In fact, this phe-omenon is hardly measurable at such low degrees
of ab-orption. The line of reasoning of Yang and Miklavcic23
isrroneous and deceptive in this matter, since Rev KM wasot
compared to measurements at an equal degree of ab-orption. Their
referred and illustrated experimental pa-ameter dependencies are
for dye grammages up tog/m2 (approximate values from the caption of
their23
ig. 4, also verified by calculations), while their illus-rated
Rev KM simulations are for dye grammages up tonly 0.2 g/m2.
To verify this, the above experimental scheme was re-eated with
ten times the absorption. This indeed gave aecrease in s for
increased k for KM, as seen in Fig. 9, buthe decrease was still not
as large as what Rev KMhowed already at the lower absorption. Of
course, thislso made KM give worse reflectance predictions than
in
ig. 8. (Color online) DORT intrinsic �s and �a dye–paper mix-ure
parameters (a) as predicted from additivity, and (b) as cal-ulated
from dye–paper mixture reflectances. The models areelf-consistent,
as (a) and (b) are very similar. (See the caption forig. 5 for
grammages and comments on noise.)
ig. 4(c). Once again both DORT tools predicted the reflec-ances
correctly without parameter dependencies, ashould be expected from
models of higher resolution. RevM overestimated the effect heavily,
did not predict the
eflectances correctly, and was clearly not self-consistent.hus,
the proposition that Rev KM convincingly repro-uces the features of
the experiments23 is based on the in-orrect comparison of the shape
of the curves of the pa-ameters measured for higher absorption
(wherearameter dependencies are present) on the one hand andf
parameters predicted by Rev KM for low absorptionwhere parameter
dependencies are actually almost notresent) on the other hand.
. Comparison with Experimental Data from Realystemsxact
quantitative comparison between simulations andeal measurements is
not possible since the exact amount
ig. 9. (Color online) Original KM s and k dye–paper
mixturearameters as calculated from the dye–paper mixture
reflec-ances with a higher dye amount. The paper grammage was
still0 g/m2, but the dye grammages were increased
to0,0.02,0.1,0.2,1.0,1.5,2.0� g/m2. Note that parameter
depen-encies (decrease in s with increased k) are now present.
ig. 10. (Color online) Measured reflectances (curves) for
hand-heets with different amounts of dye, and Monte Carlo
predic-ions (crosses). The good predictions confirm the relevance
of theheoretical experiment.
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Per Edström Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. A
555
f dye in a dyed paper sheet is in practice not known.owever, it
would still be interesting to examine how real
ystems vary from the ideal Monte Carlo model. There-ore, a
series of handsheets with various amounts of dyeas made,
reflectances were measured, and apparent
cattering and absorption parameters for the dye were es-imated,
as well as the dye grammages. The handsheetsere made to minimize
gloss and contained no fillers, toe as ideal as possible. The Monte
Carlo model was thensed to predict the reflectance of handsheets
with differ-nt dye amounts. The predictions were very good, as
seenn Fig. 10. This confirms the relevance of the
theoreticalxperiment above.
. SUMMARYhe revised Kubelka–Munk (Rev KM) theory has been
ex-mined theoretically and experimentally in this
paper.pecifically, the inclusion of the so-called scattering-
nduced-path-variation (SIPV) factor in the end results ofev KM
has been inspected.Theoretical arguments showed that the SIPV
factor
annot be used together with a differential model asroposed in
Rev KM. There, properties are derivedsing finite layers, and are
then inadequately—ithout going through a limiting process—used in
rela-
ions that are explicitly obtained using infinitesimal lay-rs.
This error was also illustrated with a geometrical
ex-mple.Simulation experiments showed that the Rev KModel yielded
significant errors in predicted mixture re-
ectances, i.e. it was not accurate, and that it was clearlyot
self-consistent. The erroneously and deceptively al-
eged correspondence of Rev KM with parameter depen-encies from
measurements did not hold when comparedt an equal degree of
absorption. The absorption was no-iceably overestimated by Rev KM,
and in no case was theodel better than the original KM.Therefore,
the main conclusion of this paper is that the
heory is wrong in the inclusion of the SIPV factor in thend
results. Consequently, Rev KM should not be used foright-scattering
calculations. Instead, KM should be usedhere its accuracy is
sufficient, and a DORT tool should besed where higher accuracy is
needed.As a concluding note, it can be noted that the purpose
f this paper is not to explain or resolve the reported
limi-ations of KM theory. However, a detailed analysis of thatssue
has been performed and will be reported elsewhere.he analysis
includes explanations and suggestions, suf-ce it to say here that
the reported problems are largelyue to the low resolution of the KM
two-flux model, andan be resolved with a radiative transfer model
of higheresolution (but not with Rev KM).
It should also be stated that the radiative transfer soft-are
DORT2002, which is adapted to light-scattering simu-
ations in paper and print, is available at no charge fromhe
author.
CKNOWLEDGMENTShe author thanks Ludovic Coppel, STFI-Packforsk,
forerforming the GRACE simulations. Ludovic Coppel and
jalmar Granberg, STFI-Packforsk, are thanked for dis-ussions and
for comments on the manuscript. This workas financially supported
by the Swedish printing re-
earch program T2F, TryckTeknisk Forskning.
Per Edström’s e-mail address is [email protected].
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