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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

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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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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 per.edstrom@miun.se.

EFERENCES1. P. Kubelka and F. Munk, “Ein beitrag zur optik der

farbanstriche,” Z. Tech. Phys. (Leipzig) 11a, 593–601(1931).

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