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Quantification of the intrinsic error of the kubelka–munk model caused by strong light absorption
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences. (FSCN - Fibre Science and Communication Network)ORCID iD: 0000-0002-0529-1009
2003 (English)In: Journal of Pulp and Paper Science (JPPS), ISSN 0826-6220, Vol. 29, no 11, 386-390 p.Article in journal (Refereed) Published
Abstract [en]

The Kubelka-Munk (KM) model is widely used within the paper industry to interpret diffuse reflectance factor measurements of paper and its components. It has been found in the literature that the addition of a dye colorant to a paper sheet not only increases its KM light absorption coefficient, but for strong absorption also decreases its KM light scattering coefficient. This effect has previously been attributed to the intrinsic error of the KM model induced by light absorption that tends to orient of the light fluxes perpendicular to the sheet. In the present work we have mapped the intrinsic error of the KM model by comparing light scattering calculations from the KM model with the more accurate Discrete Ordinate Radiative Transfer model DORT2002. We found that the models agree within 2.3% in reflectance, and that the intrinsic error in the KM model explains about 1/5 of the previously observed interdependence of the KM coefficients for heavily dyed sheets.

Place, publisher, year, edition, pages
2003. Vol. 29, no 11, 386-390 p.
Keyword [en]
light scattering, light absorption, reflectance, Kubelka-Munk, DORT2002, optical properties, errors
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-1799ISI: 000187624200006Local ID: 507OAI: oai:DiVA.org:miun-1799DiVA: diva2:26831
Available from: 2008-09-30 Created: 2008-09-30Bibliographically approved
In thesis
1. Mathematical Modelling of Light Scattering in Paper and Print
Open this publication in new window or tab >>Mathematical Modelling of Light Scattering in Paper and Print
2004 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

A problem formulation and a solution method are outlined for the radiative transfer problem in vertically inhomogeneous scattering and absorbing media, using discrete ordinate model geometry. The treatment spans from the physical problem via a continuous formulation, a discretization and a numerical analysis, to an implementation with performance evaluation and application to real-world problems. The thesis clearly illustrates how considerations in one step affect other steps, and thus provides an example of an overall treatment of mathematical modeling of a large applied problem. A selection of different steps is brought together. First all the steps necessary to get a numerically stable solution procedure are treated, and then methods are introduced to increase the speed by a factor of several thousand. The solution procedure is implemented in MATLAB under the name of DORT2002, and is adapted primarily to light scattering simulations in paper and print. A confined presentation is given of the effect of the steps that are needed, or possible, to make any discrete ordinate radiative transfer solution method numerically efficient. This is done through studies of the numerical performance of DORT2002. Performance tests show that the steps that are included to improve stability and speed of DORT2002 are very successful. Together they give an unconditionally stable solution method to a problem previously considered numerically intractable, and decrease computation time compared to a naive implementation with a factor of 1 000 � 10 000 in typical cases and with a factor up to and beyond 10 000 000 in extreme cases. It is also shown that the speed increasing steps are not introduced at the cost of reduced accuracy, and that DORT2002 converges to the true value as the discretization is made finer. It is shown by the use of DORT2002 that when a medium has a finite thickness, the light distribution deviates from the perfectly diffuse even under the theoretically ideal conditions for which the Kubelka-Munk model was created. This effect, which is in opposition to what one would intuitively expect, is caused by light escaping through the lower boundary of the medium, and causes errors in Kubelka-Munk reflectance calculations that can be up to 20% and more, even for a grammage of 80 g/m2. The magnitude of the error shows a strong dependence on the degree of absorption, with higher absorption giving greater error. This confirms previously reported problems with Kubelka-Munk for strongly absorbing media, and DORT2002 offers a partial explanation of these problems, as it can describe this effect and quantify the Kubelka-Munk errors. It is argued that DORT2002 could well be considered for increased understanding in cases where the level of accuracy of Kubelka-Munk reflectance calculations is not acceptable. A comprehensive list of advantages for the applied user of a model with higher dimensionality is supplied.

Place, publisher, year, edition, pages
Sundsvall: Mittuniversitetet, 2004
Series
Mid Sweden University licentiate thesis, ISSN 1652-8948 ; 6
Keyword
mathematical modeling, radiative transfer, solution method, numerical stability, speed, light scattering, light absorption, Kubelka-Munk, errors, reflectance calculations
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-5670 (URN)1961 (Local ID)91-87908-87-5 (ISBN)1961 (Archive number)1961 (OAI)
Presentation
(English)
Supervisors
Available from: 2008-09-30 Created: 2009-07-10 Last updated: 2009-07-10Bibliographically approved
2. Mathematical modeling and numerical tools for simulation and design of light scattering in paper and print
Open this publication in new window or tab >>Mathematical modeling and numerical tools for simulation and design of light scattering in paper and print
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This work starts with a real industrial problem - the perceived need for a moredetailed and more accurate model for light scattering in paper and print than theKubelka‐Munk model of today. A careful analysis transfers this problem into aphysical description of the phenomena involved. This is then given a mathematicalformulation, and a detailed analysis leads to numerical solution procedures forspecific sub problems. Methods from scientific computing make it possible to meetindustrial demands made on speed and stability, and implementation in computercode is then followed by analysis of accuracy and stability.A problem formulation and a solution method are outlined for the forwardradiative transfer problem. First, all necessary steps to arrive at a numericallystable solution procedure are treated, and then methods are introduced to increasethe speed by a factor of several thousands or millions compared to a naiveapproach. The method is shown to be unconditionally stable, though the problemwas previously considered numerically intractable, and systematic studies ofnumerical performance are presented.The inverse radiative transfer problem is given a least‐squares formulation, anddifferent solution methods are analyzed and compared. Specifically, a two‐phasemethod for estimation of the scattering and absorption coefficients and theasymmetry factor (σs, σa and g) is presented. A sensitivity analysis is given, and it isshown how it can be used for designing measurements with minimal impact frommeasurement noise.It is shown how the standardized use of Kubelka‐Munk and the d/0°instrument leads to errors, and that the errors arising from an over‐idealized viewof the instrument - due to the fact that instrument readings are incorrectlyinterpreted - can be larger than any errors inherent in the Kubelka‐Munk modelitself. It is argued that the measurement device and the simulation model cannot beviewed as separate instances, which is a widespread implicit practice in appliedreflectance measurements. Rather, given a measurement device, measurement datashould be interpreted through a model that takes into consideration the actualgeometry, function and calibration of the instrument.The resulting tool, DORT2002, is in all aspects the Next Generation Kubelka‐Munk, and provides a greater range of applicability, higher accuracy and increasedunderstanding. It offers better interpretation of measurement data, and facilitatesthe exchange of data between the paper and graphical arts industries. It opens forunderstanding of anisotropic reflectance and for the utilization of the asymmetryfactor to design anisotropy, and thereby for the design of different visualappearance or optical performance in new printed or paper products.

Place, publisher, year, edition, pages
Sundsvall: Mittuniversitetet, 2007. 32 p.
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 22
Keyword
mathematical modeling, radiative transfer, integro-differential equations, inverse problems, parameter estimation, solution method, numerical performance, light scattering, paper industry applications, Kubelka-Munk
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-5908 (URN)5026 (Local ID)978-91-85317-50-9 (ISBN)5026 (Archive number)5026 (OAI)
Public defence
(English)
Available from: 2008-09-30 Created: 2009-05-06 Last updated: 2009-07-13Bibliographically approved

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