Mid Sweden University

Radiative transfer theory considers radiation in turbid media, and is used in a wide range of applications. This paper outlines a problem formulation and a solution method for the radiative transfer problem in multilayer scattering and absorbing media, using discrete ordinate model geometry. A selection of different steps is brought together. The main contribution here is the synthesis of these steps, all of which have been used in different areas, but never all together in one method. First all necessary steps to get a numerically stable solution procedure are treated, and then methods are introduced to increase the speed by a factor of several thousand. This includes methods for handling strongly forward-scattering media. The method is shown to be unconditionally stable, whilst the problem was previously considered numerically intractable.

The need for optical modeling of paper is obvious to provide connections between its optical response and the actual properties of the paper. It is argued that modern solution methods from radiative transfer theory could be considered instead of the Kubelka-Munk model, and a specific example, DORT2002, is tested. It is shown that Kubelka-Munk is a simple special case of DORT2002, and the two models and their coefficients are compared. A comprehensive list of advantages for the applied user of a model with higher dimensionality is supplied. It is shown, by the use of DORT2002, that when the medium has finite thickness, the light distribution deviates from the perfectly diffuse even under the theoretically ideal conditions for which Kubelka-Munk was created. This effect causes errors in Kubelka-Munk reflectance calculations that may 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 light absorption, with higher absorption giving greater error. DORT2002 can well be considered for increased understanding in cases where the level of accuracy of Kubelka-Munk reflectance calculations is not sufficient.

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.