Kubelka-Munk model

Various models have been proposed that seek, with varying degrees of success, to quantify the reflectance and the concentration. The best known is the Kubelka-Munk model (Kubelka and Munk, 1931) which is sometimes called the... 

Yoshimura HN, Pinto MM, Lima E, Cesar PF. Optical properties of dental bioceramics evaluated by Kubelka-Munk model. In Bose S (ed.) Biomaterials Science - Processing, Properties and Applications in. Hoboken John Wiley Sons 2013. p71-79. 

Diffuse reflectance FTIR spectra of the ground Mo03/Al203 catalysts were recorded on an FTIR instrument (Nicolet, Model 740, MCT detector). The microreactor in the flow system was replaced by an FTIR cell. The cell used a Harrick diffuse reflectance accessory (DRA-2CO) fitted with a controlled environmental chamber (HVC-DRP). Spectra (500 scans, 4 cm 1 resolution) were presented in Kubelka-Munk units and recorded at RT. 

The empirical modeling element indicates an increased emphasis on data-driven rather than theory-driven modeling of data. This is not to say that appropriate theories and prior chemical knowledge are ignored in chemometrics, but that they are not relied upon completely to model the data. In fact, when one builds a chemometric calibration model for a process analyzer, one is likely to use prior knowledge or theoretical relations of some sort regarding the chemistry of the sample or the physics of the analyzer. For example, in process analytical chemistry (PAC) applications involving absorption spectroscopy, the Beer s Law relation of absorbance vs. concentration is often assumed to be true and in reflectance spectroscopy, the Kubelka-Munk or log(l/P) relations are assumed to be true. 

FIGURE 4 Model for the derivation of the Schuster-Kubelka-Munk equation. 

The many-flux theory covers applications with all levels of optical thickness from one mathematical model. By using this model to determine absolute K and S values, the software does not have to define whether white is present in the formulation. All matching is done in one database. There is no need for separate packages that would use Kubelka-Munk single-constant, Kubelka-Munk two-constant, or Lambert-Beer mathematics. 

Hecht, H.G. A comparison of the kubelka-munk, rozen-berg, and pitts-giovanelli methods of analysis of diffuse reflectance for several model systems. Appl. Spectrosc. 1983, 37, 348-354. 

The Kubelka-Munk two flux model predicted significantly different magnitudes of photon flux within the layers of a coating than a model based on the Lambert-Beer law. Equations and calculation methods are described and results are given that illustrate the effect of substrate reflectance, layer thickness and the absorption and scattering of the layer components on the photon flux and the light absorbed at various levels within the coating. 

The Kubelka-Munkli.) two flux model appears to be superior to a single flux model based on the Lambert-Beer law as it is capable of taking into account the scattering by components in the layer and the reflectance of the substrate. The model can provide estimates of the amount of light absorbed at any point within the coating and predict how this varies with film thickness, concentration of materials and reflectance of the substrate. 

In the case of a diffuse reflectance measurement, in which one measures / = R, the Kubelka-Munk radiative transfer model can be employed to extract a [2,15,16]. 

The optical measmements of diffuse reflectance are dependent on the composition of the system. Several theoretical models have been proposed for diffuse reflectance, which are based on the radiative transfer theory, and all models consider that the incident hght is scattered by particles within the medium. The most widely used theory in photometric sensors is the Kubelka-Munk theory, in which it is assumed that the scattering layer is infinitively thick, which may, in practice, be the case with the chemical transducers utilized in photometric sensors. The absolute value of the reflectance R is related to the absorption coefficient K and the scattering coefficient S by the equation... 

The Kubelka-Munk (K-M) model is applied as a linearization function to signals with scattering and absorptive characteristics as often encountered in diffuse reflectance. This relationship is given as follows (from V. P. Kubelka and F. Munk, Z. Tech. Physik 12, 593,1931) ... 

Figure 3 Relation between calculated values of transmitted light, (Kubelka-Munk hyperbolic solution) and Ti-io (multilayer model solution) in the concentration range 0 to 2 A.U.

The Kubelka-Munk equation is based on a model where the reflectance properties... 

On diffuse irradiation, Eqs. (8.10) through (8.15) become much simpler since all terms with the factor (3/m - 2) vanish, j (3/m - 2)fiod/xo = 0. Helpwise, collimated irradiation under //o = 2/3 (ao = 48.2°) has the same effect, but only for weak absorption. With increasing absorption the light fluxes inside the sample deviate more and more from the condition of diffuse irradiation. It has been often shown that the two-flux model derived first by Schuster<30) and then by Kubelka and Munk(28) has formally the same analytical solutions as the Pi-approximation under diffuse irradiation. Kubelka... 

As several researchers have shown empirically, the use of —log(reflectance) can provide, analogous to a transmittance measurement, a linear relationship between the transformed reflectance and concentration, if the matrix is not strongly absorbing as can be found for many samples studied by near-infrared spectroscopy. This issue is presented in detail below. A different approach based on a physical model was considered for UV/VIS measurements and later also applied within the mid-infrared. A theory was derived by Kubelka and Munk for a simple, onedimensional, two-flux model, although it must be noted that Arthur Schuster (1905) had already come up with a reflectance function for isotropic scattering. A detailed description of theoretical and practical aspects was given by Korttim. The optical absorption... 

A number of models have been developed to describe in quantiialive terms the iiuensily of diffuse reflected radiation. The most widely used of these models was developed by Kubelka and Munk." Fuller and (iriffilhs in their discussion of this model show that the relative reflectance intensity for a pow-der t R ) is given by - ... 

Because the simplified solufion obfained by Kubelka is a two-constant equation and therefore experimentally testable, and because so many other workers derivations are derivable from Kubelka and Munk s work, their solution is the most widely accepted, tested and used. Other workers have derived solutions to the radiation transfer equation that are more complicated than these two-constant formulas. For example, a third constant has been added to account for different fractions of forward and back scattering [36]. Ryde [37,38] included four constants since a difference in the scattering between incident light and internally diffused light is assumed, while Duntley [39] developed a model with eight constants, as a difference between both the absorption and scattering coefficients due to incident and internally diffused radiation was assumed. However, none of these theories is readily applicable in practice, and therefore the treatment of Kubelka is most often applied. 

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