Kevin Dahm

Representative Layer Theory for Diffuse Reflectance

The fractions of light absorbed by and remitted from samples consisting of different numbers of plane parallel layers can be related with the use of statistical equations. The fractions of incident light absorbed (A), remitted (R), and transmitted (T) by a sample of any thickness can be related by an absorption/remission function, A(R,T): A(R,T) = [(1 - R)2 - T2]/R = (2 - A - 2R)A/R = 2A0/R0. Being independent of sample thickness, this function is a material property in the same sense as is the linear absorption coefficient in transmission spectroscopy. The absorption and remission coefficients for the samples are obtained by extrapolating the measured absorption and remission fractions for real layers to the fraction absorbed (A0) and remitted (R0) by a hypothetical layer of infinitesimal thickness. A sample of particulate solids can be modeled as a series of layers, each of which is representative of the sample as a whole. In order for the layer to be representative of the properties of the individual particles of which it is comprised, it should nowhere be more than a single particle thick, and should have the same void fraction as the sample; further, the volume fraction and cross-sectional surface area fraction of each particle type in the layer should be identical to its volume fraction and surface area fraction in the sample as a whole. At lower absorption levels, the contribution of a particle of a particular type to the absorption of a sample is approximately weighted in proportion to its volume fraction, while its contribution to remission is approximately weighted in proportion to the fraction of cross-sectional surface area that the particle type makes up in the representative layer.

Bridging the continuum-discontinuum gap in the theory of diffuse reflectance

A system of equations described by Benford relate the absorption and remission properties of a layer of a material to the properties of any other thickness of the material. R, the fraction of light remitted from an infinitely thick sample, may be calculated from Benfords equations by increasing the sample thickness until the total remission converges to its upper limit. The fractions of light absorbed (a0) and remitted (r0) by a very thin layer may be similarly calculated. The relationship A(r,t) = [(1 – r)2 – t2] / r = (2 – a – 2r) a / r = 2 a0 / r0 = (1 – R)2 / R describes an Absorption/Remission Function for the material as a function of a, r and t, the fractions of light absorbed, remitted and transmitted by a specified layer. This is a more general expression than the widely used Kubelka–Munk equation, but gives results equivalent to it for the case of infinitesimal particles.

Test of the representative layer theory of diffuse reflectance using plane parallel samples

Equations of Benford used in the Representative Layer Theory are able to describe spectroscopic remission from layered plane parallel samples (plastic sheets) quite effectively. Losses due to reflection directly back in the direction of the incident beam are a major cause of discrepancies. Non-compositional variation and experimental errors tended to produce linear changes in the absorption coefficient, with the remission coefficient being more drastically affected. The remission coefficients obtained experimentally, in general, vary inversely to the absorption coefficient, although, as predicted by theory, front surface reflectance causes a direct variation. In transflectance, the log(1/R) spectrum of the thinnest samples is the one that is most like the absorption coefficient curve, but the shape of the Kubelka–Munk (absorption/remission) spectra are less affected by sample thickness, especially in the absence of surface reflection.

Letter: Math pretreatment of NIR reflectance data: log(1/ R ) vs F(R)

The Kubelka–Munk function, F(R), is not commonly used as a pretreatment of NIR reflectance data. Reasons cited for this in the Handbook of Near Infrared Analysis point to the limitations of the Kubelka–Munk theory in predicting diffuse scattering data, especially for highly absorbing samples. However, the case for this causing a non-linear function of concentration when using F(R) has not been adequately made. A re-examination of data presented shows the non-linearity to be largely explained by the use of methods which could not be expected to produce linear data. Thus, it is as yet uncertain if, when properly applied, the advantages of increased linearity when using the Kubelka–Munk function will outweigh the advantage of relative insensitivity to referenced photometric changes when using log(l/R) followed by a derivative.

Obtaining material absorption properties from remission spectra of directly illuminated, layered samples

The effective linear absorption coefficient, K, obtained in transflection from directly illuminated samples made up of layers of thickness d, may be approximately related to the linear absorption coefficient, k, of the material making up the layers through the following empirical equations: k = [(1 – ½ exp(–2Kd)] K and K = [(1 + exp(–4kd)] k. The fractions of incident intensity absorbed or remitted by one layer may be modeled by assuming that that the light that moves through a sample has both the characteristics of directed and diffuse radiation.

Separating the effects of scatter and absorption using the representative layer

The physical interpretation of the absorbance observed in a non-scattering sample is straightforward; it is a simple function of the concentration of the analyte(s) and its (their) ability to absorb light. In a scattering sample, the phenomena of absorption and scattering affect each other. Consequently, the measured “absorbance” is more difficult to interpret and is not suitable for direct comparison to an “absorbance” obtained from a non-scattering sample. This paper describes a strategy for separating the effects of scatter and absorption. The amount of light absorbed by a sample can be determined by measuring both the amount of light remitted and the amount transmitted by the sample. Using the mathematics of plane parallel layers, it is possible to model the sample as a series of “layers” of any thickness and calculate the absorption, remission and transmission for each of these hypothetical layers. The absorption computed for a layer having a thickness of one particle, which we term the “representative layer”, can be used to benchmark the absorbance that would be observed from the sample in the absence of scatter.

Illustration of failure of continuum models of diffuse reflectance

Continuous models are widely used to model diffuse reflection from particulate samples. They are embodied in various theories, including diffusion theory, the equation of radiation transfer, and Kubelka-Munk. Using a simple system, it is shown that such models do not simultaneously predict the fractions of incident light that are transmitted directly and diffusely. Discontinuous models do have the capability to do so, but have a variety of limitations as well.

Speaking Theoretically … Understanding Confusing Phenomena in Remission Spectra

T ake a look at Figure 1. What can you say about the relationship of the two samples? I imagine that you would say that Sample 1 (spectrum in bold) has a larger particle size than Sample 2, and that there is more of the material with characteristic absorptions around 1750 nm (marked by large arrow) in Sample 2. We’ll discuss first why we like your answer, and then we’ll tell you why you’re wrong. We are going to try to use a few different approaches in the hope of finding one you can relate to.

Comparison of the Use of Volume Fractions with other Measures of Concentration for Quantitative Spectroscopic Calibration Using the Classical Least Squares Method

Since the commercial development of modern near-infrared spectroscopy in the 1970s, analysts have almost invariably used units of weight percent as the measure of analyte concentration, due largely to the historical precedent from other analytical methods, including other spectroscopic techniques. The application of the CLS algorithm to a set of binary and ternary liquid mixtures reveals that the spectroscopic measurement sees the sample differently; that the measured absorbance spectrum is in fact sensitive to the volume fraction of the various components of the mixture. Because there is not a one-to-one relationship between volume fraction and other measures of analyte concentration, nor is the relationship linear, this has important implications for the application of both the CLS algorithm and the various other, more conventional, calibration algorithms that are commonly used.

A freshman level module in biometric systems

It is very important and challenging to teach modern topics at the freshman level. This paper describes a biometrics module that fits into any introductory freshman engineering course. The project has broad learning outcomes, namely, enhanced application of math skills, software implementation skills, interest in biometrics and comprehension of ethical issues. Assessment results based on the analysis of student surveys related to the learning outcomes and vertical integration show that the project was successful.