Near-infrared spectroscopy is based on the absorptions of molecular bonds in the infrared range (IR). The absorption bands in near infrared (NIR) ranging from 700 to 2500 nm are overtones and combinations of the fundamental vibrational transitions in the center infrared range. In the case of transmission measurements of liquids, the Lambert-Beer law is used in spectroscopy:
In near-infrared spectroscopy (NIRS), solid materials are often examined in reflectance, which is why the Lambert-Beer law is not exactly applicable. For example, there is no fixed layer thickness in reflectance measurements where the light waves are passing through the sample. Precise formulations of correlations of measurements in diffuse reflectance were made by Mie, Kubelka and Munk, for example, although the significance of these formulations for practical near-infrared spectroscopy is low. The following formulation of the Lambert-Beer law results from practical experiences:
The current instruments are used to record spectra (Fig. 3) consisting of several hundred discrete data points (monochromators, diode-array instruments, FT instruments).
Figure 3
24 spectra of Pathlength of the light through the sample
To be able to use the information contained in a spectrum for the determination of ingredients, wavelength l - where the ingredient to be determined is absorbing - is searched for at first in the simplest type of calibration development. When this wavelength has been found, the absorption coefficient e l is calculated for this ingredient at this wavelength. When l and e l are known, concentration c of the ingredient in the new sample can then be calculated from the spectra in practical applications. The conversion of model equation 1 into the so-called inverse equation has gained acceptance:
In the simplest case of only one wavelength and an ingredient absorbing at this wavelength, b in equation 3 is the reciprocal value of e l * l from equation 2.
In the case of several ingredients with overlapping NIR absorption bands, the inverse model must be expanded. This is done by recording further wavelengths. Therefore, the determination of the required number n of wavelengths is added to the specified calibration development tasks (determination of the wavelength, calculation of the extinction coefficient). A NIRS calibration equation can then be described as follows:
This reformulation results in not all the absorbing ingredients having to be known by their concentration. It is possible, however, to produce a random sample of all the samples and to use it as a calibration sample. However, the drawback of this method is that cross correlations between ingredients can markedly affect the calibrations (artifacts in the calibration) and that the calibration samples must be representative for the later application.
This reformulation also leads to a further problem: multicollinearity between the spectral data considerably increases the estimation errors of regression coefficients (b). Multicollinearity of NIR spectra means that two or several wavelengths are not independent of each other, but are highly correlated. This is always the case in adjacent wavelengths, since the absorption of adjacent wavelengths increases or decreases jointly for all samples. High estimation errors of the regression coefficients are significant particularly in such cases where samples are determined in the later application for which extrapolation beyond the calibration range is required. The request for a representative calibration selection is the direct consequence from this knowledge.
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Legende
A l = Absorption
T l = Transmission
R l = Reflexion
c = Concentration of the ingredient
e l = extinction coefficient of the ingredient for wavelength l
l = Pathlength of the light trough the sample
Legende
A l = Absorption at wavelength l
c = Concentration of the ingredient
b = Regressionskoeffizient für die Wellenlänge l
bi = Regression coefficient for wavelength l
i = 1 ... n wavelength |