Inverse Problem in Diffuse Reflectance Spectroscopy: Accuracy of the Kubelka-Munk Equations

Abstract

In diffuse reflectance spectroscopy the Kubelka-Munk equations have been used extensively. These equations provide simple solutions to the inverse problem of obtaining information on the scattering and absorption cross sections from reflected light. Proof is provided that the basic Kubelka-Munk equation

$\frac{\text{d}r(x)}{s\text{d}x}=r^2(x)-2ar(x)+1$

should be replaced by the equation

$\frac{\text{d}r(x)}{s\text{d}x}=r^2(x)-2(2a-1)r(x)+1$

and that the Kubelka-Munk function

$\frac{k}{s}=\frac{{(1-R_{\infty })}^2}{2R_{\infty }}$

should be replaced by the function

$\frac{k}{s}=\frac{{(1-R_{\infty })}^2}{4R_{\infty }}.$

Here r(x) is the reflectance; s is the scattering cross section (cm⁻¹); a = (k + s)/ s, where k is the absorption cross section (cm⁻¹); and R_∞ is the reflection coefficient of an infinitely thick sample. We note, however, that because of a redefinition of a carried out by Kubelka and Munk in the process of their calculations, the scattering cross section s calculated from their expression

$\begin{array}{l}sd=\frac{1}{\frac{1}{2}\left(\frac{1}{R_{\infty }}-R_{\infty }\right)}({coth}^{-1}\frac{\frac{1}{2}\left(\frac{1}{R_{\infty }}+R_{\infty }\right)-r(d)}{\frac{1}{2}\left(\frac{1}{R_{\infty }}+R_{\infty }\right)}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{coth}^{-1}\frac{\frac{1}{2}\left(\frac{1}{R_{\infty }}+R_{\infty }\right)}{\frac{1}{2}\left(\frac{1}{R_{\infty }}+R_{\infty }\right)})\hfill \end{array}$

is correct. But the Kubelka-Munk theory still overestimates k by a factor of two.