An Approximate Linear Relationship for the Optical Response of Turbid Media

The optical response of a homogeneous turbid medium is often expressed analytically with the aid of the simplified theory of Kubelka and Munk. The equations derived from this theory yield the response A as an explicit function of absorption Kand scatter S. In practice, however, an inverted form is frequently needed, which would display absorption as an explicit function of A and S. No closed-form rigorous inversion of the response function A(S, K) is available, though approximate solutions exist. All direct inversions, analytical or empirical, are highly non-linear, even though for measurement and instrumentation purposes a linear characteristic is highly desirable. This paper shows that this can be achieved by using a logarithmic transform for the transmittance and a reciprocal one for the reflectance mode. Graphs are given for the coefficients of the linear approximation applicable over a wide range of the parameters S and K ; also shown is the mean error incurred in this range; it is nowhere in excess of a few per cent. The values displayed were obtained numerically using the least-square method of linear approximation. It is expected that the linearized form will be useful in many areas of applied densitometry. The advantages should apply not only to instrument design and measurement procedures, but also to theoretical studies aimed at practical situations.