Appendix B

To calculate the K and S parameters for a given toner set, the exponential form of the Kubelka-Munk equation may be used. This equation predicts the reflectance R_i⁰ at a given wavelength: ³

where R_p is the reflectance of the substrate or backing, R¥ is the reflectance of an infinitely thick sample, S is the scattering coefficient at the given wavelength, and X represents the toner mass.

This can be rewritten as:

where

The mean square error of the reflectance predicted with the Kubelka-Munk model versus the measured reflectance for a given sample may be defined as follows, where R_i is the measured reflectance:

R_p and R¥ can be determined by measuring the reflectance of the bare substrate and the toner pellet, respectively, and applying the Saunderson correction.

To solve for the S value at a particular wavelength which produces the lowest overall mean square error, E², the derivative of the mean square error with respect to S is set equal to zero. That is:

Set the derivative equal to 0 to minimize E² with respect to S.

Differentiating R_i⁰ in equation 2b with respect to S:

Substituting equation 7b into equation 6b

The variables R_p, R¥ , X_i, and b are all known quantities. The scattering coefficient S, which produces the lowest overall mean square error for the entire range of toner masses, for the given wavelength, can be determined from equation 8b using an iterative numerical procedure.

The following method can be used to obtain a good starting value for the iterative procedure to determine the S coefficient. The method avoids solving Equation 8b for inaccurate S values that may be obtained from local minima with respect to E². The following method determines the S coefficient for each wavelength that produces the overall minimum error.

Start with eq.(25) from Kubelka:¹

Substitute the following equations:

Substitution yields:

At this point, a initial starting value for S may be determined from defining a y axis in the following manner:

If the above relationship is plotted as a function of X, where X is the sample TMA, the slope of the line can easily be solved for. From the slope and equation 14b, the initial guess of the S coefficient, S₀, for the wavelength of interest is as follows:

While this method produces a very good starting value, it is not suitable for the final fit, because of very low signal-to-noise ratio as R_i® R¥ . So the data has to be limited to high R_i. High R_i occur at low toner masses at absorbing wavelengths, and at all toner masses at non-absorbing wavelengths. Therefore, a criterion was set such that R_i-R¥ _³ D , where D is some arbitrary positive fraction.

When the variable S is solved for, the following relationship between K and S can be used to determine K¹:

The absorption value, K, for each wavelength is therefore defined as: