Quantifying Formation: Variance Ratio

The only well-defined reference structure for paper is a random network made from the same fibres, such as that shown in Figure 7.

The variance ratio of a paper to its random counterpart is called the formation number or variance ratio of the paper:

$$n_{f}(x) = \frac{Var(x)}{Var^{Random}(x)}.$$ (28)

Usually we find n_f(x) approximately linear with inspection zone size x, slope $\approx n_f(1mm).$ See Farnood et al. [16] for further details and the website [7] for examples of radiographs of papers.

The usefulness of n_f(x) being approximately linear with inspection zone size x lies in the fact that it is easier to characterize structures if a linearization procedure is available, which is not the case for example for formation spectra. See Norman and Wahren [29] for details of the spectral analysis of paper structure.

See Kiviranta and Dodson [23] for correlation of n_f with headbox and Fourdrinier table parameters.