 Wringing-film covariance model

A wringing-film covariance model specifies how uncertain effective contact-layer contributions vary together in a gauge block wringing measurement. Individual contact uncertainties do not determine the uncertainty of a stack unless their dependence is also specified.

Let the vector of contact differences be $\mathbf{\delta }=({\delta }_1,\dots ,{\delta }_n{)}^T$ and its covariance matrix be $\Sigma$. For a correction $\Delta ={\mathbf{c}}^T\mathbf{\delta }$, propagation givesThe coefficients encode the actual geometry and calibration subtraction. They need not all be positive. This is the general correlated-input law applied to wringing-film uncertainty.^[ref]

A simple original model writes ${\delta }_j=b+{ϵ}_j$, where one common contribution $b$ has variance $u_b^2$ and independent contact contributions have variance $u_{ϵ}^2$. ThenFor three equal-sign contacts with $u_b=3\mathrm{nm}$ and $u_{ϵ}=2\mathrm{nm}$, the variance is $93\mathrm{n}{\mathrm{m}}^2$. For a difference of two contacts with coefficients $(1,-1)$, the common contribution cancels and the variance is only $8\mathrm{n}{\mathrm{m}}^2$. These are chosen covariance structures, not universal properties of wrung surfaces.

Calibration and use can share only part of their contact history. If $n$ calibration contacts share effect $b_c$ and $n$ use contacts share effect $b_u$, their common-effect correction is $n(b_u-b_c)$. Its variance isPerfectly shared equal contributions cancel; independent preparation effects do not. Assuming either relation requires evidence about preparation, reference surfaces and measurement history.

Repeated readouts of one unchanged wring mostly probe readout scatter. Rewringing samples a different level of variability; changes of surface preparation or elapsed time may add another. The NIST handbook documents that contact layers depend on conditions and can evolve after thermal equilibration.^[ref] A nested wringing-repeatability study can help separate those levels, but repeatability data alone do not identify every shared calibration contribution.

Retain the covariance assumptions with the result and test whether the proposed matrix is a valid positive semidefinite matrix. A small combined uncertainty obtained by assigning cancelling correlations is meaningful only when those correlations describe the physical comparison.