 Shared gauge-block reference covariance

Shared gauge-block reference covariance is the dependence between calibrated length estimates caused by using the same uncertain reference. It explains how a gauge-block reference uncertainty floor can remain in individual absolute lengths while canceling from their difference.

Let corrected estimates be $X=R+d_X$ and $Y=R+d_Y$. Suppose the shared reference error has variance $u_R^2$, the comparison errors have variances $u_X^2,u_Y^2$, and these three contributions are independent. The covariance matrix isIn particular, $\mathrm{cov}(X,Y)=u_R^2$. The covariance follows from the shared input, rather than from whether the two comparison readings happened at the same instant.

For the contrast $D=X-Y$,Both reference sensitivities are exactly one in this additive measurement model, so their contributions cancel. General uncertainty propagation includes precisely this covariance term.^[ref]

An original example uses $u_R=20\mathrm{nm}$, $u_X=10\mathrm{nm}$ and $u_Y=15\mathrm{nm}$. The individual standard uncertainties are about 22.36 and 25 nm, while the difference has standard uncertainty $\sqrt{100+225}\approx 18.03\mathrm{nm}$. Incorrectly treating the absolute length estimates as independent would give $\sqrt{500+625}\approx 33.54\mathrm{nm}$ for the same difference.

For any linear combination $Z=c_{X}X+c_{Y}Y$, the reference contribution is $u_R^2(c_X+c_Y{)}^2$. It vanishes for a difference but remains $u_R^2$ in an equal-weight average, where $c_X=c_Y=1/2$. Repeatedly averaging values carrying the same reference error does not produce independent reference information.

Unequal gauge-block reference sensitivities change the cancellation. If the reference enters as $a_{X}R$ and $a_{Y}R$, a difference retains reference variance $(a_X-a_Y{)}^2{u}_R^2$. Shared temperature, comparator or wringing contributions also need their own coefficients and dependence assumptions. NIST's uncertainty procedure separates the master history from comparison-process variability.^[ref]

Retain the calibration lineage and input relationships with the estimates. A pair of certificate uncertainty numbers alone does not establish their covariance or justify setting it to zero.