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.

Unequal gauge-block reference sensitivities occur when two length estimates respond differently to the same uncertain reference value. A reference contribution cancels from their difference only when its coefficients agree. Sharing a calibration source is necessary information, but does not by itself guarantee cancellation.

Let estimates have measurement models $X=f_X(R,\dots )$ and $Y=f_Y(R,\dots )$. At the stated input values define sensitivity coefficients $a_X=\partial f_X/\partial R$ and $a_Y=\partial f_Y/\partial R$. The reference contribution to $D=X-Y$ is, to first order,This follows either by differentiating the contrast directly or by retaining shared gauge-block reference covariance in the two-output propagation. The general law includes both sensitivities and covariance.^[ref]

An original ratio-transfer model illustrates the difference: $X=q_{X}R+d_X$, $Y=q_{Y}R+d_Y$, with fixed known $q_X,q_Y$. If $R=100\mathrm{mm}$, $u(R)=20\mathrm{nm}$, $q_X=1$ and $q_Y=2$, their length difference retains a 20 nm reference contribution. The reference error is shared, but appears twice as strongly in $Y$. Treating the reference contributions as independent would instead give $\sqrt{20^2+40^2}\approx 44.72\mathrm{nm}$.

The contrast $2X-Y$ cancels this model's reference term because $2q_X-q_Y=0$. It is a different measurand from $X-Y$; cancellation is not permission to replace the quantity the experiment intended to determine. More generally, $c_{X}X+c_{Y}Y$ retains reference variance $(c_X{a}_X+c_Y{a}_Y{)}^2{u}^2(R)$.

Nearly matched sensitivities can provide strong suppression. With $a_X=1$, $a_Y=0.98$ and the same 20 nm reference uncertainty, the residual reference contribution is 0.4 nm. That does not imply the total uncertainty is 0.4 nm: comparison scatter, coefficient uncertainty, temperature and wringing contributions remain.

If the transfer ratios are themselves uncertain, their derivatives introduce terms proportional to the reference length, along with any shared covariance. Determine them from the actual calibration model rather than assigning ideal coefficients solely to obtain cancellation. NIST separates reference-master history from comparator-process variability in its uncertainty calculation.^[ref] A reference-sensitivity cancellation check should accompany a claimed low-uncertainty contrast.

A reference-sensitivity cancellation check determines whether a proposed comparison really removes the influence of a shared calibration reference. It checks the measurand, the reference's sensitivity coefficients and the range over which the claimed cancellation applies.

For estimates $X=f_X(R,\dots )$ and $Y=f_Y(R,\dots )$, a contrast $D=c_{X}X+c_{Y}Y$ has local reference sensitivityIts first-order reference variance is $s_R^2{u}^2(R)$. Checking the contrast directly and checking the full covariance propagation should agree. The GUM defines the sensitivities and retains correlations between inputs.^[ref]

An original linear model is $X=R+d_X$ and $Y=1.02R+d_Y$. For $D=X-Y$, the sensitivity is $-0.02$. Recomputing the unchanged comparison with the assigned reference value increased by $1\mu \mathrm{m}$ changes $D$ by $-0.02\mu \mathrm{m}$. With reference standard uncertainty 20 nm, its contribution to this contrast is 0.4 nm. Treating the two uses of $R$ as independent would miss this suppression.

A useful computational check perturbs one shared input consistently in every place it occurs. Perturbing its occurrences independently tests a different model. Finite-perturbation reference sensitivity also reveals whether a derivative evaluated at one point adequately describes the actual uncertainty range.

Zero first derivative is weaker than exact invariance. For an illustrative nonlinear model, take $X=R+d_X$ and $Y=R+\beta (R-R_0{)}^2+d_Y$. The derivative of $X-Y$ vanishes at $R_0$, but a reference perturbation $\delta R$ leaves a term $-\beta \delta R^2$. If $\delta R$ is zero-mean normal with standard deviation $s$, that term has mean $-\beta s^2$ and standard deviation $\sqrt{2}|\beta |s^2$. The first-order cancellation has not removed the nonlinear contribution.

With stipulated $\beta =1\mathrm{m}{\mathrm{m}}^{-1}$ and $s=0.001\mathrm{mm}$, these are $-1$ nm and about 1.41 nm. These are model coefficients, not measured comparator behaviour.

Finally, verify that the chosen contrast is the quantity the experiment intends to determine. Replacing $X-Y$ by another combination solely to cancel a reference changes the question. Retain calibration lineage and independent checks on comparator response; NIST's physical uncertainty budget distinguishes the master history from comparison-process variability.^[ref]

Finite-perturbation reference sensitivity measures the change in a comparison when one shared reference value is changed by a finite amount everywhere it occurs. It checks both local sensitivity coefficients and their usefulness over the actual reference-uncertainty range. The reference-sensitivity cancellation check is incomplete if it tests only a derivative at one point.

Let $F(R)$ be the final comparison, with all other independent inputs held fixed. A central finite difference and a second difference areFor sufficiently smooth $F$, these approach $F^{\prime }(R_0)$ and $F^{\prime \prime }(R_0)$, with errors of order $h^2$ when the required derivatives exist. Their units differ: comparison per reference unit and comparison per squared reference unit. The GUM defines sensitivity coefficients as partial derivatives and also discusses determining them by varying an input experimentally.^[ref]

An original model writes $x=R-R_0$ andIt gives $C_h=ϵ+\kappa h^2$ and $Q_h=2\beta$. Comparing $C_h$ with $C_{h/2}$ can expose cubic contamination of the local slope. But if $ϵ=\kappa =0$, every central slope is exactly zero although $F(R_0+h)-F(R_0)=\beta h^2$ is nonzero. A symmetric slope check alone misses an even-in-perturbation response.

For $\beta =1\mathrm{m}{\mathrm{m}}^{-1}$ and $h=0.001\mathrm{mm}$, the one-sided change is 1 nm, while $C_h=0$ and $Q_h=2\mathrm{m}{\mathrm{m}}^{-1}$. If the shared reference error instead has a zero-mean normal distribution with standard deviation $s=h$, the quadratic contribution has mean 1 nm and standard deviation approximately 1.414 nm. These are original model calculations. They show why exact cancellation, zero local slope and negligible finite-range influence are distinct claims.

Use step-size stability of a reference perturbation to select a sensible test range. Extremely small steps can magnify numerical subtraction errors; large steps probe a different region of a nonlinear response. Test several scales near the physical standard uncertainty and inspect one-sided changes as well as central slopes. If the model is stochastic, separate simulation noise from the response to the changed reference.

A computational perturbation changes the assigned shared reference in every dependent calculation while keeping its identity shared. Independently perturbing two uses of the same reference changes the covariance model. A physical experiment must additionally hold other influences stable, because changing a standard or its temperature may change contact or instrument behaviour. NIST separates master-block uncertainty from the comparison process in its uncertainty budget.^[ref] Retain those physical distinctions when interpreting a numerical sensitivity test.

Step-size stability of a reference perturbation checks whether a numerical reference-response estimate behaves consistently as the applied step changes. For a final comparison $F(R)$, a central slope isFor sufficiently smooth $F$, its leading truncation error is proportional to $h^2$. Comparing several steps tests that numerical behavior, while choosing steps near the physical reference uncertainty also probes whether a local derivative adequately represents the measurement model.^[ref]

For an original model, let $x=R-R_0$ and $F(R)=D_0+0.2x+\kappa x^3$, with $\kappa =100\mathrm{m}{\mathrm{m}}^{-2}$. For $h=0.010,0.005,0.0025\mathrm{mm}$, the central slopes are0.210,0.2025 and0.200625. Their successive departures from0.2 shrink by four as $h$ halves. Richardson extrapolation removes the cubic contribution exactly in this model:For a general smooth response, it removes the leading $h^2$ term rather than proving exact cancellation of every nonlinear contribution.

Small steps also amplify output error. If each evaluation has absolute error at most $q$, the error in the central slope is at most $q/h$. With $q=0.1\mathrm{nm}=10^{-7}\mathrm{mm}$, this bound is $10^{-5}$ at $h=0.01\mathrm{mm}$ but0.1 at $h=10^{-6}\mathrm{mm}$. The cubic truncation term decreases with $h$, whereas this evaluation-error bound increases. Equating their orders, $|\kappa |h^2\sim q/h$, suggests the scale $h\sim (q/|\kappa |{)}^{1/3}$, here0.001mm. This scale follows from the stipulated error model, not a universal finite-difference setting.

An unchanging central value is also insufficient to establish differentiability. For $F(R)=|R-R_0|$, every $C_h$ is zero, while the forward and backward slopes are $+1$ and $-1$. Inspecting both one-sided changes reveals the cusp. Likewise, a stable local slope does not rule out a quadratic finite-range response, as finite-perturbation reference sensitivity shows.

A meaningful test changes one assigned shared reference consistently in every dependent calculation, holds the other inputs and computational settings fixed, and records the steps, units, slopes and one-sided changes. Shared reference uncertainty and comparison-process variability are distinct physical contributions in the NIST uncertainty budget.^[ref] If evaluation includes simulation noise, separate its repeatability from the response to the changed reference.