Shared-reference cancellation in transferred blank corrections

Shared-reference cancellation in transferred blank corrections occurs when the same numerical reference enters a correction with opposite coefficients. It is an algebraic property of the measurement equation. Its uncertainty counterpart is a covariance contribution, not permission to discard arbitrary background uncertainties.

Let a specimen reading be $T$, a source blank $A$ and a target blank $B$. If the estimated transfer is $D=B-A$, the corrected result isThe source blank cancels exactly. Even when $T,A,B$ are independent, $A$ and $D$ are not: $\mathrm{cov}(A,D)=-\mathrm{Var}(A)$. Applying the correlated-input rule givesThe GUM requires correlations from shared inputs to be retained and describes equivalent formulations in which the common input is introduced explicitly.^[ref]

For stipulated independent standard deviations $u(T)=1$ mg, $u(A)=2$ mg and $u(B)=3$ mg, $u^2(D)=13\mathrm{m}{\mathrm{g}}^2$ and $\mathrm{cov}(A,D)=-4\mathrm{m}{\mathrm{g}}^2$. Thus $u^2(Y)=1+4+13-8=10\mathrm{m}{\mathrm{g}}^2$. Incorrectly treating $T,A,D$ as independent gives 18 instead. The same result follows from the covariance matrixand coefficient vector $(1,-1,-1{)}^{\mathsf{T}}$. These numbers illustrate error accounting, not a measured carbon-recovery test.

Reference identity in algebraic uncertainty cancellation is essential. If a transfer used an earlier reading $A_1$ but the correction subtracts a later reading $A_2$, then $Y=T-B+A_1-A_2$. Their difference does not vanish merely because both were called the source blank. Correlation may reduce or increase its uncertainty depending on how the readings were obtained.

Preserve reference identifiers, timestamps and coefficients in the data reduction. Simplifying the measurement equation before propagating uncertainty can expose exact cancellations; a transfer estimated independently or containing additional drift needs its own remaining uncertainty terms.

Reference identity in algebraic uncertainty cancellation asks whether two appearances of a reference are the same measured quantity or merely readings given the same descriptive name. If a transfer is $D=B-A_1$ but a correction subtracts $A_2$, thenThe last difference vanishes identically only when the same reference value is reused. Separate acquisitions may be correlated, independent, or affected by drift; sharing a label does not decide their covariance.

For independent $T$ and $B$, also independent of the reference pair, propagation givesWith equal reference standard uncertainties $s$ and correlation coefficient $\rho$, the residual reference contribution is $2s^2(1-\rho )$. It is zero for identical reused values, $2s^2$ for independent readings, and is not generally zero just because the displayed numbers agree. The GUM requires significant correlations from common inputs to be included.^[ref]

An original acquisition model separates what cancels:Here $C$ is a common uncertain calibration contribution, while ${ϵ}_1,{ϵ}_2$ are independent repeat errors. Their difference removes both $A_0$ and $C$, but leaves ${ϵ}_1-{ϵ}_2$. If $u(C)=3$ mg and each repeat error has standard deviation 2 mg, each reading has variance $13\mathrm{m}{\mathrm{g}}^2$ and their covariance is $9\mathrm{m}{\mathrm{g}}^2$. The difference therefore has variance $13+13-18=8\mathrm{m}{\mathrm{g}}^2$, not zero.

Adding independent $u(T)=1$ mg and $u(B)=3$ mg gives $u^2(Y)=18\mathrm{m}{\mathrm{g}}^2$. Reusing exactly one recorded reference value in both places instead gives $u^2(Y)=10\mathrm{m}{\mathrm{g}}^2$. These constructed cases have different data-reduction histories despite similar-looking equations.

Reference lineage in covariance models records acquisition identifiers, shared calibration inputs and coefficients in each derived quantity. A calibration change or time-dependent drift must appear in that model if significant. Algebraic simplification is valuable only when it preserves the identity of the underlying inputs; renaming two acquisitions with the same symbol does not make their errors cancel.

Reference lineage in covariance models records which primitive observations and calibration quantities enter each derived measurement, with what coefficients. Two inputs can share the same reference yet respond to it differently. A shared name alone therefore does not establish either equal errors or exact cancellation.

For a linear measurement model, collect primitive inputs in $\mathbf{z}$, derived inputs in $\mathbf{x}=A\mathbf{z}$ and their covariance matrices in $U_z,U_x$. ThensoThe matrix records the data lineage as well as the algebra. This implements the GUM approach of expressing correlated quantities in terms of their common inputs.^[ref]

An original example uses independent inputs $T,B,C,{ϵ}_1,{ϵ}_2$ with variances $1,9,9,4,4\mathrm{m}{\mathrm{g}}^2$, respectively. Let two reference readings be $A_1=C+{ϵ}_1$ and $A_2=2C+{ϵ}_2$. Correct a specimen by $Y=T-B+A_1-A_2$. Substitution givesThe shared $C$ does not cancel, because its sensitivities are one and two. Equivalently, $\mathrm{Var}(A_1)=13$, $\mathrm{Var}(A_2)=40$ and $\mathrm{Cov}(A_1,A_2)=18$ in squared milligrams. Their difference has variance $13+40-2(18)=17$, giving the same total 27. Treating the reference readings as independent would incorrectly give 63 for the total variance.

If both readings instead contained $C$ with coefficient one, that shared contribution would cancel and the total would be 18 in the same units. These are different acquisition models, not two uncertainty conventions for the same data.

A usable lineage record therefore includes acquisition identifiers, calibration versions, reuse of numerical values and sensitivity coefficients. For nonlinear data reduction, a local sensitivity matrix replaces the linear $A$ when first-order propagation is justified. Covariance must follow the actual reuse and dependence structure; assigning independence because derived columns have different labels can discard the information needed for correct cancellation.