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Unequal-noise allocation across calibration blocks distributes repetitions using block-specific variability while keeping the intended estimator fixed. Suppose independent blocks supply means ${\overline{Y}}_j$ and the target estimate is $\hat{\theta }={\sum }_j{w}_j{\overline{Y}}_j$ with specified weights $w_j$. Let one observation in block $j$ have variance ${\sigma }_j^2$ and equicorrelation $0\le {\rho }_j<1$ within that block. ThenThe first sum is a block-specific floor; only the second decreases with repetition count. Changing the estimator weights merely to improve this calculation could change the estimand, so they remain fixed here.

For continuous positive allocations with positive costs $c_j>0$, budget ${\sum }_j{c}_j{n}_j=C$ and $a_j=w_j^2{\sigma }_j^2(1-{\rho }_j)>0$, a Lagrange-multiplier calculation givesFor equal observation costs, repetitions are proportional to $|w_j|{\sigma }_j\sqrt{1-{\rho }_j}$, not to the variance itself. Minimum counts and integer rounding must subsequently respect the actual acquisition constraints.

For an original two-block example, take $w_1=w_2=1/2$, equal costs, $C=100$, independent readings within each block, and standard deviations $1$ and $3$ in the output unit. The optimal counts are $25$ and $75$, giving variance $0.04$ instead of $0.05$ from fifty each. If both within-block correlations are instead $0.1$, the same proportions give variance $0.286$, while equal counts give $0.295$. The common-error floor $0.25$ now dominates and cannot be eliminated by taking more observations inside those same blocks.

This extends optimal allocation of detector repetitions across calibration blocks, whose balanced solution used homogeneous noise. For binary detector indicators, the per-reading variance is $p_j(1-p_j)$; differing probabilities require a clearly specified weighted target. The planning variances and correlations must come from an appropriate measurement calibration model, not merely from counting files as independent blocks. NIST's uncertainty propagation retains shared-input covariance for exactly this reason.^[ref]

The variance of a corrected heterodyne overlap estimate quantifies uncertainty in $\hat{\gamma }=B/(2\sqrt{I_1{I}_2})$, where $B$ is corrected peak beat current and $I_1,I_2$ are separately measured arm currents. The consistency bound for separately measured heterodyne arm powers requires the physical overlap not to exceed one, but measured estimates fluctuate.

For small errors, let $\mathbf{e}=(\delta B/B,\delta I_1/I_1,\delta I_2/I_2{)}^T$ have covariance $R$, and put $\mathbf{v}=(1,-1/2,-1/2{)}^T$. Linearization givesThus common influences can cancel rather than contribute three independent uncertainty terms. This is the covariance form of the primary measurement-propagation rule.^[ref]

For an original model, all three current readings share a $2\%$ multiplicative calibration error, with additional independent $0.5\%$ errors on each. The common contribution to ${\mathbf{v}}^T\mathbf{e}$ is zero. The relative standard uncertainty is only $\sqrt{3/2}\times 0.5\%\approx 0.6124\%$, or $0.00490$ at overlap $0.80$. Treating the common calibration error as independent in all three readings would instead give about $2.525\%$ relative uncertainty.

The common-gain cancellation in heterodyne overlap uncertainty is exact for a single positive gain $g$: replacing $(B,I_1,I_2)$ by $(gB,g{I}_1,g{I}_2)$ leaves the ratio unchanged. A separate radio-frequency gain uncertainty affecting only $B$ does not cancel, nor does an additive dark-current offset generally have this property.

NIST's heterodyne receiver procedure corrected electrical loss and impedance effects separately from matching optical powers and modes.^[ref] Build the covariance from those actual shared and separate calibration paths. This first-order variance model also needs reconsideration when an arm current is close to zero or background subtraction makes its uncertainty comparable to its value.

Vibroscope area uncertainty quantifies uncertainty in an area inferred from a vibration model and its measured inputs. For the flexible-string vibroscope area estimator, length and frequency enter with inverse-square dependence, whereas tension and density enter to the first power. The parent article gives the full first-order propagation formula. A measurement uncertainty budget records the input standard uncertainties, their sensitivities and any correlations.^[ref]

For an original independent-input example, stipulate relative standard uncertainties of $1\%$ in tension, $2\%$ in density, $0.5\%$ in length and $0.1\%$ in frequency. After applying the sensitivity coefficients, their relative area contributions are $1\%,2\%,1\%,0.2\%$. Combining their squares givesFor a stipulated estimate $\hat{A}=80\mu {\mathrm{m}}^2$, this is $u_A\approx 1.97\mu {\mathrm{m}}^2$. Density contributes $4/6.04\approx 66.2\%$ of the combined variance. Improving the frequency readout alone would therefore make little difference to this budget. Reducing the density contribution from $2\%$ to $1\%$ instead gives $\sqrt{3.04}\%\approx 1.74\%$ overall.

These input uncertainties are assigned model values, not the readout resolutions quoted for the Favimat+ fibre testing machine. A resolution increment does not determine a standard uncertainty without a model for the associated error. Correlated inputs require covariance terms; their signed sensitivities can either reinforce or cancel contributions.

The budget also presumes an adequate measurement model. Bending-stiffness bias in the vibroscope area estimator, uncertain supports or deformation conventions may require a correction and additional uncertainty terms. Increasing the numerical precision of the frequency does not repair a wrong physical model.