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Correlated spectral endpoint error means that the errors in two estimated frequencies have nonzero covariance. Their effect on a frequency difference depends on that covariance as well as on the uncertainty of each endpoint. Shared errors can cancel in a difference without making the absolute frequencies accurately known.

Let endpoint errors be ${ϵ}_0,{ϵ}_T$, with standard uncertainties $u_0,u_T$ and correlation coefficient $\rho$. The error in the increment is ${ϵ}_T-{ϵ}_0$, soFor equal uncertainties $u$, the result is $2u^2(1-\rho )$. Independent errors give $2u^2$, perfectly shared errors give zero, and perfect anticorrelation gives $4u^2$. This is correlated-input uncertainty propagation applied to a difference.^[ref]

An original common-offset model writes ${ϵ}_0=b+{\eta }_0$ and ${ϵ}_T=b+{\eta }_T$. Suppose $b$ has variance $s_b^2$ and the two independent residual errors each have variance $s^2$, all mutually independent. Each endpoint then has variance $s_b^2+s^2$ and their covariance is $s_b^2$, butWith $s_b=0.3$ and $s=0.1$ in compatible frequency units, each endpoint has standard uncertainty about $0.316$, while the increment error has standard uncertainty about $0.141$. This spectral common-mode offset rejection follows from subtraction, rather than from knowing the common offset's value.

A shared gain error behaves differently. If $Y_t=(1+g){\omega }_t+b$, the constant offset cancels but $Y_T-Y_0=(1+g)({\omega }_T-{\omega }_0)$. A shared $1\%$ scale error therefore remains a $1\%$ increment error. Calling an error common does not establish which part cancels.

To estimate covariance, repeated measurements should preserve the reference and data-processing relationships relevant to the two endpoints. Filtering or overlapping fitting windows can introduce correlations. The spectral frequency-increment measurement noise contribution also depends on covariance between physical fluctuations and estimation errors; the endpoint formula alone does not justify dropping that separate term. A two-dimensional infrared spectroscopy statistic should be interpreted through its actual estimator and response model, not by automatically adding independent noise to two abstract frequency labels.

JCGM100:2008, GUM1995 with minor corrections; first edition September2008. Sections5.1–5.2 give first-order combination of standard uncertainties for independent and correlated inputs; covariance terms enter through the measurement-model sensitivity coefficients.

Evaluation of measurement data — Guide to the expression of uncertainty in measurement, JCGM100:2008, GUM1995 with minor corrections, first edition September2008. Sections5.1–5.2: sensitivity coefficients and propagation of uncertainty; equation(13): correlated-input covariance contribution. Sections5.2.4–5.2.5 explain common-instrument/reference-data correlations and require significant covariances to be included.

The PDF title page also identifies the corrected version2010. Sections5.1.3–5.1.4 define sensitivity coefficients and describe their experimental determination. Section5.2.2, equations13–16, propagates correlated inputs; §5.2.4 identifies common instruments, standards and reference data as correlation sources.

Official ISO-hosted text of Chapter 5: www.iso.org/sites/JCGM/GUM/JCGM100/C045315e-html/C045315e_FILES/MAIN_C045315e/05_e.html . Sections 5.2.2–5.2.4 retain covariance contributions and identify common instruments, measurement standards and reference data as sources of correlation; Annex F.1.2.3 gives practical shared-reference cases.

Official ISO-hosted Chapter 4: www.iso.org/sites/JCGM/GUM/JCGM100/C045315e-html/C045315e_FILES/MAIN_C045315e/04_e.html . Section 4.1.2 requires significant correction/influence quantities in the measurement model and discusses incomplete modeling; section 4.2.5 treats standard uncertainties of least-squares fit parameters.

Official primary PDF: www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf . JCGM 100:2008, corrected version 2010, Section 5.2 gives correlated-input uncertainty propagation; Annex F.1.2 discusses correlations from shared inputs. These supply the covariance framework for the original equal-offset cancellation and unequal-sensitivity examples in spectral common-mode offset rejection.

JCGM 100:2008, Annex E.3.3 distinguishes propagation of standard deviations from propagation of confidence intervals; the note to section 5.1.2 discusses significant nonlinearity and higher-order terms. Official primary PDF: www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf .

Spectral common-mode offset rejection is cancellation of an additive frequency-axis shift when the same shift enters two spectral estimates with equal sensitivity and the estimates are subtracted. It improves a frequency difference without establishing accurate absolute frequencies. The relevant measurement model must identify what is shared, and over which observations.

For true frequencies ${\omega }_1,{\omega }_2$, write measured estimates asThe unknown common offset $b$ disappears exactly fromFor independent residual errors of standard uncertainty $s$, the difference still has standard uncertainty $\sqrt{2}s$. Cancellation of $b$ is not cancellation of all measurement noise. With correlated residuals, the covariance term must be retained in uncertainty propagation.^[ref]

An original example uses two spectral positions at $1600$ and $1650{\mathrm{cm}}^{-1}$. An additive shift of $3{\mathrm{cm}}^{-1}$ gives measured positions $1603$ and $1653{\mathrm{cm}}^{-1}$, with the same $50{\mathrm{cm}}^{-1}$ separation. A shared gain error is different: $Y_i=1.01{\omega }_i+3$ gives separation $50.5{\mathrm{cm}}^{-1}$. Subtraction rejects the offset but retains the scale error.

The algebra generalizes to a contrast of several spectral positions. If $Z={\sum }_i{w}_i{Y}_i$ and each $Y_i$ contains the same additive $b$, its contribution is $b{\sum }_i{w}_i$. Thus weights satisfying ${\sum }_i{w}_i=0$ reject that offset. Averaging positions with weights summing to one does not. This is an original linear-model derivation of the rejection condition, not a claim that every spectral estimator has this form.

Two failures of the ideal assumption are especially informative. With unequal spectral offset sensitivity, $Y_i={\omega }_i+a_{i}b+{\eta }_i$, subtraction leaves $(a_2-a_1)b$. If the offset drifts between observations, it instead leaves $b(t_2)-b(t_1)$. For a stationary drift with variance $s_b^2$ and correlation ${\rho }_b(\tau )$ at lag $\tau$, this residual has varianceA common instrument or reference does not alone imply perfect cancellation at different times.

An apparatus-level check measures two stable reference features using the same acquisition and fitting pipeline, then deliberately shifts the assigned axis origin. Their separation should stay fixed under an additive reassignment; a scale change should alter it. Repeated reference observations test temporal drift and residual covariance. In two-dimensional infrared spectroscopy, this test concerns frequency-estimation errors; spectral diffusion also describes physical fluctuations and cannot be identified or removed merely by subtracting an instrumental offset.

Unequal spectral offset sensitivity occurs when a shared instrumental or reference offset influences two spectral estimates by different amounts. It limits spectral common-mode offset rejection: subtraction cancels a common disturbance only to the extent that its two response coefficients match.

For frequencies ${\omega }_1,{\omega }_2$, write an explicit measurement modelwhere $b$ is the shared offset, $a_i$ are dimensionless sensitivities and ${\eta }_i$ are residual errors. ThenThus even perfectly shared $b$ leaves a residual when $a_1\ne a_2$.

An original uncertainty example uses $a_1=0.98$, $a_2=1.02$ and standard uncertainty $u_b=0.50{\mathrm{cm}}^{-1}$. The offset contribution to the difference is $0.04u_b=0.020{\mathrm{cm}}^{-1}$. With independent residual standard uncertainties of $0.010{\mathrm{cm}}^{-1}$ each and no correlation with $b$, the combined uncertainty isA general calculation retains the relevant covariances; the independent-error simplification is an assumption, not a consequence of a shared instrument.^[ref]

Reweighting also changes what is measured. Subtracting $Y_1/a_1$ from $Y_2/a_2$ cancels $b$ when both sensitivities are nonzero, but estimates ${\omega }_2/a_2-{\omega }_1/a_1$, not generally the original frequency difference. Achieving cancellation is insufficient unless the resulting estimator still answers the intended physical question.

A controlled reference perturbation can estimate the slopes $a_1$ and $a_2$ while the true spectral features are held stable. Their difference predicts leakage. Spectral offset identifiability asks whether the available observations can distinguish a real change of frequency separation from this leakage; one observed difference alone cannot determine both an unknown physical change and an unknown offset. Reference features, controlled perturbations or independently established response coefficients supply additional information.