Unequal spectral offset sensitivity

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.

Spectral offset identifiability asks whether the available spectral observations can distinguish a physical frequency change from an instrumental offset. Identifiability is a property of the measurement model and its observations. Reducing random noise cannot by itself separate quantities that always enter the model in the same combination.

For an original local model, let an observed change bewhere $\theta$ is the physical change, $b$ an unknown reference offset and $a$ a known sensitivity. If $a\ne 0$, every replacementgives the same $z$. The single observation therefore does not identify both unknowns. With $z=0.10{\mathrm{cm}}^{-1}$ and $a=0.04$, $(\theta ,b)=(0.10,0)$ and $(0.02,2.0)$ in compatible frequency units are observationally indistinguishable, even without measurement noise.

Reference-assisted spectral offset separation adds an observation $r=cb$ from a stable reference with known nonzero sensitivity $c$. The combined linear system isIts determinant is $c$, so the two ideal unknowns are identifiable when $c\ne 0$. The reference must genuinely exclude the physical change represented by $\theta$; otherwise the assumed second equation is wrong.

Identifiability also differs from precision. For fixed known $a/c$, correlated-input uncertainty propagation givesA very weak reference response can make the identified estimate noisy. Uncertainty in the sensitivities requires additional terms. The primary GUM supplies the measurement-model and covariance framework; the rank and invariance arguments above are explicit derivations for this stated spectral model.^[ref]

An experiment can test the separation by measuring stable reference features and the sample while deliberately changing the assigned axis offset. Repeated spectra characterize noise; reference perturbations characterize sensitivity. Neither exercise establishes spectral offset identifiability unless the observations distinguish the competing physical and instrumental changes.

Reference-assisted spectral offset separation uses an additional reference observation to distinguish a physical spectral change from an instrumental offset. The reference must respond to the offset in a known way while excluding, or separately modelling, the sample’s physical change. Merely collecting another signal does not guarantee spectral offset identifiability.

For a local linear measurement model, writewhere $\theta$ is the desired change, $b$ the shared offset and $c\ne 0$. With known sensitivities and zero-mean residual errors,removes the offset contribution. This is a subtraction of modelled responses, not a claim that the uncorrected sample and reference frequencies are physically identical.

An original example uses $z=0.10{\mathrm{cm}}^{-1}$, $r=2.0{\mathrm{cm}}^{-1}$ and $q=0.04$, giving $\hat{\theta }=0.02{\mathrm{cm}}^{-1}$. A reference may have a much larger offset response than the sample; the response ratio supplies the necessary conversion.

The uncertainty of a reference-corrected spectral change depends on joint errors. For fixed $q$,With $u_z=0.010{\mathrm{cm}}^{-1}$, $u_r=0.20{\mathrm{cm}}^{-1}$ and correlation $0.5$, this gives approximately $0.00917{\mathrm{cm}}^{-1}$; assuming independence instead gives $0.0128{\mathrm{cm}}^{-1}$. The primary GUM supplies the correlated-input propagation rule used in this calculation.^[ref]

If $q$ is itself estimated, its uncertainty also matters. Under an additional independent-input first-order approximation, the contribution is $r^2{u}_q^2$. Large uncertainties, correlations with the fitted ratio or nonlinear responses require a more complete analysis.

A controlled experiment records stable reference features through the same acquisition and processing chain while deliberately perturbing the instrumental offset. This can estimate $a/c$ and test whether the response is linear. Sample/reference timing must also match the assumed offset: a drifting offset at different acquisition times can leave an uncorrected residual. Repeated paired data, rather than unrelated error estimates, are needed to characterize the relevant covariance.

For reference-assisted spectral offset separation, the corrected estimate $\hat{\theta }=z-qr$ depends on the measured sample change $z$, reference change $r$ and sensitivity ratio $q$. Its measurement uncertainty must include their joint variation, including uncertainty in a fitted sensitivity ratio. Subtracting a reference can reduce shared noise while adding independent noise.

The gradient of the measurement model $z-qr$ with respect to $(z,r,q)$ is $g=(1,-q,-r)$. If $\Sigma$ is their covariance matrix, first-order uncertainty propagation gives $u_{\theta }^2\approx g\Sigma g^{\mathsf{T}}$, orThe signs follow from the sensitivity coefficients, rather than a rule that all correlation reduces uncertainty. This is an application of the correlated-input propagation law in the primary GUM.^[ref]

For an original independent-input example, take $z=0.12\mathrm{c}{\mathrm{m}}^{-1}$, $r=2.0\mathrm{c}{\mathrm{m}}^{-1}$, $q=0.04$, $u_z=0.010\mathrm{c}{\mathrm{m}}^{-1}$, $u_r=0.20\mathrm{c}{\mathrm{m}}^{-1}$ and $u_q=0.010$. Then $\hat{\theta }=0.040\mathrm{c}{\mathrm{m}}^{-1}$ andTreating $q$ as exact would give only $0.01281\mathrm{c}{\mathrm{m}}^{-1}$. A large reference response amplifies uncertainty in its conversion ratio, even when that response is measured precisely.

For mutually independent random inputs, the exact product variance also includes $u_q^2{u}_r^2$:where $q,r$ on the right denote the input means. It follows by expanding $\mathbb{E}(q^2{r}^2)=(q^2+u_q^2)(r^2+u_r^2)$. This term is omitted by linearization. In the example it changes the standard uncertainty to about $0.02383\mathrm{c}{\mathrm{m}}^{-1}$.

Experimentally, use paired sample/reference spectra to estimate shared covariance, and independent sensitivity measurements to test the proposed value of $q$. If the same spectra estimate both $q$ and the correction, independence cannot simply be assumed. Drift between acquisitions or a nonlinear reference response is a measurement model problem as well as a precision problem.