= 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 model>
$$
Y_1=\omega_1+a_1b+\eta_1,\qquad Y_2=\omega_2+a_2b+\eta_2,
$$
where $b$ is the shared offset, $a_i$ are dimensionless sensitivities and $\eta_i$ are residual errors. Then
$$
(Y_2-Y_1)-(\omega_2-\omega_1)=(a_2-a_1)b+\eta_2-\eta_1.
$$
Thus 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 is
$$
u_\Delta=\sqrt{(a_2-a_1)^2u_b^2+u_{\eta_1}^2+u_{\eta_2}^2}
\approx0.0245\,\mathrm{cm}^{-1}.
$$
A general calculation retains the relevant <covariances>; the independent-error simplification is an assumption, not a consequence of a shared instrument.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, correlated-input propagation and sensitivity coefficients]{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.
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