= Reference-assisted spectral offset separation
<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>, write
$$
z=\theta+ab+\epsilon_z,\qquad r=cb+\epsilon_r,
$$
where $\theta$ is the desired change, $b$ the shared offset and $c\ne0$. With known sensitivities and zero-mean residual errors,
$$
\widehat\theta=z-qr,\qquad q=\frac ac
$$
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 $\widehat\theta=0.02\,\mathrm{cm}^{-1}$. 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 $q$,
$$
u_\theta^2=u_z^2+q^2u_r^2-2q\operatorname{Cov}(z,r).
$$
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.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, sensitivity coefficients and input covariance]{ref}
If $q$ is itself estimated, its uncertainty also matters. Under an additional independent-input first-order approximation, the contribution is $r^2u_q^2$. 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>.
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