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, 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.