= Uncertainty of a reference-corrected spectral change
For <reference-assisted spectral offset separation>, the corrected estimate $\widehat\theta=z-qr$ 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 $u_\theta^2\approx g\Sigma g^{\mathsf T}$, or
$$
u_\theta^2\approx u_z^2+q^2u_r^2+r^2u_q^2
-2q\operatorname{Cov}(z,r)-2r\operatorname{Cov}(z,q)+2qr\operatorname{Cov}(r,q).
$$
The 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.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, equations (13)–(15), correlated input quantities]{ref}
For an original independent-input example, take $z=0.12\,\mathrm{cm^{-1}}$, $r=2.0\,\mathrm{cm^{-1}}$, $q=0.04$, $u_z=0.010\,\mathrm{cm^{-1}}$, $u_r=0.20\,\mathrm{cm^{-1}}$ and $u_q=0.010$. Then $\widehat\theta=0.040\,\mathrm{cm^{-1}}$ and
$$
u_\theta\approx\sqrt{0.010^2+(0.04\times0.20)^2+(2.0\times0.010)^2}
=0.02375\,\mathrm{cm^{-1}}.
$$
Treating $q$ as exact would give only $0.01281\,\mathrm{cm^{-1}}$. 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 $u_q^2u_r^2$:
$$
\operatorname{Var}(z-qr)=u_z^2+q^2u_r^2+r^2u_q^2+u_q^2u_r^2,
$$
where $q,r$ on the right denote the input means. It follows by expanding $\mathbb E(q^2r^2)=(q^2+u_q^2)(r^2+u_r^2)$. This term is omitted by linearization. In the example it changes the standard uncertainty to about $0.02383\,\mathrm{cm^{-1}}$.
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.
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