= Spectral offset identifiability
<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 be
$$
z=\theta+ab,
$$
where $\theta$ is the physical change, $b$ an unknown reference offset and $a$ a known sensitivity. If $a\ne0$, every replacement
$$
(\theta,b)\longmapsto(\theta-aq,b+q)
$$
gives 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 is
$$
\begin{pmatrix}z\\r\end{pmatrix}
=\begin{pmatrix}1&a\\0&c\end{pmatrix}
\begin{pmatrix}\theta\\b\end{pmatrix},\qquad
\theta=z-\frac ac r.
$$
Its determinant is $c$, so the two ideal unknowns are identifiable when $c\ne0$. 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> gives
$$
u_\theta^2=u_z^2+\left(\frac ac\right)^2u_r^2-2\frac ac\operatorname{Cov}(z,r).
$$
A 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.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Sections 4.1 and 5.2, model inputs and correlated-input propagation]{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.
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