= Raman peak-fit uncertainty
{c}
<Raman peak-fit uncertainty> characterizes the uncertainty of a spectral-band position estimated by fitting a model to a <Raman spectrum>. It accompanies a <Raman peak-position map>: a colour difference between pixels need not establish a physical shift if the estimated positions are insufficiently precise. In <Raman mapping>, the instrument collects spectra at successive sample positions, and fitting extracts band parameters such as position and width.<What Raman images can reveal by Renishaw>[“How do we collect Raman images?”; “How can I analyse Raman images?”, curve-fit parameters]{ref}
For an isolated band, a <least squares> fit might estimate the center, height, width and background of a <Gaussian function>. The uncertainty assigned to its center is conditional on the chosen model and description of spectral noise. Repeating the complete acquisition and fit at one unchanged location provides another assessment: if $n$ independent center estimates have <standard deviation> $s$, the estimated <standard uncertainty> of their <sample mean> is $s/\sqrt n$. This reduction applies to independent variation; it does not eliminate an offset shared by every acquisition.<Guide to the expression of uncertainty in measurement by JCGM>[§4.2.1–4.2.3, independent repeated observations and standard uncertainty of their mean]{ref}
For an original numerical example, model two fitted positions as
$$
\widehat\nu_0=\nu_0+c+e_0,\qquad
\widehat\nu_1=\nu_1+c+e_1,
$$
where $c$ represents a shared uncertain additive correction from <measurement calibration>, while $e_0,e_1$ are independent fitting errors, also independent of $c$. Suppose their standard uncertainties are $u(c)=0.10\,\mathrm{cm^{-1}}$ and $u(e_i)=0.05\,\mathrm{cm^{-1}}$. Each position then has combined <standard uncertainty> $\sqrt{0.10^2+0.05^2}\approx0.112\,\mathrm{cm^{-1}}$. Their difference cancels the same $c$, giving
$$
u^2(\widehat\nu_1-\widehat\nu_0)
=u^2(\widehat\nu_1)+u^2(\widehat\nu_0)
-2\operatorname{cov}(\widehat\nu_1,\widehat\nu_0)
=2(0.05)^2\,\mathrm{cm^{-2}}.
$$
Here $u$ denotes <standard uncertainty>, so the difference has uncertainty about $0.0707\,\mathrm{cm^{-1}}$. The <covariance> term is essential: treating the two positions as entirely independent would instead give about $0.158\,\mathrm{cm^{-1}}$. A common additive correction cancels only insofar as it really is shared; instrument drift between measurements or a scale error need not cancel.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2, equations13–16 and §5.2.4, correlated inputs and common measurement references]{ref}
For a hypothetical stress model $\sigma=(\widehat\nu_1-\widehat\nu_0)/K$ with an exactly known illustrative $K=2\,\mathrm{cm^{-1}/GPa}$, this difference uncertainty contributes $0.0354\,\mathrm{GPa}$ to the <standard uncertainty> of stress. Uncertainty in $K$ would require its own contribution. <Raman peak-model bias>, overlapping bands, an inadequate background and laser-induced sample changes can invalidate an otherwise small fit error. Retaining spectra and repeating fits with physically defensible models therefore matters alongside quoting a numerical uncertainty.
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