= Correlated spectral residuals from a shared subtraction scale
{title2=$C_{\rm scale}=\sigma_k^2\mathbf x\mathbf x^T$}
\b[Correlated spectral residuals from a shared subtraction scale] arise when one uncertain coefficient multiplies many spectral channels. Let a fixed reference profile be $\mathbf x$ and let the estimated scale differ from its mean by $\delta k$, with variance $\sigma_k^2$. Its contribution to the subtracted spectrum is $\delta\mathbf z=-\mathbf x\,\delta k$, giving
$$
C_{\rm scale}=\mathbb E[\delta\mathbf z\,\delta\mathbf z^T]
=\sigma_k^2\mathbf x\mathbf x^T.
$$
This covariance has rank one when $\mathbf x\ne0$. It describes a coherent spectral pattern, not independent noise added separately at each wavelength. The parent <noise remaining after scaled Raman subtraction> treats a single channel and an uncertain scale.
For an original reference vector $\mathbf x=(100,200,100)^T$ and $\sigma_k=0.02$, the covariance is
$$
C_{\rm scale}=\begin{pmatrix}4&8&4\\8&16&8\\4&8&4\end{pmatrix}.
$$
The sum of the three residuals has standard uncertainty $0.02(100+200+100)=8$ in the chosen count units. Incorrectly treating the channel errors as independent gives $\sqrt{4+16+4}\approx4.90$ and understates the uncertainty of that sum.
A weighted spectral contrast $\mathbf a^T\mathbf z$ receives scale-error variance
$$
u^2_{\rm scale}=\sigma_k^2(\mathbf a^T\mathbf x)^2.
$$
For $\mathbf a=(1,-1,1)^T$ this contribution vanishes in the example. This <projection of shared-scale uncertainty onto a spectral contrast> also warns that a contrast insensitive to the reference shape can suppress a genuine signal with that same shape. Removing one error direction is not universal denoising.
Independent detector noise adds its own covariance matrix. If the reference profile is itself noisy or the scale is estimated from the same spectra, their correlations must be included too. The measurement propagation rule keeps common influences rather than discarding off-diagonal terms.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref} Matching a reference spectrum's mean therefore does not establish independent residuals or a reliable weak-band detection limit.
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