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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 $\delta \mathbf{z}=-\mathbf{x}\delta k$, givingThis covariance has rank one when $\mathbf{x}\ne 0$. 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 isThe 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 $\sqrt{4+16+4}\approx 4.90$ and understates the uncertainty of that sum.

A weighted spectral contrast ${\mathbf{a}}^T\mathbf{z}$ receives scale-error varianceFor $\mathbf{a}=(1,-1,1{)}^T$ 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.^[ref] Matching a reference spectrum's mean therefore does not establish independent residuals or a reliable weak-band detection limit.

A minimum-variance spectral contrast orthogonal to a reference shape estimates a desired signal amplitude while exactly rejecting one fixed nuisance shape. Let $\mathbf{z}=\theta \mathbf{s}+\beta \mathbf{x}+\mathbf{ϵ}$, with known shapes $\mathbf{s},\mathbf{x}$ and a positive-definite noise covariance matrix $C$. Choose $\hat{\theta }={\mathbf{a}}^T\mathbf{z}$ to minimize ${\mathbf{a}}^{T}C\mathbf{a}$, subject toThe constraints retain unit signal response and reject the reference amplitude, as in projection of shared-scale uncertainty onto a spectral contrast. The quadratic variance includes the full noise covariance.^[ref]

Remove the reference component in the inverse-covariance geometry:For nonzero $\mathbf{x}$ and $D>0$, the minimizing contrast and its variance areFor a short proof, every other feasible contrast is $\mathbf{a}+\mathbf{v}$ with ${\mathbf{v}}^T\mathbf{s}={\mathbf{v}}^T\mathbf{x}=0$. Since $C\mathbf{a}$ is a linear combination of those shapes, the cross term vanishes, leaving the added nonnegative variance ${\mathbf{v}}^{T}C\mathbf{v}$. This is a constrained best linear unbiased estimator.

In an original three-channel example, take $\mathbf{s}=(1,0,0{)}^T$, $\mathbf{x}=(1,1,1{)}^T$ and $C=\mathrm{diag}(1,4,1)$. The optimum is $\mathbf{a}=(1,-1/5,-4/5{)}^T$, with variance $1.8$. Equal subtraction of the two reference channels, $(1,-1/2,-1/2{)}^T$, has variance $2.25$. The quieter third channel receives more of the subtraction.

If the desired and reference shapes are proportional, $D=0$ and no unit-response rejecting contrast exists. The noise penalty from nearly parallel signal and reference shapes already becomes large as $D$ approaches zero. This construction assumes fixed known shapes and covariance; estimating them from the same noisy spectra requires additional uncertainty terms.