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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.

Scaled subtraction of two spatially offset Raman spectra suppresses a common surface contribution by matching its amplitude between two measurements. The spatially offset Raman spectroscopy experiment establishes that collection offset can change surface and buried-layer proportions; numerical separation then needs a model of those contributions.^[ref] The following two-component subtraction is an original simplified case, not a claim that it reproduces the paper's full analysis.

Let fixed Raman spectra $S(\nu )$ and $B(\nu )$ describe the surface and buried material. Writewith $a_0\ne 0$. ThenIf the two mixing proportions are identical, so $a_0{b}_1-a_1{b}_0=0$, this cancels the buried component too. Offset diversity must supply different mixtures, not merely a weaker copy of the same spectrum.

For an original example, take $Y_0=10S+B$ and $Y_1=4S+3B$. The correct scale $k=0.40$ gives $Z=2.6B$. Using $k=0.42$ instead gives $Z=-0.2S+2.58B$: an incorrectly subtracted surface band can produce a negative residual even when both true spectra are nonnegative. A surface-specific reference band can estimate the scale only if the assumed common shape and response are adequate.

Noise remaining after scaled Raman subtraction is distinct from the cancelled mean signal. At a surface-only channel, suppose two independent acquisitions have Poisson photoelectron counts with means $10000$ and $4000$. For fixed $k=0.40$, the expected residual is zero, butgiving a standard deviation of about $74.8$ in the scaled count units. Subtraction has not erased shot noise. Correlated acquisitions and uncertainty in $k$ add further terms. Varying shape, wavelength-dependent attenuation or an unmatched fluorescence background can invalidate this simple linear separation.