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Noise remaining after scaled Raman subtraction is the random uncertainty left when a surface contribution's mean has been cancelled. For two measured spectral channels $X,Y$ and fixed scale $k$, scaled subtraction of two spatially offset Raman spectra uses $Z=Y-kX$. Its exact variance isThe covariance term describes the actual acquisition relationship; numerical subtraction does not itself make the inputs correlated or independent. General measurement propagation retains such terms.^[ref]

A focused extension allows an uncertain subtraction scale $K$, independent of the pair $(X,Y)$, with mean $k$ and variance ${\sigma }_k^2$. Expanding the product's moments givesThe final term includes uncertainty in the reference scale, not just shot noise in the two spectra.

For an original surface-only example, let independent Poisson photoelectron counts have means ${\mu }_X=20000$ and ${\mu }_Y=6000$, with $k=0.30$. The mean residual is zero. A perfectly known scale gives variance $6000+0.30^2(20000)=7800$ and standard deviation $88.3$. If instead ${\sigma }_k=0.005$, the variance becomes $7800+0.005^2(20000^2+20000)=17800.5$, giving standard deviation $133.4$ in the scaled count units.

If $K$ is estimated from these same counts, the independence assumption needs replacement by the appropriate joint model. A scale shared across wavelengths can also produce correlated spectral residuals from a shared subtraction scale, leaving a structured band-like error rather than independent noise in each bin. The original SORS study demonstrates changing layer contributions with collection offset; the moment calculation here is an independent noise model for a simplified subtraction.^[ref]

Spatially offset Raman spectroscopy, or SORS, collects Raman scattering from a surface region laterally displaced from the illumination spot. In a diffusely scattering material, changing the offset changes the mixture of photon paths and the relative contributions of surface and buried layers. It is not simply a tightly focused measurement at a selected depth.

The2005 experiment of P. Matousek and colleagues used trans-Stilbene powder beneath a $1\mathrm{mm}$ layer of PMMA powder. Its primary report describes a19-fold improvement in buried-layer contrast before numerical separation.^[ref] The physical point is that an opaque-looking scattering overlayer need not contribute the same spectral proportion at every collection offset.

An original two-component linear combination illustrates scaled subtraction of two spatially offset Raman spectra. Let $S(\nu )$ and $B(\nu )$ be fixed surface and buried Raman spectra, and suppose dark-corrected measurements giveIf a surface-specific band determines the scale $3/8$, thenThe surface contribution cancels while a scaled buried spectrum remains. These coefficients are chosen examples, not the paper's measured mixing fractions.

A practical arrangement fixes an excitation spot, moves the collection region laterally, rejects the excitation line with an optical filter, and sends the collected light to a spectrograph. Record comparable exposure, incident radiant power, backgrounds and spectral response at each offset. If the offset changes each component's shape, rather than only its scale, the simple subtraction model fails. Strong absorption, fluorescence, overlapping bands and measurement noise also limit inference. Offset-dependent enrichment provides evidence for depth-dependent contributions; it is not an unqualified depth calibration or unique chemical identification.