= Noise remaining after scaled Raman subtraction
\b[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 is
$$
\operatorname{Var}(Z)=\operatorname{Var}(Y)+k^2\operatorname{Var}(X)
-2k\operatorname{Cov}(X,Y).
$$
The covariance term describes the actual acquisition relationship; numerical subtraction does not itself make the inputs correlated or independent. General measurement propagation retains such 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}
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 gives
$$
\operatorname{Var}(Y-KX)=\operatorname{Var}(Y)+k^2\operatorname{Var}(X)
-2k\operatorname{Cov}(X,Y)
+\sigma_k^2\bigl[(\mathbb EX)^2+\operatorname{Var}(X)\bigr].
$$
The 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.<Subsurface Probing in Diffusely Scattering Media Using Spatially Offset Raman Spectroscopy by P. Matousek and others>[Abstract, offset-dependent mixtures and numerical layer separation]{ref}
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