= Scaled subtraction of two spatially offset Raman spectra
\b[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.<Subsurface Probing in Diffusely Scattering Media Using Spatially Offset Raman Spectroscopy by P. Matousek and others>[Abstract, offset-dependent layer contributions and two-powder feasibility experiment]{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. Write
$$
Y_0=a_0S+b_0B,\qquad Y_1=a_1S+b_1B,\qquad k=a_1/a_0,
$$
with $a_0\ne0$. Then
$$
Z=Y_1-kY_0=(b_1-kb_0)B.
$$
If the two mixing proportions are identical, so $a_0b_1-a_1b_0=0$, 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, but
$$
\operatorname{Var}(Z)=4000+0.40^2(10000)=5600,
$$
giving 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.
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