= Residual-offset uncertainty in nail thickness estimates
{title2=$u(\widehat h)$}
\b[<Residual-offset uncertainty in nail thickness estimates>] is the uncertainty contributed by an imperfectly known additive background after correction of spectral signals. It differs from a known uncorrected offset's bias: subtraction can remove the best estimate of a background while leaving uncertainty about that estimate.
For an original reduced optical model, let two measured normalized signals be $Y_1,Y_2$ and subtract the same background estimate $B$. Define $Z_j=Y_j-B>0$ and suppose the attenuation contrast $D=\mu_1-\mu_2\ne0$ is known. The inferred <nail plate> thickness is
$$
\widehat h=\frac{\log(Z_2/Z_1)}{2D}.
$$
Its sensitivity coefficients are
$$
\frac{\partial\widehat h}{\partial Y_1}=-\frac1{2DZ_1},\qquad
\frac{\partial\widehat h}{\partial Y_2}=\frac1{2DZ_2},\qquad
\frac{\partial\widehat h}{\partial B}=\frac1{2D}\left(\frac1{Z_1}-\frac1{Z_2}\right).
$$
With mutually independent raw signal estimates and background estimate, first-order uncertainty propagation gives
$$
u^2(\widehat h)=\frac1{4D^2}\left[
\frac{u^2(Y_1)}{Z_1^2}+\frac{u^2(Y_2)}{Z_2^2}
+\left(\frac1{Z_1}-\frac1{Z_2}\right)^2u^2(B)\right].
$$
This follows the sensitivity-and-<covariance> approach in the measurement guide.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1–5.2, propagation of input uncertainties and correlated quantities]{ref}
For stipulated $Z_1=0.30$, $Z_2=0.50$, $D=0.50\,\mathrm{mm^{-1}}$, $u(Y_1)=u(Y_2)=0.003$ and $u(B)=0.010$, the estimated thickness is approximately 0.5108 mm and its combined <standard uncertainty> is 0.0177 mm. The common-background contribution alone is 0.0133 mm. Averaging 100 independent repetitions of each raw signal while reusing the same background estimate reduces the total only to about 0.0134 mm; it does not reduce that shared contribution by ten.
An equivalent calculation treats corrected signals as correlated: $\operatorname{Cov}(Z_1,Z_2)=u^2(B)$ under the stated independence assumptions. Omitting this covariance would describe a different experiment with independent background errors. Near zero corrected signal, the <logarithm> also becomes strongly nonlinear, so this local approximation needs further checking.
The primary nail experiment included a <fluorescence> correction in its optical transmission measurement.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract, fluorescence correction]{ref} Background emission and <dark current> must be characterized separately when they have different spectral responses. The equations above are an illustrative uncertainty budget, not a validated thickness calibration or reported precision from that experiment.
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