Offset sensitivity of log-ratio nail models ($\partial \hat{h}/\partial \beta$)

Offset sensitivity of log-ratio nail models describes the bias produced when an additive background enters a ratio used to estimate a nail plate parameter. A shared multiplicative gain cancels from an ideal ratio, whereas a shared additive term generally remains.

Consider the original simplified spectral model $y_j=r{e}^{-2{\mu }_{j}h}$, with plate thickness $h$, known attenuation coefficients ${\mu }_j$ and wavelength-independent effective reflectance $r$. Write $\Delta \mu ={\mu }_1-{\mu }_2\ne 0$. The ideal thickness is $h=\log (y_2/y_1)/(2\Delta \mu )$. If measured signals are $s_j=g{y}_j+b$ with $g>0$, define the input-equivalent offset $\beta =b/g$. Ignoring that offset givesBoth offset-adjusted signals must be positive. Equal additive errors do not cancel inside this logarithm.

With stipulated $r=0.8$, $h=0.5$ mm, ${\mu }_1=1\mathrm{m}{\mathrm{m}}^{-1}$ and ${\mu }_2=0.5\mathrm{m}{\mathrm{m}}^{-1}$, the signals are about 0.2943 and 0.4852. Adding $\beta =0.020$ gives an apparent thickness of about 0.4746 mm. Positive background pulls the ratio towards one and underestimates thickness in this example. The numbers are model calculations, not an experimentally validated nail-thickness calibration.

The local sensitivity isAt zero offset in the example it is approximately $-1.337\mathrm{mm}$ per unit normalized signal. A residual-offset standard uncertainty of 0.005 therefore contributes approximately 0.00668 mm to the local thickness uncertainty. If that same uncertain background correction is reused across channels, its errors are correlated; the covariance terms belong in the propagation model.^[ref]

Residual-offset uncertainty in nail thickness estimates remains even when repeated photon measurements reduce random fluctuations. A useful test measures dark current and examines optical background separately, since fluorescence need not have the same spectral dependence as either. The primary nail-transmission study explicitly included a fluorescence correction.^[ref] That motivates identifying the background component; it does not validate this reduced thickness model.

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 $D={\mu }_1-{\mu }_2\ne 0$ is known. The inferred nail plate thickness isIts sensitivity coefficients areWith mutually independent raw signal estimates and background estimate, first-order uncertainty propagation givesThis follows the sensitivity-and-covariance approach in the measurement guide.^[ref]

For stipulated $Z_1=0.30$, $Z_2=0.50$, $D=0.50\mathrm{m}{\mathrm{m}}^{-1}$, $u(Y_1)=u(Y_2)=0.003$ 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: $\mathrm{Cov}(Z_1,Z_2)=u^2(B)$ 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.^[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.