= Offset sensitivity of log-ratio nail models
{title2=$\partial\widehat h/\partial\beta$}
\b[<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 $y_j=r e^{-2\mu_jh}$, with plate thickness $h$, known attenuation coefficients $\mu_j$ and wavelength-independent effective <reflectance> $r$. Write $\Delta\mu=\mu_1-\mu_2\ne0$. The ideal thickness is $h=\log(y_2/y_1)/(2\Delta\mu)$. If measured signals are $s_j=gy_j+b$ with $g>0$, define the input-equivalent offset $\beta=b/g$. Ignoring that offset gives
$$
\widehat h=\frac{\log[(y_2+\beta)/(y_1+\beta)]}{2\Delta\mu},\qquad
\widehat h-h=\frac{\log(1+\beta/y_2)-\log(1+\beta/y_1)}{2\Delta\mu}.
$$
Both offset-adjusted signals must be positive. Equal additive errors do not cancel inside this <logarithm>.
With stipulated $r=0.8$, $h=0.5$ mm, $\mu_1=1\,\mathrm{mm^{-1}}$ and $\mu_2=0.5\,\mathrm{mm^{-1}}$, 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 is
$$
\frac{\partial\widehat h}{\partial\beta}
=\frac1{2\Delta\mu}\left(\frac1{y_2+\beta}-\frac1{y_1+\beta}\right).
$$
At 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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation 13, correlated input quantities]{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.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract, fluorescence correction]{ref} That motivates identifying the background component; it does not validate this reduced thickness model.
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