= Spectral-contrast conditioning in nail optics
\b[Spectral-contrast conditioning in nail optics] describes how measurement errors are amplified when thickness or tissue parameters are recovered from spectral differences. A model can be identifiable in exact arithmetic while its parameter estimates are too sensitive to small signal or calibration errors to be useful.
For the original simplified plate model $y_j=r e^{-2\mu_jh}$ with known coefficients and wavelength-independent effective <reflectance> $r$, thickness is
$$
h=\frac{\log(y_2/y_1)}{2\Delta\mu},\qquad\Delta\mu=\mu_1-\mu_2.
$$
Let each log signal have <standard uncertainty> $\sigma$ and their correlation be $\rho_e$. The first-order thickness uncertainty is
$$
u(h)\simeq\frac{\sigma\sqrt{2(1-\rho_e)}}{2\lvert\Delta\mu\rvert}.
$$
The <covariance> term belongs to the uncertainty calculation; treating every spectral error as independent can be misleading.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation 13]{ref}
With stipulated $\sigma=0.01$ and independent errors, coefficient differences $0.5$ and $0.05\,\mathrm{mm^{-1}}$ give thickness uncertainties about 0.0141 and 0.141 mm, respectively. A tenfold loss of spectral contrast produces tenfold error amplification. At the larger contrast and correlation 0.9, the modeled uncertainty is approximately 0.00447 mm. These are local model calculations, not validated precision claims for nail measurements.
Coefficient calibration also matters. If signal-ratio error and $\Delta\mu$ error are independent,
$$
u^2(h)\simeq\frac{u^2(\log(y_2/y_1))}{4(\Delta\mu)^2}+h^2\frac{u^2(\Delta\mu)}{(\Delta\mu)^2}.
$$
Collecting more signal averages does not eliminate the second term when the same uncertain calibration is reused.
<Offset sensitivity of log-ratio nail models> distinguishes error types. A common multiplicative factor cancels from $gy_2/(gy_1)$; a common additive offset generally does not cancel from $(y_2+b)/(y_1+b)$. Background light or unseparated <fluorescence> can therefore bias an otherwise precise ratio. The primary nail-transmission experiment explicitly included a <fluorescence> correction.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract, correction in the monochromator/integrating-sphere measurement]{ref}
Choose spectral channels using both sensitivity and a defensible optical model. A strong contrast is not sufficient if the assumed plate coefficients, collection geometry or underlying spectral behaviour are wrong.
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