= Plate-bed optical identifiability
\b[Plate-bed optical identifiability] asks whether the measured optical data uniquely separate the covering <nail plate> from the underlying <nail bed> within a specified model. Additional measurements help only if their dependence on the unknown quantities supplies independent information. This is an application of <identifiability>, beyond obtaining a good numerical fit.
For an original two-pass model without surface-return background, $y_\lambda=T_\lambda^2R_\lambda$, where $y$ is normalized measured intensity, $T$ plate <transmittance> and $R$ effective underlying <reflectance>. With an independently unknown $T$ and $R$ at each wavelength, $n$ spectral values give $n$ equations for $2n$ unknowns. Choosing an admissible $T$ determines a corresponding $R=y/T^2$; collecting a denser spectrum alone need not remove the ambiguity.
A deliberately restrictive example assumes known attenuation coefficients $\mu_1,\mu_2$, plate thickness $h$, and one wavelength-independent effective <reflectance> $r$. Taking $T_j=e^{-\mu_jh}$ gives
$$
\log y_j=\log r-2\mu_jh,\qquad
h=\frac{\log(y_2/y_1)}{2(\mu_1-\mu_2)},\qquad
r=y_1e^{2\mu_1h}.
$$
If $\mu_1\ne\mu_2$, the ideal equations identify two parameters. At stipulated $\mu_1=1\,\mathrm{mm^{-1}}$, $\mu_2=0.5\,\mathrm{mm^{-1}}$, $h=0.5$ mm and $r=0.8$, the signals are approximately 0.2943 and 0.4852 and inversion recovers the chosen parameters. This exponential example is not a validated transport law for a living nail.
<Spectral-contrast conditioning in nail optics> distinguishes mathematical uniqueness from usable precision. With independent errors in the logarithmic signals,
$$
\operatorname{Var}(\widehat h)\simeq\frac{\operatorname{Var}(\log y_1)+\operatorname{Var}(\log y_2)}{4(\mu_1-\mu_2)^2}.
$$
Nearly equal coefficients amplify noise. At equal coefficients, the ideal inversion fails altogether. An unknown spectral shape of $R_\lambda$, surface return or additional <multiple scattering> introduces further unknowns.
A primary transmission study measured isolated autopsy nails using a <monochromator>, <integrating sphere> and <photodiode>, including a <fluorescence> correction.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract, measurement apparatus]{ref} Isolated-plate measurements illustrate how an independent experiment can constrain one component. They do not by themselves identify living nail-bed <blood flow>. State which quantities are independently measured and which are assumed before interpreting a fitted tissue parameter.
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