= Aperture-scan separation of direct and diffuse nail transmission
{title2=$Y_j=T_d+\eta_jT_s$}
\b[<Aperture-scan separation of direct and diffuse nail transmission>] estimates the directly collected and diffusely transmitted fractions of light by changing detector acceptance while keeping the <nail plate> and illumination fixed. Separation depends on a stated angular model: a change in aperture is an experimental control, not a direct label attached to every photon.
In the original constant-diffuse-<radiance> model, let $Y_j=T_d+\eta_jT_s+\epsilon_j$, where $T_d$ is the fully accepted direct fraction, $T_s$ the total diffuse fraction and $\eta_j$ the known diffuse acceptance. For a coaxial circular opening, the geometric approximation gives $\eta_j=a_j^2/(a_j^2+r^2)$. Two distinct acceptances give
$$
\widehat T_s=\frac{Y_2-Y_1}{\eta_2-\eta_1},\qquad
\widehat T_d=\frac{\eta_2Y_1-\eta_1Y_2}{\eta_2-\eta_1}.
$$
The distinction between these components is identifiable within this model only when the acceptances differ.
Suppose $\eta_1=0.20$, $\eta_2=0.50$, $Y_1=0.056$ and $Y_2=0.080$. The fitted fractions are $\widehat T_s=0.08$ and $\widehat T_d=0.04$. If independent signal errors each have <standard uncertainty> $\sigma=0.002$, with acceptance uncertainty temporarily neglected, linear propagation gives
$$
u^2(\widehat T_s)=\frac{2\sigma^2}{(\eta_2-\eta_1)^2},\qquad
u^2(\widehat T_d)=\frac{(\eta_1^2+\eta_2^2)\sigma^2}{(\eta_2-\eta_1)^2},
$$
and
$$
\operatorname{Cov}(\widehat T_d,\widehat T_s)
=-\frac{(\eta_1+\eta_2)\sigma^2}{(\eta_2-\eta_1)^2}.
$$
The resulting uncertainties are about 0.00943 and 0.00359. Their negative <covariance> matters when adding the components: the total-transmission uncertainty is about 0.00629, rather than the value obtained by incorrectly treating the fitted components as independent.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation 13, covariance in propagation]{ref}
Additional apertures give a straight-line fit of $Y$ against $\eta$, with intercept $T_d$ and slope $T_s$. <Weighted least squares> can use independently characterized signal uncertainties. Systematic curvature would challenge the assumed angular law or fixed direct-beam acceptance. A plausible line alone does not prove that a finite source, changing geometry or residual <fluorescence> has been modeled correctly.
The primary nail-transmission study collected transmitted light using an <integrating sphere>, <monochromator> and <photodiode>.<The transmission of optical radiation through human nails by S. G. Parker and B. L. Diffey>[Abstract, collection apparatus]{ref} The aperture scan here is an original proposed comparison, not a procedure or angular scattering law reported in that experiment.
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