= Layer-thickness tradeoff in erosion-density sampling
The \b[layer-thickness tradeoff in erosion-density sampling] balances spatial averaging bias against uncertainty in the small mass and volume of a sampled surface layer. A thinner layer can better represent boundary <density> while being harder to measure precisely. The preferred thickness depends on the measurement model and the required inference, rather than being a material constant.
In an original locally planar model, layer density is estimated as $\widehat\rho=\Delta M/(A\delta)$, with area $A$ and thickness $\delta$. For boundary density $\rho_s$ and outward density gradient $g$, <finite-shell bias in boundary-density calibration> gives leading bias $-g\delta/2$. Suppose the standard uncertainties $u_M$ and $u_\delta$ are independent and approximately constant over the tested thickness range. First-order propagation gives
$$
\operatorname{Var}(\widehat\rho)\simeq
\frac{(u_M/A)^2+(\rho_su_\delta)^2}{\delta^2}
+\rho_s^2\frac{u_A^2}{A^2}.
$$
The final term assumes independent area uncertainty; correlations require additional terms. The GUM supplies the general correlated-input propagation rule.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, Eq. 13]{ref}
Write $b^2=(u_M/A)^2+(\rho_su_\delta)^2$ and $c^2=\rho_s^2u_A^2/A^2$. The resulting squared-error planning objective is
$$
E(\delta)=\frac{g^2\delta^2}{4}+\frac{b^2}{\delta^2}+c^2.
$$
For nonzero $g,b$, differentiation gives $\delta_*=(4b^2/g^2)^{1/4}=\sqrt{2b/|g|}$. The area term raises the error floor in this approximation without changing the optimum; thickness uncertainty changes the optimum through $b$.
For stipulated $A=1\,\mathrm{cm^2}$, $\rho_s=1\,\mathrm{g\,cm^{-3}}$, $|g|=0.2\,\mathrm{g\,cm^{-4}}$ and $u_M=0.0001$ g, neglecting thickness uncertainty gives $\delta_*\approx0.316$ mm. Including $u_\delta=0.0001$ cm, or 1 micrometre, gives $b=\sqrt2\times10^{-4}\,\mathrm{g\,cm^{-2}}$ and $\delta_*\approx0.376$ mm. These are original planning examples, not prescribed section thicknesses.
<Thickness-uncertainty limit in erosion-density sampling> becomes important when small layers approach the dimensional readout's resolution. If the feasible interval excludes the calculated optimum, optimize over the actual interval and test whether the local-gradient and planar approximations remain adequate. If $g=0$, this simplified model has no finite interior optimum; other constraints determine the usable thickness.
A proposed experiment uses matched conditioning, several layer thicknesses and recorded mass-difference correlations. Sectioning damage or density changes during drying can invalidate the assumed error model. Kureha's <polyglycolic acid> experiment measures filtered, dried retained solids after water exposure.<Kuredux Polyglycolic Acid Resin technical brochure by Kureha>[PDF p. 3, Degradability test method]{ref} Local density sampling is an additional measurement, and its tradeoff should be assessed separately from the retained-mass curve.
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