 Layer-thickness tradeoff in erosion-density sampling

The 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 $\hat{\rho }=\Delta M/(A\delta )$, 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 givesThe final term assumes independent area uncertainty; correlations require additional terms. The GUM supplies the general correlated-input propagation rule.^[ref]

Write $b^2=(u_M/A{)}^2+({\rho }_s{u}_{\delta }{)}^2$ and $c^2={\rho }_s^2{u}_A^2/A^2$. The resulting squared-error planning objective isFor nonzero $g,b$, differentiation gives ${\delta }_{\ast }=(4b^2/g^2{)}^{1/4}=\sqrt{2b/|g|}$. The area term raises the error floor in this approximation without changing the optimum; thickness uncertainty changes the optimum through $b$.

For stipulated $A=1\mathrm{c}{\mathrm{m}}^2$, ${\rho }_s=1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, $|g|=0.2\mathrm{g}\mathrm{c}{\mathrm{m}}^{-4}$ and $u_M=0.0001$ g, neglecting thickness uncertainty gives ${\delta }_{\ast }\approx 0.316$ mm. Including $u_{\delta }=0.0001$ cm, or 1 micrometre, gives $b=\sqrt{2}\times 10^{-4}\mathrm{g}\mathrm{c}{\mathrm{m}}^{-2}$ and ${\delta }_{\ast }\approx 0.376$ 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.^[ref] Local density sampling is an additional measurement, and its tradeoff should be assessed separately from the retained-mass curve.