 Finite-shell bias in boundary-density calibration

Finite-shell bias in boundary-density calibration is the difference between a density averaged through a layer of nonzero thickness and the local density at an erosion boundary. Making a layer thinner reduces spatial averaging, but can increase relative uncertainty in the small removed mass and volume. Boundary-density calibration for erosion fronts must manage both effects.

For an original spherical model with outer radius $R$, sampled thickness $\delta$ and smooth radial density $\rho (r)$,Write $s=R-r$. Expanding $\rho (R-s)=\rho (R)-s{\rho }^{\prime }(R)+O(s^2)$ and the volume weight $(R-s{)}^2$ givesThe leading bias is set by the local density gradient. Curvature changes higher-order terms; a uniform density has no such averaging bias, even for a thick shell.

A first-order polymer erosion front speed conversion uses $v=-M^{\prime }/(4\pi R^2{\rho }_s)$, with ${\rho }_s=\rho (R)$. Replacing ${\rho }_s$ by ${\rho }_{\delta }$ givesA density increasing toward the surface therefore leads to overestimated speed when the sampled layer has lower mean density. This calculation assumes a genuine moving front and negligible independent interior density change.

An original local planning model makes the thickness tradeoff explicit. Approximate the sampled area by $A$, the density-gradient magnitude by $|g|$, and the standard uncertainty of removed mass by a thickness-independent $u_M$. Neglect other uncertainties. Squared bias plus variance is approximatelyMinimization givesFor stipulated $A=1\mathrm{c}{\mathrm{m}}^2$, $|g|=0.2\mathrm{g}\mathrm{c}{\mathrm{m}}^{-4}$ and $u_M=0.0001$ g, ${\delta }_{\ast }\approx 0.0316$ cm, or 0.316 mm. At that thickness, bias magnitude and measurement standard deviation are each about $0.00316\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. This is a conditional planning result, not a universal required slice thickness.

Layer-thickness tradeoff in erosion-density sampling also depends on uncertain layer thickness, geometry, conditioning and the accuracy of the gradient model. Shared balance corrections can correlate successive mass differences; the GUM requires those covariance contributions to be included.^[ref]

A proposed check compares several layer thicknesses and looks for a stable extrapolation toward zero thickness, with matched drying and dimensional protocols. A changing gradient or damage caused by sectioning can invalidate the extrapolation. Kureha's polyglycolic acid experiment measures filtered, dried retained solids after water exposure.^[ref] A local shell-density calibration would be an additional measurement, not a quantity already determined by that residue curve.