= Finite-shell bias in boundary-density calibration
\b[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 $\rho(r)$,
$$
\rho_\delta=\frac{\int_{R-\delta}^R\rho(r)r^2\,dr}{\int_{R-\delta}^Rr^2\,dr}.
$$
Write $s=R-r$. Expanding $\rho(R-s)=\rho(R)-s\rho'(R)+O(s^2)$ and the volume weight $(R-s)^2$ gives
$$
\rho_\delta-\rho(R)=-\frac{\rho'(R)}2\delta+O(\delta^2).
$$
The 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 $v=-M'/(4\pi R^2\rho_s)$, with $\rho_s=\rho(R)$. Replacing $\rho_s$ by $\rho_\delta$ gives
$$
\frac{v_{\rm inferred}}v-1=\frac{\rho_s}{\rho_\delta}-1
\simeq\frac{\rho'(R)\delta}{2\rho_s}.
$$
A 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 approximately
$$
\operatorname{MSE}(\rho_\delta)=\left(\frac{g\delta}{2}\right)^2+
\left(\frac{u_M}{A\delta}\right)^2.
$$
Minimization gives
$$
\delta_* =\sqrt{\frac{2u_M}{A|g|}}.
$$
For stipulated $A=1\,\mathrm{cm^2}$, $|g|=0.2\,\mathrm{g\,cm^{-4}}$ and $u_M=0.0001$ g, $\delta_*\approx0.0316$ cm, or 0.316 mm. At that thickness, bias magnitude and measurement standard deviation are each about $0.00316\,\mathrm{g\,cm^{-3}}$. 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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, Eq. 13]{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.<Kuredux Polyglycolic Acid Resin technical brochure by Kureha>[PDF p. 3, Degradability test method]{ref} A local shell-density calibration would be an additional measurement, not a quantity already determined by that residue curve.
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