 Positive-thickness bounds in density sampling

Positive-thickness bounds in density sampling constrain the removed layer's thickness away from zero before converting its mass and area into density. Positivity alone, $\delta >0$, is weaker than a measured lower bound $\delta \ge d_{\min }>0$. The distinction matters because the density model contains the reciprocal $1/\delta$.

Suppose positive input intervals describe mass $M\in [M_{-},M_{+}]$, area $A\in [A_{-},A_{+}]$ and thickness $\delta \in [d_{-},d_{+}]$. Monotonicity gives the enclosing boundsThis is an original deterministic propagation, not a confidence interval unless the joint coverage of the input intervals is separately established. Correlations may make some corner combinations unattainable, in which case this box bound can be conservative. The GUM explicitly distinguishes propagation of standard uncertainties from propagation of confidence intervals.^[ref]

For stipulated $M\in [0.99,1.01]$ mg, $A\in [0.98,1.02]\mathrm{c}{\mathrm{m}}^2$ and $\delta \in [8,12]$ micrometres, the resulting density lies between approximately $0.809$ and $1.288\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. The nominal inputs give $1\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, but the exact enclosing interval is asymmetric.

A probabilistic subtlety is finite inverse moments in thickness models. If a positive-thickness probability density function obeys $p(\delta )\ge c>0$ throughout an interval $0<\delta <\epsilon$, thenThus merely conditioning a normal distribution on positive thickness does not necessarily make mean inferred density finite. A physical lower bound or a justified distribution with sufficiently vanishing near-zero density changes that conclusion; clipping simulated values at a convenient minimum changes the model.

An experiment should record the two surface positions, their common reference and which geometric constraint supplies the lower bound. Rejecting impossible geometry and concealing poorly resolved geometry are different operations. Repeated mass readings cannot by themselves establish a positive thickness bound.