Revisiting Pierre Gy's formula (TOS) - A return to size-density classes for applications to contaminated soils, coated particular aggregates and mixed material systems.

For some real-world material systems, estimations of the incompressible sampling variance based on Gy's classical s(FSE) formula from the Theory of Sampling (TOS) show a significant discrepancy with empirical estimates of sampling variance. In instances concerning contaminated soils, coated particular aggregates and mixed material systems, theoretical estimates of sampling variance are larger than empirical estimates, a situation which does not have physical meaning in TOS. This has led us to revisit the development of estimates of s(FSE) from this famous constitutional heterogeneity equation and explore the use of size-density classes for mixed material systems (mixtures of both analyte-enriched and coated particles), an approach which has been mostly unused since Gy's original derivation. This approach makes it possible to avoid taking into account the granulometric and liberation factors from Gy's classical treatment, and present grounds for criticising the use of 'standard' input values of critical parameters such as f: = 0.5, and g: = 0.25. But, as always, the "liberation factor" (l) issue still plays an important role, which is paid due attention. The constitutional heterogeneity formula based on size-density classes is presented in a form that allows for easy implementation in practice, within specified limitations. We present extensive experimental results from real-world systems. Using the "SDCD model" with published data reproduced the relative sampling variances calculated for the standard "mineral-like matrices", but more importantly corrected the relative sampling variance calculated for real contaminants by several orders of magnitudes. In all cases, the recalculated relative sampling variances were decreased to below their corresponding experimental measurements, now fully as expected from TOS, substantiating our development.