For some real-world material systems, estimations of the incompressible sampling variance based on Gy’s classical s²(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 s²(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 new development.

对于某些现实世界的材料系统，基于采样理论 (TOS)中 Gy 的经典 s ^{2 (FSE) 公式的不可压缩采样方差的估计与采样方差的经验估计存在显着差异。}在涉及受污染的土壤、涂层特定聚集体和混合材料系统的情况下，抽样方差的理论估计值大于经验估计值，这种情况在 TOS 中没有物理意义。这使我们重新审视了从这个著名的本构异质性方程对s ² (FSE) 的估计的发展，并探索了混合材料系统的尺寸-密度类别的使用（富含分析物的颗粒和包被颗粒的混合物），自 Gy 最初的推导以来，这种方法大部分未被使用。这种方法可以避免考虑 Gy 经典处理中的粒度和释放因素，并提出批评使用诸如f := 0.5 和g := 0.25 等关键参数的“标准”输入值的理由。但是，一如既往，“解放因素”（$l$) 问题仍然发挥着重要作用，受到了应有的重视。基于尺寸-密度等级的构成异质性公式以允许在特定限制内在实践中轻松实施的形式呈现。我们展示了来自现实世界系统的大量实验结果。使用带有已发布数据的“SDCD 模型”再现了为标准“类矿物基质”计算的相对抽样方差，但更重要的是，将实际污染物计算的相对抽样方差修正了几个数量级。在所有情况下，重新计算的相对抽样方差都降低了 低于他们相应的实验测量值，现在完全符合 TOS 的预期，证实了我们的新发展。