# A.6.5.2 Homogeneous contaminant concentration

In a homogeneous bulk material DU, all particulates are assumed to be roughly of same size and shape (e.g., silt and clay DU, soils at a golf course, spill area), and those particulates are uniformly and evenly distributed throughout the DU. This assumption implies that CH and small-scale DH are negligible. However, large-scale DH can be present in such bulk material homogenous DUs; that is, the mean concentration of the contaminant can be different at various sampling locations within the DU.

It should be noted that, in practice, the occurrence of such bulk material homogeneous DU is highly unlikely. This concept is discussed here to illustrate the differences between smoothed DUs (with one point at each location) obtained via spatial interpolations on gridded data and simulated heterogeneous bulk material DUs potentially consisting of multiple nondistinct particles (points) at various sampling locations. Occurrence of multiple points (particulates) at sampling locations gives rise to GSE (and also FE), whereas in smoothed DUs (e.g., Scenario M3-B), GSE (and, therefore, FE) is already addressed, as one and only one distinct particle is present at each location of the DU.

The following statements can be made about homogeneous bulk material DUs:

- Since all particulates are of the same size and shape and are evenly distributed within the DU, material CH and GSE are minimal; therefore, FE (relative bias) is also minimal. As a result, ISM on such DUs yields a fairly accurate (unbiased) estimate of the arithmetic mean (AM) of the contaminant present in the DU. Several examples in support of these statements are considered in other sections (dealing with lognormal distribution) of this appendix.
- Depending on the contamination variability (e.g., low, high) and patterns (e.g., spatial, temporal, plumes) within the DU (e.g., spill area DU), the mean contaminant loadings of particulates at various sampling locations can be significantly different, giving rise to large-scale DH. Typically, DUs with large-scale DH have higher variability and skewness. For such DUs with higher variability, ISM-based
*t*-95% UCL (Student’s-*t*-statistic-based 95% UCL) does not provide the specified 95% coverage to DU mean (e.g., Singh, Singh, and Engelhardt 1997). For such DUs, a 95% UCL based on Chebyshev inequality may be used in risk assessment applications (Singh, Singh, and Iaci 2002, USEPA 2010b) ) to address uncertainties associated with ISM replicates. Scenario M3-D considered later in this section illustrates the issues described in this paragraph in the context of ISM. - For bulk material homogeneous DUs, the ISM yields an unbiased estimate of DU mean provided increments are collected using a simple random sampling pattern; however, spatial/ temporal patterns (including hot spots) potentially present in the DU could not be identified.