# A.5.1 DU Generation and ISM Implementation Method

Simulation experiments with three additional maps (scenarios M3-A to M3-C) were performed on hypothetical DUs consisting of bulk material particulates with CH and DH. Section A.1.2 provides a detailed description of the generation of M-3 DUs and implementation of ISM on those DUs. The simulations address the following concepts relative to bulk material sampling and sample support:

- DUs can consist of bulk material (surface soils) with varying degrees of heterogeneities.
- ISM sampling designs can be described in terms of Gy sampling principles (see Section A.6.7).
- Simulations with ISM yield performance metrics, including bias in the mean, standard deviation of the relative bias, and coverage of the 95% UCL.

Issues related to the optimal increment sampling pattern and optimal number of increments and replicates needed to collect representative ISM samples from heterogeneous bulk material DUs are also discussed. When sampling for bulk material, it is not possible to collect single particles, creating the potential to introduce GSE. Presence of GSE and FE in bulk material tends to yield biased samples. Gy proposed the use of incremental sampling to address GSE and then combining all increments to address FE. A key concept associated with bulk material increment sampling is that each increment is associated with a specific sampling location of a DU represented by a 2-D map (e.g., Figures A-16, A-18), and a typical collected IS of specified SS consists of all contaminated particles as well as other potentially uncontaminated items (including trash, twigs, pebbles, dead creatures, etc.) found at that location using an appropriate sampling tool (e.g., pogo stick). In bulk material increment sampling, one is collecting increments of "equal mass" with SS of same size. Each increment potentially consists of many contaminated particles as well as other uncontaminated items which are discarded during ISM sample preparation process (e.g., drying, sieving). Simulations were implemented using software called MIS (Singh, Maichle, and Armbya 2009), which generates 2-D DUs representing surface bulk materials (e.g., soils) and implements alternative ISM sampling designs. Gy sampling concepts, terms, and equations relevant to ISM are defined at the end of this appendix (see Section A.6.1).

In all simulation experiments (PD, M-1, M-2, and M-3) considered in this document, a concentration can be associated with the sample, but the scenarios represent very different perspectives regarding the heterogeneities (and sampling errors) that are captured by the calculation of variance. If each sampling location is represented by a single particle (with some concentration), GSE is not present because each location of the DU consists of one and only one data value, analogous to sampling a single item from a batch of discrete items (Smith 2006). In addition, for DUs with analytical results generated using a population described by a single distribution (e.g., some of the scenarios presented in prior sections), small-scale and large-scale DH does not factor into the results. The scenarios presented in this section are intended to reinforce the following observations regarding bulk material sampling:

- When there is minimal small-scale DH (heterogeneity in particle distribution), there is negligible GSE.
- An important performance metric with ISM sampling is FE in the mean estimate. If GSE is not present (e.g., as in M-1 and M-2 DUs), bias or FE in the mean estimate is negligible using all sampling patterns (e.g., serpentine, simple random, random within grids) and lowest with simple random sampling.
- Bias or FE in the mean estimate cannot be reduced by increasing the number of ISM replicates.
- For heterogeneous DUs with CH and DH, one can obtain an unbiased estimate of DUs mean provided increments are collected using appropriate sample support following simple random sampling pattern. These facts and observations are illustrated by bulk material DU example used in DU M3-A (see Section A.5.2).
- ISM cannot identify spatial or temporal patterns present in any DU (especially if the DU is not divided into multiple SUs), as shown in Example M3-C (see Section A.5.2) by using a large real radium-226 data set of over 15,000 points

For homogeneous bulk material DUs with all contaminant particulates of same size and shape which are distributed evenly (e.g., one and only one particulate) throughout the DU (concentration distribution can be highly skewed), all sampling patterns yield unbiased estimates DU mean. For such homogeneous DUs, the size of the sample support does not matter much in reducing the bias in the mean estimate. These observations are illustrated by using a homogeneous bulk material DU in Example M3-B (see Section A.5.2). Just as heterogeneities are difficult to quantify in practice, they are difficult to represent in maps used to define hypothetical DU scenarios. For example, in practice, not all locations in a DU can be sampled due to obstacles such as trees, buildings, boulders, and water; therefore, a truly random sampling design is difficult to implement in practice. For the maps used to represent DUs in this section, accessible sampling locations are represented by points/particulates and inaccessible locations are represented by empty spaces. When an empty space (inaccessible location) is encountered, an increment of the same SS is collected from a neighboring location. These issues are discussed earlier in M-3 simulation section of Section A.1.2. Simulated ISM increments located at empty spaces yield nonzero results by drawing values from a local neighborhood of results. Therefore, the process used to simulate a DU map (e.g., smoothing/interpolation) has important implications as far as the extent to which the modeling results may inform real-world conditions.

In contrast to the M-3 maps, the M-1 and M-2 maps do not include areas that are inaccessible because the maps were generated in a way that interpolated between the original points. Thus, the true means of those DUs actually represent estimated means. Those estimated "true" means are then estimated using ISM. It is a well-known fact that a sample (e.g., consisting of 36 increments) obtained using simple random sampling from a discrete data set (e.g., PD DUs or M-2 DUs) yields an unbiased estimate of the mean of the population represented by that data set.