# A.5.4 Simulation Results for Scenario M3-C

The DU in Figure A-20 shows spatial patterns in radium-226 based on a data set of 15,356 discrete readings (kindly provided by Dr. Robert Johnson of Argonne National Laboratory). Samples were collected using walk-over gamma detectors with a sample support of approximately 1 m^{2} and 6 deep. This process yields a single observation (point) per sampling location; consequently, small-scale DH and GSE are already addressed. Since data are collected using a gamma detector (without collecting physical bulk material samples), it is implicitly assumed that bulk material CH is not present. Therefore, without appreciable CH, this scenario represents a compositionally homogeneous bulk material DU.

The mean background measurement and regulatory action level are 800 and 1800 cpm, respectively. Large-scale DH is present as the average (mean) reading is not the same at all locations in the DU. In addition, there are several pockets of highly elevated radium-226 readings (significantly exceeding the action level), as can be seen in the post plot of the site shown in Figure A-20.

Figure A-20. Post plot of radium-226 readings obtained from a real site (Scenario M3-C)**.**

The (true) DU mean and standard deviation are 848 and 1074 cpm, respectively (CV = 1.3). Since small-scale DH and material CH have already been addressed, this DU scenario with specified (x,y) coordinates is similar to that of the M-1 map. As expected, in the absence of GSE, ISM yields an unbiased estimate of the DU mean using the three sampling patterns; however, simple random sampling yields the least bias in the mean, consistent with statistical theory.

This example demonstrates that the use of ISM correctly accounts for the proportion of the DU with areas of high and low values; however, ISM does not identify the spatial resolution that allows for the identification of hot spots and large-scale DH. If pockets with elevated radium-226 at a scale smaller than the DU are of interest, then the CSM should be revisited to determine whether the DU should be further subdivided. The use of modified ISM described below is better suited to identify large-scale DH. Table A-18 summarizes population parameters for the DU. Tables A-19 to A-22 summarize simulation results for standard ISM, while Tables A-23 and A-24 summarize results for modified ISM.

In practice, it is not realistic to expect/assume that a DU is homogeneous with respect to bulk particulates present in the DU, as the occurrence of particulate CH is inevitable at environmental sites. It is also not practical to assume that heterogeneities (CH, DH—small scale and large scale) are not present in an environmental SU/DU.

**Table A-18. Summary statistics (population parameters) for the DU given by scenario M3-C, entire area and individual quadrants**

Statistic | DU | Quadrant 1 | Quadrant 2 | Quadrant 3 | Quadrant 4 |
---|---|---|---|---|---|

Percent of total | 100% | 24.1% | 21.8% | 28.9% | 24.8% |

Number of cells | 15,356 | 3,699 | 3,347 | 4,442 | 3,813 |

Arithmetic mean | 848 | 748 | 911 | 898 | 837 |

Median | 705 | 663 | 743 | 731 | 689 |

Minimum | 308 | 432 | 437 | 308 | 393 |

Maximum | 51,187 | 26,748 | 20,712 | 42,171 | 51,187 |

Standard deviation | 1,074 | 714.8 | 892 | 1,293 | 1,221 |

CV (RSD) | 1.27 | 0.96 | 0.98 | 1.44 | 1.46 |

The DU mean is 848 cpm; however, several patches of elevated radium-226 are present in each of the four quadrants of the DU. The regulatory not-to-exceed threshold is 1800 cpm.

ISM samples based on increments consisting of even single points yield a fairly unbiased estimate of the DU mean, as shown in Tables A-19 (36 increments) and A-21 (64 increments). ISM results based on SS of 0.05 units are shown in Tables A-20 (36 increments) and A-22 (64 increments). The use of increased SS reduces the FE in the mean by less than 1%.

Table A-19. Summary statistics of ISM applied to M3-C using 36 increments, each with no sample support (i.e., single points)

Statistics | 3 Replicates | 5 Replicates | ||||
---|---|---|---|---|---|---|

Serpentine | Systematic random | Simple random | Serpentine | Systematic random | Simple random | |

Minimum | 674 | 710 | 723 | 699 | 730 | 732 |

Maximum | 1374 | 1626 | 1573 | 1245 | 1169 | 1391 |

Sample mean | 833 | 829 | 858 | 841 | 828 | 852 |

Bias | –15.1 | –19.7 | 9.3 | –7.1 | –20.3 | 3.7 |

Relative bias (FE) | 0.018 | 0.023 | 0.011 | <0.01 | 0.024 | <0.01 |

Student's-t-UCL95 coverage |
78% | 70% | 80% | 76.5% | 66% | 74.8% |

Student's-t-UCL95 (average) |
1,053 | 1,004 | 1,068 | 988 | 935 | 978 |

Chebyshev UCL95 coverage | 85.7% | 81.7% | 87.7% | 90.5% | 86% | 91.8% |

Chebyshev UCL95 (average) | 1161 | 1090 | 1171 | 1141 | 1047 | 1110 |

RMSE | 109.1 | 117.1 | 117.8 | 88.7 | 71.4 | 85.7 |

SD of FE | 0.129 | 0.138 | 0.139 | 0.105 | 0.084 | 0.101 |

CV Bar (mean of CVs) | 0.14 | 0.11 | 0.13 | 0.17 | 0.13 | 0.15 |

Table A-20. Summary statistics of ISM applied to M3-C using 36 increments, each with a sample support of 0.05 units

Statistics | 3 Replicates | 5 Replicates | ||
---|---|---|---|---|

Systematic random | Simple random | Systematic random | Simple random | |

Minimum | 713 | 718 | 733 | 727 |

Maximum | 1336 | 1349 | 1108 | 1353 |

Sample mean | 844 | 845 | 828 | 846 |

Bias | –4.8 | –3.8 | –20.7 | –2.7 |

Relative bias (FE) | <0.01 | <0.01 | 0.02 | <0.01 |

Student's-t-UCL95 coverage |
73.7% | 80% | 66% | 79.5% |

Student's-t-UCL95 (average) |
1,039 | 1,040 | 932 | 966 |

Chebyshev UCL95 coverage | 81.3% | 89.3% | 84.8% | 94.5% |

Chebyshev UCL95 (average) | 1135 | 1136 | 1041 | 1092 |

RMSE | 106.2 | 100.6 | 70.1 | 76.4 |

SD of FE | 0.125 | 0.119 | 0.082 | 0.090 |

CV Bar | 0.123 | 0.125 | 0.126 | 0.141 |

Table A-21. Summary statistics of ISM applied to M3-C using 64 increments, each with no sample support (i.e., single points)

Statistics | 3 Replicates | 5 Replicates | ||||
---|---|---|---|---|---|---|

Serpentine | Systematic random | Simple random | Serpentine | Systematic random | Simple random | |

Minimum | 747.5 | 724.9 | 749.3 | 748.9 | 743.3 | 757.2 |

Maximum | 1210 | 1104 | 1400 | 1068 | 1098 | 1047 |

Sample mean | 836 | 823 | 857 | 835 | 831 | 848 |

Bias | –12.1 | –25.7 | 8.9 | –13.5 | –17.7 | –0.4 |

Relative bias (FE) | 0.014 | 0.030 | 0.010 | 0.0160 | 0.021 | <0.01 |

Student's-t-UCL95 coverage |
73% | 71.3% | 84.3% | 69.3% | 68.3% | 81% |

Student's-t-UCL95 (average) |
986 | 935 | 1030 | 932 | 920 | 950 |

Chebyshev UCL95 coverage | 83% | 80% | 90.7% | 88.3% | 87.8% | 95% |

Chebyshev UCL95 (average) | 1060 | 990 | 1115 | 1033 | 1013 | 1057 |

RMSE | 78.3 | 63.9 | 85.6 | 57.1 | 54.7 | 57.0 |

SD of FE | 0.092 | 0.075 | 0.101 | 0.067 | 0.065 | 0.067 |

CV Bar | 0.10 | 0.08 | 0.11 | 0.12 | 0.11 | 0.12 |

Table A-22. Summary statistics of ISM applied to M3-C using 64 increments, each with a sample support of 0.05 units

Statistics | 3 Replicates | 5 Replicates | ||
---|---|---|---|---|

Systematic random | Simple random | Systematic random | Simple random | |

Minimum | 736 | 737 | 749 | 748 |

Maximum | 1126 | 1415 | 1035 | 1126 |

Sample Mean | 830 | 851 | 831 | 851 |

Bias | –18.6 | 2.9 | –17.3 | 3.1 |

Relative Bias (FE) | 0.02 | <0.01 | 0.02 | <0.01 |

Student's-t-UCL95 coverage |
76% | 79.7% | 68% | 81% |

Student's-t-UCL95 (average) |
961 | 1017 | 914 | 955 |

Chebyshev UCL95 coverage | 82.7% | 89.3% | 90.3% | 93.8% |

Chebyshev UCL95 (average) | 1027 | 1099 | 1001 | 1063 |

RMSE | 70.9 | 85.5 | 52.4 | 61.3 |

SD of FE | 0.084 | 0.100 | 0.062 | 0.072 |

CV Bar | 0.089 | 0.108 | 0.101 | 0.122 |

Results summarized in Tables A-19 and A-22 provide the following insights:

- For smoothed DUs (without small-scale DH and GSE), all sampling patterns yield fairly unbiased estimates of the DU mean as supported by sampling theory (Cochran 1977). In the present case, FE is <3% for all sampling methods.
- The size of the SS (mass) does not matter as GSE is not present. In other words, increasing SS does not decrease the bias in the mean estimate.
- Being a homogeneous DU without material CH and small-scale DH, the bias does not decrease significantly with increased number of increments and sample support (e.g., Tables A-19 and A-21 for 36 increments; Tables A-20 and A-22 for 64 increments).
- Due to the presence of large-scale DH, the variability is moderate throughout the DU (CV = 1.3), and as a result, the
*t*-UCL95 does not provide the specified 95% coverage for the DU mean. - Chebyshev UCL95 does not provide 95% coverage when only three replicates are used.
- Based on ISM replicate data alone, it is difficult to identify pockets of elevated radium-226.