# A.5.3 Simulation Results for Scenario M3-B

The DU shown in Figure A-18 represents a homogeneous DU with a mean concentration of about 200 units at each sampling location. The particulates are evenly distributed throughout the DU without any spatial patterns (i.e., a stationary process) with the same mean concentration of 200 mg/kg at each location within the DU. The concentrations of the contaminant present in bulk material follow a skewed gamma distribution with scale parameter of 1000 and shape parameter of 0.2. Figure A-19 shows the normal distribution Q-Q plot.

Figure A-18. Post plot for Scenario M3-B representing a homogeneous DU with mean of 200 mg/kg.

Concentrations follow a gamma distribution (shape = 0.2, scale =1,000).

Figure A-19. Scenario M3-B normal distribution Q-Q plot of concentrations.

This kind of DU can be generated by applying a spatial interpolation method (e.g., kriging, inverse distance weighting [IDW]) to a data set generated from discrete samples. A similar concept was demonstrated in the generation of the map used in Scenario M1, which was generated using IDW. This smoothing process is equivalent to assuming that all particles in the DU are of the same size, shape, and mass and that those particulates are evenly and homogeneously distributed throughout the DU *with one particle per sample location*. Although the particles are homogeneously distributed within the DU, the concentrations of a contaminant can vary from location to location.

In the absence of material CH, small-scale DH, and GSE, any random sampling method (e.g., discrete, composite, or ISM) yields a fairly representative, unbiased estimate of the DU mean. The smoothing process does not address large-scale (long-range) DH that may arise due to contamination patterns potentially present in the DU. The presence of large-scale DH implies that the mean concentrations of the contaminant at different locations can be different. As mentioned previously, ISM is not expected to address large-scale (long-range) DH. ISM masks large-scale distributional (e.g., present due to spatial/temporal patterns) heterogeneity present in a DU.

With DUs that are homogeneous with respect to particles as well as with respect to concentration contents of those particles, although any random sampling scheme (discrete or ISM) yields an unbiased estimate of DU mean, this fact does not guarantee that the UCL provides adequate coverage for the mean. If the concentration distribution is highly skewed, the distribution of sample means may also be asymmetric, and the uncertainty in the overall mean of replicate ISM samples can be high (with any sampling method). Specifically, the coverage provided by Student’s-*t* 95% UCL could be less than 95%.

Due to homogeneity of particles present in the DU, the size of the SS (mass) does not matter much in reducing the bias in the mean estimate. The simulations demonstrated with M3-B illustrate how increasing the SS (e.g., 0.05 units instead of a single point) introduces a marginal change (<2%) in FE and SD (FE). This observation reiterates the importance of using an appropriate SS for heterogeneous bulk material DUs.

Table A-15 summarizes the population parameters for the M3-B scenario and Tables A-16 and A-17 summarize the simulation results with alternative ISM sampling methods.

Table A-15. Summary statistics (population parameters) for the DU given by scenario M3-B, entire area and individual quadrants

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

Percent of total | 100% | 25.0% | 24.9% | 25.2% | 24.8% |

Number of cells | 60,000 | 15,026 | 14,963 | 15,115 | 14,896 |

Arithmetic mean | 200.7 | 203 | 197.5 | 202.5 | 199.8 |

Median | 21.2 | 21.2 | 20.5 | 21.6 | 21.9 |

Minimum | 1 | 1 | 1 | 1 | 1 |

Maximum | 11,675 | 7,568 | 6,179 | 11,675 | 5,857 |

Standard deviation | 449.5 | 445.9 | 438.3 | 465.6 | 447.7 |

CV (RSD) | 2.24 | 2.27 | 2.22 | 2.30 | 2.24 |

Table A-16. Summary statistics of ISM applied to M3-B 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 | 79.7 | 105.4 | 101.0 | 113.9 | 118.0 | 121.5 |

Maximum | 325.0 | 321.3 | 350.6 | 339.1 | 309.8 | 323.5 |

Sample mean | 202.2 | 200.9 | 201.1 | 203.5 | 201.7 | 202.8 |

Bias | 1.5 | 0.14 | 0.43 | 2.8 | 1.0 | 2.1 |

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

Student's-t-UCL95 coverage |
92.7% | 93.7% | 91.3% | 93.0% | 92.7% | 92.0% |

Student's-t-UCL95 (average) |
314.8 | 312.5 | 310.4 | 272.9 | 267.1 | 267 |

Chebyshev UCL95 coverage | 96% | 98.7% | 95.3% | 99% | 99% | 99% |

Chebyshev UCL95 (average) | 370.2 | 367.6 | 364.2 | 345.4 | 335.3 | 334.2 |

RMSE | 42.5 | 42.6 | 44.1 | 35.8 | 33.9 | 36.2 |

SD of FE | 0.211 | 0.212 | 0.220 | 0.178 | 0.169 | 0.180 |

CV Bar | 0.330 | 0.331 | 0.321 | 0.357 | 0.340 | 0.333 |

Table A-17. Summary statistics of ISM applied to M3-B 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 | 107.1 | 108.0 | 122.9 | 138.6 | 136.0 | 133.9 |

Maximum | 320.2 | 291.0 | 305.3 | 274.1 | 289.1 | 275.3 |

Sample mean | 202.7 | 200.3 | 205.1 | 205.2 | 200.7 | 202.4 |

Bias | 2.03 | -0.42 | 4.41 | 4.49 | 0.003 | 1.70 |

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

Student's-t-UCL95 coverage |
94% | 93.7% | 94% | 95.3% | 89.3% | 93% |

Student's-t-UCL95 (average) |
290.1 | 283.7 | 290.9 | 255.8 | 251.1 | 253 |

Chebyshev UCL95 coverage | 98.3% | 97% | 96% | 98.3% | 98.3% | 99.3% |

Chebyshev UCL95 (average) | 333.2 | 324.8 | 333.2 | 308.7 | 303.8 | 305.8 |

RMSE | 34.4 | 32.0 | 33.6 | 25.0 | 27.4 | 26.0 |

SD of FE | 0.171 | 0.159 | 0.168 | 0.125 | 0.136 | 0.130 |

CV Bar | 0.26 | 0.25 | 0.25 | 0.26 | 0.26 | 0.26 |