# A.5.2 Simulation Results for Scenario M3-A

Figure A-16 shows a hypothetical simulated heterogeneous target DU with CH and DH. This DU represents a typical training target scenario in which the deposition (and density) of particles (with contaminant loading) decreases as one moves farther from the center of the target. Therefore, the concentrations are highest in the center—the mean concentration around the target central area is about 614 mg/kg, compared to a mean outside the target area (i.e., background conditions) of approximately 3 mg/kg. The overall mean in the DU is approximately 492 mg/kg. Tables A-11 and A-12 give summary statistics for population parameters.

Figure A-16. Post plot for Scenario M3-A representing a shooting range with a central target area (mean concentration = 492 mg/kg).

Table A-11. Summary of population parameters for the DU given by scenario M3-A, entire area and individual quadrants

Percent of total 100% 25.13% 24.93% 24.95% 25.00%
Number of cells 60,000 15,077 14,957 14,969 14,997
Arithmetic mean 491.8 493.6 491.9 492.7 489.1
Median 516.6 519.1 516.5 516.8 513.6
Minimum 1 1 1 1 1
Maximum 1,996 1,996 1,627 1,732 1,802
Standard deviation 325.2 326.8 324.3 325.5 324.2
CV (RSD) 0.661 0.662 0.659 0.661 0.663

Table A-12. Summary of population parameters for target area and background area in the DU given by scenario M3-A

Statistic Target Background
Percent of total 80% 20%
Number of cells 48,000 12,000
Mean 614 3.1
Median 593.3 2.7
Minimum 1 1
Maximum 1,996 15.7
Standard deviation 240 1.6
CV (RSD) 0.39 0.52

The points shown in Figure A-16 represent particulates present at accessible sampling locations. Some empty locations have zero particles and represent inaccessible locations. The mean concentration of the contaminants in the particulates is not the same at all accessible sampling locations, implying that large-scale DH is also present in the DU (i.e., a nonstationary process). Figure A-17 shows the probability distribution of particulates throughout the DU represented graphically on a normal distribution quantile-quantile (Q-Q) plot. The DU is heterogeneous with a CV of 0.86, which is relatively low. This example is a good reminder that a DU can exhibit a range of heterogeneities and yet the distribution need not have a high CV. CV is calculated based on concentration values, and heterogeneities are present in bulk material within the DU. DUs used in these sections represent bulk material DUs with one and only one point at each sampling location. There is no GSE present in such DUs, and all sampling patterns resulted in unbiased estimate of DU (data set) mean.

Figure A-17. Scenario M3-A normal distribution Q-Q plot of concentrations of particulates.

The M3-A DU represents a DU with both CH and DH. Being a target, the center of the DU consists of more bulk material particles than other areas of the DU, giving rise to small-scale DH and resulting in GSE. The GSE (and therefore FE) can be addressed by collecting and combining increments of adequate sample support. Being a heterogeneous DU, the use of an appropriate SS helps in collecting a representative sample that yields unbiased estimate of DU mean. Simulations were conducted using 36 and 64 increments each of SS of 0.05 units collected using systematic random grid sampling pattern and simple random sampling pattern. Tables A-13 and A-14 provide summary statistics relevant to various performance metrics.

Table A-13. Summary statistics of ISM applied to M3-A using 36 increments, each with a sample support of 0.05 units

Statistics 3 Replicates 5 Replicates
Systematic (with random start) Simple random Systematic random Simple random
Minimum 337.5 405.5 348.5 423.6
Maximum 457.0 558.4 439.3 559.2
Sample mean 395.7 476.1 395.9 476.5
Bias –96.1 –15.7 –95.9 –15.4
Relative bias (FE) 0.20 0.032 0.20 0.031
Student's-t-UCL95 coverage 14.3% 91% 0.3% 91%
Student's-t-UCL95 (average) 450.9 555.7 430.3 524.2
Chebyshev UCL95 coverage 33.7% 95.3% 17.7% 99.3%
Chebyshev UCL95 (average) 478.1 594.9 466.1 574.1
Root mean square error (RMSE) 98.7 31.9 97.4 26.4
Standard deviation of FE 0.200 0.065 0.198 0.054
CV Bar 0.083 0.099 0.091 0.106

Table A-14. Summary statistics of ISM applied to M3-A 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 345.8 403.0 348.2 431.4
Maximum 433.2 552.0 421.7 521.7
Sample mean 390.7 478.1 389.6 476.1
Bias –101.2 –13.8 –102.2 –15.8
Relative bias (FE) 0.21 0.028 0.21 0.032
Student's-t-UCL95 coverage 1.7% 89.7% 0% 82.7%
Student's-t-UCL95 (average) 433.4 536.5 414.5 510.7
Chebyshev UCL95 coverage 16.3% 96% 1.7% 97.7%
Chebyshev UCL95 (average) 454.5 565.3 440.4 546.8
RMSE 102.5 27.2 103.1 23.1
Standard deviation of FE 0.208 0.055 0.210 0.047
CV Bar 0.065 0.073 0.067 0.076

Results summarized in Tables A-13 and A-14 provide the following insights:

• ISM collected using simple random sampling (with 3 or 5 replicates) with appropriate SS resulted in an unbiased estimate of DU mean, an observation supported by statistical theory.
• ISM based on a systematic random grid pattern (and also serpentine pattern shown in Figure A-21) yields an estimate of the DU mean with relative bias of 19.5% using 36 increments (3 and 5 replicates) and relative bias of about 21% using 64 increments (3 and 5 replicates); ISM based on an simple random sampling pattern yields DU means with relative bias of only 3% using 36 increments (3 and 5 replicates) and 2.8% to 3.2% using 64 increments (3 and 5 replicates, respectively). The observation that simple random sampling tends to yield unbiased estimates of DU mean is supported by statistical theory.SD (FE) based on a systematic random grid pattern is about 0.20 using 36 increments (with 3 and 5 replicates) and 0.21 using 64 increments (with 3 and 5 replicates). SD (FE) based on simple random sampling is 0.06 (36 increments with 3 and 5 replicates) and 0.05 (64 increments with 3 and 5 replicates).
• The coverage is directly related to bias and FE. Since bias and FE in the mean estimate based on a systematic random grid pattern (and also serpentine pattern, not included here) is high (for 36 as well as 64 increments), the associated coverages provided by both UCL methods are poor (i.e., much lower than the nominal 95% coverage).
• The coverage by a UCL tends to improve (come closer to 95%) as FE decreases. For the Student’s-t-UCL95, coverage decreases marginally (from 91% to 89%) when increments are increased from 36 to 64 (using simple random sampling). For the Chebyshev UCL95, results vary within the margin of stochastic error of the simulation, but all of the coverages exceed the nominal 95% target. It is well known that Chebyshev UCL95 method tends to provide conservative values of UCLs (USEPA 2010b), especially when data are mildly skewed with CV <1.

When selecting a sampling design, it is important to weigh the advantages and disadvantages of various options. These simulations demonstrate how the coverage of the 95% UCL may actually decrease when a greater number of increments (using the same sample support per increment) are collected from a DU with moderate CH and DH. However, while the impact on the statistical concept of coverage is marginal, it is likely offset by the considerable improvement in the spatial coverage afforded by nearly doubling the density of the sampling network within the DU.

Is should be pointed out that for this DU, simulations were performed by collecting increments consisting of a single point (and also using SS of 0.01 units) from each selected sampling location. These simulations resulted in biased estimates of DU mean using all three sampling patterns. These observations reiterate the importance of using adequate SS in collecting incremental samples. Typically appropriate SS is calculated based on the particle size and other properties of the sampling medium (Pitard 1993). Section A.6 provides some details about selecting an appropriate SS size.

The above statements are verified further by using a real raidum-226 DU (where GSE is already addressed just like in M-1 and M-2 simulations) with large-scale DH (Scenario M3-C) and a hypothetical homogenous DU with a constant mean concentration within the DU (Scenario M3-B).