**Contents**

E.4.1 Using the unity rule;

E.4.2 Radio-nuclide concentrations with fixed ratios;

E.4.3 Unrelated radio-nuclide concentrations;

E.4.4 Example application of WRS Test to multiple radio-nuclides.

There are two cases to be considered when dealing with multiple radio-nuclides: 1) the radionuclide concentrations have a fairly constant ratio throughout the survey unit, or 2) the concentrations of the different radio-nuclides appear to be unrelated in the survey unit. In statistical terms, we are concerned about whether the concentrations of the different radio-nuclides are correlated or not. A simple way to judge this would be to make a scatter plot of the concentrations against each other, and see if the points appear to have an underlying linear pattern. The correlation coefficient can also be computed to see if it lies nearer to zero than to one. One could also perform a curve fit and test the significance of the result. Ultimately, however, sound judgement must be used in interpreting the results of such calculations. If there is no physical reason for the concentrations to be related, they probably are not. Conversely, if there is sound evidence that the radionuclide concentrations should be related because of how they were treated, processed or released, this information should be used.

### E.4.1 Using the unity rule

In either of the two above cases, the unity rule described in Section 3.3.6.3 is applied. The difference is in how it is applied. Suppose there are n radio-nuclides. If the concentration of radio-nuclide i is denoted by C_{i}, and its DCGL_{W} is denoted by D_{i}, then the unity rule for the n radio-nuclides states that:

C_{1} / D_{1} + C_{2} / D_{2} + C_{3} / D_{3} + … + C_{n} / D_{n} = 1 …………………………………………………… (E-11)

This will ensure that the total dose or risk due to the sum of all the radio-nuclides does not exceed the release criterion. Note that if D_{min} is the smallest of the DCGLs, then

(C_{1} + C_{2} + C_{3} + … + C_{n})/D_{min} = C_{1}/D_{1} + C_{2}/D_{2} + C_{3}/D_{3} + + C_{n}/D_{n} ………………………….. (E-12)

so that the smallest DCGL may be applied to the total activity concentration, rather than using the unity rule. While this option may be considered, in many cases it will be too conservative to be useful.

### E.4.2 Radio-nuclide concentrations with fixed ratios

If there is an established ratio among the concentrations of the n radio-nuclides in a survey unit, then the concentration of every radio-nuclide can be expressed in terms of any one of them, e.g., radio-nuclide #1. The measured radio-nuclide is often called a surrogate radio-nuclide for the others.

If

C_{2} = R_{2} C_{1}, C_{3} = R_{3} C_{1},…, C_{i} = R_{i} C_{1}, …, C_{n} = R_{n} C_{1}

then

C_{1}/D_{1} + C_{2}/D_{2} + C_{3}/D_{3} + … + C_{n}/D_{n} = C_{1}/D_{1} + R_{2} C_{1}/D_{2} + R_{3} C_{1}/D_{3} + … + R_{n} C_{1}/D_{n}

= C_{1} [1/D_{1} + R_{2}/D_{2} + R_{3}/D_{3} + … + R_{n}/D_{n}]

= C_{1}/D_{total} ………………………………………….. (E-13)

where

D_{total} = 1/(1/D_{1} + R_{n}/D_{n} + R_{3}/D_{3} + … + R_{n}/D_{n}) ………………………………………………. (E-14)

Thus, D_{total} is the DCGL_{W} for the surrogate radio-nuclide when the concentration of that radio-nuclide represents all radio-nuclides that are present in the survey unit. Clearly, this scheme is applicable only when radio-nuclide specific measurements of the surrogate radio-nuclide are made. It is unlikely to apply in situations where the surrogate radionuclide appears in background, since background variations would tend to obscure the relationships between it and the other radio-nuclides.

Thus, in the case where there are constant ratios among radio-nuclide concentrations, the statistical tests are applied as if only the surrogate radio-nuclide were contributing to the residual radioactivity, with the DCGL_{W} for that radionuclide replaced by D_{total}. For example, in planning the final status survey, only the expected standard deviation of the concentration measurements for the surrogate radionuclide is needed to calculate the sample size.

For the elevated measurement comparison, the DCGL_{EMC} for the surrogate radio-nuclide is replaced by

E_{total} = 1/(1/E_{1} + R_{2}/E_{2} + R_{3}/E_{3} + … + R_{n}/E_{n} ) …………………………………………….. (E-15)

where E_{i} is the DCGL_{EMC} for radio-nuclide i.

### E.4.3 Unrelated radio-nuclide concentrations

If the concentrations of the different radio-nuclides appear to be unrelated in the survey unit, there is little alternative but to measure the concentration of each radio-nuclide and use the unity rule. The exception would be in applying the most restrictive DCGLW to all of the radio-nuclides, as mentioned later in this section.

Since the release criterion is

C_{1}/D_{1} + C_{2}/D_{2} + C_{3} /D_{3} + … + C_{n}/D_{n} < 1 …………………………….. (E-16)

the quantity to be measured is the *weighted sum*,

T = C_{1}/D_{1} + C_{2} D_{2} + C_{3} /D_{3} + … + C_{n}/D_{n}

The DCGL_{W} for *T* is one. In planning the final status survey, the measurement standard deviation of the weighted sum, *T*, is estimated by

Σ^{2}(T) = (σ(C_{1})/D_{1})^{2} + (σ(C_{2})/D_{2})^{2} + (σ(C_{3}) /D_{3})^{2} + … + (σ(C_{n})/D_{n})^{2} ………… (E-17)

since the measured concentrations of the various radio-nuclides are assumed to be uncorrelated.

For the elevated measurement comparison, the inequality

C_{1}/E_{1} + C_{2}/E_{2} + C_{3} /E_{3} + … + C_{n}/E_{n} < 1 …………………………….. (E-18)

is used, where E_{i} is the DCGL_{EMC} for radio-nuclide i. For scanning, the most restrictive DCGL_{EMC} should generally be used.

When some of the radio-nuclides also appear in background, the quantity

T = C_{1}/D_{1} + C_{2} D_{2} + C_{3} /D_{3} + … + C_{n}/D_{n}

must also be measured in an appropriate reference area. If radionuclide i does not appear in background, set C_{i} = 0 in the calculation of T for the reference area.

*Note:* that if there is a fixed ratio between the concentrations of some radio-nuclides, but not others, a combination of the method of this section with that of the previous section may be used. The appropriate value of *D _{total}* with the concentration of the measured surrogate radio-nuclide should replace the corresponding terms in equation E-17.

### E.4.4 Example application of WRS Test to multiple radio-nuclides (^{60}Co and ^{137}Cs)

This section contains an example application of the non-parametric statistical methods in this report to sites that have residual radioactivity from more than one radio-nuclide. Consider a site with both ^{60}Co and ^{137}Cs contamination. ^{137}Cs appears in background from global atmospheric weapons tests at a typical concentration of about 3.7 10^{-2}Bq (1 pCi/g). Assume that the DCGL_{W} for ^{60}Co 7.4 10^{-2}Bq (2 pCi/g) and for ^{137}Cs is 5.2 10^{-2}Bq (1.4 pCi/g). In disturbed areas, the background concentration of ^{137}Cs can vary considerably. An estimated spatial standard deviation of 1.9 10^{-2}Bq (0.5 pCi/g) for ^{137}Cs will be assumed. During remediation, it was found that the concentrations of the two radio-nuclides were not well correlated in the survey unit. ^{60}Co concentrations were more variable than the ^{137}Cs concentrations, and 2.6 10^{-2}Bq (0.7 pCi/g) is estimated for its standard deviation. Measurement errors for both ^{60}Co and ^{137}Cs using gamma spectrometry will be small compared to this. For the comparison to the release criteria, the weighted sum of the concentrations of these radio-nuclides is computed from:

Weighted sum = (^{60}Co Concentration)/( ^{60}Co DCGL_{W}) + (^{137}Cs Concentration)/( ^{137}Cs DCGL_{W})

= (^{60}Co Concentration)/(7.4) + (^{137}Cs Concentration)/(5.2)

The variance of the weighted sum, assuming that the ^{60}Co and 137Cs concentrations are spatially unrelated is

σ² = [(^{60}Co Standard deviation)/( ^{60}Co DCGL_{W})]² + [(^{137}Cs Standard Deviation)/( ^{137}Cs DCGL_{W})]²

= [(2.6)/(7.4)]² + [(1.9)/(5.2)]² = 0.25.

Thus σ = 0.5. The DCGL_{W} for the weighted sum is one. The null hypothesis is that the survey unit exceeds the release criterion. During the DQO process, the LBGR was set at 0.5 for the weighted sum, so that Δ = DCGL_{W} – LBGR = 1.0 -0.5 = 0.5, and Δ/σ = 0.5/0.5 = 1.0. The acceptable error rates chosen were α = β = 0.05. To achieve this, 32 samples each are required in the survey unit and the reference area.

The weighted sums are computed for each measurement location in both the reference area and the survey unit. The WRS test is then performed on the weighted sum. The calculations for this example are shown in Table E.11. The DCGL_{W} (i.e., 1.0) is added to the weighted sum for each location in the reference area. The ranks of the combined survey unit and adjusted reference area weighted sums are then computed. The sum of the ranks of the adjusted reference area weighted sums is then compared to the critical value for n = m = 32, α = 0.05, which is 1162 (see formula following Table E.4). In Table E.11, the sum of the ranks of the adjusted reference area weighted sums is 1281. This exceeds the critical value, so the null hypothesis is rejected. The survey unit meets the release criterion. The difference between the mean of the weighted sums in the survey unit and the reference area is 1.86 – 1.16 = 0.7. Thus, the estimated dose or risk due to residual radioactivity in the survey unit is 70% of the release criterion.

Table E.11 Example WRS test for two radio-nuclides

Reference Area |
Survey Unit |
Weighted Sum |
Ranks |
||||||

number |
^{137}Cs |
^{60}Co |
^{137}Cs |
^{60}Co |
Ref |
Survey |
Adj Ref |
Survey |
Adj Ref |

1 | 2 | 0 | 1.12 | 0.06 | 1.43 | 0.83 | 2.43 | 1 | 56 |

2 | 1.23 | 0 | 1.66 | 1.99 | 0.88 | 2.18 | 1.88 | 43 | 21 |

3 | 0.99 | 0 | 3.02 | 0.56 | 0.71 | 2.44 | 1.71 | 57 | 14 |

4 | 1.98 | 0 | 2.47 | 0.26 | 1.41 | 1.89 | 2.41 | 23 | 55 |

5 | 1.78 | 0 | 2.08 | 0.21 | 1.27 | 1.59 | 2.27 | 9 | 50 |

6 | 1.93 | 0 | 2.96 | 0 | 1.38 | 2.11 | 2.38 | 37 | 54 |

7 | 1.73 | 0 | 2.05 | 0.2 | 1.23 | 1.56 | 2.23 | 7 | 46 |

8 | 1.83 | 0 | 2.41 | 0 | 1.3 | 1.72 | 2.3 | 16 | 52 |

9 | 1.27 | 0 | 1.74 | 0 | 0.91 | 1.24 | 1.91 | 2 | 24 |

10 | 0.74 | 0 | 2.65 | 0.16 | 0.53 | 1.97 | 1.53 | 27 | 6 |

11 | 1.17 | 0 | 1.92 | 0.63 | 0.83 | 1.68 | 1.83 | 13 | 18 |

12 | 1.51 | 0 | 1.91 | 0.69 | 1.08 | 1.71 | 2.08 | 15 | 32 |

13 | 2.25 | 0 | 3.06 | 0.13 | 1.61 | 2.25 | 2.61 | 47 | 63 |

14 | 1.36 | 0 | 2.18 | 0.98 | 0.97 | 2.05 | 1.97 | 30 | 28 |

15 | 2.05 | 0 | 2.08 | 1.26 | 1.46 | 2.12 | 2.46 | 39 | 58 |

16 | 1.61 | 0 | 2.3 | 1.16 | 1.15 | 2.22 | 2.15 | 45 | 41 |

17 | 1.29 | 0 | 2.2 | 0 | 0.92 | 1.57 | 1.92 | 8 | 25 |

18 | 1.55 | 0 | 3.11 | 0.5 | 1.11 | 2.47 | 2.11 | 59 | 35 |

19 | 1.82 | 0 | 2.31 | 0 | 1.3 | 1.65 | 2.3 | 11 | 51 |

20 | 1.17 | 0 | 2.82 | 0.41 | 0.84 | 2.22 | 1.84 | 44 | 19 |

21 | 1.76 | 0 | 1.81 | 1.18 | 1.26 | 1.88 | 2.26 | 22 | 48 |

22 | 2.21 | 0 | 2.71 | 0.17 | 1.58 | 2.02 | 2.58 | 29 | 62 |

23 | 2.35 | 0 | 1.89 | 0 | 1.68 | 1.35 | 2.68 | 3 | 64 |

24 | 1.51 | 0 | 2.12 | 0.34 | 1.08 | 1.68 | 2.08 | 12 | 33 |

25 | 0.66 | 0 | 2.59 | 0.14 | 0.47 | 1.92 | 1.47 | 26 | 5 |

26 | 1.56 | 0 | 1.75 | 0.71 | 1.12 | 1.6 | 2.12 | 10 | 38 |

27 | 1.93 | 0 | 2.35 | 0.85 | 1.38 | 2.1 | 2.38 | 34 | 53 |

28 | 2.15 | 0 | 2.28 | 0.87 | 1.54 | 2.06 | 2.54 | 31 | 61 |

29 | 2.07 | 0 | 2.56 | 0.56 | 1.48 | 2.11 | 2.48 | 36 | 60 |

30 | 1.77 | 0 | 2.5 | 0 | 1.27 | 1.78 | 2.27 | 17 | 49 |

31 | 1.19 | 0 | 1.79 | 0.3 | 0.85 | 1.43 | 1.85 | 4 | 20 |

32 | 1.57 | 0 | 2.55 | 0.7 | 1.12 | 2.17 | 2.12 | 42 | 40 |

Average |
1.62 | 0 | 2.28 | 0.47 | 1.16 | 1.86 | 2.16 | sum = |
sum = |

Std Dev |
0.43 | 0 | 0.46 | 0.48 | 0.31 | 0.36 | 0.31 | 799 | 1281 |