(Compliance Sampling to Assess the Success of Remedial Actions at UXO Sites)
Soil at some large military bases in the United States is known to contain unexploded ordnance (UXO). Confirmation surveys using geophysical detectors and the digging of anomalies are often conducted after UXO cleanup operations to provide confidence that the UXO removal effort has achieved cleanup goals. Accept on zero attribute compliance sampling (AOZ-ACS) (Squeglia, 1994; Bowen and Bennett, 1988) is a statistical approach for establishing confidence in the cleanup effort.
Suppose a UXO cleanup operation has been conducted in a defined area of land (a lot). If the military base contains more than one lot then a separate decision on the success of the UXO cleanup effort can be made for each lot. The use of AOZ-ACS requires that a lot be divided into \(N\) equal-size units (areas), which are usually transects of a specified width and length. The width corresponds to the particular size (width) and type of geophysical detector system that will be used to survey the lot for targets of interest (TOI). Targets of interest are defined as the items that should have been removed during the initial remediation.
For a given lot, this AOZ-ACS module in VSP can be used to determine the number of units (transects), \(n\), that should be surveyed to establish \(Y\%\) confidence that at least \(X\%\) of the units in the lot do not contain TOI, where \(N\), \(Y\) and \(X\) are specified by the VSP user. Once \(n\) is determined, the \(n\) units are selected from the \(N\) units using a form of simple random sampling and then surveyed using the same or equivalent detector system used during the remediation. All detected anomalies within the \(n\) units are then dug to determine if they are TOI. If none of the \(n\) units contain TOI then it can be stated that there is \(Y\%\) confidence that at least \(X\%\) of the \(N\) units in the lot do not contain TOI. If one or more of the \(n\) units are found to contain TOI, then the lot is rejected, which means that the required \(Y\%\) confidence is not achieved. In that case, some sort of action is usually required, such as surveying additional units and conducting further remediation as necessary to achieve the \(Y\) and \(X\) specifications.
The following assumptions are needed to assure the validity of AOZ-ACS and the confidence statements:
1. All \(N\) units in the lot are approximately the same size (area)
2. The site owners and regulators have agreed on the value of the three input parameters (\(N\), \(Y\) and \(X\)) of AOZ-ACS. That is, they have agreed on the size of the units and hence know the value of \(N\) for the site, and they have agreed on the level of confidence required that at least \(X\%\) of the \(N\) units do not contain TOI.
3. Random sampling is used to select the \(n\) units that will be inspected using the detector system.
4. The prior belief should hold that the likelihood of any specific transect from the lot containing TOI is equivalent to the remaining transects. Any transects believed to have a much higher likelihood of containing TOI should be examined in a separate study and excluded from the \(N\) transects (reducing the value of \(N\)) before using AOZ-ACS.
5. The geophysical detector system and evaluation of anomalies (targets) used for inspection will reliably identify any TOI contained within a transect.
\(N\) |
is the total number of units (transects) of the specified size in the lot. |
\(n\) |
is the number of units that are selected from the \(N\) units using simple random sampling that will be surveyed for TOI. |
\(P_d\) |
is the maximum fraction of the \(N\) units that can contain TOI without the lot being rejected. \(P_d\) is typically a small fraction, e.g., 0.01 (1 percent). |
\(1- P_d\) |
is the minimum acceptable fraction of the \(N\) units that do not contain TOI. |
\(H_0\) |
is the Null Hypothesis, which is believed and assumed to be true prior to measuring any of the \(n\) units. The Null Hypothesis used with AOZ-ACS is |
|
\(H_0\): The fraction of the \(N\) units that contain TOI is 0 (note that \(\alpha\), the probability of a Type I error, is 0). |
\(H_a\) |
is the Alternative Hypothesis, which is accepted as being true if the null hypothesis is rejected. The Alternative hypothesis used with AOZ-ACS is |
|
\(H_a\): The fraction of the \(N\) transects that contain TOI is greater than \(P_d\) |
\(\beta\) |
is the acceptable Type II decision error rate. That is, \(\beta\) is the probability that can be tolerated of falsely accepting the Null Hypothesis based on the \(n\) surveyed units. |
\(1- \beta\) |
is the required probability of correctly accepting the Null Hypothesis, i.e., the required probability of correctly concluding that the fraction of the \(N\) units that are defective is less than or equal to \(P_d\). |
Using the above notation, the statement "\(Y\%\) confident that at least \(X\%\) of the \(N\) units in the lot do not contain TOI" can be restated as "\(100(1-\beta)\%\) confident that at least \(100(1 - P_d)\%\) of the \(N\) units in the lot do not contain TOI."
The VSP user is asked to specify the values of \(N\), 100(1- \(\beta\)) and 100(1 - \(P_d\)). Then VSP uses \(N\), \(P_d\) and \(\beta\) in the following equation to compute \(n\):
\begin{equation} n \cong 0.5\Big(1-\beta^{\frac{1}{N \times P_d}}\Big)(2N -N \times P_d+1) \end{equation}
This equation is from Bowen and Bennett (1988, page 887, Equation 17.8). It was originally derived by Jaech (1973, page 327). Equation (1) gives the same values of \(n\) as obtained using the table look-up method described in Schilling (1982).
Example 1: If \(N\) = 600 transects in the lot, \(\beta\) = 0.05 and \(P_d\) = 0.01, that is, if we require 95% confidence that at least 99% of the \(N\) transects in the lot do not contain TOI, then
\(n \cong 0.5\Big(1-0.05^{\frac{1}{600 \times 0.01}}\Big)(2 \times 600-600 \times 0.01+1) = 234.84\)
\(\cong235\)
Hence, 235 transects should be selected and surveyed for TOI.
Example 2: If \(N\) = 600, \(\beta\) = 0.01 and \(P_d\) = 0.01, that is, if we require 99% confidence that at least 99% of the \(N\) transects in the lot do not contain TOI, then
\(n \cong \Big(1-0.01^{\frac{1}{600 \times 0.01}}\Big)(2*600-600 \times 0.01+1) = 320.17\)
\(\cong 321\)
Hence, 321 transects should be selected and surveyed for TOI.
Bowen, M.W. and C.A. Bennett. 1988. Statistical Methods for Nuclear Material Management, NUREG/CR-4604, U.S. Nuclear Regulatory Commission, Washington, DC.
Jaech, J.L., 1973. Statistical Methods in Nuclear Material Control, TID-26298, NTIS, Springfield, Virginia.
Schilling, E.G. 1982. Acceptance Sampling in Quality Control, Marcel Dekker, Inc, New York.
Squeglia, N.L., 1994. Zero Acceptance Number Sampling Plans. ASQ Quality Press, Milwaukee, WI.
Total Possible Number of Non-overlapping Transects at the Site
Minimum % of Transects that Do Not Contain TOI
Transect Placement page