(Transect sampling to meet percent clean objectives)
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 can be conducted to support site remediation assumptions. Specifically, at times, there are areas where prior belief supports the decision that no munitions use occurred however, some stake holders require additional sampling to support this prior belief. Accept on zero attribute compliance sampling (AOZ-ACS) (Squeglia, 1994; Bowen and Bennett, 1988) is a statistical approach that supports this objective.
Suppose a remedial investigation is being conducted and there are large portions of the investigation site (a subarea) that are presumed to have never been used for explosive munition detonations. If there are additional facts that support stratifying presumptively clean areas then separate designs can be completed for each. The use of AOZ-ACS requires that each of the \(N\) samples from the subarea are equal-sized units (parcels). This VSP methodology requires the user to specify the parcel size of concern. VSP determines the number of parcels (acreage) that should be sampled and then translates that to the number of transects of a specified width and length that are required to cover the parcel-equivalent sample acreage . The transect width corresponds to the particular footprint that the type of geophysical detector system that will be used to survey the lot for targets of interest (TOI) will see as it moves along the transect. Presumptively clean designs require a pre-specified definition of TOI and, depending on site objectives, the TOI definition could range from any munitions fragment to intact ordnance.
For a given sample area, this AOZ-ACS module in VSP can be used to determine the number of units (parcels or parcel equivalent transects), \(n\) , that should be surveyed to establish \(X\%\) confidence that at least \(Y\%\) of the units in the sample area do not contain TOI, where \(N\) , \(X\) and \(Y\) are specified by the VSP user. Once \(n\) is determined, the required survey acreage is calculated and sufficient transects are placed to cover that acreage using a form of random sampling and then surveyed. All detected anomalies within the transects are then dug to determine if they are TOI. If none of the transects contain TOI then it can be stated that there is \(X\%\) confidence that at least \(Y\%\) of the total acreage does not contain TOI. If one or more of the transects are found to contain TOI, then the required \(X\%\) confidence is not achieved. In that case, some sort of action is usually required, such as additional surveying to achieve the \(X\) and \(Y\) specifications and/or performing a root cause analysis.
The following assumptions are needed to assure the validity of AOZ-ACS and the confidence statements:
1. All \(N\) units (possible parcel-sized equivalent transects) from the sample area are approximately the same size (area).
2. The site owners and regulators have agreed on the value of the four input parameters (Parcel size of concern, transect size , \(X\) and \(Y\) ) of AOZ-ACS. That is, they have agreed on the size of the units (parcel equivalent transects) and hence know the value of \(N\) for the site, and they have agreed on the level of confidence required that at least \(Y\%\) of the \(N\) possible units (transects) do not contain TOI.
3. Random sampling is used to select the \(n\) units (parcel equivalent transects) that will be surveyed using the detector system.
4. The prior belief should hold that the likelihood of any specific transect from the sample area 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 possible parcels (units or parcel equivalent transects) of the specified size in the sample area. |
\(n\) |
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 sample area being rejected. P d is typically a small fraction, e.g., 0.05 (5 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 \(N\) parcels or parcel equivalent 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 contain TOI is less than or equal to \(P_d\). |
Using the above notation, the statement "\(X\%\) confident that at least \(Y\%\) of the \(N\) units in the sample area do not contain TOI" can be restated as "\(100(1-\beta)\%\) confident that at least \(100(1-P_d)\%\) of the \(N\) parcels in the sample do not contain TOI." The figure below shows how the parameters of the statistical equation relate to the VSP dialogue.
The VSP user is asked to specify the parcel size, site size (both used to calculate \(N\)) , 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).
If we have a 500 acre site and our parcel size of concern is a quarter acre lot, then there are a total of 2000 possible quarter acre lot areas (\(N\) = 2000). With \(\beta\) = 0.10 and \(P_d\) = 0.05, that is, if we require 90% confidence that at least 95% of the \(N\) parcel equivalent transects in the sample area do not contain TOI, then
\(n \cong 0.5\Big(1-0.10^{\frac{1}{2000 \times 0.05}}\Big)(2 \times 2000-2000 \times 0.05+1) \cong 45\)
Hence, 45 quarter acre parcels or equivalent acreage (11.25 acres) should be surveyed. If the proposed transects are 1000 by 3 foot transects, then 164 must be selected using simple random sampling and surveyed for TOI to achieve the required 11.25 acres.
The VSP user can also specify that they want to account for their prior belief regarding the likelihood of unacceptable items (TOI) in the sample area. The number of required transects will decrease as the prior becomes stronger. For more explanation on the methodology behind this Bayesian option, see the Presence / Absence Bayesian sampling design.
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.