This page is designed for help with the RI TOI Estimation page. Help is also available for the Costs page and the Post-Survey Analysis page.
Remedial investigations (RI) are conducted at Department of Defense (DoD) sites to identify regions with munitions and explosives of concern (MEC). This module in VSP allows for the estimation of the rate (or count) of unacceptable items, referred to as Targets of Interest (TOI), (e.g. UXO, MEC, etc.) on a site and shows that this rate (or count) is no more than some pre-specified limit.
This design assumes that a transect survey will be performed and determines the survey area required and the location of randomly placed transects needed to confidently demonstrate that the rate (or count) of interest is no more than a pre-specified limit.
The assumptions associated with the formula for computing the number of transects are:
the distribution of the true number of unacceptable items given a sampling area follows a Binomial\((n,p)\) distribution, where \(n =\) true number of unacceptable items and \(p = \) proportion of site area that is surveyed,
the transect survey locations will be selected randomly and the size of transects is known,
prior to surveying, the likelihood of a transect containing TOI is equivalent across all possible transects,
the method used for inspection of anomalies (often digging and visual inspection) will reliably identify TOI.
no unacceptable items are found during surveying. If unacceptable items are found, the design conclusions presented are not valid and other analyses are needed (see post-survey analysis tab).
\( A \) |
is the area (user specified) of the site. |
\( N \) |
is the true (unknown) number of unacceptable items on the site. |
\( N_1 \) |
is the maximum count of unacceptable items (user specified). If a rate, \(r_1\), is specified, \(N_1=A*r_1\). |
\( p \) |
is the proportion of the site to be surveyed. |
\(k\) |
is the number of unacceptable items discovered during surveying (assumed to be 0). |
\( (1 - \alpha)\times 100\% \) |
is the desired confidence (user specified) that the true rate (or count) of unacceptable itmes is no more than the specified threshold. |
\(\mu\) |
is the true (unknown) rate of unacceptable items on the site, |
\(a\) |
is the shape parameter of the Gamma prior distribution for \(\mu\) |
\(b\) |
is the rate parameter of the Gamma prior distribution for \(\mu\) |
Given that \(N\) unacceptable items are present on the site, the probability of observing \(k\) unacceptable items when surveying \(p \times 100\% \) of the site is assumed to follow a binomial distribution:
\begin{equation} P(N; p) = \left(\begin{array}{c}N\\ k \end{array}\right) p^k (1 - p)^{N-k}. \end{equation}
Using Equation (1), the probability of the true number of unacceptable items being equal to some number \(j\), given that no unacceptable items are discovered (\(k = 0\)) during surveying is:
\begin{equation}P(N = j \mid k = 0) = (1 - p)^j \end{equation}
An informative prior distribution for \(N\) is assumed to follow a Poisson distribution with a mean equal to \(A\mu\):
\begin{equation} f_0(N = j \mid \mu) = \frac{exp\lbrace -A\mu \rbrace \left(A\mu\right)^j}{j!} \end{equation}
Additionally, the rate \(\mu\) is assumed to follow a Gamma distribution with shape parameter \(a\) and rate parameter \(b\):
\begin{equation} f_0(\mu \mid a,b) = \frac{b^a}{\Gamma(a)}\mu^{a-1}\exp\lbrace-b\mu\rbrace \end{equation}
Unconditionally, the prior distribution of \(N\) is:
\begin{equation}f_0(N=j)=\int_0^\infty f_0(N=j\mid \mu) f_0(\mu \mid a,b) d\mu = \frac{\Gamma(a+j)}{\Gamma(a)j!} \left(\frac{b}{b+A}\right)^a \left(\frac{A}{b+A}\right)^j, \end{equation}
The values for hyperparameters \(a\) and \(b\) are determined based on the user's prior knowledge. The user provides knowledge about the maximum number of unacceptable items on the site with a confidence level:
I am quite sure (with probability \( \beta \times 100\% \)) that the maximum number of unacceptable items on this site is no more than \(M\).
Additionally, users are asked to specify the shape of the prior distribution choosing one of three options in the statement:
There is a(n) equal/lesser/greater chance of having \(0.5\times M\) or fewer unacceptable items than there is of having \(>0.5\times M\).
The value of \(a\) is set equal to 0.5, 1, or 2 when the user specifies greater, equal, and lesser in the above statement, respectively. Then, the value of \(b\) is determined by using Equation (5). The value is determined by solving:
\begin{equation} 1 - \beta = \sum_{j=0}^M \frac{\Gamma(a+j)}{\Gamma(a)j!} \left(\frac{b}{b+A}\right)^a \left(\frac{A}{b+A}\right)^j, \end{equation}
for \(b\).
The posterior distribution of \(N\) given no unacceptable items are discovered is derived as:
\begin{equation}f(N=j \mid k=0) = \frac{P(N=j \mid k=0)f_0(N=j)}{\sum_{j=0}^\infty P(N=j \mid k=0)f_0(N=j)} = \frac{\Gamma(a+j)}{\Gamma(a)j!}\left(\frac{b+pA}{b+A}\right)^a \left(\frac{A-pA}{b+A}\right)^j. \end{equation}
The probability of the true number of unacceptable items on the site being no more than \(N_1\), given that no unacceptable items are discovered (\(k = 0\)), during surveying is:
\begin{equation}P(N \leq N_1 \mid k = 0) = \sum_{j=0}^{N_1} f(N=j \mid k=0) \end{equation}
Using Equation (8) the required proportion of the site to be surveyed is determined by solving the following equation for \(p\):
\begin{equation} 1 - \alpha = \sum_{j=0}^{N_1} \frac{\Gamma(a+j)}{\Gamma(a)j!}\left(\frac{b+pA}{b+A}\right)^a \left(\frac{A-pA}{b+A}\right)^j, \end{equation}
and sample area, \(S\), is given by \(S = A*p\). If a desired rate of unacceptable items, \(r_1\), is provided (rather than a desired count) calculations are done using the above equations and \(N_1 = A*r_1\). The number of transects necessary will depend on the pre-specified transect dimension and can be calculated by dividing the total sample area by the dimension of one transect.
To provide as much spatial coverage as possible, survey transects should be as small as can be practically implemented.