Maximum likelihood estimation with poisson (counting) statistics for waste drum inspection Page: 4 of 19
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transformation of the data that gets rid of this problem (see Ref. 1). This would possibly
allow use of existing least-squares software.
For the particular problem we are considering, two measurements are taken at each
detector, say the kth. The first is meant to be an observation of background radiation
counts; the second is an observation of activity radiation counts. We denote the counts
observed in the first measurement by the random variable Mk, and the counts in the second
measurement by Nk. The first measurement is taken to account for background effects in
the second measurement as it is impossible to measure activity counts alone. Complicating
the problem further is the fact that some activity counts occur in the first measurement
as well. Each detector measures the counts in spectral bins. It is assumed that the bins
used for the first measurement are disjoint from the bins used for the second measurement.
This allows us to assume that the two measurements are statistically independent. This
is only an approximation, but it permits a reasonably simple model. The mean total
background count at the kth detector is denoted by wk. This total is split between the
first measurement and the second measurement. For some 0 < T1 < 1 we assume that the
mean background count for the first measurement is T1wk and the mean background count
for the second measurement is T2wk where T2 = 1 - Ti.
The mean activity count at the ktn detector is denoted by zk. The activity counts
depend on the location of the activity in the drum. Let x be a T-vector representing
the unknown activities in the T voxels. Then we write z (x) to denote this dependence,
which is assumed to be of the form z = Ax, where A is a K x T matrix. Some small
fraction of the activity counts appear in the first measurement; we denote the mean of
this contribution by T3zk(x), where 0 < T3 1. The mean activity count in the second
measurement is T4zk(x) where T4 = 1 - 73.
It follows that the mean total count in the first measurement is
E{M} = T1wk + T3zk(x), (5a)
and the mean total count in the second measurement is
E{N} = T2wk +T4zk(x), (5b)2
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Goodman, D. Maximum likelihood estimation with poisson (counting) statistics for waste drum inspection, report, May 1, 1997; California. (https://digital.library.unt.edu/ark:/67531/metadc696835/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.