Measurement of airborne 218Po--a Bayesian approach

The standard mathematical treatment of the buildup and decay of airborne radionuclides on a filter paper uses the solutions of the so-called Bateman equations adapted to the sampling process. These equations can be interpreted as differential equations for the expectation of an underlying stochastic process, which describes the random fluctuations in the accumulation and decay of the sampled radioactive atoms. The process for the buildup and decay of airborne 218Po can be characterized as an "immigration-death process" in the widely adopted, biologically based jargon. The probability distribution for the number of 218Po atoms, accumulated after sampling time t, is Poisson. We show that the distribution of the number of counts, registered by a detector with efficiency epsilon during a counting period T after the end of sampling, is also Poisson, with mean dependent on epsilon, t, T, the flowrate and N(o), the number of airborne 218Po atoms per unit volume. This Poisson distribution was used to construct the likelihood given the observed number of counts. After inversion with Bayes' Theorem we obtained the posterior density for N(o). This density characterizes the remaining uncertainty about the measured number of 218Po atoms per unit volume of air.