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This paper describes and compares two approaches for the problem of determining the number of radioactive point sources that potentially exist in a designated area and estimating the parameters of these sources (their locations and strengths) using a small number of noisy radiological measurements provided by a radiation sensor. Both approaches use the Bayesian inferential methodology but sample the posterior distribution differently: one approach uses importance sampling with progressive correction and the other a reversible-jump Markov chain Monte Carlo sampling. The two approaches also use different measurement models for the radiation data. The first approach assumes a perfect knowledge of the data model and the average background radiation level, whereas the second approach quantifies explicitly the uncertainties in the model specification and in the average background radiation level. The performances of the two approaches are compared using experimental data acquired during a recent radiological field trial.

There is growing concern in recent years regarding the risk of the smuggling and illicit trafficking of stolen radiological material. This state of affairs has heightened the spectre of radiological terrorism involving the use of an improvised radiological dispersal device (dirty bomb) to disperse radiological material over a large area using the force of conventional explosives (see, Panofsky [

Various researchers have focussed on the problem of recovering the physical location and strength of radioactive sources using measurements obtained from a number of radiological detectors. Howse et al. [

In this paper, we describe two approaches for radiological source localisation for the case when the number of sources is unknown

The experimental data for evaluating the two approaches were acquired using a Geiger-Müller detector in the presence of multiple radiological point sources of unequal strengths. Background radiation data in the absence of these radiation sources were also acquired to provide an estimate of the average background radiation level.

The organisation of the paper is as follows. The radiological source localisation problem is formulated in Section

The problem of the detection and estimation of the characteristics (e.g., location, activity) of an unknown number of static radiological point sources from a finite number of “noisy” measurements (data) obtained from radiation sensors is addressed using the Bayesian formalism. This formalism involves the application of Bayes' rule:

Here, we assume that a number

its location

its intensity rate (source strength),

The parameter vector of source

If we assume the logical independence of the various source parameters (or equivalently, the components of

The prior distribution of

Two alternative functional forms are used for the prior distribution of

Finally, the prior distribution for the source location

To prescribe a functional form for the likelihood function, we need to relate the hypotheses of interest about the unknown source(s)^{1}

The radiation counts from nuclear decay obey Poisson statistics (see, Tsoulfanidis [

The measurements of the radiation field are made using a Geiger-Müller (GM) counter. Let

The likelihood function is simply the joint density of the measurement vector

The constant

The mean signal count rate

The key assumptions in the formulation of the measurement model of (

Towards this objective, the following model is assumed for the (true though unknown) mean signal count rate

The measurement model of (

To complete the specification for the likelihood function in (

To accomplish this, we utilize a Bayesian solution for the extraction of weak signals in strong background interference described by Loredo [

We will assign a least informative prior probability for

The likelihood function in (

Given the posterior probability of

For the case

Finally, the likelihood function of (

This section describes the computational procedures that were used for extracting the source parameter estimates for source reconstruction. To this purpose, two different methodologies for sampling from two different forms of the posterior distribution for the source parameters are described.

Importance sampling using progressive correction is a computational methodology for sampling from a posterior distribution. In this paper, this sampling procedure is applied to a form of the posterior distribution for the source parameters in which the number of sources

Because the posterior PDF

Because the prior distribution will often be more diffuse than the likelihood, many samples drawn from the prior may fall outside the region of parameter space favoured by the likelihood function. Therefore, a straightforward application of (

Let

(1)

(2)

(3) Draw

(4) Compute

(5)

(6)

(7)

(8) Select

(9)

(10)

(11) Weights:

(12)

(13) Resample

(14)

(15)

(16)

(17)

The performance of the procedure depends somewhat on the number of steps

The case

The PC estimate of (

The importance sampling algorithm using progressive correction described above assumes that the number of sources is known

For this purpose, we use the minimum description length (MDL) criterion which chooses the value of

To estimate the number of sources using the MDL criterion, we first run the importance sampling using progressive correction for all values of

In Section

Towards this objective, we apply a reversible-jump MCMC (RJMCMC) algorithm. The formalization of RJMCMC algorithms for dealing with variable dimension models has been described in the seminal work of Green [

In the first category of moves (which are dimension-conserving), the parameter vector

In the second category of moves (which are dimension-changing), the source distribution is modified by

In practical terms, the RJMCMC algorithm as applied to the problem of source reconstruction can be set up as follows.

Specify values for the hyperparameters

Set the iteration counter

Starting from

Change the counter from

In Step 3,

To improve convergence of the Markov chain, we have implemented a simulated annealing procedure. This procedure is very similar to the progressive correction that is used in conjunction with the importance sampling (as described in Section

The parameter

We initialize the stochastic sampling scheme at

Next, a resample is drawn from the discrete distribution concentrated at the samples

There are many ways to implement this resampling (with replacement) from

It is necessary to specify the sequence of

When

A radiological field trial was conducted on a large, flat, and open area without obstacles within a military site in Puckapunyal, Victoria, Australia. An AN/PDR-77 radiation survey meter equipped with a gamma probe containing two Geiger-Müller tubes to cover both low and high ranges of dose rates was used to collect the radiation data. When connected to the AN/PDR-77 radiation survey meter, this gamma probe is capable of measuring gamma radiation dose rates from background to 9.99 Sv

Three radiation sources were used in the field trial: two cesium sources (

Radiation sources used in the field trial.

Source | Type | Activity (MBq) |
---|---|---|

1 | ||

2 | ||

3 |

The sources were mounted at the same height above the ground as the gamma probe. To ensure that radiation sources appear as isotropic in the horizontal plane, they were placed in a vertical configuration so that the handling rods were pointing up. Radiation dose measurements were collected on a grid of surveyed points that were carefully measured and marked beforehand in the local Cartesian coordinate system on the asphalt surface of the airfield. The data were acquired when the trolley-mounted gamma probe was positioned over individual grid points. During data collection at each grid point, the gamma probe remained stationary until sixty measurements were acquired. The exposure time for each radiation dose measurement (effectively the sampling interval) was kept constant at about

Data sets were collected with one, two, and three radiation sources emplaced, respectively. These data sets are referred to as test sets 1, 2 and 3. The sources used when collecting these data sets and their locations in the local Cartesian coordinate system are listed in Table

The locations of radiation sources in the field trial.

Test set | Source 1 Location | Source 2 Location | Source 3 Location |
---|---|---|---|

1 | — | — | |

2 | — | ||

3 |

An areal picture of the Puckapunyal airfield site where the field trial was conducted is shown in Figure

Aerial image of the Puckapunyal airfield site where the field trial was conducted. The red stars located at coordinates

In order to estimate the background radiation level in the field trial, measurements were collected at a number of grid positions in the

The importance sampling algorithm using progressive correction was applied to the experimental field trial data for source reconstruction. Towards this purpose, this algorithm was used to sample from the following posterior distribution of the source parameters:

The importance sampling algorithm using progressive correction is applied to the experimental field trial data using the following values for the hyperparameters that define the prior distribution in (

The PC algorithm is initialized by drawing a random sample in

The expansion factors and the actual number of stages of the PC algorithm are data dependent. We have tuned the PC algorithm so that on average the number of stages is directly proportional to the number of sources. Parameter

The reversible-jump MCMC algorithm (applied in conjunction with simulated annealing) was used to infer an unknown number of radiological point sources from the experimental field trial data. This involves drawing samples of radiological point source distributions from the following posterior distribution for the source parameters

which combines the likelihood function given by (^{2}

The stochastic sampling algorithm was executed with

The samples of radiological point source distributions drawn from the posterior distribution of (

In addition to these statistical quantities, the RJMCMC algorithm applied in conjunction with simulated annealing permits the normalization constant (or evidence) ^{2}^{3}

The evidence

To test the usefulness of the two approaches for source reconstruction, we have applied them to test data sets 1, 2, and 3 corresponding to one-, two-, and three-source examples, respectively. However, owing to space limitations, we will show only the results of application of the two approaches for test case 3. This test is arguably the most difficult case, but the results from the test are representative also for the other two test cases.

The results for test case 3 will be shown for two different applications: namely, the radiation measurements at the grid positions (see Figure

Firstly, we show the results of the two applications of the importance sampling algorithm using PC for test case 3. For each of these two applications, the algorithm (summarized in Algorithm

The results of the source recovery for the two applications of the importance sampling algorithm are summarized in Table

The posterior mean, posterior standard deviation, and lower and upper bounds of the 95% HPD interval of the parameters ^{2}

Parameter | Mean | Standard deviation | 95% HPD | Actual |
---|---|---|---|---|

| ||||

10.7 | 0.2 | 11.0 | ||

9.9 | 0.2 | 10.0 | ||

1930 | 43 | 1912 | ||

| ||||

2.1 | 0.6 | 3.0 | ||

49.8 | 0.3 | 50.0 | ||

330 | 31 | 392 | ||

| ||||

40.9 | 0.1 | 41.0 | ||

5.0 | 0.3 | 5.0 | ||

92.9 | 15.2 | 98.1 | ||

| ||||

10.9 | 0.01 | 11.0 | ||

9.6 | 0.01 | 10.0 | ||

1906 | 1.3 | 1912 | ||

| ||||

2.3 | 0.01 | 3.0 | ||

49.6 | 0.01 | 50.0 | ||

336 | 0.6 | 392 | ||

| ||||

40.8 | 0.00 | 41.0 | ||

5.3 | 0.01 | 5.0 | ||

83.6 | 0.73 | 98.1 |

Inference of the parameters of the three radiological point sources obtained from samples drawn from the posterior distribution

Source 1

Source 2

Source 3

Inference of the parameters of the three radiological point sources obtained from samples drawn from the posterior distribution

Source 1

Source 2

Source 3

For the application with

Next, we show the results for source reconstruction for test case 3 using the RJMCMC algorithm (applied in conjunction with simulated annealing) for the two applications with

Trace plot (a) of the number of radiological point sources

Figure

Given the fact that our best estimate for the number of sources is

The posterior mean, posterior standard deviation, and lower and upper bounds of the 95% HPD interval of the parameters ^{2}

Parameter | Mean | Standard deviation | 95% HPD | Actual | |
---|---|---|---|---|---|

| |||||

8.7 | 0.8 | 11.0 | |||

10.6 | 0.8 | 10.0 | |||

1404 | 152 | 1912 | |||

| |||||

3.0 | 0.6 | 3.0 | |||

49.8 | 0.6 | 50.0 | |||

385 | 81 | 392 | |||

| |||||

40.9 | 0.5 | 41.0 | |||

5.0 | 0.9 | 5.0 | |||

194 | 46 | 98.1 | |||

| |||||

9.3 | 0.9 | 11.0 | |||

10.2 | 0.8 | 10.0 | |||

1433 | 156 | 1912 | |||

| |||||

3.8 | 0.6 | 3.0 | |||

49.8 | 0.6 | 50.0 | |||

382 | 80 | 392 | |||

41.1 | 0.5 | 41.0 | |||

4.8 | 0.8 | 5.0 | |||

204 | 52 | 98.1 |

From Table

An examination of Table

A comparison of Tables

Histograms of the source parameters (approximating the marginal probability distributions for the parameters), associated with each of the three identified radiological point sources, are exhibited in Figure _{s}- and y_{s}-directions is generally very good. The recovery of the source intensity rates Q_{s} is quite good, although both the accuracy and precision in this recovery are poorer than the recovery of the source locations.

Inference of the parameters of the three radiological point sources obtained from samples drawn from the posterior distribution

Source 1

Source 2

Source 3

Finally, the information gain ^{4}

This paper presented two approaches for estimating an unknown number of point radiation sources and their parameters employing radiation measurements collected using a gamma radiation detector. The first approach used importance sampling with PC to sample the posterior distribution, and the second approach used the reversible-jump MCMC technique in conjunction with simulated annealing for this purpose. The two approaches were compared by applying them to experimental data collected in a recent field trial.

It was demonstrated that both approaches perform well on the experimental data: in general, the number of sources is correctly determined and the parameters (e.g., location and intensity rates) that characterize each radiological point source are reliably estimated, along with a determination of the uncertainty (e.g., standard deviation, credible intervals) in the inference of the source parameters. However, it was shown that application of the first approach with

The results of this paper suggest that Bayesian probability theory is a powerful tool for the formulation of methodologies for source reconstruction. It should be stated that the application of importance sampling using PC to the posterior distribution of (

From this perspective, it should be noted that if the Bayes factor is used instead to determine the number of sources (namely, to determine which model of a distribution of radiological point sources to use), then the methodology would be fully consistent with a Bayesian formalism (see, Morelande and Ristic [

Importance sampling using PC may also be carried out over the joint space of

The authors would like to thank A. Eleftherakis, D. Marinaro, P. Harty, and I. Mitchell for their involvement in the field trial and A. Skvortsov, R. Gailis, and M. Morelande for useful technical discussions. This work has been partially supported by Chemical Biological Radiological-Nuclear and Explosives Research and Technology Initiative (CRTI) program under project number CRTI-07-0196TD.

This hypothesis is of the form “the unknown value of the source parameters is

The evidence

Actually, the technique of thermodynamic integration dates back to Kirkwood [

The relative entropy is measured in natural units (nits) rather than the more usual binary units (bits) because a natural logarithm was used in the definition of the relative entropy, rather than a logarithm to the base 2 (see (