Information driven localisation of a radiological point source

doi:10.1016/j.inffus.2007.06.004

Information Fusion

Volume 9, Issue 2, April 2008, Pages 317-326

https://doi.org/10.1016/j.inffus.2007.06.004 Get rights and content

Abstract

The paper presents an algorithm for detection and a subsequent information gain driven control of the observer for the purpose of parameter estimation of an unaccounted point source of relatively low-level gamma radiation. The source parameters to be estimated are its location and intensity. Source detection and parameter estimation are carried out jointly in the Bayesian framework using a particle filter. The observer motion and the radiation exposure time are controlled by the algorithm. Initially the observer control vectors take predefined values until the source is positively detected. After detection, the control vectors are selected sequentially for the purpose of reduction in the observation time and consequently the radiation exposure. The selection of control vectors is carried out via a multiple-step ahead maximisation of the Fisher information gain.

Introduction

Since the end of the Cold War, the risk of nuclear proliferation has increased dramatically due to the relative ease of acquiring radioactive materials [19]. Of growing concern is that numerous accidents have already been reported involving a loss or theft of radioactive sources, which could potentially be used to improvise nuclear devices for high-impact spectacular attacks [19]. A dirty bomb, for example, is a radiological weapon which consists of a conventional explosive packaged with radioactive materials, aimed to kill or injure through the initial blast of the conventional explosive and by airborne radiation and contamination.

This paper proposes a sequential Monte Carlo technique for detection and a subsequent information driven control of the observer for the purpose of parameter estimation of an unaccounted point source of relatively low-level gamma radiation. The search for nuclear materials in a similar context has been considered earlier in [7], [9]. In [7], the authors apply a sequence of statistical hypothesis tests for a survey on a predetermined route. Our goal, however, is to perform an on-line feedback of detection and estimation results into the search plan and thus control the observer motion and the radiation exposure times. Another fundamental difference between our problem formulation and those presented in [7], [9] is that we exploit prior knowledge of the propagation properties of gamma radiation, experimentally verified to obey the inverse distance square law.

The problem can be described informally as follows. Suppose that based on an intelligence report, an area has been identified where an unaccounted radiological source is likely to be found. Our goal is twofold. First, by searching the area, the presence of the source has to be firmly established (this is the detection part). Second, if the source has been detected, it has to be localised in an optimal manner (for example, by minimising the number of measurement acquisition steps, or by minimising the total search time). This second part involves both estimation and resource allocation strategies. Our study is restricted to the case where the point source, to be detected and localised, is static and placed in an open field.

The paper is organised as follows. A formal problem description is presented in Section 2. Section 3 describes the conceptual solution based on the particle filter, for detection and estimation, and the information gain control of the observer. Section 4 details two versions of the particle filter. Section 5 is devoted to the observer control for the purpose of fast, accurate and safe estimation of the source parameters. The numerical simulation results are presented in Section 6 and the findings of this study are summarised in Section 7.

Section snippets

Problem formulation

Suppose an area has been identified in which the potential presence of a radiological point source has to be confirmed. For simplicity, let us assume that this area is flat and rectangular with limits x_min and x_max along axis x and similarly y_min and y_max along axis y.

Equipped with a radiation survey instrument, mounted on a vehicle or carried by a person, the task is to quickly confirm the presence of the source and in the case of positive detection, estimate its location and its level of

Conceptual solution

Let us for a moment disregard the sensor control aspect of the problem (i.e. how the measurements of radiation intensity are collected). The goal is to jointly detect and estimate the source parameter vector using a cumulative set of recorded measurements. The problem has many similarities to the recursive track-before-detect described in [21, Chapter 11], except that the likelihood functions are different and, more importantly, here we treat the problem as a static parameter estimation

Particle filter for integrated detection and estimation

The recursive Bayesian hybrid state estimator, which performs jointly detection and parameter estimation (Steps 5.d and 5.f.(i.)), is implemented as a particle filter (PF). The main idea of the PF is to represent the posterior distribution p(x, E∣Z_k) through a finite set of random samples (particles). When a new observation z_k+1 is received, the particles are updated in order to represent the new posterior p(x, E∣Z_k+1). An additional problem in our context is that both x and E are static

Sensor control

The radiological survey instrument is controlled automatically for the purpose of detection and parameter estimation of a radioactive source. While P_k is below a certain threshold value P^∗, in the absence of any prior information on the source location, the measurements are taken along a predefined path that scans the area in a uniform manner (known as the parallel sweeps search [10]). An example of a path scanning the area is shown in Fig. 1, where Δ is the distance between consecutive

Simulation parameters

A radiological point source of an equivalent intensity I_s = 18 × 10³ cts/s is placed at x_s = 240 m, y_s = 532 m in a 2D Cartesian plane. Our prior knowledge is as follows: x_min = 100 m, x_max = 600 m, y_min = 300 m, y_max = 800 m, I_min = 8 × 10³ cts/s and I_max = 33 × 10³ cts/s. The mean count-rate of the background radiation is μ_b = 1 cts/s. The count measurements were generated using a Poisson random number generator and the propagation model (4).

The parameters of the proposed algorithm are: N_p = N/10, N = 30,000, h = 0.005, P^∗ = 0.75, Δ = 50

Summary

The paper presented an algorithm for joint detection and parameter estimation of a radiological point source, where the emphasis was on post-detection observer control by maximisation of the Fisher information gain. The numerical simulations show a remarkably good performance of the algorithm under various conditions in the open field environment. There are many possibilities for future research and improvements of the basic algorithm described in this paper. First, a more efficient

Acknowledgments

The authors thank the anonymous reviewers for valuable comments and Mark Rutten (DSTO) and Mark Morelande (The University of Melbourne) for useful technical discussions.

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Information driven localisation of a radiological point source

Abstract

Introduction

Section snippets

Problem formulation

Conceptual solution

Particle filter for integrated detection and estimation

Sensor control

Simulation parameters

Summary

Acknowledgments

Automatica

Comput. Math. Appl.

Improved particle filter for non-linear problems

IEE Proc. F

Novel approach to nonlinear/non-Gaussian Bayesian state estimation

IEE Proc. F

Statistical data evaluation in mobile gamma spectrometry: an optimisation of on-line search strategies in the scenario of lost point sources

Health Phys.

Monte Carlo filter and smoother for non-Gaussian non-linear state space models

J. Comput. Graphical Stat.

Efficient strategies for low-level nuclear searches

IEEE Trans. Nucl. Sci.

Search and Scanning

A particle algorithm for sequential Bayesian parameter estimation and model selection

IEEE Trans. Signal Process.

Combined parameter and state estimation in simulation-based filtering

Sequential Monte Carlo methods for dynamical systems

J. Am. Stat. Assoc.