On-line experiments on rapid detection of radionuclides based on sequential Bayesian analysis

Abstract

A rapid detection method based on sequential Bayesian analysis provides a new perspective on national security in preventing the smuggling and illegal transportation of nuclear materials. In this paper, a sequential Bayesian analysis system, which mainly consists of a LaBr₃(Ce) scintillator detector, a pulse analyzer based on a field-programmable gate array technique, and a sequential Bayesian analysis processor, is developed to directly validate the feasibility of sequential Bayesian analysis. The detection ability of ⁶⁰Co, ¹³⁷Cs, ¹³³Ba, and ¹⁵²Eu for a specific radioactivity is studied and quantified using the maximum detection distance and the equivalent minimum detection activity.

Appendices

Appendix energy distribution and estimator

The full-peak energy of the jth monoenergetic source of the mth radionuclide under the observations X _m,j (n−1) is typically a Gaussian distribution as following

$$ f_{1} (\varepsilon_{m,j} (n)|X_{m,j} (n - 1)) = \frac{1}{{\sqrt {2\pi } \sigma_{{\epsilon_{m,j}^{t} }} }}\exp \left\{ { - \frac{{(\varepsilon_{m,j} (n) - \epsilon_{m,j}^{t} )^{2} }}{{2\sigma_{{\epsilon_{m,j}^{t} }}^{2} }}} \right\} $$

(9)

$$ f_{0} (\varepsilon_{m,j} (n)|X_{m,j} (n - 1)) = \frac{1}{{\sqrt {2\pi } \sigma_{{\epsilon_{m,j}^{ \wedge } }} (n)}}\exp \left\{ { - \frac{{(\varepsilon_{m,j} (n) - \epsilon_{m,j}^{ \wedge } (n))^{2} }}{{2\sigma_{{\epsilon_{m,j}^{ \wedge } }}^{2} (n)}}} \right\} $$

(10)

If H ₁ is true, the mean and standard deviation are the true parameters denoted by superscript t, and if H ₀ is true, the mean and standard deviation are estimated parameters (denoted by superscript ^) which should be sequentially innovated by evaluating the measurement up to the nth event. The energy estimator proposed [5] is base on the Gauss-Markov representation as

$$ {\text{State prediction:}}\;\epsilon_{m,j} (n) = \epsilon_{m,j} (n - 1) + \omega_{{\epsilon_{m,j} }} (n - 1) $$

$$ {\text{Measurement model:}}\;\upepsilon_{m,j} (n) = \epsilon_{m,j} (n) + \nu_{{\epsilon_{m,j} }} (n) $$

where ω _ε  ~ N(0, R _ωω ) is an uncorrelated zero-mean, Gaussian noise, mainly depending on the shield system uncertainty and ν _ε  ~ N(0, R _νν ) is the measurement (instrumentation) noise. For the Kalman filter achieves a minimum variance estimate in the linear Gauss-Markov model case so that it is utilized as the optimal estimator in energy estimation. The algorithms of the Kalman filter are

• (1) Initialize: $\epsilon_{m,j}^{ \wedge } (0|0) =\epsilon _{m,j}^{t} ,R_{{\epsilon_{m,j}^{ \wedge } }} (0|0) = = \sigma_{{\epsilon_{m,j}^{t} }}^{2}$

• (2) Prediction: $\epsilon_{m,j}^{ \wedge } (n|n - 1) = \epsilon_{m,j}^{ \wedge } (n - 1|n - 1)$

• (3) $R_{{\epsilon_{m,j}^{ \wedge } }} (n|n - 1) = R_{{\epsilon_{m,j}^{ \wedge } }} (n - 1|n - 1) + R_{\omega \omega } (n - 1)$

• (4) Innovation: $e_{m,j} (n) = \varepsilon_{m,j} (n) - \epsilon_{m,j}^{ \wedge } (n|n - 1),R_{ee} (n) = R_{{\epsilon_{m,j}^{ \wedge } }} (n|n - 1) + R_{\nu \nu } (n)$

• (5) Gain coefficient: $K_{m,j} (n) = R_{{\epsilon_{m,j}^{ \wedge } }} (n|n - 1) \times R_{ee}^{ - 1} (t)$

• (6) Correction: $\epsilon_{m,j}^{ \wedge } (n|n) = \epsilon_{m,j}^{ \wedge } (n|n - 1) + K_{m,j} (n) \times e_{m,j} (n)$

  $$ R_{{\epsilon_{m,j}^{ \wedge } }} (n|n) = [1 - K_{m,j} (n)] \times R_{{\epsilon_{m,j}^{ \wedge } }} (n|n - 1) = \sigma_{{\epsilon_{m,j}^{ \wedge } }}^{2} (n) $$

Interarrival time distribution and estimator

The interarrival time for the jth monoenergetic source of the mth radionuclide under the observations X _m,j (n−1) and current event’s energy ε _m,j (n) is exponentially distributed as

$$ f_{1} (\Delta \tau_{m,j} (n)|\varepsilon_{m,j} (n),X_{m,j} (n - 1)) = \lambda_{m,j}^{t} \exp ( - \lambda_{m,j}^{t} \times \Delta \tau_{m,j} (n)) $$

(11)

$$ f_{0} (\Delta \tau_{m,j} (n)|\varepsilon_{m,j} (n),X_{m,j} (n - 1)) = \lambda_{m,j}^{ \wedge } \exp ( - \lambda_{m,j}^{ \wedge } \times \Delta \tau_{m,j} (n)) $$

(12)

If H ₁ is true, the mean interarrival time is theoretically calculated based on the specific model of radionuclide detection, and if H ₀ is true, the mean interarrival time is also estimated by evaluating the measurement up to the nth event using particle filter rather than the Kalman filter because of the nonlinear property of the interarrival time estimation. The interarrival time estimation model is

$$ {\text{State prediction:}}\;\;\Delta \tau_{m,j} (n) = \omega_{{\Delta \tau_{m,j} }} (n) $$

$$ {\text{Measurement model:}}\;\Delta \tau_{m,j} (n) = \Delta \tau_{m,j} (n) + \nu_{{\Delta \tau_{m,j} }} (n) $$

The noise source $\omega_{{\Delta \tau_{m,j} }} \sim f_{\omega } (\omega_{{\Delta \tau_{m,j} }} ) = \lambda_{m,j}^{t} \exp ( - \lambda_{m,j}^{t} \times \omega_{{\Delta \tau_{m,j} }} ),\lambda_{m,j}^{t} = 1/\Delta \tau_{m,j}^{t}$ and $\nu_{{\Delta \tau_{m,j} }}$ is exponential measurement (instrumentation) noise which depends on the time measurement accuracy of experimental instruments with the probability density as $f_{\nu } (\nu_{{\Delta \tau_{m,j} }} ) = \lambda_{\nu } \exp ( - \lambda_{\nu } \times \nu_{{\Delta \tau_{m,j} }} )$. The corresponding conditional probability with the state prediction $\Delta \tau_{m} (n)$ is$f_{\nu } (\Delta \tau_{m} (n)|\Delta \tau_{m} (n)) = \lambda_{\nu } \exp [ - \lambda_{\nu } \times (\Delta \tau_{m} (n) - \Delta \tau_{m} (n))]$. In this project, the particle filter is utilized to estimate the maximum a-posterior (MAP) distribution of interarrival time. The algorithms of particle filter are

• (1) Initialize: $\Delta \tau_{m,j}^{ \wedge } (0) = \Delta \tau_{m,j}^{t} = 1/\lambda_{m,j}^{t}$, sampling N _p pairs of {particle, weight}

  $$ \Delta \tau_{m,j}^{(i)} = \omega_{{\Delta \tau_{m,j} }} (0)\sim \lambda_{m,j}^{t} \exp ( - \lambda_{m,j}^{t} \times \omega_{{\Delta \tau_{m,j} }} ),W_{m,j}^{(i)} (0) = 1/N_{p} ,\quad i = 1,2, \ldots ,N_{p} $$

• (2) Prediction: $\Delta \tau_{m,j} (n) = \Delta \tau_{m,j}^{(i)} + \nu_{{\Delta \tau_{m,j} }} (n), \quad i = 1,2, \ldots ,N_{p}$

• (3) ${\text{Likelihood: }} f_{\nu } (\Delta \tau_{m,j} (n)|\Delta \tau_{m,j}^{(i)} ) = \lambda_{\nu } \exp [ - \lambda_{\nu } \times (\Delta \tau_{m,j} (n) - \Delta \tau_{m,j}^{(i)} )], \quad i = 1,2, \ldots ,N_{p}$

• (4) Weight update: $\begin{aligned} W_{m,j}^{(i)} (n) & = W_{m,j}^{(i)} (n - 1) \times f_{\nu } (\Delta \tau_{m,j} (n)|\Delta \tau_{m,j}^{(i)} ) \\ & =W_{m,j}^{(i)} (n - 1) \times \lambda_{v} \exp [ - \lambda_{v} \times (\Delta \tau_{m,j} (n) - \Delta \tau_{m,j}^{(i)} )], \quad i = 1,2, \ldots ,N_{p} \\ \end{aligned}$

• (5) Weight normalization: ${{W_{m,j}^{(i)} (n) = W_{m,j}^{(i)} (n)} \mathord{\left/ {\vphantom {{W_{m,j}^{(i)} (n) = W_{m,j}^{(i)} (n)} {\sum\nolimits_{i = 1}^{{N_{p} }} {W_{m,j}^{(i)} (n)} ,i = 1,2, \ldots ,N_{p} }}} \right. \kern-0pt} {\sum\nolimits_{i = 1}^{{N_{p} }} {W_{m,j}^{(i)} (n)} , \quad i = 1,2, \ldots ,N_{p} }}$

• (6) MAP estimate: $\Delta \tau_{m,j}^{ \wedge } (n) = \Delta \tau_{m,j}^{(k)} ,W_{m,j}^{(k)} (n) = \mathop {\hbox{max} }\limits_{{1 \le i \le N_{p} }} W_{m,j}^{(i)} (n) \Rightarrow \lambda_{m,j}^{ \wedge } (n) = 1/\Delta \tau_{m,j}^{ \wedge } (n)$

Normalized detection probability estimator

According to the definition mentioned above in Eq. (5), the normalized detection probability as the function of detection efficiency, photoelectric effect probability and emission probability corresponding to all monoenergetic sources of the mth radionuclide can be described as

$$ L_{m,j}^{t} = \frac{{p_{m,j} \eta_{m,j} \alpha_{m,j} }}{{\sum\nolimits_{j = 1}^{{K_{m} }} {p_{m,j} \eta_{m,j} \alpha_{m,j} } }} $$

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Meanwhile, a sequential counting technique is used to obtain the estimated normalized detection probability. The sequential counting estimator for the jth monoenergetic source of the mth radionuclide is given by

$$ L_{m,j}^{ \wedge } (n) = \frac{{M_{m,j} (n)}}{{\sum\nolimits_{j = 1}^{{K_{m} }} {M_{m,j} (n)} }} $$

(14)

where M _m,j (n) is the total counts of full energy events for the jth monoenergetic source of the mth radionuclide at arrival time n.