A Bayesian Approach to Spectrum UnfoldingHaonan Zhu(1) , Yoann Altmann(2), Angela Di Fulvio(1), Alfred Hero(1), Sara Pozzi(1)

(1) University of Michigan, Ann Arbor, MI, (2) Heriot-Watt University, Edinburgh, U.K.

Contact: Angela Di Fulvio (difulvio@umich.edu), PIs: Prof. Alfred Hero and Prof. Sara Pozzi

Consortium for Verification Technology (CVT)

This work was funded in-part by the Consortium for Verification Technology under Department of Energy National Nuclear Security Administration award number DE-NA0002534

Motivation and Introduction

Neutron spectra unfolding

Conclusions and Future Work

• Neutron spectrometry without time-of-flight can be extremely useful in

safeguards and nonproliferation applications, e.g. neutron imaging for

material accountancy and verification (Fig. 1), to discriminate between fissile

material and other neutron emitting sources.

Fig. 1 Radiation Inspection System [1].

• Improved algorithms are needed

to successfully recover the

neutron energy spectrum from

the observed light output

response, which is an ill-

conditioned inverse problem.

References[1] “Technology R&D for Arms Control”, Office of Nonproliferation Research and Engineering, Spring 2001.[2] Mark A. Norsworthy, Alexis Poitrasson-Rivière, Marc L. Ruch, Shaun D. Clarke, Sara A. Pozzi, Evaluation of neutron light output response functions in EJ-309 organic scintillators,Nuclear Instruments and Methods in Physics Research Section A, Volume 842, 2017, https://doi.org/10.1016/j.nima.2016.10.035. [3] A. C. Kaplan, et al., “EJ-309 pulse shape discrimination performance with a high gamma-ray-to-neutron ratio and low threshold,” Nucl. Instr. Meth. A, 729, (2013)[4] M. Reginatto, “Overview of spectral unfolding techniques and uncertainty estimation,” Radiation Measurements, vol. 45, no. 10, pp. 1323–1329, Dec. 2010[5] Reginatto, M. “Spectrum unfolding, sensitivity analysis and propagation of uncertainties with the maximum entropy deconvolution code MAXED,” Nucl. Instr. Meth. A, 476, 242 (2002).[6] Harmany, Z. T. et al., “This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction Algorithms—Theory and Practice”, IEEE Trans. Image Process., 2012

Fig. 4 Unfolded neutron energy spectrum from measured Cfsource

zi light output spectrum

M is the number of detection channels

Ri(E) is the detector response

Φ(E) is the Neutron spectrum flux cm-2

ei accounts for the observation noise and

modelling error

New Simulated RESPONSEMATRIX [2]

MEASURED LIGHT OUTPUT

UNFOLDINGALGORITHM

NEUTRON SPECTRUM • A general framework of Bayesian method for spectrum unfolding is

presented, which is a generalization of traditional least-square based

methods

• A Gaussian prior is being proposed, and a state-of-art Monte Carlo

method is being applied to exploit the posterior distribution. Our

proposed method showed better unfolding results, and it provides a

quantitatively way to evaluate the uncertainty

• Further investigation on measured data with simulated response

matrix on various measured data

• Incorporate the pulse shape discrimination to the unfolding

algorithm to improve fidelity at low energies

UNFOLDING

Bayesian Method

𝑑𝐿

𝑑𝑥=𝐴𝑑𝐸𝑑𝑥

1 + 𝐵𝑑𝐸𝑑𝑥

energy deposited per unit distance𝑑𝐸

𝑑𝑥

scintillation photons emitted per unit distance𝑑𝐿

𝑑𝑥

A, B: material-specific coefficients

Fig. 3 Unfolded neutron energy spectrum from simulated

continuous source: Cf

𝒛𝒊 =

𝒋

𝑹𝒊𝒋𝜱𝒋 + 𝒆𝒊 (𝒊 = 𝟏…𝑴)

𝑓 𝜙|𝑧 =𝑓 𝑧 𝜙 𝑓(𝜙)𝑓(𝑧)

∝ 𝑓(𝑧|𝜙)𝑓(𝜙) (Bayes Formula)

f(Φ|z) is called the Posterior (the conditional distribution of neutron energy spectrum

given the light output)

f(z|Φ) is called the Likelihood

f(Φ ) is called the Prior which reflect our prior belief about the neutron energy spectrum

𝒇 𝒛 𝝓 =

𝒊

( 𝒋𝑹𝒊𝒋𝝓𝒋)𝒛𝒊

𝒛𝒊 !𝐞𝐱𝐩(−

𝒋

𝑹𝒊𝒋𝝓𝒋)

The light output follows Poisson statistics:

•Standard summary statistics (e.g., maximum a posterior probability (MAP) estimate)

can be extracted from the posterior distribution obtained. Alternatively, sampling

methods can be used to fully exploit this distribution

•Most existing optimization-based methods can be seen as special case of the maximum

a posteriori estimation strategies. If a uniform prior distribution is used, the maximum a

posteriori and maximum likelihood estimators coincide

•In this work, we use a state of art sampling method, namely a Hamiltonian Monte

Carlo method, to draw samples from the posterior distribution

•Previous published work uses prior distributions derived from data, which is known as

Empirical Bayes approaches[4,5]. Since such methods can highly depend on the data

used, they can heavily bias the estimation

•In this work, we proposed to use a flexible Gaussian prior with nearly identity

covariance matrix to reflect our prior belief that the spectrum is smooth

Unfolding Results

Estimation Strategy:

Empirical Bayes vs Bayes

Fig. 2 Neutron energy spectrum from measured AmBe source