Keywords: gamma-ray spectra, acquisition, 2, 2, 2 Photon interactions with matter, 2, 2, 3 Gamma-ray spectrum contributions, spectrometry, spectrum analysis, 2, 4, 2 Full spectrum analysis, gamma-ray spectrum analysis

Je tiens tout d'abord à remercier les rapporteurs de la thèse Jean-François Giovannelli et Michel Paindavoine, pour leur lecture attentive de ma thèse. Je tiens à remercier Thomas Rodet d'avoir accepté d'être président du jury. Je remercie également tous les membres du jury d'avoir accepté d'assister à la présentation de ce travail.

Je tiens à remercier mon directeur de thèse, Jérôme Bobin et ma tutrice de thèse, Anne De Vismes Ott, qui m'ont apporté un encadrement exceptionnel tout au long de ces trois ans de thèse. Les discussions que nous avons eues ainsi que leurs conseils sont pour beaucoup dans le résultat final de ce travail. Je remercie également Christophe Bobin et Rémi André pour leurs aide et

Résumé

Dans le cadre de la mission de surveillance radiologique de l'environnement de l'Institut de Radioprotection et de Sûreté Nucléaire (IRSN), le Laboratoire de métrologie de la radioactivité dans l'environnement (LMRE) effectue des mesures de la radioactivité des échantillons prélevés dans l'environnement afin d'identifier et quantifier les radionucléides, naturels et artificiels. La radioactivité dans les échantillons de l'environnement est, entre autres techniques, mesurée par spectrométrie gamma, mesure rapide, non-destructive et multi-élémentaire. L'objectif de la thèse est d'améliorer les performances de cette technique, en particulier en termes de limites de détection, en proposant de nouvelles méthodes d'analyse, qui permettent de rendre la détection des radionucléides plus sensible. Cette sensibilité accrue permet d'atteindre les niveaux traces des radionucléides artificiels rencontrés dans les mesures de routine, mais également d'avoir des mesures plus courtes, ce qui est particulièrement intéressant en situation de crise.

Contexte de la thèse

Le contexte des mesures de la radioactivité de l'environnement par spectrométrie gamma est présenté dans le chapitre 2. Un spectre est la distribution en fréquence des dépôts d'énergie dans le détecteur due aux photons incidents. Il est composé, pour un photon d'énergie E, d'un pic d'absorption totale à l'énergie E, ainsi qu'un fond continu, appelé fond Compton, à plus basse énergie. Un radionucléide pouvant émettre plusieurs photons, le spectre individuel du radionucléide est composé de plusieurs pics et fonds associés. Enfin, un échantillon de l'environnement contient plusieurs radionucléides induisant un spectre complexe qui est la somme des spectres individuels des radionucléides.

L'analyse d'un spectre gamma est habituellement basée sur l'étude des pics : les énergies des pics pour identifier les radionucléides et le nombre d'événements dans les pics pour quantifier l'activité de chaque radionucléide. Cependant, cette approche présente des limitations, car elle n'exploite que les informations des pics et elle ne prend pas en compte la statistique de Poisson du processus physique de la détection. Dans le cadre de la thèse, nous proposons de prendre en compte le spectre de chaque radionucléide dans sa globalité, qui permet d'utiliser l'ensemble de l'information disponible dans toute la gamme d'énergie, ainsi que la statistique de Poisson du modèle.

Démélange spectral

Dans le chapitre 3, nous proposons d'analyser des spectres gamma par le démélange spectral, qui consiste à séparer un spectre gamma en spectres individuels des radionucléides. Considérant le spectre comme un vecteur, l'estimation des activités des radionucléides consiste à estimer les poids de mélange associés aux signatures spectrales des radionucléides. D'un point de vue mathématique, ce problème peut s'écrire comme un problème inverse régularisé, les signatures spectrales étant connues, le démélange spectral est d'abord étudié avec une contrainte de non-négativité.

Les expériences sur les spectres simulés démontrent que, comparé aux estimateurs des moindres carrés, le démélange spectral basé sur la statistique de Poisson permet d'améliorer la précision de l'estimation avec des biais d'estimation et des incertitudes plus faibles. L'analyse d'un spectre dans sa globalité est plus efficace que les méthodes basées sur les pics, en particulier pour les radionucléides dont les spectres sont fortement corrélés. La méthode présente également une amélioration de la sensibilité lors de l'analyse des mesures d'échantillons de filtres d'aérosols, où la méthode standard présente toujours des difficultés pour la détection de 137 Cs, un radionucléide présent à l ' é t a t d e t r a c e d a n s l ' e n v i r o n n e m e n t . L e d é m é l a n g e s p e c t r a l p e r m e t a i n s i de diminuer le temps nécessaire à sa détection de 8 jours à 4 jours.

En pratique, l'ensemble des radionucléides présents dans un échantillon de l'environnement n'est jamais parfaitement connu. Le démélange avec un ensemble supposé de radionucléides plus grand ou plus petit que ceux réellement présents peut générer des biais de l'estimation, ainsi que de fausses identifications de radionucléides qui ne sont pas présents dans l'échantillon. Ceci nécessite la sélection de modèle de la combinaison linéaire des signatures spectrales. Pour ce faire, dans le chapitre 4, nous proposons d'estimer conjointement l'ensemble des radionucléides actifs et leur poids de mélange. Cette approche appelée démélange spectral parcimonieux est étudiée afin de trouver le plus petit ensemble de radionucléides qui permet d'expliquer le spectre mesuré. Pour ce faire, nous proposons un nouvel algorithme OMP (Orthogonal Matching Pursuit) basé sur la statistique de Poisson. Cet algorithme sélectionne séquentiellement le radionucléide qui maximise la vraisemblance de Poisson et estime les poids de mélange des radionucléides sélectionnés avec l'algorithme de démélange spectral.

Nous montrons que l'algorithme du démélange spectral parcimonieux proposé permet d'améliorer la précision de l'estimation en limitant les fausses identifications et diminue les biais de l'estimation pour les radionucléides actifs.

Utilisation métrologique du démélange spectral

Dans les chapitres 3 et 4, les études sont effectuées sur l'analyse des spectres gamma par le démélange spectral. En général, l'utilisation métrologique d'un algorithme d'analyse nécessite l'évaluation des limites caractéristiques pour la prise de décision et l'étalonnage du détecteur pour l'analyse quantitative des résultats.

Nous nous concentrons essentiellement sur deux limites caractéristiques dans le chapitre 5: le seuil de décision et les limites de l'intervalle de confiance. i), Le seuil de décision (SD) permet de décider si un radionucléide est présent ou non. Le SD associé à un certain taux de faux positif pour un radionucléide est déterminé par un test statistique basé sur le fond équivalent estimé dans le modèle de démélange spectral. ii), Nous proposons également d'estimer les intervalles de confiance à partir de la matrice de Fisher. La variance de la distribution de l'estimation est approximée par l'inverse de la matrice de Fisher. Ces approches sont évaluées avec les spectres gamma simulés, ceci permet de valider leurs applications pour analyser les spectres de mesure.

Dans le chapitre 6, l'étalonnage est étudié pour un détecteur du laboratoire en utilisant le démélange spectral. L'étalonnage en rendement de détection et en résolution consiste à ajuster le modèle de simulation pour que les signatures spectrales simulées reproduisent les réponses expérimentales. Le ré-étalonnage en énergie permet de corriger le décalage en énergie des signatures spectrales simulées. Ces étapes d'étalonnage sont évaluées et validées avec une source d'étalonnage dont les activités sont connues.

Les résultats obtenus pour les mesures expérimentales montrent que la nouvelle méthode démélange spectral est plus sensible que l'analyse standard, plus particulièrement pour la détection et la quantification des radionucléides à f a i b l e n i v e a u . the i th entry of x X matrix X ij the [i, j] th entry of X x j the j th column of X diag (x)d i a g o n a l m a t r i x w i t h d i a g o n a l e l e m e n t s : x 1 ,...x n X † pseudo inverse matrix of X Specific notations of gamma-ray spectrum model

List of Abbreviations

Introduction

This thesis focuses on the development of analysis tools for gamma-ray spectrometry. Involved in the radioactivity monitoring work, gamma-ray spectrometry measurements need to be as sensitive and rapid as possible to provide timely and reliable information to the population and the authority. In this context, the study to improve the gamma-ray spectrum analysis has attracted more and more attention in the scientific community. The key contribution of this work is the solution it provides for the gamma-ray spectrum analysis.

In the first part of the thesis, we thoroughly investigate the spectral unmixing methods for gamma-ray spectrum analysis, as well as their applications to different types of measurements. Such methods have been evaluated with both synthetically generated data and experimental data. They present significant advantages in the task of providing more sensitive analysis than standard methods.

In the second part, we explore the metrological use of the investigated spectral unmixing methods. The accurate quantitative analysis requires the proper instrumentation calibrations and needs to cope with the standards in radioactivity measurements. Different metrological aspects of the quantitative analysis have been investigated and further applied to analyze experimental measurements performed in IRSN (French Institute for Radiation Protection and Nuclear Safety) laboratory/LMRE. The contribution of this thesis is as follows:

• In Chapter 2, we firstly present the environment radioactivity measurements with the gamma-ray spectrometry technique. Next, we introduce the gamma-ray spectrometry instrumentation, including the main components of a gamma spectrometer and specific installations in the laboratory. To better understand the features of a gamma-ray spectrum (i.e., data to be analyzed), we briefly discuss the photon interactions that characterize a gamma-ray spectrum. We end up with the review of the state-of-the-art studies in the field of gamma-ray spectrum analysis, where we present the existing analysis methods and their limitations.

• In Chapter 3, we start by overcoming the activity estimation with spectral unmixing techniques, which allows accounting for the full spectrum analysis of a gamma-ray spectrum. More precisely, we formulate the problem as a regularized inverse problem, where activities appear as mixing weights related to individual spectra (i.e., spectral signatures) and the non-negativity constraint of the radionuclides' activities is taken into account. We investigate different approaches to solve the underlying spectral unmixing problem:

-Standard least squares regression and re-weighted least squares regression that can better account for the Poisson statistics.

-Poisson based spectral unmixing that takes into account the precise physical model underlying the detection process.

Their estimation performances are evaluated as follows:

-The algorithms are first applied to the experiments on gammaray spectra simulations of two radioactive sources, which enables us to compare the proposed approaches and better understand the limitations of standard gamma-ray spectrum analysis methods. As a result, the Poisson-based full spectrum analysis presents significant advantages with lower estimation errors and lower uncertainties.

-Next, we further evaluate the estimation performances with realistic simulations of real data, and particularly the ability to estimate the low-level artificial radionuclides as a challenging issue in real data analysis.

-Finally, the study on real data contains two aspects: i), assessing the impact of the spectral signatures' dictionary. ii), comparing the sensibility of the Poisson-based spectral unmixing to the standard method used in the laboratory. The main conclusion that can be drawn is that the Poisson-based spectral unmixing allows reducing the time to detect the low-level 137 Cs in environmental samples from 8 days to 4 days.

While providing promising results on both simulated and real data, the proposed Poison-based spectral unmixing analysis is however sensitive to the spectral signatures' dictionary, which requires the identification of active radionuclides present in the measurement.

• In Chapter 4, the identification of the spectral signatures' dictionary is further investigated, which amounts to finding the subset of active radionuclides present in a gamma-ray spectrum measurement. For this purpose, the spectral unmixing is extended to jointly estimate the subset of active radionuclides and their activities, which requires adding a model complexity penalty in the spectral unmixing.

To identify the smallest subset of active radionuclides that best explains the measured spectrum, we investigate the spectral unmixing with a sparsity constraint. By reviewing available approaches that enforce the sparsity constraint in optimization problems, we propose the Orthogonal Matching Pursuit (OMP) algorithm that sequentially selects active radionuclides from a measured spectrum. However, the Poisson statistics of the gamma-ray spectrum data makes the selection procedure difficult. We propose a novel Poisson-based OMP algorithm. Such a greedy sparse regression method sequentially selects the radionuclide that maximizes the Poisson likelihood and stops selecting new radionuclides using a statistical deviance test.

Experimental results on both simulated and measured spectra are presented and compared to the standard OMP algorithm, the proposed Poisson based OMP algorithm presents the following advantages:

-Comparing to the OMP algorithm that not fully accounts for the Poisson statistics, the Poisson-based OMP algorithm is shown to improve the gamma-ray spectrum analysis while limiting the false identification of radionuclides that not present in the measurement, and reducing the estimation bias of active radionuclides.

-The proposed deviance-based stopping rule is compared to other information criteria. As a result, the deviance-based stopping rule provides more accurate radionuclides' identification.

The Poisson-based spectral unmixing approach for gamma-ray spectrum analysis is thoroughly investigated in Chapter 3 and 4. The main focus of the second part of the thesis is the metrological use of the proposed algorithms. The aim is to provide accurate quantitative analysis of gamma-ray spectra measurements that cope with standards of the radioactivity measurements.

• In Chapter 5, we present the assessment of two characteristic limits: the decision threshold and the limits of the confidence intervals. These characteristic limits are related to statistical limits for decision making purposes in experimental data analysis.

The chapter starts with the definitions of the characteristic limits, where we present how these limits are related to the statistical hypothesis framework. Next, we discuss the quantification of the exact characteristic limits with Monte Carlo simulations, which can not be used in practice due to the massive computation. Therefore, we propose alternative approaches and compare the results to those carried out with the Monte Carlo simulations.

-The decision threshold of the Poisson-based spectral unmixing algorithm is firstly investigated, we propose a statistical test-based approach that enables accounting for the full spectrum information.

-Next, we explored the use of Fisher information matrix to derive the confidence intervals of the results.

The above methods are evaluated with realistic data of the routine measurements, which confirms the validation of the procedures to analyze experimental data performed in the laboratory.

• In Chapter 6, we focus on the quantitative analysis of a gamma-ray spectrum, which depends on not only the spectrum analysis algorithm, but also the proper calibrations of the instrument. As the standard calibration procedures in standard peak-based analysis can not be used in the spectral unmixing analysis, we propose novel calibration methods that adapt to the full spectrum analysis.

The main calibration steps include the energy calibration, the resolution calibration, and the efficiency calibration. The evaluation of these calibration steps with a standard gamma source (i.e., with known activities) allows validating a pipeline to analyze gamma-ray spectra of routine aerosol filter measurements performed with the same detection system.

Finally, the calibrated detection system is applied to analyze experimental data. The results show a significant improvement of the sensitivity to detect low-level artificial radionuclides.

CHAPTER 2

Context of the radioactivity measurements with gamma-ray spectrometry In this chapter, we introduce the basics of the gamma-ray spectrometry and its application to radioactivity measurements. In Section 2.1, we present the radioactivity measurements, as well as the quantitative analysis of radioactivity. At the end of the section, we present the gamma-ray spectrometry measurements in the laboratory and the procedure of typical aerosol measurements investigated in this thesis. In Section 2.2, we present the instrumentation of the gamma-ray spectrometry, where we firstly present the detection principle of different detectors and their data acquisition system. Next, we focus on the detection mechanisms, how photons transfer energy to electrons, and the major features of a photon spectrum, which enables us to understand the contributions of a gamma-ray spectrum. Section 2.3 provides the studies to improve the detection limits with gamma-ray spectrometry. The chapter ends up with the state of the art of the gamma-ray spectrum analysis in Section 2.4.

Environment radioactivity measurements

The radioactivity phenomenon

The radioactivity is the phenomenon in which a nucleus, unstable due to an excess of protons and/or neutrons, disintegrates into another nucleus. We begin with some basic nuclear notations:

• A chemical element is determined by its number of electrons Z (i.e., atomic number) related to its name and its chemical symbol. e.g.,t h e chemical symbol "C" for carbon that has an atomic number of Z =6. The elements and their corresponding symbols can be found in the periodic table of elements, which is a tabular display of the chemical elements arranged by atomic number (see Appendix A).

• A nuclide is characterized by the number of protons (equal to the number of electrons Z), the number of neutrons (N )a n dt h ee n e r g ys t a t e of the nucleus. It is noted as A Z X, where X is the chemical symbol, Z is the number of protons, and A = Z + N is the mass number. e.g., 12 6 Cw i t h6p r o t o n sa n d6n e u t r o n s ,u s u a l l ys i m p l yn o t e dw i t hi t sm a s s number 12 C.

• Isotopes of a given element have the same number of protons but different numbers of neutrons. For instance, the nuclides 12 C, 13 Cand 14 C are isotopes of carbon. 12 Ci ss t a b l e ,w h i l e 13 Ca n d 14 Ca r eu n s t a b l e isotopes of carbon.

• A radionuclide is an unstable nuclide that has excess of nuclear energy and decays by the emission of nuclear radiation to achieve the stability (e.g., 60 Co).

The disintegration of a radionuclide (called "parent") into another nuclide (called "daughter") is accompanied by the emission of alpha radiation or beta radiation. The alpha decay emits an alpha particle identical to 4 2 He with two protons and two neutrons. The beta decay transforms a neutron into a proton through the emission of an electron ( ), or transforms a proton into an e u t r o nt h r o u g ht h ee m i s s i o no fapo s i t r o n( + ).

↵:

A Z X ! A 4 Z 2 Y+ 4 2 He : A Z X ! A Z+1 Y+e + ⌫ + : A Z X ! A Z 1 Y+e + + ⌫
The daughter nuclide can be created in its fundamental energy level, but more often in an excited level. The decay is thus followed by the deexcitation of the daughter radionuclide accompanied by the emission of a photon (i.e., electromagnetic radiation) called "gamma-ray". Some other physical phenomena lead to the emission of other photons, called "X -ray". The difference between gamma-ray and X-ray is that they are of different origin, while the gamma-ray originates from the deexcitation of the nucleus and the X-ray originates from the deexcitation of the atom from an excited level to al e v e lo fl o w e re n e r g y . T h ee m i t t e dp h o t o ne n e r g yi se q u a lt ot h ed i fference between the energies of the initial excited state and the final state.

The photon energy according to the emission of a photon is:

E = hc (2.1)
where h is the Planck constant, c is the speed of light in vacuum, and is the wavelength of the photon.

The commonly used unit of photon energy is the electronvolt (eV): 1eV = 1.602176634 ⇥ 10 19 J 2.1.2 Why measuring radioactivity in the environment ?

Environment radioactivity measurements, which are performed in worldwide laboratories, play a central role in the field of radiation protection.

Within the French Institute for Radiation Protection and Nuclear Safety (IRSN), the Laboratory of Environment Radioactivity Metrology (LMRE) is in charge of environmental radioactivity measurements for several purposes:

• Monitoring the environmental radioactivity1 in France to ensure that the activity level in the environment is consistent with regular radioactive releases from industrial and human activities.

• Rapid detection and rapid characterization of sources under emergency conditions, such as increasing radioactivity levels due to an incident or an accident with radioactive releases.

• Radioecology studies for a better understanding of the behavior and transfer mechanisms of the radionuclides in the environment, e.g.,modeling the dispersion of radionuclides in the environment can help us to predict the atmospheric dispersion of artificial radionuclides in case of incident or accident. The transfer model of radionuclides in the environment can also be used to assess the human health effects resulting from the radioactive contamination in the environment.

For instance, the monitoring of activity concentration of 137 Cs (Half-life =3 0 . 1 7y e a r s )i nt h ea i rm e a s u r e di nt h ee n v i r o n m e n ti nF r a n c ei ss h o w n in Figure 2.1. It is an artificial radionuclide resulting mainly from postatmospheric nuclear weapon tests and the Chernobyl accident.

Measuring radioactivity in the environment requires tackling the quantification of the radionuclides' activity in environmental samples, e.g., aerosols, sediments, biological samples (fauna and flora), etc. It is a challenging problem since a given radionuclide can occur more than one mode of decay and more than one radionuclide can be present in the sample to be measured. Next, we will present the quantitative analysis of gamma-ray emitting radionuclides.

Quantitative analysis with gamma-ray spectrometry

Modeling radioactivity measurements first necessitates accounting for the time evolution of the radioactivity phenomenon. The radioactive decay process can be described by:

dN dt = N (2.2)
where N is the number of radioactive nuclei, (s1 )isthedecayconstant that is specific for each radionuclide. emitted photons of different energies is directly related to the radionuclide's activity by their emission probabilities, also called intensities. The emission probability of a photon energy is defined as the number of emitted photons of this energy per 100 disintegrations.

The decay process of each radionuclide is summarized by its decay scheme, which contains its "daughter nuclide" and the ↵ or decay, as well as the gamma-ray emissions. Taking the example of 60 Co for which the simplified decay scheme is shown in Figure 2.2. 1 . Figure 2.2 -Simplified decay scheme of 60 Co.

The 60 Co disintegrates by 1 emission to excited levels of 60 Ni, mainly to the 2505.7 keV energy level (99.88 % of the disintegrations of 60 Co). As illustrated in the decay scheme, this decay is mainly accompanied by the emission of two photons with their according intensities:

• Photon of 1173 keV due to the deexcitation of the daughter nuclide 60 Ni from its excited level at 2505.7 keV to another level at 1332.5 keV, intensity = 99.85 %.

• Photon of 1332.5 keV due to the deexcitation of 60 Ni from the excited level 1332.5 keV to its fundamental level, intensity = 99.9826 %.

In general, for a given radionuclide, the number of emitted photons per second of an energy level, noted N , with its according intensity, noted I, enables calculating the activity of the radionuclide:

Activity (Bq) =
N (number of emitted photons of energy E/s) I(intensity of energy E) (2.6)

In this context, measuring an energy spectrum of a gamma-emitting source allows the identification and the quantification of radionuclides. For instance, the detection of gamma rays of 1173 keV and 1332.5 keV in such an energy spectrum reveals the presence of 60 Co in the sample, and the number of photons (1173 keV and 1332.5 keV) observed in the spectrum related to the number of emitted photons by the detection efficiency, provides the activity of 60 Co in becquerel (Bq).

Gamma-ray spectrometry measurements in the laboratory

In the framework of radioactivity measurements, different environmental samples are measured in IRSN/LMRE. The samples of different origins, such as waters, aerosol filters, mineral samples (soils, sediments), and biological samples (fauna and flora) are collected, prepared (e.g., dried, freeze-dried, evaporated, calcined, ground, sieved, homogenized) and packed into cylindrical polyethylene containers. Next, we measure the activities of radionuclides in a sample (i.e., gamma-ray and X-ray source), which are:

• Naturally occurring radionuclides including cosmic ray induced radionuclides (e.g., 7 Be, 22 Na) and telluric radiation (e.g., 40 K, 210 Pb, 228 Ac).

• Artificial radionuclides that are:

present at trace levels in the environment due to the normal discharges of the nuclear facilities (e.g., 129 I, 60 Co, 110m Ag), and the global fallout due to the atmospheric nuclear weapon tests and the Chernobyl accident (e.g., 137 Cs).

potentially released in case of incident or accident (e.g., 131 I, 134 Cs).

My investigations focused mainly on aerosol filter measurements intending to detect releases due to low-magnitude incident or accident with higher magnitude but farther away. This is particularly interesting as the first contamination vector for a release is the air transfer. The main steps to measure aerosol samples in the laboratory are described in Figure 2.3.

In the analysis of environmental radioactivity using gamma-ray spectrometry, the activity estimation of low-level radionuclides is particularly required

State of the art of gamma-ray spectrum analysis

In the previous sections, we introduced the context of gamma-ray spectrum measurements. Thanks to the spectral features of radionuclides characterized by their decay schemes, a measured gamma-ray spectrum allows identifying and quantifying radionuclides. In this section, we review the state of the art of the gamma-ray spectrum analysis methods.

Peak-based gamma-ray spectrum analysis

The peak-based method (e.g., Genie 2000 software from Canberra1 )i s conventionally used in the gamma-ray spectrum analysis, where radionuclides are identified thanks to their characteristic photon energy peaks and quantified from the observed counts in the peaks, which are proportional to the radionuclides' activities. This approach is usually based on Region of Interest (ROI) (see simplified illustration in 2.8 and [Gilmore, 2008] for more details).

In brief, a radionuclide is firstly identified from its characteristic photon energy peaks. Then the method estimates the background counts n b (i.e., Continua of radionuclides) from the average of two regions (see Figure 2.8) nearby with:

n b = p 2b (n b1 + n b2 )( 2 . 7 )
The net counts n N can be then calculated from:

n N = n g n b n N 0 (2.8)
where the gross number in the ROI noted n g (i.e., the total counts in the ROI), and n N 0 is the net counts of the background radiation spectrum to be subtracted if the peak is also detected in the background radiation spectrum.

The peak-based analysis has been further extended to account for the exact Poisson statistics of the measurements in [START_REF] Kirkpatrick | Poisson statistical methods for the analysis of low-count gamma spectra[END_REF]. In this paper, the authors show that the ROI analysis taking into account the Poisson statistics provides improved accuracy comparing to traditional Gaussian methods. However, the analysis only relies on the photon peaks. It is of interest to further account for the entire spectrum that provides more information.

Full spectrum analysis

To overcome the limitation of the gamma-ray spectrum analysis due to the overlapping of individual spectra of radionuclides, the problem can be addressed by the full spectrum analysis. This approach aims at determining the radionuclides' activities by using the observed data in all the energy range of the measured spectrum, including the full energy peaks and the Compton continuum of each radionuclide.

Full spectrum analysis (FSA) has been studied in [START_REF] Hendriks | Full-spectrum analysis of natural -ray spectra[END_REF], [START_REF] Caciolli | A new fsa approach for in situ ray spectroscopy[END_REF], [START_REF] Jeong | Gamma-ray Full Spectrum Analysis for Environmental Radioactivity by HPGe Detector[END_REF] and their references. In these papers, the FSA is applied to analyze different gamma-ray spectra measurements, it addresses the analysis of a spectrum of M channels, 8i 2 [1,...M], while the linear mixing of individual spectra can be modeled with:

y i = N X j=1 a j j (i)+b i (2.10)
where y i is the sum of the standard spectra ( 1 ,..., j )o fr a d i o n u c l i d e s multiplied by their activity concentrations a j and the background spectrum b j in the i th channel, and N is the number of radionuclides.

The gamma-ray spectra detection follows the Poisson distribution:

x i ⇠ Poisson (y i )( 2 . 1 1 )
where x i is the counts observed in the i th channel. The solution of this Poisson regression problem is the least squares weighted by the variance:

min M X i=1 (x i y i ) 2 2 i
(2.12) while the variance of the Poisson distribution is given by 2 i = y i , which is the unknown linear mixing model.

The investigations of FSA make use of the weighted least squares method that determines the activities of radionuclides by minimizing the least squares errors weighted by the observed data (i.e., measured spectrum). min

M X i=1 (x i y i ) 2 x i (2.13)
These studies show that using the full spectrum information provides better counting statistics thus a better estimation accuracy. However, this approach uses the least squares weighted by the measured spectrum is not fully adapted to account for the Poisson statistics of the data, particularly when the mean/ variance value y i is small.

Machine learning algorithm in gamma-ray spectrum analysis

Other contributions of the activity estimation in the field of machine learning algorithms were also applied to gamma-ray spectral analysis.

In [START_REF] Yoshida | Application of neural networks for the analysis of gamma-ray spectra measured with a ge spectrometer[END_REF], an artificial neural network (ANN) algorithm is applied to identify radionuclides from gamma-ray spectra by using peak energy data. The performance is however limited when the peaks are overlapped to the Compton continua of other radionuclides.

In [START_REF] Sharma | Anomaly detection in gamma ray spectra: A machine learning perspective[END_REF], authors have presented the application of machine learning to the anomaly detection in gamma-ray spectra. The purpose in this paper is to apply a supervised classification framework to measured gamma-ray spectra by labeling them with normal or certain class of anomaly events.

However, the stated approaches based on neural networks addresses the gamma-ray spectrum analysis problem with measured spectra, which do not allow to precisely account the physical model underlying the detection.

Conclusion

The stated gamma-ray spectrum analysis methods and their limitation can be summarized in These studies reveal that the standard peak-based gamma-ray spectrum analysis can be improved by accounting for the Poisson statistics or the full spectrum analysis. This thesis addresses a more systematic study on the development of alternative spectrum analysis tools, from the mathematical modeling of the gamma-ray detection physics to the metrological aspects for the activity determination in the real data analysis.

CHAPTER 3

Spectral unmixing From a general perspective, the problem of identifying and estimating the activity of radionuclides from gamma-ray spectra can be tackled as a traditional inverse problem in signal processing. While spectral unmixing is now standard in other fields of research, such as remote sensing [START_REF] Bioucas-Dias | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches[END_REF], [START_REF] Keshava | Spectral unmixing[END_REF], it has seldom been investigated in gamma-ray data analysis. This chapter first reports on the development of algorithms for spectral unmixing, allowing to carefully account for the underlying mixing model and the statistics of the measure. Next, we apply the spectral unmixing algorithms to gamma-ray spectra and assess their performances to analyze both simulated and experimental data. The contribution of this chapter is as follows:

• The problem of estimating radionuclides' activities with spectral unmixing is formulated in section 3.1. The proposed approach tackles the unmixing problem as a regularized linear inverse problem involving Poisson-distributed measurements.

• Section 3.2 explores the use of unmixing algorithms for activity estimation. To that end, we propose a novel regularized maximum Poisson likelihood estimator. Meanwhile, algorithms based on least squares estimators are also presented for a comparison purpose.

• To evaluate the prop osed sp ectral unmixing approaches, we first investigate the application of spectral unmixing to synthetic data (i.e., simulated gamma-ray spectra).

-In contrast to standard unmixing problems that involve additive Gaussian noise, the Poisson nature of the measurements' statistics makes the noise highly dependent on the actual mixing. Therefore, the ability to precisely estimate the mixing weight of a given radionuclide will strongly depend on the others' contributions. We evaluate this impact in Section 3.3 with simulated gamma-ray spectra that contain two radioactive sources in both the HPGe detector and NaI detector settings.

-Next, we focus on realistic data of HPGe gamma-ray measurements in Section 3.4, where simulations with realistic activities of radionuclides are analyzed with the proposed algorithms to assess the ability to analyze real data.

• Experimental results on real spectra are presented in Section 3.5, where the implementation of spectral unmixing algorithms is assessed with sequential data of an aerosol sample measured with HPGe detector.

In this chapter, we consider that the spectral signatures Φ and the background b are known in beforehand. The estimation of the mixing weights a from the measured spectrum x can be addressed by minimizing some distance (i.e., data error associated to noise, noted n)be t w e e nt h ed a t ax and the model Φa + b.

The activity estimation can be therefore viewed as an inverse problem that minimizes an objective function, in which the error term n is data dependent in the Poisson statistics model. In such context, to account for the precise Poisson statistics of the spectroscopic measurement, we make use of an estimator that maximizes the likelihood related to the Poisson statistics. In this setting, the probability to observe a given number of counts x i in the i th channel is given by:

P ⇣ X i = x i [Φa] i + b i ⌘ = x i i e i x i ! (3.3) where i =[ Φa] i + b i .
Thanks to the statistical independence of the channels, the joint probability or likelihood for the different channels is then given by:

P ⇣ X = x Φa + b ⌘ = Y i x i i e i x i ! (3.4)
Maximizing the likelihood is then equivalent to minimizing the neg-loglikelihood, which leads to the following Poisson statistics-based activity estimator:

âPoisson

2 argmin a Φa + b x log (Φa + b)( 3 . 5 )
where is the Hadamard product. This optimization problem can be addressed as a generic inverse problem ( [START_REF] Bertero | Introduction to Inverse Problem in Imaging[END_REF]) of the form:

â 2 argmin a f (a)+g(a)( 3 . 6 )
where the objective function consists of two terms:

• f (a): data fidelity term related to the minimization of data error (e.g., f (a)=Φa + b x (Φa + b)f o rd e s c r i be dP o i s s o n -e s t i m a t o r ) .

• g(a): regularization term, which penalizes the solution of a with some prior information.

More generally, the problem can be formulated as a convex optimization problem of the form:

â 2 argmin a f (a)+g 1 (a)+... + g n (a)( 3 . 7 )
where g i (a), 8i 2 [1,...n] for different regularization terms. In the gammaray spectrum analysis, we can impose physical constraints in the minimization problem, such as the non-negativity of activities and the number of active radionuclides present in the measurement. The solution of a can also be penalized by data-driven constraints extracted from an archive of measurements, for instance, the interval of radionuclides activities that are commonly present.

Spectral unmixing algorithms

In the gamma-ray spectrum problem formulation, the mixing weights a is an array with non-negative entries. We firstly focus on overcoming the optimization problem by adding the non-negativity constraint.

â 2 argmin a f (a)+i. a 0 (3.8)
where i. a 0 is the characteristic function of the convex set (i.e. nonnegative orthant) {a 0}. It is precisely defined as follows:

i. a 0 = ( 0, if a 0 1, otherwise (3.9)

Least squares unmixing algorithm

A classical least squares (LS) approach consists in finding the solution a that minimizes the least squares error. In the current unmixing problem, this can be recast as:

âLS 2 argmin a 1 2 kΦa + b xk 2 + i. a 0 (3.10)
We tackle this optimization problem with Forward-Backward Splitting (FBS) algorithm ( [START_REF] Combettes | Signal recovery by proximal forward-backward splitting[END_REF]). An accelerated version of the FBS algorithm coined FISTA [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF] is used and summarized in Algorithm 1. (Details can be found in Appendix B). In the algorithm:

• the gradient of the data fidelity term f

(a)= 1 2 kΦa + b xk 2 is: rf (a)=Φ T (Φa + b x)( 3 . 1 1 )
• the proximal operator of the non-negativity constraint Eq.(3.9) is defined as the orthogonal projection onto the non-negative orthant:

prox i. a≥0 = ( 0, if a < 0 a, otherwise (3.12)
The algorithm is guaranteed to converge when the gradient step  1/kΦ T Φk 2 . In practice, the algorithm stops when the relative variation of a between two consecutive iterations is lower than 10 12 .

Algorithm 1 Pseudocode of FISTA with constant stepsize Input: Fix the step size 0 < < 1/kΦ T Φk 2

Initialization:

y (1) = a (0) , t (1) =1 while ka (k) a (k-1) k ka (k-1) k > 10 12 do a (k) =pro x i. a≥0 y (k) Φ T (Φy (k) + b x) t (k+1) = 1+ p 1+4t (k) 2 2 y (k+1) = a (k) + ⇣ t (k) 1 t (k+1) ⌘ (a (k) a (k 1) ) end while
From a statistical perspective, the least squares approach is equivalent to a maximum likelihood estimate assuming that the underlying noise is additive, white and Gaussian: n ⇠ N (0, σ 2 ). However, because of the Poisson statistics of the spectroscopic measurement, the noise variance is data dependent. It is well known that the variance of the Poisson distribution is equal to the expected value, which leads to

2 i =[Φa] i +b i , for 8i 2 [1,...,M].
The weighted least squares (WLS) is indeed the generic form of the least squares, which leads to model the minimization of the WLS error related to the Poisson statistics model with:

âWLS 2 argmin a (Φa + b x) T W 1 (Φa + b x)+i. a 0 (3.13)
where W is a diagonal matrix with diagonal elements specified by Φa+b:

W = 0 B B B B B B @ [Φa] 1 + b 1 [Φa] 2 + b 2 . . [Φa] M + b M 1 C C C C C C A
However, the linear mixing model Φa + b can not be known in advance. Previous studies (e.g., Full spectrum analysis (FSA) reviewed in Chapter 2) considers W =d i a g( x)( i.e., the measured spectrum), which is not fully adapted to account for the Poisson statistics.

We further propose to tackle this optimization problem with iterative estimation of the true model with an iterative re-weighted least squares algorithm. More precisely, as described in Algorithm 2, the main step of the algorithm consists in estimating the mixing weights, noted â and updating the weights with the estimated model: Φâ + b.

Each estimation step makes application of the aforementioned FISTA algorithm by considering the gradient of the data fidelity term:

rf (a)=Φ T W 1 (Φa + b x)( 3 . 1 4 )
Algorithm 2 Pseudo code of iterative re-weighted squares algorithm Initialization: W (0) = Φâ ls + b, where âls is the least squares solution.

Estimate the mixing weights ! â(0)

for k<k max do Update: W (k+1) = Φâ (k) + b Re-estimation with W (k+1) ! â(k+1) end for
While the iterative re-weighted least squares algorithm allows better accounting for the Poisson statistics, the algorithm still leads to estimation bias of the true model. An approach that takes into account the actual Poisson maximum likelihood estimation is therefore further required.

Poisson statistics-based unmixing algorithms

Along with the non-negativity penalization, the Poisson statistics-based estimator can be formulated as follows:

âPoisson 2 argmin a Φa + b x log (Φa + b)+i. a 0 (3.15)
We now present two algorithms to solve this optimization problem, Multiplicative update algorithm [START_REF] Lee | Algorithms for nonnegative matrix factorization[END_REF] and the Chambolle-Pock algorithm [START_REF] Chambolle | A firstorder primal-dual algorithm for convex problems with applications to imaging[END_REF].

Multiplicative update algorithm

The multiplicative update rule with respect to the KL (Kullback-Leibler) divergence loss enables solving the non-negativity regularized Poisson based optimization problem. The multiplicative update algorithm introduced in [START_REF] Lee | Algorithms for nonnegative matrix factorization[END_REF]] is applied to gamma-ray spectral unmixing in [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF], which is summarized in Algorithm 3.

a (k+1) j = a (k) j P M i=1 Φ ij x i /M k i P M i=1 Φ ij with: M k = Φa k + b end while
This algorithm addresses the non-negativity regularized Poisson estimator. However, the convergence can be slow for low statistic regimes. Moreover, it is less generic due to the difficulty of adding other regularization terms, whereas the proximal algorithms can provide solutions with simplicity. Therefore, we investigate the Primal-dual proximal algorithm for solving the minimization problem in Eq.(3.15).

Primal-dual proximal algorithm

We propose to solve the optimization problem in Eq.(3.15) with primaldual proximal algorithms such as the one introduced by Chambolle and Pock in [START_REF] Chambolle | A firstorder primal-dual algorithm for convex problems with applications to imaging[END_REF] (Details can be found in Appendix B). The pseudo-code of the Chambolle-Pock algorithm is given in Algorithm 4, in which the proximal operator of the joint Poisson distribution of the measurement is:

prox ⇢f (y)= y + b ⇢ + q (⇢ y b) 2 +4⇢x 2 b (3.16)
where x and b stand for the measured spectrum and the background spectrum.

The convergence of the algorithm is ensured with ⌧ < 1/kΦ T Φk 2 and ✓ =1 . T h es t e pp a r a m e t e r s and ⌧ are chosen with with respect to the total number of counts in the measured spectrum and ⌧ =0 .9/( ⇤ kΦ T Φk 2 ) for a better convergence rate. Similarly to the FISTA algorithm, the algorithm stops when the relative variation of a between two consecutive iterations is lower than 10 12 .

Algorithm 4 Pseudocode of Chambolle-Pock algorithm

Input:

Fix the parameters: , ⌧ > 0a n d ⌧ < 1/kΦ T Φk 2 . Initialization: ā(0) = a (0) , u (0) = Φa (0) while ka (k) a (k-1) k ka (k-1) k > 10 12 do v = u (k) + Φā (k) u (k+1) = v prox (1/ )f v a (k+1) =pro x i. a≥0 a (k) ⌧ Φ T u (k+1) ā(k+1) = a (k+1) + ✓(a (k+1) a (k) )
end while

Experiments on the combination of two radioactive sources

To assess the impact of the Compton contribution of a given radionuclide on the determination of another radionuclide, we focus on simulations of simple mixing scenarios that are composed of two sources. The experiments are performed for simulated spectra of HPGe detector and NaI detector, where the spectral signatures used in the simulation process are:

• For HPGe detector, simulations performed with the Monte Carlo Nparticle (MCNP) Transport Code, a software package for simulating radiation transport developed by the Los Alamos National Laboratory [Briesmeister, 2000]. We make use of the MCNPX (MCNP eXtended), which simulates the gamma-ray spectrum of sources that emits photons at one or more specific energies with given weights. It provides simulated spectra consisting of given energy peaks and the associated Compton continua.

• For NaI detector, measured spectra with a 3"x3" NaI(Tl) detector without shielding using point sources placed at a distance of 1 m (see details in [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF]).

Experiments on simulations of HPGe detector

We consider the mixture of two radioactive sources at 500 keV, noted φ 1 , and 200 keV, noted φ 2 . The measured spectra are defined as follows:

x ⇠ Poisson (φ 1 a 1 + φ 2 a 2 + b)( 3 . 1 7 )
where φ 1 and φ 2 are simulated with MCNPX, a 1 and a 2 stand for the mixing weights of the sources. b is the spectrum of the background radiation.

As shown in Figure 3.2, we generate simulations as follows: a 1 for the source with energy peak at 500 keV kept fixed and we change a 2 for the one at 200 keV. For each linear combination level, we simulate 100 gamma-ray spectra by random Poisson process as described in Eq.(3.17). Experiments are carried out for these simulated spectra with:

• Poisson unmixing using Chambolle-Pock algorithm. To further highlight how much full spectrum knowledge can help improving unmixing, we further consider two distinct implementations:

af u l ls pe c t r u mb a s e dv e r s i o n

The median values and confidence intervals between percentile 25 and percentile 75 are displayed. Similarly, the relative errors obtained with different estimators for experiments of NaI detector are compared in Figure 3.5 for both 57 Co and 152 Eu.

From these results, we can draw the following conclusions:

• Advantage of using full spectrum analysis. As we can see in Figure 3.2, the spectrum of 200 keV is below the Compton continuum of the spectrum of the source of energy 500 keV. The precision of the estimated mixing weight of the 200 keV source is limited, especially when the number of counts is low (see Figure 3.4-a). Nevertheless, the full spectrum analysis using Poisson, LS, WLS estimators provide lower estimation bias than the peak-based analysis.

• Advantage of Poisson statistics based estimation. In experiments from both HPGe detector and NaI detector, the Poisson estimator and the WLS estimator provide lower estimation bias and uncertainties, which is shown in Figure 3.4 and Figure 3.5. It should be noted that, in case of mixtures of 57 Co and 152 Eu, the relative estimation errors for 152 Eu is significantly lower for the Poisson estimator and the WLS estimator, as well as the estimation uncertainties. In conclusion, accounting for the Poisson statistics has significant advantages to tackle overlapped spectra of NaI measurements.

• Poisson estimator and WLS estimator. The WLS estimator is shown to yield similar estimation bias and error bars than the Poisson estimator. However, the WLS estimator requires iterative estimation of the mixing weights to update the weight matrix. Moreover, the matrix W 1 =d i a g 1 Φa+b is usually ill-conditioned, which leads to slow convergence rate. The speeds of convergence will be compared in the next section.

Realistic simulations of routine aerosol samples

In this work, the spectral signatures are simulated with the MCNP-CP code (A Correlated Particle Radiation Source Extension of a General Purpose Monte Carlo N-Particle Transport) [Berlizov, 2006], which allows simulating physics of nuclear decay and the subsequent emissions.

An example of MCNP-CP simulation for 60 Co is shown in Figure 3.6. The MCNP-CP code simulates spectral components due to photon interactions refer to Chapter 2, including two absorption peaks E 1 = 1173 keV, E 2 =

Comparisons of the unmixing algorithms in different counting regimes

Different statistical regimes are considered so as to simulate different measurement counting times. This is done by introducing a multiplicative factor by:

x ⇠ Poisson ([Φa 0 + b] ⇥ )( 3 . 2 1 )
Experiments are carried out with =0 .1, 0.16, 0.3, 0.6, 1, where =1 according to realistic number of counts for which the simulated spectrum is shown in Figure 3.7. We generate 100 simulations for each mixture using the random Poisson process in Eq.(3.21).

The Poisson unmixing is compared to LS unmixing by using the relative estimation error in Figure 3.9. The median values of Monte-Carlo simulations are given. The confidence intervals are defined by the first quartile (25% percentile) and last quartile (75% percentile).

Results

• Practical advantage for realistic data analysis. In practice, the role of is similar to changing the counting time of a measurement. The relative error bars of Figure 3.9 shows that the estimation performances of each algorithm improve when increases, which confirms the limitation in the low statistics regime. The realistic levels in experimental measurements correspond to mixtures between =0.6and = 1, where the Poisson unmixing algorithm provides more accurate estimation with lower relative errors and lower uncertainties. The method is therefore of special interest to reduce the counting time of the measurements.

• Impact of Compton continua. Referring to the spectral contribution shown in Figure 3.7, the results obtained for 210 Pb indicate that both estimators have similar performances for estimating a spectrum which consists of dominant peak when the count number is large. The Poisson unmixing has significant advantage for the estimation of 22 Na, 40 K, 137 Cs at low statistics and composed of peaks and significant continuum contributions.

Evaluation of unmixing algorithms for low-level artificial radionuclide

The ability to estimate the low-activity artificial radionuclide plays a central role in the field of rapid detection. In such a context, we focus on

Experimental results with real spectra from aerosol samples' measurements

In this section, the proposed approaches are applied to real aerosol samples, which are routinely collected as a part of the surveillance mission of the laboratory. To better focus on the rapid detection of radionuclides, the different unmixing methods are applied to short-time counting statistics. To this end, a dedicated scenario of measurements has been set up for an aerosol sample. These measurements are performed half an hour after the collection, in a continuous manner, with pre-defined counting times for 8 days (see Table 3.2). Figure 3 The experiment was carried out with an 10 mL cylindrical counting geometry (h =5mm,; = 50mm) measured with an HPGe detector (60% relative efficiency). The measured spectra are subsequently analyzed with the proposed spectral unmixing algorithms and Genie 2000, which is traditionally used in the laboratory.

In this section, we first investigate how the choice of the spectral signatures impacts the activity estimation quality. This is particularly important as the subset of active radionuclides is generally unknown in practice. Next, the results are compared with the standard peak-based Genie 2000 algorithm.

Moreover, the statistical deviance ([DasGupta, 2008]) is displayed in Figure 3.16, which provides a measure of the goodness-of-fit for both estimators, for either simulated or measured spectra.

More precisely, the statistical deviance for the Poisson distribution is defined by:

D =2 ⇣ x log( x Φâ + b ) x + Φâ + b ⌘
where â stands for the estimated mixing weights.

Results

As shown in Figure 3.14-a and Figure 3.15-a, making use of the small dictionary Φ 5 leads to significant biases in the first measurements, where short-lived radionuclides can hardly be neglected. The results obtained with alargerspectraldictionaryΦ 10 leads to much more accurate results as shown in Figure 3.14-b and Figure 3.15-b. These results are similar to the ones we obtained with real spectra (see Figure 3.13). Poisson unmixing tends to overestimate the activities comparing to the least squares unmixing. One can observe that Poisson unmixing tends to be more sensitive to the choice of the spectral dictionary. The origins of this phenomenon is the Poisson unmixing aims to precisely fit the measured spectrum with the full spectrum, which yields increased errors when the spectral dictionary cannot fit the actual set of radionuclides, since the logarithmic scaling in the likelihood term induces lower weights on channels in peak regions.

Figure 3.16 shows that, in the first measurements, the spectral unmixing with Φ 10 provides lower deviance than with Φ 5 for both simulated and measured spectra. In this case, the estimated models with Φ 5 tend to fit for the short-lived radionuclides that are not in the spectral signatures dictionary, with the ones contained in the dictionary. In the last measurement, shortlived radionuclides having significantly decayed, making use of Φ 5 leads to lower deviance since this dictionary already provides an accurate description of the measurements. Moreover, lower deviance is achieved with Poisson unmixing for each experiment, which confirms the advantage of Poisson unmixing to analyze these data, whether they are simulated or real measurements.

The results obtained using different choices of spectral dictionaries highlight that the accurate identification of spectral signatures is of key importance for the spectral unmixing to work efficiently.

Comparisons with peak-based analysis

We compare the results of the Poisson unmixing estimator with those of Genie 2000, which is the workhorse method in a large number of radioactivity measurement laboratories. As discussed previously, these comparisons have been carried out with the 10-radionuclides dictionary, which is suitable for the analysis of measured spectra. We first focus on the long-lived radionuclide 7 Be and the short-lived radionuclide 212 Bi. The estimated mixing weights are compared in Figure 3.17. Secondly, the results for 137 Cs are displayed in Figure 3.18.

Results

As shown in Figure 3.17, the Poisson unmixing and Genie 2000 algorithms provide similar results for the estimation of 7 Be and 212 Bi, while the estimated mixing weights are of the same order in the last two measurements. This is related to the larger counting rates obtained with long counting times (3 days for s10 and 4 days for s11).

Figure 3.18 shows that the Poisson unmixing algorithm detects 137 Cs with similar level to that obtained with Genie 2000 from the measurement s10, which is evaluated from day 2 to day 4. To validate this result, simulations are performed with the final estimated mixing weights obtained with the Poisson unmixing algorithm (â =0 .016/s for 137 Cs) to mimic the measurement s10. 1000 Monte Carlo simulations are generated to compute a confidence interval. 1000 extra Monte Carlo simulations are carried out with 137 Cs = 0 to quantify the false positive rate. In Figure 3.19, the distribution of the estimated mixing weights is presented for both experiments.

These results allow to draw the following conclusions:

• The detection of 137 Cs is statistically significant for the significance level of ↵ =0.05 in the null hypothesis.

• The confidence interval derived from the simulations also confirms that the activity level measured from the real measurement s10 is significant.

This confirms that the Poisson unmixing algorithm allows identifing 137 Cs four days before the usual method Genie 2000.

In this section, we can first conclude that when full spectrum unmixing is used, the lack of knowledge of the actual active radionuclides may lead to under-fitting or over-fitting effects, which eventually lead to biased activity estimation. Secondly, Poisson unmixing yields significant improvements for the estimation of low-activity radionuclides, which is a key advantage for the rapid detection of an anomaly in the air.

Meanwhile, the experimental results on successively measured gamma-ray spectra carried out in Chapter 3 show that the spectral unmixing is sensitive to the choice of the spectral signatures dictionary. Using a larger/smaller dictionary can lead to under/over fitting effects that affect the estimation accuracy.

In this chapter, we further investigate how the set of spectral signatures of the actually active radionuclides in a measured gamma-ray spectrum can be identified. In such a context, the spectral unmixing problem is reformulated so as to jointly estimate the dictionary of the spectral signatures and the corresponding mixing weights. The contribution of this chapter is as follows: we start by recasting the spectral signatures identification task as a model selection problem in Section 4.1, where we reformulate the spectral unmixing along with an additional constraint on the model complexity. This brings us to consider a sparsity-based approach for spectral unmixing in Section 4.2. Next, to impose the sparsity constraint in the spectral unmixing problem, we explore the sparse spectral unmixing algorithms in Section 4.3, where a novel Poisson-based greedy algorithm is proposed. The evaluation of sparse spectral unmixing is then carried out in Section 4.4 and Section 4.5, where we focus respectively on experiments on both NaI and HPGe measurements. The Section 4.6 summaries the benefits of the sparse spectral unmixing and the perspectives of this work.

A model selection approach

The spectral unmixing addresses the problem of gamma-ray spectrum analysis by decomposing a measured spectrum into individual spectra of radionuclides. The problem has been tackled as a non-negativity regularized inverse problem to estimate the mixing weights of spectral signatures. In other words, the spectral unmixing aims at explaining the measured spectrum with the radionuclides in a given dictionary. Recall that we consider the following mixing model with N radionuclides:

x ⇠ Poisson N X j=1 φ j a j + b ! (4.1)
Nevertheless, the set of active radionuclides that are actually present is not known in advance. A badly chosen dictionary of spectral signatures leads to increased activity estimation errors. To alleviate this problem, an alternative consists in using a large dictionary, which contains the radionuclides that are commonly present in certain type of of measurement (e.g., the aerosol candidate models, which allows penalizing the number of parameters in the model to avoid the over-fitting.

It has to be noticed that other more recent information criteria such as the Deviance Information Criterion (DIC - [START_REF] Spiegelhalter | Bayesian measures of model complexity and fit[END_REF]) could be envisaged, but they would require resorting to MCMC-based solvers, which is computational difficult to be implemented. In the following, we will mainly focus on the sparsity as a model complexity penalization.

Sparse solution

The objective of the model selection in the spectral unmixing is to find the smallest subset of columns of Φ c that best explains the measured spectrum. It is commonly done by adding a sparsity regularization in the objective function that enforces the number of active radionuclides to be low while still explaining the data.

One can tackle the sparsity constraint with:

• `0 -"pseudo-norm", which is defined with: kak 0 =t h en u m be ro fn o n -z e r oe l e m e n t si na.

The solution a is said to be k-sparse if kak 0  k, which naturally induces the sparsity constraint.

• As a relaxation of `0-"norm", `1-"norm", noted kak 1 , is the sum of absolute values of elements in a. It is easier to solve due to the convexity of kak 1 .

However, the `1 that imposes a threshold of the solution is not suitable in the Poisson noise case, it considers an additive noise independent to the data, that is not true in the Poisson model. The `0 sparsity regularization is therefore needed in the Poisson-based spectral unmixing. Fortunately, the dimension of spectral signatures dictionary is low in the gamma-ray spectrum analysis. The straightforward search of the subset of active radionuclides is feasible, which allows solving the `0 minimization with greedy sparse regression methods.

The sparse spectral unmixing can be formulated as follows:

â 2 argmin a f (a)+i. a 0 s.t. kak 0  k (4.3)
where the sparsity constraint enforces the mixing weights vector a to have exactly k non-zero elements. Greedy algorithms are involved in the `0 optimization problem. These algorithms have been proposed to find the sparse solution in least squares problem, in which the selection of columns of Φ c starts by an empty support, then the support is updated step by step with the element that optimally solve the problem. Algorithms in such category includes the Matching Pursuit (MP), [START_REF] Mallat | Matching pursuits with time-frequency dictionaries[END_REF] and Orthogonal Matching Pursuit (OMP), [START_REF] Pati | Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition[END_REF], which allows sequentially selecting the signatures in the dictionary Φ c that has the largest correlation with the current residual.

Let's now investigate the use of OMP algorithm to the spectral unmixing problem. Recall the gamma-ray spectrum model:

x = Φa + b + n (4.4)
In such OMP algorithm, we make use of the iterative re-weighted least squares mentioned in Chapter 3 to estimate the mixing weights with selected active radionuclides. The pseudo-code of the OMP algorithm for solving the sparse solution of Eq.( 4.4) is stated in Algorithm 5.

However, such algorithm have been developed to find the sparse solution to an underdetermined system (i.e., Φ 2 R M ⇥N : M<N ). This is not the case of the gamma-ray spectral unmixing problem. Furthermore, the main objective of the sparse spectral unmixing here is to select the active radionuclides, which requires that the capacity to accurately identify radionuclides when the noise is low. Therefore, a Poisson-based OMP algorithm is particularly needed.

The Poisson OMP Algorithm

Only a few studies have been carried out in Poisson-based greedy algorithms. An extension to the Poisson denoising case has been published in [START_REF] Dupé | A greedy approach to sparse poisson denoising[END_REF]. The algorithm selects new variables based on the minimization of the gradient of the Poisson negative log-likelihood. This method requires the number of active radionuclides k to be known, which is however not true in gamma-ray spectrum analysis since the number of active radionuclides is unknown for a measured spectrum.

We rather extend the OMP algorithm to P-OMP (Poisson-based OMP) described in Algorithm 6. It selects forward the radionuclides in Φ c by sequentially adding the radionuclide that maximizes the Poisson likelihood. More precisely, the model is initialized with the background spectrum b,a t while the stopping condition is not achieved do

i = i +1
Find the column of Φ c that solves:

argmax j kr i 1 Φ c k
Add the j th column to the selected radionuclides:

I i = I i 1 [ j Estimate âi with selected radionuclides (Φ c [I i ]
) and compute the new residual with:

r i = x Φ c [I i ]â i b Φ c [I i ]â i + b
end while each step, each of the non-selected radionuclides is added into the spectral signatures for activity estimation using the Poisson unmixing algorithm described in Chapter 3. At the end of the iteration, it selects the radionuclide that minimizes the Poisson-based cost function. The algorithm stops when certain stop condition is achieved.

Stopping criteria for radionuclides identification

The OMP and P-OMP algorithms sequentially select the radionuclides at each step. Once a radionuclide has been added into the spectral signatures, a stopping condition needs to be applied to compare two candidate models:

• M 0 : the model with selected radionuclides

Initialization:

The selected indices:

I 0 = ;
The indices to check:

I c =[1...N ]
The neg-log-likelihood (with background spectrum):

L 0 = b x log(b)
while the stopping condition is not achieved do

i = i +1
for j 2 I c do Add a radionuclide to the subset of spectral signatures:

I test I i 1 [ j Estimate mixing weights a new with Φ c [I test ]
Compute the neg-log-likelihood:

Φ c [I test ]a new + b x log(Φ c [I test ]a new + b)
end for

Find j ⇤ that minimizes the neg-log-likelihood Add the j ⇤th column to the selected radionuclides:

I i = I i 1 [ j ⇤
Remove the selected radionuclide j ⇤ from I c . end while

To use these information criteria as stopping rule, the

AIC =2⇤ k 2lnL,B I C = k ⇤ N 2lnL
need to be calculated at each straightforward selection step, where L is the likelihood function, k is the number of parameters to be estimated (i.e., the number of selected radionuclides), and N is the sample size (i.e., the number of channels in the spectrum). The algorithm stops when the information criteria increases.

However, these model selection methods can not provide an idea about the false positive rate. To consider the false positive rate of adding a new radionuclide to the model, we propose to apply a stopping criteria with a statistical test under the following hypotheses:

• H 0 : the model M 0 is true • H 1 : the model with an extra active radionuclide M 1 is true
For this purpose, the statistical deviance test between M 0 and M 1 ([Das-Gupta, 2008]) is well-suited, which is defined by the difference of their neglog-likelihood:

D = 2(L 1 L 0 )( 4 . 5 )
where L 0 and L 1 stand for the neg-log-likelihood calculated with model M 0 (selected radionuclides) and M 1 (selected radionuclides + an extra radionuclide) respectively. More precisely, as we presented in Chapter 3, for the OMP algorithm with weighted least squares estimation, the neg-loglikelihood function takes the form of:

L (x|a)=(Φa + b x) T W 1 (Φa + b x)
For the Poisson based OMP algorithm, the Poisson neg-log-likelihood takes the form of:

L (x|a)=Φa + b x (Φa + b)
When M 0 is nested within M 1 and M 1 is the true model, the statistical deviance asymptotically follows a 2 distribution with d degrees of freedom (d is the difference of the number of radionuclides in the two models). However, the models are not nested in the straightforward search of P-OMP algorithm, since among the radionuclides to be tested, the true model may be a combination of radionuclides. In such a multiple hypothesis testing problem, the Bonferroni correction [START_REF] Benjamini | Controlling the false discovery rate -a practical and powerful approach to multiple testing[END_REF] is commonly used, which allows to control the number of false positives based on a p-value corrected by the number of hypotheses.

The selection of extra radionuclides stops whenever the deviance is consistent with a 2 distribution with the critical p-value:

↵ = ↵ 0 n (4.6)
where n is the number of radionuclides to be tested at this step and ↵ 0 is the desired false positive rate, while ↵ 0 =0 .01 is used in the experiments of this chapter.

Indeed, the results obtained with Genie 2000 is according to Genie 2000 + a post-analysis procedure, while some false identifications of radionuclides are rejected. Furthermore, the results of Genie 2000 are given based on certain decision threshold level which implies the significance of the radionuclides' presence (see details in Chapter 5).

However, one limitation for applying P-OMP on real measurements of HPGe detectors is the bias of the Poisson spectral unmixing. Firstly, the spectral signatures of radionuclides are simulated with MCNP-CP, which provides bias on gamma-ray responses. Secondly, background spectra are measured every two months, b used in the spectral unmixing process is generated from a previous measurement of background spectrum, which is not the actual background at the time of the measurement. This bias on background spectrum leads to over/under estimations since the background consists of several radionuclides which also participate in the measured sample.

Application of P-OMP to NaI measurements 4.5.1 Description of the data

The spectral unmixing applied to the activity estimation from NaI gammaray measurements have been studied in [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF] based on the multiplicative update algorithm described in Chapter 3. The investigations in this paper consider that the set of active radionuclides is known. To further overcome the radionuclides' identification, [André et al., ] makes use of the P-OMP algorithm to impose the sparsity constraint into the spectral unmixing algorithm proposed in [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF]. In this section, we aim to further investigate how the P-OMP algorithm behaves when the correlation of radionuclides' spectra between the spectra of the spectral dictionary increases, which is more challenging in NaI spectra analysis.

It is indeed expected that larger correlation will hamper the estimation bias of the activities, and more importantly the detection capabilities of the sparse spectral unmixing algorithm. For that purpose, we apply P-OMP to simulated measurements of a 3"x3" NaI(Tl) detector without shielding using point sources placed at a distance of 1 m [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF]. The spectral dictionary is composed of ten spectral signatures, which are made of 1024 channels. These signatures correspond to the detector response of ten gamma-emitting radionuclides with photon emissions covering a range of energies between 40 keV and 2 MeV:

57 Co, 60 Co, 88 Y, 133 Ba, 134 Cs, 137 Cs, 152 Eu, 207 Bi, 241 Am, 237 Np. Simulations of different mixing scenarios are performed with the following choice of radionuclides (i.e., active radionuclides present in the spectrum).

The choice of 57 Co, 60 Co, 133 Ba, 137 Cs and 241 Am is particular relevant as they correspond to the main radionuclides that need to be identified for illicit radioactive material monitoring [IEC 62484:2010[IEC 62484: , 2010]]. Furthermore, 152 Eu has been chosen as its signature covers a large range of energies, 120 keV to 1410 keV, which is largely correlating with other radionuclides. The following scenarios have been considered :

• Scenario I: following [André et al., ], the scenario considers measurements where only four radionuclides are present in the mixture: 57 Co, 133 Ba, 152 Eu, 241 Am. A natural background spectrum is further added. This scenario allows to compare the precision of the unmixing methods for both estimation quality and detection capability.

• Scenario II: in these experiments, 137 Cs is added to the mixture of Scenario I. This allows to evaluate the ability to detect low-activity radionuclides such as 137 Cs, as well as the impact of the estimation on other radionuclides. These experiments are reminiscent of the 137 Cs estimation experiment we performed in Chapter 3.

• Scenario III: 60 Co is added to the standard mixture Scenario I. This set-up allows to highlight how radionuclides' identification can be hampered when their spectral signatures have a significant overlap, which is the case of 57 Co, 60 Co and 152 Eu.

For a comparison purpose, the OMP algorithm based on the iterative re-weighted least squares estimation and the P-OMP algorithm based on the Poisson estimation presented in Section 4.3 are applied to analyze the simulated spectra of the above three scenarios. At the end of the section, we compare the proposed deviance-based stopping criteria to those based on information criteria.

Application to NaI measurements -scenario I

This scenario has been investigated in [André et al., ] to simulate measurements similar to illicit radioactivity monitoring with NaI-based detectors. Four radionuclides and a background spectrum are considered in the measurements, their proportion can be found in Table 4.1). To better illustrate the correlations between individual spectra of the measurements, the spectral signatures of the above 4 active radionuclides are shown in Figure 4.5.

We quantify the estimation bias of the mixing weights (i.e., the activities) of these radionuclides as well as the detection performances of the P-OMP and OMP algorithms, with an empirical evaluation of the false positive and These results first show that AIC systematically over-estimates the number of active radionuclides, leading to a detection rate of a few percents for the 5 non-active radionuclides. This holds true for the three different levels of Europium that were tested. In contrast, BIC is more conservative and leads to detection results that are closed to the deviance-based criterion, a slightly large false positive rate.

It has to be emphasized that controlling the FPR is key for tackling spectral unmixing problem, which favor the use of the deviance-based criterion along with the sparsity penalization for model selection.

FNR of active radionuclides FPR of non-active radionuclides

Summary of the sparse spectral unmixing

In this chapter, the spectral unmixing is reformulated with a sparsity constraint so as to estimate the subset of active radionuclides present in a measured gamma-ray spectrum. We proposed a novel Matching Pursuit algorithm, named P-OMP, which enables to find the sparsest solution of mixing weights that best fits the measured spectrum according to the Poisson statistics. We evaluated the performance of this P-OMP algorithm by comparing to standard OMP algorithm based on the iterative re-weighted least squares estimation. As results, the P-OMP is shown to provide more accurate estimation of the active radionuclides' subset, as well as their according mixing weights. This further highlights the advantage of accounting for the Poisson data error in spectral unmixing rather than weighted least squares error. While providing promising results, assuming that the radionuclides' spectral signatures and the background spectrum are accurately known is an important limitation in real data analysis, particularly in case of aerosol measurements that we investigate, since the spectral signatures need to be simulated. In practice, the variability of spectral signatures might increase false identifications as well as estimation bias of the mixing weights. The next chapters will focus on the quantitative analysis of radionuclides' activities with experimental data, where we will discuss the error estimation with characteristic limits and propose calibrations for quantitative analysis.

CHAPTER 5

Characteristic limits in spectral unmixing The precise analysis of gamma-ray spectra requires two key ingredients: the first one is an efficient spectral unmixing algorithm, which we thoroughly investigated in the previous chapters. The second ingredient, which is key to reach a truly metrological analysis quality, is the quantification of the errors of the estimation and the detection procedures. As detailed in-depth in this chapter, this requires evaluating the so-called characteristic limits, which are essential for decision making purposes in the quantitative analysis of the radioactivity measurements. The assessment of two major characteristic limits: the decision threshold and the limits of the confidence interval will be thoroughly evaluated in this chapter.

While being well studied in traditional peak-based analysis, the main goal of this chapter is to introduce and evaluate a new approach to derive such characteristic limits for the spectral unmixing algorithms we introduced previously. More precisely, the contribution of this chapter is as follows:

• We first focus on the concept of characteristic limits for radioactivity measurements in Section 5.1, where we present their definitions.

• In Section 5.2, we review how the characteristic limits can be calculated from Monte Carlo simulations, which were used to determine the significance of results provided with spectral unmixing in Chapter 3.

• The assessment of the decision threshold in spectral unmixing is investigated in Section 5.3.

• The evaluation of the confidence interval of the activity estimation is then carried out in Section 5.4.

The proposed approaches to derive characteristic limits have been evaluated and validated with realistic simulations of HPGe measurements and NaI measurements. Section 5.5 finally discusses their metrological use in the gamma-ray spectrum analysis.

Characteristic limits in radioactivity measurements

Refer to [ISO 11929, 2010], the notations below are used in the description of the characteristic limits:

• Y : Measurand, the quantity of interest.

• y: Determined value of the measurand Y (i.e., the estimate of Y ).

• ỹ: True value of the measurand.

We first present the classical statistical hypothesis framework used for decision making in gamma-ray spectrum analysis. It is commonplace to consider testing hypotheses with the two alternatives (associated with type Ie r r o ra n dt y peIIe r r o rd e s c r i be di nT a b l e5 . 1 ) :

• H 0 : the null hypothesis, where a given radionuclide is not "active".

• H 1 : the alternate hypothesis, where the radionuclide is present in the mixture. The standardization document [ISO 11929, 2010] defines the determination of the characteristic limits, namely the decision threshold, the detection limit, and limits of the confidence interval for ionizing radiation measurements. It provides a framework for the computation of the characteristic limits. Referring to [START_REF] Weise | Determination of the detection limit and decision threshold for ionizing-radiation measurements: fundamentals and particular applications[END_REF], [Michel, 2016], the definition and interpretation of the characteristic limits for some estimate y of a measurand Y are as follows:

H 0 is
• Decision threshold (DT) allows a decision to be made on whether or not the physical effect quantified by the measurand is present.

The determination of DT is related to the Type I error described in Table 5.1. When the quantity y exceeds the critical value (DT), the null hypothesis H 0 should be rejected with respect to a given false positive rate (FPR). It can be described with:

↵ = P (y DT |ỹ =0) (5.1)
where ỹ is the true value of the measurand and ↵ is the desired critical FPR.

• Detection limit (DL) indicates the smallest true quantity value of the measurand, which can still be detected with the applied measurement procedure.

The determination of DL is related to the Type II error described in Table 5.1. It is selected with resp ect to a desired false negative rate (FNR) based on the decision threshold level.

More precisely, the detection limit (DL) is the smallest value that provides a desired Type II error probability :

= P (y  DT |ỹ = DL)( 5 . 2 )
where the DT is given and ỹ is the true value of the measurand.

• The confidence interval for the estimate y is an interval that has a probability of containing the true value ỹ.

In this work, we mainly focus on the determination of the decision threshold and the confidence interval, since in practice, these two ingredients allow us to determine whether the resulting activity of a radionuclide is significant. Firstly, we can evaluate the measurement uncertainty of each radionuclide from the combination of the confidence interval and other metrological uncertainties. Secondly, for the decision making purpose, it is sufficient to compare the low boundary of the measurement uncertainty to the decision threshold. In practice, the detection limit can be used as reference values, which is however not useful in case of the aerosol measurements that will be analyzed.

Quantification with Monte Carlo simulations

A traditional approach to quantify the characteristic limits for some estimation method is to make use of Monte-Carlo simulations. Indeed, we pointed out in Chapter 3 that the significance level of the radionuclide's presence and the confidence interval of the activity estimation with spectral unmixing can be derived from Monte Carlo simulations.

In this chapter, the evaluation of characteristic limits are carried out with the radionuclides' realistic mixture of aerosol measurements (see Chapter 3). Recall that the mixture consists of 10 radionuclides: 7 Be, 22 Na, 40 K, 137 Cs, 210 Pb, 208 Tl, 212 Bi, 212 Pb, 214 Bi, 214 Pb. It should be noted that the evaluations from this realistic mixing scenario are meaningful since the counting statistics is similar to typical aerosol routine measurements. In this context we focus on the assessment of characteristic limits for 4 radionuclides: 7 Be, 22 Na, 137 Cs, 212 Pb, since these radionuclides cover the whole energy range and different statistic regimes. The simulation model of 10 radionuclides and the contribution of these 4 radionuclides are displayed in Figure 5.1.

Decision threshold from Monte Carlo simulations

Monte Carlo simulations that mimic the mixture under the null hypothesis of a radionuclide allows quantifying the false positive rate of this radionu-the first ones obtained with simulations under the null hypothesis can be used to quantify the significance of the radionuclide's presence, which allows computing a decision threshold with respect to some false positive rate from the according percentile of the distribution. The second ones obtained with the true mixture allows computing a confidence interval from the according percentiles of the distribution that contains the true value.

In practice, Monte Carlo simulations are seldom used to analyze gammaray spectra in routine analysis procedures. The main drawback is their massive computational cost since Monte Carlo simulations are needed for each new spectrum to be analyzed. In the next sections, we focus on computationally cheaper and yet precise alternatives to derive the characteristic limits without resorting to Monte-Carlo simulations.

Quantifying the decision threshold

Decision threshold in peak-based analysis

In gamma-ray spectrum analysis, the decision threshold is usually derived from some statistical test based on the measured spectrum. This amounts to evaluating how much the estimated quantity associated with a radionuclide's activity departs from the background (i.e., other contributions composed in the measured spectrum) and is therefore statistically consistent or not with this background.

In the peak-based analysis, the activity is associated with the net counts (see details in Chapter 2). Recall that it is the total number of counts measured in a given ROI, which further corrected by the average number of counts of the background:

N n = N g N 0 (5.3)
where N n is the net counts number associated to the activity, N g is the observed gross number of counts and N 0 is the number of background counts. Recall the definition of DT in Section 5.1, the DT level of the measurand (i.e., net counts number) is derived from:

↵ = P(N n DT | Ñn =0) (5.4)
where ↵ and Ñn stands for the desired FPR and the true value of the net number of counts respectively. The estimated number of N g and N 0 , noted Ng and N0 respectively. In practice, the DT is derived from some statistical test of the quantity Ng N0 under the hypothesis of Ñn =0,whic hhas:

• mean value equal to zero.

• variance according to 2 N0 , since both of N g and N 0 follow the Poisson statistic thus their mean value are equal to their variance, and N g = N 0 under the hypothesis of Ñn =0.

Decision threshold in spectral unmixing analysis

The DT determination in peak-based analysis considers that the background spectrum N 0 is well estimated and provides a mean value of the distribution under the null hypothesis, from which the DT can be derived based on a desired FPR. Now, we aim to investigate the DT in the spectral unmixing approach, where the measurand associated with a radionuclide's activity is the number of counts in the full spectrum range. The DT can be derived with the same idea of statistical test based on a "background", but adapted to the full spectrum analysis.

Recall that the spectral unmixing decomposes a gamma-ray spectrum into individual spectra of radionuclides. To determine the decision threshold of a single radionuclide indexed by j in the unmixing model, we reformulate the true linear mixing model with this radionuclide and an equivalent background:

Φa + b ! φ j a j + m (5.5) where φ j a j represents the individual spectrum of the j th radionuclide, while the other radionuclides and the background spectrum b compose an equivalent background:

m = l6 =j X φ l a l + b
In the spectral unmixing analysis, the activity of the j th radionuclide is associated to the mixing weight a j . Recall the definition in Section 5.1, the DT level of the measurand (i.e., a j )i sd e r i v e df r o m :

↵ = P (a j DT |ã j =0) (5.6)
Recall that, the measured spectrum x is composed of M channels, 8i 2 [1,...,M], the observed counts in each channel of the spectrum follows a Poisson distribution with mean value:

i =[φ j a j ] i + m i .
The DT can be derived from a standard hypothesis testing procedure test under the null alternative of H 0 :ã j =0,whic hleadsto i = m i for 8i 2 M . This can be generally formulated with some statistical test T as follows:

↵ = P T T ( )|8i, i = m i (5.7)
For this purpose, we propose to make use of statistical test based on different assumptions as follows:

a. Test based on the sum of observed counts: as i m p l es t a t i s t i c a l test to consider is based on the total number of counts, as measured by the sum of observed counts under the null hypothesis H 0 . Thanks to the statistical independence of each channel, this quantity should follow a Poisson distribution with mean value m (i.e., estimated equivalent background).

X i x i ⇠ Poisson X i mi ! (5.8)
Therefore, the DT (noted a ⇤ j )oftheestimatedactivityforthej th radionuclide with a given false positive rate ↵, can be derived from the cumulative distribution function (CDF) of the following distribution:

↵ = P ⇣ X i2C x i X i2C [φ j a ⇤ j ] i + X i2C m i ⌘ (5.9)
where C is some set of observed channels. The total number of counts from the full spectrum is a special case where C defines all the observed channels. While it allows to account for the full information carried out by the spectrum, it however poorly distinguishes the radionuclide to be tested from the background. We rather use the pre-specified channels in a region of interest, where the equivalent background is better distinguished from the j th radionuclide. (e.g, peak region of the radionuclide).

b. Test based on sum of weighted observed counts: In order to better distinguish between the radionuclide to be tested and the equivalent background, we further investigate statistical test derived from the sum of weighted counts in different channels written as P i w i m i with the following choice of w i :

• Let Ψ = h φ j m i
, the least squares solution of the mixing vector of Ψ can be written as:

â 2 argmin a 1 2 kx Ψak 2 (5.10)
for which the solution is â = Ψ † x, where Ψ † =( Ψ T Ψ) 1 Ψ T is the pseudo inverse matrix.

We make use of the the component of Ψ † related to φ j as the weights matrix, noted as w 1 = Ψ † . As graphically illustrated in Figure 5.3, it allows projecting onto the span of φ j parallelly to m.

Next, we evaluate the above statistical tests to assess the decision threshold for HPGe and NaI spectra, while the results are compared to those carried out with Monte Carlo simulations.

Evaluation of the decision threshold determination

For a comparison purpose, by fixing the false positive rate to ↵ =2 .5%, we calculate the DT for Poisson-based spectral unmixing with the following approaches:

• For each radionuclide, the accurate DT level quantified with respect to 1 ↵ percentile of the distribution of estimated value from Monte Carlo simulations under the null hypothesis of this radionuclide.

• the decision threshold assessment with different statistical tests for each Monte Carlo simulation. The comparisons are carried out for:

the Poisson statistical test based on the sum of counts in peak region. This is used only for HPGe measurements since using the peak regions to analyze NaI measurements is not interesting due to the correlations of spectra, for which it is better to take into account the full spectrum information.

the Gaussian statistical test based on two choices of weighted sum of counts, noted w 1 and w 2 respectively.

The evaluation are carried out for:

• Experiments on the described mixing scenario (see Figure 5.1) of an HPGe measurement, while the peak region, noted ROI, used are: 7 Be at 477 keV, 22 Na at 1274 keV, 137 Cs at 661 keV and 212 Pb at 238 keV.

• Simulations of NaI gamma-ray spectra measurements. By considering the spectral signatures used in [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF] where I(✓) is the Fisher information ( [Fisher, 1956]) defined as:

I(✓)=E ✓  @ 2 log f (x|✓) @✓ 2 (5.15)
The estimation standard error can be obtained by replacing the unknown true value ✓ by the estimated value ✓.

In our activity estimation problem, the investigated Poisson-based spectral unmixing provides a maximum likelihood estimate of the mixing weights, noted â. According to the Poisson likelihood, the Fisher information matrix can be written as:

I(â)=Φ T diag x ↵ (Φâ + b) 2 Φ (5.16)
We propose to assess the confidence interval of the estimated mixing weights â by the diagonal elements of p I(â) 1 , which approximates the standard deviation of the distribution.

Evaluation of confidence interval in spectral unmixing

In this paragraph, the confidence interval assessment with Fisher information matrix is evaluated with simulations of HPGe and NaI measurements described in the previous section. More precisely, we calculate the standard deviation from the Fisher information matrix and evaluate the results with Monte Carlo simulations.

Firstly, we make use of the Q-Q (quantile-quantile) plots, which compares the distribution of estimated activities of Monte Carlo simulations to the distribution generated from the standard deviation carried out with Fisher information. The aim is to i), test the normality of the estimator, which allows validating the confidence interval with a standard uncertainty in terms of Normal distribution. ii), compare the distribution to those obtained with Fisher information.

More precisely, for each radionuclide, we show the Q-Q plot for two data samples noted A 1 and A 2 :

• A 1 for estimated activity values of Monte Carlo simulations.

• A 2 for Normal distribution generated from the Fisher information of the estimation: N a 0 , 2 f where a 0 is the expected mixing weight of the radionuclide and f is the standard uncertainty calculated from the Fisher information matrix.

7 Be 22 Na 137 Cs 212 Pb percentage within a 0 ± f 68.0 66.1 68.6 68.4 percentage within a 0 ± 2 f 96.4 93.9 95.7 95.0 percentage within a 0 ± 3 f 99.9 99.8 99.8 99.9

Table 5.3 -Standard deviation from Fisher information matrix comparing to Monte Carlo simulations (HPGe measurements).

60 Co 134 Cs 137 Cs 152 Eu percentage within a 0 ± f 68.1 66.0 67.5 66.2 percentage within a 0 ± 2 f 95.3 95.0 94.0 95.5 percentage within a 0 ± 3 f 99.7 99.5 99.9 99.8

Table 5.4 -Standard deviation from Fisher information matrix comparing to Monte Carlo simulations (NaI measurements).

confidence interval are evaluated and validated for simulations of HPGe and NaI measurements. It should be noted that the simulations of HPGe measurement mimic realistic activities of aerosol measurements; therefore, the decision making with the proposed decision threshold and the confidence interval (i.e., statistical uncertainty) assessment with the Fisher information matrix can be used in the quantitative analysis of HPGe gamma-ray spectra measurements, this will be presented in Chapter 6.

CHAPTER 6

Analysis of HPGe gamma-ray spectra measurements Contents In this chapter, we investigate the application of spectral unmixing methods to analyze spectra of HPGe gamma-ray measurements. We focus on the evaluation and validation of the spectral analysis pipeline for gamma-ray measurements performed with a detection system (i.e., detector and source geometry) used in the laboratory, which can be virtually applied to other detection systems.

Applying the spectral unmixing to real measurements is not straightforward as the data need to be calibrated for a precise analysis. The general questions of calibrations in gamma-ray spectrum analysis are described in Section 6.1; we describe how these calibration tasks are carried out in standard peak based analysis and propose the calibration procedures in the spectral unmixing approach. Next, three calibrations steps are investigated and evaluated with a standard multi-gamma source of known activities:

• The calibrations required in the simulation process of spectral signatures are presented in Section 6.2. In this framework, the efficiency and the resolution are adjusted to mimic the response of the detection system.

• In Section 6.3, we discuss the complexity of the spectral signatures, which amounts to adding spectral contributions due to the lead shielding system into the spectral signatures.

• Section 6.4 presents the energy calibration for the Poisson based spectral unmixing algorithm.

The calibrations are subsequently validated with the spectrum analysis of the standard source in Section 6.5. Next, the proposed quantitative pipeline is applied to analyze past measurements of environmental samples (aerosol filters) in Section 6.6, where the results are compared to those obtained with the standard method. Finally, conclusions and perspectives can be found in Section 6.7.

Challenges of calibrations in gamma-ray spectrum analysis

The qualitative analysis in gamma-ray spectrometry consists of: i), the identification of radionuclides in the sample. ii), the quantification of each radionuclide by determining the activity in becquerel (Bq), as well as its characteristic limits and the uncertainties of the measurement.

In practice, the accurate quantitative analysis in gamma-ray spectrometry strongly depends on the proper calibration of the measurement instrument (i.e., the detector and the sample geometry), which relates the detected counts in a measured spectrum to the activity of radionuclides. The main calibration tasks are: energy calibration for identifying radionuclides, efficiency calibration and resolution calibration for the quantification of radionuclides.

In this section, we detail these calibrations and how they are performed in standard peak based analysis. The calibrations in spectral unmixing analysis need to be performed differently; they are introduced at the end of this section.

Calibrations in peak based analysis

Energy calibration

We would like to recall that a gamma-ray spectrum measurement provides the energy response of emitted photons (gamma-ray or X-ray) with multichannel analyzer (MCA), such a spectrum is the histogram of detected events as a function of the voltage ranges (i.e., channel number). More precisely, the number of channels in MCA used for gamma-ray spectrometry is fixed: 1024 channels (i.e., NaI measurements [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF]) up to 16384 channels (i.e., HPGe measurements used in the laboratory). Referring to Chapter 2, the MCA collects pulses in all voltage ranges (depending on the deposited energies), noted 0 V max , while each pulse height increments the count in a corresponding channel. For instance, when V max takes 12 V and the number of channels is set to 16384, the number of counts of 5 V is according to channel 8192. The radionuclides are characterized by energies, the energy calibration is thus needed to identify radionuclides, which allows to relate the channel number to the corresponding energy:

E = f (N Ch )( 6 . 1 )
where N Ch and E are the channel number and the corresponding energy. As shown in Figure 6.1, the energy calibration is performed by localizing peaks of known energies, which allows fitting the calibration function introduced in Eq.(6.1).

Efficiency calibration

Efficiency calibration defines a function that relates the number of observed photons and the disintegration rate. In peak based analysis, the detection efficiency of full energy peak is defined as:

✏ = N p N s (6.2)
As illustrated in Figure 6.2, it relates the number of counts in the full energy energy peak, noted N p , and the number of photons emitted by the source for this energy, noted N s , with N s = A⇥I ⇥t (A: activity, I: emission function to describe the peak shape versus the corresponding energy:

FWHM = a + b p (E + cE 2 )( 6 . 3 )
where E is the peak energy and FWHM represents the full width at half maximum of the peak.

Towards calibrations for spectral unmixing

In the spectral unmixing setting, the activities of radionuclides are proportional to the estimated mixing weights â in the mixing model: Φâ + b. While we are concerned with measurements of environmental samples, the spectral signatures (i.e., each column of Φ) need to be simulated, since the radionuclides contained in environmental samples are rarely present in standard sources. To this end, we make use of simulated spectral signatures based on the MCNP particle transport code [Briesmeister, 2000], which is used to perform gamma-ray spectra simulations throughout this thesis. The experiments in this thesis use two versions of the MCNP code (see Appendix Cf o rd e t a i l so ft h es i m u l a t i o nc od e ) .

• MCNPX (MCNP eXtended) [Pelowitz, 2011] for the gamma-ray spectrum simulation of source that emits photons at specific energies.

• MCNP-CP (A Correlated Particle Radiation Source Extension of a General Purpose Monte Carlo N-Particle Transport Code) [Berlizov, 2006] for the gamma-ray spectrum simulation of radionuclides, which simulates the physics of nuclear decay and the subsequent emissions.

Quantitative analysis with simulated spectral signatures

The MCNP simulates gamma-ray spectrum and finally normalizes the spectrum by the number of source-particle histories run in the simulation process. Accordingly, a simulated spectral signature corresponds to the energy response with unit particle (i.e. one disintegration). The estimated mixing weight â of a radionuclide is therefore the number of disintegrations. Recall that activity unit becquerel (Bq) is defined as the number of disintegrations per second. This leads to quantify the radionuclides' activity with:

Activities (Bq) = â t (6.4)
where â is the estimated mixing weights and t is the counting time of the measurement. To validate the quantification in Eq.(6.4), the simulation where N p is the number of observed counts in this full energy peak E, A is the activity of the radionuclide that emits photons at this energy, which is known for the standard source, I is the intensity of this energy peak (i.e. the probability of emission of the corresponding gamma-ray for 100 disintegrations of the radionuclide) and t is the counting time of the measurement.

• Efficiency calibration in simulation configurations: the objective is to calibrate the simulation detection system so that simulated detection efficiencies are close to ✏ exp values. In the settings of MCNP-CP, F8 tally (pulse height tally) specifies the energy distribution of pulses, which allows us to simulate the photon energy peak as a Dirac by default (see details in Appendix C).

Recall that a radionuclide emits photons at several energy peaks. Simulations are performed individually for each radionuclide of the source. Knowing that the simulated response is normalized by the number of particles run in the simulation, the detection efficiency of a full energy peak E of the simulated detection system is:

✏ simu (E)= H(E) I(E) (6.6)
where H is the peak height of this peak and I is the intensity according to E.

The simulation configurations have been changed so as to provide better agreement of ✏ exp and ✏ simu , while the details can be found in Appendix C. By comparison, the full energy peak efficiency of radionuclides in the reference source, noted ✏ exp and the full energy efficiency calculated with MCNP-CP simulations, noted ✏ simu ,t h er a t i o ✏ simu ✏ exp at each peak energy are displayed in Figure 6.5.

As a result, the full energy peak efficiency ratio shows a difference within 5% for all the peaks, which will be taken into account in the uncertainty evaluation of the measurement.

Resolution calibration in spectral unmixing

The detection resolution is an important feature of gamma-ray measurements, for which the calibration is particularly key in the analysis of HPGe measurements since the number of counts is large in the few channels of the peaks. In the MCNP-CP input file, the resolution can be specified with the option GEB card, the parameters in the empirical model Eq. ( 6.3) need to be calibrated in this step.

• Simulated spectral signatures need to be interpolated to the measurement energy bins.

• Estimation bias can be caused by the shift of energy between a measured spectrum and simulated spectral signatures.

To overcome these problems, we prop ose to use interp olations of simulated spectral signatures in a high resolution domain corresponding to responses in smaller energy bins. It can be formulated as follows:

Φ = HΦ hr , Φ hr = LΦ (6.7)
where Φ and Φ hr represent spectral signatures respectively in actual energy domain and high resolution domain. Note that E and E hr are energy bins of Φ and Φ hr . The operator H and L correspond to the interpolations of E hr ! E and E ! E hr respectively.

More precisely, the energy re-calibration for the analysis of the mentioned standard source is described as follows:

Step 1: Interpolation of simulated spectral signatures In the MCNP simulation process, the energy bin scheme of the simulated spectrum is set to interval of 0.1 keV (i.e., simulation of energy responses at each 0.1 keV). As illustrated in Figure 6.7, simulated spectral signatures Φ simu are firstly interpolated into high resolution energy bins (see illustration in Figure 6.7a), with 10 energy bins in one channel (i.e., energy responses at each 0.01 keV). Then the high resolution spectral signatures Φ hr are interpolated into measurement energy bins (see illustration in Figure 6.7-b), noted Φ r .

Step 2: First activity estimation By using Φ r , knowing that x (measured spectrum) and b (background). The spectral unmixing provides a first estimation of activities, which is denoted as âr . The energy shift is illustrated in Figure 6.8 for a single peak.

Step 3: Correction of the energy shift for peaks of known energies The energy shift present in the first estimation for one of the energy peaks is illustrated in Figure 6.8, we aim to fit an energy shift correction function from k peaks, noted [e 0 ,...e k ]. In such context, to determine the energy shift of each peak, we define a maximum number of shift step in the high resolution domain, which is fixed to N max =1 5( i.e., shift the spectra to left or right with maximum 15 energy bins in the high resolution domain, which is according to 0.15 keV in our simulated response). The energy shift of a peak can be therefore obtained with the shift step that minimizes the estimation residual. Details are given in Algorithm 7. re-calibrate the spectral signatures' energy by solving the following weighted least squares problem that aims to fit a polynomial function with c 0 ,c 1 ,c 2 ,c 3 .

min k X i=1 w i e i + ∆ i e (c 0 + c 1 e i + c 2 e 2 i + c 3 e 3 i ) 2 (6.8)
where e i is the photon energy of the i th peak, e i + ∆ i e is the target energy of the i th peak, and k is the number of peaks used in this calibration step. The weight of each peak is computed as follows:

w i = 1 P channels in the i th peak m (6.9)
m is the equivalent background with respect to other spectral contributions with the exception of the radionuclide that emits photons at the i th peak.

To overcome this problem, we can write the co efficients with a vector c =[c 0 ,c 1 ,c 2 ,c 3 ], by considering the following matrix form:

•

c = V T Σ 1 V 1 V T Σ 1 (e + ∆ e )( 6 . 1 1 )
Finally, the corrected spectral signatures Φ can be obtained by interpolating the corrected energy bins (High resolution domain) into energy bins of the measurement.

Φ f = H c 0 + c 1 E hr + c 2 E 2 hr + c 3 E 3 hr
where H is the interpolation operator described in Eq.(6.7), and Φ f is the final re-calibrated spectral signatures.

Evaluation of the energy re-calibration The estimation residual is compared for results that we computed before and after energy correction. These residuals, which are normalized by the observed counts, are displayed for peaks in Figure 6.9. This result highlights that the energy shift correction improves the spectral unmixing, which has significant benefits by reducing the estimation residual in peak regions of the spectrum. For the analysis of HPGe gamma-ray spectrum, where the counts in peaks regions are dominant, we can especially benefit from a significant improvement from this energy re-calibration step.

Validation with the standard source analysis

In this section, we aim to validate the proposed calibrations with the standard source. For this purpose, the standard source is measured with the calibrated detection system. The measured spectrum is analyzed with the investigated spectral analysis pipeline, which includes the calibration steps presented in previous sections and characteristic limits assessment introduced in Chapter 5.

Uncertainties of the standard source analysis

Before proceeding with the spectral analysis, the main practical problem comes from the uncertainties of the actual measurements. It is well known that the measurement uncertainties contain not only statistical uncertainties of the estimation, but also metrological uncertainties related to the material and the analysis method. In practice, the uncertainties of the results are determined from the probability that the estimated activity is contained in ac o n fi d e n c ei n t e r v a lb a s e do nag i v e np -v a l u e , where =0 .05 is usually taken into account. The uncertainty is according to 2 for the Normal distribution assumption (i.e. k=2). For this purpose, we make use of the relative uncertainties defined with: u = 2 â where â is the estimated activity. We establish an uncertainty budget by using the spectral unmixing, which contains the following uncertainty terms:

• For a given radionuclide, the statistical uncertainty with the Fisher information matrix as described in Section 5.4, noted:

u 1 = 2 f â
where f and â stand for the standard deviation calculated from Fisher information matrix and the estimated activity respectively.

• As previously investigated, spectral signatures Φ are calibrated with a relative uncertainty of u 2 =5%.

• The uncertainty of the standard source that used for the calibration: u 3 =5%.

• The uncertainty to take into account the variation of the sample position, for which we take the same value as considered in Genie 2000 analysis in the laboratory u 4 =2%.

• The uncertainty associated withthe variation of the detector u 5 =5%.

The relative uncertainty of the estimated activities can be assessed with the combination of the above uncertainty terms:

u = q u 2 1 + u 2 2 + u 2 3 + u 2 4 + u 2 5 (6.12)
As a result, the activity estimation carried out with the Poisson based spectral unmixing are compared to the reference activities given by the source in Table 6.1. To further evaluate the results, we make use of the ⇣ scores [ISO 13528, 2015] derived with a standard normal distribution centered at the reference activity:

⇣ scores = â a ref q 2 + 2 ref (6.13)
where â and are the estimated activity and the standard uncertainty (according to k =1o fE q . ( 6 . 1 2 ) ) ,a ref and ref represent the activity and uncertainty (k =1)ofthereferencesource.

As reported in Table 6.1, the ⇣ scores assessed for radionuclides of the standard source satisfies the following criteria:

• |⇣ scores |  2i n d i c a t e s" s a t i s f a c t o r y "

• 2  |⇣ scores |  3i n d i c a t e s" q u e s t i o n a b l e "

• |⇣ scores | 3i n d i c a t e s" u n s a t i s f a c t o r y "

The evaluation of calibrations in spectral unmixing with the standard source allows drawing the following conclusion: the activity and uncertainty estimation with the spectral unmixing method is able to provide quantitative analysis such that close to reference values. It can be used to analyze the aerosol measurements performed with the same detection system. 6.6 Experiments on aerosol measurements 6.6.1 Uncertainty budget of the aerosol measurements

The uncertainty terms to analyze spectra performed with the calibrated detection system has been assessed in the previous section. To analyze aerosol measurements, another uncertainty term associated with the variation of the pressed filter needs to be taken into account in the uncertainty budget. More precisely, due to the preparation process of aerosol samples, the thickness and the density of the pressed filter samples are different than those of the standard source, which leads to variations of the detection efficiency.

This uncertainty term is evaluated for the first time during the thesis. To this end, the thickness and the density of 1500 filters of past gamma-ray measurements in the laboratory are measured and the results are summarized in table 6.2.

The spectral analysis procedure is calibrated with the standard source of thickness = 4.64 mm and density = 1.15 g/cm 3 , which is however far from the median values of the actual sample geometries. Therefore, to analyze the aerosol measurements, we make use the spectral signatures simulated with 6.6.2 Re-analysis of aerosol measurements

The challenging problem to analyze environment radioactivity measurements is to estimate the activity of low-level radionuclides with high accuracy and high sensitivity. We make use of the Poisson based spectral unmixing to re-analyze past measurements of environmental samples (aerosol filter) performed with the calibrated detection system. The results are carried out with the characteristic limits introduced in Chapter 5 and the evaluated uncertainty budget.

The spectral unmixing is applied to analyze 67 aerosol measurements during the past two years. They have been measured with the calibrated HPGe detector. The results of the estimated activities consist of:

• Activity of radionuclides.

• Assessment of uncertainties.

• Decision threshold of radionuclides derived from statistical test introduced in Chapter 5 with ↵ =0.025.

For a comparison purpose, we focus on the activity estimation of 7 Be, 22 Na, 40 K, 137 Cs and 210 Pb resulting from the Poisson-based spectral unmixing and Genie 2000. Figure 6.11 illustrates the distribution of the ratio of activities estimated with the two methods:

r = â1 â2
where â1 for Poisson based spectral unmixing and â2 for Genie 2000.

The ratio of activity estimations of the two methods show that:

• For 7 Be and 210 Pb, the estimated activities with the spectral unmixing are systematically lower than those obtained with Genie 2000. Recall that Genie 2000 determines radionuclides' activities with an efficiency curve calibrated from the standard source. However, as we can see in Figure 6.10, the efficiency performed with the source geometry is systematically less efficient than routine measurements, which leads to an over-estimate of the activities. In the proposed spectral unmixing analysis, the spectral signatures are simulated with median values of the geometries' height and density (see Table 6.2 for details).

• For the analysis of low-level radionuclides, the spectral unmixing provides larger activities than Genie 2000. This can be explained by some over-estimation with spectral unmixing or under estimation with Genie 2000 at low statistics. For further validation of low-level radionuclides' quantification, low activity sources of known activities are needed.

Furthermore, we compare the activity estimation of the two methods by considering the uncertainties. Figure 6.10 displays the activities estimated with Genie 2000 as a function of those estimated with Poisson based spectral unmixing, where the error bars represent the uncertainties of the results.

In Figure 6.10, the error bars box (red) includes the uncertainties of the results is comparable with the line of y = x (blue) for 7 Be, 40 Kand 210 Pb. For 22 Na and 137 Cs of very low-level activities, the error bars boxes are under the line of y = x. The results confirm that the spectral unmixing tends to systematically provides larger activity estimation comparing to the Genie 2000. To interpret the results, the Genie 2000 is likely to over estimate the background when the number of counts in the peak region is low. However, the quantification of low activity radionuclides needs to be further investigated with reference sources.

Next, we focus on the sensibility of the Poisson spectral unmixing analysis to detect the 137 Cs at trace level. The results reported in Table 6.3 are illustrated in Figure 6.11. The detection of 137 Cs of these 6 measurements are shown to be significant comparing to the decision threshold, whereas the results are non-significant with Genie 2000, This further confirms the sensibility of the spectral unmixing analysis.

Conclusion

In this chapter, we investigated the quantitative analysis of experimental aerosol measurements with HPGe gamma-ray spectrometry. Instrumentation calibration procedures of the spectral unmixing are proposed to analyze HPGe measurements using MCNP-CP simulated spectral signatures. To summarize, the quantitative analysis with a given detection system is performed with the following calibration tasks: i), in the simulation process, the efficiency and the resolution are calibrated so that the simulated spectral signatures can reproduce the actual energy response of the detection system, ii) the measured spectrum need to cope with the spectral contributions due to the specific installations of the detection system (e.g., lead shielding) by adding a spectrum into the spectral signatures' dictionary, iii), an energy recalibration step is proposed to correct the energy shift between energy bins of the simulations and the measured spectrum.

The proposed calibration procedures are evaluated and validated for a detection system with a multi-gamma standard source of known activities. The aerosol measurements performed with this calibrated detection system are subsequently analyzed with the proposed analysis pipeline (characteristic limits + calibrations). The results are compared to those obtained with Genie 2000 analysis. In conclusion, the Poisson based spectral unmixing significantly improves the sensitivity of radionuclides' identification, which is particularly required by the rapid detection and rapid characterization of artificial radionuclides (e.g., 137 Cs) under emergency conditions. Future investigations with low-level standard source are necessary to validate the quantification of low-level radionuclides in real data analysis.

CHAPTER 7 Conclusion and perspectives

Conclusion

The thesis addresses the gamma-ray spectrum analysis problem, which covers both the development of analysis methods and metrological aspects of the radioactivity determination with gamma-ray spectrometry technique. We investigated the spectral unmixing to tackle the identification and quantification of radionuclides with the aim of finding the spectral contributions from a measured gamma-ray spectrum. The proposed spectral unmixing was formulated as a regularized inverse problem that makes use of the observed data in the full energy range and takes into account the Poisson statistics of the underlying physical process.

The spectral unmixing aims at decomposing a measured spectrum into the radionuclides' spectral signatures and a background spectrum. We first tackled the problem in a supervised framework in Chapter 3, where the spectral signatures and the background spectrum are considered as known. Experiments on both simulated and measured gamma-ray spectra indicated the significant advantages of using the Poisson-based full spectral unmixing. The time to detect the low-level radionuclide 137 Cs in aerosol filters was reduced to 4 days after the sampling, whereas we need 8 days with standard analysis method.

In the spectral unmixing problem, the identification of the spectral signatures is usually a challenging problem, it is as well an important issue that we need to cope with in gamma-ray spectrum analysis, since the active radionuclides present in a measurement is actually unknown in advance. For this purpose, we presented the sparse spectral unmixing in Chapter 4. A novel Poisson-based Greedy algorithm has been proposed, it enables to accurately identify the active radionuclides and improves the accuracy of the activity estimation.

Next, the quantitative analysis of the radioactivity measurements by using spectral unmixing methods has been investigated by focusing on different metrological problems. In the field of radioactivity measurements, the characteristic limits need to be assessed. In Chapter 5, we presented the characteristic limits calculation for the spectral unmixing analysis tool investigated in this thesis. In Chapter 6, we proposed the spectrum analysis procedure for experimental gamma-ray measurements, in which the instrumentation calibrations are thoroughly investigated and further evaluated with a detection system used in the laboratory.

The novel spectral unmixing approach has been applied to analyze gammaray spectra measurements and compared to standard analysis method, the spectral unmixing is shown to be more sensitive in the task of detecting low-level radionuclides.

In this thesis, we focused on the spectrum analysis of aerosol measurements performed with HPGe detectors. The investigated methods can be further applied to other detectors and other radioactivity measurement domains:

• The measurements with NaI(Tl) detectors are performed by the TRI-LATAC device designed by IRSN for the radioactivity monitoring in food and feed. The objective of these measurements is to control if the activity in the food and feed samples exceed the maximum permitted levels of radioactive contamination following a nuclear accident or any other case of radiological emergency laid down by the Council Regulation (Euratom) 2016/52. Due to the large number of samples to be measured in this case, fast measurements are particularly required while keeping good identification performance.

• In the framework of controlling illegal nuclear material trafficking, the investigated spectral unmixing method can be applied to measurements using radiation portal monitors with scintillation detectors (plastic, NaI(Tl)) to improve the detection limits.

Perspectives

Spectral unmixing with temporal signatures

In the thesis, we focused on the analysis of individual gamma-ray spectra, where only information about the signatures in energy of the radionuclides can be used in the unmixing process. In Chapter 3, we investigated the analysis of a temporal sequence of 11 HPGe spectra, which corresponds to a non-standard data acquisition procedure at IRSN. However, for the rapid detection of artificial radionuclides, it is expected that further accounting for the temporal information carried out by the radioactive decay will be beneficial for the accuracy and sensitivity of the spectral unmixing.

More precisely, as presented in Chapter 2, the activity of a given radionuclide at time t can be obtained by an exponential decay function:

a[t]=a[0]e t (7.1)
where a[0] is the activity at t[0]. Each radionuclide has a specific decay constant , which can further help discriminating between distinct radionuclides. The Poisson unmixing algorithm we investigated in this thesis can be therefore extended to jointly take into account the spectral signatures and the temporal signatures of radionuclides to estimate their activities.

As depicted in Figure 7.1, for a given measurement between two different times t i and t j , a radionuclide's activity, noted a j , is defined as the integral of the activity decay between t i and t j :

a j = Z t j t i a[0]e t dt (7.2) = a[0] (e t i e t j ) (7.3)
In the experiments described in Section 3.5, the time intervals are contiguous, the starting time of some measurement is exactly the ending time of the previous one.

Let's now define by S the matrix defined as the row-wise concatenation of L temporal measurements, where the j th column of S defines the measurement ending at time t j . The mixing model can now be extended to the are considered in the simulations, where the decay constant of the parent radionuclide is used for its daughter radionuclide (see Table 7 The temporal/spectral-based unmixing algorithm is compared with our standard Poisson unmixing algorithm based on the relative estimation error. The comparisons are carried out for the 5 radionuclides and featured in Figure 7.2, where the median values and the first and last quartiles are displayed.

Unsurprisingly, further accounting for the temporal information featured by the difference in activity decay of different radionuclides, help improving the estimation accuracy to a large extent. This is especially true for early measurements, where the increased complexity of the measured spectra dramatically hampers the estimation of the low-activity radionuclides. It is also expected that accounting for both temporal and spectral information can help improving the sensitivity and detection abilities of the unmixing process.

Future work will also have to deal with the full complexity of gamma-ray measurements by also accounting for the measurements in which radionuclides are not in equilibrium (e.g., the measurements corresponding to s1-s5 in experiments of Section 3.5). In this case, the activity decay is not known for certain radionuclides, which either requires an hybrid mixture model or aj o i n te s t i m a t i o no ft h e i ra c t i v i t yd e c a y .

The preliminary results carried out with the temporal/spectral-based unmixing opens the perspectives of the following applications:

(timeline of the measurements) and the spectral information (counting statistics of each measurement).

• Online analysis of aerosol filters collected with high volume air samplers measured with HPGe detectors in LMRE. These samplers, part of the OPERA-Air network of IRSN, are localized all over France and the filters are thus received at LMRE at least 24 hours after sampling; the radon progenies are therefore at equilibrium and the future algorithm taking into account the decay information should improve the radioactivity analysis in terms of sensibility, rapidity and accuracy (bias).

Making profit of the past processed data, learn to unmix

Involved in the radioactivity monitoring work, the laboratory is in charge of gamma-ray spectra measurements with 20 HPGe detectors. 400-500 aerosol filter measurements have been performed per year during the last decade. The unmixing algorithms we introduced during this thesis apply to a single measurement without accounting for the knowledge accumulated from the processing of past measurements. In other words, they are agnostic to the available archive of past measurements, which bring highly valuable information to perform accurate radionuclide activity estimation, especially for the challenging low-statistics regime.

Extracting information from the archive of past processed measurements can be done in several ways. Following a traditional statistical approach would boil down to deriving a statistical model for the past measured radionuclides' activities to build a prior distribution, which can be plugged in our Poisson unmixing algorithm. However, the relationship between the activities of several radionuclides may be intricate and therefore hard to model with simple statistical models.

In contrast, machine learning is a potentially interesting framework to design a model for the sought-after radionuclides' activities that could better account for their complexity. Interestingly, deep unfolding techniques have attracted a growing interest during the last five years [START_REF] Diamond | Unrolled optimization with deep priors[END_REF], [START_REF] Monga | Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing[END_REF], allowing to combine standard solvers for linear inverse problems with prior model learning. In a nutshell, such methods consist in building a recurrent network that reproduces that inversion process of common optimization algorithms (e.g. Forward-Backward Splitting [START_REF] Gregor | Learning fast approximations of sparse coding[END_REF], Adler and Öktem, 2017, Andrychowicz et al., 2016] and proximal primal-dual [START_REF] Adler | Learned primaldual reconstruction[END_REF] where is the Hadamard product. The term J stands for a penalization term based on some prior knowledge about a, e.g., non-negativity investigated in Chapter 3. The second term is the neg-log-likelihood of the Poisson distribution. Beyond the non-negativity of a, one could benefit from the available archive of hundreds to thousands of routinely taken measurements to further learn a data-driven prior J ( . ), which should improve both the estimation bias/variance.

In [START_REF] Bobin | Learning to unmix from Poisson measurements with application to -spectroscopy[END_REF], we introduced an unrolled version of the ADMM algorithm to tackle the following equivalent of the problem in Eq.( 7 where u is an extra variable, v is the dual variable related to the constraint u = Φa + b and ⇢ is a positive scalar. The regularization term J Θ now depends on some parameters Θ to be learned from the available training set.

In the present context, the advantage of ADMM is that it allows to split the inversion of spectral signatures dictionary and the application of the regularization in two distinct steps. A single iteration k of the proposed Learned-ADMM, and subsequently each layer of the resulting recurrent network, reads as:

• Update of u : minimizing Eq.(7.5) with respect to u leads to: u (k+1) = prox 1/⇢P ⇣ Φa (k) + b 1/⇢v (k)

⌘

, where prox 1/⇢P is proximal operator of the Poisson neg-likelihood with scaling parameter 1/⇢ [START_REF] Combettes | A douglas-rachford splitting approach to nonsmooth convex variational signal recovery[END_REF], the proximal operator of the Poisson neg-likelihood is calculated in Appendix B.

• Update of a : updating a for fixed parameters Θ is done as follows:

a (k+1) =argmin a J Θ (a)+ ⇢ 2 kΦa + b +1/⇢v (k) u (k+1) a (k) k 2 2
Since the dictionary of spectral signatures Φ is not orthogonal, this problem does not admit a closed-form solution. Instead of resorting to an u m e r i c a le v a l u a t i o nw i t ha ne x t r ai t e r a t i v ep r o c e d u r e ,i ti sr a t h e r approximated with a projected least-square estimate of the form:

a (k+1) = R Θ Φ † u (k+1) b +1/⇢v (k)
clides of interest in aerosol filters have been weekly measured at LMRE with HPGe for several decades (see Figure 7.4 for 2-years period). The learning algorithm allows extracting knowledge of the average levels and the variability of their activity concentrations. Future investigations can also focus on the rapid detection of anomaly events with information learned from these routine measurements.

• Use of high frequency in situ measurements. In situ air measurements are performed by gamma probes, part of the OPERA-Air network of IRSN, for environmental radiation surveillance and early warning purposes. They measure continuously the environmental gamma dose rate. Replacing this probes by new spectrometric probes with NaI(Tl) (e.g. SpectroTracer, Bertin Instruments) would allow to use air measurements by NaI detectors every 10 minutes to improve the analysis tools. A similar approach could be applied to the in situ measurements of the aerosols filters collected by air samplers. For instance, the air samplers of the international network managed by the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) performed one sampling/measurement per day. Using the acquisition of the HPGe daily measurement could improve the analysis tools.

Open questions

• Dealing with the variability of the spectral signatures: It is a question of future research to investigate the variability of the spectral signatures. The detection efficiency varies from one detector to another, using spectral signatures simulated for a detector to analyze spectra measured with other detectors can provide significant estimations bias. For a given detector, the detection efficiency varies when measuring the actual geometries of different dimensions (e.g., 10 mL and 30 mL used in the laboratory).

We investigated the instrument calibrations in Chapter 6 for quantitative analysis of HPGe gamma-ray measurements. In such context, the efficiency and the resolution of a given detection system (detector + source geometry) are adjusted thanks to a standard source. However, the characteristics of the detector vary over time and lead to variable spectral signatures. Moreover, as discussed in Chapter 6, the dimension and the density of pressed aerosol samples has an impact on the detection efficiency. e.g., Figure 7.8 illustrates the simulated spectra of 40 Kbyslightlychangingtheconfigurationsofthedetectororgeometry .

estimation error of the background b, which leads to over/under estimate of radionuclides' activities.

-Moreover, the background spectrum may consists of several radionuclides, which also participate in the measured sample, e.g., 40 K, which hampers the estimation accuracy of these radionuclides.

Therefore, it is of interest to jointly estimate the background spectrum in the spectral unmixing problem, which would allow to improve the estimation accuracy of the radionuclides' activities.

Publications

Papers

• Xu, J., [START_REF] Paradis | Spectral unmixing applied to fast identification of -emitting radionuclides using NaI(Tl) detectors[END_REF]. Sparse spectral unmixing for activity estimation in -ray spectrometry applied to environmental measurements. Applied Radiation and Isotopes, 156:108903.

• André, R., Bobin, C., Bobin, J., Xu, J., and de Vismes Ott, A. Metrological approach of -emitting radionuclides identification at low statistics: application of sparse spectral unmixing to scintillation detectors. Metrologia in revision.

Papers in preparation

• Analysis of -ray spectra with spectral unmixing: determination of characteristic limits. In Prep.

• Calibrations for quantitative analysis of HPGe -ray measurements. In Prep.

Presentations

• Sparse spectral unmixing for activity estimation in gamma-ray spectrometry. Oral presentation, International Conference on Radionuclide Metrology and its Applications (ICRM), Salamanca, Spain. 2019.

• Learning to unmix from Poisson measurements with application tospectroscopy. Poster, Signal Processing with Adaptive Sparse Structured Representations (SPARS) workshop, Toulouse, France. 2019.

• learning to unmix in gamma-ray spectrometry. Oral presentation, Journées Machine Learning et Physique Nucléaire, Orsay, France. 2019.

Open source code

The methods developed in this thesis have been tested on both simulated and experimental data. The code will be available at https://github.com/ IRSN-LMRE.

APPENDIX A

Nuclear data of radionuclides More precisely, the above minimization problem in the gamma-ray spectral unmixing problem can be described as follows:

• The data fidelity term f (a) associated to least squares error is smooth with a Lipschitz gradient, it can be solved with gradient descent algorithms. However, along with non-differentiable regularization terms such as the non-negativity constraint used in the gamma-ray spectral unmixing, it requires the use of non-smooth optimization algorithms, more specifically the use of proximal algorithms [START_REF] Parikh | Proximal algorithms[END_REF].

• In case that we consider the exact Poisson statistics based data fidelity term, which is not differentiable. Along with the non-negativity constraint, none of the two terms is differentiable. Fortunately, both terms admit a proximal operator, which also makes the application of the proximal algorithms.

Involved in solving convex non-smooth optimization problems, the overall advantage of using proximal algorithms is the flexibility and simplicity it provides to add regularization terms. Since the optimization steps in these with Lipschitz constant L = kΦ T Φk 2 , where k . k 2 stands for the spectral norm of a matrix (i.e., its largest eigenvalue).

Similarly, when weighted least squares error is considered, in which: • the regularization term g (a)=i. a 0 is convex and admits a proximal operator.

Also known as the proximal gradient algorithm, the minimization step of the FBS algorithm can be described as:

a (k+1) =pro x g a (k)
rf a (k) (B.9)

where the "forward" referring to the gradient step on the differential function f and the "backward" to the proximal step of the function g. The algorithm is guaranteed to converge to the unique minimum when the gradient step  1/L. In the above optimization problem, none of the two terms is differentiable. Hence, different algorithmic strategies can be considered to design a minimizer. This includes the Alternating Direction Method of Multipliers (ADMM -see [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]) or primal-dual proximal algorithms such as the one introduced by Chambolle and Pock in [START_REF] Chambolle | A firstorder primal-dual algorithm for convex problems with applications to imaging[END_REF]. In this thesis, we chose the latter as it requires few algorithmic hyperparameters to be tuned, which can further be fixed in a quite easy manner.

Primal-dual algorithm

More generally, the Chambolle-Pock algorithm aims at tackling optimization problems of the form: 3. Refer to [START_REF] Parikh | Proximal algorithms[END_REF], by Moreau decomposition, the proximal operator of the convex conjugate of the function f :

prox ⇢f * (v)=v ⇢prox (1/⇢)f ✓ v ⇢ ◆ (B.21)
• SDEF card is used to specify the source and its position, in which:

the keyword "ERG" in MCNPX specifies the energies for which we aim to simulate the detector response.

the keyword "ZAM" in MCNP-CP specifies the radionuclides for which we aim to simulate the detector response with their respective atomic and mass numbers Z and A, M indicates an isomeric state. It simulates the statistical processes accomplished following the disintegration of the given radionuclides, including the cascade emissions, the escape peaks, and the peak sum, with the nuclear data from the Evaluated Nuclear Structure Data File1

The spectral signatures used in the experiments in Chapter 6 are performed with MCNP-CP code. To reproduce the actual energy response of a detection system, the simulation input file is configured for the specification of the detector illustrated in Figure C.1.

For a better agreement of the simulated spectral signatures and the actual energy response of the detection system (i.e.,efficiency calibration), the configurations of the Ge crystal dimension (see Figure C.2) can be slightly changed with the following principles:

• changing the crystal diameter has an impact of the detection efficiencies on the whole energy range.

• changing the dimension at the bottom of the Ge crystal has an impact of the detection efficiency for source that emit photons at high energy.

• the thickness of the dead layer on the top of the Ge crystal has an impact of the detection efficiency for source that emit photons at low energy.

In summary, the configurations of the Ge crystal dimension is changed by removing 3mm thickness from the bottom of Ge Crystal to decrease the efficiency of sources that emit photons at high energy and changing the dead layer of 4µm to 13µm in the top of Ge Crystal to decrease the efficiency of sources that emit photons at low energy.

  3.3.3 Results on two radioactive sources mixture . . . . . . . 3.4 Realistic simulations of routine aerosol samples . . . . . . . . 3.4.1 Convergence rate of algorithms . . . . . . . . . . . . . 3.4.2 Comparisons of the unmixing algorithms in different counting regimes . . . . . . . . . . . . . . . . . . . . . 3.4.3 Evaluation of unmixing algorithms for low-level artificial radionuclide . . . . . . . . . . . . . . . . . . . . . . 3.5 Experimental results with real spectra from aerosol samples' measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Dimension of the spectral dictionary . . . . . . . . . . 3.5.2 Comparisons with peak-based analysis . . . . . . . . . 3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Fisher information to compute confidence intervals . . 5.4.2 Evaluation of confidence interval in spectral unmixing . 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

  M ⇥ 1c o l u m nv e c t o r( m e a s u r e ds pe c t r u m ) b M ⇥ 1c o l u m nv e c t o r( b a c k g r o u n ds pe c t r u m ) Φ M ⇥ N matrix (spectral signatures) a N ⇥ 1c o l u m nv e c t o r( m i x i n gw e i g h t s ) M number of channels N number of radionuclides f (a)d a t a fi d e l i t y t e r m g(a)r e g u l a r i z a t i o n t e r m life of a radionuclide becquerel (Bq) unit of activity List of operators kxk 0 norm `0: number of non-zeros elements in x kxk 2 spectral norm of x, its largest eigenvalue Hadamard product rf (.)g r a d i e n t o f f u n c t i o n f prox ⇢h (y)p r o x i m a l o p e r a t o r o f t h e f u n c t i o n h defined in Appendix B. L (x|✓)c o s t f u n c t i o n , n e g -l o g -l i k e l i h o o d ( f (x|✓)f o rl i k e l i h ood ) I x (✓) Fisher information

Algorithm 3

 3 Pseudo code of spectral unmixing with multiplicative update algorithm Input: Measured spectrum: x Spectral signatures: Φ and Background: b while stopping condition not achieved do Update 8j =[1,...,N]:

4. 3

 3 Greedy algorithm to identify active radionuclides 4.3.1 Greedy algorithms in least squares problem

Algorithm 5

 5 Pseudo-code of OMP algorithm Input: Measured spectrum: x and background radiation spectrum b The spectral signatures: Φ c = 1 ... N ([1,...N]: each index for a radionuclide in Φ c ) Initialization: The selected indices: I 0 = ;, the residual r 0 = x b

• M 1

 1 : the model with an extra active radionuclide As mentioned in Section 4.1, model selection methods such as AIC and BIC are commonly used to penalize the model complexity. Algorithm 6 Pseudo-code of Poisson-based OMP algorithm Input: Measured spectrum: x and background radiation spectrum b The spectral signatures: Φ c = 1 ... N ([1,...N]: each index for a radionuclide in Φ c )

Contents 5 . 1 4

 514 Characteristic limits in radioactivity measurements . . . . . . . . . . . . . . . . . . . . . . . Confidence interval . . . . . . . . . . . . . . . . . 95 5.4.1 Fisher information to compute confidence intervals 95 5.4.2 Evaluation of confidence interval in spectral unmixing . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . 98

•

  Vector e =[e 1 ,...e k ] is the photon energies of the k peaks • Matrix of variables in the polynomial function noted as: Vector energy shift of peaks: ∆ e =[∆ 1 e ,...∆ k e ] • Weights matrix Σ =diag(w 1 ,...w k ) The coefficients can be calculated by solving the following problem: argmin c Σ 1 ke + ∆ e Vck 2

  Contents 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . 127 7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . 129 7.2.1 Spectral unmixing with temporal signatures . . . . 129 7.2.2 Making profit of the past processed data, learn to unmix . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.3 Open questions . . . . . . . . . . . . . . . . . . . . 140 7.4 Publications . . . . . . . . . . . . . . . . . . . . . . 142

  algorithms or Neumann Network -NN-[Gilton et al., 2019]) while allowing to learn a data-driven regularization from some training set. An unrolled version of the ADMM min a J (a)+Φa + b x log (Φa + b) , (7.5)

  a)+u x log (u)+v T (u Φa b)+ ⇢ 2 ku Φa bk 2 2

  1 -S = second, M = minute, D = day, Y = year, RN = radionuclide, Z=a t o m i cn u m be r . APPENDIX B Proximal algorithms for spectral unmixing B.1 Optimization problem underlying the spectral unmixing As mentioned in Chapter 3, the spectral unmixing aims to solve the minimization of an objective function of the form: â 2 argmin a f (a)+g 1 (a)+... + g n (a)( B . 1 )

f

  (a)=(Φa + b x) T W 1 (Φa + b x)( B . 7 ) the gradient is: rf (a)=Φ T W 1 (Φa + b x)( B . 8 ) with Lipschitz constant L = kΦ T W -1 Φk 2 .

  Recall the Poisson statistics-based activity estimator with non-negativity constraint presented in Chapter 3: âPoisson 2 argmin a Φa + b x log (Φa + b)+i. a 0 (B.10)

  argmin a f (Φa)+g (a)( B . 1 1 ) 2. The negative log-likelihood of the Poisson estimation: Poisson (x| )of the form: -ray spectrum formulation, we need to calculate the proximal operator of the function:f (Φa)=(Φa + b) x (Φa + b)( B . 1 9 )According to the basic operations of the proximal operator (see[START_REF] Parikh | Proximal algorithms[END_REF]), the proximal operator of Eq.(B.19) is:prox ⇢f (y)= y + b ⇢ + b stand for the measured spectrum and the background spectrum in Eq.(B.19).

6

  Analysis of HPGe gamma-ray spectra measurements 6.1 Challenges of calibrations in gamma-ray spectrum analysis . . 6.1.1 Calibrations in peak based analysis . . . . . . . . . . . 6.1.2 Towards calibrations for spectral unmixing . . . . . . . 6.2 Efficiency calibration and resolution calibration in simulation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Efficiency calibration . . . . . . . . . . . . . . . . . . . 6.2.2 Resolution calibration in spectral unmixing . . . . . . . 6.3 Complexity of the spectral signatures . . . . . . . . . . . . . . 6.4 Re-calibration of energy in spectral unmixing . . . . . . . . . 6.5 Validation with the standard source analysis . . . . . . . . . . 6.6 Experiments on aerosol measurements . . . . . . . . . . . . . 6.6.1 Uncertainty budget of the aerosol measurements . . . 6.6.2 Re-analysis of aerosol measurements . . . . . . . . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Making profit of the past processed data, learn to unmix133 7.3 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization problem underlying the spectral unmixing . . . . B.2 Proximal operators and proximal algorithms . . . . . . . . . . B.2.1 Definition of the proximal operator . . . . . . . . . . . B.2.2 Proximal algorithms used in this work . . . . . . . . . B.2.3 Proximal operators calculation . . . . . . . . . . . . . .

	viii	CONTENTS
	List of notations
	General notations
	x	vector
	x i
	A Nuclear data of radionuclides
	B Proximal algorithms for spectral unmixing
	B.1 C The MCNP simulation code

7 Conclusion and perspectives 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Spectral unmixing with temporal signatures . . . . . . 7.2.2

Table 2

 2

	.3.

  .11 displays the counting rate of these measured spectra as a function of the time after sampling.

	start time	ending time counting time(s)
	s1 19/04 09:10:53	0.5h	1h	1800
	s2 19/04 09:41:42	1h	1.5h	1800
	s3 19/04 10:12:30	1.5h	2h	1800
	s4 19/04 10:44:11	2h	3h	3600
	s5 19/04 11:45:03	3h	4h	3600
	s6 19/04 12:46:59	4h	5.5h	5400
	s7 19/04 14:20:23	5.5h	8.5h	10800
	s8 19/04 17:22:17	8.5h	23.5h	54000
	s9 20/04 08:41:41	23.5h	1day and 7h	28000
	s10 20/04 16:29:59 1day and 7h 4day and 2 h	240000
	s11 23/04 11:44:44 4day and 2 h 7day and 19h	320000
	Table 3.2 -Measurements of an aerosol filter sampled on 19/04/2018 8:46:00,
	Orsay, France. The start time and ending time of each measurement are
	considered from the sampling.

Table 4 .

 4 4 -False negative rate of 57 Co, 60 Co and 152 Eu (%) (Scenario III). different values for the average total number of counts of 152 Eu: 25, 100 and 200.

	Counts 152 Eu	25	50 100 150 200 250 300 400 500
	57 Co	OMP (WLS) P-OMP	0 0	0 0	0 0	0 0	0 0	0 0	0 0	0 0	0 0
	60 Co	OMP (WLS) 79.2 78.4 76.4 78.4 80.4 79.2 68 76.4 77.2 P-OMP 0 0 0 0 0 0 0 0 0
	152 Eu	OMP (WLS) 10 P-OMP 16.4 10.4 2.4 9.2 3.2	1.2 0	0.4 0	0 0	0 0	0 0	0 0

Table 4 .

 4 57 Co 60 Co 133 Ba 152 Eu 241 Am 88 Y 134 Cs 137 Cs 207 Bi 237 Np 5 -Comparison of different model complexity penalizations. Average total number of counts for 152 Eu of 25.

	Dev.	0	0	0	16.8	0	0	0	0.4	0	0.8
	AIC	0	6.8	0	19.2	0	0	6.0	5.2	3.2	5.6
	BIC	0	0	0	16.8	0	0	0.8	0.4	0	0.8

Table 4

 4 Poisson Orthogonal FNR of active radionuclides FPR of non-active radionuclides 57 Co 60 Co 133 Ba 152 Eu 241 Am 88 Y 134 Cs 137 Cs 207 Bi 237 Np Co 60 Co 133 Ba 152 Eu 241 Am 88 Y 134 Cs 137 Cs 207 Bi 237 Np

	Dev.	0	0	0	1.6	0	0	0	0	0	0
	AIC	0	6.8	0	2.8	0	0	4.0	8.8	7.2	7.2
	BIC	0	0.8	0	1.6	0	0	0.4	0	0.8	0.8
	Table 4.6 -Comparison of different model complexity penalizations. Average
	total number of counts for 152 Eu of 100.
			FNR of active radionuclides		FPR of non-active radionuclides
	57 Dev.	0	0	0	0	0	0	0	0	0	0
	AIC	0	9.6	0	0	0	0	6.4	5.2	6.8	7.2
	BIC	0	0	0	0	0	0	0	0.4	1.2	0.4

.7 -Comparison of different model complexity penalizations. Average total number of counts for 152 Eu of 200.

  , simulations are performed with 60 Co, 134 Cs, 137 Cs, 152 Eu with a background spectrum. As shown in Figure 5.4, 1000 simulations are performed for the mixture with respect to number of counts shown in Table 5.2. Radionuclide 60 Co 134 Cs 137 Cs 152 Eu background

	Number of counts 1500 500	1500 2500	4000
	Table 5.2 -Number of counts of NaI simulations.

  Conclusion . . . . . . . . . . . . . . . . . . . . . . 122

	. 107
	6.2.2 Resolution calibration in spectral unmixing . . . . 108
	6.3 Complexity of the spectral signatures . . . . . . 109
	6.4 Re-calibration of energy in spectral unmixing . 110
	6.5 Validation with the standard source analysis . . 116
	6.6 Experiments on aerosol measurements . . . . . . 118
	6.6.1 Uncertainty budget of the aerosol measurements . 118
	6.6.2 Re-analysis of aerosol measurements . . . . . . . . 120
	6.7

6.1 Challenges of calibrations in gamma-ray spectrum analysis . . . . . . . . . . . . . . . . . . . . 102 6.1.1 Calibrations in peak based analysis . . . . . . . . . 103 6.1.2 Towards calibrations for spectral unmixing . . . . 105 6.2 Efficiency calibration and resolution calibration in simulation process . . . . . . . . . . . . . . . . 106 6.2.1 Efficiency calibration . . . . . . . . . . . . . . . .

  Table 6.3 reports the activity estimation of 137 Cs for 6 of the 67 measurements, for which the 137 Cs is not detected with Genie 2000 but with the Poisson based spectral unmixing.

	Date 09/07/18 11/10/19 21/03/19 18/10/19 25/10/19 17/12/19
	Activity	0.136	0.043	0.045	0.058	0.040	0.054
	Uncertainty 0.063	0.025	0.016	0.022	0.019	0.021
	DT	0.040	0.016	0.009	0.012	0.011	0.011
	Table 6.3 -Activity (µBq/m 3 )o f 137 Cs analyzed with spectral unmixing for
	measurements when 137 Cs is not detected with Genie 2000. Uncertainties are
	assessed with k=2 (2 ), DT = decision threshold.

  .1).

		Half-life time		decay chain
	10.64 h ( 212 Pb) 26.916 min ( 214 Pb)	212 Pb ! 212 Bi ! 208 Tl 214 Pb ! 214 Bi
	Table 7.1 -Decay constant considered in the equilibrium state.
		start time from collecting counting time
	s7	5.5h		10800
	s8	8.5h		54000
	s9	23.5h		28000
	s10	1day and 7h	240000
	s11	4day and 2 h	320000
	Table 7.2 -Simulation of measurements for spectral unmixing with temporal
	signatures.

Table A .

 A

	RN	Element Z	Half-life	RN	Element Z	Half-life
	7 Be	Beryllium 4	53.22 D	134 Cs	Cesium 55	2.0644 Y
	22 Na	Sodium 11	2.6029 Y	137 Cs	Cesium 55	30.05 Y
	40 K Potassium 19 1.2504E9 Y	139 Ce	Cerium 58 137.641 D
	51 Cr Chromium 24	27.704 D	152 Eu	Europium 63	13.522 Y
	54 Mn Manganese 25	312.19 D	207 Bi	Bismuth 83	32.9 Y
	57 Co	Cobalt 27	271.81 D	208 Tl	Thallium 81	3.058 M
	60 Co	Cobalt 27	5.2711 Y	210 Pb	Lead 82	22.23 Y
	85 Sr Strontium 38	64.850 D	212 Bi	Bismuth 83	60.54 M
	88 Y	Yttrium 39	106.63 D	212 Pb	Lead 82	10.64 H
	106 Rh	Rhodium 45	30.1 S	214 Bi	Bismuth 83	19.8 M
	109 Cd	Cadmium 48	461.9 D	214 Pb	Lead 82	26.916 M
	110m Ag	Silver 47	249.78 D	228 Ac	Actinium 89	6.15 H
	113 Sn	Tin 50	115.09 D	237 Np Neptunium 93 2.144E6 Y
	129 I	Iodine 53	16.1E6 Y	234 Th	Thorium 90	24.10 D
	131 I	Iodine 53	8.0233 D 241 Am Americium 95	432.6 Y
	133 Ba	Barium 56	10.539 Y

https://www.mesure-radioactivite.fr

http://www.nucleide.org/Laraweb/index.php

https://www.mirion.com/products/genie-2000-gamma-analysis-software

https://www.iaea.org/resources/databases/evaluated-nuclear-structure-data-file

Remerciements

where Φ † is the pseudo-inverse of Φ.

The operator R Θ stands for a shrinkage operator that depends on parameters Θ that are updated during training procedure so as to minimize the mean estimation bias of P p kâ p a ? p k, from a training set of T samples {(x p , a ? p )} p=1,••• ,T . The processing of routine aerosol measurements generally leads to activities that do not vary to a large extent for natural radionuclides (see Figure 7.4). Therefore, the goal of learning R Θ is to capture this standard regime as well as the intricate correlation between the radionuclides' estimated activities.

• Gradient ascent on v : the dual variable is updated as: k+1) b . The parameter ⇢ is also trained along with the inversion procedure. As displayed in Figure 7.5, each recurrent block is composed of a main block where u and a are updated sequentially. The update of the dual variable v appears externally and is the only variable that explicitly cumulates information in the recurrence; this is reminiscent of skip connections. Several values for the number of recurrent layers L have been tested. In these experiments, L =5p r o v i d e st h eb e s tr e s u l t s ;m o r el a y e r sd i dn o tp r o v i d e significant improvements. algorithms involve the proximal operators of the terms in the objective function independently, such as the non-negativity regularization investigated in Chapter 3 and data-driven regularization terms discussed in Chapter 7.

B.2 Proximal operators and proximal algorithms B.2.1 Definition of the proximal operator

The proximal operator ( [START_REF] Parikh | Proximal algorithms[END_REF]) of a closed proper and lower semi-continuous convex function h : R n [1 is defined by:

where k.k 2 is the usual Euclidean norm. The function is strongly convex and not everywhere infinite, so it has a unique minimizer for y 2 R n .

The proximal operator of the scaled function ⇢h with the scale parameter ⇢ > 0i sm o r ec o m m o n l yu s e d ,w h i c hc a nbee x p r e s s e da s :

For more interpretations of the proximal operators, refer to [START_REF] Parikh | Proximal algorithms[END_REF].

B.2.2 Proximal algorithms used in this work

Forward-Backward splitting

The Forward-Backward splitting(FBS) algorithm ( [START_REF] Combettes | Signal recovery by proximal forward-backward splitting[END_REF]) tackles the minimization problem of the form:

with f differentiable and g admits a proximal algorithm. In the nonnegativity regularized least squares problem described in Chapter 3:

• the data fidelity term:

is differentiable, its gradient is:

Recall the convex conjugate of the function f :

The dual problem of the minimization of f (Φa)i s :

The primal-dual problem of Eq.(B.11) can be therefore written as:

In the framework of the Chambolle-Pock algorithm, each iteration alternates the minimization on the dual variable u and the primal variable a by calculating the proximal gradient descent, since we can easily get the proximal operators of the function f ⇤ and g. The main minimization step is described as follows:

The convergence of the algorithm is ensured with ⌧  1/L and ✓ =1.

B.2.3 Proximal operators calculation

The proximal operators of functions used in this work are given as follows:

1. The non-negativity indicator function defined by:

Its proximal operator is defined as the orthogonal projection onto the non-negative orthant:

APPENDIX C

The MCNP simulation code

Gamma-ray spectrum simulations can be performed with Monte Carlo simulation by using different software refer to [START_REF] Lépy | A benchmark for monte carlo simulation in gamma-ray spectrometry[END_REF]. In this thesis, we make use of the MCNP simulation process based on the random number generators by taking into account the interaction of particles with materials. The MCNPX and MCNP-CP codes configure the detection system information in the same manner with an input file, which allows us to specify:

• the three-dimensional geometrical cells according to the detection system.

• the material densities and the interaction of particles with materials.

The MCNP can be used for neutron, photon, electron, or coupled neutron/photon/electron transport. The specific configurations for the gammaray spectrum simulations are:

• Tally F8 enables to obtain pulse height per emitted particle in the source, which can provide the energy distribution of pulses created in ad e t e c t o r ,i.e., a spectrum.

• The E8 card specifies the energy bins of the spectrum.

• By default, the simulation code provides the detector response with photon energy peak as a Dirac. With GEB card option, the parameters of the empirical function:

can be set to reproduce the actual resolution of the experimental measurements (see Section 6.1 for details).