Raphaël Bajou

Marina Rosas-Carbajal

Jacques Marteau

A new versatile method for the reconstruction of scintillator-based muon telescope events

Keywords: Geophysics, Earth sub-surface imaging, Volcanic hazard, Hydrothermal System, Tomography, Muons, Three-dimensional modelization, Simulation muon radiography, track reconstruction, volcano imaging, RANSAC

Progress in three-dimensional muon tomography: towards a better characterization of the hydrothermal activity of La Soufrière de Guadeloupe volcano

Version Française

Titre: Progrès de la tomographie muonique en vue d'une meilleure caractérisation de l'activité hydrothermale du volcan de La Soufrière de Guadeloupe Résumé: Le développement de méthodes de télédétection pour visualiser la structure interne et la circulation des fluides à la surface de la Terre est une quête constante des géophysiciens pour affiner leur compréhension des processus dynamiques qui se produisent à l'intérieur d'édifices naturels tels que les volcans. Parmi ces méthodes, la tomographie muonique, ou muographie, a prouvé au cours des deux dernières décennies qu'elle permettait d'obtenir des informations uniques sur la structure interne de la densité des volcans.

La muographie est le point de rencontre de la physique des particules et des géosciences, car elle repose sur la détection de particules élémentaires appelées muons qui irradient abondamment la surface de la Terre, provenant des interactions entre rayons cosmiques de haute énergie et noyaux de l'atmosphère. Grâce à leurs propriétés physiques, les muons peuvent se propager sur des distances kilométriques dans un milieu rocheux, en étant affectés le long de leur trajectoire par les changements de densité. C'est pourquoi la muographie est souvent présentée comme un analogue à grande échelle de l'imagerie à rayons X.

Cette thèse se concentre sur l'application de la muographie au volcan actif de La Soufrière de Guadeloupe, dans les Petites Antilles françaises. L'une des principales composantes de l'évaluation du risque volcanique à La Soufrière est la stabilité de l'édifice. Celle-ci est affectée par la circulation de fluides hydrothermaux chauds et acides à l'intérieur du dôme de lave du volcan, ce qui aggrave la probabilité d'un effondrement partiel des flancs. Pour étudier la stabilité mécanique des volcans, la structure de la densité de ces édifices est d'une importance capitale et est traditionnellement déduite par des études gravimétriques, qui estiment les variations de densité par des mesures de l'intensité du champ de gravité à la surface de l'édifice. Cependant, outre la difficulté d'obtenir une bonne couverture de prises de données à la surface de l'édifice, l'inversion des données gravimétriques est hautement non unique et mal posée et nécessite des contraintes préalables sur la structure interne.

Le travail a d'abord consisté à développer de nouvelles méthodes de traitement et d'analyse des données pour la campagne de prise de mesures muoniques en cours depuis 2015 sur les flancs du dôme de lave de La Soufrière. Ce nouveau traitement s'est accompagné du développement d'outils de simulation Monte-Carlo pour mieux comprendre les performances du détecteur de muons, et évaluer la capacité de rejet du bruit de fond et la contamination du flux transmis par les muons sortant du volcan par des sources de bruit de fond irréductibles. Suite au développement de ces outils, l'analyse de quatre ensembles de données du télescope a été réalisée. Cette étude fournit une excellente couverture spatiale du dôme de lave. Une attention particulière a été portée à l'estimation et à la propagation des incertitudes statistiques et systématiques afin de mieux contraindre les estimations de ii densité déduites des mesures de flux. Une inversion linéaire bayésienne des données du télescope à quatre muons a été réalisée sur cette base.

Le nouveau modèle de densité met en évidence une anomalie de faible densité et de volume significatif dans la région la plus active du volcan, où celui-ci présente les manifestations les plus soutenues d'activité hydrothermale depuis le début de sa période de réveil en 1992. Cette observation est cohérente avec la connaissance actuelle de la circulation des fluides hydrothermaux, obtenue grâce à d'autres études géophysiques telles que la tomographie électrique. Des modèles de densité comme celui présenté dans ce travail, ainsi que des mesures sur le terrain pour évaluer précisément les propriétés mécaniques de la roche, sont les principales données d'entrée pour poursuivre la modélisation numérique de la stabilité de l'édifice et affiner l'évaluation du risque d'effondrement de flanc.

Mot-clés: Géophysique, Imagerie de la surface de la Terre, Risque volcanique, Système hydrothermal, Tomographie, Muons, Modélisation tri-dimensionnelle, Simulation iii

Résumé substanciel

Le développement de méthodes de télédétection pour visualiser la structure interne et la circulation des fluides à la surface de la Terre est une quête constante des géophysiciens pour affiner leur compréhension des processus dynamiques qui se produisent à l'intérieur d'édifices naturels tels que les volcans. Parmi ces méthodes, la tomographie muonique, ou muographie, dont traite l'essentiel de cette thèse, a prouvé au cours des deux dernières décennies qu'elle permettait d'obtenir des informations uniques sur la structure interne de la densité des volcans.

La Soufrière de Guadeloupe, située sur l'île de Basse-Terre dans les Petites Antilles françaises, est un archétype de volcan hautement instrumenté, doté d'un vaste réseau de stations mise en place afin de surveiller son activité (stations sismiques, GNSS, extensomètres, mesures de température). Sa dernière éruption, de nature phréatique, a lieu en 1976. Avec son activité hydrothermale intense, et en raison de sa proximité avec une zone densément peuplée, il est l'un des volcans les plus dangereux au monde Non seulement impliquée dans des éruptions phréatiques, cette dynamique des fluides hydrothermaux à l'intérieur des édifices volcaniques favorise les processus physico-chimiques tels que l'altération des roches et exerce un impact significatif sur leur stabilité, en affaiblissant la résistance mécanique des roches et en influençant la probabilité d'effondrement des flancs, l'une des principales composantes de l'évaluation du risque volcanique.

Pour répondre au besoin d'une meilleure caractérisation de la stabilité de l'édifice volcanique, les estimations de densité figurent parmi les observables les plus importantes pour développer des simulations numériques d'effondrement de flan dans les zones les plus affectées par l'activité hydrothermale. Ces mesures de densité sont traditionnellement effectuées en mesurant les anomalies du champ de gravité, causées par des différences dans la distribution de la masse à l'intérieur de l'édifice. Mais ces méthodes ont leurs limites, notamment en termes de couverture de données et de résolution spatiale.

La muographie est le point de rencontre de la physique des particules et des géosciences, car elle repose sur des particules élémentaires appelées muons, produites dans les interactions des rayons cosmiques de haute énergie dans l'atmosphère. L'histoire de leur première identification va de pair avec la compréhension du rayonnement cosmique à la surface de la Terre au début du XXe siècle et la naissance de la physique des particules. Leur abondance à la surface de la Terre et leurs propriétés physiques en ont fait une source unique pour l'imagerie d'objets denses et aussi volumineux que les volcans.

Outre cette application, les propriétés des muons en font d'excellents outils de calibration pour les expériences de physique des particules, afin d'évaluer les performances des détecteurs. Nous rapportons dans le chapitre 2 de cette thèse la première année de travail, réalisée au laboratoire APC, à Paris, sur l'analyse de données de muons dans le cadre de la mise en service d'un prototype de détecteur de neutrinos pour l'expérience DUNE.

Capable de sonder un milieu rocheux sur des distances de l'ordre du kilomètre, les muons ont permis l'essor de cette technique d'imagerie qu'est la muographie. A l'instar de l'imagerie médicale par rayons X, celle-ci consiste à mesurer le flux de particules sortant de l'objet imagée, ou flux transmis. De la même façon qu'un flux artificiel de rayons X, la propagation des muons est également directement influencée par les variations de densité du milieu traversé. Les applications se sont par la suite multipliées en archéologie et en géosciences, et commencent aujourd'hui à se développer en dehors du monde de la recherche, iv dans les domaines de la sûreté nucléaire et du génie civil.

Ces trois dernières décennies ont vu naître différentes études menées sur des volcans actifs d'abord au Japon, puis en Italie et en France. Celles-ci ont mis en évidence le potentiel élevé de cette méthode d'imagerie et sa complémentarité avec d'autres observables géophysiques, telle que la gravimétrie. Quelques-unes de ces campagnes sont présentées dans le chapitre 3.

L'étude muonique de La Soufrière de Guadeloupe est en cours depuis les premiers travaux et études de faisabilité commencés en 2008. Cette campagne représente la première application de la muographie à un volcan actif en France. Cette thèse s'inscrit ainsi dans la continuité des travaux doctoraux de Nolwenn Lesparre (2009[START_REF] Lesparre | Design and operation of a field telescope for cosmic ray geophysical tomography[END_REF] et Kevin Jourde (2012[START_REF] Jourde | Un nouvel outil pour mieux comprendre les systemes volcaniques : la tomographie par muons, application à la Soufrière de Guadeloupe[END_REF]. Ces études substantielles ont notamment posé les bases de la conception des matrices de scintillateurs et de la structure des détecteurs, opérants aujourd'hui sur les flancs de La Soufrière, appelés télescopes à muons. La conception d'un télescope de terrain pouvant être transporté sur le terrain, assemblé, et opérer à distance pour des périodes d'acquisition de données, de l'ordre minimal du mois afin de s'affranchir de fluctuations statistiques, dans l'environnement tropical de La Soufrière représentait en effet un véritable défi, relevé avec succès grâce à ces études.

Les télescopes de terrain visent à détecter le très faible flux transmis de muons sortant du volcan. Sans obstacle, près de l'horizon, le flux est en effet plus de deux ordres de grandeur inférieur au flux vertical. Le flux diminue à nouveau de deux ordres de grandeur après avoir traversé 1 km de roche. En plus de ces conditions de terrain difficiles, de nombreuses sources de bruit de fond provenant de l'atmosphère viennent polluer le signal de flux de muons transmis et conduisent finalement à la sous-estimation de la densité. A ce bruit de fond physique s'ajoute le bruit aléatoire généré dans le détecteur, lié par exemple au système de photo-détection et à l'émission thermoïonique de photo-électrons, ou au bruit corrélé entre les différents canaux d'amplification et de lecture électroniques.

Le bruit de fond, le volume conséquent des jeux de données actuels (plus de 10 8 événements enregistrés au cours de différentes campagnes entre 2015 et aujourd'hui), ainsi que l'ajout d'une quatrième matrice de détection sur le télescope situé au Sud-Ouest du dôme ont motivé la révision du cadre de traitement des données. Cette nouvelle configuration de télescope à quatre matrices permet l'amélioration de la rejection du bruit de fond atmosphérique et la résolution angulaire sur la partie la plus active du dôme. Ainsi une nouvelle méthode de reconstruction de trajectoires de muons basée sur un algorithme de RANSAC (random sampling consensus) a été implémentée. Cette dernière est conçue pour éliminer l'influence des points aberrants dans les événements enregistrés et l'analyse des trajectoires reconstruites. Les performances de cet algorithme ont été comparés à une autre méthode de reconstruction, appelée transformée de Hough, plus communément employée en physique des hautes énergies, pour la reconnaissance de traces de particules.

Ces travaux sont allés de pair avec le développement d'outils de simulation Monte-Carlo pour évaluer et comparer les performances des deux algorithmes de reconstruction, comme décrit au chapitre 5. Les deux algorithmes ont notamment été testés sur un échantillon de 10 6 muons simulés dans le logiciel Geant4 et interagissant avec une reproduction numérique du détecteur à quatre plans de scintillateurs. Les performances des algorithmes ont été étudié en fonction d'une pollution croissante du jeu de données de muons, simulant un bruit de fond aléatoire de nature instrumentale, artificiellement ajouté. Les résultats montrent un avantage pour RANSAC pour des proportions réalistes de bruit de fond (20v 30% des points) en termes d'efficacité, de résolution angulaire, et de temps de calcul. Ces développements sont essentiels pour avoir une compréhension solide de la réponse du détecteur, notamment de sa capacité de rejection du bruit de fond, et préparent le terrain pour d'autres études basées sur des simulations, notamment une meilleure caractérisation de la principale source de bruit de fond liée à la diffusion de particules à basse énergie (< 1 GeV) sur les flancs du volcan, imitant le signal fourni par les muons de haute-énergie sortant du dôme.

Le chapitre 6 présente les résultats de l'analyse des données muoniques de quatre télescopes, qui ont enregistré des données de tomographie entre septembre 2016 et août 2019 sur quatre sites d'observation différents, offrant une couverture spatiale sans précédent du volume du dôme de lave de La Soufrière de Guadeloupe. Pour chaque télescope, l'estimation de l'acceptance expérimentale sur la base d'une acquisition de calibration conduite à ciel ouvert précède et corrige l'estimation des flux transmis des effets instrumentaux, telles que des différences d'efficacité de détection entre barreaux scintillateurs. Le flux estimé pour chaque ligne de visée est ensuite comparé à un modèle de flux transmis préalablement calculé en fonction de la densité intégrée de roche, appelée opacité. Cette relation permet ainsi l'interpolation de l'opacité. La connaissance des distances de parcours des muons dans le dôme permet l'estimation d'une image en deux dimensions de la densité moyenne depuis chacun des quatre sites d'observation autour du dôme.

Dans le but de mieux contraindre les estimations de densité déduites des mesures de flux, une attention détaillée a été apportée à l'estimation et à la propagation des incertitudes statistiques et systématiques. Ces incertitudes systématiques peuvent être simplement expérimentales, liées par exemple à l'orientation du télescope, ou physique liée au modèle de spectre en énergie des muons, utilisé pour déduire le flux de muons transmis, et à la méconnaissance du milieu rocheux traversé par les muons, notamment concernant son contenu en eau.

Elles constituent des informations préalables essentielles pour la modélisation tridimensionnelle de la densité du dôme de lave. Cette modélisation s'est faite sur la base d'un modèle numérique de terrain du dôme de lave obtenu par télédétection laser (LiDAR). Le volume couvert par ce modèle a ensuite été divisé en voxels de 16 m de côté, tout en adaptant leur géométrie en surface afin de restituer la topographie du dôme. Grâce à cette discrétisation du volume du dôme, une inversion linéaire bayésienne des données des quatre télescopes à muons a pu être réalisée. Avant d'être appliquée aux vraies estimations de densité moyenne des télescopes, les paramètres de cette inversion ont préalablement été estimés sur des jeux de données synthétiques générés dans les quatre détecteurs.

Le nouveau modèle de densité obtenu met en évidence une anomalie contrastée de faible densité et de volume significatif, de l'ordre de 7% du volume total du dôme, dans la région la plus active, où le volcan présente les manifestations les plus soutenues d'activité hydrothermale, notamment depuis le début de sa période de regain d'activité observée depuis 1992. Cette observation est cohérente avec les connaissances actuelles sur la circulation et la distribution des fluides hydrothermaux dans cette région peu profonde du volcan, obtenue grâce à d'autres études géophysiques comme la tomographie électrique, menée sur le dôme de lave. Des modèles de densité comme celui présenté dans ce travail, ainsi que des mesures sur le terrain pour évaluer précisément les propriétés mécaniques de la roche, sont les principales données d'entrée pour poursuivre la modélisation de la stabilité de l'édifice et affiner l'évaluation du risque d'effondrement des flancs. vi • À la direction de l'école doctorale STEP'UP pour avoir permis ma transition de sujet de thèse, et financé deux mois supplémentaires de prolongation de mon contrat. Sur ce point, merci également à Michel Diament d'avoir permis une seconde prolongation de deux mois en débloquant des fonds de l'ANR Megamu,

• Aux membres de l'ANR MYGALE pour cette exaltante mission de terrain sur La Soufrière de Guadeloupe au mois de Mai 2022.

• À Jacques Marteau pour m'avoir partagé les données des robustes et performants télescopes à muons, fruits de longues années de R&D et d'acclimatation aux conditions imposées par cette chère Soufrière. Merci d'avoir pris le temps de répondre à mes questions. Merci également à Amélie Cohu et Matías Tramontini, également doctorants amateurs de muons pour les multiples échanges d'information sur le sujet.

• À mes encadrantes Alessandra Tonazzo et Marina Rosas-Carbajal pour leurs conseils avisés, suivi régulier de mon travail, commentaires et retours dans la rédaction d'article et de mon manuscrit, ainsi que pour leurs encouragements. Plus particulièrement, merci à Alessandra et Jaime d'avoir entendu et accepté mon souhait de réorientation, en autorisant cette transition de la physique des particules vers la géophysique (des particules). Un grand merci à Marina pour sa confiance, pour avoir eu la souplesse d'esprit et l'audace d'accepter de travailler avec moi après notre rencontre en Septembre 2020, tout en participant à ma formation en géophysique.

• À mes collègues de l'équipe Systèmes Volcaniques à l'IPGP (et extension), pour leur accueil et pour tous les chaleureux moments partagés qui ont installé et alimenté un précieux climat d'échange et d'humour au quotidien.

• Au fabuleux groupe de doctorants/post-docs, principalement résidents du 4ème étage de l'IPGP, que j'ai eu l'immense plaisir de côtoyer pendant ma thèse et que je peux désormais compter parmi mes amis. Merci pour ces belles années.

• À mes proches, amis d'enfance ou d'études, et à ma famille pour leur soutien indéfectible.

vii viii Contents [START_REF] Neddermeyer | Note on the Nature of Cosmic-Ray Particles[END_REF] with its original caption that describes the different trajectories of the incident and produced particles including muon tracks along other products of an electromagnetic shower that are ionizing the detection gaseous mixture. .

2.13

Charge deposit along tracks dQ/dx as a function of electrons drift time, plotted for three cosmic runs taken in October, November and January. . . . .

2.14 (a) Schematic diagram of a purity monitor. The photocathode receives photons from a xenon source light and emits electrons by photoelectric effect. These "photo-electrons" Q c are driven by an electric field towards the anode, where a charge Q a is collected. The measurement of the ratio Qc Qa is proportional to the charge loss factor exp ( -t τe ) where τ e is the electrons lifetime. (b) Photo of the short purity monitor in the bottom of the cryostat.

2.15 Measurements of the two short purity monitors (bottom and middle TPC), superimposed to cosmic tracks analysis results (red markers) shown in Fig. 2.13. This comparison shows consistent results between the two short purity monitors and the cosmic analysis outputs. . . . . . . . . . . . . . . . . 3.9 Sketch of a plastic scintillator bar, featuring its embedded WLS fiber, shifting the collected "blue" scintillation photons to "green" photons conducted towards the PMT. Taken from [START_REF] Lesparre | Mise au point d'une methode de tomographie utilisant les muons d'origine cosmique. Applications au laboratoire souterrain du Mont Terri et au volcan de la Soufrière de Guadeloupe[END_REF]

Introduction

The development of remote sensing methods to visualize the internal structure and the circulation of fluids inside the Earth's surface is a constant quest among geoscientists to refine their understanding of the dynamic processes occurring inside natural edifices like volcanoes. Among these methods, muon tomography, or muography, has proven to give unique insights into the inner density structure of volcanoes. and shallow fluid dynamics, over the last three decades. As each volcano is a unique case of study, these methods are associated with numerous sources of information among field investigations, laboratory experiments, geochemical and geophysical observables, and numerical modeling. La Soufrière de Guadeloupe, in the French Lesser Antilles, is an archetype of this type of highly-instrumented volcano, featuring a wide network of stations to monitor its activity (seismic, GNSS stations, extensometers, temperature measurements). With its intense hydrothermal activity, it is one of the most hazardous volcanoes in the world due to its proximity to a densely populated area. This hydrothermal activity favors physicochemical processes such as rock alteration and has a significant impact on volcano stability, weakening the rock's mechanical strength, and influencing the proneness of flank collapse, one of the main components of the volcanic risk evaluation. To answer the need for a better characterization of the volcano edifice stability, density measurements are of the most importance. These measurements are traditionally carried out by measurements of the gravity field anomalies, caused by differences in mass distribution inside the edifice. But these methods have their limits notably in terms of achievable data coverage and spatial resolution.

Muography is the meeting point of particle physics and geosciences, as it relies on elementary particles called muons, produced in the interactions of high-energy cosmic-ray in the atmosphere. The history of their first identification goes along with the comprehension of cosmic radiation at Earth's surface in the early XX th century and the birth of particle physics. As we will see in this study, their abundance at Earth's surface and physical properties have turned them into a unique source for imaging dense and very large objects like volcanoes, introduced in chapter 1. Besides this application, the muons' properties make them excellent calibration tools for particle physics experiments, to assess detector performances. We report in chapter 2 of this thesis the first year of work, carried out at the APC laboratory, on muon data analysis in the context of a neutrino detector prototype commissioning for the DUNE experiment.

Being able to probe up to kilometric distances of dense rock medium, muography is often presented as a large-scale analog to X-ray imaging, as muon propagation is also directly influenced by the traversed medium density variations. Applications have grown in archaeology, and geosciences, and nowadays are starting to flourish outside of the research world in nuclear safety and civil engineering. The studies reviewed in chapter 3 throughout campaigns carried out first on active volcanoes in Japan, Italy, and France will emphasize the high potential of this imaging method and its complementarity with other geophysical observables. After a more in-depth introduction to the geological and geophysical context at La Soufrière de Guadeloupe in chapter 4. We will present the muon survey, detection technology, and methodology employed to obtain mean density. This survey is ongoing since the early work and feasibility studies started in 2008 [131]. This campaign represents the first muography xxiii application to an active volcano in France.

This thesis takes place in the continuation of the work of Nolwenn Lesparre (2009-2012) [START_REF] Lesparre | Mise au point d'une methode de tomographie utilisant les muons d'origine cosmique. Applications au laboratoire souterrain du Mont Terri et au volcan de la Soufrière de Guadeloupe[END_REF] and Kevin Jourde (2012[START_REF] Jourde | Un nouvel outil pour mieux comprendre les systemes volcaniques : la tomographie par muons, application à la Soufrière de Guadeloupe[END_REF] [START_REF] Jourde | Un nouvel outil pour mieux comprendre les systemes volcaniques : la tomographie par muons, application à la Soufrière de Guadeloupe[END_REF]. These substantial studies have notably set the ground for the design of the scintillator-based detectors, operating today on the volcano flanks, referred to as muon telescopes. The conception of a field telescope being able to be carried on the field, assembled, and operate remotely for month-long periods of data acquisition in the La Soufrière tropical environment was a real challenge that has been successfully tackled thanks to these studies. The field telescopes aim at detecting the very low transmitted flux of muon outgoing from the volcano. Without any obstacle, close to the horizon, the flux is indeed more than two orders of magnitude smaller than the vertical flux. The flux decreases again by two orders of magnitude after traversing 1 km of rock. In addition to these harsh terrain conditions, numerous sources of background coming from the atmosphere come to pollute the transmitted muon flux signal and lead to the underestimation of density. This pollution along with the consequent volume of today's data sets (more than 10 8 recorded events through different campaigns between 2015 and today) and the upgrades of muon telescopes have motivated the revision of the data processing framework, including new tracking methods such as RANSAC, designed to discard the influence of outlier points in recorded events and analysis of the reconstructed trajectories. This work went along with the development of Monte-Carlo simulation tools to assess and compare different tracking algorithms' performances as described in chapter 5. These developments are crucial to have a solid comprehension of the detector response, notably its background rejection ability, and are setting the ground for further simulation-driven studies, notably a better characterization of the main source of background linked to low-energy particle scattering. Chapter 6 presents the results of the muon data analysis of four telescopes, which recorded tomography data between September 2016 and August 2019 at four different observation sites, offering unprecedented spatial covering of the lava dome volume of La Soufrière de Guadeloupe. With the aim of better constraining the density estimates inferred from flux measurements, detailed attention has been brought to the estimation and propagation of statistical and systematic uncertainties. These uncertainties are linked to the transmitted flux model, used to infer and to the unknown medium composition crossed by muons. They constitute critical prior information for performing three-dimensional density modeling of the lava dome. On this basis, a Bayesian linear inversion of the four telescopes' data has been conducted. The obtained new density model highlights a contrasting lowdensity anomaly of significant volume in the most active region of the dome, where the volcano is showing the most sustained manifestations of hydrothermal activity since the beginning of its reawakening period in 1992. This observation is consistent with the actual knowledge of the circulation of hydrothermal fluid in this shallow region of the volcano, obtained through other geophysical surveys like electrical tomography, conducted on the lava dome. Density models like the one presented in this work along with field measurements to precisely assess the rock's mechanical properties are the main inputs for pursuing further modelization of the edifice stability and refining the assessment of flank collapse hazard.

Introduction

The muon discovery takes part in the rich and sinuous historical path that saw the birth of particle physics, along the first observation of extraterrestrial radiations, the cosmic rays, and their identification as the source of the atmospheric ionization. The latter was a main breakthrough of the early XX th century and opened a whole new world of questions for physicists about the nature of those cosmic rays and their abundance on Earth, in which muons have a key role. Twenty years after their identification, muons started crossing the boundary of fundamental research and becoming more of a use, first in underground physics experiments, for rock overburden characterization. The journey toward muography had begun. In this chapter, alongside a historical review to emphasize the main steps that lead to today's muon imaging application in geosciences and modern volcanology, we will present the physical processes that govern the muon propagation in matter and lead to Earth subsurface muography, as well as the description of the atmospheric flux detected on Earth, in which are produced muons.

A first misty footprint

In 1936, Carl Anderson and Seth Neddermeyer, two physicists at CalTech, identified a new particle in their cloud chamber whose trajectory curvature in the applied magnetic field laid between an electron and a proton (fig. 1.1). At the time, only protons, neutrons, electrons, and photons were known. This article was preliminary baptized by Anderson mesotron, using the Greek prefix "meso-", for "mid-" in English [START_REF] Neddermeyer | Note on the Nature of Cosmic-Ray Particles[END_REF]. The existence of such a particle was confirmed in 1937 by J. C. Street and E. C. Stevenson in another cloud chamber experiment [START_REF] Street | New Evidence for the Existence of a Particle of Mass Intermediate Between the Proton and Electron[END_REF]. While the parentage of the first muon discovery is debatable, this first official identification did go to the first two CalTech physicists. Nevertheless, three years before Anderson and Neddermeyer's measurement, the first muon track appeared in the detector of Paul Kunze [START_REF] Kunze | Untersuchung der Ultrastrahlung in der Wilsonkammer[END_REF] (cf. Fig. 1.5) as a "particle of uncertain nature" during his study of the composition of cosmic rays, but this misfit track was then classified as a nuclear reaction product. In 1940, E.J. Williams et G.E. Roberts pursuing their research on the mesotron, assumed by Hideki Yukawa in his theory of nuclear forces and β-disintegration, observed the first direct muon decay into an electron [START_REF] Williams | Evidence for Transformation of Mesotrons into Electrons[END_REF]. Later on, the identified particle was found much more closer to the electron behavior, and insensitive to the strong force, thus very different from the other known mesons (composite particles, formed by an even number of quarks and anti-quarks). The term meson was thus abandoned in favor of the name that is used today: "muon" µ. The first real mesons, pions π and kaons K, were identified a decade after in 1947.

Origin of the discovery

The first muon observation in a cloud chamber, and more generally the premises of particle physics, cannot be separated from the history of cosmic ray detection from which muons emanate. By the time of the first muon footprints in cloud chambers, the atmosphere ion- [START_REF] Neddermeyer | Note on the Nature of Cosmic-Ray Particles[END_REF] with its original caption that describes the different trajectories of the incident and produced particles including muon tracks along other products of an electromagnetic shower that are ionizing the detection gaseous mixture. ization was already understood to be of extraterrestrial origin, and caused by a penetrating flux of cosmic rays, whose atmospheric interaction products were raining down on Earth. This comprehension of the existence of proper radiation that does not originate from the Earth's natural radioactivity, discovered by Henri Becquerel in 1896, was the product of numerous observations that can at least be traced back to the XVIII th century, with the first recorded observation by Abbé Nollet of the spontaneous discharge of his golden-foil electroscope. In the very beginning of the XX th century, motivated by the visionary insight that spontaneous discharge might be caused by an external source other than groundoriginating radiation, C. Wilson [START_REF] Thomson | On the ionisation of atmospheric air[END_REF] proposed to completely shield the electroscope to protect it from any radioactive material. In 1909, Theodor Wulf equipped with an upgraded electroscope (using quartz wires) installed his apparatus at the top of the Eiffel tower (330m high). As expected if the radiation would come from the ground, at the top of the tower the ionization rate inside the electroscope vessel would be quasi-null. Although he did measure an important decrease from 6 ions.cm -3 at ground level to 3.5 ions.cm -3 at the summit, the rate was still too high to discard the existence of an external source of ionization coming from the atmosphere. In 1910-1911, Domenico Pacini (shown above in Fig. 1.2) compared measurements of the discharge of electroscope on the ground and on the sea in the gulf of Genova to remove the effect of the continental crust radioactivity. He also designed an experimental apparatus to perform underwater measurements, at 3 m depth, with which he observed a significant decrease in the ionization rate (20% lower than at the surface). He finally concluded in his article [START_REF] Pacini | La Radiazione Penetrante[END_REF] "that a significant proportion of the pervasive radiation that is found in the air had an origin that was independent of the direct action of the active substances in the upper layers of the Earth's surface. ", interpreting this decrease by the "absorbing power of water [...] of radiation coming from the outside". A detailed account of this significant contribution to the history of cosmic ray detection can be found in [START_REF] Carlson | Nationalism and internationalism in science: the case of the discovery of cosmic rays[END_REF]. At the same time, the Austrian physicist Victor Hess demonstrated that the quantity of the atmospheric ions was clearly increasing with altitude, performing ten balloon flights including five by night, reaching altitudes of 5200 m. No difference between day and night data was found, which lead Hess to conclude on the non-solar origin of this radiation. Two years later, Hess's results were corroborated by W. Kolhörster that led another high-altitude balloon survey up to an altitude of 9000 m.

Following those major observations, the nature of the cosmic rays was becoming a hot topic (cf. Fig. 1.3a), disputed between partisans of a photonic nature of the radiation due to their highly penetrating property, and partisans of their charged nature, hypothesis defended by A. Compton. In 1927, the Dutch physicist J. Clay accidentally discovered the dependence of cosmic radiation on geomagnetic effect, measuring the cosmic rays intensity (analog to a flux in m -2 s -1 sr -1 ) with an ionization chamber during a boat trip to the island of Java. As he was crossing the equator, he measured a decreased in the intensity of 10%. Following this intriguing observation, many surveys were organized, among which the ones planned by A. Compton (cf. Fig. 1.3b), in the early 1930s, to study the dependence on the latitude of cosmic-ray intensity. After numerous observations, it became clear that cosmic rays were sensitive to variations in the Earth's magnetic field through latitude, socalled the geomagnetic effect, which lead to conclude on their charged nature.

The invention of the Geiger-Müller counter in 1928 [START_REF] Geiger | Elektronenzählrohr zur Messung schwächster Aktivitäten[END_REF] allowed the detection of single particles and lead Pierre Auger and his collaborators ten years later to perform coincidence measurements at several altitudes of particles of cosmic origin. By performing these simultaneous detections in distant counters they witnessed the avalanche of secondary particles produced by a very high-energy primary particle entering the atmosphere, which later re- 1913,1914). Taken from [START_REF] Carlson | Nationalism and internationalism in science: the case of the discovery of cosmic rays[END_REF].

ceived the name of air showers. Nowadays, Auger has given his name to an observatory [20] in the western side of the Argentinian Patagonia region that dedicated a 3000 km 2 surface dotted with 1660 detectors (water Cherenkov tanks) for the study of ultra-high energy cosmic rays whose origins still remain undetermined. Those rare events (∼1 km 2 /year) involve kinetic energy that goes above 10 18 eV and a produced atmospheric shower that can contain up to the order of 10 billion particles, spreading on Earth over a typical 20 km 2 surface.

In the second half of the XX th century, the distinction between the worlds of cosmic- Even if the highest measured energies remain in the domain of cosmic ray observation ((3.2 ± 0.9) × 10 20 eV for the most energetic cosmic ray ever measured [START_REF] Bird | Detection of a Cosmic Ray with Measured Energy Well beyond the Expected Spectral Cutoff due to Cosmic Microwave Radiation[END_REF]), modern kilometer-scale accelerators like the worldwide famous Large Hadron Collider (LHC) at CERN, Switzerland, running seven orders of magnitude lower in energy (1.3.10 13 eV for LHC maximal collision energy) remain the most relevant tools to pursue the development of particle physics and tackle unsolved questions, allowing to reach energies where deviations to the successful Standard Model are expected to occur.

Muons in the current research world 1.2.1 A door to new physics?

Muons have played an important role in the development of the Standard Model of particle physics (SM) throughout the XX th century, that constitutes nowadays the solid mathematical framework used by particle physicists to describe all the known particles and their interactions. Recently, they may have opened the door to new physics, delivering a long-term awaited experimental result: an "anomalous" magnetic moment, deviating from the SM prediction. The latter quantifies the particle's behavior in an electromagnetic field, which in the frame of the SM is exactly equal to two for a lepton. However, the real value is expected to be slightly shifted away from the latter theoretical prediction due to interactions with virtual particles in a vacuum. The neutrino or a potential dark matter particle is identified thanks to the so-called daughter particles produced when an interaction occurs in the detection medium. To correctly tag this primary particle of interest and measure its kinetic properties, experiments rely on the finest reconstruction and identification of those daughter particles. Hence to estimate these detection abilities, muons have a leading role to play. As they represent a natural well-characterized source of charged particles, the signal they leave in the detector can serve as a benchmark to assess in real data-taking conditions the detector and dedicated algorithms' performances in reconstructing incident ionizing particle tracks. In chapter 2, we present the first contribution of this thesis, in muon data analysis carried out on a prototype of neutrino detector built for the international collaboration DUNE (Deep underground neutrino experiment) [7], [6]: the ProtoDUNE-Dual Phase time projection chamber. We thoroughly describe the scientific context of this experiment along with the employed detection technology in the concerned chapter.

Towards muography

As we will see in detail throughout the next chapters, nowadays like gamma X-rays for medical imaging, muons have found their own end-use in the field of imaging. As muons are sensitive to the amount of matter encountered along their path, they have allowed to map out the density distribution of very large systems such as pyramids, or volcanoes. Thus, muon-based imaging, often shorten as muography, has given birth to a wide and growing spectrum of applications notably: archaeology, nuclear safety, civil engineering, and more particularly in geosciences, and applied volcanology, in which this thesis takes part.

For the latter domain, if the muon imaging method still holds a niche status in the current volcanic activity monitoring toolkit, it is more and more recognized thanks to the pivotal information it can bring to volcano mechanical structure characterization.

In chapter 3, we will highlight the main achievements through past and current studies led in Japan, Italy, and France, where muon studies have shown their complementarity with other geological and geophysical observables, and discuss its current status and potential in applied volcanology. This review will be further completed and illustrated throughout this work carried out on the La Soufrière de Guadeloupe volcano lava dome. Through the results of muographic data analysis, we will show how muon-based observations can bring some important and reliable information on the density distribution of a complex natural system, Chapter 1

and how it could be incorporated into a larger volcanic hazard surveillance framework.

Firstly, we shall introduce through the next sections the physical properties that make muons great candidates to probe dense matter.

Muon physics: propagation in matter

Muons are unstable elementary charged particles with a mean lifetime of 2.2 µs, of spin 1/2, that belong in the Standard Model of particle physics to the lepton family. The latter includes all elementary particles sensitive to electromagnetic, and weak interactions but not subjected to strong force. The muon is often presented as the heavier cousin of the electron with a rest mass m µ ≃ 105.7 MeV, 207 times heavier than the mass of the electron m e ≃ 0.511 MeV, and 17 times lighter than the third cousin, the tau lepton τ of mass m τ ≃ 1777.8 MeV.

In a dense medium, several interaction processes contribute to slowing down the incident penetrating muon. These processes depend on the kinetic energy T , but most interestingly for muographers: on the medium density ρ (almost directly proportional), and to a lesser extent, they also depend on its chemical composition expressed in terms of Z/A ratio where Z is the atomic number and A the atomic mass. Through this section, we frequently refer to the exhaustive review made by Groom et al. [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF]. This paper along with the Particle Data Group [START_REF] Workman | Review of Particle Physics[END_REF] and their reference atomic and nuclear properties database 1 , provide all requested information for computing energy deposits for muons in many types of medium, and mixtures of mediums. In the following, we will give a brief summary of the main points about muon interaction processes and their associated energy deposit contributions in a dense medium. The latter are commonly gathered in two categories:

• ionization process, considered as a continuous succession of Coulomb interactions with atomic electrons, dominant below ∼ 500 GeV for muons. At high energies, the fact that the ionization contribution for muons is fairly constant and low, of the order of 2.0 MeVcm2 /g (cf. Fig. 1.9). This makes muons extremely penetrating particles that can cross the whole atmosphere from their emission in the high-atmosphere (cf. Sec. 1.4.1) to the ground, and still propagate through a km-thick rock edifice.

• radiative processes, of stochastic nature, which includes Bremsstrahlung, e -/e + pair production, and photonuclear interactions. Those processes start taking over after a certain energy value, called critical energy, referenced as E µc (in standard rock E SR µc ≃ 693 GeV ).

-Bremsstrahlung 2 : Electromagnetic interaction between incident charged particle and atom nuclei. The induced electromagnetic field deformation slows down the incident particle which results in continuous photons emission, proportional to Z 2 /m 2 µ ;

e -/e + pair production: electromagnetic interaction produced by high energy Bremsstrahlung photons (E > 2m e ), gives the main radiative contribution, as seen in Fig. 1.6, in first approximation proportional to Z/m µ ;

-Photo-nuclear interaction occurs between Bremsstrahlung photons and nuclei, which results in nucleon or photon emission.

To quantify the overall impact of those interaction processes on the muon energy loss crossing a medium, particle physicists use a quantity called stopping power. The latter, expressed in GeV cm -1 , accounts for the energy deposition of the particle along its path in a given medium. Its mean value is defined in its most generic form as:

- dE dx = a(E) + b(E)E (1.1)
where a(E) represents the ionization (or electronic) component that accounts for interactions between the incident particle and electrons, and the coefficient b(E) represents the total energy-scaled radiative processes contribution, i.e the sum of e + /e -pair production, Bremsstrahlung, and photonuclear interactions: b ≡ b brems + b pair + b nucl . All the detailed expressions for radiative components expressions can be found in [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF]. If particles propagate in a medium where density is not homogenous, it is more convenient to write Eq. 1.1 in terms of opacity unit ϱ, a parameter introduced hereafter in Eq. 1.5. We then talk about the mean mass stopping power ⟨-dE/dϱ⟩, expressed in GeVcm 2 /g:

- dE dϱ = 1 ρ - dE dx (1.2)
where ρ is the density of the traversed medium. For moderately relativistic particles (0.1 ≤ βγ ≤ 1000) and intermediate Z material, the loss of energy due to atomic electrons along the muon path is well-described by the Bethe-Bloch equation. We present here this equation in its form given by Groom et al.:

a(E) = - dE dϱ elec = K Z A 1 β 2 1 2 ln 2m e c 2 β 2 γ 2 Q max I 2 -β 2 - δ 2 + 1 8 Q 2 max (γM c 2 ) 2 + ∆ dE dx (1.3)
with m e the mass of the electron, A the atomic mass g mol -1 , I the mean excitation energy eV, K = 0.3071 MeVg/cm 2 which denotes for the medium polarization and chemical composition. Q max is the maximum possible electron recoil energy during a collision between particles of mass M , its expression can be found in [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF], and δ(E) is a density correction term discussed hereafter, that has a significant impact above 1 GeV. The term ∆ dE dx represents a high-energy correction, accounting for photon emission by Bremsstrahlung from atomic electrons. Standard rock Unless other geological data are available, for practical reasons a simplified representation of a "typical rock" is widely used to describe the geology of scanned objects, alongside its tabulated energy deposition [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF]. Though the definition of its physical parameters (density ρ, atomic mass A, the atomic number Z) have evolved through time, as described in [START_REF] Lechmann | The effect of rock composition on muon tomography measurements[END_REF], this generic rock nowadays refers to a material with a density of crystalline quartz ρ sr = 2.65 g cm -3 , A and Z respectively set to 22 g mol -1 and 11.

Density effect

The δ/2 term in the Bethe-Bloch equation (cf. Eq. 1.3) corrects from the dielectric polarization of the medium. It has naturally its importance when studying the impact of different rock compositions on energy deposition and the subsequent muon flux attenuation. The effect is shown in Fig. 1.8: it is insignificant below the minimum of energy deposition, but gets an increasing contribution below the critical energy, up to the order of 24% at 100 GeV in standard rock. Extensive studies were led by Sternheimer et al. [START_REF] Sternheimer | Density effect for the ionization loss of charged particles in various substances[END_REF], that introduced a parametrization to account for this effect in Eq. 1.3. A review of this correction can be found in [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF].

Minimal crossing energy How can we estimate the flux of muons outgoing from a rock body structure? The aforementioned mean stopping power plotted in Fig. 1.9 allows us to derive the minimal energy E min that muons need for crossing a given thickness of rock. In the frame of the continuous slowing down approximation (CSDA), which implies that the energy deposition is a continuous process and fluctuations are neglected, one can derive

E min = ϱ(L) 0 - dE dϱ dϱ + E 0µ (1.4)
where E 0µ =105.6 MeV is the muon mass energy, ϱ(L) is the opacity in g/cm 2 (also called density-length) defined as the integrated density along a given travel length L inside the object to probe:

ϱ(L) = L ρ(ξ) dξ (1.5)
This estimation of E min though meets some limitations. Indeed if the ionization energy loss can easily be considered continuous and well described by the Bethe-Bloch equation (cf. Eq.1.3), the integration of the muon mean mass stopping power (cf. Eq. 1.2) over the opacity range to retrieve E min is neglecting the importance of stochastic fluctuations of energy loss from radiative effects that dominate at high energies (in standard rock E ≥ E SR µc ≃ 693 GeV). The computation is then said to be performed in the continuous slowing-down approximation (CSDA), for which the true energy loss dE/dϱ at each point of the muon path is assumed to be equal to the mean stopping power.

N.B To echo a cosmic-ray attenuation standard, the opacity ϱ is often times expressed in terms of meter water equivalent referred as mwe: 1 mwe is equivalent to 100 g cm -2 , and thus both units can be indistinctly used. in Fig. 1.10, we can see the computed minimal energies obtained from the integration of the mean mass stopping power (Eq.1.2), for water and three density values of standard rock. Once minimal crossing energies are known on a given opacity (or obstacle thickness) range-of-interest (cf. Fig. 1.10), the muon energy spectrum at Earth's surface Φ(E, ϱ), can be integrated for each ϱ over [E min (ϱ), +∞[ to obtain the expected transmitted muon flux :

I(ϱ, θ) = +∞ E min (ϱ) Φ(E, θ) dE (1.6)
In Fig. 1.11, we can see the output of the muon flux spectra integration using Eq.1.6 for minimal energies computed for standard rock. Below 500 mwe, the flux is higher for low zenith values, whereas at higher opacity the flux close to the horizon (high zenith) predominates by almost one order of magnitude compared to vertical flux above 2.10 3 mwe. This dependency on θ is due to muon decay and stronger absorption probabilities, which increase at low energy for inclined directions, as the traveled distances in the atmosphere grow. Today's definition of primary cosmic rays refers to charged nuclei and elementary particles of non-solar origin. Primary cosmic rays are stable charged particles (protons, electrons, photons) and nuclei with a lifetime superior to one million years (carbon, oxygen, iron, and some other products of nucleosynthesis in stars). Free protons and helium nuclei represent the main component of this primary flux, respectively 79% and 15% of the total flux. Their associated energy spectrum measured on Earth's surface ranges from sub-Gev to at least 10 20 eV (about 1 particle per km 2 and century). Originating from galactic systems (or supposedly extra-galactic for the most energetic part of the spectrum), cosmic rays are accelerated in the vicinity of astrophysical objects (e.g. supernova remnants [START_REF] Spurio | The Cosmic Rays and Our Galaxy[END_REF]). The general description of the cosmic-ray spectrum follows a power-law spectrum and can be presented by nucleons per energy-per-nucleon E, as plotted in Fig. 1.12. The differential flux (also called intensity) of primary nucleons is hence approximately described by [START_REF] Workman | Review of Particle Physics[END_REF]:

dN dE = k (E/1 GeV) -γ nucleons GeVm 2 s sr (1.7)
with k = 1.8 GeV 1.7 /(m 2 s sr) and spectral index γ = 2.7 for particles with energy E ∈ [1, 10 5 ] GeV.

Figure 1.12: Primary differential fluxes as a function of kinetic energy per nucleus. Free protons (H) represent 74% of the total flux, whereas nucleons bound in helium account for 70% of the other nucleons. Adapted from [START_REF] Workman | Review of Particle Physics[END_REF].

The high-energy collision between a primary cosmic ray on an upper-atmosphere nucleus (mostly nitrogen and oxygen) triggers the production of an extensive air shower (EAS). As it is energetic enough to be sensitive to an atmosphere nuclei structure, this first collision is an hadronic3 interaction that usually occurs at altitudes between 20 and 30 km. The products, so-called secondary particles, are composed of other nuclei fragments and mesons 4 , mostly pions π and kaons K. For example, one 10 6 GeV primary cosmic ray produces of the order of 10 6 secondary particles in the shower development process [START_REF] Kampert | Cosmic Rays and Particle Physics[END_REF]. Among these products, we distinguish three components: hadronic, electromagnetic, and muonic.

• Hadronic: constitutes the core of the shower that includes all the nucleons and mesons produced through the primary interaction, e.g a free proton p on nuclei X: p + X → p, n, Y, π 0 , π ± , K 0 S , K 0 L , K ± , .... Nucleons continue to trigger hadronic interactions as their interaction length is roughly 10 times inferior to the size of the atmosphere, whereas the decays of pions and kaons start initiating the other two components of the shower.

• Electromagnetic : initiated through neutral pions π 0 decay (very short-living particle τ π 0 ∼ 10 -17 s) in two photons π 0 → γ + γ which in turn decay in electron-positron pair γ + X → X + e -+ e + . The produced electrons interact with atmosphere nuclei through Bremsstrahlung, emitting further more photons: e ± + X → e ± + X + γ. At ground level, muon decays represent the dominant source of low-energy electrons.

• Muonic component: initiated in the high atmosphere at altitudes between 10 and 15 km, as shown in Fig. 1.13 through the decay of the first showers products via the following decay channels :

-charged pions π ± (τ π ± ∼ 10 -8 s): * π ± → µ ± + ν µ (ν µ ) -long-lived neutral kaon K 0 L (τ K 0 L ∼ 5.10 -8 s) and charged kaons K ± (τ K ± ∼ 1.10 -8 s): * K 0 L → π ± + µ ∓ + ν µ (ν µ ) * K ± → µ ± + ν µ (ν µ )
-A tiny contribution comes from charmed mesons, and heavier flavored mesons.

This source is often neglected. N.B Given their origin from a primary cosmic ray interaction, muons produced through EAS are very often referred to in the literature as cosmic muons.

The following part will present how this muon flux of great interest is modeled at Earth's surface.

Muon flux on Earth

As shown in Fig. 1.13, muons µ ± , alongside their associated neutrinos ν µ (ν µ ), represent the most abundant source of particles reaching Earth's surface: 63% of the total flux (primary and secondary particles combined), in front of secondary neutrons (21%), electrons (15%), protons and pions (less than 1%) [START_REF] Allkofer | Cosmic rays on earth[END_REF]. The flux reaching ground level is the result of a convolution between the muon production in the high-atmosphere in EAS, their energy loss, their decay, and atmospheric effects. Taking into account the energy loss, 2.4 GeV muons have a decay length of the order of 8.7 km, and down to the ground the mean energy is ⟨E µ ⟩ ≃ 4 GeV.

For "back-of-the-envelope" estimations, the muon flux on Earth is often summarised to some main figures: I ≃ 1 cm 2 /min, the overall angular dependence on zenith angle θ is proportional to cos 2 θ, characteristic at ⟨E µ ⟩. The whole journey through the atmosphere, Figure 1.13: Vertical flux given as a function of atmosphere depth, i.e its opacity (see lexicon), and altitude. Data points show the flux measurements for negative muons µ -. Taken from [START_REF] Workman | Review of Particle Physics[END_REF]. typically 1000 g cm -2 thick (= 10 mwe), costs in average about 2 GeV to muons through ionization process [START_REF] Workman | Review of Particle Physics[END_REF].

Like the primary flux, the muon spectrum at Earth's surface also follows a power law. In 1990, Gaisser introduced an analytic model for describing the muon spectrum at the Earth's surface based on the muon production from mesons in the high atmosphere [START_REF] Gaisser | Cosmic Rays and Particle Physics[END_REF]. This parametrization is convenient when neglecting muon decay for energy at sea level E 0 ≥ 100/ cos(θ) GeV as well as Earth's curvature i.e for θ ≤ 70 • :

Φ G (E 0 , θ) = A E -γ 0 1 1 + E P cos(θ) Eπ + B 1 + E P cos(θ) E K + r c [GeV -1 cm -2 s -1 sr -1 ] (1.8)
where E 0 is the muon energy at sea level, E P is the initial muon energy at production at the production point in the atmosphere. The two terms in parenthesis reflect the produc-Chapter 1

Figure 1.14: Illustration of the formation of an electromagnetic shower triggered by a primary cosmic ray interaction in the high atmosphere. Cosmic muons, represented in magenta continuous lines, are typically produced at altitudes around 15 km. Taken from [START_REF] Kotera | Cent ans de rayons cosmiques[END_REF] tion through charged pions and kaons. E π = 115 GeV, E K = 850 GeV are respectively the critical energies of pions and charged kaons, B is the branching ratio between kaonproduced and pion-produced muons. A, γ, and r c refer respectively to an overall scaling factor, a spectral index, and to the ratio of prompt muons, which accounts for the small portion of muons coming from charmed meson decays with respect to pions and kaons production (cf. Sec. 1.4.1). The original Gaisser parametrization sets E 0 = E P , A = 0.14 GeV -1.7 cm -1 s -1 sr -1 , γ = 2.7 and r c = 0.

Modified parametrizations of Eq. 1.8 thus focus on two main points :

• energy loss and muon decay in the atmosphere, by introducing an additional term W (E, θ) that accounts for the probability for muons to reach sea level.

• Earth's curvature, by defining an emission angle θ * at production in the atmosphere, rather than only considering the angle θ.

Hence, they took the following form :

Φ(E 0 , θ) = A E -γ P P (E P , θ * ) W (E P , θ * ) (1.9)
where E P = E 0 + ∆E is the muon energy at the production point in the atmosphere, and ∆E is the total energy loss in the atmosphere. Assuming that ∆E is independent of the energy (relevant for minimum ionizing particle), it can be parameterized in the following way :

∆E = a atm ϱ F cos θ * -ϱ P (1.10)
The term a atm = 2.06 × 10 -3 GeV/mwe is accounting for energy loss per ionization in the atmosphere for 50 GeV muons, where radiative loss are under 1%. ϱ F and ϱ P respectively represent the mean atmosphere opacity crossed by muons and the mean opacity between the cosmic primary and the muon production point.

Muon flux parametrizations

We will here introduce three different models of the muon spectrum at ground level. The first two follow a modified Gaisser spectrum (cf. Eq. 1.9) and propose different parametrizations of energy and incident angle θ to achieve the best fit to data over extended ranges, whereas the third one takes a more phenomenological approach, depending primarily on reference flux data for model parameters best estimates. Along with the Gaisser model and its modified versions, other muon spectra have been developed like the one by Bugaev et al. [START_REF] Bugaev | Atmospheric muon flux at sea level, underground, and underwater[END_REF] (with Reyna [START_REF] Reyna | A Simple Parameterization of the Cosmic-Ray Muon Momentum Spectra at the Surface as a Function of Zenith Angle[END_REF] adjustment). Comparison to data can be found in the literature, e.g. in the review paper of Hebbeker and Timmerman [START_REF] Hebbeker | A Compilation of High Energy Atmospheric Muon Data at Sea Level[END_REF].

Tang model

As the initial Gaisser model does not consider Earth's curvature thus limiting the incident zenith angle to θ < 70 • , Tang et al. [START_REF] Tang | Muon simulations for Super-Kamiokande, KamLAND, and CHOOZ[END_REF] introduced a modified parametrization of Eq. 1.8. They corrected the incident angle θ based on the zenith correction introduced by Chirkin [START_REF] Chirkin | Fluxes of Atmospheric Leptons at 600[END_REF] (cf. Sec. 1.4.5).

Three segments of energy are defined with their following transformations to be applied in Eq. 1.9 :

• At high energy E 0 > 100/ cos θ * : the standard Eq. 1.8 is used with its original parametrization.

• At moderate energy 1/ cos θ * < E 0 ≤ 100/ cos θ * GeV :

-The prompt ratio of muons is fixed to r c = 10 -4 .

-The muon energy at production is E P = E 0 + ∆E with energy loss ∆E corresponding to Eq. 1.10 with a atm = 2.06 × 10 -3 GeV/mwe, ϱ F = 950 g cm -2 and ϱ P = 90 g cm -2 .

-The interaction and decay probability W (E P , θ) = 1.1 the following energy transformation is made on E 0 : E 0 → 3E 0 +7 sec θ * 10 where sec θ * = 1/ cos θ * . Then the transformed E 0 is injected in previous equations at moderate energy.

Chapter 1

Guan model

Also using the zenith correction introduced by Chirkin [53], Guan et al. [START_REF] Guan | A parametrization of the cosmic-ray muon flux at sea-level[END_REF] proposed a modified Gaisser parametrization, defining the following decay probability for muons :

W (E 0 , θ * ) = 1 + 3.64GeV E 0 cos(θ * ) 1.29 -γ
(1.11) with parameters 3.64 GeV and 1.29 obtained from the fit of cosmic ray reference flux measurements detailed here [START_REF] Guan | A parametrization of the cosmic-ray muon flux at sea-level[END_REF]. As the standard Gaisser parametrization, the Guan spectrum sets r c = 0 as well as E P = E 0 and thus does not include any term for energy loss in the atmosphere. Nonetheless, the authors manage to improve the fit to sea level data over an extended range of energy and all zenith angles.

Shukla model

If Tang and Guan are modified parametrizations of the initial Gaisser spectrum, Shukla [START_REF] Shukla | Energy and angular distributions of atmospheric muons at the Earth[END_REF] proceeds in a different approach introducing two terms to respectively account for energy ξ(E) and angular D(θ) distributions. This gives a versatile model valid for E ≥ 0.5 -1GeV, that relies on the best fit to data at a given altitude of observation and geographical coordinates and thus requires reference measured flux :

Φ(E, θ) = I 0 ξ(E) D(θ) -(n-1) ξ(E) = N (∆E + E) -n 1 + E ϵ D(θ) = R 2 E H 2 cos 2 θ + 2 R E H + 1 - R E H cos θ (1.12)
where I 0 is the vertical flux (θ = 0) integrated over energy, N = (n -1)

(∆E + E c ) (n-1)
is the normalization factor where E c is the cut-off value of the data, ∆E is the total energy loss (both hadronic and ionization) in the atmosphere, n is the power law index, R and H atm are respectively Earth radius and characteristic altitude of muon production, defined in Fig. 1.16. The angular distribution D(θ), as Eq. 1.14, corrects the effect of Earth's curvature and provides a closed expression for column density for an inclined trajectory of muons. It approximates to 1/ cos θ for flat Earth. The parameter ϵ is a free parameter of the fit to flux data.

Comparison

The comparison between models and referenced muon spectrum data (datasets used in [START_REF] Workman | Review of Particle Physics[END_REF]) is shown in Fig. 1.15. The standard Gaisser parametrization only fits data at the right end of the spectrum for E > 100GeV. The largest variations between models and discrepancy to data occur for the low energy component (E ≤ 1 GeV) of the spectrum and close-to-the-horizon incident angles, which modified Gaisser parametrizations (e.g [START_REF] Tang | Muon simulations for Super-Kamiokande, KamLAND, and CHOOZ[END_REF] and [START_REF] Guan | A parametrization of the cosmic-ray muon flux at sea-level[END_REF]) or other phenomenological models ( [START_REF] Shukla | Energy and angular distributions of atmospheric muons at the Earth[END_REF]) allow to mitigate and thus extend the range of applicability of models. A caveat though may lie in the extent of datasets acquired through the past 50 years (e.g [START_REF] Ayre | Precise measurement of the vertical muon spectrum in the range 20-500 GeV/c[END_REF], [START_REF] B C Rastin | An accurate measurement of the sea-level muon spectrum within the range 4 to 3000 GeV/c[END_REF], [START_REF] De Pascale | Absolute spectrum and charge ratio of cosmic ray muons in the energy region from 0.2 GeV to 100 GeV at 600 m above sea level[END_REF], [START_REF] Kremer | Measurements of Ground-Level Muons at Two Geomagnetic Locations[END_REF], [START_REF] Haino | Measurements of primary and atmospheric cosmic-ray spectra with the BESS-TeV spectrometer[END_REF], [8]) chosen to tune model parameters, that often cover only vertical or rarely a limited number of incident θ angular values as for the L3 experiment (e.g [START_REF] Haino | Measurements of primary and atmospheric cosmic-ray spectra with the BESS-TeV spectrometer[END_REF], [8]). Nonetheless, when experiments are interested in estimating the muon flux that crosses shallow depths, corrections at low-energy to mild energy provided by modified parameterizations are the first resort to consider. We will see in Sec. 1.4.5 that the muon spectrum is in fine subjected to several effects, which makes its parametrization geographically and timely dependent.

Corrections to the muon flux

As the initial Gaisser's formula restrains the description of the muon flux on a given range of energy and zenith angle, several corrections have been proposed to formula 1.8 to extend its range of applicability. These corrections are pivotal for muography studies oftentimes targeting objects at high zenith angles. An exhaustive review of all the phenomena causing muon flux modulations can be found in [131]. We propose hereafter a condensed summary of these effects and some associated corrections.

Earth's curvature

The first correction is geometrical and is made to correct zenith angle θ from Earth's curvature. To do so, an angle at the muon production point in the atmosphere is introduced θ * [202], which differs from the angle of detection θ at ground level, as shown in Fig. 1.16.

Assuming that the altitude of the production point in the atmosphere is known, θ * can be obtained geometrically:

sin(θ * ) R E = sin(π -θ) R E + H = sin(θ) R E + H (1.13)
Chirkin [START_REF] Chirkin | Fluxes of Atmospheric Leptons at 600[END_REF] proposed a different treatment for this angle at production point θ * , solving the following equation on interaction length for each θ bin, X(θ) = X(0) to retrieve the altitude of first interaction. This leads to the following parametrization for θ * :

cos(θ * ) = cos(θ)

Geomagnetic effect

As Clay first observed in 1927 when the latitude of detection is getting closer to the magnetic equator, the particle flux decreases. Indeed the deflection caused by Earth's magnetic field imposes a minimum momentum for charged particles of cosmic origin to reach the ground. This effect that filters out low-energy particles (below 10 GeV) is known in the literature as the geomagnetic cut-off and depends on geographical coordinates. On the contrary to the equator, at the magnetic poles, the magnetic barrier is the lowest, and particles are only stopped by their energy loss in the atmosphere. The spectrum of charged cosmic particles can also be visibly altered by regional geomagnetic anomalies [START_REF] Hebbeker | A Compilation of High Energy Atmospheric Muon Data at Sea Level[END_REF].

Altitude

Since EAS development and the energy loss depends on the thickness of the atmosphere that muons cross, the altitude of observation has thus significant importance in the incoming muon flux. An empirical correction valid for momenta above 10 GeV and altitudes below 1000 m was introduced by De Pascale et al. [START_REF] De Pascale | Absolute spectrum and charge ratio of cosmic ray muons in the energy region from 0.2 GeV to 100 GeV at 600 m above sea level[END_REF]:

Φ(h) = Φ(0) exp(h/L) (1.15)
where h is the altitude of observation, and L represents a characteristic height of production. Hebbeker & Timmermans [START_REF] Hebbeker | A Compilation of High Energy Atmospheric Muon Data at Sea Level[END_REF] introduced the following empirical parametrization: h 0 = 4900 + 750p where p is the muon momentum in GeV. As a result, observers at an altitude of 1000 m will measure a 5% increase in flux compared to detection at sea level.

Other parametrizations for L based on Monte-Carlo simulation results may also be introduced: Nishiyama et al. in [START_REF] Nishiyama | 3D Density Modeling with Gravity and Muon-Radiographic Observations in Showa-Shinzan Lava Dome, Usu, Japan[END_REF] extend the fit to data up to 4000 m altitude for momenta above 3 GeV/c introducing L = 3400 + 1100 p cos(θ).

Influence of temperature and pressure conditions and seasonal changes Atmospheric temperature and pressure play a key role in the development of particle showers. In the stratosphere, the interaction length of the main muon producers -charged pions and kaons (from now on referred to as mesons) -is indeed affected by changes in density induced by temperature-pressure variations. For example, an increase in temperature will lower the density which in turn decreases the probability of interaction for mesons. As a result, the proportion of mesons decay to muons increases [START_REF] Grashorn | The atmospheric charged kaon/pion ratio using seasonal variation methods[END_REF], and so does the muon flux reaching ground level.

The muon production is thus affected by the variation of temperature and a seasonal dependence on temperature can experimentally be observed. In order to pursue muon-based monitoring (cf. 3), this effect has to be taken into account. In order to study atmospheric temperature changes on muon flux (e.g [START_REF] Tramontini | Middle-Atmosphere Dynamics Observed With a Portable Muon Detector[END_REF]) at a given observation site, relative variation with respect to average I flux can be approximated in terms of pressure and temperature variations [START_REF] Paul | Interpretation of Cosmic-Ray Measurements Far Underground[END_REF]:

∆I I = β∆P + α T ∆T eff T eff (1.16)
where β is the barometric coefficient, ∆P is the pressure variation from the average pressure that is measured at the point of observation, α T is the temperature coefficient that depends on the amount of traversed materials (i.e opacity), and T eff is the weighted average of temperatures measured under different atmospheric pressures [START_REF] Grashorn | The atmospheric charged kaon/pion ratio using seasonal variation methods[END_REF].

Solar modulations

The low energy component of the primary cosmic ray spectrum is affected by variations in solar wind velocity. This effect can be both time-dependent and not, based on particle momentum. For muon momenta below 10 GeV/c, solar wind modulation significantly influences the muon intensity at ground level. The effect of long-term solar modulation on Chapter 1 secondary nucleon and muon spectra has been studied by Allkofer and Dau [13] at different atmospheric depths. With their phenomenological model, Bhattacharyya [START_REF] Bhattacharyya | Effect of Solar Modulation on the Low Energy Sea Level Muon Spectrum Near the Geomagnetic Equator[END_REF] managed to obtain experimental evidence that solar activity could explain an observed enhancement of muon intensity below 5 GeV near the geomagnetic equator. Bhattacharyya and Hebbeker & Timmermans [START_REF] Hebbeker | A Compilation of High Energy Atmospheric Muon Data at Sea Level[END_REF] observed that solar activity modulation impact on primary flux results in a muon flux uncertainty of 1 (10) GeV/c at momenta of 10(1) GeV/c GeV. Above 10 GeV/c, the uncertainty decreases proportionally with momenta.

We will introduce in the following section the calculation of a custom muon energy spectrum at the geographical coordinates of the observation site using the CORSIKA toolkit, which allows taking into account some main corrections discussed above.

Modeling the muon flux at La Soufrière de Guadeloupe

CORSIKA [START_REF] Heck | CORSIKA: A Monte Carlo Code to Simulate Extensive Air Showers[END_REF] is a Monte-Carlo framework originally built in FORTRAN (newest version in C++ [102]) to simulate the generation and propagation of particle showers in the Earth's atmosphere. Monte-Carlo simulations are the common standard in high-energy physics to confront theoretical model developments with experiment results. These probabilistic tools are indeed required to describe the complex interaction of a cosmic-ray proton or nucleus colliding with an atmosphere nucleus, the production of secondary hadrons, and their decays that will give birth to muons (cf. Sec. 1.4.1). The spectrum of parameters and options of the CORSIKA toolkit is vast and the use requires careful reading of the provided documentation. For further details on the parameterization of CORSIKA, we refer to the work of Amélie Cohu (cf. [START_REF] Cohu | Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications[END_REF], [START_REF] Cohu | Analyse structurelle et fonctionnelle par tomographie muonique d'un haut fourneau[END_REF]). Among the main features, the user is invited to choose during the software compilation:

• the high-energy (HE) hadronic model, for describing the primary cosmic ray interaction on an atmosphere nucleus and the early stage of shower development. The chosen model is QGSJET-II [START_REF] Ostapchenko | Monte Carlo treatment of hadronic interactions in enhanced Pomeron scheme: QGSJET-II model[END_REF], which is built on the latest developments of hadronnuclei interactions with parameters tuned accordingly to recent accelerator data.

• the low-energy (LE) hadronic model, the boundary with HE being generally located around 80 GeV [START_REF] Fedynitch | Influence of hadronic interaction models and the cosmic ray spectrum on the high-energy atmospheric muon and neutrino flux[END_REF]. For this, the FLUKA code was adopted here to describe this part of the energy spectrum for its reliability, being jointly developed for years both at CERN and INFN to cover from the MeV to the TeV energy range all elementary particles and ions simulations for energy deposition, radiation damages, but also actively used for low-energy particle propagation calculations in the Earth's atmosphere [START_REF] Böhlen | The FLUKA Code: Developments and Challenges for High Energy and Medical Applications[END_REF].

• the geometry of the detector, either: flat, volumic, or vertical. This will set the corresponding acceptance accordingly. Our muon detector is simulated as flat.

The curvature of the atmosphere is taken into account for zenith angles larger than 60 • , as it significantly increases atmosphere depth and column density and thus the shower development. For more vertical directions, below 60 • , the atmosphere is defined as a succession of planar layers with decreasing density. Besides the curvature, the toolkit offers the possibility to take into account various atmospheric parameters. Among these parameters, the observation altitude, geomagnetic coordinates, and pressure-temperature conditions are of significant impact, as mentioned in Sec. 1.4.5

We respectively show in Figs. 1.17 and Fig. 1.18, the differential muon flux Φ(E, θ) as a function of the energy, and its integral I(ϱ, θ) on [E min (ϱ), +∞[ (cf. Eq.1.6) for three zenith angles θ = {0, 60, 80} • . The simulation was run at the geographical and geomagnetic coordinates of our subject of study: La Soufrière de Guadeloupe (cf. Tab. 1.1). The muon flux was computed over 12 energy bins E ranging from [10 -1 , 10 7 ] GeV each sampled with N = 10 9-log(E) number of showers, empirically defined in [START_REF] Cohu | Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications[END_REF]. The output of CORSIKA is compared to the Guan model described in Sec. 1.4.3. We first see in Fig. 1.17, that the CORSIKA differential flux is lower than the Guan model for the three zenith angles shown on the whole energy spectrum. The largest relative deviations to the model are located in the lowest and highest regions of the spectrum. The low-energy muons in the sub-GeV domain are not relevant in muography applications, but above 100 GeV, the deviations for all zenith angles θ start to be significant of the order of 25 -30%. Important fluctuations are seen for vertical θ = 0 above 3.10 2 GeV, while for θ = 60 • , the deviations remain quite constant around 25%, while the deviation for highest θ = 80 slightly increase and reaches up to 45% close to 10 4 GeV. This deviation between both differential fluxes propagates to the integrated flux in Fig. 1.18 computed for a standard rock medium. The maximum deviations around 20% are measured for the lowest opacity below 10 m w e, which is not the order of magnitude of interest in volcano muography. In the [10 1 , 10 3 ] m w e, for θ = {60, 80} • the deviations are in overall constant at 15 ± 1%, with respective maxima of 17% and 18% reach at 645 m w e and 340 m w e. These calculations illustrate the significant variations that can occur between an analytical flux model and a Monte-Carlo simulated flux, accounting for the geographical and atmospheric effects mentioned above.

Altitude z [m]

Geographical coordinates (WGS84) (lat., long.)

[ • ] Geomagnetic field components (B x , B z ) [µT] 1.2.

Conclusion

We have seen in this chapter the properties that make muons ideal candidates to probe large and dense objects: small energy losses mainly by ionization, absence of strong nuclear interactions, negligible probability of producing electromagnetic showers, and broad energy spectrum at ground level. The latter is a crucial aspect, as only high energy muons (E ≥1 TeV) can travel through kilometer distance without being absorbed, and thus allow to scan objects such as mountains and volcanoes (cf. chapter 3).

Muon spectrum heuristic models allow us to get a first grasp on the incident flux at ground level, and through the computation of minimal crossing, energies to get estimates on the transmitted number of muons traversing a given opacity range. We have presented how dedicated Monte-Carlo tools such as CORSIKA allow performing the simulation of primary cosmic ray interaction and EAS propagation in the atmosphere, thus refining the muon spectrum calculation by accounting for significant atmospheric effects at the observation site. Finally, to get estimates of the transmitted muon flux from a given medium on a given opacity range, two ingredients are needed: 1) the energy loss in the crossed medium, approximated by the mean stopping power, and 2) a reliable description of the muon spectrum. This operation represented in Fig. 1.19 is referred to as the forward problem solving, i.e. what muon flux exits a given quantity of matter. We will see in chapter 3, how to go the opposite way and manage to get opacity estimates of a given scanned object from the number of traversing muons recorded in a detector operating for a steady and sufficiently-long data acquisition time, one main challenge of muography surveys. Scheme summarizing the relationships between the scanned object with its associated density distribution, its impact on the propagation of traversing muons, and the resulting transmitted integral flux. In the case of the forward problem, the latter is estimated from a given density distribution, whereas for the inverse problem, the density is modeled from estimates of the muon flux.

Scanned object

Forward problem

Muon physics

Density distribution

Propagation

Chapter

Preliminary cosmic muons analysis in ProtoDU Dual Phase In this chapter, we will present the work done during the first year of the thesis from October 2019 to September 2020, in the context of the DUNE experiment at the APC laboratory, Université Paris-Cité. The study followed the commissioning on the CERN Neutrino Platform of a novel neutrino detector prototype named ProtoDUNE-Dual Phase, which was based on an innovative design, relying on a liquid phase of argon as a detection medium, overhung by a gaseous argon phase to amplify the electronic signal, unique at such scale. In the context of this thesis, the focus was on the methodology used to reconstruct charged particle tracks, more specifically cosmic muons, and on the extraction of preliminary results of the detector's performance for such reconstruction. The scientific context of the DUNE experiment is briefly introduced.

Introduction to DUNE and its scientific goals

The Deep Underground Neutrino Experiment (DUNE) is a next-generation neutrino experiment that aims to study the neutrino oscillation over a very long baseline of 1300 km with a muonic neutrino (or anti-neutrino) beam, from its emission at the Fermi National Accelerator Laboratory (FNAL) in Illinois, and its detection at Underground Research Facility (SURF), located in South-Dakota ( [7], [6]). During this two-decades-long experiment, DUNE will rely on the Long-Baseline Neutrino Facility (LBNF) project which aims to build a very high-intensity beamline in both muonic neutrinos and anti-neutrinos channels, respectively ν µ and ν µ . One of the main LBNF requirements is to provide a 1.2 MW beam over a wide energy band from 0.5 GeV to 8.0 GeV to allow DUNE to be sensitive both to matter effects predominant beyond 3 GeV and to an expected leptonic CP symmetry violation below 1.5 GeV. With this unprecedented spectral information, DUNE will be able to improve precision on known parameters that drive oscillations and give the first 5σ measurements of unknown features:

• Mass hierarchy, meaning the mass eigenstates order 1, 2, 3 (normal ordering) ou 3, 1, 2 (inversed ordering),

• CP-symmetry violation in leptonic sector, measuring the parameter δ CP ̸ = 0 or π of the PMNS matrix,

• the mixing angle θ 23 of the PMNS matrix and its associated octant, i.e its deviation to maximum mixing angle value 45 • : lower octant for θ 23 < 45 • , or upper octant for θ 23 > 45 • (cf. [START_REF] Das | Determination of the θ 23 octant in long baseline neutrino experiments within and beyond the standard model[END_REF]). DUNE will also be able to detect galactic supernovae neutrino bursts, and probe phenomena beyond the Standard Model of particle physics such as proton decay. In order to achieve these ambitious goals, the DUNE experiment is composed of two main detectors located on the LBNF beamline :

• the Near Detector (ND) [6], located at a few hundred meters of the neutrinos beam source at FNAL. The role of the ND is notably to allow initial unoscillated beam state characterization, measure cross sections, and constraint systematics. It will perform energy spectra measurements for both ν µ and ν e and their corresponding anti-neutrinos, giving important information on the shape of the expected oscillation signal in the Far Detector (see below).

• the Far Detector (FD) [1], composed of four 17-kiloton modules (10 kt fiducial mass of liquid argon each). The DUNE FD complex is under construction at the Sanford research facility (South Dakota), 1300 km away from the beam source, located 1.5 km underground. The liquid Argon Time Projection Camber (LArTPC) technology was chosen to reconstruct neutrino interactions thanks notably to its ability to perform both millimeter-scale 3D particle tracking and precise calorimetric measurements.

For the design of the future modules of the FD, two LArTPC technologies (Single Phase and Dual Phase, described below) were proposed and studied at the neutrino platform complex at CERN, in the context of the ProtoDUNE experiments.

The ProtoDUNE Dual Phase detector 2.2.1 Introduction

The two ProtoDUNE detectors were built to tackle the technical challenges and performance requirements of the future DUNE far detector modules. The Dual-Phase (DP) LArTPC detector -ProtoDUNE-DP (the NP02 experiment at CERN) visible in Fig. 2.1, including the largest vertical drift (6 meters) built so far and main detector components (e.g cathode, extraction grid, charge readout planes) at full scale with a FD module, was conceived to demonstrate and validate the production, assembly, and operation for DUNE [START_REF]The DUNE Far Detector Interim Design Report[END_REF].

Dual-Phase design Overview

The DUNE far detectors will use liquid Argon (LAr at 87.3 K) both as a target for neutrino interactions and a sensitive medium for the detection of the particles produced in neutrino interactions. The interactions of charged particles in LAr produce ionization and excite the argon atoms. De-excitation results in the emission of a UV (127 nm) scintillation signal, which provides the timestamp of the event. The ionization charge is drifted under the action of a uniform electric field (500 V/cm) to the readout electronics plane.

The DP LArTPC design is based on the presence of a thin gaseous argon layer on top of the liquid argon detection volume, where the multiplication of the ionization charge and collection at the anode takes place. This amplification area aims at significantly improving the signal-to-noise ratio in the charge readout, making longer drift distance achievable for LArTPC and therefore larger homogeneous fully active detection volume. The DP design also involves fully accessible and replaceable readout electronics for the whole duration of

Main features

Extraction from liquid argon to gas In the DP design depicted in Fig. 2.2, primary ionization charges are drifting vertically under the effect of a homogeneous electric field E drif t with a nominal value of 500 V cm -1 towards the Charge Readout Planes (CRP) area at the top of the detector, that they enter after getting through an extraction grid in the LAr phase just below the liquid-gas interface, where an extraction field E extraction ≃ 2 kV cm -1 is applied.

Amplification After extraction, charges drift towards Large Electron Multipliers (LEM).

Each CRP is divided on its underside into thirty-six LEMs of surface 50 × 50 cm 2 . Those LEMs had been mechanically drilled of 5.10 5 holes of 500 µm diameter. Inside those LEM holes the amplification electric field reaches 30 kV cm -1 , which triggers Townsend charge avalanches.

Collection Lastly, the secondary charges produced inside LEM holes are driven by an induction E field in order to be collected on a single multi-layer Printed Circuit Board (PCB) anode, patterned into two orthogonal sets of gold-plated copper strips, called views, each with 3.125 mm readout pitchapter This design allows a 2D readout, measuring both orthogonal directions Y and Z [START_REF] Scarpelli | Performance studies of the dual-phase Liquid Argon TPC for the DUNE experiment and analysis of the 4-tonne prototype detector data[END_REF]. The design of the anode (see Fig. 2.5) is such that it allows an equal sharing of the collected charge between the two views.

To measure the depth of the interaction in the LAr, and obtain full 3D reconstruction, the 2D readout is combined with the measurement of the drift time between the time of primary ionization charges emission obtained with scintillation light detected by PMTs underneath the cathode and the time of charge collection on the anode views.

Commissioning of ProtoDUNE-DP

During summer 2019, after the purging and cooling phases, ProtoDUNE-DP was filled with liquid argon from July 5 th to August 9 th . Once the filling was completed, ProtoDUNE-DP started operating on August 10 th .

HV extender fault

The ProtoDUNE-DP High-Voltage (HV) power supply was initially designed to deliver -300 kV at the cathode (located at the bottom of the TPC) and thus ensure a nominal electric field of 500 V cm -1 over 6 m drift. To do so, the HV is driven from its insertion at the top of the cryostat to the cathode thanks to the HV extender (prolongation of the HV feedthrough) which covers the entire drift height.

Just after the argon filling was completed, a progressive HV ramping-up procedure was followed in order to reach the expected cathode HV but failed to go beyond HV∼ -168 kV. Further investigation showed that the fault came from the HV extender. A short circuit was identified between the inner conductor of the HV extender and the first field-shaping ring connected to the field cage (cf. Fig. 2.2), intended to prevent E-field distortions between the extender and the cage.

Consequences

In the end, this short has resulted in significant E-field inhomogeneities, especially in the area close to the field cage. It prevents the HV extender from delivering nominal HV to the cathode. On the initial six meters drift with an expected homogeneous drift field of 500 V cm -1 between cathode and anode, only the upper first quarter of the total drift volume maintained a sufficiently uniform electric drift field (away from the field cage) and close to the nominal value at HV∼ -120 kV.

ProtoDUNE-DP data

The available data for ProtoDUNE-DP are from the interaction of cosmic rays and contain essentially ionization tracks left by atmospheric muons in the liquid argon. These raw data are stored both at CERN and Fermilab, taking advantage of Grid computing resources for offline event reconstruction. From August 28 th , 2019 to the end of January 2020, cosmic runs have been taken in stable detector conditions. During this period, 1.9M cosmic events were recorded using a random trigger at 10 Hz, for a total of 66k data files accounting for 200 TB of data volume storage. In Fig. 2.6, an example of a cosmic event is displayed. Data coming from the charge readout system are sampled at a frequency of 2.5 MHz and are contained in a 4 ms drift time window. Information on those runs, their event content, and configuration parameters (the high-voltage setting for cathode and CRPs, pressures and temperatures, level meters values, purity monitors measurements, etc.) are collected in an offline database np02run at CERN.

Cosmic muon analysis

The work presented below has been done in the context of the ProtoDUNE experiment and the commissioning of the ProtoDUNE-DP detector. This section aims to present the analysis of the first cosmic muon data in ProtoDUNE-DP between October 2019 and the summer of 2020. The analysis was performed using the LArSoft framework and is presented in the next section (Sec. 2.3.1).

Event reconstruction 2.3.1 LAr experiments common framework: LArSoft

Like other neutrino experiments such as MicroBooNE [5] or the Short Baseline Neutrino Detector (SBND) [START_REF] Majumdar | Review of Liquid Argon Detector Technologies in the Neutrino Sector[END_REF] sharing LAr as detection medium, the event simulation, reconstruction and analysis inside DUNE detectors are performed with the Liquid Argon Software (LArSoft) which provides versatile tools and a shared framework between LAr experiments [START_REF] Snider | LArSoft: toolkit for simulation, reconstruction and analysis of liquid argon TPC neutrino detectors[END_REF]. For simulation studies, particle Monte-Carlo generators such as CORSIKA [START_REF] Heck | CORSIKA: A Monte Carlo Code to Simulate Extensive Air Showers[END_REF] for cosmic ray showers and GENIE [START_REF] Costas | The GENIE Neutrino Monte Carlo Generator: Physics and User Manual[END_REF] for neutrinos events are interfaced with LArSoft, as well as GEANT4 [11] tools in order to simulate the interaction of particles in LAr and the propagation of the signals up to the detector response. In the analysis presented in Sec. 2.4, we have relied on cosmic muons generation with CORSIKA. The ProtoDUNE-DP geometry and its response were already implemented in the DUNE LArSoft framework. The event reconstruction follows the workflow presented in the following Sec. 2.3.2.

Track reconstruction main steps

1. Signal pre-processing: This step aims to identify the region of interest (ROI) on the raw waveform based on a threshold logic and also includes several noise-removal tools.

2. Hit finding: After pre-processing, one needs to identify individual charge deposits within waveforms, that are called hits. From those hits, we notably aim to measure two important features: the integral charge which is the area under the fit function f h (t) introduced below, and the hit peaktime. Those two quantities respectively give a measurement of the collected charge, and the depth of the interaction in the LAr. Until recent times, the default tool used to find and fit those hits were based on an asymmetric time-dependent function of the form:

f h (t) = exp((t -t 0 )/τ 1 ) 1 + exp((t -t 0 )/τ 2 ) (2.1)
where the parameters of the fit are t 0 (the hit peaktime), and τ 1 and τ 2 , the fall and rise time constants [START_REF] Scarpelli | Performance studies of the dual-phase Liquid Argon TPC for the DUNE experiment and analysis of the 4-tonne prototype detector data[END_REF]. The data analysis that will be presented in part 2.4 has been conducted on reconstructed data using this hit finding tool called DPraw hit finder inside the LArsoft framework. In the current reconstruction workflow, the hit finder algorithm has been modified to match the requirements of the track reconstruction algorithm updates (cf Sec. 2.3.3). It is now preceded by the deconvolution of the raw waveform and uses a Gaussian function which provides a more accurate fit of the waveform hits than the DPraw hit finder, and thus improves the precision on the measurement of the charge integral and peak time for each identified hit. We can see in Fig. 2.7 an illustration of the different outputs of both hit finders for the same raw waveform signal.

Pandora

The Pandora Software Development Kit [START_REF]The Pandora multi-algorithm approach to automated pattern recognition of cosmic-ray muon and neutrino events in the MicroBooNE detector[END_REF] is a pattern-recognition multi-algorithm. It is integrated into LArSoft via the LArPandora module. As for LArSoft, it is a shared tool among LAr experiments to reconstruct neutrino and cosmic ray events and is now commonly adopted inside the DUNE experiment, notably for the ProtoDUNE events reconstruction (both SP and DP).

For the DP LArTPC though, the Pandora implementation in the event reconstruction workflow was challenging due to the specific dual-phase detector geometry, which was not part of the LArPandora interface module until recently. After the dual-phase geometry integration to the reconstruction framework, efforts have been made in the context of the thesis of Etienne Chardonnet [START_REF] Chardonnet | Study of the response of the dual-phase liquid argon TPC, WA105 at CERN, and the development of tools improving the detector performances for the DUNE experiment at Fermilab[END_REF] to develop the Pandora tools for dual-phase, which improve significantly the track reconstruction efficiency in the dual-phase LArTPC.

Electron lifetime measurement with cosmics

The ProtoDUNE-DP design involves a six meters long drift for charges, at scale 1:2 with the dual-phase FD module design. Ionization electrons may be captured on electronegative contaminants during the drift, resulting in a loss of the ionization signal at the anode plane. One main requirement of the DP detector is then to guarantee a minimal lifetime for electrons of about 4 milliseconds (corresponding to a 6 m drift at the velocity of 1.5 mm/µs with a 500 V/cm E field), by maintaining a high argon purity with contaminants such as oxygen, nitrogen, water, or carbon dioxide at a ppt concentration level.

As primary ionization electrons drift along E drif t lines towards the top of the TPC and its GAr amplification phase, those charges have a non-null probability to be attached to these contaminants. Due to this attachment probability, the total amount of charge Q(t) is then decreasing exponentially with time t at which electrons are collected on anode views (hit peak time). This loss of charge in the LAr takes an exponential decay law form:

Q(t) = Q 0 e -(t-t 0 ) τe (2.2)
where Q 0 is the initial ionization charge, emitted at time t 0 of the interaction in LAr, and τ e is the electron lifetime which is the quantity that we aim to measure here.

In a LArTPC located at the Earth's surface, one can efficiently retrieve the electron lifetime τ e using the tracks of atmospheric muons, called most often "cosmic" muons.

Muon tagging

For this analysis, in order to select well-reconstructed MIP-like tracks in the LAr, several cuts on data have been made. First, only hits formed on the anode of the two fully instrumented CRPs area have been considered. Then we selected only data from LEM whose HV stability has been sustained during the run duration. Indeed, HV can be damaged by sparks occurring on LEM or on the extraction grid. Then, we select reconstructed tracks that have entered the LAr volume by its top surface (i.e. incident particles that have first crossed the top amplification region) and which start after the trigger, so that the first recorded hit on the anode wires corresponds to a null drift time. Lastly, we want to ensure the quality of reconstructed tracks. Indeed, the Pandora reconstruction algorithm (see Sec. 2.3.3) may have failed to reconstruct particle tracks that are broken due to gap crossings (cf. Fig. 2.9). In this case, the algorithm does not manage to match the different track segments and reconstruct the 3D trajectory as a single particle track. Therefore, those tracks more likely to be segments of a long broken particle trajectory are removed from the MIP selection. In the end, we choose to select tracks: a) with length > 50 cm (∼ 65% of tracks) to remove the shortest segments of broken tracks, b) which start at least 3 cm away from dead areas which refer to non-instrumented regions located at the interface between CRPs and between two consecutive LEMs, so-called LEM gaps. This breaking of trajectories translates in a peak of reconstructed tracks starting close to a LEM gap or CRP border ≤ 3 cm, visible in Fig. 2.10. This cut downsizes the number of remaining tracks by ∼ 45%.

Charge deposit versus drift time

The exponential loss of electrons due to attachment to electronegative contaminants and the associated electron lifetime τ e can be retrieved in the collected charge at the anode per unit length usually referred to as dQ/dx. The charge dQ stands for the hit charge integral, and the dx is a differential step computed along the ionization track (not referring to the X drift vertical coordinate). For this analysis, the dQ/dx has been plotted as a function of the drift time ∆t = tt 0 . For each drift time bin of size 100 µs the most probable value (MPV) of dQ/dx is computed from the fit of the distribution with a Landau function convoluted with a Gaussian as shown in Fig. 2.11. Finally, the fit of the dQ/dx MPVs with a decreasing exponential function gives us τ e .

The lifetime measurement has been first tested using existing simulation tools for Dual Phase in LArSoft (cf Sec. 2.3.1). The result is shown in Fig. 2.12. A constant CRP effective gain (charge multiplication factor) was set by default to 6 (current estimation gives values for effective gains slightly higher than 1, see Sec. 2.4.2), and we choose to set the lifetime for electrons to 4 ms.

Results on data for three different cosmic runs taken in October, November, and January can be seen in Fig. 2.13. As the LAr purity improves between October and January, the electron lifetime τ e is also increasing by a factor ∼ 4 to reach the order of magnitude required for τ e for a long drift dual phase module.

Comparison with purity monitors data

Like the ProtoDUNE Single-Phase detector, Dual-Phase is equipped with three small ionization chambers called purity monitors represented in Fig. 2.14, located between the cryostat wall and the field cage: two "small" ones with a drift distance of 17 cm, one at the field cage bottom and one at half drift height, and one with a long drift of 48 cm at the top of the TPC.

The use of cosmic tracks to measure τ e directly inside the detection volume remains crucial since those three purity monitors measure electron lifetimes at their own location inside the cryostat but outside the field cage [2]. In Fig. 2.15 we give the purity monitors τ e measurements as a function of time starting from detector commissioning in august. Those

Gain measurements Definition

Primary ionization charges multiplication occurs in LEM holes through Townsend avalanches triggered by an amplification field E amp ∼ 30 kV cm -1 that results from a voltage difference applied between the two LEM faces top and bottom ∆V LEM = V LEM top -V LEM bottom . This charge multiplication is characterized by an intrinsic amplification gain G LEM which is not directly measurable by looking at the collected charge at the anode. What is actually measured is an effective gain G ef f that results from the convolution of three detector effects: the liquid-to-gas extraction described by an efficiency ϵ extraction , charge multiplication inside LEMs holes (G LEM ), and charge collection on the two anode views with ϵ collection :

G ef f = ϵ extraction × G LEM × ϵ collection (2.3) Chapter 2
This effective gain G ef f is measurable by retrieving for cosmic tracks the most probable value of the deposited charge by unit length distributions dQ/dx i=0,1 , where i stands for anode view 0 or 1. One can define G ef f as follows:

G ef f = dQ/dx 0 + dQ/dx 1 dQ/dx M IP (2.4)
where dQ/dx M IP is the average charge deposition for a minimum ionizing particle (MIP) in LAr. In first approximation, considering a homogeneous E drif t , the recombination factor R(ρ, E drif t ), which represents the fraction of ionization charges that survive Ar + -e - recombination (model-dependent quantity), is constant. Using the Birk's law ICARUS parameterisation [4], at E drif t = 166 V/cm, we obtained R = 0.54. Hence, one can estimated the expected charge deposit dQ/dx M IP ∝ R × dE/dx M IP /W ion ≃ 8 fC/cm, where dE/dx M IP is predicted by the Bethe-Bloch formula, and W ion is the argon ionization energy. Due to the significant E field inhomogeneities mentioned before, especially at CRP borders, this value should be taken with caution: a more precise mapping of the recombination factor R for correction is required. Another solution is to compute for each individual i LEM ⟨dQ/dx M IP,i a reference MIP charge deposit obtained from the dual-phase simulation including a realistic E field map.

Results

Fig. 2.16 shows a map of the LEM gains, measured as described in the previous section, on two runs taken at different times. A significant loss of amplification effective gain between October and January is visible. This decrease in G ef f with time is most likely due to the LEM charging-up effect, a charge accumulation on the insulator inside and around the LEM hole. The reader could refer to previous studies done on LEM stability by Cantini et al. (cf. [START_REF] Cantini | Long-term operation of a double phase LAr LEM Time Projection Chamber with a simplified anode and extraction-grid design[END_REF] and [START_REF] Cantini | Performance study of the effective gain of the double phase liquid Argon LEM Time Projection Chamber[END_REF]) for a more complete description of this effect, and its impact on LEM effective gain.

Charge collection

Charge sharing between anode views The anode in Dual-Phase is designed in such a way that it allows an equal 50:50 charge sharing between the two collection views. This key requirement greatly facilitates the reconstruction of charge signals, and thus physical quantities measurements such as the effective gain (cf. [12] and [START_REF] Scarpelli | Performance studies of the dual-phase Liquid Argon TPC for the DUNE experiment and analysis of the 4-tonne prototype detector data[END_REF]). To study this sharing one can define a charge asymmetry coefficient A to be plotted for several polar angles ϕ defined in a horizontal (anode) plane :

A = dQ/dx 0 -dQ/dx 1 dQ/dx 0 + dQ/dx 0 (2.5)
where dQ/dx i=0,1 are the MPVs of the dQ/dx distributions fitted with a Landau function convoluted with a Gaussian smearing which takes into account the collected charge fluctuations [START_REF]Performance study of a 3×1×1 m3 dual phase liquid Argon Time Projection Chamber exposed to cosmic rays[END_REF]. in Fig. 2.17, we see that the charge asymmetry coefficient A does not exceed 5% except for the central bin around ϕ = 90 • and the last extreme bin just below 180 • , where A ∼ 20%. For these bins, the reconstructed tracks are respectively nearly parallel to wires of view 1 and view 0. This configuration is a disputed case from the point of view of hits reconstruction since in one view the amount of collected charge is shared on just a few wires which makes it therefore more difficult for the hit finding algorithm to reconstruct Chapter 2 single charge deposits and disentangle them from noise within the waveform.

dE/dx measurement for a MIP

One main feature, and key advantage of a LArTPC is its ability to perform precise calorimetric measurements along particle tracks. The energy deposit dE/dx reconstruction is a major part of the particle identification (PID) process which leads to final state interaction characterization. Before being able to measure accurately the energy deposits, and identify particles involved in neutrino interactions, a calorimetric calibration step of the TPC, using cosmic muons as a benchmark, is required. This calibration consists in converting the collected charge on anode wires from ADC counts to energy in MeV. Starting from the measured charge deposit (dQ/dx) measured , one has first to convert the charge dQ from ADC to the number of collected electrons at the anode n e -, taking into account the measured effective gain G ef f from the amplification phase, and a calibration factor C cal (expressed in ADC/fC) that accounts for the readout electronics response. This factor C cal was measured at IP2I Lyon, using a dedicated calibration system, that allows the injection of known charge signals into the front-end electronics and measures the output in ADC counts.

(dQ/dx) converted = 1 C cal G ef f (dQ/dx) measured (2.6)
where (dQ/dx) measured in ADC/cm and (dQ/dx) converted is in fC/cm.

The dE/dx is then derived from (dQ/dx) converted using the so-called modified box model which takes into account the electron-cation recombination effect inside LAr, introduced by the ArgoNeut experiment to describe this phenomenon in their LArTPC (see [4]).

dE/dx = ρE drif t β exp βW ion ρE drif t (dQ/dx) converted -α (2.7)
where ρ(T ) is the liquid argon density at temperature T , E drif t is the drift electric field (estimated at 166 V/cm in ProtoDUNE-DP homogeneous field region), W ion is the argon ionization energy, i.e the average energy to form an electron-ion pair in argon. The recombination coefficient α = 0.93 and the fit parameter β = 0.30 g/(MeV.cm 2 ).kV/cm in the dE/dx expression were obtained by the ArgoNeut experiment, following the work of the ICARUS experiment that was the first experiment to explore recombination as a function of E drif t and dE/dx over the range of interest for a LArTPC experiment. Hence, ArgoNeut was able to provide a consistent global description for all data, both at low and high dE/dx regimes [4], and assess the best agreement between the canonical Box and Birks, calling this new α and β parameterization modified box model. The reader could refer to [4] for further information on the study of Ar + -e -recombination.

Conclusion

In this chapter we presented preliminary measurements done in ProtoDUNE-DP using cosmic muons data collected from August 2019 to April 2020. The measured electron lifetime was consistent with the short purity monitors measurements and with the detector specifications; the charge sharing between the two anode views was found to reflect the detector geometry; the effective LEM gain was measured and its evolution with time between October and January due to the charging-up effect was observed; finally, the dE/dx distribution for MIPs was analyzed.

At the time when the analysis was performed (Fall 2019-Spring 2020), the development of a realistic Dual Phase LArTPC simulation was ongoing: the main features of the data could be reproduced, but the width of the simulated dE/dx distribution could not reproduce the one observed in real data. Since then, more detailed studies have ben performed and the simulation has been improved, mainly with the inclusion of details on the charge extraction and collection. A paper on the performance of the ProtoDUNE-DP detector will soon be published by the DUNE Collaboration.

In the past two years, the Dual Phase has evolved towards the so-called Vertical Drift design [START_REF] Sacerdoti | A LArTPC with Vertical Drift for the DUNE Far Detector[END_REF], which will be adopted for the second module of the DUNE Far Detector. The amplification in the gaseous phase has been abandoned, but a similar technology will be used for the ionization charge collection and readout. Some of the analysis techniques and results developed in this chapter will most likely be reused.

In the context of this thesis, this chapter represents a preliminary approach to the reconstruction of cosmic muon tracks and the evaluation of the performance of a detector for this purpose.

Chapter

The era of muography: application to volcanology As presented in chapter 1, the muon flux observation on Earth has progressively migrated from fundamental physics through the exploration of the cosmic ray spectrum, to a research tool used for particle detector calibration (cf. chapter 2). Nowadays, a whole blooming sector is evolving in parallel to these fields: muon-based imaging so-called muography. The latter neologism has now infused the literature, as it mirrors the medical X-ray imaging "radiography", whose analogy is often used to illustrate the basic principle of the method, the photon beam, and photographic film being respectively replaced by the atmospheric muons and a particle tracking detector, or the so-called muon telescope. The wide spectrum of applications seen today ranging from geosciences, archaeology, nuclear safety, and homeland security to civil engineering testifies to the versatility of this novel imaging method, relying on the wide energy spectrum of atmospheric muons raining down to Earth, and their associated high-penetrating power. We will introduce throughout the chapter the basic principle of muography, and see how physicists and geophysicists dotted with a reliable muon telescope can take advantage of the penetration of muons through matter to detect density anomalies in large rock volumes such as volcanoes. To this day, two distinct muographic imaging methods have flourished to allow the probing of (very) large and/or dense objects :

• the scattering muography, which relies on the deviation of muons trajectories inside the crossed medium influenced by chemical composition variation, more precisely atomic number Z variation.

• the absorption or transmission muography that relies on muon absorption during the crossing of a dense medium like a rock structure, influenced by density variation.

In the two following sections, we will first briefly introduce scattering muography, to then focus on the method used for geoscientific applications: absorption muography.

Scattering muography

Due to the intense nuclei electric fields, muons transmitted matter are deflected from their original straight trajectory by multiple Coulomb interactions. When the thickness of the crossed material is extremely small and the probability of having more than one single scattering is negligible, the probability distribution followed by the angular deviation ∆θ is described by Rutherford's law:

P (∆θ) ∝ 1 sin(∆θ/2) 4 (3.1)
In the case of a thicker target, the successive small deviations make the overall probability distribution converge to a Gaussian distribution (central limit theorem) with zero means but whose standard deviation is a function of muon momentum and can be expressed as follows: where p is the muon momentum, l is the path length from entry to exit, and X 0 is the radiation length. For elements other than helium, the latter can be approximated by the following [START_REF] Tsai | Erratum: Pair production and bremsstrahlung of charged leptons[END_REF]:

σ(∆θ) = 13.6 MeV βcp l X 0 1 + 0.038 ln l X 0 β , (3.2)
1 X 0 ∝ Z(Z + 1) 716.4A log 287 √ Z (3.3)
where Z and A are respectively the atomic number and atomic mass. The Z dependence in equations 3.2 and 3.3 allows to distinguish contrasts of composition between low-Z and high-Z materials inside the scanned object (e.g identify a metallic structure in a stone statue). This constitutes the goal of scattering muography. Tracking the direction of the muon before and after it crosses the object allows measuring the scattering angle ∆θ along with the spatial displacement ∆x, represented in Fig 3 .2. The scattering position inside the object is ideally at the intersection of the incoming and transmitted measured muon path directions. However, this is true only for the ideal scenario where muons undergo only one scattering. But experimentally, due to detector resolution and multiple scattering occurring inside the object, the initial scattering position is "blurred" and other methods based on scattering angle and displacement need to be used to infer this position, as presented in S. Bouteille thesis (chapter 5) [START_REF] Bouteille | Development of Micro-Pattern Gaseous Detectors for Muography[END_REF]. Another challenge lies in accurately measuring the muon momentum, which requires in addition of a muon tracker, a calorimeter detector, or a track-bending magnet. Thus, a mean momentum value is often used for experiments using a simple tracking layout. Finally, the knowledge of both scattering angles and scattering positions can lead to mapping out in 3-D the atomic number distribution of the target in a single acquisition. This muonimaging method is devoted to probing smaller objects than in absorption muography, and thus requires less amount of exposure time to achieve statistically significant estimates. To this day, typical applications are found in the fields of nuclear safety and security, in order to track the smuggling of radioactive material [START_REF] Borozdin | Radiographic imaging with cosmic-ray muons[END_REF]. "Muon portals" prototypes are currently developed to exploit the high-penetrating power of muons, as a backup tool to standard "radiation portals" that could fail to detect a well-shielded radioactive source [START_REF] Riggi | The Muon Portal Project: Commissioning of the full detector and first results[END_REF].

Absorption muography 3.2.1 Goal and principle

The goal of absorption imaging is to retrieve the density distribution of an object from estimates of the muon flux leaving the latter. As in classical medical X-ray, the method relies on the screening effect created by the object and its associated density distribution on the incident muon flux at ground level Φ 0 . For estimating this transmitted muon flux, a muon absorption experiment can be composed of one or several detectors around the object. As detected muons come exclusively from above the horizon, detectors need to be positioned downstream of the investigated object, with a positive elevation angle. Common muography results either display the flux estimated from Eq.3.15 as a function of the incident direction (θ, ϕ), or the ratio between detected transmitted particles and expected "open-sky" muons derived from Φ 0 , i.e the probability for a muon to be absorbed by the medium referred to as the muon transmission. Both can be further interpreted in terms of the opacity variation, involving the comparison with a semi-analytic model spectrum as presented in Sec. 1.4 or Monte-Carlo simulation. Due to the detector pixelization, objects are probed along discrete lines of sight in which are recorded the number of detected muons. These lines of sight correspond to different muon paths in the target, characterized by their travel lengths or thicknesses L in the traversed medium. If the travel lengths for each line of sight are known thanks to a digital elevation model (DEM) (cf. Fig. 3.5) mapping the object surface, a mean density can be inferred: In the i-th line-of-sight,

ρ i = ϱ i L i (3.4)

Number of expected muons

The number of expected muons to be detected in the i-th line-of-sight (or i-th bin) and in a certain amount of time can be defined in its most general form as a function of the muon spectrum at Earth's surface Φ(θ, E), discussed in Sec. 1.4:

N i = ∆T S i Ω i +∞ E min (ϱ i ) Φ(θ, E) dT dS dΩ dE , (3.5)
where ∆T is the acquisition time, S i is the surface of the i-th bin, dS is the surface element looking into solid angle Ω i . These parameters are determined by the experimental setup and geometry of the detector. E min (ϱ i ) is the minimum energy required for crossing the opacity targeted by the i-th line of sight ϱ i .

Detectors employed in muography are generally not conceived to measure the energy of the muons. The flux to estimate is therefore the muon integral flux I(θ, ϱ) (also called muon intensity, cf. Eq. 1.6), that leaves the object after transmitted through different paths L, and thus different opacity values ϱ (cf. Eq. 1.5). From the number of detected muons, N , the first objective is thus to infer I(θ, ϱ), that is the energy spectrum Φ(θ, E) integrated over [E min (ϱ), +∞ [. Considering that the muon spectrum Φ is only a function of energy and zenith angle, after integration on energy and time, Eq.3.5 can be written:

N i = ∆T S i Ω i I(θ, ϱ i ) dS dΩ (3.6)
Even if the detector pixels used to reconstruct the particle track have the same pitch, the effective surface "seen" by incident muons is however not constant for all the directions over the detector's angular aperture. A function called acceptance A, or directional response function in [START_REF] Sullivan | Geometric factor and directional response of single and multi-element particle telescopes[END_REF], is thus introduced, which depends on angular coordinates while taking into account detector geometrical parameters (e.g pixel width) of the experiment as:

A(θ i , φ i ) = S i dS (3.7)
where S i is the surface associated to the i-th detector line-of-sight, i.e the total surface of the rear matrix pixel [START_REF] Sullivan | Geometric factor and directional response of single and multi-element particle telescopes[END_REF].

Considering that the muon spectrum Φ is only a function of energy and zenith angle, after integration on energy and time, Eq. 3.5 can be written:

N i = ∆T Ω i I(θ, ϱ i ) A i (φ, θ) dΩ. (3.8)
Hence, to solve Eq. 3.8, one needs to integrate the acceptance function over the solid angle. We then introduce a last parameter called integrated geometrical acceptance T geo expressed in cm 2 sr, namely the total angular response of the detector:

T geo i = Ω i A i (φ, θ) dΩ (3.9)

Detector acceptance

We introduce hereafter the computation of the geometrical acceptance in the case of a detector featuring planar, coaxial, and rectangular detection planes, arranged as two-dimensional pixel matrices. The lines-of-sight are defined by a given front matrix pixel (x f , y f ) -rear matrix pixel (x r , y r ) combination, i.e (∆x, ∆y) = (x f -x r , y f -y r ). The i-th line-of-sight of coordinates (θ i , φ i ) is thus repaired as (∆x i , ∆y i ). Given the detector geometry, multiple front-rear pixel couples correspond to the same (∆x i , ∆y i ).

Point-like approximation For a very large scanned object like a volcano, it is commonly accepted to consider a point-like telescope, given its negligible size compared to the object. This approximation implies that each (∆x i , ∆y i ) corresponds to a unique (θ, φ) angular direction.

For front and rear squared matrices, featuring an equal number of scintillators in X and Y directions N x,y , the number of couples targeting the same i is given by [START_REF] Jourde | Un nouvel outil pour mieux comprendre les systemes volcaniques : la tomographie par muons, application à la Soufrière de Guadeloupe[END_REF] :

C i = N x,y (N x,y -(|∆x i | + |∆y i |)) + |∆x i ∆y i | (3.10)
We pursue by introducing the geometrical factor G i , that we define as the portion of integrated acceptance carried by a single pixel couple associated with line i, such that :

T geo i = C i × G i (3.11)
To compute this geometrical factor G i , the methods employed by Sullivan [START_REF] Sullivan | Geometric factor and directional response of single and multi-element particle telescopes[END_REF] and Thomas and Willis [START_REF] Thomas | Analytical derivation of the geometric factor of a particle detector having circular or rectangular geometry[END_REF] can be used. They both derived an analytical expression of G, assuming a detector composed of two infinitely thin, plane, coaxial, parallel areas. For our present case, those two planes represent a front-rear pixel couple.

The geometrical factor defined in [START_REF] Sullivan | Geometric factor and directional response of single and multi-element particle telescopes[END_REF] explicitly takes into account the dependence on θ of flux intensity I :

G i = S i Ω i F (θ)dSdΩ, (3.12)
where F (θ) denotes the angular component of I. Assuming an angular dependence of muon intensity in F (θ) ∝ cos 2 θ and considering infinitesimal surfaces on front and rear pixels, δS f = δx f δy f and δS r = δx r δy r , the infinitesimal geometry factor δG can be written [START_REF] Thomas | Analytical derivation of the geometric factor of a particle detector having circular or rectangular geometry[END_REF]:

δG = cos 4 θ δS f δS r d 2 (3.13)
where d = L 2 + ∆x 2 + ∆y 2 is the distance between the two pixels and L is the distance between the two detection planes, thus cos θ = L/d, represented in Fig. 4.11.

Hence the total geometric factor is obtained by integrating over the surfaces of front and rear pixels:

G i = L 4 S f Sr dx f dy f dx r dy r [L 2 + ∆x 2 + ∆y 2 ] 3 (3.14)
The result of this integral is given in [START_REF] Thomas | Analytical derivation of the geometric factor of a particle detector having circular or rectangular geometry[END_REF] (as in [START_REF] Sullivan | Geometric factor and directional response of single and multi-element particle telescopes[END_REF] upon a different form). We refer the reader to those references for details about the computation. We show in Fig. 3.4 an example of the final integrated acceptance displayed in the (∆x i , ∆y i ) plane, computed for a telescope with distance L = 180 cm between front and rear matrices, each featuring N x = N y = 16 scintillator bars forming pixels with widths w x = w y = 5 cm. The twodimensional function presents a symmetry around a maximum on the central detector bin, where the largest number of muons is thus expected; and decreases towards the edges, i.e for the most inclined muon directions, where the acceptance tends towards 0.

Number of detected muons

Experimentally, if the size of the detector i-bin is kept relatively small (of the order of one degree) it is commonly admitted to average over its angular range and directly consider the center coordinates (θ i , φ i ), which gives the following approximation:

N i = I(θ i , ϱ i ) × T geo i × ∆T. (3.15)
Regarding terminology and definition choices made by the different experiments, the definition of acceptance can be extended to include efficiencies related e.g to detection medium, event reconstruction, and selection efficiencies, respectively ϵ det , ϵ reco , ϵ sel . In order to be consistent with previous studies, we choose here to refer to the experimental integrated acceptance T exp that can be introduced as:

T exp i = T geo i × ϵ det × ϵ reco × ϵ sel [cm 2 sr]. (3.16)
In Sec. 3.3.3, we provide an estimation of T exp thanks to an open-sky calibration flux. Hence, the experimental number of detected muons becomes:

N exp i = I(θ i , ϱ i ) × T exp i × ∆T
(3.17)

Towards 3-D density modeling

By performing muographies at several positions around an object of known surface, the final goal is to combine the 2-D muon images of mean density to compute a three-dimensional (3-D) density model, as reported for volcanoes in [START_REF] Rosas-Carbajal | Three-dimensional density structure of La Soufrière de Guadeloupe lava dome from simultaneous muon radiographies and gravity data[END_REF], [START_REF] Nishiyama | 3D Density Modeling with Gravity and Muon-Radiographic Observations in Showa-Shinzan Lava Dome, Usu, Japan[END_REF], and [START_REF] Barnoud | Robust Bayesian Joint Inversion of Gravimetric and Muographic Data for the Density Imaging of the Puy de Dôme Volcano (France)[END_REF]. We show in Fig. 3.5 a representation of one detector in absorption mode targeting the La Soufrière de Guadeloupe lava dome (cf. chapter 4). As we will see in chapter 3.3.3, this combination of 2-D images sets a mathematical problem that lies in the category of the so-called inverse problems, inherent to geophysical 3-D tomography.

In geosciences, the term muon tomography has thus made its appearance when performing a 3-D inversion of 2-D muographies [START_REF] Lechmann | Muon tomography in geoscientific research -A guide to best practice[END_REF].In order to illustrate the principle, the comparison to medical X-ray imagery is relevant as the latter is also sensitive to density variations in the crossed medium. Just like 2-D muographies obtained from one single detector are analog to X-ray radiography, the 3-D density modeling from multiple 2-D images could thus be compared to a computed tomography (CT) scan.

What limits muography 3.3.1 Detector position

As mentioned in Sec. 3.2, the first straightforward boundary to muography is that it can only scan objects from below. Besides, targeting the object from a single point-of-view is of course not sufficient to obtain the location of an existing anomaly shown by muon count deficit (negative anomaly) or excess (positive anomaly), as the latter could be located at any position along the concerned travel lengths in the medium. Multiple detection positions are thus required to raise the ambiguity on the anomaly position. Hence, if 3-D modeling is foreseen one should dispose of an array of detectors, or relocate the same one on positions distributed as much as possible all around the investigated object, to achieve the better possible spatial coverage and constrain the anomaly region.

Acquisition time

Both absorption and scattering methods require large samples of muons to achieve a recognizable image. As the operator cannot act on the muon source, the time to record a sufficient amount of particles can be quite significant depending on the size of the object, especially for absorption muography. To quantify the minimum acquisition time in absorption needed to detect an opacity change in a given detector line of sight, Lesparre et al. in [131] proposed a first-order expression based on the following equation for the muon count N recorded at a given opacity ϱ (cf. Eq.3.15). Hence, for an opacity variation of δϱ at zenith θ (we hereafter omit the dependence on θ), and under the Gaussian assumption holding for a large enough number of detected muons N , the acquisition time should be:

∆T > γ 2 T I(ϱ) ∆I(ϱ, ∆ϱ) 2 (3.18)
where γ is the number of Gaussian standard deviations σ, i.e. the statistical significance associated to the measured flux variation, T is the acceptance, I(ϱ) is the integrated muon flux over [E min (ϱ), +∞[, and ∆I(ϱ, ∆ϱ) = I(ϱ + ∆ϱ) -I(ϱ), i.e is the variation of flux linked to a ∆ϱ shift in opacity. In Fig. 3.6, we show the minimum acquisition time at γ = 1σ confidence level as a function of relative opacity variation ∆ϱ/ϱ along three travel lengths L in the range of interest of a volcano, and for three mean density values of standard rock. For this study the muon energy spectrum Φ(θ, E) (Guan parametrization [START_REF] Guan | A parametrization of the cosmic-ray muon flux at sea-level[END_REF]) was integrated on [E min (ρ L),+∞[, for three different angle θ. One can also notice that the acquisition time is smaller for high zenith angle θ as the differential flux above 10 2 GeV increases with θ. We note from Fig. 3.6, that for a typical 1 km thickness, to reach up to a 10% resolution on opacity variation, the muographer needs to wait for between ∆T min ∈ [0.2, 1]10 2 days, depending on the apparent mean density. In the case of a temporal monitoring study, muographers thus need to adapt the detector acceptance T consequently to the characteristic timescale of the studied phenomenon, i.e. magma or hydrothermal fluid ascent. This acceptance tuning is a serious matter as it basically implies increasing the surface of detection to enlarge the solid angle Ω cone, hence rethinking the whole mechanical structure of the detector along with its electrical consumption. To cope with large acquisition time, one heuristic solution is to merge neighboring line-of-sight to increase the statistics. The counterpart is obviously a loss of spatial resolution on the investigated object. This strategy has been studied in [START_REF] Jourde | Muon dynamic radiography of density changes induced by hydrothermal activity at the La Soufrière of Guadeloupe volcano[END_REF], using principal component analysis (PCA), to optimize the merging of telescope lines-ofsight and identify several zones of interest relative to muon flux variation for a dynamic hydrothermal fluid circulation study.

Deflection from straight trajectory

The absorption muographies are blurred by the angular resolution of the detector and also by the multiple Coulomb scattering (cf. Sec. 3.1) occurring inside the scanned object. However, as the angular deflection is inversely proportional to the muon momentum (cf. Eq. 3.2), in the case of a huge rock body (from 100 m to 1 km scale) traversed only by high-energy muons, the angular deflection of trajectory due to multiple scattering is often negligible compared to the instrument angular resolution. In standard rock, the deflection quickly decreases as a function of rock thickness and reaches a plateau below 10 m, similar in order of magnitude to most detector angular resolution [START_REF] Schouten | Muon Tomography for Underground Resources[END_REF]. As presented in [START_REF] Jourde | Improvement of density models of geological structures by fusion of gravity data and cosmic muon radiographies[END_REF], we show in Sec. 6.1.2 additional simulation results that illustrate this negligible deflection effect in the case of La Soufrière volcano.

The design of a muon telescope must encompass these limitations and as a result consists in a balance between many factors: its acceptance (cf. Sec. 3.2.3); its transportability on the field; its cost; the acquisition time needed to acquire enough muons, which needs to be reasonable when aiming at performing a temporal study (e.g. monitoring subsurface fluid dynamics inside a volcano); and the spatial resolution that needs to be achieved on the inner structure of the scanned object.

The different technologies of muographers

At the cornerstone of the muography survey: the muon detector or so-called "telescope", able to count and measure the direction of incident muons that traversed the scanned structure. The full telescope setup consists of several detection planes, that can provide twodimensional XY position measurements of where the incident muon has crossed. Being a heritage of high-energy physics experiments (e.g. accelerator, neutrino, or cosmic-ray experiments), we propose hereafter a condensed review of the three main detection technologies that have migrated towards muon imaging. The reader might complete the following information on this matter with existing review papers, such as in [START_REF] Procureur | Muon imaging: Principles, technologies and applications[END_REF] and [START_REF] Bonechi | Atmospheric muons as an imaging tool[END_REF], from which this section is partly inspired.

Nuclear emulsion detectors

Deployed on the flanks of volcanoes, e.g on Mt Asama, Japan [START_REF] Hiroyuki | High resolution imaging in the inhomogeneous crust with cosmic-ray muon radiography: The density structure below the volcanic crater floor of Mt. Asama, Japan[END_REF], and Stromboli [START_REF] Tioukov | First muography of Stromboli volcano[END_REF], nuclear emulsion plates present significant assets for volcanology applications, but as any detection technology also carries their downsides.

A nuclear emulsion is a mix of gelatin and silver halides (e.g. bromite AgBr). The principle of detection is analog to photography. Ionization caused by muons reduces the silver ions (Ag + ) into metallic silver. After irradiation, the emulsion plate is developed to force the reduction reactions around the sites already reduced. This improves the optical contrast around these sites. During the last step of fixation, the reduction reaction is stopped and the remaining silver halides dissolved, leaving a plate darkened along particle tracks, as seen in Fig. 3.7.

With respect to the other two main technologies (gas-filled, and scintillator-based detectors), emulsion plates provide the best angular and spatial resolutions, respectively of the order of a few mrad for GeV particles and 1 µm. The plate thickness is also helpful to reconstruct the muon track in 3-D. Furthermore, one main practical point that draws towards this technology is that emulsion plates are completely autonomous: as the detection process neither requires an external trigger nor any sort of readout, no power supply is involved, which significantly reduces the complexity of the deployment on the field. However, this technology also presents downsides such as its sensitivity to harsh terrain conditions (low temperature below 10 • C and humidity) that limits its lifetime to a few months [START_REF] Procureur | Muon imaging: Principles, technologies and applications[END_REF]. High temperatures above 30 • C provoke the melting of the emulsion [START_REF] Bozza | Nuclear emulsion techniques for muography[END_REF]. Besides, after exposure, the plates require a very specific automated readout system to scan and identify all the muon track candidates, using a pattern recognition algorithm (that does not operate faster than a few cm 2 per hour). The absence of any electronic readout also implies the absence of a clocking system to timestamp incident particles, which prevents any real-time study of muon count variation.

Gas-filled detectors

The historical record of gas-filled detectors can be traced back from the Wilson's cloud chamber [START_REF] Thomson | On the ionisation of atmospheric air[END_REF] mentioned in Sec. 1.1, at the origin of pivotal particle identification including the discovery of the muon. One century later, these detectors are still widely popular in HEP experiments and rely on the same primary principle, that is the ionization of a gaseous medium along an incident charged particle track. Nowadays, equipped with the latest electronic readout, gaseous detectors exist under several sub-families of design and layout, which differ from e.g the geometry, the detection medium, the presence of an amplification volume, and the ionization charge readout. Regarding current muography studies, the focus points towards the development of Micro-Pattern Gaseous Detectors (MPGD) [START_REF] Bouteille | Development of Micro-Pattern Gaseous Detectors for Muography[END_REF], initiated at the end of the XX th century, to cope with high spatial and time resolution demanded by HEP experiments.

Among the different versions of MPGD, the Micromegas (MICRO-MEsh GAseous Structure) [START_REF] Giomataris | MICROMEGAS: a high-granularity position-sensitive gaseous detector for high particle-flux environments[END_REF], developed at CEA Saclay, have notably made their proof during the ScanPyramids campaign [181] (cf. Sec. 3.5.2). Micromegas are composed of two distinct volumes separated by a metallic micro-mesh: one first volume, the drift volume, in which occurs the primary ionization and one secondary volume, the amplification volume, sketched in Fig. 3.8. From the incident muon track in the drift volume, the primary ionization charges are conducted along an electrical field of 5 × 10 4 V m -1 towards the amplification volume. After being filtered through the micro-mesh grid, primary electrons trigger the development of so-called charge avalanches under the acceleration provided by a much stronger electric field of 4 × 10 6 V m -1 . At the bottom of the amplification volume, induced electric signals from avalanche charges develop into a readout anode, whose XY segmentation into strips (or pads) allows the reconstruction of the initial crossing position of the muon. Highly Figure 3.8: Sketch of a Micromegas detector. Taken from [START_REF] Bouteille | Development and applications of micro-pattern gaseous detectors for muon tomography[END_REF] sensitive to gas pressure and temperature conditions, the amplification process can set a limitation for the long-run operation of such detectors during field data-taking campaign. Nevertheless, in steady acquisition conditions, this detector offers thanks to the fine anode segmentation high spatial resolution O(100µm), with the possibility of conducting a temporal variation study. Besides, ongoing readout development notably on multiplexing, the method offers the possibility to reduce the amount of readout electronics [START_REF] Bouteille | Development of Micro-Pattern Gaseous Detectors for Muography[END_REF], [START_REF] Filosa | Tomographie muonique: du développment de détecteurs à la résolution du problème inverse[END_REF].

Another category of gas-filled detectors that has gained popularity in geoscientific muography is the Resistive Plate Chamber(RPC), featuring a sub-millimeter spatial resolution with time resolution comparable to scintillators. [START_REF] Cârloganu | Towards a muon radiography of the Puy de Dôme[END_REF] [START_REF] Giammanco | Resistive Plate Chambers in Muography[END_REF]. They consist of two parallel plates of high-resistivity (commonly composed of Bakelite and glass with 10 10 -10 12 Ωcm) plastic material that contains a thin detection layer of gas. A strong electric drift field is applied between the outer conductive coating of both resistive plates. Contrary to the aforementioned Micromegas, the charge avalanche takes place directly inside the detection volume, and drifting secondary electrons induce signals on the outer readout anode segmented into conductive strips or pads, which yields XY position measurements. If the provided spatial resolution is a main asset, but this kind of detector, unfortunately, presents a sensitivity to temperature and humidity (Aielli et al. 2004), which so far limits its use in well-controlled laboratory environments [START_REF] Bonechi | Atmospheric muons as an imaging tool[END_REF].

Scintillator-based detectors

This last family of tracking detectors might be classified as the most robust and insensitive to harsh terrain conditions. This durability along with practical parameters -cost, ease of assembly, modularity [166] -put scintillators as ideal candidates to conduct up to year-long continuous study in a climatically hostile environment such as a tropical volcanic island, like the Basse-Terre island, in Guadeloupe, France. It exists several kinds of scintillators (liquid, solid, organic or inorganic) integrated into HEP tracking detectors e.g for MINOS at Fermilab [9], OPERA at the Gran Sasso National Laboratory [10], or the recent upgrade of the Pierre Auger Observatory detectors [START_REF] Castellina | AugerPrime: the Pierre Auger Observatory Upgrade[END_REF] (Auger Prime). In muography, the most used type remains the plastic scintillator (e.g [START_REF] Tanaka | Development of a two-fold segmented detection system for near horizontally cosmic-ray muons to probe the internal structure of a volcano[END_REF], [START_REF] Marteau | Muons tomography applied to geosciences and volcanology[END_REF], [START_REF] Peña-Rodríguez | Design and construction of MuTe: a hybrid Muon Telescope to study Colombian Volcanoes[END_REF]), whose popularity is explained by the aforementioned practical parameters. One can indeed easily mold, and extrude the plastic as necessary to fit the shape of the detection volume to experimental requirements [START_REF] Marteau | Development of Scintillator-Based Muon Detectors for Muography[END_REF]. As this technology is used for the muography campaign at La Soufrière de Guadeloupe, we give further detail on the technical layout and performances of the scintillator-based telescope in the concerned section in chapter 4 (cf. Sec. 4.4.1). The basic detection element is thus a plastic scintillator bar (with a rectangular or triangular shape) in which is embedded a wavelength shifting fiber (WLS) to collect produced scintillation light, represented in Fig. 3.9. The principle is the following: the "small" deposited energy by the incident muon excites the scintillator molecules, which, upon deexcitation to the ground state, emit photons in the UV range (fluorescence mechanism, extremely fast O(ns)). The resulting light pulses (∼ 10 5 photons) are conducted through the WLS fiber towards the light and electronics readout system, whose first component is a photo-detection system e.g a photo-multiplier tube (PMT) optically connected to the fiber end.

Figure 3.9: Sketch of a plastic scintillator bar, featuring its embedded WLS fiber, shifting the collected "blue" scintillation photons to "green" photons conducted towards the PMT. Taken from [START_REF] Lesparre | Mise au point d'une methode de tomographie utilisant les muons d'origine cosmique. Applications au laboratoire souterrain du Mont Terri et au volcan de la Soufrière de Guadeloupe[END_REF].

Early days of muography

First density measurement in Australia

In 1953, twenty years after the first muon track identification in Neddermeyer and Anderson's chamber came the first recorded attempt by E.P. George [START_REF] George | Cosmic rays measure overburden of tunnel[END_REF], in Australia, to measure the areal rock density overlaying the tunnel of Guthega-Munyang in the Snowy Mountains region of New South Wales. For this first shot, no imaging was possible, but nonetheless, he successfully measured a density matching referenced core measurements with the use of a Geiger counter (cf Fig. 3.10).

Exploring the pyramids

In the 1970s came a second pivotal moment for cosmic-ray muon applications, with the first in situ experiment led by L.W. Alvarez, on the Chephren pyramid [START_REF] Alvarez | Search for Hidden Chambers in the Pyramids[END_REF], one of the three pyramids of Giza, in Egypt. This interdisciplinary project involving cosmic ray physicists and archaeologists was designed to probe the inner structure of the pyramid and search for hidden chambers. Alvarez et al. benefited from the technological development of spark chambers associated with a digital readout. With this technology, they were able to measure the incident angles of penetrating cosmic-ray muons over a large sensitive area. This detection setup was located in the Belzoni chamber at the basis of the pyramid, below an average of about 100 meters of limestone. More than a million events were detected for an exploration of 19% of the total volume of the pyramid, showing no evidence in the flux measurements for a hidden room of volume compared to one of the four known chambers in the other two pyramids of Giza.

Almost 50 years after Alvarez's pioneer experiment, from 2015 to 2017, a muography survey of the great Khufu pyramid, the largest of the Giza complex, was organized, in the framework of the ScanPyramids project [181], an international collaboration coordinated by the HIP institute and the Engineering University of Cairo [169]. Three experiments were led using different detection technologies: nuclear emulsion films from Nagoya University, Japan; scintillator hodoscopes from KEK, Japan; and gaseous detectors from CEA, France. Emulsion films and hodoscopes were placed inside the Queen's chamber, underneath the Great Gallery, while gaseous detectors were installed outside of the pyramid. This complementarity between detectors along with the comparison to Monte-Carlo simulations allowed us to confirm with a 5.8 σ statistical significance the existence of an unknown void with a corridor-like structure of estimated length of 30 m with a cross-section similar to the Great Gallery [START_REF] Morishima | Discovery of a big void in Khufu's Pyramid by observation of cosmic-ray muons[END_REF], The muon excess observed in outdoor CEA gaseous detectors that led to this discovery is represented in Fig. 3.12. The most recent study was published during the writing time of this work (March 2023) presenting the characterization of another significant identified void during the muography campaign started in October 2019, located in the North face of the pyramid [START_REF] Procureur | Precise characterization of a corridor-shaped structure in Khufu's Pyramid by observation of cosmic-ray muons[END_REF]. This void is referred to as the North Face Chevron (NFC), thought to be an architectural setting whose function during the building of the pyramid is still uncertain. The length of this anomaly was estimated to be close to 9 m with a transverse section of about 2 m. At the same time, an endoscopic camera (nondestructive) infiltration was conducted to the estimated location of the NFC thanks to the muography results, and successfully unveils the existence of such a void in the pyramid structure

Image Earth's subsurface and first probing of mountains

Theoretical and feasibility studies were conducted by Malmqvist in 1979 [START_REF] Malmqvist | Theoretical studies of in-situ rock density determinations using underground cosmic-ray muon intensity measurements with application in mining geophysics[END_REF] in the context of geophysics mining and boreholes to demonstrate the interest of cosmic ray muons to locate density anomalies and perform absolute density measurement, as a complementary method to gravimetric surveys. In the early 1990s, motivated by Alvarez's pioneer experiment, K. Nagamine [START_REF] Nagamine | Method of probing inner-structure of geophysical substance with the horizontal cosmic-ray muons and possible application to volcanic eruption prediction[END_REF] explored the possibility to probe kilometer rock distance with the highest-energy part of the cosmic ray muon flux, almost parallel to the ground surface. The ambition was to image the inner region of Mt. Tsukuba. With the use of a plastic-scintillator detector, they managed to demonstrate the feasibility of such a probing method, detecting a significant portion of transmitted muons from the mountain, consistent with calculated fluxes.

Application to volcanology 3.6.1 Achievements and current projects

Volcanoes have become the main field of application for absorption muography in geosciences. As it targets directions above the horizon, muography has only access to shallow regions of volcanoes and thus cannot image the deeper magma plumbing system. Nevertheless, this information is highly valuable and can be interpreted at many levels: providing hints on the eruptive style of volcanoes, on the shallow magmatic or hydrothermal fluid dynamics through changes in density, and lastly on the mechanical strength of the edifice. As of this day, it exists several detailed reviews and books (e.g. [START_REF] Hiroyuki | Japanese volcanoes visualized with muography[END_REF], [START_REF] Leone | Muography as a new complementary tool in monitoring volcanic hazard: implications for early warning systems[END_REF], [START_REF] Lechmann | Muon tomography in geoscientific research -A guide to best practice[END_REF], [START_REF] Oláh | Muography: Exploring Earth's Subsurface with Elementary Particles[END_REF]) dedicated to muography applications in geosciences. We propose hereafter a condensed non-exhaustive summary of the main highlights of the last two decades of muography applied to volcanology in Japan, Italy and France.

In Japan H.K.M. Tanaka and collaborators [START_REF] Tanaka | Development of the Cosmic-Ray Muon Detection System for Probing Internal-Structure of a Volcano[END_REF], [START_REF] Tanaka | Development of a two-fold segmented detection system for near horizontally cosmic-ray muons to probe the internal structure of a volcano[END_REF], [START_REF] Tanaka | Radiographic measurements of the internal structure of Mt. West Iwate with near-horizontal cosmic-ray muons and future developments[END_REF], [START_REF] Hiroyuki | High resolution imaging in the inhomogeneous crust with cosmic-ray muon radiography: The density structure below the volcanic crater floor of Mt. Asama, Japan[END_REF], [START_REF] Hiroyuki | Detecting a mass change inside a volcano by cosmicray muon radiography (muography): First results from measurements at Asama volcano, Japan[END_REF], [START_REF] Tanaka | Cosmic-ray muon imaging of magma in a conduit: Degassing process of Satsuma-Iwojima Volcano, Japan[END_REF] were the spearhead of the implantation of muography in the scope of modern applied volcanology. A decade after the first pioneered studies on Mt. Tsukuba, Tanaka et al. [START_REF] Hiroyuki | High resolution imaging in the inhomogeneous crust with cosmic-ray muon radiography: The density structure below the volcanic crater floor of Mt. Asama, Japan[END_REF], showed the first muographic of the inner structure of Mt. Asama volcano, imaging the rock density below the bottom of the crater with resolution O(10 m). They manage to identify several lowdensity anomalies under solidified magma interpreted as potential pathways for magma ejection.

In 2009, Tanaka et al. published the first muography of Mt Iwodake (cf. Fig. 3.13a from [START_REF] Tanaka | Cosmic-ray muon imaging of magma in a conduit: Degassing process of Satsuma-Iwojima Volcano, Japan[END_REF]). This volcano is located on the Satsuma-Iwojima island. It is one of the active cones in the northeastern of the Kikai caldera, formed during the Akahoya eruption 7300 BP, one of the largest eruptions of the Holocene with a VEI1 of 7. Tanaka's muography became a main illustration of what can be achieved in volcanology with muography. Their results were consistent with the known magma fluid ascent model for this volcano: the density anomaly observed at 400 m a.s.l, roughly 12% below average density, was interpreted as degassing magma, in the context of a convection model: the magma is ascending through a conduit connected to a deep magma chamber. During the ascent, as the pressure decreases the concentration of bubbles formed by dissolved gas in the magma increases. When a certain altitude is reached, magma becomes saturated in volatile elements and gas exsolution happens. Several observations done on Mt. Iwodake supported this interpretation: given the rhyolitic nature of the magma and the measured volcanic gases, such degassing was found to happen at very low-pressure conditions and thus to occur at shallow altitudes in the conduit [START_REF] Kazahaya | Degassing process of Satsuma-Iwojima volcano, Japan: Supply of Volatile components from a deep magma chamber[END_REF], consistent with the measured altitude around 200 m below the crater floor on the muographic image. In addition, a gravimetric survey showed that a low-density anomaly was located in the crater region (Komazawa et al. 2005).

Since the pioneer works, several research groups remain active in Japan using different muon detector technologies to study the Japanese volcanoes. For example, in 2022, Nagahara et al. [START_REF] Nagahara | Three-dimensional density tomography determined from multi-directional muography of the Omuroyama scoria cone, Higashi-Izu monogenetic volcano field, Japan[END_REF] conducted a multi-directional muography campaign from 11 sites around Omuroyama scoria cone, Japan. The detection was based on nuclear emulsion films. With this array of detectors, they manage to perform a 3-D inversion of the data with an estimated spatial resolution of 100 m, which has notably allowed identifying a high-density anomaly cylindrical region just below the summit crater in the central part of the scoria cone. Their work has participated in highlighting the eruptive processes occurring inside the scoria cone, particularly the lava flows at the foot of the cone and the formation of flank craters. [START_REF] Tanaka | Cosmic-ray muon imaging of magma in a conduit: Degassing process of Satsuma-Iwojima Volcano, Japan[END_REF]). Only density anomalies below 95% of the average density are mapped;(b): Topographic map of Statsuma-Iwojima island (Kyushu, Japan) featuring a representation in the upper left of the Kikai caldera. Taken from [START_REF] Kazahaya | Degassing process of Satsuma-Iwojima volcano, Japan: Supply of Volatile components from a deep magma chamber[END_REF].

In Italy

Italian volcanism is notoriously famous for its sustained activity and hazardousness, illustrated by Holocene eruptions like the famous eruption of Mt. Vesuvius in 79 CE, which destroyed the towns of Pompeii and Herculaneum; or more ancient high-magnitude eruptions, for example, the Campanian Ignimbrite (39280 ± 110 years BP [START_REF] Fedele | The Campanian Ignimbrite Eruption, Heinrich Event 4, and Palaeolithic Change in Europe: A High-Resolution Investigation[END_REF]) and Neapolitan Yellow Tuff (15600 ± 800 BP [START_REF] Deino | The age of the Neapolitan Yellow Tuff caldera-forming eruption (Campi Flegrei caldera -Italy) assessed by 40Ar/39Ar dating method[END_REF]), that shaped the vast Phlegrean Fields region (13 km wide), now a densely populated urban area (3.10 6 inhabitants). The monitoring of volcanic activity conducted by the the Istituto Nazionale di Geoficica e Vulcanologia (INGV) and the Protezione Civile is thus of high social importance and has pushed the development of numerous projects to better characterize the eruptive dynamic and internal structure of volcanoes. This context has given rise to muography applications on Mt. Vesuvius, Mt. Stromboli, and Mt. Etna. In the following, we describe in their main lines the different muography surveys that were or still are conducted on those three volcanoes :

• The three scintillator trackers of the MURAVES project (MUon RAdiography of VESuvius) are installed inside a protective container at the foot of the Mt.Vesuvius volcano on the southwest slope. These detectors include nowadays four panels arranged as XY matrices of triangular scintillating bars, each matrix covering a surface of 1 m 2 [START_REF] Macedonio | Muography of the Volcanic Structure of the Summit of Vesuvius, Italy[END_REF]. The detection pattern created by the combination of adjacent triangular scintillators allows for increasing the spatial resolution by weighting the amount of light detected in each bar [START_REF] Saracino | The MURAVES muon telescope: technology and expected performances[END_REF]. The 3 detectors benefit from a 60 cm-thick lead wall positioned between the third and the fourth detection panel, that acts as passive shielding to reject soft muons (with cut-off energy at 1 GeV). The data-taking campaign is still in progress. The size and configuration of the Vesuvius set though some limits on the sensitivity to density contrast. From the detection site, above a certain horizontal angle (15 • in elevation) the rock thickness becomes too large, and a feasibility limit is reached in terms of reasonable acquisition time needed to reach a minimal 1σ confidence level. The current project is thus focused on the shallow crater region, whose density resolution could reach 5% after two years of data acquisition [START_REF] Raffaello | Volcanoes in Italy and the role of muon radiography[END_REF].

• On Stromboli volcano, Tioukov et al. [START_REF] Tioukov | First muography of Stromboli volcano[END_REF] conducted a muography survey between November 2011 and March 2012 with nuclear emission plates (0.96 m 2 detection surface). They detected a density anomaly of 30 to 40% negative difference relative to bedrock in the crater terrace region, shaped by 2002-2003 and 2007 effusive eruptions. After this campaign, the focus was drawn towards a multi-approach characterization of the crater structure. The observed density distribution was compared with seismic data, and topographic change studies were obtained through high-resolution digital elevation maps provided by airborne Light Detection and Ranging (LiDAR) and ground deformation (ground-based InSAR technique). This comparative study has led to better estimating the crater terrace volume scanned by the muon detector, and refining the interpretation of muon results. The volume where the anomaly is localized could then be interpreted as describing the growth of a cone within a filling depression left by the 2007 eruption. This result illustrated the ability of muography to image the shallower portion of a volcano supply system [START_REF] Tioukov | Structure of the Shallow Supply System at Stromboli Volcano, Italy, through Integration of Muography, Digital Elevation Models, Seismicity, and Ground Deformation Data[END_REF].

• The Muography of Etna Volcano (MEV) project started in 2016 and was initially thought as a pilot project to install a permanent muographic research facility at the foot of Mount Etna [133], [START_REF] Gallo | Three years of muography at Mount Etna: results and perspectives[END_REF]. The developed scintillator detector was installed on the slope of the northeast crater and ran from August 2017 to October 2019. Even if the project was still in its R&D phase, the main result obtained through this single detector was the detection of an expanding underground cavity months before the collapse of the crater floor (December 2017). Significant variations in muon transmission map (up to 45-50% with respect to the surrounded regions assimilated to standard rock) were recorded with a 7-m spatial resolution on the scanned region. This result has sparked several possible interpretations for this anomaly: the presence of numerous fractures, and a bell-shaped cavity 200m below the surface, but the most likely is the continuous weakening of the mechanical structure through the uprising of very hot magmatic gases which eventually leads to the 2017 collapse [133]. This result obtained in the harsh terrain condition of Mt Etna (3.100 m a.s.l. for the observation site) motivates further investigation with the building of a second detector to probe the crater region from another point-of-view, and eventually the installation of a permanent monitoring system. Besides, the project has led to the development of physical background rejection tools such as a time-of-flight module to separate the signal provided by transmitted muons, from background particles entering by the rear side of the detector [START_REF] Gallo | Three years of muography at Mount Etna: results and perspectives[END_REF].

In France

The volcanic activity in France takes place in its insular overseas territories around densely populated areas, and thus carries many stakes in terms of natural hazard prevention: in Guadeloupe and Martinique in the Lesser Antilles arc, Atlantic Ocean; La Reunion and Mayotte, in the Indian Ocean. The latter saw in May 2018 the appearance of a submarine volcanic edifice that gave birth to the largest underwater eruption ever documented [START_REF] Feuillet | Birth of a large volcanic edifice offshore Mayotte via lithospherescale dyke intrusion[END_REF].

In this context, two volcano muography projects have aroused :

• The TOMUVOL project was born in 2009 to conduct a data-taking campaign on the Puy de Dôme, dormant volcano of the chaîne des Puys, in the Massif Central of France [START_REF] Cârloganu | Towards a muon radiography of the Puy de Dôme[END_REF], [START_REF] Menedeu | RPC application in muography and specific developments[END_REF]. The volcano dormancy and location made it an ideal "standard candle" for testing the imaging method [START_REF] Giammanco | Resistive Plate Chambers in Muography[END_REF]. Several campaigns of data-taking took place from January 2011 to April 2016 at three different locations (one in the South-West, two on the Eastern side). RPCs were used as muon trackers to estimate the transmitted flux. 17 million events were recorded, giving the first density maps in 2012. The latter showed structural contrasts in the summit region but suffered from irreducible background pollution in the most horizontal directions [START_REF] Cârloganu | Towards a muon radiography of the Puy de Dôme[END_REF]. Cross-comparisons with other geophysical observables were further conducted in [START_REF] Portal | Inner structure of the Puy de DÔme volcano: cross-comparison of geophysical models (ERT, gravimetry, muon imaging)[END_REF], using electrical resistivity tomography (ERT) and gravity data. Recent works explored various jointinversion approaches with muon data to perform 3-D density modeling on an unstructured volume mesh of the Puy-de-Dôme (cf. [START_REF] Peter G Lelièvre | Joint inversion methods with relative density offset correction for muon tomography and gravity data, with application to volcano imaging[END_REF] and [START_REF] Barnoud | Robust Bayesian Joint Inversion of Gravimetric and Muographic Data for the Density Imaging of the Puy de Dôme Volcano (France)[END_REF]).

• Finally, the DIAPHANE project started in 2008 was the first muography project in France to be conducted on an active volcano: La Soufrière de Guadeloupe, in the Lesser Antilles, where the volcanic hazard is at high stake. As the La Soufrière volcano is our main case of study, we provide an extended description of the former DIAPHANE and current MEGAMU project along with their main results in chapter 4.

Adapting muography to volcanic monitoring

The latter applications have illustrated the ability of muography to image density spatial variations, notably linked to the presence of a degassing magma, or magma pathway ( [START_REF] Hiroyuki | High resolution imaging in the inhomogeneous crust with cosmic-ray muon radiography: The density structure below the volcanic crater floor of Mt. Asama, Japan[END_REF], [START_REF] Tioukov | First muography of Stromboli volcano[END_REF]) and cross-check its results with other observables. Indeed, the field of volcano subsurface imaging commonly involves geophysical surveys like seismic tomography, electrical resistivity tomography, magnetotellurics, and gravimetry. Every method has its boundaries, either in terms of maximal depth reached, spatial resolution, and/or difficulty achieving good data coverage on complex topography. Muography, which also has its boundaries (cf. Sec. 3.3), balances its drawbacks with some key assets: the ability to remotely perform month-long runs of data taking in harsh terrain conditions (e.g for scintillator detectors), and provide a unique spatial resolution on structural features and geofluids subsurface dynamics in shallow volcano regions. When discussing volcanic monitoring, several main questions arise among which: can muons help in forecasting an eruptive event? Rather than seeking the truth in one unique signal, volcanic activity monitoring relies on crosscomparisons between geological, geophysical, and geochemical observables. Correlations between multiple signals originating from different sources have to be actively studied: e.g seismic activity, ground deformation, gas emission (e.g. SO 2 , H 2 S and CO 2 ).

Even if temporal density variations measured by muon detectors are not today considered as a possible eruption precursor in the same sense as e.g a volcanic Tremor could be, they still convey a unique set of information on the evolution of the internal fluid and heat dynamics in the shallow volcano subsurface. In 2009, the first time-sequential muographic images were obtained by Tanaka et al. [START_REF] Tanaka | Cosmic-ray muon imaging of magma in a conduit: Degassing process of Satsuma-Iwojima Volcano, Japan[END_REF] on Mt Asama during an unrest period that led to a magmatic explosive eruption (Vulcanian style). The latter occurred on February 2 th , 2009, offering a unique time set of muon data to analyze: one month before and after the eruption. A significant low-density region below the crater floor was previously detected by geophysical imaging methods (e.g survey by synthetic aperture radar (SAR) before and after the 2004 eruption) and later supported by muographic images [START_REF] Hiroyuki | High resolution imaging in the inhomogeneous crust with cosmic-ray muon radiography: The density structure below the volcanic crater floor of Mt. Asama, Japan[END_REF] (cf. Fig. a). This anomaly was then interpreted as the main magma pathway plugged by older deposits of higher density. From January to March 2009, no significant changes of mass were detected within this low-density region that could have been related to magma ascent. Nevertheless, a significant variation of the shape of the crater related to the disappearance of old magma deposits from the 2004 eruption was identified, as depicted in Fig. b. In June 2013, muography snapshots at a rate of 10 frames per month were obtained during an eruptive period of the Satsuma-Iwojima volcano. Those muographies showed density variations on a time scale of two days (1σ CL) caused successively by the ascent and descent of magma [START_REF] Hiroyuki | Radiographic visualization of magma dynamics in an erupting volcano[END_REF]. Taken from [START_REF] Hiroyuki | Japanese volcanoes visualized with muography[END_REF] Muographic monitoring is obviously bound to the acquisition time limit raised in Sec. La Soufrière de Guadeloupe is an active volcano, located on the Basse-Terre island in the French Lesser Antilles. With its last eruption in 1976, La Soufrière has shaped the modern history of volcanology and more globally the history of the Lesser Antilles, becoming a pivotal case of study for volcanologists and local authorities in the context of volcanic crisis management. As a reaction to this major past event, and to the current phase of unrest ongoing since 1992, La Soufrière progressively became a very-well instrumented volcano, under the surveillance of the French national observatories, in charge of monitoring wide seismological, geochemical, and meteorological networks providing information on the ongoing unrest period. This period is characterized by sustained hydrothermal activity, classed as one of the most hazardous on the planet regarding the proximity of the nearby populations to the volcanic edifice [START_REF] Siebert | [END_REF]. Not only responsible for sudden violent phreatic eruptions without juvenile material involved (the last major phreatic eruption occurred in 1976-1977), hydrothermal activity (e.g. [START_REF] Boudon | Guadeloupe[END_REF], [START_REF] Moretti | The 2018 unrest phase at La Soufrière of Guadeloupe (French West Indies) andesitic volcano: Scrutiny of a failed but prodromal phreatic eruption[END_REF], [START_REF] Moune | Gas Monitoring of Volcanic-Hydrothermal Plumes in a Tropical Environment: The Case of La Soufrière de Guadeloupe Unrest Volcano (Lesser Antilles)[END_REF]) and its resulting rock alteration processes play a major role in the destabilization of the structure, which increases the risk of slope failure and partial flank collapse [START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF][START_REF] Ball | The hydrothermal alteration of cooling lava domes[END_REF].

First of their kind in France, several muography campaigns applied to this active volcano have taken place since the early work and feasibility studies started in 2008 [131] and are still ongoing to this day. The ANR Domoscan project in 2012 followed by the ANR Diaphane granted in 2015 allowed the development and deployment of up to six muon detectors on the flanks of La Soufrière. Started in 2018, and currently ongoing at the time of writing, the ANR MegaMu is carrying out the development of a multi-physics approach to volcanic activity monitoring, combining muography with geophysical observables such as gravimetric surveys, self-potential monitoring, as well as gas temperature measurements.

Throughout this chapter, we will present the geological and geophysical context at La Soufriere and the subsequent natural hazard implications in which the different muon imaging surveys have taken part. We will give a review of the muon imaging achievements at the La Soufrière de Guadeloupe we will discuss its current state and evolution perspectives in which this thesis takes part.

4.1 La Soufrière de Guadeloupe's geological history

Formation of La Grande Découverte-Soufrière

The Guadeloupe archipelago belongs to the lesser Antilles region (cf. Fig. 4.1), which results from the subduction of the Atlantic oceanic crust under the Caribbean tectonic plate. Based on the volcanic arc geometry that shapes the Lesser Antilles, two distinct regions can be identified:

• the Eastern part, which includes the Grande-Terre, Petite-Terre and Marie-Galante islands. These islands share a similar Pleistocene reef limestone composition and belong to the oldest outer arc dated at 35 My. They do not present signs of volcanic activity.

• the Western part, which includes the Basse-Terre and Les Saintes islands. They are composed entirely of Tertiary and Quaternary volcanic rocks and result from the activity of the inner Lesser Antilles volcanic arc, active since the upper Pliocene, about 4 to 3.5 Ma ago [START_REF] Boudon | Guadeloupe[END_REF].

Figure 4.1: The lesser Antilles volcanic arc, featuring inset a detailed map of the Guadeloupe archipelago. From [START_REF] Boudon | A new scenario for the last magmatic eruption of La Soufrière of Guadeloupe (Lesser Antilles) in 1530 A.D. Evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF].

Only the Basse-Terre island hosts volcanic activity with its Grande Découverte-Soufrière (GDS) complex, which culminates at 1467 m a s l, the highest summit of the Lesser Antilles. Located in the south of the island, the GDS's complex geological history has been summed up in three main eras. The reader is invited to refer to [START_REF] Boudon | Guadeloupe[END_REF] and [START_REF] Boudon | A new scenario for the last magmatic eruption of La Soufrière of Guadeloupe (Lesser Antilles) in 1530 A.D. Evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF] for in-depth reviews on the history of these constructional steps, and/or the original papers of Boudon et al. [START_REF] Boudon | Magma and hydrothermally driven sector collapses: The 3100 and 11,500 y. B.P. eruptions of la Grande Decouverte (la Soufrière) volcano, Guadeloupe, French West Indies[END_REF], [START_REF] Georges Boudon | I.)[END_REF] to find precise information on the different dome-forming and flank-collapse eruptive events at GDS. 1. Grande Découverte phase (0,2 MA BP -42,000 BP) : This first phase accounts for the emergence of the Grande Découverte strato-volcano from a series of effusive eruptions producing andesitic lava flows reaching from 5 to 10 km away from the main vent. This resulted in the formation of an edifice with a basis of 2 km radius. During this period, the volcano went through three major explosive caldera-forming eruptions whose latest one, known as the Pintade eruption, dated at 42.000 BP, marked the end of this formation phase. This eruption left a 3 km diameter wide caldera in which the whole Basse-Terre volcanic activity had ( [START_REF] Georges Boudon | I.)[END_REF], [START_REF] Boudon | Guadeloupe[END_REF]);

2. Carmichaël phase (42,000 BP -11,500 BP): This second phase gave birth to the Carmichaël volcano within the Grande Découverte caldera, and is marked by another major eruptive event, the St-Phy eruption (26,000 BP). The composite Carmichaël volcano collapsed in the aftermath of a sustained hydrothermal activity phase between 18,000 BP and 14,000 BP, which leads to flank collapses during two eruptions dated at 13,500 BP ( [START_REF] Boudon | Guadeloupe[END_REF]) and 11,500 BP ( [START_REF] Boudon | Magma and hydrothermally driven sector collapses: The 3100 and 11,500 y. B.P. eruptions of la Grande Decouverte (la Soufrière) volcano, Guadeloupe, French West Indies[END_REF], [START_REF] Georges Boudon | I.)[END_REF]).

3. Soufrière phase (11,500 BP -present) : This last and current era was marked by a prolonged period of magmatic activity as well as periods of phreatic activity. The current Soufrière lava dome was formed in 1530 AD during the last magmatic eruption [START_REF] Boudon | A new scenario for the last magmatic eruption of La Soufrière of Guadeloupe (Lesser Antilles) in 1530 A.D. Evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF].

The edifice also went through the Amic crater formation 3100 years ago [START_REF] Boudon | Magma and hydrothermally driven sector collapses: The 3100 and 11,500 y. B.P. eruptions of la Grande Decouverte (la Soufrière) volcano, Guadeloupe, French West Indies[END_REF], [START_REF] Boudon | A new scenario for the last magmatic eruption of la soufrière of guadeloupe (lesser antilles) in 1530 a.d. evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF]. Two additional domes feature the present GDS complex, and were formed about 1700 years ago: l'Échelle and la Citerne.

Past volcanic crises

Since its last lava dome formation, la Soufrière underwent six major phreatic eruptions: in 1690, 1797 -98, 1812, 1836-37, 1956, and 1976-77 ([112], and [START_REF] Legendre | Reconstruction fine de l'histoire éruptive et scenarii éruptifs à la soufrière de Guadeloupe : vers un modèle intégré de fonctionnement du volcan[END_REF]). La Soufrière's history also features an impressive catalog of partial flank collapses at least 8 in the past 9150 years.

Collapses revealed hydrothermally altered rock material in debris avalanche deposits ( [START_REF] Boudon | Magma and hydrothermally driven sector collapses: The 3100 and 11,500 y. B.P. eruptions of la Grande Decouverte (la Soufrière) volcano, Guadeloupe, French West Indies[END_REF], [START_REF] Boudon | Guadeloupe[END_REF], [START_REF] Le | Potential Flank-Collapse of Soufrière Volcano, Guadeloupe, Lesser Antilles? Numerical Simulation and Hazards[END_REF], [START_REF] Peruzzetto | Modeling of partial dome collapse of La Soufrière of Guadeloupe volcano: implications for hazard assessment and monitoring[END_REF]). The last phreatic eruption of 1976-1977 turned into a volcanic crisis that resulted in the evacuation of 72,000 inhabitants for several months causing significant social and economic upheavals for the island [START_REF] Feuillard | The 1975-1977 crisis of la Soufriere de Guadeloupe (F.W.I): A still-born magmatic eruption[END_REF], [START_REF] Boudon | Guadeloupe[END_REF], [START_REF] Komorowski | Improving crisis decision-making at times of uncertain volcanic unrest (Guadeloupe, 1976)[END_REF], [START_REF] Hincks | Retrospective analysis of uncertain eruption precursors at La Soufrière volcano, Guadeloupe, 1975-77: volcanic hazard assessment using a Bayesian Belief Network approach[END_REF]: the total cost of the damages have been estimated as 60% of the total annual per capita Gross Domestic Product of Guadeloupe in 1976, which establishes it among the most costly eruptions of the XX th century [START_REF] Komorowski | Improving crisis decision-making at times of uncertain volcanic unrest (Guadeloupe, 1976)[END_REF].

Several interpretations of the cause of the eruption have been formulated, at different epochs of time in the few months/years following the crisis until the 2000s. They all converge towards pressurization of the hydrothermal system but differ on its origin either (a) due to a magma intrusion at shallow depth (∼ 3 km) ( [START_REF] Villemant | The memory of volcanic waters: Shallow magma degassing revealed by halogen monitoring in thermal springs of La Soufrière volcano (Guadeloupe, Lesser Antilles)[END_REF], [START_REF] Boichu | Degassing at La Soufrière de Guadeloupe volcano (Lesser Antilles) since the last eruptive crisis in 1975-77: Result of a shallow magma intrusion?[END_REF], [START_REF] Villemant | Evidence for a new shallow magma intrusion at La Soufrière of Guadeloupe (Lesser Antilles)[END_REF]), (b) a pure hydrothermal eruption as suggested in the model of Zlotnicki et al. 1992, caused by an over-pressurization of the deeper aquifers close to the main magma chamber (∼ 6 km) Using noble-gas flux data Ruzie et al. (2012) propose a supply of the existing magma reservoir without any migration involved, followed by a rupture and propagation of the pressure pulse to the surface, which could notably explain the seismicity migration observed at the time by Hirn and Michel (1979).

After the 1976-1977 eruption the global volcanic activity settled down. But in 1992, manifest signs of unrest were detected among which an increase of hydrothermal activity (e.g apparition of fumaroles, steaming ground, and hot springs) and seismic swarms (Allard et al., 1994; [START_REF] Boudon | Guadeloupe[END_REF] From this date, the volcano has experienced a ramp-up period, whose apex of fumarolic and seismic activity increase was reached in April 2018 when the largest felt volcano-tectonic earthquake since the 1976-1977 eruption (M L = 4.1 or M W = 3.7) was recorded. No evidence of upward direct magma migration was detected, and this event was interpreted as a failed phreatic eruption [START_REF] Moretti | The 2018 unrest phase at La Soufrière of Guadeloupe (French West Indies) andesitic volcano: Scrutiny of a failed but prodromal phreatic eruption[END_REF]. The current unrest phase is characterized by the continuous growth of the outgassing area, along with the appearance in July 2014 and February 2016 of new steam-dominated fumaroles with a deep magmatic gas component.

The lava dome morphology

The lava dome was formed during the 1530 CE magmatic eruption. The latter began with a partial flank collapse that triggered a phreatic explosive phase, followed by a sub-plinian explosive phase [START_REF] Boudon | Guadeloupe[END_REF] [38] [START_REF] Legendre | Reconstruction fine de l'histoire éruptive et scenarii éruptifs à la soufrière de Guadeloupe : vers un modèle intégré de fonctionnement du volcan[END_REF]. As of today, the dome presents a basis and a top plateau with respective diameters of 900 m and 400 m, with height around 300 m, for a total volume of 5.10 -2 km 3 [START_REF] Boudon | A new scenario for the last magmatic eruption of La Soufrière of Guadeloupe (Lesser Antilles) in 1530 A.D. Evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF]. It is mainly composed of basaltic-andesitic lava and pyroclastic deposits from past eruptions. Past geophysical surveys have estimated an average bulk density of 1800 kg m -3 [START_REF] Gailler | Crustal structure of Guadeloupe Islands and the Lesser Antilles Arc from a new gravity and magnetic synthesis[END_REF]. Since its formation, phreatic eruptions and regional tectonics have carved some main structural features such as radial fractures, represented in Fig. 4.3. During the 1976-1977 eruptive event, one main fault appeared on the south-east Flank, in the aftermath of the violent phreatic eruption that occurred the 30 th of August 1976. This fault nowadays hosts a muon telescope that is still recording data at the time of writing. The dome can be structured along two main regions traversed by a main fault of tectonic origin, the Ty fault [START_REF] Feuillet | Arc parallel extension and localization of volcanic complexes in Guadeloupe, Lesser Antilles: EXTENSIONAL ACTIVE FAULTING IN GUADE-LOUPE[END_REF].

• the Northern region: along which we found the crater Dupuy, followed by the Northern Fault (Fente du Nord), which hosts the entrance of one large cavity of the dome: the Spallanzani cave, whose dimensions are often casually compared to a "cathedral" based on its impressive volume 55x35x11m 3 as described in [120];

• the South-East region : where are located the 1956 and 30 th of August 1976 faults. We can also locate the Napoleon and Southern Crater regions, which host most of the present fumarolic activity. Recent phreatic and hydrothermal eruptions have brutally illustrated the hazardousness that well-developed volcanic hydrothermal systems represent. As referenced in [START_REF] Montanaro | Phreatic and Hydrothermal Eruptions: From Overlooked to Looking Over[END_REF], 30 of these eruptions happened during the last decade (2011-2021). The terms phreatic and hydrothermal can sometimes get mixed up in the literature, as they both refer to explosive non-magmatic eruptive events. According to [START_REF] Montanaro | Phreatic and Hydrothermal Eruptions: From Overlooked to Looking Over[END_REF], the difference between these two types lies in the involvement of a magma intrusion in the destabilization of the shallow volcanic aquifers. Phreatic eruptions are caused by the rapid arrival of heat and gas from intruding magma or magmatic fluids, whereas hydrothermal eruptions are caused by the flashing and expansion of hydrothermal water without the need for any magma input. These explosive events can trigger laterally-directed blasts that eject ballistic blocks. Like magmatic eruptions, they can also give rise to pyroclastic density currents as depicted in Fig. 4.5. Even if they present many similarities to a magmatic event, they remain more sudden and difficult to forecast due to a lack of universal precursor signals, which remains a pivotal research field in today's volcanology. As in many scientific fields, the focus is notably drawn towards the use of deep-learning methods (e.g random forest classifier) to manage to identify eruptive precursors (e.g volcanic Tremor [218]) that may be unnoticed without the use of modern algorithms (cf. applications at Whakaari, New Zealand [START_REF] Dempsey | Automatic precursor recognition and real-time forecasting of sudden explosive volcanic eruptions at Whakaari, New Zealand[END_REF], and La Soufrière de Guadeloupe [START_REF] Falcin | A machine-learning approach for automatic classification of volcanic seismicity at La Soufrière Volcano, Guadeloupe[END_REF]).

The Mt Ontake hydrothermal eruption that occurred on the 27 th of September 2014 passed under the radar of conventional monitoring using seismic and ground deformation signals and caused 63 fatalities among the hundreds of hikers present at the time of the eruption. Posterior studies have shown that this hydrothermal eruption was caused by a slow pressurization of the volcanic conduit by a 10-year increase in volatile elements [START_REF] Sano | Ten-year helium anomaly prior to the 2014 Mt Ontake eruption[END_REF]. A more recent eruption that took place at Whakaari/White Island, New Zealand, on the 9 th of December 2019 ( [START_REF] Dempsey | Automatic precursor recognition and real-time forecasting of sudden explosive volcanic eruptions at Whakaari, New Zealand[END_REF]; [START_REF] Burton | Insights into the 9 December 2019 eruption of Whakaari/White Island from analysis of TROPOMI SO2 imagery[END_REF]) is also emblematic of the suddenness of this phenomenon: pyroclastic density currents produced after the rise of a 3-4 km-high ash plume killed 21 people and wounded 26. The explosion occurred 40 min after a brutal increase in seismicity and SO 2 flux [START_REF] Montanaro | Phreatic and Hydrothermal Eruptions: From Overlooked to Looking Over[END_REF].

At La Soufrière

Six eruptions of phreatic type have been identified since the European colonization of the island in 1630 AD. Hydrothermal activity rises from the interaction between meteoric water (or rainwater) infiltration, and in-depth magma heat and fluid fluxes. As the temperature of the hydrothermal fluids increases, so decreases their density. As a result, through fractures and faults, fluids can reach the surface in the form of hot springs, or circulate inside the superficial part of the volcanic edifice. This hot and acid fluid circulation involves several physicochemical processes that impact the mechanical stability of the whole edifice.

Constant efforts are made to better characterize the geometry and dynamics of the La Soufrière hydrothermal system, its connections to the main structural features of the edifice [START_REF] Brothelande | Fluid circulation pattern inside La Soufrière volcano (Guadeloupe) inferred from combined electrical resistivity tomography, self-potential, soil temperature and diffuse degassing measurements[END_REF], such as its large radial fractures formed during past eruptive events, represented in Fig. 4.3. The main stake lies in the fine understanding of the flank destabilization and risk of partial collapse due to hydrothermal fluid circulation and pressurization [START_REF] Peruzzetto | Modeling of partial dome collapse of La Soufrière of Guadeloupe volcano: implications for hazard assessment and monitoring[END_REF]. Indeed in addition to triggering explosive eruptions, hydrothermal dynamics involves physicochemical processes, such as rock alteration, responsible for creating mechanically weakened zones inside the lava dome.

Risk of partial flank collapse

As mentioned above, La Soufriere owns an impressive record of past flank collapses. Flank instabilities can have multiple sources: magma or hydrothermal pressurization, intense volcanic seismicity, external earthquake (e.g. Ontake volcano, 1984, Japan), or heavy rainfalls (e.g. Casita volcano, 1998, Nicaragua). The current unrest phase represents a serious hazard in terms of small-volume partial dome collapses for the nearby population in the Eastern and Northern of the closest town Saint-Claude, located 5 km south of the volcano, with a population of 10 000 inhabitants. Particularly when magma bodies are involved like for La Soufrière at depths ranging from 6 to 7 km, flank collapse episodes have the potential to depressurize the volcanic plumbing and hydrothermal systems and to produce lateral explosions, so-called blasts. Such explosions can cause damage across a far larger region than the avalanche of debris they cause. At La Soufrière de Guadeloupe, the 3100 BP [START_REF] Boudon | Flank failure-directed blast eruption at Soufriere, Guadeloupe, French West Indies: A 3,000-yr-old Mt. St. Helens?[END_REF] event was estimated to have covered an area superior to 60 km 2 , hence ravaging the southern region of the Basse-Terre island [START_REF] Le | Potential Flank-Collapse of Soufrière Volcano, Guadeloupe, Lesser Antilles? Numerical Simulation and Hazards[END_REF]. Current research aims thus to input that information into numerical modeling and improve the likelihood calculation of the different eruptive/collapse scenarios. Simulations have shown that the impact of a major dome collapse in the worst-case scenario would be much more widespread over the Basse-Terre island region ( [START_REF] Le | Potential Flank-Collapse of Soufrière Volcano, Guadeloupe, Lesser Antilles? Numerical Simulation and Hazards[END_REF], [START_REF] Peruzzetto | Modeling of partial dome collapse of La Soufrière of Guadeloupe volcano: implications for hazard assessment and monitoring[END_REF]).

Rock alteration

Among the understudied physicochemical processes to draw the causal links between the hydrothermal activity and the repetitive flank collapse events in active volcanoes, rock alteration is recognized to be of major importance [START_REF] Ball | The hydrothermal alteration of cooling lava domes[END_REF]. Caused by the persistent flow of hot acid fluids through the rock matrix, rock alteration can be defined as the fraction of secondary minerals (e.g. smectite clays, kaolinite, and alunite) found in a volcanic rock [134]. Indeed, hydrothermal alteration can significantly modify the physical and mechanical properties of the rock, e.g density, strength, porosity, Young's modulus). To study the rock failure mechanism, laboratory tests conducted by geologists involve compression tests to measure the aforementioned quantities. The most common test is a uniaxial compression test used to assess Poisson's ratio, Young's modulus, and uniaxial compressive strength (σ p in Pa), i.e the maximum axial compressive stress that a right-cylindrical rock sample can sustain before macroscopic failure [START_REF] Heap | The mechanical behaviour and failure modes of volcanic rocks: a review[END_REF], defined as σ p = P max /A, where P max is the maximum load applied on the rock sample measured, and A is the area of the cross-section of the cylindrical sample.

Heap et al. [START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF] studied the rock mechanical strength and its degree of alteration at La Soufrière. They conducted measurements over a collection of variably-altered blocks sampled at different locations on the lava dome to determine the range of porosity and hydrothermal alteration in the dome. The results show that uniaxial compressive strength decreases as a function of increasing hydrothermal alteration. They also point out a correlation between porosity and alteration but the causal relation between the two remains difficult to disentangle, as high porosity provides a higher surface for rock-fluid interaction. Besides laboratory tests, rock alteration at the field scale can be studied through geophysical methods that measure proxies to this parameter. For example, a higher electrical conductivity is associated with higher clay content. In the case of muography, density is directly linked to porosity, which is a proxy for rock strength and alteration as just mentioned.

In Fig. 4.6, we show the empirical power-law relation inferred by Heap et al. relating the measurements of uniaxial compressive strength σ p and bulk density ρ b at La Soufrière. The latter is linked to rock mineral, fluid content, and porosity, as defined through the following formula [START_REF] Zhang | Multi-turn track fitting for the COMET experiment[END_REF]:

ρ b = (1 -ϕ)ρ m + ϕρ f (4.1)
where ρ m is the matrix or mineral density; ϕ is the porosity of the rock; and ρ f is the fluid density in the rock (e.g meteoric water, a mixture of hydrothermal-magmatic gases).

Chapter 4

To best represent the natural state of the lava dome, the assumption was to consider the bulk density measured on water-saturated samples, referred to as ρ W , rather than dry samples. This empiric relation allowed deriving a first approximation rock strength model for the La Soufrière lava dome using density estimates from gravimetry and muon data joint-inversion, performed by [START_REF] Rosas-Carbajal | Three-dimensional density structure of La Soufrière de Guadeloupe lava dome from simultaneous muon radiographies and gravity data[END_REF]. This model notably highlighted the presence of a very low strength region (<5 MPa), in the southern flank of the volcano where the main surface signs of hydrothermal activity locate. This low-strength region is also coincidental with the high conductivity anomaly presented in [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF] (cf. Sec. 4.3.4). This work highlights the importance of associating laboratory measurements performed on in situ rocks with three-dimensional geophysical models to study flank collapse hazards.

Volcanic geophysical monitoring 4.3.1 A well-instrumented volcano

In the aftermath of the 1976-1977 crisis, volcanic activity monitoring has dramatically improved, including continuous seismic and global navigation satellite system (GNSS) monitoring, gas plume monitoring, meteorological networks, periodic extensometers surveys of lava dome fractures, regular fumarole gas and thermal spring sampling and thermal monitoring. All those observables, which are potential precursors of an eruptive event, are mon-itored by the volcanological and seismological observatory of Guadeloupe (OVSG) which is part of the French observatory network operated by the Institut de Physique de Globe de Paris. The different developments done in the surveillance network to extend the data coverage over the whole GDS complex, along with improvements in analysis and simulation tools (e.g probabilistic tools like Bayesian-Belief Network [START_REF] Hincks | Retrospective analysis of uncertain eruption precursors at La Soufrière volcano, Guadeloupe, 1975-77: volcanic hazard assessment using a Bayesian Belief Network approach[END_REF] to provide decision support) have turned La Soufrière into an ideal natural laboratory to improve knowledge on the geological history, internal dynamics inside, and the associated hazards of an archetype of an active andesitic volcano. Besides, the relatively easy access to the dome (roughly 40 min by feet from its basis) is another advantage of this natural site to conduct field campaigns and test novel methods.

Seismic monitoring

Recurrent clusters of seismic events occurring over a short time span of a few days to a few weeks, referred to as seismic swarms are recorded by the seismic network monitored by the OVSG (cf. Fig. 4.7). The latter releases a report every month to the general public and local authorities. Most of the swarms occur at shallow depths between 300 m to 1 km and are not felt by the surrounding population, as they remain at a magnitude M w ≤ 1. The OVSG monitors a fleet of 27 seismic stations on the Basse-Terre island (network code GL), including 7 stations in the direct vicinity of the dome as shown in Fig. 4.7, close enough to detect the low energy released during the volcano-tectonic (VT) earthquakes.

As mentioned in Sec. 4.2.1, to this day unambiguous and measurable precursors of phreatic or hydrothermal eruptions have not yet been identified. In order to refine the characterization of the hydrothermal system dynamics, dedicated studies of this shallow seismo-volcanic activity provide important hints. Extension of the microseismicity towards the surface has indeed been detected after the April 2018 still-born eruption, interpreted as the subvertical main fracture extension at ∼1 km depth, and possibly new fracture opening. This investigation was conducted thanks to the data recorded by a dense, temporary array of 65 geophones deployed from November 13, 2017, to January 20, 2018, above the dome high-conductivity anomaly identified in [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF] (cf. Sec. 4.3.4). Studies are ongoing to investigate the current evolution of the activity triggering recurrent seismic swarms, and moreover their link to the hydrology inside La Soufrière dome, and potential destabilization of the hydrothermal system (Pantobe et al., 2022 in rev.).

Ground Deformation monitoring

Ground deformations are key measurements recorded by observatories, which can be thoroughly studied over decades-long periods of time to monitor volcano dynamics. Indeed, along with the seismic signal, they constitute the two essential geophysical tools for monitoring volcanoes [START_REF] Scarpa | Monitoring and Mitigation of Volcano Hazards[END_REF]. The geodetic data sets can contain precursors for an upcoming eruption relating to magma intrusion through the supply system, which manifests at the surface through volcano inflation e.g [START_REF] Cayol | Monitoring Volcano Deformation[END_REF]. Moreover, they also can be studied to gain insights on hydrothermally-driven processes, providing valuable inputs for parameter estimation of the source of unrest (e.g. its location, depth, volume), e.g [START_REF] Fournier | Understanding volcano hydrothermal unrest from geodetic observations: Insights from numerical modeling and application to White Island volcano, New Zealand[END_REF], [START_REF] Moretti | The 2018 unrest phase at La Soufrière of Guadeloupe (French West Indies) andesitic volcano: Scrutiny of a failed but prodromal phreatic eruption[END_REF].

There exist various methods to monitor ground deformation, from ground-based surveys like Electronic Distance Measurements, Electronic Tiltmeters, and Global Navigation Satellite System (GNSS), to remote methods such as deformation images from Interferometric Synthetic Aperture Radar (InSAR). Among these, GNSS is established as one of the most popular methods [START_REF] Battaglia | Monitoring Volcanic Deformation[END_REF]. A network of ground-based antennae allows tracking signals transmitted from a constellation of satellites to provide an absolute location at a given time. Permanent GNSS stations can characterize displacements at the mm level.

La Soufrière is equipped with a permanent GNSS network composed of nine stations (cf. Figs 4.7 and 4.8) allowing continuous ground deformation monitoring. These data sets are completed with reiterated campaign points displayed in Fig. 4.8. This whole network has allowed since 2015 to reach an estimated sensitivity in volume change detection ∆V of the order of 1.10 5 m 3 at 4-5 km depth [START_REF] Moretti | The 2018 unrest phase at La Soufrière of Guadeloupe (French West Indies) andesitic volcano: Scrutiny of a failed but prodromal phreatic eruption[END_REF]. In Fig. 4.8, the horizontal velocity field inferred from permanent stations and campaign measurements are plotted for the 1995-2018 period, both in the La Soufrière vicinity and in the southern region of Basse-Terre. On the summit of the lava dome, the distribution of velocities can be considered at first approximation radial centered between the Tarissan crater and Napoléon crater. At the scale of the La Soufrière dome, the displacement rate remains relatively small (3 to 7 mm/yr), except in the southwestern region (represented as a yellow-dashed area) where the displacement rate gradient reaches up to 9 mm/yr.

Geophysical imaging

To understand the extent and evolution of the physicochemical processes and the structural features like faults or detachment planes that threaten the volcano stability, earth subsurface imaging methods such as seismic tomography, gravimetry surveys, electrical resistivity tomography, and last but not least muon tomography, are of high value. We here summarize some important applications to La Soufrière volcano.

Gravimetry

Gravity acceleration can provide information on the inner mass distribution of volcanoes. Mapping out variations in acceleration from measurements at the surface of volcanoes and calderas reveal information on their internal structure. As for ground deformation signal, monitoring variations provides information on the displacement of mass associated with volcanic unrest [START_REF] Carbone | Review of Microgravity Observations at Mt. Etna: A Powerful Tool to Monitor and Study Active Volcanoes[END_REF], [START_REF] Van Camp | Geophysics From Terrestrial Time-Variable Gravity Measurements[END_REF], [START_REF] Okubo | Advances in gravity analyses for studying volcanoes and earthquakes[END_REF]. Gravimetry thus consists of retrieving variations linked to mass distribution inhomogeneities and identifying anomalies. A gravity anomaly is defined as a deviation of the observed acceleration from reference gravity acceleration, so-called normal gravity, which corresponds to the global trend of the Earth's gravity field. A positive anomaly is linked to a mass excess, while a negative anomaly is caused by a mass deficit. As variations are very small (0.5% of g), a common unit used in geophysics is the mGal (10 -3 cm s -2 ). In order to measure gravity acceleration, two generic classes of instruments, exist: 1) absolute gravimeters, which measure the acceleration of a test mass during free-fall in a vacuum chamber (thus suited for laboratory conditions); 2) relative or spring-type gravimeters, mostly used on the field which measure the tiny variation ∆g thanks the displacement x of the test mass m tied to spring with constant k (Hooke's law: m∆g = k∆x). Absolute and relative gravimeters determine the magnitude of the gravity acceleration along the local plumb line with errors ranging from 1 to 10 µGal. The aforementioned deviation from normal gravity is denoted ∆g f ree-air and can be decomposed as such [START_REF] Nishiyama | Joint Inversion of Muography and Gravity Data for 3D Density Imaging of Volcanoes[END_REF]:

∆g free-air = ∆g target + ∆g terrain + ∆g trend (4.2)
where ∆g target is the gravitational effect of interest, i.e from the masses within the target volume that we wish to retrieve density from inversion analysis, ∆g terrain is the effect of topographic mass outside the target volume and above the geoid surface (reference gravity equipotential surface at the given observation point), ∆g trend is the regional trend which originates from the deeper parts of the Earth such as lithology, tectonics, and mantle dynamics. The latter two terms must be properly understood and removed to extract the first one of interest.

One main experimental challenge with gravimetry lies in achieving satisfying data coverage during a measurement campaign over a system with complex topography such as a volcano, needed to perform 3-D density modeling. At La Soufrière, gravimetric data led to three-dimensional density models shown in [START_REF] Barnoud | 3-D linear inversion of gravity data: method and application to Basse-Terre volcanic island, Guadeloupe, Lesser Antilles[END_REF], following a methodology compatible with seismic tomography data, hence favoring joint inversion.

Seismic tomography

Seismic tomography consists in reconstructing the spatial distribution of acoustic wave parameters, e.g longitudinal wave (P-wave), and shear wave (S-wave) velocities, or their ratio V P /V S , to obtain information about the structures, the content of fluids and melts in the traversed mediums. These waves can either originate from natural sources (earthquakes, seismic noise), so-called passive sources, or human-made active sources (hence with known epicenter). The analysis of their propagation is also valuable to obtain information on anisotropies in the traversed mediums related to changes in the stress and deformation fields, and migration of fluids, reaching up to depths where the main magma sources are located [START_REF] Koulakov | Seismic Tomography of Volcanoes[END_REF].

At La Soufrière de Guadeloupe, the overall low seismicity level makes it challenging to apply such a tomography method according to Coutant et al. [START_REF] Coutant | Joint inversion of P wave velocity and density, application to La Soufrière of Guadeloupe hydrothermal system[END_REF]. The latter have managed to overcome the problem of low resolution by using active sources and performing a joint gravity/travel time inversion to obtain density and P-wave velocity images of La Soufrière hydrothermal system. They exhibit the existence of dense bodies, interpreted as andesite spines from 3100 B.P. or 1530 A.D eruptions, impacting the stability of the edifice.

Electrical resistivity imaging

Electrical resistivity tomography (ERT) can be used to image the presence of fluids and their motion in the subsurface by detecting conductivity anomalies associated with the presence of fluid-saturated rock. Indeed, fluids can significantly boost the electrical conductivity of the medium by several orders of magnitude, compared to the dry host rock matrix [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF]. The bulk electrical conductivity of a rock depends strongly on the rock porosity, the pore connectivity, and the saturating fluid conductivity, which in turn varies with temperature and salinity. Therefore with this ERT method, it is possible to map out structural features (e.g faults, craters) along which the most permeable and porous rock are found, and identify the main pathways for fluid circulation.

The principle consists of ERT in directly injecting a current through a couple of current electrodes plugged in the soil, and measuring the voltage V between another pair of potential electrodes, to measure soil conductivity σ in S/m) (or resistivity in Ω m). The method's complexity increases when multi-electrode arrays are built to infer a three-dimensional resistivity model of the ground through data inversion.

Using ERT data collected during different surveys between 2003 and 2011 on the Soufrière dome, Rosas-Carbajal et al. [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF] studied the structure and dynamics of the GDS hydrothermal system. They inferred through an inversion method introduced by [103], a threedimensional conductivity model of the lava dome (cf. Fig. 4.9). This model highlights the presence of significant conductivity gradients ranging from 10 -4 S/m for the resistive bulge region composed of non-altered and dense rock to 1 S/m for the most-conductive regions connecting the southern hot thermal springs at the dome basis, to the dome summit below the South Crater region (CS). The model is divided in two sub-regions delimited by iso-conductivity contours: a less conductive "A1" sub-region (σ ∈ [0.1, 1] S/m), that contours a highly conductive "A2" region (σ > 1 S/m) closer to the vertical conduit and following the main pathway of chlorine-rich gases ascent from the deep magmatic reservoir. The electrical conductivity decreases between those two sub-regions from A2 to A1 and is inputted to lower temperatures (30 to 100°) that result from the magmatic fluids mixing with meteoric water. Although the conductivity is lower in A1, the latter still presents signs of sustained hydrothermal activity, illustrated by the presence at the surface of Galion thermal springs of hydrothermally altered rocks. The smaller sub-region labeled "D" is interpreted as a remnant of past hydrothermal activity, as it does not present current surface manifestation.

MYGALE project

A current campaign to study how rock alteration is caused, distributed, and how it affects volcano stability is ongoing within the MYGALE project started in 2022. The latter is a multidisciplinary and collaborative project, which combines laboratory measurements, geophysical and geochemical techniques, and numerical modeling. Laboratory experiments are expected to provide physical property data for degrees and timescales of alteration, while time-lapse field geophysics such as electrical resistivity tomography (ERT) measurements will map the extent and evolution of alteration over the following 4 years. Large-scale 4D Figure 4.9: Three-dimensional representation of the lava dome electrical conductivity expressed in S/m as a blue-to-red color scale. The slices are oriented south-north and eastwest when viewed from the southeast. Reddish areas indicate more conductive material, which translates to more liquid and higher temperatures. Taken from [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF]. numerical modeling will assess the influence of alteration on volcano stability by combining the geophysical and upscaled laboratory data. At the time of the writing of this work, the first campaign has been carried out in May 2022 on the La Soufrière dome summit and basis. A substantial quantity of data has been acquired along 100 m-long electrodes transects, set as an XY grid pattern overlapping the zone of transition between the degassing area with intense fumarolic activity and (including the Southern Crater region) and the zone with reduced degassing where persists the vegetation.

Muography survey at La Soufrière

As introduced in chapter 3, muography represents a rather novel tool in the scope of modern volcanology compared to established geophysical imaging methods. Through more than two decades of research, development, and application, it has though demonstrated at some pivotal levels its complementarity on volcanic systems with other geological and geophysical results, its ability to provide information on the density distribution of edifices, and in some cases perform monitoring over large regions of subsurface fluid dynamics (cf. Sec. 3.6 and 3.6.2).

As mentioned in the introduction, the muography of La Soufrière developed in the framework of the Diaphane1 project, a French inter-disciplinary collaboration started in 2009 that involves physicists of Institut de Physique des deux infinis (IP2I), Lyon and geophysicists of Géosciences Rennes and Institut de Physique du Globe de Paris (IPGP).

The La Soufrière muon survey became one of the largest in the world in terms of detectors deployed on the field: in 2015 four telescopes were already assembled and recording data. Their locations are depicted with their respective acceptance cones in Fig. 4.10. Since 2015, two more locations were investigated: one in the North Fault (Fente du Nord), and one in the South-West region (Savane à mulets). In the following sections, we will present the design of these detectors, along with their optoelectronics readout system. We will end this part by reviewing the main results obtained during the different phases of the muography survey at La Soufrière.

The muon telescope

The muon telescopes are robust particle trackers specifically designed to meet all the constraints imposed by tropical climate conditions at La Soufrière de Guadeloupe (e.g tolerance to humidity, low energy consumption, remote access.

Layout

As a particle tracker, the goal of a muon telescope is to reconstruct the incident muon trajectory (or track) using the small amounts of deposited energy (so-called hits) that fire detection matrix pixels along its path, without stopping the particle. Plastic scintillators were chosen as detection technology to face the harsh terrain conditions, and to match cost and ease-of-assembly requirements, as exposed in Sec. 3.4.3. The telescope design shown in Fig. 4.11 is conceptually simple: it consists of an assembly of multiple coaxial, equidistant, and parallel detection panels (at least three). The panels are conceived as independent, modular and interchangeable detection units that can be easily put together during detector montage on site or replaced during maintenance operations. They are composed of two planes plastic scintillator bars, placed in orthogonal directions to provide a squared XY pixel matrix. The use of at least three matrices allows us to discard fake matrix coincidences, that would mimic a muon event. Since 2019, a telescope configuration with an extra detection panel is operated. Not only does it decrease even more the probability of fake coincidences this new configuration also improves the spatial resolution on the inner dome investigated region providing a "zoomed" muography. The 3-panel and 4-panel configurations respectively provide angular resolutions of 12 and 8 mrad. In addition to this detection panel layout, the telescope's mechanical structure hosts a 7-cm thick shielding panel (one 5-cm thick lead layer, plus 2-cm thick steel layer) with the same dimensions as the detection panels for background rejection. This passive shielding is positioned behind the middle matrix.

Autonomy

As well as being designed to face tropical volcanic field conditions, telescopes have been conceived to be autonomous and remotely operated over year-long periods of time. Hence, attention has been brought to their global electrical power consumption, whose supply is provided by solar panels. The total power of an operating three-panel telescope is estimated to reach 50 W [START_REF] Jourde | Un nouvel outil pour mieux comprendre les systemes volcaniques : la tomographie par muons, application à la Soufrière de Guadeloupe[END_REF]. The ability to remotely perform operations like data transmission, acquisition interruption and relaunch, software updates, and trigger tuning, was also a pivotal requirement and enabled thanks to a Wi-Fi link that connects them to the OVSG network [START_REF] Lesparre | Design and operation of a field telescope for cosmic ray geophysical tomography[END_REF]. Their assembly relies on a circular basis to easily set the telescope azimuth. The horizontality is ensured thanks to adjustable scaffolding feet. The mechanical structure is designed to additionally host a shielding panel on its central plane (not depicted here). The control box (in blue) that hosts the onboard computer is positioned between the middle and rear matrices. Taken from [START_REF] Marteau | Development of Scintillator-Based Muon Detectors for Muography[END_REF].

Photo-detection

The scintillator bars are 80-cm long and made of polystyrene with adjuvants like PPO (2,5-diphenyloxazole) and POPOP (1,4-bis(5-phenyloxazole-2-yl)benzene) [START_REF] Marteau | Development of Scintillator-Based Muon Detectors for Muography[END_REF]. They are coated with a reflective paint (TiO2) to prevent scintillation signal contamination. To collect and guide the UV scintillation photons, each bar has one embedded wave-length shifting (WLS) fiber (cf Fig. 3.9) of 1-1.5 mm diameter, to re-emit the light in the green, for which the maximum quantum efficiency2 of the photo-cathode is reached.

To mitigate light attenuation along fibers, and gain mechanical flexibility and resistance, a transition from WLS fiber to a clear fiber is made before reaching the photo-detection system. Clear fibers thus end on a multi-anode photo-multiplier (MAPM), in which light-toelectronic signal conversion and amplification are performed. A schematic view is depicted in Fig. 4.12. The current model of MAPM is a Hamamatsu 8804-300 featuring an upgraded bialkali photo-cathode. An MAPM features 8×8 channels, each one optically connected to a WLS fiber end. One MAPM can be used for the optical readout of either one or two panels, depending on the number of scintillator bars involved.

Along one channel, charge emission from the photo-cathode (photo-electrons), multi-plication, and collection proceeds as follows [START_REF]Hamamatsu Photonics Photomultiplier Tubes Handbook[END_REF]:

1. The photocathode receives light passing through the glass window and emits electrons by photoelectric effect (photoelectrons) in the tube vacuum.

2. Photoelectrons are accelerated and focused by the focusing electrode and impinge on the first dynode where they are multiplied by means of secondary electron emission.

3. The secondary electron emission is repeated at each dynode in the electron multiplier.

4. A cluster of secondary electrons emitted from the last dynode are finally multiplied up to 10 6 to 10 7 times and are extracted from the anode. This multiplication factor is called the gain. For the current MAPM, the gain is of the order of 3.10 6 at nominal high-voltage.

Among the phenomenons that affect the light readout by the MAPM at different levels, noticeable effects are: Thermionic-electron emission Also called "dark current". It is the current associated with the random emission of photo-electrons from the heated photo-cathode or dynodes.

Crosstalk MAPM are subject to correlated noise such as "crosstalk" among neighboring channels, which limits how accurately the light (signal) incident on a certain position of the photocathode is detected while still retaining the position information. In photomultiplier tube operation, crosstalk is mainly caused by the broadening of the electron flow when light is converted into electrons and those electrons are multiplied by the dynode section. The incident light spread within the faceplate is another probable cause of crosstalk. For telescope matrix MAPM, the crosstalk contribution on the readout signal is estimated to ∼ 2% [START_REF] Gibert | Bayesian dual inversion of experimental telescope acceptance and integrated flux for geophysical muon tomography[END_REF].

Spread The cluster of secondary electrons has a temporal width of typically 1 ns. Hence the collection is ultimately limited by the spread in charges arrival time at the anode. Such a spread is caused by the slightly different trajectories the produced charges take to reach the anode from the first dynode. This spread is expected to cause a factor 1:3 dispersion on the pixel gains [START_REF] Gibert | Bayesian dual inversion of experimental telescope acceptance and integrated flux for geophysical muon tomography[END_REF].

Data Acquisition system

The readout system of the telescope shown in Fig. 4.14 has been thoroughly described in [131], [START_REF] Gibert | Bayesian dual inversion of experimental telescope acceptance and integrated flux for geophysical muon tomography[END_REF], [START_REF] Marteau | The OPERA global readout and GPS distribution system[END_REF]. We present here its structure and roles of its principal components in the main guidelines and refer the reader to these references for further details. The data acquisition (DAQ) system has been designed to perform signal pre-processing, event building, and data transfer to the outside world, following the concept of "Ethernet capable smart autonomous sensors" [START_REF] Marteau | The OPERA global readout and GPS distribution system[END_REF]. Each detection panel (i.e a sensor) has its autonomous readout electronics and represents a node on this Ethernet network, centered on an onboard eventbuilding computer. The latter is protected in a robust, opaque, and waterproof "control box", generally positioned behind the middle matrix (cf. Fig. 4.11).

On a detection panel, the readout electronic layout, schematized in Fig. 4.13 is shared between a Front-End board (FE) and a motherboard that hosts three modules:

• the power unit required to apply the high-voltage (HV) to the MAPM, preventing the need for an external HV line,

• the Ethernet controller: the core of the readout system,

• the clock system to synchronize all sensors with a common clock (Master Clock) whose set the basic timing resolution to 10 ns [START_REF] Marteau | Implementation of sub-nanosecond time-to-digital convertor in field-programmable gate array: applications to time-of-flight analysis in muon radiography[END_REF].

The MAPM output signal is thus first readout on the FE by two multi-channel chips, one for each direction X and Y. Several logical operations are performed including moderate amplification applied to correct individually the gain in each MAPM channel from signal spreading. The latter are preliminary calibrated thanks to a LED pulse system. The FE chip design corresponds to an ASICS (Aplication-Specific Integrated Circuit) originally developed at LAL (Laboratoire de l'Accélérateur Linéaire, Orsay, France) for the OPERA experiment. The FE connections to the motherboard are made through two 26-pin flat cables that host the core of the readout system: the Ethernet controller module (ECM). On the latter is located: an FPGA (Field-Programmable Gateway) that controls the FE readout, an external FIFO (i.e a data buffer system), and a microprocessor with an Ethernet interface. The signal is digitized by an Analog-to-digital converter (ADC), and transmitted to the FIFO thanks to the microprocessor. The buffered data in the FIFO are accessed by the microprocessor and transferred every second via the ECM switch to an onboard computer. The latter further communicates to the outside world and transfers data to the OVSG server through the WiFi antenna.

The DAQ trigger follows a dual level scheme:

• L0 (level 0) directly applied on the individual plane sensors required 1 channel above 1 photo-electrons (p.e) and 1 channel above zero suppression set at 0.2 p.e, that is 1 XY coincidence required on the panel in a time of 10 ns,

• L1 (level 1): 2 L0 triggers in time coincidence within a typical 100 ns gate. This trigger is applied online by the DAQ onboard computer.

Event-building is thus performed by looking at signal coincidences between all detection matrices and consists in gathering simultaneous hits on matrices with their respective timestamps within the L1 time window. In the end, each event is a timestamped (10 ns precision) collection of the triggered matrix signals. Each matrix signal is composed of all the hit intensities (in ADC counts) and channels, associated with the X and Y coordinates of the fired pixels.

!ht Figure 4.14: Photo of the telescope readout system designed in IP2I, Lyon. The MAPM is connected to the Front-End (FE) board that features two chips, one for each direction X and Y, for trigger generation and signal readout. The motherboard contains the Ethernet Controller Module (ECM), the High Voltage (HV), and the LED pulse system for calibration.

Taken from [START_REF] Gibert | Bayesian dual inversion of experimental telescope acceptance and integrated flux for geophysical muon tomography[END_REF].

From event counts in telescope to mean density estimates

After operating the telescope in tomography mode on a given period of time and collecting large datasets through signal pre-processing, and event building granted by the previously described DAQ system (cf. Sec. 4.4.2), how is the raw event count converted into mean density estimates?

In this section, we will present the generic methodology that we follow to convert the muon counts recorded in each detector line of sight, to mean density values. The resulting two-dimensional muon radiographies can be either interpreted as such or treated as inputs for further three-dimensional modeling (inverse problem solving). The overall scheme is depicted in Fig. 4.15, and quantities of interest are discussed in the following lines. We will not address here the event reconstruction and muon selection step that precedes transmitted flux estimation (cf. chapter 5).

Experimental acceptance T exp

We have seen in Sec. 3.2.3 that the telescope has an intrinsic angular efficiency that depends on its geometrical layout and impacts the number of muons recorded in each line of sight. This is quantified by a function called acceptance T geo which depends on the geometrical parameters of the telescope, which can be analytically computed for simple detector geometry (cf. [START_REF] Sullivan | Geometric factor and directional response of single and multi-element particle telescopes[END_REF], [START_REF] Thomas | Analytical derivation of the geometric factor of a particle detector having circular or rectangular geometry[END_REF]). However, correcting only for geometrical effects is not sufficient to estimate the transmitted muon flux. Indeed, an estimation of instrumental detection effects (e.g a defective optical coupling between a fiber end and the associated MAPM channel), reconstruction efficiency linked to the tracking algorithm, and muon-candidate selection through further processed data filtering. For this purpose, as introduced in Sec. 3.2.3, we proceed with the estimation of a global experimental acceptance T exp during an open-sky calibration run, taken with telescope central axis oriented at zenith θ = 0 • .

T exp i = N cal i ∆T cal × I os (θ i ) (4.3)
where the index i stands for a given line of sight of zenith angle θ i . N cal i is the number of events recorded during calibration, ∆T cal is the calibration run duration. I os (θ i ) is the open-sky integrated muon flux over E ∈ [0, +∞[ of a reference muon spectrum Φ(θ i , E) parameterized by an analytic model (cf. Sec. 1.4) or estimated through Monte-Carlo (MC) simulation, such as CORSIKA [START_REF] Heck | CORSIKA: A Monte Carlo Code to Simulate Extensive Air Showers[END_REF] (cf. chapter 1, Sec. 1.4.6). The latter option is privileged to infer the most realistic muon flux at the observation site. Thorough studies on the parameterization of such simulation and the influence of geomagnetic and atmospheric effects were for example led by [START_REF] Cohu | Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications[END_REF].

Transmitted flux I

After the T exp estimation, the transmitted integrated muon flux is estimated using Eq.3.15, which gives along the i-th line of sight:

I i = N tomo i ∆T tomo × T exp i (4.4)
where N tomo i is the number of detected events, ∆T tomo is the tomography acquisition duration, and T exp i the experimental acceptance in i.

Opacity ϱ

The opacity estimation implies a comparison between the estimated transmitted flux and an expected transmitted flux derived over a given range of opacity (typically for volcanoes, ϱ ∈ [5.10 2 ; 3.10 3 ] mwe), in a given medium (e.g standard rock SR), and for each vertical angle θ so that opacity values can be interpolated through the obtained relation I(ϱ) SR .

Mean density ρ

Once the opacity is estimated from data-model/MC flux comparison, a mean density value can be obtained for each line-of-sight:

ρ i = ϱ i /L i (4.5)
where L i is the travel length (or rock thickness) computed along the i-th line of sight thanks to the dome terrain model, as shown in Fig. 4.4. We show in Fig. 4.16, the resulting two mean density images obtained during the field campaign of 2011 from the Southern and Eastern sites around the La Soufrière dome [START_REF] Lesparre | Density muon radiography of La Soufrière of Guadeloupe volcano: comparison with geological, electrical resistivity and gravity data[END_REF], presented in the following section Sec. 4.4.4.

Achievements on La Soufrière

After having demonstrated the feasibility of such a survey applied to the La Soufrière lava dome and the telescope performances notably through test studies led in Monte-Terri underground laboratory (cf.

[131], [START_REF] Gibert | Bayesian dual inversion of experimental telescope acceptance and integrated flux for geophysical muon tomography[END_REF]). The first telescope version developed in 2009 and sent to the field included only two detection panels which made it exposed to a significant rate of fake coincidences with respect to the transmitted muons signal. The first muographies of La Soufrière were acquired in 2011 deploying two 3-panel telescopes, assembled in the South (Ravine Sud) and East (Roche Fendue) of the lava dome base (cf. Fig. 4.16).

In the following years, further upgrades on the mechanical structure were conducted, reducing the overall weight, improving the stability, and the mechanical strength in order to host a shielding panel of volume similar to a detection panel, composed of one 5 cm-thick lead layer plus 2-cm thick steel layer, to be positioned behind the middle detection panel. In May 2015, four upgraded 3-panel telescopes were sent to Guadeloupe. Three of them - Nain Jaune (NJ), Géant Vert (GV), Baron Rouge (BR) -shared a similar structure, shown in Fig. 4.11, three panels containing 16 × 16 scintillators XY matrices covering each a detection surface of 80 × 80cm 2 , separated by a 70 cm distance. These three were assembled on the flanks of the La Soufrière dome on the two previous sites and on an additional position (South-West Parking). The fourth telescope -MiniTel MT -was a compact, lighter version of this 3-panel layout (total length L = 1m) with reduced matrix size and inter-distance, down to 10 × 10 scintillators, and thus only 60 channels to be readout by one MAPM. This telescope was carefully installed at the bottom of the 30 th of August fault, geographically closest to the main active vents of the dome. Through these different survey phases, muographic images have shown the highly heterogeneous constitution of the dome, presenting density anomaly regions reflecting the wide network of fractures, faults, and cavities shaped during the six last phreatic eruptions, described in [120]. In parallel to mean density imaging, background studies have been led in [START_REF] Jourde | Experimental detection of upward-going cosmic particles and consequences for correction of density radiography of volcanoes[END_REF] showing the existence of an upward going flux contamination, and the existence of scattering on dome surface impacting the low-energy muons (E ≤ 1 GeV) [START_REF] Gómez | Forward scattering effects on muon imaging[END_REF]. The work in [START_REF] Jourde | Improvement of density models of geological structures by fusion of gravity data and cosmic muon radiographies[END_REF] first presented conceptual and methodological studies led with simulated data and synthetic tests, demonstrating the usefulness of gravity surveys to delimit the density anomalies in case of incomplete spatial coverage by the telescopes. Then, based on real gravity and muon data joint-inversion was performed in [START_REF] Rosas-Carbajal | Three-dimensional density structure of La Soufrière de Guadeloupe lava dome from simultaneous muon radiographies and gravity data[END_REF] where density anomalies were highlighted consistent with precedent estimates and moreover consistent with identified conductivity anomaly regions in Fig. 4.9 [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF].

Further studies explored the feasibility of continuously monitoring muon flux and detecting huge mass changes related to hydrothermal activity, as presented in Sec. 3.6.2. Crossanalysis with seismic noise signal backed up with vent temperature measurements in order to detect sudden destabilization in the hydrothermal activity on time scales of hours and days [START_REF] Gonidec | Abrupt changes of hydrothermal activity in a lava dome detected by combined seismic and muon monitoring[END_REF]. The latter study suggests strong correlations between the three signals over a 3day period, making it the first muon-based hydrothermal activity identification supported by complementary geophysical observations.

Conclusion

We have presented the overall geological and geophysical context at La Soufrière de Guadeloupe, the muography surveys carried on its lava dome, and described the general methodology to obtain two-dimensional mean density muographic images. We have seen that La Soufrière de Guadeloupe is an established natural laboratory for the development of new volcano studies, in particular muography. In turn, muography applications at La Soufrière have shifted from methodological and feasibility tests to a novel source of information to understand the complex structure and dynamics of this hazardous volcano.

In the following chapter, we will focus on the first part of the work conducted during this thesis, which first consisted in developing algorithms able to track muons in a telescope and filter out as much as possible the spurious noisy hits either related to the optoelectronics readout system (e.g dark photons generated from the photo-cathode) and irreducible physical background (e.g soft electromagnetic component, secondary electrons, and photons), along with simulation tools to characterize the reconstruction performances, first pivotal steps towards the estimation of the transmitted muon flux.

We have presented in chapter 4, Sec. 4.4.2 how the events that formed our raw data set are built by the telescope DAQ system, operating from the individual detection matrix light and electronic readout to the selection of XY hit coincidences on the triggered matrices. This chapter will introduce the methodology applied for the reconstruction of muon trajectories in telescope data. Next, we will present the Monte-Carlo simulation tool that allows us to assess the performances of the tested algorithms in terms of reconstruction efficiency, angular resolution, and processing time. The main goal of this processing is to optimize the number of reconstructed true muon events while discarding the impact of secondary particles like electrons and photons, and instrumental noise originating from the readout system.

Tracking methods

In particle detectors, the reconstruction of particle trajectories, so-called tracking, is analog to the detection of shapes (straight-line, curves, circles), a recurrent problem that occurs in noisy image processing. For scintillator-based telescopes described in Sec. 4.4.1, the muon tracking, remains in principle fairly simple given the planar geometry of the detection matrices, the limited number of fired pixels, and the absence of magnetic field that could affect the track curvature. The goal is to seek in each built event a set of colinear hits distributed in at least three fired matrices. Their fit to a straight-line model allows for measuring the incident muon direction. As for noisy image processing, we are confronted with outlier points (hits) in the raw data. Indeed, despite the trigger requirements (i.e at least one MAPM pulses coincidence in X and Y above 1 photo-electron), and the redundancy of the three matrices that discard fake coincidences, some spurious hits are formed by instrumental effects like a deficient optical coupling between WLS and clear fibers, dark current arising from the photo-cathode and dynodes in MAPM, electronic cross-talk between MAPM channels. Secondary electrons (e.g delta ray) and photons formed in the detection volume are also part of this outlier set. Hence, the impact of these hits needs to be mitigated during the track reconstruction process, to correctly measure the incident muon direction. So far the reconstruction of muon incident direction through telescope raw data processing was based on a simple test on the collinearity of the most-energetic hits formed in each panel. But this high-energy hit selection overlooks the presence of particles like atmospheric or secondary electrons that may deposit more energy in the scintillator medium as muons, which are for most of them minimum ionizing particles. Furthermore, the addition of a fourth telescope detection matrix in 2019, to increase angular resolution (by ∼ 49% compared to the 3panel configuration) on the highly active region of the La Soufrière lava dome (mostly its Southern region), has motivated the usage of a more precise tracking procedure. In the following, we describe two tracking algorithms and compare their performances in terms of reconstruction efficiency and angular precision on the reconstructed track direction:

1. the Hough Transform [START_REF] Hough | METHOD AND MEANS FOR RECOGNIZING COMPLEX PAT-TERNS[END_REF], commonly used to perform three-dimensional reconstruction of particle tracks in detectors like time-projection-chambers (TPC) [START_REF] Kouzinopoulos | Performing track reconstruction at the ALICE TPC using a fast Hough Transform method[END_REF],

2. the random sampling consensus (RANSAC) [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF], the method that we have adopted for the muon tomography data processing of the different telescope runs, whose results are presented in chapter 6.

Hough transform

One method for the detection of colinear points can be built on the Hough transform [START_REF] Hough | METHOD AND MEANS FOR RECOGNIZING COMPLEX PAT-TERNS[END_REF]. The latter was first applied in 1959 to read out photographic plates of bubble chamber experiments and reconstruct particle trajectories [START_REF] Paul | Machine Analysis of Bubble Chamber Pictures[END_REF]. During this process, three-dimensional image point cloud {(x, y, z)} i are transformed into lines in a parameter space called the Hough space (θ, φ, ρ). One can mathematically show that in this parameter space, intersecting lines correspond to colinear points in the image space [START_REF] Duda | Use of the Hough transformation to detect lines and curves in pictures[END_REF], with the intersection coordinates giving the line parameters passing through the colinear points. For example, two points in the image space transform into two lines in (θ, φ, ρ), whose intersection coordinates give the line parameters in the image space of the single straight line that crosses the two points. The problem of checking the colinearity between {(x, y, z)} i image points thus lies in finding concurrent curves in the parameter space.

The basic working principle involves mapping the image points into their representing curves in the Hough space. Depending on the application, this computation can be efficiently restricted to some regions containing all line representations e.g. to half-space -π ≤ θ ≤ +π, and avoid redundancy. A pivotal step consists in defining an appropriate discretization of the Hough space (θ, φ, ρ). The "transform" of the original point cloud in the image space to the Hough space is represented as an array-like object (e.g. a twodimensional histogram) called an accumulator. Each cell or bin of the accumulator represents a set of line parameters that records the count of curves passing through it, referred to as the voting process. After processing all the image points, the local maxima of votes in the accumulator indicate the parameters of the different straight lines crossing colinear points. This discretization of the parameter space is the main limitation of the classical Hough transform as introduced in [START_REF] Duda | Use of the Hough transformation to detect lines and curves in pictures[END_REF], which cannot be overcome by a finer granularity as points belonging to the same line would eventually vote for different lines [START_REF] Dalitz | Iterative Hough Transform for Line Detection in 3D Point Clouds[END_REF].

To test the tracking efficiency of the Hough transform for muons crossing the 4-panel telescope, we have applied the algorithm developed by Dalitz et al. [START_REF] Dalitz | Iterative Hough Transform for Line Detection in 3D Point Clouds[END_REF], written in C++. The latter is an application of the Hough transform to a three-dimensional point cloud {(x, y, z)} i , as introduced above. Depending on the number of lines to be found, the algorithm iteratively applies the Hough transform on {(x, y, z)} i = ⃗ x i and only focuses on the highestvoting cell to find the best-guest (⃗ a, ⃗ b) line parameters where ⃗ a is a reference point on the line, ⃗ b is the direction vector. It then selects points in ⃗ x i according to a distance threshold criteria dx and performs a least-square fit of the model to these selected points to refine the estimation of (⃗ a, ⃗ b). At the end of each iteration, selected points are removed from the set ⃗ x i . The algorithm either stops when a minimal number of points remaining in ⃗ x i or the maximal number of lines are reached. The input parameters of the algorithm are the following:

• dx: the distance threshold for a point ⃗ x to belong to a detected line ⃗ a + t ⃗ b in vector form, such as :

|| ⃗ a + t ⃗ b || ≤ dx, (5.1)
where t = ⟨ ⃗ b, ⃗ x -⃗ a⟩.

• nlines: the maximal number of returned lines, that we set to 1,

• minvotes: the minimal number of points imposed per line, that we set to 3, equal to the required number of triggered detection matrix.

The algorithm outputs the fitted n lines in vector form

{(⃗ a, ⃗ b)} i ∈ [1,n] .

Random Sampling Consensus

The Random Sampling Consensus (RANSAC) [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF] belongs to the category of robust algorithms designed to process data sets polluted by a significant amount of outliers. This is a procedure that initiates on a small data set sample and tries to enlarge this data set finding consistent data points tagged as inliers when possible, based on deviation or error tolerance (also named loss value) to a model. The initial data sample s should be large enough to estimate model parameters (i.e. at least 2 points for a straight-line model). Operating directly on the {(x, y, z)} i data set (i.e image space), contrary to the Hough transform introduced in Sec. 5.1.1, the algorithm takes as input the following main arguments:

1. the initial sample size s, which defines the size of the randomly sampled subset of data points used to compute a model hypothesis for each try (each iteration), 2. the residual threshold t, which will allow the inlier/outlier tagging for the remaining ns data points at each try. A given i-th data point will have a deviation d i to the hypothesized model. This point is tagged inlier if d i ≤ t, else outlier.

3. the maximum number of iterations N max , as a stopping criterion for the algorithm. As we will see below, this parameter can be set accordingly to the sample size s, and a hypothesized proportion of outliers in the original data.

The processing of a raw event starts by randomly sampling in the identified XY points coordinates, a subset of points of size s, perform a line least-squared fit to this subset. It then evaluates which hits of the whole dataset are consistent with this hypothesized model, computes a fit error, and counts the number of hits whose orthogonal distance (i.e the loss value) to the model is below a given distance threshold t (set to one scintillator width). These points form the consensus set. After N max iterations, the algorithm outputs the model that minimizes the fit error and maximizes the size of the consensus set. The hits belonging to the consensus are tagged as inliers and the rest of the dataset as outliers.

In order to estimate which number of iterations is required, the following reasoning can be applied: Considering that the probability to get at least one outlier in the subset of s points sampled in the dataset after N iterations is:

q = (1 -ϵ s ) N (5.2)
where ϵ is the inlier fraction. The converse probability p = 1q to select at least one outlier-free subset in N is:

p = 1 -(1 -ϵ s ) N (5.3)

Event filtering

To mitigate the impact of fake coincidences and electron contamination, a timely coincidence criteria between three detection matrices is imposed by the onboard event-building software. The RANSAC tracking algorithm is applied after this pre-processing step. Before the identification of the best linear trajectory among all the hits formed in at least three triggered detection matrices, we choose to filter the event based on hit multiplicity and reject it if more than 10 points were formed on one detection matrix. We recall that one XY point is the combination of two hits, coincidentally detected on the X and Y scintillator layers of one matrix. As illustrated in Fig. 5.4b, electrons tend to trigger electromagnetic showers in the telescope after crossing the lead shielding panel. Hence this hit multiplicity criteria is of huge help for discarding such background. As RANSAC will output for each reconstructed event the tagging inlier/outlier of each point depending on the distance to the best model (cf. Sec. 5.1.2), further multiplicity cuts can be performed after tracking, based on for instance the number of outlier points, especially on the third and/or fourth (rear) matrices, behind the shielding panel. We present in Fig. 5.2 the general workflow that we followed for the telescope data reconstruction. The tracking algorithm that we are using was written in Python, with the use of the ransac module available in the scikit-image library [START_REF] Van Der | scikit-image: image processing in Python[END_REF].

Development of simulation tools

Event generation

To test the performances of the two tracking algorithms, we have simulated a run of 10 6 muons, with the CRY [START_REF] Wright | Cosmic-ray shower generator (CRY) for Monte Carlo transport codes[END_REF] Monte-Carlo generator. This code is favored here for its rapidity in the generation process (i.e O(1min) for 10 6 primary muons on a standard 8 GB CPU). Indeed, CRY takes advantage of a library of pre-computed atmospheric particle fluxes (muons, neutrons, protons, electrons, photons, and pions) obtained from cosmic-ray shower simulations initiated by primary protons, transported into the atmosphere modeled as a series of 42 constant-density flat layers. The fluxes are available at three altitudes (sea level, 2100 m, and 11300 m). The user may also set in the generator the latitude of observation to account for the geomagnetic effect (i.e reproducing the energy cutoff imposed by Earth's magnetic field that ranges from ∼15 GeV at the equator towards zero at the poles). As a result, the muon flux of interest generated by CRY is consistent at sea level with Rastin et al. data [START_REF] B C Rastin | An accurate measurement of the sea-level muon spectrum within the range 4 to 3000 GeV/c[END_REF] in the [4, 3.10 4 ] GeV range. It has a broad energy spectrum [1, 10 7 ]MeV distributed around a mean value of 4 GeV, with a uniform spatial distribution, that can be sampled over a flat square surface. Though, as it uses a pre-computed library, the energy sampling is thus affected by "discretization" effects compared to a generation from an analytical formula (e.g from Guan or Shukla spectra, cf. Sec. 1.4).

Particle transport with GEANT4

GEANT4 [11] can be considered as the main Monte-Carlo tool used in particle physics to coincidentally simulate the detector assembly, the particle interaction in a defined medium, the tracking, hit management, and detector responses. It is designed as a flexible objectoriented C++ framework that allows tailored applications mainly in high-energy physics, medical physics, and accelerator physics.

GEANT4 allows the three-dimensional reproduction of the experiment's geometry layout. During this implementation, the user is expected to define the "sensitive" parts of the detector in which the physical information on the traversing particles needs to be recorded. The most common information might be the sum of all energy deposits during one event, gathered to form an individual "hit" in the traversed sensor. A particle is represented in terms of a "track" object (snapshot of the particle) transported "step-by-step". Each "step" object is defined by given spatial and time paths between two well-located points: the prestep and post-step points. The step object owns the 'delta' physical information available on a particle (energy loss over the step, time-of-flight, direction, current traversed object, etc.). During this transportation at the step level, the particle depending on its nature can interact with matter through different physical "processes", also represented as individual objects in GEANT4 (e.g accounting for ionization, e+/e-pair production, photoelectric effect, Compton scattering). These processes can apply continuously along the i-th step while moving from pre-step to post-step point (e.g. ionization). They also can be discrete, occurring at the post-step point (e.g. Compton scattering), or can occur at rest i.e. when the kinetic energy of the particle gets to zero (e.g. decay and production of daughter particles). Hence depending on the needs, to retrieve all the required physical information, the user may access a peculiar step of interest along the particle track(s) transport, i.e when a track crosses a solid object geometry boundary (e.g entering the detector), or look at a whole collection of steps belonging to a given traversed part of the geometry setup (e.g path into a lava dome numerical model). In our case, the geometry and materials of the different detection elements (i.e the scintillator layers arranged as XY matrices) and passive shielding elements (lead shielding panel between the second and third matrices, plus the steel protecting matrix) are reproduced as different solids positioned inside a three-dimensional volume, so-called "mother" volume which stands for the muon telescope volume itself. This mother volume is positioned inside the "World" volume, defined in GEANT4 as the whole simulation space, i.e. containing all volumes and transported particles.

Limitations

The current simulation framework was specifically designed to study the tracking efficiency of atmospheric particles in the muon telescope developed for tomography at La Soufrière. It is a simplified representation of a real muon telescope as it does not reproduce the whole light propagation and readout process driven by the instrumental characteristics of scintillator detectors like the light yield -that is the amount of ionization energy the scintillator can re-emit as scintillation photons -associated with the specific plastic scintillation This information is however not used in the tracking algorithm. medium (cf. Sec. 4.4.1) as well as effects like light attenuation during transport along optical fibers, the variation in gains and electronic cross-talks between photo-multiplier channels.

Tracking performances

In the following section, we compare the performances of the tracking algorithms described above: the Hough transform and RANSAC. We mention that an analog comparison between these two tracking methods was performed in [125], though applied to a completely different detection technology (a 2D multiplexed gaseous detector designed as Time Projection Chamber), currently under development for muography applications at CEA Saclay, France. The two algorithms' performances are here compared in terms of reconstruction efficiency, angular residuals, and processing time. This comparative study is led in a simulated 4-panel muon telescope whose geometry is described in Sec. 4.4.1. This numerical telescope was irradiated with a 10 6 muon sample generated with CRY. We recall that this 4-panel geometry is decomposed into three-panel configurations : These configurations are referred to as 3p1 (Front -Middle1 -Middle2 panels), 3p2 (Middle1 -Middle2 -Rear panels), and 4p (Front -Middle1 -Middle2 -Rear panels) (cf. Fig. 5.4a).

Noise impact

To compare the quality of the muon track reconstruction of the Hough transform and RANSAC in the presence of an increasing proportion of noise relative to the number of true muon hits, we have synthetically polluted our MC muon sample with random outlier hits, sampled uniformly on the scintillator bars of the numerical 4-panel detector. We do not assign any intensity to these outlier hits as this parameter is not taken into account in the current tracking workflow. In the following, quantities are plotted as a function of the noise proportion defined as p n = n out /n tot µ , where n out is the number of synthetic outlier hits, and n tot µ is the total number of true muon hits. The range of noise proportion tested here is much wider than what is observed for real data, where the proportion of outlier hits in reconstructed events does not exceed 40% in both open-sky calibration and tomography acquisition.

Reconstruction efficiency

The muon reconstruction efficiency is simply defined as the ratio between the number of reconstructed muons and the number of generated muons (or primary muons) in each pixel (or line of sight) of the detector. In the i-th pixel:

ε i r = N i r /N i 0 (5.5)
where N i r is the number of reconstructed muons, and N i 0 is the number of primary muons generated.

In Fig. 5.5, we plot the mean reconstruction efficiency computed over all pixels of each configuration as a function of the noise proportion. We observe that the RANSAC efficiency is at least by 10% higher than the Hough reconstruction for all panel configurations. Nevertheless, we notice that the two extreme cases tested at n p = 150% and n p = 200%, i.e. when the noise proportion overcomes the muon signal the Hough transform presents better reconstruction efficiency than RANSAC.

Angular residuals

The quality of the track reconstruction can be quantified in terms of angular residuals, which we define for both polar angles as follows:

∆θ = θ 0 -θ reco , ∆φ = φ 0 -φ reco , (5.6)
where (θ 0 , φ 0 ) are the primary angles, i.e the angles that define the primary muon initial direction at generation, and (θ reco , φ reco ) are the angles given by the reconstructed trajectory model direction obtained after tracking. Angular residual estimates obtained for n p = 50% are shown in Fig. 5.6 for each telescope configuration. If the RANSAC residuals distributions seem to be Gaussian distributed around 0, the Hough distributions feature prolonged tails on the distribution edges, which would need further investigation not carried out here. We thus decide to consider the root mean square error (RMSE) of histograms as an estimate of the angular resolution. Fig. 5.7 is showing the evolution of angular residuals RMSE as a function of noise proportion. Below the equal proportion of outlier and muon hits (i.e. here 100% of noise proportion), RANSAC has the lowest RMSE, hence the best trajectory fit resolution. The 3-panel and 4-panel configurations have respective 41±11 and 28±8 mrad angular pitch (angle between two consecutive scintillator bars). We notice that RANSAC RMSE exceeds these values in all three configurations starting from 90% of outliers. Below this noise level, RANSAC is therefore suited for the telescope's angular resolution. On the contrary, the Hough transform RMSE is above these resolution values for the two 3-panel configurations.

Processing time

To conclude this comparison between the Hough transform and RANSAC tracking, we simply compare both algorithm performances in terms of processing time (cf Fig. 5.8). The latter is an important parameter in the context of muography monitoring of volcanic activity, and in the processing of month-long to year-long tomography acquisition, as the size of muon data sets keeps increasing. The study was conducted on a 10 5 MC muon sample processed on an 8 GB CPU standard machine. RANSAC is showing a linear increase in processing time with noise proportion, whereas Hough is constant around 700 s. In the context of realistic noise proportion below 80%, RANSAC presents the best trade-off.

Scintillator efficiency estimation

Besides comparing the performances of the two tracking algorithms, this simulation framework built in GEANT4 allowed us to infer a first estimation for the scintillator bar efficiency using a comparison to real data. This efficiency is linked to the intrinsic light yield proper to the scintillation medium but also linked to the light attenuation across the optical fiber that conducts the light towards the MAPMT.

In order to estimate this scintillator efficiency for real detector matrices, we compared in Appendix A), the RANSAC reconstruction efficiency as a function of a parameter associated with the scintillator efficiency ε scint . The latter was defined in the tracking algorithm as a probability for each hit in a scintillator bar to give a signal above the threshold, that we assumed to be uniform for all scintillator bars of all four matrices. The reconstruction efficiency estimated on simulated muons ε MC reco was compared with estimations of the real reconstruction efficiency using data recorded during an open-sky calibration acquisition in the 4-panel telescope at La Soufrière (22 to 24 Feb. 2019). This real data reconstruction efficiency was estimated as ε DATA reco = T exp /T geo , that is the ratio between the experimental telescope acceptance T exp , introduced in Sec. 4.4.3, and the geometrical acceptance T geo , derived analytically (cf. Sec. 3.2.3). In Fig. 5.9, we show how ⟨ε MC reco ⟩ varies as we choose different values for ε scint . This relation allows interpolating estimates for telescope scintillator efficiency, from the ⟨ε DATA reco ⟩ estimates displayed on the right vertical axis. We respectively obtained the following scintillator efficiencies for two 3-panel sub-configurations and 4-panel configuration: ε 3p1 scint = 0.899 ± 0.012, ε 3p2 scint = 0.900 ± 0.010 and ε 4p scint = 0.868 ± 0.015. These estimates are consistent with the instrumental efficiency of the order of 0.9 usually considered for the technology of plastic scintillator bars employed here (further details can be found in [START_REF] Lesparre | Design and operation of a field telescope for cosmic ray geophysical tomography[END_REF]).

Energy cut-off estimation

Surface muon tomography experiments are exposed to different sources of atmospheric background, that pollute the low transmitted muon flux signal coming out of the scanned object. Besides electrons, whose pollution can be mitigated by imposing cuts on the hit multiplicity, low-energy background encompasses also low-energy muons that can undergo multiple Coulomb scattering on the scanned object surface and be re-directed towards the muon telescope acceptance cone, mimicking the signal produced by the transmitted muons. We present and discuss in more detail this background source in the next chapter (cf. chapter 6, Sec. 6.1.3) In the absence of any calorimeter device that can allow a rough estimation of the incident particle energy, only passive shielding can prevent the low-energy background. To estimate the energy cut-off provided by the current shielding panel whose thickness is composed of 5cm-lead plus 2cm-steel, we plot in Fig. 5.10, the RANSAC reconstruction efficiency as a function of primary energy for muons. This plot shows an energy cut-off at (1.7 ± 0.1) × 10 -1 GeV, which turns out to be too low for an optimal rejection of the soft atmospheric background, typically around 1 GeV [START_REF] Nishiyama | Monte Carlo simulation for background study of geophysical inspection with cosmic-ray muons[END_REF], [START_REF] Gómez | Forward scattering effects on muon imaging[END_REF], [START_REF] Bonechi | Atmospheric muons as an imaging tool[END_REF]. ] GeV. We compare histograms for both telescope configurations with (green) and without (blue) a shielding panel (5 cm lead + 2 cm steel). The latter histogram is fitted with an erf function (solid green line), which gives a cut-off energy estimate of E c = (1.7 ± 0.9).10 -2 GeV.

Conclusion

This work has allowed the building of a processing code for the reconstruction of muon events in the telescope's raw data including pre-processing and particle tracking. The processing code was written in Python and could be easily adapted for including further event selection criteria or other tracking procedures (e.g. Kalman Filter [START_REF] Ghosh | Track fitting by Kalman Filter method for a prototype cosmic ray muon detector[END_REF]). A GEANT4 simulation framework was developed for assessing the tracking performances in the new muon telescope configuration at 4-panel, developed for La Soufrière dome tomography, which sets the ground for further simulation-driven studies, notably a characterization of the atmospheric background and its impact on the muon-transmitted flux estimates. A comparative study of two tracking algorithms, the Hough transform widely spread in particle physics experiments and a newly applied RANSAC-based algorithm designed to reject outlier hits in data sets, allowed us to choose in favor of the latter. We have seen though that the RANSAC performances are strongly driven by the initial proportion of outliers in the data set. The Hough transform seems to overtake RANSAC performances in terms of reconstruction efficiency, angular residuals, as well as processing time, when the number of outliers is superior to the number of inliers (i.e true muon hits), but regarding the event filtering applied during the pre-processing of raw data (time coincidence between matrices, and hit multiplicity filtering) these noise proportions represent irrealistic experimental situations. Hence, for the simple layout of our muon telescope, we have shown that under the assumption that the maximal pollution, i.e the number of noisy hits (outlier) in a built-event does not exceed 50% of the total number of hits, RANSAC offers the highest reconstruction efficiency, lowest angular residuals, but also the best trade-off in terms of processing time. This tracking procedure has thus been chosen as the most robust method to reconstruct the large muon data sets for the tomography analysis presented in the next chapter.

Throughout this chapter, we will present the main muography data analysis that we have conducted at La Soufrière. The two-dimensional muographies acquired at four different positions around the La Soufrière lava dome constitute our valuable experimental inputs to perform three-dimensional modeling. A novel tracking procedure based on RANSAC that improves the rejection of background events described in chapter 5, was applied to large muon data set acquired since 2015 at four different locations around the lava dome, leading to new estimates for performing three-dimensional density modeling of the lava dome. As presented in chapter 3, the main goal of muography applications is to detect density anomalies through precise estimates of the transmitted flux, which further leads to rock opacity estimates, assuming a reference medium for the composition of the scanned system. One muography two-dimensional image remains nonetheless insufficient and ambiguous to draw fine interpretations regarding the internal structure of the system. In geosciences applications, three-dimensional density modeling of the natural system under study is often foreseen, which requires sufficient data coverage to achieve a reasonable spatial resolution, only provided through several detectors distributed around the scanned system or through combinations of density-sensitive observables, like muon and gravimetry data [START_REF] Nishiyama | 3D Density Modeling with Gravity and Muon-Radiographic Observations in Showa-Shinzan Lava Dome, Usu, Japan[END_REF], [START_REF] Rosas-Carbajal | Three-dimensional density structure of La Soufrière de Guadeloupe lava dome from simultaneous muon radiographies and gravity data[END_REF], [START_REF] Nagahara | Three-dimensional density tomography determined from multi-directional muography of the Omuroyama scoria cone, Higashi-Izu monogenetic volcano field, Japan[END_REF]. Besides the resolution, a clear understanding of the different sources of uncertainties that affect the muon signal is required to perform realistic density modeling. As often in Earth's subsurface imaging, this modeling refers to a so-called inverse problem. After a rapid contextualization on this matter, we will present the Bayesian inversion procedure that we have followed and the required synthetic tests (or toy experiments) that are used to optimize the trade-off between prior model information and data fitting. We have performed these synthetic tests on a toy density model for the La Soufrière dome. This part will be followed by the presentation of the results of the real data three-dimensional inversion that allowed deriving a novel three-dimensional density model of the lava dome. We end this chapter by putting this model in comparison with previous density models, followed by a discussion of the possible interpretation that this density model can offer and the implications in terms of better characterization of the rock mechanical behavior and partial flank collapse hazard at La Soufrière de Guadeloupe.

Uncertainty on flux measurement

During the month-long data-taking period that is required to overcome the statistical fluctuations in the number of detected events (cf. Sec. 3.3.2), the muon telescope is not only exposed to sometimes rough terrain conditions that can affect its long-run operation (as we have seen for the OM telescope) but also to different kinds of atmospheric backgrounds that will bias the low transmitted muon flux from the scanned object. Close to the horizon, the muon flux is indeed more than two orders of magnitude smaller than the muon vertical flux. The flux decreases again by two orders of magnitude after traversing 1 km of standard rock. For a simple estimate of opacity values along each line of sight, linear variations of transmitted flux I with opacity ϱ can be assumed: I(ϱ) = a + b × ϱ where a = I(ϱ + dϱ) and b = | ∂I ∂ϱ |. As we know from flux computation discussed in Sec. 1.4 as a function of ϱ, this linearization constitutes a strong approximation that introduces a systematic error ∆ϱ lin :

∆ϱ lin = b -1 × ∆I (6.1)
where ∆I is the combination of the statistical error made on the muon detection σ stat and the error that results from the transmitted flux computation through the opacity range of interest in a given medium (generally standard rock) σ model . Since muon detection is a random process, that occurs at a constant rate and whose events are independent of the previous one, the statistical fluctuations in the number of muons are governed by a Poisson distribution with associated standard deviation √ N where N is the number of detected muons, hence the statistical Poissonian error associated to flux σ stat is proportional to √ N . Hence ∆I = σ 2 stat + σ 2 model . The transmitted flux, that is the integral on [E min (ϱ), +∞[ of an analytical muon spectrum model (e.g modified Gaisser muon spectrum like the Guan parameterization) or a muon spectrum derived from Monte-Carlo simulations like CORSIKA (cf chapter 1, Sec. 1.4.6), is impacted by the estimation of the minimal crossing energy E min (ϱ) (cf. Sec. 1.3). As discussed in [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF] and recalled in [START_REF] Lechmann | The effect of rock composition on muon tomography measurements[END_REF], the uncertainty on energy loss by ionization a(E) for a multiple compound medium is of the order of σ a = ±6%, while the uncertainty on radiative losses b(E) derive from uncertainties on Bremsstrahlung, pair production and photonuclear cross-sections (respectively σ b brems = ±1%, σ b pair = ±5%, σ bpn = ±30%). These uncertainties allow deriving around E min the 1σ lower and upper bounds [E low min ; E As discussed in Sec. 1.3, the integration of the mean stopping power to retrieve E min is valid upon the continuous slowing-down approximation (CSDA) and neglects fluctuations in energy loss along the muon path. This assumption is typically valid below the cut-off energy E µc where ionization loss dominates radiative effects (E SR µc ≃ 693 GeV [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF]). The propagation of error on the ionization loss and radiative coefficients described above is a first approximation to account for the fluctuations in energy loss that start to be significant for muons above 1 TeV, that can cross up to 3.10 3 mwe rock thickness. The muography of La Soufrière lava dome takes place just below this range of opacity. The interpolation of opacity values in the present study is hence limited to rock thicknesses smaller than 1.5 km, which correspond to the more horizontal telescope's lines of sight. Details regarding the fluctuations of energy loss for high-energy muons through standard rock can be found in [START_REF] Lipari | Propagation of multi-TeV muons[END_REF]. In the end, these energy loss fluctuations can be trustfully accounted for using a Monte-Carlo simulation framework like GEANT4, described in Sec. 5.2.2.

We show in Fig. 6.2 the relative opacity error ∆ϱ lin /ϱ that in the (ϱ, θ) plane. This error directly propagates to mean density value uncertainty as: ∆ρ lin = 1/L (∆L/L ϱ + ∆ϱ lin ), where ∆L is equal to the spatial resolution of the digital terrain model, i.e 1 m. In Fig. 6.2, for a σ stat uniform, we observe that the error is globally decreasing with the opacity for all θ as the decrease of b (flux derivative) with ϱ slows down and σ model also decreases with ϱ (cf. Fig. 6.1). For real data, the statistical error σ stat obviously depends on the vertical angle θ, as for a given exposure time, the number of detected muons can vary from three orders of magnitude in the θ ∈ [START_REF] Dalitz | Iterative Hough Transform for Line Detection in 3D Point Clouds[END_REF][START_REF] Guan | A parametrization of the cosmic-ray muon flux at sea-level[END_REF] • angular range.

Choice of the flux model

The opacity estimation relies on the choice of a realistic parameterization of the muon spectrum Φ(θ, E) at the observation geographical coordinates and altitude (cf. Sec. 1.4). For this, the muon spectrum can be derived from air shower simulations, which are computationally intensive, or using a model flux that is based on the muon production physics, mainly through pions and kaons decays in the high atmosphere (e.g Gaisser [START_REF] Gaisser | Cosmic Rays and Particle Physics[END_REF]), or that is purely empirical, parameterized to provide the best fit over a certain range of energy covered by a chosen set of reference data, often recorded only at vertical incidence θ = 0 (e.g Bugaev et al. [START_REF] Bugaev | Atmospheric muon flux at sea level, underground, and underwater[END_REF], Hebbeker & Timmermans [START_REF] Hebbeker | A Compilation of High Energy Atmospheric Muon Data at Sea Level[END_REF]). As described in [131], this model choice introduces a bias that needs to be accounted for in the transmitted flux estimation. For example, Lesparre et al. [131] found a maximum 10% discrepancy between the transmitted flux of nearly horizontal muons in the 50 to 500 m range of rock thicknesses. These fluxes were derived from three different muon spectrum models: modified Gaisser [START_REF] Kudryavtsev | Muon simulation codes MUSIC and MUSUN for underground physics[END_REF], Reyna-Bugaev [START_REF] Reyna | A Simple Parameterization of the Cosmic-Ray Muon Momentum Spectra at the Surface as a Function of Zenith Angle[END_REF], Reyna-Hebbeker [START_REF] Hebbeker | A Compilation of High Energy Atmospheric Muon Data at Sea Level[END_REF]. In practice, the measurement of the incident flux at the observation site during an open-sky acquisition is expected to avoid the bias introduced by the model choice through the correction of the geometrical telescope acceptance. This correction is contained in the experimental acceptance, introduced in Sec. 3.2.3, that allows to re-normalize the transmitted muon flux and thus compensates for this model-dependent effect in the opacity estimation.

Influence of the propagation medium

Nature of the medium Standard rock constitutes a common representation of the geology of the scanned object (cf. Sec. 1.3), which is widely adopted in muography applications or muon background characterization for underground experiments. Nevertheless, the interaction of muons depends on the nuclear properties of the different compounds in the crossed medium and thus on the rock's chemical composition. When the geology of the object is known, the derivation of muon interaction cross-sections is achievable for single to multiple compounds medium, where the effect of each compound 123 Chapter 6 is weighted accordingly to its atomic number over atomic mass ratio (Z/A). This allows deriving the energy deposit even for rock medium with a complex mineral structure. One main pitfall for this computation may lie in determining a realistic mean excitation energy ⟨I⟩ for an un-tabulated multiple compounds medium, that can differ from the weighted average of the known compounds, as electrons are more tightly bound than in each constituent elements taken individually. Using the methodology presented in [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF], Lechmann et al. [START_REF] Lechmann | The effect of rock composition on muon tomography measurements[END_REF] studied the impact of rock composition on the flux propagation and derived relative variations to standard rock for volcanic rocks (e.g andesite, basalt) and sedimentary rocks (e.g limestone). He observed that for all types of rock, the deviation from standard rock remains below 2.5% for rock thicknesses under 300 m. Furthermore, according to these estimations, for la Soufrière lava andesitic dome, along 1 km thicknesses the standard rock assumption would result in transmitted flux overestimates by 5%, which remains below the statistical and systematic errors that affect density estimates. If standard mineral rock composition variations to standard rock are not likely to have a huge impact on the transmitted flux, the water content of the crossed medium, specifically for hydrothermally-altered material of the La Soufrière dome, might have a larger effect on the transmitted muon flux (cf. [START_REF] Lesparre | Mise au point d'une methode de tomographie utilisant les muons d'origine cosmique. Applications au laboratoire souterrain du Mont Terri et au volcan de la Soufrière de Guadeloupe[END_REF], [START_REF] Jourde | Un nouvel outil pour mieux comprendre les systemes volcaniques : la tomographie par muons, application à la Soufrière de Guadeloupe[END_REF]). Fig. 6.3 shows for three zenith angles θ = {0, 60, 80} • the fractional transmitted flux to standard rock, for (a) a pure liquid water medium and (b) a mixed medium of half water -half standard rock. In the range of opacity [1, 3.10 3 ] mwe, the maximal deviations from standard rock for {60, 80} • angles are comprised between 25% to 20% and 10% for ϱ ∈ [5.0, 5.5].10 2 mwe, depending on the zenith angle θ value. For this range of opacity, the transmitted flux is dominated by muons with energy below the critical energy E µc (cf. Sec. 1.3), i.e with muons who preferentially lose their energy through ionization. As for muons, the mean ionization energy deposit is larger in water than in standard rock [START_REF] Groom | MUON STOPPING POWER AND RANGE TABLES 10 MeV-100 TeV[END_REF] (⟨∆E water ⟩/⟨∆E SR ⟩ ≃ 1.18), fractional fluxes shown in Fig. 6.3 are below one. Beyond 3.10 3 mwe, stochastic radiative losses start to dominate in standard rock, which thus becomes more absorbent than water.

Multiple scattering inside the medium

On average, high-energy muons (E > 100 GeV) travel in matter along straight paths, meaning that the mean scattering angle is null ([131], [START_REF] Ambrosino | Joint measurement of the atmospheric muon flux through the Puy de Dôme volcano with plastic scintillators and Resistive Plate Chambers detectors[END_REF], [START_REF] Procureur | Muon imaging: Principles, technologies and applications[END_REF]). To verify that the deviation from a null scattering angle is negligible for transmitted muons with respect to the angular resolution of the muon telescope, we use a muon tomography simulation in GEANT4 that includes the numerical telescope (described in Sec. 5.2.2) and a dome model built from the La Soufrière DEM, with uniform standard rock composition, of density 1.8 g cm -3 . The rock thickness range is [10 2 , 10 3 ] m. Muons were generated using the Guan spectrum, sampling (θ, E) ∈ ( [START_REF] Carlson | Nationalism and internationalism in science: the case of the discovery of cosmic rays[END_REF][START_REF] Guan | A parametrization of the cosmic-ray muon flux at sea-level[END_REF] • , [10, 10 4 ] GeV). In Fig. 6.4, the total vertical angle deviation is computed between the entry point angle (θ i ) in the scanned object and the exit point (θ f ): ∆θ = θ iθ f . The deviation is plotted as a function of muon initial kinetic energy. The measured standard deviation above E > 100 GeV is about one order of magnitude lower than the angular resolution of the telescope. The deviation from the muon straight path due to multiple scattering inside the rock is therefore negligible.

Atmospheric background contamination

Contrary to underground experiments, that benefit from natural rock shielding, a surface muon tomography experiment is exposed to significant contamination from atmospheric shower products (muonic, electromagnetic and hadronic components) that reach the Earth's surface. These different sources of contamination globally cause an overestimation of the transmitted muon flux signal, which further leads to an underestimation of the traversed rock opacity. The atmospheric background was studied using dedicated simulations in [START_REF] Jourde | Experimental detection of upward-going cosmic particles and consequences for correction of density radiography of volcanoes[END_REF], [START_REF] Nishiyama | Monte Carlo simulation for background study of geophysical inspection with cosmic-ray muons[END_REF] and [START_REF] Gómez | Forward scattering effects on muon imaging[END_REF]. With dedicated time-of-flight measurements, Jourde et al. [START_REF] Jourde | Experimental detection of upward-going cosmic particles and consequences for correction of density radiography of volcanoes[END_REF] showed that a source of contamination lies in the muon flux produced in the atmosphere below the telescope altitude. Hence, if the topography of the terrains allows it, a backward muon flux can enter the telescope from its rear matrix. This effect mainly impacts the most horizontal lines-of-sights above θ > 80 • and can significantly blur muon radiography close to the horizon, accounting for up to 50% of the detected events. In [START_REF] Nishiyama | Monte Carlo simulation for background study of geophysical inspection with cosmic-ray muons[END_REF], Nishiyama et al. studied the origin of the background flux and found significant contamination from sub-GeV electrons, muons and protons. They notably obtained consistent results with [START_REF] Jourde | Experimental detection of upward-going cosmic particles and consequences for correction of density radiography of volcanoes[END_REF] measuring an upward-going muon flux of 44% of the generated atmospheric muon background (E ∈ [1; 10 4 ] GeV) for the most horizontal lines of sight θ < 80

• . This effect was shown to rapidly decrease with θ, accounting account for 23% of the muon atmospheric background in θ ∈ [START_REF] Fedele | The Campanian Ignimbrite Eruption, Heinrich Event 4, and Palaeolithic Change in Europe: A High-Resolution Investigation[END_REF][START_REF] Gallo | Three years of muography at Mount Etna: results and perspectives[END_REF] • . Both articles emphasize the need for a clock system with ns precision for incident particle time-of-flight measurements between the front and rear matrices. Upward-going particles would hence be tagged by negative time-of-flight and ideally rejected.

In addition to the upward-going flux, the contamination by low-energy muons was proven to be particularly important if the geometry of the scanned object favors the forward scattering of these "soft" muons towards the acceptance cone of the telescope, as sketched in Fig. 6.5 (upper left) and simulated in Fig. 6.6. This forward scattering occurs along the flanks of the scanned object, by multiple Coulomb scattering (cf. Sec. 3.1) for particles that have a very large incident angle with respect to the surface flank normal (almost parallel). The effect is thus particularly significant for large targets with smooth-conical geometry as volcanoes. In La Soufrière, the two South-West detectors located at the basis of the dome were thus particularly exposed to this contamination as they were facing a large unobstructed volcano flank without any rock shielding in their direct environment. Gomez et al. [START_REF] Gómez | Forward scattering effects on muon imaging[END_REF] have shown for a simplified cone-shaped dome geometry that the forward scattering of muons increases with the zenith angle θ, exceeding 50% of the total number of reconstructed events for θ superior to 85°. As the scattering is affected by the surrounding detector environment, terrain topography and by the distance between the scanned object and the telescope, this background noise is highly location-dependent and thus requires specific estimation for each muon imaging study. To cope with this irreducible background, only simulations specifically tailored for the geometrical and topographic layout of the experiment, provide the right framework to accurately estimate the fraction of forward-scattered to transmitted muons. In the end, both a ns clocking system and detector shielding are respectively required for tagging upward-going particles and rejecting sub-GeV background muons and electrons, The current time resolution (10 ns) of La Soufrière telescopes does not allow such tagging and the mechanical constraint on their structure only allowed a passive 7 cm-thick lead- For the present study, the low-energy background systematic uncertainty could unfortunately not be estimated properly due to a lack of time. Hence, the present analysis does not account for the associated transmitted flux corrections that would be required. However, to mitigate the impact of this bias, we do not consider the most horizontal lines of sight for θ > 83 • where it is likely to be of the order of 50% of the flux in the acceptance cone of the telescope, as presented from [START_REF] Nishiyama | Monte Carlo simulation for background study of geophysical inspection with cosmic-ray muons[END_REF] and [START_REF] Gómez | Forward scattering effects on muon imaging[END_REF].

Tomography datasets

In Tab.6.1, we present the four muon telescopes, their configuration and their position and orientation towards the lava dome, along with data acquisition-related information. Acceptance cones of telescopes are represented in Fig. 6 Table 6.1: Telescope configurations, location and their respective run acquisition period. SNJ is the only telescope featuring 4 detection panels. Hence it encompasses three muography configurations: the main 4-panel configuration (4p), along with two 3-panel subconfigurations (3p). passed by each telescope given in Tab.6.1, the dome volume (whose surface is represented as a red hashed region in Fig. 6.7) was discretized in cubic voxels of side 16 m. The surface voxels were cut according to the topography, obtained from the dome digital terrain model 1 . The union of the four scanned volumes is of the order of 1.5.10 7 m 3 , i.e 30% of the total volume of the dome 5.0.10 7 m 3 [START_REF] Boudon | A new scenario for the last magmatic eruption of La Soufrière of Guadeloupe (Lesser Antilles) in 1530 A.D. Evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF]. Their intersection corresponds to the volume encompassed by OM, which is 3.8.10 6 m 3 . Hence to get maximal coverage for the threedimensional density modeling presented in Sec. 6.5.3, we choose to consider the union, rather than the intersection.

The SB and SNJ telescopes were located in the South-West of the dome encompassing the two largest dome volumes in their respective acceptance cones. As a result, they also target the highest rock thicknesses. The detectors are installed at the lowest altitudes on a smooth unobstructed slope, facing a large portion of the dome flank as seen in Figs. 6.8a and 6.8b. As we will discuss in the next sections, due to their location, these detectors are the most exposed to atmospheric background contamination (cf. Sec. 6.1.3). With its increased acceptance, SNJ 3-panel configuration event rate in Fig. 6.9 is between three to four times higher than in SB, or BR. The latter in the East appears in Fig. 6.8c. This is the telescope for which we have the longest and most steady run muography run (640.4 days), as shown in Fig. 6.9. The latter is less exposed than the two South-West telescopes to background contamination, benefiting from the natural shielding of the col de l'Echelle in its back, in the South-East direction (summit altitude z = 1391 m). The OM telescope is located in the North fault (fente du Nord) of the dome in Fig. 6.8d. The event rate is thus lower than for the other three detectors but also filled with gaps which denote more unstable run operation conditions.

Two-dimensional muographies

As explained previously, estimating the transmitted flux in each telescope along with the computation of muon ray paths in the dome allow us to obtain two-dimensional maps of mean density estimates in each telescope angular sight plane, as shown in Figs. 6.10 and 6.11. These density maps reflect the highly heterogeneous inner structure of the dome. The precise location of structural anomalies from the sole interpretation of these two-dimensional maps is not possible as we can only obtain information on their direction, relative to the telescope observation point. However, the telescope paths overlap provided by this specific configuration of four detectors reduces the ambiguity on the location of potential density anomalies, notably in the most active southern region of the dome (cf. Each density map features this large low-density anomaly that spans over the closest region to the surface, for almost the whole azimuth range. BR is the closest to the most active fumarole region located between the Crater Sud (CS), Gouffre 56 (G56) and the Tarissan crater (TAR) while encompassing in its acceptance cone all the main craters and pits of the dome. This proximity to the dome's main structural features is very likely to explain the size of the low-density region that spans from azimuth 270 • to 330 • . This anomaly takes apparent density values of the order 1 g cm -3 thus physically unrealistic, especially in the OM detector (d). However, being located in the fault, this same detector, is also the closest to the rock and to a known important cavity, the Spallanzani cave of volume 55x35x11m 3 [120], whose entrance is also located in the north fault of the dome (FNO on Fig. 6.7). This volume is part of a wide network of cavities beneath the dome surface whose geometry is not taken into account when computing the dome thicknesses L traversed by the muons. These void spaces add to an average bulk density significantly lower than the average value of actual rocks. Furthermore, the impact of a systematic error in telescope orientation would have a larger impact on the computation of the shortest thicknesses in this most superficial region of the dome, where locate the density anomalies. We estimated for all telescopes that an error in azimuth of 0.5 • would cause less than a 1% error in thickness computation. On the contrary, the same error in telescope elevation is causing ⟨∆ L ⟩ ∼ 20% for the closest directions to the surface (computed on a band of three pixels wide below the topography) and ⟨∆ L ⟩ ∼ 5% below this superficial band (cf. Tab. 6.2).

In Figs. 6.12 and 6.13 are respectively shown the relative statistical and systematic uncertainties. The statistical uncertainty dominates the edge regions of the acceptance cone for all telescopes. For SB (a), high statistical uncertainties span over a whole angular region θ > 70 • where mean relative uncertainty ⟨∆ stat ⟩ SB = 0.30 ± 0.17, higher by more than 10% than for the other three telescopes (cf. Tab.6.2). The large statistical uncertainties seen in SB are notably caused by the event filtering we impose on the reconstructed events, to mitigate the atmospheric background contamination, we impose a cut on the outlier hit multiplicity (cf. Sec. 5.1.2) at m = 2 in the rear panel after the crossing of the shielding panel. We thus expect to filter out electrons with a mean rejection efficiency estimated at 59 ± 6% for 3-panel and 55 ± 5% for 4-panel configuration, (measured with the GEANT4 simulation, described in Sec. 5.2.2). The mean rejection efficiency of this cut on muon tracks does not exceed 10% for both configurations in each pixel, from which we compensate in the transmitted flux computation. In real data, this cut has a significant impact, reducing the event rate by 78% in SB. In the other telescopes, the event rate is reduced by 68% in BR, 50% in OM, 25% in SNJ. This cut has a limited impact on BR statistical uncertainty due to the consequent number of reconstructed events in the data set (an order of magnitude larger than SB). Further studies should be led in order to optimize the selection of muon candidates, avoiding such cuts that can result in an increased uncertainty on estimates from short data sets like SB.

The total systematic uncertainties are shown in Fig. 6.13. This calculation accounts for uncertainties linked to muons physics ∆ µ sys and for uncertainty linked to the telescope elevation angle known with an error of ±0.5 • . As mentioned previously, this error propagates to the thickness computation, especially for the lowest zenith angles and thus to the final density estimates. This systematic is referred to as ∆ θ sys . All mean uncertainties are shown in Tab.6.2. The calculation of ∆ µ sys accounts here for the uncertainty σ model related to the differential flux calculation with CORSIKA and to the computation of E min (cf. Sec. 6.1) that propagate to the integral muon flux. We also take into account an uncertainty linked to the unknown nature of the traversed medium σ medium which influences muon absorption. For this, we consider the relative error to standard rock transmitted flux for an equally mixed water-rock medium flux (cf. Fig. 6.3b), knowing the presence of liquid saturated zones within the volcano as well as dry zones [START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF]. Their distribution is still poorly constrained to this day. Further studies could lead to estimates of standard rock flux deviation for realistic andesitic rock-altered mineral composition, obtained from sampling campaigns on the dome, but regarding the calculations done in [START_REF] Lechmann | The effect of rock composition on muon tomography measurements[END_REF], we do not expect the effect to be larger than a few percent for the opacity range of La Soufrière, thus lower than the impact of water content in the rock medium. The estimation of these systematic uncertainties is key for setting realistic constraints during data inversion and performing three-dimensional density modeling of the dome, as we will present in Sec. 6.5.3. As we have mentioned at the end of Sec. 6.1.3, the systematic uncertainty linked to the flux contamination by low-energy particles is not taken into account here, since at the time of writing we unfortunately miss a full characterization study of the atmospheric background, which represents a source of bias for transmitted flux estimation in surface muography experiments ( [START_REF] Nishiyama | Monte Carlo simulation for background study of geophysical inspection with cosmic-ray muons[END_REF], [START_REF] Gómez | Forward scattering effects on muon imaging[END_REF]).

Inverse problem approach

To better understand Earth's physical processes, numerous fields in geosciences rely on subsurface numerical modeling which provides clearer descriptions of the Earth's interior structure and composition. The main challenge lies in the combination of data sets coming from multiple sources of information: geophysical, geological, geochemical and petrophysical data. Assuming a physical model describing the properties of interest of a system (e.g. density, electrical conductivity, seismic wave speeds), numerical modeling of expected geophysical data can be performed. This operation is referred to as a forward problem. The latter thus consists in predicting outcomes for experimental data d from the knowledge of the physical model M described by parameters m. The forward problem solution can be presented as:

d = f(m) + e, (6.2)
where d corresponds to the expected data vector; f is an operator (linear or nonlinear) called the forward operator expressing here the true physical model parameterized by m.

The vector e contains the measurement errors and any errors in the physical model or its numerical implementation. Regarding muography applications, f would theoretically describe the muon flux attenuation through a complete physics description of the muon flux at the observation site and the muon absorption in matter, accounting for the traversed medium physicochemical properties along all possible paths, in the set of parameters m. The vector e would stand for the statistical and systematic errors regarding e.g. the fluctuations in energy loss during muon propagation and instrumental detection effects, while d would contain the number of detected muons in each detector pixel. For example, a forward problem is solved when assessing the experimental feasibility of a muography experiment -namely estimating the data that would be produced given a certain detector acceptance, resolution and exposure time -from a hypothesized flux model, along with density and composition of the scanned object. After a data-taking campaign, the situation is reversed and the estimation of the unknown parameters m describing the system's properties of interest from a set of observations d is foreseen. This type of problem is then referred to as an inverse problem since it schematically involves the inversion of the aforementioned f operator to retrieve m. Various strategies exist to tackle inverse problems depending on their specifics, along with the instability and non-uniqueness of the inverse solutions, which we will not cover here. Indeed, several questions are raised when tackling an inverse problem. First, what are the model parameters of interest? Is the problem linear or non-linear? Should it be considered continuous or discrete? If discrete, how should the system be modeled (e.g voxelized volume, nodes, tetrahedral mesh)? Which mathematical approach should be used (e.g deterministic, Bayesian)? What cost function should be minimized? Each inverse problemsolving strategy goes with its criteria to define the "best" estimate of the model parameters.

When the relation between the physical model parameters and the data is assumed to be linear, Eq. 6.2 becomes:

d = F m + e, (6.3)
where d ∈ R n is the data vector, F ∈ R n×p is a linear operator, m ∈ R p is the model parameter vector and e ∈ R n represents the stochastic and systematic error vector. N.B F -1 (de) is not a solution for m, since F is generally not directly invertible. Even if F coincidentally happens to be a squared matrix then it is generally rank-deficient2 .

A deterministic approach consists in seeking a unique set of model parameters, minimizing the residuals between the experimental data and the model response. As they represent a distance measurement, these residuals are most commonly computed as the l2-norm (i.e the Euclidean distance) between experimental data and forward data inferred from the model response.

In the case of a linear inverse problem, when the data vector d is produced, the parameter estimates m est are given by the least-squares solution:

m est = (F t C -1 D F) -1 F t C -1 D d (6.4)
The data uncertainties are contained in the covariance matrix C D . If all errors are assumed to be uncorrelated :

C D = diag{σ 2 1 , ..., σ 2 n }.
Another strategy consists in following a Bayesian probabilistic approach where parameters are described in terms of probability density functions. This approach relies on prior information on the model parameters and data set uncertainty. Widespread in geoscience applications, this formalism was thoroughly developed by Albert Tarantola (cf. [START_REF] Tarantola | Three-dimensional inversion without blocks[END_REF] and [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]), who introduced the combination of states of information paradigm, gathering the experimental information about data, the prior information about parameters and the theoretical information.

Gaussian linear inverse problem

Under the assumption that prior information, data and associated noise probability densities, are Gaussian distributed, the least-squares minimization method mentioned above can apply and the inverse problem can be solved in a deterministic way. For further details on the Gaussian linear inverse problem solving, we invite the reader to refer to [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. The prior information that model parameters m follows are Gaussian distributed around mean m prior with model covariance matrix C M , also referred to as the regularization matrix (cf. Sec. 6.4.2):

ρ M (m) = const × exp - 1 2 (m -m prior ) t C -1 M (m -m prior ) (6.5)
The data density distribution is also assumed to be described by a Gaussian probability function accounting for data uncertainties in covariance matrix C D :

ρ D (d) = const × exp - 1 2 (d -Fm) t C -1 D (d -Fm) (6.6)
.

In the Gaussian linear case, and under the hypothesis that the uncertainties on observed data d dominate the modelization uncertainties [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF], the posterior probability density that describes the model estimates m is simply inferred through the combination of states of information carried by prior functions in Eqs. 6.5 and 6.6:

σ M (m) = 1 k ρ M (m) ρ D (d) (6.7)
where k is a normalization constant k = M dm ρ M (m) ρ D (d), M being the model space.

Eq.6.7 then becomes:

σ M (m) = const × exp - 1 2 ϕ(m) (6.8)
where ϕ(m) is introduced as the objective or cost function [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]. The goal is then to retrieve the model estimates by minimizing the function ϕ(m):

ϕ(m) = (d -Fm) t C -1 D (d -Fm) + (m -m prior ) t C -1 M (m -m prior ) (6.9)
where m prior are the prior model parameters, C D and C M are respectively the data and prior model covariance matrices. We assume that the data uncertainties are uncorrelated and thus the covariance matrix can be written

C D = diag{σ 2 1 , ..., σ 2 N d }.
The minimization of ϕ(m) is achieved with the following expression that is the center (and the mean) of the Gaussian posterior model density function σ M (cf. Eq. 6.8) [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF]:

m = m prior + (F t C D F + C -1 M )F t C -1 D (d -Fm prior ) (6.10)
with the posterior covariance matrix:

C M = (F C D F t + C -1 M ) -1 (6.11)

Model regularization

To tackle the non-uniqueness of the inverse problem solution, and balance the impact of data in the posterior model parameter estimation, it is necessary to define a regularization scheme [START_REF] Menke | Geophysical Data Analysis: Discrete Inverse Theory[END_REF]. This model regularization can take the form of a smoothing constraint that can be introduced on the prior covariance matrix C M parameterized as an exponential correlation function, first introduced in [START_REF] Tarantola | Three-dimensional inversion without blocks[END_REF]:

C M (i, j) = σ 2 M exp (-r(i, j)/l c ), (6.12)
where σ M is the prior error made on the density which represents the allowed deviation to the prior density ρ prior . This parameter sets the trade-off between the prior information and the fit of input data d. Indeed as we can see from Eq.6.9, the higher is σ M , the less importance will be allowed to the model prior information. The parameter l c is the correlation length and r(i, j) is the distance between the i-th and j-th voxels. The larger l c is, the farther the smoothing constraint propagates to neighboring voxels. This particular scheme hence favors smoothed solutions on the defined l c , allowing adjustment of the data inversion on different scales and mitigating the impact of the arbitrary volume discretization of the system. We consider here l c to be isotropic. This smoothing procedure was already employed in the context of muography data inversion in [START_REF] Nishiyama | Integrated processing of muon radiography and gravity anomaly data toward the realization of high-resolution 3-D density structural analysis of volcanoes: Case study of Showa-Shinzan lava dome, Usu, Japan[END_REF], [START_REF] Nishiyama | 3D Density Modeling with Gravity and Muon-Radiographic Observations in Showa-Shinzan Lava Dome, Usu, Japan[END_REF], [START_REF] Barnoud | Bayesian joint muographic and gravimetric inversion applied to volcanoes[END_REF], [START_REF] Nagahara | Three-dimensional density tomography determined from multi-directional muography of the Omuroyama scoria cone, Higashi-Izu monogenetic volcano field, Japan[END_REF].

Inversion of muography data

The problem of muography data inversion is generally tackled as a linear inverse problem [START_REF] Nishiyama | Integrated processing of muon radiography and gravity anomaly data toward the realization of high-resolution 3-D density structural analysis of volcanoes: Case study of Showa-Shinzan lava dome, Usu, Japan[END_REF], [START_REF] Rosas-Carbajal | Three-dimensional density structure of La Soufrière de Guadeloupe lava dome from simultaneous muon radiographies and gravity data[END_REF] and [START_REF] Peter G Lelièvre | Joint inversion methods with relative density offset correction for muon tomography and gravity data, with application to volcano imaging[END_REF]. The latter is set on a discrete volume encompassed by the muon telescopes acceptance cones (cf. Fig. 6.7 and Tab. 6.1) divided into n v cells, nodes, or voxels, described by parameters m = {ρ 1 , ..., ρ nv } = {ρ} j , which are the density estimates in each j-th cell. The data to be inverted are represented as a vector of n d elements accounting for mean density values ρ i recorded in each i-th detector line of sight, which is directly derived from opacity ϱ i estimation along travel length

L i , such that d = {ϱ 1 /L 1 , ..., ϱ l /L n d } = {ρ 1 , ..., ρ n d } = {ρ} i .
Hence, for the i-th line of sight Eq.6.3 gives:

d i = nv j=1 F i,j ρ j , (6.13)
where F is the matrix operator that performs the mapping of the data component d i on the dome voxels, of index j, referred to as the data kernel [START_REF] Menke | Geophysical Data Analysis: Discrete Inverse Theory[END_REF].

Synthetic test

To optimize the model regularization described in Sec. 6.4.2 for the inversion of the real muon data, we have designed a specific synthetic test on a density toy model of the La Soufrière lava dome. As we are solving a discrete inverse problem, we have constructed a voxelized dome model, with each cubic voxel of side 16 m assigned to a true density value.

The three-dimensional toy model is shown in Fig. 6.14, where the shape of the surface voxels is adjusted to fit the terrain topography. The dome model features a total of 16 032 voxels, shared between 14 416 bulk voxels of density ρ b = 2.0 g cm -3 and 1616 anomaly voxels of density ρ a = 1.0 g cm -3 . The anomaly is cylindrical (with radius 100 m and height 300 m) coaxial with the dome's central axis.

In the following section, we will present the results of the inversion of synthetic telescope datasets. The datasets are Gaussian distributed around the true density model ρ true described above, with a standard deviation set accordingly to the real uncertainty measured for mean density estimates in Sec. , where ∆ tot i is the For a Bayesian inversion scheme as used in this study, the choice of regularization parameters introduced in Sec. 6.4.2 highly depends on the geometry and data coverage of the structure. The tuning of these parameters consists in finding the optimal trade-off between the size of the regularized solution and the quality of the fit to the data [START_REF] Per | Analysis of Discrete Ill-Posed Problems by Means of the L-Curve[END_REF]. This operation is classically carried out with the means of the so-called L-curve, displaying the size of the regularized solution as a function of the data misfit. The identification of the maximum curvature on the L-curve, i.e. the "corner", is expected to give the optimal set of regularization parameters [START_REF] Per | The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[END_REF].

Following the methodology described in [START_REF] Barnoud | Bayesian joint muographic and gravimetric inversion applied to volcanoes[END_REF], we performed 8 × 8 inversions of the four telescope synthetic data set over a parameter space ranging between σ M ∈ [0.1; 0.8] and l c ∈ [50; 400]. The final result is displayed in Fig. 6.15 where the posterior standard deviation σ M (diagonal terms of the posterior covariance matrix C M , in Eq.6.11) as a function of the data misfit ϕ d defined as :

ϕ d = 1 n d n d i=1 d syn i -d post i σ syn i 2 (6.14)
where n d is the number of data points, d syn is the synthetic data vector, and d post = Fρ post where ρ post is the final dome density model i.e. the posterior density estimates vector. In Using these regularization parameters, the synthetic density model shown in Fig. 6.16 with the full four-detector array was compared to additional synthetic tests respectively conducted with one detector data set (SNJ), and three detectors (SNJ, SB, BR) as used in past studies (e.g. [START_REF] Rosas-Carbajal | Three-dimensional density structure of La Soufrière de Guadeloupe lava dome from simultaneous muon radiographies and gravity data[END_REF]), in order to see the impact of the current 4-telescope configuration. We first observe that the anomaly fails to be reconstructed in the case of a single telescope data set, as low-density voxels distribution is oriented along the central sight axis of the telescope. Whereas in the three-telescope case, the cylindrical anomaly is reconstructed but tends however to be artificially blurred over a scale ranging from 1.5 to 2 cylinder radius and this blurring effect significantly increases with depth, as noticed comparing reconstruction at altitudes 1300 m and 1400 m (Fig. 6.18 (b) and (d)) which could be explained by the reduced coverage. Besides density artifacts appear, especially in the direction of the SB and BR central axis. The four data sets allow for mitigating this blurring effect and deeper propagation of the cylindrical anomaly, even though the OM acceptance cone just covers altitudes above 1350 m. This first illustrates the advantage of distributing as many detectors as possible around the scanned object to optimize the data coverage of the whole structure. The 4-telescope model also presents some small artifacts on its edges close to detector positions, but also in the northeast region of the dome. However, these artifacts seem to have a limited impact on the posterior detector data distribution, shown in Fig. 6.19, as the data misfit in each detector is also distributed around one. The latter displays for each detector the posterior mean density computed over all the crossed voxel posterior density values for each line-of-sight, e.g for the i-th line-of-sight of SNJ detector:

ρ SNJ i = F SNJ i ρ post = 1/n SNJ d n i vox j ρ post i,j
, where n i vox is the number of traversed voxels, and ρ post is the posterior density vector, containing the final density model estimates.

Regularization parameters optimization

As stated in the above section, the choice of the regularization parameters highly depends on the geometrical layout of the muography experiment (number and spatial distribution of muon telescopes). To optimize the tuning of these parameters and prevent any potential bias, cross-validation methods can be employed as in [START_REF] Barnoud | Bayesian joint muographic and gravimetric inversion applied to volcanoes[END_REF]. We give an overview of this work in the following.

Cross-validation consists in splitting the original data set into k subsets. Among these, one subset will be tagged as a validation subset or test subset, while the remaining k -1 will serve as input for model computation with a given set of inversion parameters (σ M , l c ). The task is repeated k times, each for a different test subset In the end, all of the data subsets have once served as a validation test and the k inversion performances are quantified using for example the Mean Square Error (MSE) as an estimator between test model data and test true data. Hence, the optimal set (σ M , l c ) that minimizes the MSE can be robustly estimated. The procedure is known as a kf old cross-validation procedure. The particular case where k equals n, the size of the whole data set, is referred to as the Leave-one-out cross-validation (LOO). For both muon data inversion and muon-gravimetry joint-inversion, Barnoud et al. [START_REF] Barnoud | Bayesian joint muographic and gravimetric inversion applied to volcanoes[END_REF] thoroughly studied the potential of Bayesian inversion on different inversion scenarii. Both cross-validation methods mentioned above were compared for the tuning of regularization parameters. The regularization method was based on smoothing constraints applied in covariance matrix C M . The smoothing was parameterized with a Gaussian correlation function of parameters (σ M , l c ) analog to what is presented in Sec. 6.4.2. The synthetic density model was set on a homogeneous volcano model (Puy de Dôme, Chaîne des Puys, France), generating muon data sets from three existing detector viewpoints, along with 650 gravimetry data points. Among the different inversion scenarii that were explored e.g. one or three muon viewpoints, joint-inversion of muon and gravimetry data, the cross-validation methods obtained the best results in the muon- gravimetry joint-inversion case. Besides, it turns out that for less than three muon detector viewpoints, the gravimetry data help to better constrain the surface anomalies identified with muography, but also constrain the uncrossed regions' muography line of sights or the regions where the flux statistical uncertainties dominate in the most horizontal directions.

Inversion of real data

Using the inversion scheme introduced in Sec. 6.4.2 and the tuned parameters obtained from synthetic tests, we can now perform density modeling from the inversion of real data, using the two-dimensional mean density estimates shown in Sec. 6.3 as data inputs. In Fig. 6.20, the three-dimensional density estimates (cf. Eq. 6.10) at four different altitude slices are shown, with the associated posterior error in Fig. 6.21. The model shows a distribution of the largest density values (1.6 ≤ ρ ≤ 2.7 g cm -3 ) along a band crossing the dome from the South-East region to the North-West. Within this denser region, a low-density anomaly is located in the direct vicinity of the dome center, in the South-East direction, particularly visible in (b) and (c) (X ∈ [0, 150] m and Y ∈ [-150, 0] m). The Fig. 6.22 highlights the overlapping of this anomaly with the most active fumarolic region, where the most advanced signs of hydrothermal alteration are registered ( [START_REF] Moretti | The 2018 unrest phase at La Soufrière of Guadeloupe (French West Indies) andesitic volcano: Scrutiny of a failed but prodromal phreatic eruption[END_REF], [START_REF] Moune | Gas Monitoring of Volcanic-Hydrothermal Plumes in a Tropical Environment: The Case of La Soufrière de Guadeloupe Unrest Volcano (Lesser Antilles)[END_REF]). We also emphasize that this anomaly appears in the region covered by all four telescope acceptance cones, i.e. on the best-resolved region of our model. Besides, the model features the largest low-density region in the South-South-West region at the lowest altitudes, closest to the SB and SNJ detectors. As previously mentioned, the topography of this region of the dome is assumed to particularly favor the forward scattering of low-energy particles towards the telescope acceptance cone which causes muon flux overestimation and thus could explain the presence of this anomaly. Through the inversion of SB, SNJ and BR data solely, it has been shown that OM density estimates (cf. Fig. 6.10 (d)), which feature the largest lowdensity region, have no impact on the presence of this South-South-West large anomaly. However, OM estimates are more likely to cause the distribution of low-density voxels distributed around the telescope position in the North Fault. We recall the presence of one of the largest cavities in the dome, the Spallanzani cave, whose entrance is located at roughly 70 m of OM observation site, almost aligned with the central observation axis of the telescope, along the fault. The new density model derived through muon data inversion introduced in Sec. 6.5.3 and shown in Fig. 6.20 can be compared to the previous model obtained in [START_REF] Rosas-Carbajal | Three-dimensional density structure of La Soufrière de Guadeloupe lava dome from simultaneous muon radiographies and gravity data[END_REF]. The latter was computed through the inversion of muography data sets recorded between June and August 2015 in three 3-panel detectors, for a total of 931 muon data points (line of sights). These data were coupled with gravity measurements including 103 data points, recorded between March 2014 and February 2015. This former density model already featured low-density anomalies in the Southern region of the dome. However, the contrast between low density and high density of ∆ρ ∼ 0.5 g m -3 in the most active Southern region was only visible in this area between 1372 and 1404m a.s.l depths. The presented model has been obtained thanks to an improved muon tomography layout, featuring one extra telescope in the Northern fault and an upgraded 4-panel detector that gives three muography images. Besides this improved geometry, the telescopes were operated for much longer periods (cf. Fig. 6.9). In the Southern region, the found low-density anomaly was propagated down to 1300 m, deeper than the previous model in this critical area highlighted in the Fig. 6.22, between the Tarissan pit and the South Crater where the degassing activity is intense and the most pronounced signs of rock alteration are visible. Further study should determine to what extent this deepest anomaly propagation is due to the smoothing regularization we impose on the model covariance matrix, by assessing the model spatial resolution.

Spatial resolution

The synthetic test conducted in Sec. 6.5.1 has shown that the current data coverage with four muon telescopes allows reconstructing a cylinder-shaped anomaly radius 100 m (the dome summit plateau has an approximated 400 m-long diameter [START_REF] Boudon | A new scenario for the last magmatic eruption of La Soufrière of Guadeloupe (Lesser Antilles) in 1530 A.D. Evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF]) upon the whole height of the volume encompassed by telescope acceptance cones union, with increasing blurring with depth. For comparison with our case of study, Naghara et al. [START_REF] Nagahara | Three-dimensional density tomography determined from multi-directional muography of the Omuroyama scoria cone, Higashi-Izu monogenetic volcano field, Japan[END_REF] applied a similar linear inversion, with smoothing regularization to the inversion of muography data recorded in 10 emulsions detectors, surrounding the scoria cone of Omuroyama, Japan, which has larger dimensions than the La Soufrière dome with a basis diameter of 1.1.103 m and 5.8.10 2 m. Through different simulations of density anomalies distributed as e.g cylinder and dyke geometries inside a homogeneous cone, they assessed a spatial resolution 100 m, with model regularization applying on a smaller smoothing scale (l c = 70 m, σ ρ = 0.3 g cm -3 ). Indeed, beyond the simple voxel size of the dome volume, the spatial resolution of our model is driven by the large-scale smoothing (l c = 140 m) we impose for model regularization. Among the methods available to assess the impact of each voxel on the spatial resolution, the main way is to derive the resolution matrix linked to our model [START_REF] Menke | Geophysical Data Analysis: Discrete Inverse Theory[END_REF]. The analysis of these matrix columns can provide information on the variations of the spatial resolution in vertical and horizontal directions but the computation of this matrix is computationally intensive and in practice, only key points from the model can be extracted and used for interpolation [START_REF] Alumbaugh | Image appraisal for 2-D and 3-D electromagnetic inversion[END_REF]. Another approach is based on the resolving kernels, by adopting a functional formulation of the inverse problem [START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF], where the muon data kernel elements (cf. Eq. 6.13) and the 3-D density distributions (posterior, true and prior) are considered as continuous functions 3 . Resolving kernels are functions that can be seen as spatial filters defined on the investigated volume that are the linear combination of the data kernel elements. Applied to a true density function, they can render the achievable resolution on the posterior density distribution. Using this inversion scheme, Jourde et al. [START_REF] Jourde | Improvement of density models of geological structures by fusion of gravity data and cosmic muon radiographies[END_REF] have led a study on the interest in combining gravimetry data points with muography data through joint-inversion. This study showed that for regions scanned by at least two muon detectors, gravity data does not bring any additional constraint on density values, but does help in deeper regions, not covered by muon paths. We highlight that this conclusion converges with the more recent results obtained by Barnoud et al. in [25], previously mentioned in Sec. 6.5.2.

Refining volcanic rock mechanical behavior

In chapter 4, Sec. 4.2.3, we introduced the role of hydrothermal alteration in the potential weakening of volcanic rock strength. For volcano flank stability studies, the interest in three-dimensional density modeling notably lies in setting better constraints on the spatial distribution in the edifice of some main mechanical properties such as the rock strength, the porosity and the strain rate, that are driving the stability of the whole edifice. The density estimates obtained through muography are in fact estimates of the rock bulk density. As introduced in Sec. 4.2.3, bulk density is linked to rock mineral, fluid content and porosity. Furthermore, bulk density can be considered a proxy for rock strength. The latter along with the porosity represent key inputs for pursuing further numerical modeling of edifice stability and refining the characterization of partial flank collapse hazard [START_REF] Peruzzetto | Modeling of partial dome collapse of La Soufrière of Guadeloupe volcano: implications for hazard assessment and monitoring[END_REF], which is a main component of volcanic risk analysis.

The study conducted by Heap et al. [START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF] on the degree of alteration in different regions of the La Soufière lava dome leads notably to measurements of the uniaxial compressive strength as a function of wet bulk density. Like in the original paper, the empirical powerlaw relation between uniaxial compressive strength and bulk density allows converting the new density model presented here into a first-order model of uniaxial compressive strength for the lava dome shown in Figs. 6.24. This model is a simplified representation of the strength distribution in the dome as it does not account for example for the increase of rock strength with depth. We estimated that the South-East low-density/low-strength anomaly that appears in the vicinity of the dome center with strength below 3 MPa represents a volume of 3.547.10 6 m 3 (24% of the model volume, corresponding to 7% of the total dome volume), while the surrounding high-strength volume that appears in blue shades (σ p > 10MPa) accounts for roughly 9.425.10 6 m 3 (63% of the model volume, 18% of the total dome volume). Hence about 37% of investigated volume can be considered to have low strength, a lower percentage than the 55% presented in [START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF]. However, the precedent model was considering a larger inversion volume up to 1200 m at depth than this current model by accounting for density estimates from gravimetry data to cover the volume portion below the muon telescope horizon. This new strength estimate is highlighting the large volume encompassed by the south anomaly and thus its potentially significant impact on a flank destabilization in this dome region, mechanically weakened by the hydrothermal fluid circulation [START_REF] Brothelande | Fluid circulation pattern inside La Soufrière volcano (Guadeloupe) inferred from combined electrical resistivity tomography, self-potential, soil temperature and diffuse degassing measurements[END_REF], [START_REF] Tamburello | Spatio-Temporal Relationships between Fumarolic Activity, Hydrothermal Fluid Circulation and Geophysical Signals at an Arc Volcano in Degassing Unrest: La Soufrière of Guadeloupe (French West Indies)[END_REF].

Comparison with other geophysical observables

Besides laboratory tests, rock alteration at the field scale can be studied through geophysical methods that measure proxies to this parameter. Electrical resistivity tomography (ERT) models (cf. chapter 4, Sec. 4.3.4) give insights into the distribution in the investigated volume on the rock fluid content, but also on conductive materials like clay content. However, the correlation between resistivity and density is not easy to establish. We plot in Figs. 6.25 and 6.26 the projections of the electrical conductive regions presented by Rosas-Carbajal et al. in [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF]. These regions referred to as A1 and A2 respectively correspond to conductivity values of σ A1 ∈ [0.1, 1] S/m and σ A2 > 1 S/m. The A1 anomaly is assumed to be composed of moderately altered weak rocks, while the high conductivity A2 anomaly is interpreted as a region of highly altered weak rock, saturated with hot acid fluids. Even if the A2 region stops at z=1300m, as seen on the vertical slice of Fig. 6.26 we see that the anomaly head locates in the same pathway drawn by the south density anomaly of the present model. This pathway separates two high-density regions above 2 1. Previous results [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF] were pointing toward the presence of a dense unaltered and low conductive superficial region in the South-South-East flank referred to as a 'bulge' [START_REF] Boudon | A new scenario for the last magmatic eruption of La Soufrière of Guadeloupe (Lesser Antilles) in 1530 A.D. Evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF] (cf. chapter 4, Fig. 4.3), overlying the conductive anomalies A1 and A2. Even though the density model presents unphysical density (< 1g cm -1 ) for the closest voxels to the surface, we can interpret the presence of this Southern high-density region as a manifestation of this dense bulge region. , superimposed with the isocontour points of electrical conductive anomalies A2 and A1 presented in [START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF]. N.B The northern part of the A1 region shown here is not represented in Fig. 6.25.

Conclusion

Throughout this chapter, we have presented the tomography analysis of the consistent muon data sets recorded at four locations around the lava dome between September 2016 and August 2019 for a total number of 4.10 7 reconstructed events. These data sets have allowed estimating in total 6 two-dimensional mean density maps (5 with 3-panel telescope configurations and 1 with 4-panel configuration). We have discussed the implications of the different sources of uncertainty on the transmitted muon flux signal and assessed some of their significant contributions (e.g. muon transmitted flux estimation, unknown nature of the propagation medium). We have made a detailed estimation of these uncertainties to maximize the quantity of prior information input to numerical modeling. However, the characterization of a main source of background originating from the low-energy particle scattering is still missing. The two-dimensional density estimates have allowed further three-dimensional modeling in the framework of a linear inversion relying on a prior density distribution and on a smoothing regularization scheme for the model covariance matrix. A fine assessment of the model spatial resolution needs to be carried out and methods to optimize the tuning of smoothing parameters could be further employed, notably through cross-validation. Despite the resolution affected by the imposed smoothing regularization, the obtained posterior density model features a low-density anomaly whose location overlaps very well with the most active region of the lava dome in the Southern region. This result echoes the outcomes of recent studies that have emphasized the importance of crossanalysis between geological, geochemistry and geophysical observables (cf. [START_REF] Gonidec | Abrupt changes of hydrothermal activity in a lava dome detected by combined seismic and muon monitoring[END_REF], [START_REF] Tamburello | Spatio-Temporal Relationships between Fumarolic Activity, Hydrothermal Fluid Circulation and Geophysical Signals at an Arc Volcano in Degassing Unrest: La Soufrière of Guadeloupe (French West Indies)[END_REF], [START_REF] Moretti | The 2018 unrest phase at La Soufrière of Guadeloupe (French West Indies) andesitic volcano: Scrutiny of a failed but prodromal phreatic eruption[END_REF]), specifically through a combination of laboratory studies on rock mechanical properties and field measurements provided by muography [START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF]. This work has illustrated the advantage allowed by the four-muon telescope tomography setup that encompasses a significant fraction of the dome volume ( 30%) and provides unique insights into the shallow dome density distribution where the hydrothermal activity is the most intense. The muon density estimates are thus precious inputs for achieving a fine characterization of the dome flank stability, whose potential collapse is one of the main volcanic hazards.

Chapter 7

Conclusions and Perspectives

The La Soufrière de Guadeloupe constitutes an ideal natural laboratory to apply and develop muography. As this active volcano is one of the most instrumented in the world, cross-analysis with different geological, geochemical, and geophysical observables is made possible which allows refining the development of this unique imaging tool and consolidates its use in the scope of modern volcanology. This environment goes with many technical challenges, either regarding the good operation of the telescope, but also the numerous sources of background that come to pollute the low transmitted muon flux signal, outgoing from the volcano. During this thesis, the data processing developed along with the Monte-Carlo simulation tools has allowed a better grasp of the quality of the muon reconstruction in La Soufrière telescopes. These developments were motivated by the irreducible contamination of the transmitted muon flux, either due to atmospheric particles background or due to instrumental effects that can randomly occur in scintillator detectors and especially in field telescopes that need to withstand different sources of environmental stresses such as variations in temperature, intense tropical rain, and humidity that alter their performances. The detection efficiency of the scintillator matrices is taken care of by the estimation of the experimental acceptance that corrects the transmitted flux estimation. This acceptance relies on a unique open-sky calibration run taken before the beginning of the tomography acquisition and on a model flux derived with CORSIKA, accounting for main corrections to the muon flux linked to the altitude and to the geomagnetic effect at La Soufrière geographical coordinates. Even though scintillator detector performances are trusted for being particularly stable in time, monitoring of the open-sky flux, using the portion of sky visible during the tomography could allow correction for potential variations of efficiency due to the aforementioned stresses. This work has hence allowed the building of a full processing pipeline for the reconstruction of large muon data sets, along with routines to perform the tomography data analysis up to the three-dimensional density modeling as presented in chapter 6. It has been designed to be versatile, and easy to adapt, either for implementing new objects such as additional telescope geometry with different detection matrices layouts or including new tracking procedures. The developments made in the GEANT4 simulation framework in which telescope perfor-mances are assessed can be the basis for further simulation-driven studies. As the framework encompasses the possibility of including a dome mesh model, further studies can be conducted for achieving the low-energy muon scattering background characterization, one of the main background sources that affect muon flux signal, notably in the South-West region of the dome where several factors favor this contamination: the two detectors are the most distant to the dome with the largest acceptance cones and facing a smoothed topography. On La Soufrière telescopes, a passive lead shielding panel in the middle of the detector allows the rejection of low-energy particles, but the provided energy cut-off is too low for optimal rejection. Developments and feasibility studies on a new layout of a transportable field scintillator telescope are ongoing at IP2I, Lyon. The idea is to insert a water-Cherenkov detector tank, between the front and rear detection matrices, replacing the current shielding panel. These detectors are designed to improve the suppression of background noise such as electrons by detecting the amount of Cherenkov light emitted in the water, linked to the energy of the incident particle. With an ideal design, this detector would also tag back-scattered muons that can enter the telescope by the rear side. For this source of background, they offer an alternative to time-of-flight measurements whose resolution is affected in scintillator detectors by several parameters such as the propagation of light driven by the length of the optical fibers.

A Bayesian linear inversion of the four muon telescope data has been conducted. In order to maximize the quantity of input prior information, we have made a detailed assessment and propagation of uncertainties linked to mean density estimation. These uncertainties are notably linked to the muon transmitted flux model derived with CORSIKA and to the unknown medium composition, whose water content significantly impacts the muon propagation. The obtained three-dimensional density model features a low-density anomaly whose volume was estimated to account for 7% of the dome's total volume. Its location at the surface overlaps with the fumarolic region in the Southern region and seems to correspond to the actual knowledge of the hydrothermal fluid circulation pathway inside the dome, notably highlighted by ERT tomography. The density model was interpreted in terms of rock strength, thanks to laboratory measurements carried out on a specific range of altered rock samples of La Soufrière (cf. [START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF]) Further studies of the model spatial resolution need to be carried out to refine the characterization of the anomaly geometry and volume. Besides the Bayesian inversion scheme, Monte-Carlo methods are also widespread in geosciences thanks to their versatility. They can allow dealing with non-linear problems thanks to constraints on model parameters set by robust uncertainty estimation, as foreseen here. The sampling of the posterior parameter probability densities can take the form of a random walk in Markov chain Monte-Carlo algorithm like Metropolis-Hastings, where at each step the likelihood of the sampled model is derived regarding the fit to data. In our case, this stochastic exploration of the model space could optimize the spatial resolution of the density distribution but involves significant work on algorithm parameter optimization to achieve reasonable processing time due to the large size of the investigated dome volume, and the high statistical uncertainties that affect the horizontal telescope line of sights.

In addition to the interest in achieving a better characterization of the La Soufrière dome rock mechanical properties distribution through density modeling, to assess the risk of partial flank collapse, one burning question concerns the application of muography to "real-time" volcanic activity monitoring, and the detection of hydrothermal system fluid circulation and fluid pressurization, possibly related to magmatic fluids ascent. As we have presented throughout this work, muography is subjected to main physical limits, the most obvious ones being the imposed horizon and its very low rate of muon events. At La Soufrière, the most horizontal lines of sight corresponding to kilometer rock thicknesses requires between one to two months of data acquisition to reach a 1-σ statistical significance on a 10% opacity variation. For the detection of early warning signals, the question hence shifts toward the compatibility of these acquisition times and the typical timescales involved in the hydrothermal system dynamics. The suddenness of some hydrothermal eruptions is extreme, giving volcano-tectonic seismic precursor signals less than an hour before the eruption (e.g. Mt.Ontake, Japan, September 2014, Whakaari/White Island, December 2019). But the variability of timescales is in fact quite large depending on the nature of the perturbation of the hydrothermal system (up to timescales of years for slow heating and of aquifers [START_REF] Montanaro | Phreatic and Hydrothermal Eruptions: From Overlooked to Looking Over[END_REF]) and obviously proper to the characteristics of each volcano. To be able to tell precisely what would an early-warning signal look like in terms of muon flux variation is hence difficult. It requires both input on the involved hydrothermal fluid volume and realistic timescales, which could then be interpreted in terms of opacity change seen in each telescope line-of-sight. In the end, to cope with the acquisition time limitation, the requested solution is the merging of neighboring telescope pixels at the expense of the spatial resolution on the scanned volume to increase the number of detected muons and overcome the statistical fluctuations ( [START_REF] Jourde | Muon dynamic radiography of density changes induced by hydrothermal activity at the La Soufrière of Guadeloupe volcano[END_REF], [START_REF] Gonidec | Abrupt changes of hydrothermal activity in a lava dome detected by combined seismic and muon monitoring[END_REF]). Such operation has successfully allowed detection variation up to the 3-day scale period, correlated with seismic noise and vent temperature time series.

Today, the network at la Soufrière comprises two muon telescopes that were recently upgraded in 2022 to four detection matrices: on the Eastern side (Roche Fendue site), and in the 30 th of August fault (South flank). This telescope benefits from optimal rock shielding and is the closest geographically to the southern active region. These data will allow for achieving an unprecedented resolution on the density anomaly highlighted during this work in this critical region by the intense circulation of hot and acid hydrothermal fluids.

studies and turned into an in situ experiment, by [START_REF] Alvarez | Search for hidden chambers in the pyramids[END_REF], during an imaging campaign performed on the Chephren pyramid of the Giza ancient Egyptian complex. This pioneer field measurement and feasibility demonstration lead the path towards the applications that are used these days in volcanic structure imaging, with the first measurements led by Japanese teams in the 1990s [START_REF] Nagamine | Method of probing inner-structure of geophysical substance with the horizontal cosmic-ray muons and possible application to volcanic eruption prediction[END_REF]. Repeating muon measurements at several positions around a volcano, the final goal is to combine the 2-D mean density images to compute a three-dimensional To this day, muon telescope data at La Soufrière were reconstructed using a fast and simple tracking principle: a straight-trajectory check of the most-energetic hits on the 3 scintillator panels. But muons are not the only particles generating a signal in scintillator panels, and often many hits per panel are observed in what is registered as a single "event". Muons are indeed expected to form a straight track in the detector, and they are minimum ionizing particles, which implies that their associated energy deposit in the scintillator medium -of the order of 2 MeV/(g.cm -2 ) -is not the highest possible on each panel. Therefore this kind of selection is already biased by the presence of other types of particles generated in atmospheric showers. Furthermore, depending on the relative angle between the muon trajectory and the detector orientation, a single muon can traverse more than one scintillator "pixel" per panel. A solid fit procedure of the hits is thus necessary to retrieve the correct muon trajectory and cope with contamination from other atmospheric particles and spurious hits formed by instrumental noise (e.g. optical or electrical cross-talks, dark counts arising in time coincidence).

In this paper, we present a new method to process the raw muography data, which improves the reconstruction of detected muons. Our tracking algorithm is based on a random sampling consensus procedure RANSAC (Fischler & Bolles, 1981), which provides the best fit to the particle trajectory con-sidering all the panel hits involved in a single event. To evaluate the performance of our algorithm, we use the GEANT4 Monte-Carlo simulation framework for particle propagation [START_REF] Agostinelli | Geant4-a simulation toolkit[END_REF], and assess the reconstruction efficiency. Finally, we apply our algorithm to novel data acquired at La Soufrière de Guadeloupe by two muon detectors. We provide the relative density muographies and interpret them in terms of the main geological features observed at the volcano.

METHODOLOGY

Muon detection

Scintillator-based particle detectors are well-established tools for muography (Tanaka et cm long, 1 cm thick, and 2.5 cm wide, while its middle matrix ensures signal coincidence without affecting the total number of lines-of-sight and thus presents a lower pixel resolution with a 16x16 bar configuration, similar to SNJ matrices. All the matrices have the same detection area of 80x80 cm 2 .

Each scintillator bar has its own wavelength-shifter (WLS) fiber embedded within the scintillator medium to conduct the light towards a multi-anode photomultiplier (MAPMT), with 64 channels.

One MAPMT per detection matrix takes care of the light signal conversion to electrons and ampli-fication before processing by the readout electronics described in [START_REF] Marteau | Muons tomography applied to geosciences and volcanology[END_REF]. A simple straight-line interpolation of the signals in the first and last plane would provide an angular resolution of 12 mrad (3p) or 8 mrad (4p) for SNJ and of 6 mrad for BR.

In addition to the detection matrices, a 100 mm thick lead shielding panel has been installed on both telescopes behind the middle matrix, compatible with mechanical constraints on the structure. This passive shielding allows for stopping low-energy particles. We estimated its cutoff energy to be around 170 MeV, using the numerical simulations explained later on in this work. The shielding also

Rock density estimation

In muon absorption tomography, the observable is the number of reconstructed particle tracks N detected by the muon telescope during a period of time ∆T , for each line-of-sight r i,j where (i, j) represents a given front-rear pixels combination. Assuming a point-like telescope in comparison to the size of the scanned object, each (i, j) corresponds to a unique (θ, ϕ) angular direction. The detector is characterized by a direction-dependent experimental acceptance T exp , which accounts for the telescope geometry, as well as instrumental and reconstruction effects. To estimate T exp , the telescope is placed in open-sky mode during a calibration run, in a vertical position, that is, the central axis at zenith θ = 0 • , and the following quantity is derived: Once the acceptance has been calculated, the telescope is oriented towards the volcano and the transmitted muon flux noted I data can be estimated for each line-of-sight r i,j with the following expression:

T exp (r i,j ) = N sky (r i,j ) ∆T calib × I sky (r i,j ) [cm 2 .sr] (1
I data (r i,j ) = N (r i,j ) ∆T × T exp (r i,j ) [cm -2 .s -1 .sr -1 ] (2)
where N (r i,j ) is the number of transmitted muon candidates along r i,j , ∆T is the tomography run duration, and T exp (r i,j ) is the experimental acceptance in r i,j , defined in Eq. 1.

In order to infer density values from I data (r i,j ), a comparison with reference muon flux after attenuation in rock is necessary. The reference transmitted flux noted I calc ( , θ) depends on the zenith angle θ and on the crossed amount of matter, quantified in muography with a parameter called opacity , defined as the integral of density ρ along rock thickness L:

(L) = ˆL ρ(ξ) dξ [g.cm -2 ] (3)
As for the experimental acceptance computation (cf. Eq. 1), I calc ( , θ) is estimated from Monte-Carlo simulation e.g. CORSIKA, computationally intensive but more accurate at the observation site for the reasons mentioned before than semi-analytical models (e.

I calc (θ, ) = ˆ+∞ E min ( ) Φ(E, θ) dE [cm -2 .s -1 .sr -1 ] (4)
Once the opacity is estimated from data-model flux comparison, a mean density value can be obtained for each line-of-sight r i,j : ρ i,j = i,j /L i,j .

Data processing

Along with the energy deposits (so-called hits) generated by muons crossing the detector and used to reconstruct their trajectories, other sources induce signals in the scintillator bars: other atmospheric particles from electromagnetic and hadronic components of atmospheric showers, secondary background particles (electrons and photons) produced inside the detection volume, or instrumental noise caused notably by random emission of electrons from the MAPMT photo-cathode and cross-talks. To mitigate their impact on the track reconstruction process, we present a new track reconstruction algorithm that considers the presence of these "outlier" hits. The tracking algorithm was written in Python, with the use of the ransac module available in the scikit-image library (van der Walt et al., 2014). The fit procedure is also useful in the context of the 4-panel telescope configuration, to account for detection inefficiencies: for muons forming hits in only 3 panels, but contained in the 4-panel angular region, the trajectory model allows to extrapolate the muon position in the untouched panel.

Our event-by-event track fitting procedure lies on an hypothesize-and-test iterative process, called random sample consensus RANSAC (Fischler & Bolles, 1981). The different steps of the reconstruction workflow are represented in Fig. 2. First, we filter out events presenting a number of hits per panel larger than 10, which are likely to be due to an electromagnetic shower, or impacting less than 3 detection panels. The process starts by randomly sampling within the hit coordinates, a subset of points of a given size, and fitting a hypothesized line model to this subset. It then evaluates which hits of the whole dataset are consistent with this hypothesized model, computes a fit error, and counts the number of hits whose orthogonal distance to the model is below a given distance threshold t (set to one scintillator width). Those hits form the consensus set. The size of the initial sample s and the maximal number of iterations N max were set to maximize the number of muon hits tagged as inliers, using the simulation of an open-sky muon run with the framework described in Section 2.4. After N max iterations, the algorithm outputs the model that minimizes the fit error and maximizes the size of the consensus set. The hits belonging to the consensus are tagged as 'inliers' and the rest of the hits as 'outliers'. This method manages to increase the number of reconstructed muons and to improve the angular precision on the incident particle direction, compared to the former reconstruction algorithm used, relying on the check of the alignment of the most-energetic hits, i.e a check of the "straightness" of the track. A comparison of the two methods has been studied in [START_REF] Bajou | A New Versatile Method for the Reconstruction of Scintillator-Based Muon Telescope Events[END_REF].

Numerical simulations

Developing simulations for particle detectors in muography, as in all particle physics experiments, represents a crucial step to a better understanding of the detector and its response to particles of interest. More precisely, it allows estimating detection efficiency, a key parameter for quantifying the performance of our tracking algorithm. The most widespread tool for this purpose, the GEANT4 toolkit [START_REF] Agostinelli | Geant4-a simulation toolkit[END_REF] offers a framework with an exhaustive set of physics models to simulate particle interactions and tracking through a detailed numerical replica of the detector. The simulated signals can then be formatted as real data to test the reconstruction algorithm. Examples of simulated 10 GeV/c electron and muon events are shown in Fig. 3. The electron event in Fig. 3a is filtered out thanks to the 7-cm thick shielding panel between the middle 1 and middle 2 panels. The muon track in Fig. 3b is reconstructed with RANSAC. Along with the physical muon hits, Fig. 3b features outlier hits that were randomly added to the muon event signal. These "synthetic" hits were not assigned any intensity value (i.e. deposited energy) as this parameter is not considered in the current tracking workflow. This reconstructed event display illustrates the ability of the RANSAC procedure to fit the correct muon trajectory from a data set polluted by hits generated by instrumental effects like a deficient optical coupling between optical fibers, dark current arising from the photo-cathode and dynodes in MAPM, electronic cross-talk between MAPM channels.

RESULTS

Reconstruction performances

In the following section, we characterize the performance of our muon tracking algorithm for a 4panel telescope. For this purpose, we use the GEANT4 simulation setup described in Section 2.4. We simulate an open-sky acquisition and write the muon hits recorded in the scintillator panels following the real telescope raw data format. Then, we use our algorithm to reconstruct the particle tracks following the reconstruction outline depicted in Fig. 2. Because we know the exact particle trajectories from the GEANT4 simulations, we can assess the tracking algorithm's performance. The results presented hereafter were obtained on a pure-muon sample of 10 7 events simulated in GEANT4 with energies E µ ∈ [1, 10 6 ] MeV. Within this sample, we estimated that 7% of the generated muon tracks contained in the 4-panel acceptance cone are being absorbed in the detector shielding, before im- pacting the rear detection matrix. The RANSAC tracking algorithm allows extrapolating the muon intersection position on this rear matrix.

Influence of secondary particles

Secondary electrons and photons produced by high-energy muons inside the detection volume, notably after crossing the shielding panel, impact the quality of the reconstruction of the incident muon trajectory. The RANSAC tracking allows for mitigating the impact of those spurious hits. In Fig. 4 we show the comparison between the real particles obtained in GEANT4 and the hit tagging performed by the algorithm. For initial sample size s = 2 and N max ≥ 20 the true muon hit inlier fraction reaches a plateau at 98.4 ± 0.2%. Besides, the muon tracking result also shows relatively low contamination from electrons and photons: these secondary hits represent 14.3 ± 0.7% of the total hits, and nearly half of this fraction (6.7 ± 0.6%) have been tagged as outliers.

Reconstruction efficiency

To assess the reconstruction efficiency ε reco of our tracking algorithm, we computed for each pixel corresponding to a given line of sight r i,j , the ratio between the number of RANSAC-reconstructed primary muons and the number of generated primary muons in GEANT4 : ε MC reco (r i,j ) = N reco (r i,j ) N 0 (r i,j )

where N reco is the number of reconstructed muons, and N 0 is the number of primary muons.

The reconstruction efficiency obtained with the GEANT4 simulation ε MC reco (cf. Eq. 5) can be compared with the reconstruction efficiency estimated with open-sky calibration data, which we defined as the following:

ε DATA reco (r i,j ) = T exp (r i,j ) T geo (r i,j ) (6)
that is the ratio between the experimental detector acceptance T exp (cf. Eq. 1), and the geometrical acceptance T geo , whose analytical expression is only a function of the telescope geometrical parameters: distance between the front and rear detection matrices, number, length, and width of the scintillator bars in each matrix (Sullivan, 1971;Thomas & Willis, 1972). Examples of efficiency maps obtained with the simulation and with real open-sky calibration data are shown in Fig. 5.

The reconstruction efficiency depends on the detection efficiency of the scintillator bars ε scint , that is, the probability for a given hit in a scintillator bar to give a signal above the threshold. We assume ε scint to be the same for all X and Y bars over each panel, although small differences from one bar to another occur in reality due to instrumental effects such as a deficient optical coupling between fibers and the light readout system (cf. Sec. 2.1). In Fig. 6, we show how the mean efficiency ε MC reco as a function of the ε scint parameter. From this relation, and using the mean data reconstruction efficiency ε DATA reco , it is straightforward to interpolate the real telescope scintillator efficiency. We consider here the acceptance measurements performed on a 4-panel telescope (see Fig. 1b). As a result, we obtained the following scintillator efficiencies for the 3-panel and 4-panel events respectively: ε 3p1 scint = 0.899 ± 0.012, ε 3p2 scint = 0.900 ± 0.010 and ε 4p scint = 0.868 ± 0.015. These are consistent with the instrumental efficiency of the order of 0.9 usually considered for the technology of plastic scintillator bars with optical fiber readout employed in our detectors (Lesparre et al., 2012a).

Tomography data analysis

We present in this section the results obtained by applying our tracking algorithm (cf. Sec. 2.3) to real muon tomography data acquired by two muon telescopes (cf. Sec. 2.1): SNJ and BR, respectively located at South-West and East of the La Soufrière lava dome (cf. Fig. 1a).

The SNJ tomography dataset contains 1.9 × 10 7 reconstructed events acquired over a total run duration of 145 days in steady operation conditions (excluding dead time superior to one hour), between February and August 2019. The BR acquisition period ran from February 2018 to August 2019 over 547 steady days and contains 1.2 × 10 7 events. Both tomography runs were preceded by a 2-day long open-sky calibration run. Following the methodology explained in Sec. 2, the detector experimental acceptance is first estimated from calibration runs, then the incoming transmitted muon flux for each detector can be calculated, including the 3-and 4-panel configurations of detector SNJ.

To quantify the agreement between the data-estimated and modeled transmitted fluxes for different density values we use a Pearson χ 2 -test :

χ 2 (θ, ρ) = N los i,j I data (r ij ) -I calc (θ, ρL ij ) σ I (r ij ) 2 (7)
where N los is the total number of lines of sight at a given θ, and σ I is the statistical uncertainty on the estimation of I data .

We have chosen to present the following analysis results for both telescopes in terms of relative mean density variations rather than absolute density whose estimation suffers from irreducible background pollution (see Sec. 4). We then consider the quantity ∆ρ defined with respect to the average value ρ 0 = ρ from each telescope muography: ∆ρ = (ρρ 0 )/ρ 0 . Figure 7 shows on a given azimuth range [ϕ c -2.5, ϕ c + 2.5] • (where ϕ c is the central telescope azimuth) how the χ 2 varies as a function of ∆ρ and zenith angle θ for each muon telescope. The chosen azimuth range targets for both tele- The color coding corresponds to the density radiography and the thickness of rock traversed for each line of sight, respectively. When observing these figures, it should be noted that the relative density variations correspond to an average value through the whole trajectory inside the volcano, and not to the point where the circle is represented. We can observe in Fig. 8 c) that the high-density anomaly thus corresponds to lines of sight that have traversed the dome flanks in a region directly south of the summit, usually referred to as the "bulge" in the literature [START_REF] Boudon | A new scenario for the last magmatic eruption of la soufrière of guadeloupe (lesser antilles) in 1530 a.d. evidence from stratigraphy radiocarbon dating and magmatic evolution of erupted products[END_REF][START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF]Heap et al., 2021). This region is thought to consist of relatively fresh and dense rocks. Focusing on the lines of sight close to the South Crater and the Tarissan acid pond, we can interpret from Figs.

8 b) and c) that Tarissan is located in a region of overall lower bulk density than the South-Crater (CS). Finally, if we compare the relative density distribution to the rock thickness distribution, it is interesting that the color spatial distributions are not the same, suggesting that the density anomalies observed do not correspond to changes in the amount of rock traversed by the muons.

Figure 9 shows the equivalent tomography and bird view representations for the BR detector.

Although this is a 3-panel detector, the spatial resolution is increased with respect to SNJ because the scintillator bars are smaller. The largest density anomalies are observed for θ < 80 • and φ ∈ [260, 290],

that is, for lines of sight that cross the same "bulge" region as in the SNJ radiography (cf.

DISCUSSION

In this section, we discuss the results obtained regarding the performance of our particle tracking algorithm, the implications of the density radiographies provided, and possible improvements in future works.

Our RANSAC-based muon tracking algorithm was already shown to outperform the straightness check tests used previously by our team in terms of the number of events reconstructed and the precision on the inferred trajectory [START_REF] Bajou | A New Versatile Method for the Reconstruction of Scintillator-Based Muon Telescope Events[END_REF]. Here, we did a more exhaustive performance test using GEANT4 simulations to test our algorithm against perfectly known particle trajectories.

Moreover, our numerical simulations consider the secondary particles produced as a result of the interaction with the lead panel located at the center of our detector (cf. Fig. 4). The processing of the GEANT4-generated data is encouraging in the sense that our algorithm successfully tagged true muon hits as inliers and significantly mitigated secondary hits.

The comparison of real open-sky acquisitions with simulated data in equivalent conditions provides comparable results (cf. Fig. 6) and allows us to estimate the scintillator bars' efficiency. Our tracking algorithm is flexible in that it can be adapted to different geometries of scintillator panels.

The main drawback is that it takes more time to run that the straightness check routines: O(10 3 s) for 10 6 events on an 8-core CPU. Still, these are not prohibiting computation times considering that the analyses so far are not performed in real-time, and the significant improvement in data quality they provide.

The density spatial variations inferred from the real data acquisitions in La Soufrière de Guadeloupe can be qualitatively compared with a high-resolution electrical resistivity tomography of the dome performed by Rosas-Carbajal et al. (2016), keeping in mind that rock conductivity is also influenced by other parameters such as rock porosity and fluid content. This work highlighted the presence of two distinct electrical conductivity regions: a low-conductivity shallow region named "bulge" on the south flank composed of non-altered dense andesitic rock, overlying a highly conductive region composed of mechanically weakened rock by hydrothermal activity. This bulge region scanned by both telescopes appears as the reddish high-density regions on the three muographies, for ϕ > 30 • in Fig. 8a and 8b, and ϕ < 290 • in Fig. 9a. On the other hand, the observed low-density region on the south flank is thought to correspond to the superficial part of the high-conductivity anomaly rising up to the dome summit.

As of today, this high electrical conductivity anomaly is understood to be the main pathway for hydrothermal acid fluids circulation, connecting southern thermal springs at the base of the lava dome to the dome summit. Surface manifestation of the arrival of these fluids includes continuous emission of sulfur and chlorine-rich gas through the fumarolic vents around Southern and Tarissan craters (cf. Fig. 1a) [START_REF] Moune | Gas Monitoring of Volcanic-Hydrothermal Plumes in a Tropical Environment: The Case of La Soufrière de Guadeloupe Unrest Volcano (Lesser Antilles)[END_REF]. The continuous circulation of these fluids greatly contributes to the alteration and mechanical weakness of the dome rocks (Heap et al., 2021). Mechanical weakness is in part related to an increase in porosity, and thus it is reasonable that our muographies observe low-density values in highly altered regions as compared to fresh rock zones.

Among the structural features, several cavities of large volume inside the lava dome could have a significant impact on the observed densities. The BR muography featuring a broad low-density anomaly in Fig. anomaly that appears in Fig. 8a and b may also be caused by the existing network of cavities, expanding to the west. Nevertheless, the cavities delimitation is not well-documented in this area of the dome. As this low-density anomaly concerns the most superficial part of the dome and thus the shortest thicknesses L (cf. Fig. 8d) it is also likely to be affected by a systematic error on telescope orientation, whose impact on density estimation is important on this thinnest part of the dome. For the shallowest lines of sight of Fig. 8a and 8b covering a band of 3 pixels wide below the topographic profile (black line), we estimate that a 1-degree shift in both telescope azimuth and elevation gives a mean relative error ∆L/L = 0.21 ± 0.12, while for the rest of deeper lines of sights, this error reaches down to 0.09 ± 0.04. In the future, we intend to further develop the simulation tools to reproduce the geometry of the main cavities and provide a better characterization of their impact on the mean density estimates.

The telescopes are ideally located to scan what is thought to be the most-altered and collapseprone part of the lava dome on the south flank. A downside of their closeness to the volcano flanks is that for lines of sight close to the horizon, that is, for large zenith values, there is a high probability of exposition to a particular source of noise contamination. This contamination comes from irreducible background sources such as the forward scattering effect of low-energy muons. structed events for θ superior to 85°. This background noise is also location-dependent and thus requires specific estimation for each muon imaging study. To cope with this irreducible background noise and infer realistic absolute density values, the simulation setup presented in Sec. 2.4 is expected to provide the right framework to accurately estimate the percentages of forward-scattered to outgoing muons. We leave these developments for a future contribution.

CONCLUSION

We presented a new muon tracking algorithm to process data from 3-and 4-panel scintillator-based muon detectors. Our algorithm successfully captures muon hits and rejects outlier hits formed by secondary particles in the telescope data. We have also developed simulation tools that show that our new algorithm outperforms previously used ones based on straight-line analyses. Furthermore, we can use these simulations to reconstruct our scintillator's efficiency and study the experimentally calibrated acceptance against the simulation predictions.

Our algorithm was successfully applied to estimate muon density radiographies of La Soufrière with data acquired by a new generation of muon telescopes. The density variations have an improved spatial resolution with respect to previous works and are overall in agreement with previous studies.

Although the absolute mean density estimates are biased by irreducible background sources, the relative mean density radiographies still reflect the heterogeneous structure of the dome, thanks to long and steady runs of data acquisition obtained in the two different detectors.

Future work will focus on correcting the two-dimensional muographies from systematic effects such as muon forward scattering, and in the following, on performing data inversion on the corrected density maps to obtain a realistic three-dimensional density model of the dome. The forward scattering correction will be a natural improvement using the GEANT4 simulations developed in this study.

The topography of the terrain surrounding each detector has key importance in this matter. It can now be reproduced with the implementation of the digital terrain model in the presented GEANT4 framework. As an alternative to GEANT4, we also mention the PUMAS library [START_REF] Niess | The PUMAS library[END_REF]) that could also be used for scattering correction studies. This code is based on backward Monte-Carlo to perform muon transport in matter and was initially developed for muography applications. Furthermore, implementing the geometry of the main known geological structural features of the lava dome, fractures, faults, and craters is also possible and will greatly improve the quality of our radiographies and therefore of the three-dimensional density models obtained with them.

B Appendix B

This appendix is a conference proceeding written after the International Workshop on Cosmic-Ray Muography that took place in November 2021, in Ghent, Belgium. The proceeding was published in the Journal of Advanced Instrumentation in Science, in January 2022.

INTRODUCTION

The intense hydrothermal activity occurring at La Soufrière de Guadeloupe, in the French Lesser Antilles, is a major concern for volcanologists at the Volcanological and Seismological Observatory of Guadeloupe (OVSG) and for the local population that lives nearby. Partly hosted within the andesitic lava dome, this hydrothermal system, continuously fueled with meteoric waters, is indeed responsible for both violent phreatic eruptions (last major event occurred in 1976-1977), and fast rock alteration due to hot and acid fluid circulation that worsens the risk of a partial volcano flank collapse. Since 2015, the deployment of particle trackers developed at IP2I, Lyon, to perform dynamic muon imaging of the lava dome structure, in addition to other geophysical methods such as seismic monitoring, has allowed to increase the knowledge of the hydrothermal system dynamics (see [1]). Nowadays two new generation particle trackers equipped with 4 scintillator panels are installed at two different locations: South, shown on figure 1a, and East of the volcano. This 4-panel configuration allows to increase precision on the 2D density radiography of the central part of the scanned region and offers two more 3-panels sub-configurations, which lead finally to three different muon images. Previously acquired data by our group were based on 3-panel detectors, and processed using a direct straightness check on the scintillator bars hits in each panel. Here, we propose a more complex particle reconstruction algorithm, which takes into account the presence of outlier events and allows to improve the precision on the reconstructed trajectories.

DETECTION TECHNOLOGY

The detection panels are composed of an array of 16x16 plastic-scintillator bars of 5 cm width (detection area 80x80 cm 2 ) (fig. 1b). Each bar has its own Wavelength Shifting Fibers (WLS) embedded within the scintillator medium (fig. 1c), to collect the light emitted along the charged particle path, and guide the signal towards multi-anodes photomultipliers. In addition to the four detection matrices, a 100 mm thick lead shielding panel has been installed between the third and the fourth (rear) matrices, satisfying mechanical constraints on the telescope structure. This passive shielding allows to stop low-energy particles and also trigger electromagnetic showers inside our detection volume initiated by non-muonic events characterized with high hit multiplicity on the rear panel caused by secondary shower particles. For more information about the detector assembly, we invite the reader to refer to [2] and [3]; for details about the light and electronics readout system to [4] and [5].

DATA PROCESSING AND PERFORMANCES

Within telescope data, the detected muon events outgoing from the volcano are polluted with noise resulting from the electromagnetic component of atmospheric showers, and secondary electrons from interactions inside the detector, as well as an important contribution from low-energy muons scattered on volcano flanks [6]. To mitigate the noise contribution on radiographies, raw data are processed event-by-event with a preliminary filter on the minimum number of impacted detection panels, and hit multiplicity. Events with a multiplicity larger than 10 on one panel are rejected, to remove electromagnetic shower-like events triggered after shielding panel crossing. After filtering, to minimize the impact of the noisy hits on the measurement of the incident muon direction, we have chosen to reconstruct each track with a random sampling consensus (RANSAC) fit procedure.

The RANSAC procedure

The RANSAC algorithm [7] is based on an hypothesize and test iterative process. The process starts by randomly sampling within the input dataset, i.e. the hits XYZ coordinates, a subset of points of given size and fit an hypothesized line model to this subset. It then evaluates which hits of the whole dataset are consistent with this hypothesized model, computing a fit error, and counting the number of hits whose orthogonal distance to the model is below a certain distance threshold (set to one scintillator width). Those hits forms the consensus set. After N iterations, the algorithm outputs the model that minimizes the fit error, and maximizes the size of the consensus set. The hits belonging to the consensus are tagged as 'inliers' and the rest of the dataset as 'outliers'. N is set so that the probability of getting a pure-signal consensus, i.e. only composed of true inliers, at the end of the process is 0.99.

Numerical simulation

To measure the performances of this RANSAC-based reconstruction, we have developed a dedicated GEANT4 [8] simulation with our 4-panel detector geometry, illustrated on figure 2, interfaced with the MC generator CRY [9] to simulate the incident flux from atmospheric showers (muonic, hadronic and electromagnetic components). CRY appeared well-suited for processing performance tests, since it is very fast: O(1min) for 10 5 MC events.

Performances

Using the simulation framework described in 3.2, we first compared the RANSAC processing output of pure-muon and pureelectron samples of 10 6 particles each. We then measured, for the muon sample, the reconstruction efficiency and the angular resolution of the estimated trajectories. We compared these measurements with the results obtained from a simpler processing algorithm relying on a track straightness criteria, without any trajectory fit involved. This former processing was based on the check of the alignment of the most energetic hit formed in each impacted panel, which is a significant bias since muons are minimum ionizing particles and therefore would not be expected to systematically form the highest energy deposit in each detection matrix.

2 . 9 (

 29 . . . . . . . . . . . . . . . . . . . . . . . 1.2 Domenico Pacini measuring the discharge of its electroscope in 1910. Taken from [62]. (Image credit: Pacini's family) . . . . . . . . . . . . . . . . . . . 1.3 (a) New York Times headline in 1932 relating the fierce debate between American physicists A. Compton and R. Millikan upon the nature of cosmic radiation. The latter was convinced that cosmic rays were particles of light, photons. (b) Worldwide measurement campaign of cosmic rays intensity to study their dependence on geomagnetic effect [57]. . . . . . . . . . . . . . 1.4 (a) Famous photo of Victor Hess in 1912 aboard his air balloon, preparing his expedition (American Physical Society). (b) Ionization rate as a function of altitude. Left panel: Final ascent by Victor Hess (1912), carrying two ion chambers. Right panel: Ascents by Werner Kolhörster (1913, 1914). Taken from [50]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 First muon track observed in the Wilson cloud chamber of Paul Kunze [119], but wrongly classified at the time as a nuclear reaction product at the time. 1.6 Radiative coefficients b as a function of energy for standard rock and water. 1.7 Ionization a(E) (dash-dotted line) and radiative components b(E)E (dashed lines) of the mass stopping power (solid line), computed for standard rock (Z = 11, A = 22, ρ = 2.65 g cm -3 ). Critical energy E µc = a(E)/b(E) is marked in grey vertical dotted line. For standard rock, E SR µc = 693 GeV. . . 1.8 Ionization component a(E) with and without density effect correction δ(E), here computed for standard rock composition (Z = 11, A = 22) and four different density values ρ = 0.8, 1.2, 1.8, 2.65 g cm -3 . . . . . . . . . . . . . 1.9 Total mean stopping power plotted as a function of kinetic energy for water and three different rock densities: ρ = 1.0, 1.8, 2.65 gcm -3 . . . . . . . . . 1.10 Minimal crossing energy E min as a function of obstacle thickness, plotted for water and three different rock densities: ρ = 1.0, 1.8, 2.65 gcm -3 . . . . 1.11 Integrated muon flux through standard rock as a function of opacity in units of meter water equivalent (mwe), plotted for several zenith angles. . . . . . 1.12 Primary differential fluxes as a function of kinetic energy per nucleus. Free protons (H) represent 74% of the total flux, whereas nucleons bound in helium account for 70% of the other nucleons. Adapted from [216]. . . . . . . 1.13 Vertical flux given as a function of atmosphere depth, i.e its opacity (see lexicon), and altitude. Data points show the flux measurements for negative muons µ -. Taken from [216]. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Illustration of the formation of an electromagnetic shower triggered by a primary cosmic ray interaction in the high atmosphere. Cosmic muons, represented in magenta continuous lines, are typically produced at altitudes around 15 km. Taken from [114] . . . . . . . . . . . . . . . . . . . . . . . . xiii 1.15 Muon spectrum models as a function of energy, compared to reference flux data: at vertical θ = 0 • , data points come from ♦ [63], ▼ [172], ▲ [21], ×, + [117], • [91], • [8], and at θ = 75 • from ♢ [104]. . . . . . . . . . . . . . . 1.16 Definition of θ and θ * zenith angles. H is the height of interaction, P is the muon production point, and D the detection point. . . . . . . . . . . . . . . 1.17 Differential muon flux as a function of energy derived from CORSIKA tuned for geographical coordinates of La Soufrière de Guadeloupe, with its associated 1σ uncertainty band. The flux is put in comparison to the Guan muon flux model [90]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Integrated flux as a function of opacity in standard rock, derived from the CORSIKA and Guan differential fluxes shown in Fig. 1.17. . . . . . . . . . . 1.19 Scheme summarizing the relationships between the scanned object with its associated density distribution, its impact on the propagation of traversing muons, and the resulting transmitted integral flux. In the case of the forward problem, the latter is estimated from a given density distribution, whereas for the inverse problem, the density is modeled from estimates of the muon flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Photos of (a) the twelve-meter high cryostat that hosts the dual-phase LArTPC of ProtoDUNE-DP at CERN neutrino platform; (b) the inside of the ProtoDUNE-DP LArTPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Schematic of the Dual Phase detector, featuring the representation of a cosmic muon interaction (in green). The produced ionization charges (in red) along the track in the liquid argon drift towards the upper amplification area. Scintillation light is detected by photomultiplier tubes at the bottom, and the charge tracks are imaged at the top by the LEM/Anode structure. . 2.3 Three-dimensional exploded-view drawing of the ProtoDUNE-DP detector, with its principal components. . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Electric field map of the CRP region showing the charge extraction from liquid to gas, amplification in LEM holes, and collection in the Dual Phase design, with a zoomed picture on LEM holes, and anode view. . . . . . . . 2.5 (left) Picture of the anode PCB, (right) schematic view of the two orthogonal views: view 0 is in white, view 1 appears in red. Adapted from [22]. . . . . 2.6 An example of a cosmic event displayed on drift time (1 sample = 400 ns) vs anode view channels window, showing ionization tracks in the LAr left by cosmic muons including a high-energy particle that has ended up its travel forming an electromagnetic shower. . . . . . . . . . . . . . . . . . . . . . . 2.7 Comparison between the outputs of two hit finder algorithms (a) DP raw hit finder on the raw waveform and (b) Gaussian hit finder on a deconvoluted waveform. 1 time tick = 400 ns . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Example of a reconstructed cosmic event in ProtoDUNE-DP, shown in LAr-Soft event display window split between the two anode views. 3D reconstructed tracks by Pandora are shown in the same color in the two anode views, and identified vertices by the algorithm are numbered. The most distorted track segments due to E field inhomogeneities seen on view 0 between wires 400 and 550 fail to be reconstructed. N.B 1 tick = 400 ns . . . . xiv Left) Broken particle track due to LEM gaps, (Right) Fully reconstructed particle trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Distance between the starts of track and the closest LEM gap in Y direction (without any cut on data here). . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Charge deposit per unit length dQ/dx distributions. Four distributions at different drift time bins of width ∆t = 100 µs are shown here. All LEMs at a nominal voltage of ∆V LEM = 2.9 kV for this run 1333 taken in November are included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Charge deposit dQ/dx vs electron drift time from simulated cosmic events. The red markers are the MPVs of dQ/dx in each time bin, and the red curve is the exponential fit of those values. In purple, we plot the decreasing exponential function with τ e set to 4 ms. . . . . . . . . . . . . . . . . . . .

2. 16 3 . 2

 1632 Effective gain maps in the Charge Readout plane with each individual LEM G ef f value, for two cosmic runs taken in (a) October 2019 and in (b) January 2020. Only the High-Voltage LEMs that have sustained a potential difference ∆V LEM at +/-100V around the initial nominal voltage during the whole run duration is shown here . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Charge asymmetry coefficient A as a function of polar angle ϕ. . . . . . . 2.18 dE/dx for both collection views computed with modified box model . . . 3.1 Illustrated principle of the two muographic imaging methods. The blue lines represent the incident muons on the upper coat of the investigated object, depicted in light grey. Red spheres symbolize the medium anomaly inside the object. (left) Two-dimensional absorption muography only relies on one detector (dark grey) to record variations in the number of transmitted muons after encountering a density anomaly. Here the red sphere is a high-density anomaly, as the number of muons decreases. (right) Scattering muography relies on two detectors to record the scattering angle before and after object crossing. Variations of the scattering angle are caused by chemical composition variation (here a high-Z anomaly) inside the object. Taken from [109]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Scattering muography observables: scattering angle ∆θ and spatial deviation ∆x. The detectors (not depicted here) are positioned above (before particles cross) and below (after crossing) the scanned object. Adapted from [185]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Geometry of a simple detector layout featuring squared coaxial and parallel XY matrices, front and rear, separated by a distance L. The line-of-sight of incident angle θ depicted in the solid red line is associated with a given front-rear pixel couple (∆x, ∆y) separated by a distance d. . . . . . . . . . 3.4 Integrated geometrical acceptance of a 4-panel scintillator muon telescope, described in Sec. 4.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Representation of the muon detector's lines-of-sight (orange lines), i.e the individual muon paths scanned by the detector, converging towards the detector (sphere), targeting the Soufrière de Guadeloupe lava dome (cf.chapter 4) from a South-South-West position. The line's directions are defined by the detector pixelization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Minimum acquisition time ∆T min as a function of relative opacity variation ∆ϱ/ϱ. The acceptance T is set to 10 cm 2 sr. . . . . . . . . . . . . . . . . . . 3.7 Developed nuclear emulsion image in transmitted light. Muons appear as the longest and straightest tracks. Taken from [41]. . . . . . . . . . . . . . 3.8 Sketch of a Micromegas detector. Taken from [39] . . . . . . . . . . . . . . 3.10 Pioneer experiment in Australia led by E.P. George to measure the rock overburden. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  . . . . . . . . . . . . . . . . . . . . . . . 3.11 (a) Scheme of a cross-section of the Chephren pyramid. The Belzoni chamber is located near the center of the basis of the edifice (point B); (b) Photo of the full detection apparatus in the Belzoni chamber, with the spark chamber panels in the background of the photo. Taken from Alvarez et al., 1970 [16] 3.12 (a) Scheme illustrating the muography obtained with gaseous detectors measuring the transmitted muon flux from the pyramid. The yellow and red cones represent two angular areas where were detected unexpected muon excesses; (b) Number of muons as a function of the horizontal angle recorded in the yellow area. The black line represents the number of muons expected from a model for a homogeneous medium. Both figures are taken from [148] 3.13 (a): Mean density muon image showing the crater floor of Mt. Iwodake (taken from [197]). Only density anomalies below 95% of the average density are mapped;(b): Topographic map of Statsuma-Iwojima island (Kyushu, Japan) featuring a representation in the upper left of the Kikai caldera. Taken from [111]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 3.14 (a) Muography of the shallow crater region of Mt. Asama. Black arrows indicate the lower-density (in blue shades) and higher-density regions (in red), respectively corresponding to the head of the magma pathway, plugged by solidified magma that composes the crater floor. The white frame delimits the region observed on (b) the muon images before (upper image) and after (lower) the 2009 eruption. The dotted line represents the shape of the crater floor before the eruption.Taken from [198] . . . . . . . . . . . . . . . . . . 3.15 (a) (upper) picture of the lava dome of La Soufrière de Guadeloupe featuring the identified four regions of interest for flux variations. (lower) The associated relative flux variations are plotted as a function of time. (b) The pixel spatial maps feature in color scale the eigenvalues derived by PCA which allowed identification of the regions of interest in (a). The solid black line represents the apparent topography of the dome summit from the detection site. The red triangles and associated dotted lines depict the azimuth of the main active zones to be seen on the dome summit. Taken from [108]. . . . 4.1 The lesser Antilles volcanic arc, featuring inset a detailed map of the Guadeloupe archipelago. From [38]. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Photos of the La Soufrière volcano eruption in 1976-1977 (copyright IPGP): (a) Intense fumarolic activity view from the north (b) Phreatic explosion and dense ash cloud, 4 th of October 1976. . . . . . . . . . . . . . . . . . . . 4.3 The lava dome schematic map and its main sites: A: La Soufrière lava dome; B: bulge region (convex outward dome slope profile); C: superficial hydrothermal altered material. Taken from [38]. . . . . . . . . . . . . . . . . 4.4 Three-dimensional elevation distribution of the La Soufrière volcano. The digital elevation model (DEM) was realized by LiDAR by the French Marine hydrographic and oceanographic service (SHOM) in the framework of the Litto3D © project [138]. The fraction of the DEM shown here covers a surface of 4 km 2 , including the lava dome and its surroundings with a spatial resolution of 1 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Illustration of common phreatic and hydrothermal eruption types in volcanic and geothermal environments, highlighting initiating factors such as magma/fluid injection, landslides, sulfur sealing, and lake drainage with respective examples of eruption: Te Maari/Tongariro 2012, Poás 2014, Mt. Ontake 2014, and Kverkfjöll/Gengissig 2013. The figure was taken from [146].

4. 12 5 . 3 6 . 1

 125361 Scheme of a multi-anode photo-multiplier (MAPM), showing along one channel the photo-electron emission, acceleration, multi-stage multiplication through the dynodes, and collection on the anode. Taken from[START_REF]Hamamatsu Photonics Photomultiplier Tubes Handbook[END_REF]. . . . . 4.13 Schematic diagram of the data acquisition chain on a detection panel. . . . 4.14 Photo of the telescope readout system designed in IP2I, Lyon. The MAPM is connected to the Front-End (FE) board that features two chips, one for each direction X and Y, for trigger generation and signal readout. The motherboard contains the Ethernet Controller Module (ECM), the High Voltage (HV), and the LED pulse system for calibration. Taken from[START_REF] Gibert | Bayesian dual inversion of experimental telescope acceptance and integrated flux for geophysical muon tomography[END_REF]. . . . . . 4.15 Schematics of the different steps with their causal links to assess mean density estimates ρ of a given investigated dome region whose topography and thus travel lengths inside the rock L(θ, φ) are known thanks to a digital elevation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Mean density radiographies obtained in 2011 from (a) Southern site Ravine Sud (RS) (b) Eastern site Roche Fendue (RF). Variations shown in blue-to-red color scale are relative to the reference value in the image's radiography. Taken from [128]. N.B These maps result from the smoothing of original pixeled maps, through some convolution with spatial filters like Hamming windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 RANSAC number of iterations N as a function of the inlier proportion ϵ. The probability of getting at least a free-outlier output data set is fixed at p = 0.99. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Data processing algorithm flowchart based on the RANSAC method. . . . xviii Results of RANSAC tracking on simulated muon sample: (a) fraction of muon hits tagged as inliers as a function of maximal number of iterations N max , for different sizes of initial sample s; (b) composition of the XY hits formed by muons µ, and secondary electromagnetic particles: electrons e and photons γ, resulting from the interaction of muons with the detector. The inner circle represents the recorded hit composition from GEANT4 particle identification. The outer circle represents the proportion of hits tagged by the tracking algorithm as inliers and outliers. The RANSAC parameters have been set to t = 50 mm, N max = 20, and s = 2. . . . . . . . . . . . . . 5.4 (a) and (b) are GEANT4 event displays respectively showing a 10 GeV/c muon (in solid blue line) and 10 GeV/c electron (in solid red line), both emitted at the same position, 50 cm above the detector with an incident angle θ = 10 • to the z-axis. In (b) the incident electron triggers an electromagnetic shower (secondary gammas in yellow and electrons in red) after crossing the lead shielding panel at the center of the detector. The scintillator bars (sensitive parts) are depicted in green, and the passive shielding (both steel and lead materials) appears in red. (c) and (d) below show the same events after the RANSAC track reconstruction. We show in color scale the energy deposit dE recorded for each XY hit, around 4 MeV in ionization in each detection matrix (2cm thick) for the muon in (c), and a lot more widespread between secondary hits for the electromagnetic shower in (d). This information is however not used in the tracking algorithm. . . . . . . 5.5 Mean reconstruction efficiency computed over all angular bins as a function of noise proportion, for each of the three 4-panel configurations. . . . . . . 5.6 Angular residuals estimated with a 10 6 MC muons sample, with energy E 0 ∈ [10 -2 , 10 2 ] GeV for a proportion of 50% of outlier hits. . . . . . . . . 5.7 RMSE of angular residuals histograms ∆θ (left column) and ∆φ (right column) as a function of noise proportion. Each line is a given telescope configuration of the 4-panel telescope. . . . . . . . . . . . . . . . . . . . . . . . 5.8 Processing time for the reconstruction of the 10 5 MC muon sample for both RANSAC and the Hough transform. . . . . . . . . . . . . . . . . . . . . . . 5.9 Mean reconstruction efficiency ⟨ε reco ⟩ as a function of scintillator efficiency parameter ε scint , obtained from the muon MC simulation in GEANT4 (colored bands). The efficiency is derived for each of the three 4-panel telescope configurations: 3p1, 3p2, and 4p. The figure features calibration open-sky data points for each of the configurations. The hatch regions correspond to the intersection of the MC efficiency bands with the data uncertainty bands of the same configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 RANSAC reconstruction efficiency as a function of muon kinetic energy, computed over a 10 6 MC muons sample, with energy E 0 ∈ [10 -2 , 10 2 ] GeV. We compare histograms for both telescope configurations with (green) and without (blue) a shielding panel (5 cm lead + 2 cm steel). The latter histogram is fitted with an erf function (solid green line), which gives a cut-off energy estimate of E c = (1.7 ± 0.9).10 -2 GeV. . . . . . . . . . . . . . . . xix Mean integral flux as a function of opacity, linearized around opacity ϱ = 5.10 2 mwe. The bandwidth displays the uncertainty related to the transmitted flux computation in the standard rock σ model . The model of flux here is Guan's parametrization [90] of the Gaisser formula. . . . . . . . . . . . . . 6.2 Relative linear interpolation error on opacity from the modeled integral flux in the plane opacity/vertical incident angle (ϱ, θ). . . . . . . . . . . . . . . 6.3 Fractional transmitted muon flux through (a) a pure water medium over standard rock and (b) a mixed medium composed of half water -half standard rock over standard rock, for three zenith angle values θ = {0, 60, 80} • . The color band covers the 1σ uncertainty range. . . . . . . . . . . . . . . 6.4 (a) Histogram of the number of transmitted muons in a standard rock medium in the vertical angle deviation ∆θ and kinetic energy E plane; (b)∆θ distribution at energy bin E = 150 GeV, with its estimated standard deviation. 6.5 Sketches of the main physical background sources that enter the muon telescope. The upper left and the bottom right sketch represent a low-energy muon scattered respectively on the object flanks and on the rear side of the detector. The upper right and bottom left sketches show the electron background flux that arises from the electromagnetic component of atmospheric showers. Taken from [165]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 GEANT4 event display of two "soft" muons of energy E µ = 200 MeV (straight solid blue lines starting in the figure upper left) that are entering the standard rock volume (here of planar surface in brown) and propagate over a few cm in the rock before being scattered back towards the muon telescope volume (in yellow). Along the muon tracks are produced secondary electrons (in red) and photons (in yellow). The muons have a very large incident angle with respect to the surface normal (θ > 85 • ). . . . . . 6.7 The four telescopes with their respective cones of sight. The SNJ telescope (upgraded NJ telescope) features two acceptance cones corresponding to its 3-panel (the lightest and widest yellow-shade cone) and its 4-panel configuration encompassing a narrower region of the dome. . . . . . . . . . . . . 6.8 Photos of (a) SB, (b) SNJ telescopes, both located in the South-West region of the dome, with respective acceptance cones in blue and yellow in Fig.6.7; (c) BR telescope (red acceptance cone), (c) OM telescope, respectively located in the East region and in the north fault (Fente du Nord). The latter appears in the bottom right of the picture, very close to the rock. . . . . . . . . . . 6.9 Event rate as a function of time in the four muon telescopes. SNJ event rates are shown for the first 3-panel configuration (3p1) and its full 4-panel configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Two-dimensional mean density estimates in each 3-panel telescope. The blue markers on the topography (black solid line) represent angular coordinates of the main craters and pits angular, shown in Fig.6.7. . . . . . . . 6.11 Two-dimensional mean density estimate of the 4-panel configuration of the SNJ telescope, with increased resolution on the inner angular region of the 3-panel muography shown in Fig.6.10 (b). . . . . . . . . . . . . . . . . . . . 6.12 Statistical uncertainty associated with mean density estimates shown in Fig.6.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx 6.13 Systematic uncertainty associated with mean density estimates shown in Fig.6.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Three-dimensional toy density model of the La Soufrière lava dome where density voxel values are set to ρ b = 2.0 g cm -3 excepted along the cylindrical anomaly in its center at density ρ a = 1.0 g cm -3 . The lines of sight of the SNJ telescope are displayed as yellow continuous lines. Both SB (blue square marker) and SNJ (yellow square marker) telescope positions are also displayed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Posterior standard deviation σ M as a function of the data misfit ϕ d . The color coding represents the prior density error σ M , and the marker size represents the correlation length l c as indicated in the legend. The star marker points out the optimal set of regularization parameters whose values are displayed on the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Density estimates obtained from the inversion of synthetic data generated in the four detectors. The model is shown for dome horizontal slices at four altitudes z={1250, 1300, 1350, 1400} m a.s.l. The voxel density values correspond to the posterior three-dimensional density distribution ρ post . The telescope positions appeared in magenta. The figure insets show the input density toy model ρ true . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.17 Density estimates obtained from the inversion of synthetic data generated in one detector: the SNJ telescope. The model is shown for dome horizontal slices at four altitudes z={1250, 1300, 1350, 1400} m a.s.l. The voxel density values correspond to the posterior three-dimensional density distribution ρ post . The telescope positions appeared in magenta. The figure insets show the input density toy model ρ true . . . . . . . . . . . . . . . . . . . . . . . . 6.18 Density estimates obtained from the inversion of synthetic data generated in three detectors: SB, SNJ and BR telescopes. The model is shown for dome horizontal slices at four altitudes z={1250, 1300, 1350, 1400} m a.s.l. The voxel density values correspond to the posterior three-dimensional density distribution ρ post . The telescope positions appeared in magenta. The figure insets show the input density toy model ρ true . . . . . . . . . . . . . . . . . . 6.19 Posterior synthetic data, shown in each of the four telescopes' sight planes. The color scale is showing the posterior mean density computed over all the crossed voxel posterior density values for each line of sight. The inset plot in each subplot is displaying the detector position and a slice (at altitude z=1300 m) of the true dome density model. . . . . . . . . . . . . . . . . . . 6.20 Density estimates ρ post from real muography data inversion from the four telescopes. Four dome horizontal slices are shown at altitudes z = {1250, 1300, 1350, 1400} m a.s.l. The telescope positions appeared in squared markers with their associated color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.21 Standard deviation associated with density estimates presented in Fig. 6.20. 6.22 Horizontal slice of Fig.6.20 (b), superimposed with the main structural features (in blue lozenge markers) and the main fumaroles (purple cone-shaped markers), already shown in Fig. 6.7. . . . . . . . . . . . . . . . . . . . . . . 6.23 Vertical slices of the density model in the (Northing, Altitude) plane. The slice in (a) is taken for the Easting coordinate of the dome center. . . . . . xxi

Chapter 1 Muons

 1 : from fundamental physics to imaging tools 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 A first misty footprint . . . . . . . . . . . . . . . . . . . . 1.1.2 Origin of the discovery . . . . . . . . . . . . . . . . . . . . 1.2 Muons in the current research world . . . . . . . . . . . . . . . . 1.2.1 A door to new physics? . . . . . . . . . . . . . . . . . . . 1.2.2 Muons as a calibration tool . . . . . . . . . . . . . . . . . 1.2.3 Towards muography . . . . . . . . . . . . . . . . . . . . . 1.3 Muon physics: propagation in matter . . . . . . . . . . . . . . . . 1.4 Modeling the muon flux: from the source to the detection . . . . . 1.4.1 Birth and development of atmospheric showers . . . . . . 1.4.2 Muon flux on Earth . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Muon flux parametrizations . . . . . . . . . . . . . . . . . 1.4.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Corrections to the muon flux . . . . . . . . . . . . . . . . 1.4.6 Modeling the muon flux at La Soufrière de Guadeloupe . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: A cloud chamber picture taken by Anderson and Neddermeyer and featuring in their article of 1937[START_REF] Neddermeyer | Note on the Nature of Cosmic-Ray Particles[END_REF] with its original caption that describes the different trajectories of the incident and produced particles including muon tracks along other products of an electromagnetic shower that are ionizing the detection gaseous mixture.

Figure 1 . 2 :

 12 Figure 1.2: Domenico Pacini measuring the discharge of its electroscope in 1910. Taken from [62]. (Image credit: Pacini's family)

Figure 1 . 3 :Figure 1 . 4 :

 1314 Figure 1.3: (a) New York Times headline in 1932 relating the fierce debate between American physicists A. Compton and R. Millikan upon the nature of cosmic radiation. The latter was convinced that cosmic rays were particles of light, photons. (b) Worldwide measurement campaign of cosmic rays intensity to study their dependence on geomagnetic effect [57].

Figure 1 . 5 :

 15 Figure 1.5: First muon track observed in the Wilson cloud chamber of Paul Kunze [119], but wrongly classified at the time as a nuclear reaction product at the time.

Figure 1 . 6 :

 16 Figure 1.6: Radiative coefficients b as a function of energy for standard rock and water.

Figure 1 . 7 :

 17 Figure 1.7: Ionization a(E) (dash-dotted line) and radiative components b(E)E (dashed lines) of the mass stopping power (solid line), computed for standard rock (Z = 11, A = 22, ρ = 2.65 g cm -3 ). Critical energy E µc = a(E)/b(E) is marked in grey vertical dotted line. For standard rock, E SR µc = 693 GeV.

Figure 1 . 8 :

 18 Figure 1.8: Ionization component a(E) with and without density effect correction δ(E), here computed for standard rock composition (Z = 11, A = 22) and four different density values ρ = 0.8, 1.2, 1.8, 2.65 g cm -3 .

Figure 1 . 9 :

 19 Figure 1.9: Total mean stopping power plotted as a function of kinetic energy for water and three different rock densities: ρ = 1.0, 1.8, 2.65 gcm -3 .

Figure 1 . 10 :igure 1 . 11 :

 110111 Figure 1.10: Minimal crossing energy E min as a function of obstacle thickness, plotted for water and three different rock densities: ρ = 1.0, 1.8, 2.65 gcm -3 .

1. 4

 4 Modeling the muon flux: from the source to the detection 1.4.1 Birth and development of atmospheric showers

Figure 1 . 15 :

 115 Figure 1.15: Muon spectrum models as a function of energy, compared to reference flux data: at vertical θ = 0 • , data points come from ♦ [63], ▼ [172], ▲ [21], ×, + [117], • [91], • [8], and at θ = 75 • from ♢ [104].

( 1 .Figure 1 . 16 :

 1116 Figure 1.16: Definition of θ and θ * zenith angles. H is the height of interaction, P is the muon production point, and D the detection point.

Figure 1 . 17 :

 117 Figure 1.17: Differential muon flux as a function of energy derived from CORSIKA tuned for geographical coordinates of La Soufrière de Guadeloupe, with its associated 1σ uncertainty band. The flux is put in comparison to the Guan muon flux model [90].

Figure 1 . 18 :

 118 Figure 1.18: Integrated flux as a function of opacity in standard rock, derived from the CORSIKA and Guan differential fluxes shown in Fig. 1.17.

Figure 1 .

 1 Figure1.19: Scheme summarizing the relationships between the scanned object with its associated density distribution, its impact on the propagation of traversing muons, and the resulting transmitted integral flux. In the case of the forward problem, the latter is estimated from a given density distribution, whereas for the inverse problem, the density is modeled from estimates of the muon flux.

Chapter 2 theFigure 2 . 1 :

 221 Figure 2.1: Photos of (a) the twelve-meter high cryostat that hosts the dual-phase LArTPC of ProtoDUNE-DP at CERN neutrino platform; (b) the inside of the ProtoDUNE-DP LArTPC

Figure 2 . 2 :

 22 Figure 2.2: Schematic of the Dual Phase detector, featuring the representation of a cosmic muon interaction (in green). The produced ionization charges (in red) along the track in the liquid argon drift towards the upper amplification area. Scintillation light is detected by photomultiplier tubes at the bottom, and the charge tracks are imaged at the top by the LEM/Anode structure.

Figure 2 . 3 :

 23 Figure 2.3: Three-dimensional exploded-view drawing of the ProtoDUNE-DP detector, with its principal components.

Figure 2 . 4 :

 24 Figure 2.4: Electric field map of the CRP region showing the charge extraction from liquid to gas, amplification in LEM holes, and collection in the Dual Phase design, with a zoomed picture on LEM holes, and anode view.

Figure 2 . 5 :

 25 Figure 2.5: (left) Picture of the anode PCB, (right) schematic view of the two orthogonal views: view 0 is in white, view 1 appears in red. Adapted from [22].

Figure 2 . 6 :

 26 Figure 2.6: An example of a cosmic event displayed on drift time (1 sample = 400 ns) vs anode view channels window, showing ionization tracks in the LAr left by cosmic muons including a high-energy particle that has ended up its travel forming an electromagnetic shower.

Figure 2 . 7 : 4 .

 274 Figure 2.7: Comparison between the outputs of two hit finder algorithms (a) DP raw hit finder on the raw waveform and (b) Gaussian hit finder on a deconvoluted waveform. 1 time tick = 400 ns

Figure 2 . 8 :

 28 Figure 2.8: Example of a reconstructed cosmic event in ProtoDUNE-DP, shown in LArSoft event display window split between the two anode views. 3D reconstructed tracks by Pandora are shown in the same color in the two anode views, and identified vertices by the algorithm are numbered. The most distorted track segments due to E field inhomogeneities seen on view 0 between wires 400 and 550 fail to be reconstructed. N.B 1 tick = 400 ns

Figure 2 . 9 :

 29 Figure 2.9: (Left) Broken particle track due to LEM gaps, (Right) Fully reconstructed particle trajectory.

Figure 2 . 10 :

 210 Figure 2.10: Distance between the starts of track and the closest LEM gap in Y direction (without any cut on data here).

Figure 2 . 11 :

 211 Figure 2.11: Charge deposit per unit length dQ/dx distributions. Four distributions at different drift time bins of width ∆t = 100 µs are shown here. All LEMs at a nominal voltage of ∆V LEM = 2.9 kV for this run 1333 taken in November are included.

Figure 2 . 12 :

 212 Figure 2.12: Charge deposit dQ/dx vs electron drift time from simulated cosmic events. The red markers are the MPVs of dQ/dx in each time bin, and the red curve is the exponential fit of those values. In purple, we plot the decreasing exponential function with τ e set to 4 ms.

Figure 2 . 13 :

 213 Figure 2.13: Charge deposit along tracks dQ/dx as a function of electrons drift time, plotted for three cosmic runs taken in October, November and January.

Figure 2 .

 2 Figure 2.14: (a) Schematic diagram of a purity monitor. The photocathode receives photons from a xenon source light and emits electrons by photoelectric effect. These "photoelectrons" Q c are driven by an electric field towards the anode, where a charge Q a is collected. The measurement of the ratio Qc Qa is proportional to the charge loss factor exp ( -t τe ) where τ e is the electrons lifetime. (b) Photo of the short purity monitor in the bottom of the cryostat.

Figure 2 . 15 :

 215 Figure 2.15: Measurements of the two short purity monitors (bottom and middle TPC), superimposed to cosmic tracks analysis results (red markers) shown in Fig. 2.13. This comparison shows consistent results between the two short purity monitors and the cosmic analysis outputs.

Figure 2 . 16 : 1 Figure 2 . 17 :

 2161217 Figure 2.16: Effective gain maps in the Charge Readout plane with each individual LEM G ef f value, for two cosmic runs taken in (a) October 2019 and in (b) January 2020. Only the High-Voltage LEMs that have sustained a potential difference ∆V LEM at +/-100V around the initial nominal voltage during the whole run duration is shown here

Figure 2 . 18 :

 218 Figure 2.18: dE/dx for both collection views computed with modified box model

Figure 3 . 1 :

 31 Figure 3.1: Illustrated principle of the two muographic imaging methods. The blue lines represent the incident muons on the upper coat of the investigated object, depicted in light grey. Red spheres symbolize the medium anomaly inside the object. (left) Two-dimensional absorption muography only relies on one detector (dark grey) to record variations in the number of transmitted muons after encountering a density anomaly. Here the red sphere is a high-density anomaly, as the number of muons decreases. (right) Scattering muography relies on two detectors to record the scattering angle before and after object crossing. Variations of the scattering angle are caused by chemical composition variation (here a high-Z anomaly) inside the object. Taken from [109].

Figure 3 . 2 :

 32 Figure 3.2: Scattering muography observables: scattering angle ∆θ and spatial deviation ∆x. The detectors (not depicted here) are positioned above (before particles cross) and below (after crossing) the scanned object. Adapted from [185].

Figure 3 . 3 :

 33 Figure 3.3: Geometry of a simple detector layout featuring squared coaxial and parallel XY matrices, front and rear, separated by a distance L. The line-of-sight of incident angle θ depicted in the solid red line is associated with a given front-rear pixel couple (∆x, ∆y) separated by a distance d.

Figure 3 . 4 :

 34 Figure 3.4: Integrated geometrical acceptance of a 4-panel scintillator muon telescope, described in Sec. 4.4.1.

Figure 3 . 5 :

 35 Figure 3.5: Representation of the muon detector's lines-of-sight (orange lines), i.e the individual muon paths scanned by the detector, converging towards the detector (sphere), targeting the Soufrière de Guadeloupe lava dome (cf.chapter 4) from a South-South-West position. The line's directions are defined by the detector pixelization.

Figure 3 . 6 :

 36 Figure 3.6: Minimum acquisition time ∆T min as a function of relative opacity variation ∆ϱ/ϱ. The acceptance T is set to 10 cm 2 sr.

Figure 3 . 7 :

 37 Figure 3.7: Developed nuclear emulsion image in transmitted light. Muons appear as the longest and straightest tracks. Taken from [41].

Figure 3 . 10 :

 310 Figure 3.10: Pioneer experiment in Australia led by E.P. George to measure the rock overburden.

Figure 3 .Figure 3 .

 33 Figure 3.11: (a) Scheme of a cross-section of the Chephren pyramid. The Belzoni chamber is located near the center of the basis of the edifice (point B); (b) Photo of the full detection apparatus in the Belzoni chamber, with the spark chamber panels in the background of the photo. Taken from Alvarez et al., 1970[START_REF] Alvarez | Search for Hidden Chambers in the Pyramids[END_REF]

Figure 3 .

 3 Figure 3.13: (a): Mean density muon image showing the crater floor of Mt. Iwodake (taken from[START_REF] Tanaka | Cosmic-ray muon imaging of magma in a conduit: Degassing process of Satsuma-Iwojima Volcano, Japan[END_REF]). Only density anomalies below 95% of the average density are mapped;(b): Topographic map of Statsuma-Iwojima island (Kyushu, Japan) featuring a representation in the upper left of the Kikai caldera. Taken from[START_REF] Kazahaya | Degassing process of Satsuma-Iwojima volcano, Japan: Supply of Volatile components from a deep magma chamber[END_REF].

Figure 3 .

 3 Figure 3.14: (a) Muography of the shallow crater region of Mt. Asama. Black arrows indicatethe lower-density (in blue shades) and higher-density regions (in red), respectively corresponding to the head of the magma pathway, plugged by solidified magma that composes the crater floor. The white frame delimits the region observed on (b) the muon images before (upper image) and after (lower) the 2009 eruption. The dotted line represents the shape of the crater floor before the eruption.Taken from[START_REF] Hiroyuki | Japanese volcanoes visualized with muography[END_REF]

3. 3 .Figure 3 .

 33 Figure 3.15: (a) (upper) picture of the lava dome of La Soufrière de Guadeloupe featuring the identified four regions of interest for flux variations. (lower) The associated relative flux variations are plotted as a function of time. (b) The pixel spatial maps feature in color scale the eigenvalues derived by PCA which allowed identification of the regions of interest in (a). The solid black line represents the apparent topography of the dome summit from the detection site. The red triangles and associated dotted lines depict the azimuth of the main active zones to be seen on the dome summit. Taken from [108].

Figure 4 . 2 :

 42 Figure 4.2: Photos of the La Soufrière volcano eruption in 1976-1977 (copyright IPGP): (a) Intense fumarolic activity view from the north (b) Phreatic explosion and dense ash cloud, 4 th of October 1976.

Figure 4 . 3 :

 43 Figure 4.3: The lava dome schematic map and its main sites: A: La Soufrière lava dome; B: bulge region (convex outward dome slope profile); C: superficial hydrothermal altered material. Taken from [38].

4. 2

 2 Current unrest at La Soufrière 4.2.1 Hydrothermal activity Phreatic and hydrothermal eruptions

Figure 4 . 4 :

 44 Figure 4.4: Three-dimensional elevation distribution of the La Soufrière volcano. The digital elevation model (DEM) was realized by LiDAR by the French Marine hydrographic and oceanographic service (SHOM) in the framework of the Litto3D © project [138]. The fraction of the DEM shown here covers a surface of 4 km 2 , including the lava dome and its surroundings with a spatial resolution of 1 m.

Figure 4 . 5 :

 45 Figure 4.5: Illustration of common phreatic and hydrothermal eruption types in volcanic and geothermal environments, highlighting initiating factors such as magma/fluid injection, landslides, sulfur sealing, and lake drainage with respective examples of eruption: Te Maari/Tongariro 2012, Poás 2014, Mt. Ontake 2014, and Kverkfjöll/Gengissig 2013. The figure was taken from [146].

Figure 4 . 6 :

 46 Figure 4.6: Uniaxial rock strength (σ p ) as a function of wet bulk sample density (ρ W ). Taken from [96].

Figure 4 . 7 :

 47 Figure 4.7: Current network of extensometers, GNSS, and seismic stations deployed on La Soufrière de Guadeloupe lava dome, monitored by the OVSG.

Figure 4 . 8 :

 48 Figure 4.8: Map of the GNSS horizontal velocities from 1995 to 2018 with respect to the Guadeloupe archipelago. Error ellipses are 95% of confidence. The same scale has been used for south Basse-Terre Island (a) and La Soufrière volcano (b). Permanent stations are located in orange markers. Taken from[START_REF] Moretti | The 2018 unrest phase at La Soufrière of Guadeloupe (French West Indies) andesitic volcano: Scrutiny of a failed but prodromal phreatic eruption[END_REF]

Figure 4 . 10 :

 410 Figure 4.10: Map featuring sight cones of the four detectors whose datasets were analyzed during the 2015-2019 period, targeting the main structural features of the dome. The "MT" detector is located in a fault (F30) shaped on the 30 th of August 1976, during the phreatic eruption.

Figure 4 . 11 :

 411 Figure 4.11: Layout of the muon telescope featuring three detection panels (in yellow).Their assembly relies on a circular basis to easily set the telescope azimuth. The horizontality is ensured thanks to adjustable scaffolding feet. The mechanical structure is designed to additionally host a shielding panel on its central plane (not depicted here). The control box (in blue) that hosts the onboard computer is positioned between the middle and rear matrices. Taken from[START_REF] Marteau | Development of Scintillator-Based Muon Detectors for Muography[END_REF].

Figure 4 . 12 :

 412 Figure 4.12: Scheme of a multi-anode photo-multiplier (MAPM), showing along one channel the photo-electron emission, acceleration, multi-stage multiplication through the dynodes, and collection on the anode. Taken from [92].

Figure 4 . 13 :

 413 Figure 4.13: Schematic diagram of the data acquisition chain on a detection panel.

Figure 4 . 15 :

 415 Figure 4.15: Schematics of the different steps with their causal links to assess mean density estimates ρ of a given investigated dome region whose topography and thus travel lengths inside the rock L(θ, φ) are known thanks to a digital elevation model.

Figure 4 . 16 :

 416 Figure 4.16: Mean density radiographies obtained in 2011 from (a) Southern site Ravine Sud (RS) (b) Eastern site Roche Fendue (RF). Variations shown in blue-to-red color scale are relative to the reference value in the image's radiography. Taken from [128]. N.B These maps result from the smoothing of original pixeled maps, through some convolution with spatial filters like Hamming windows.

Figure 5 . 2 :

 52 Figure 5.2: Data processing algorithm flowchart based on the RANSAC method.

Figure 5 . 3 :

 53 Figure 5.3: Results of RANSAC tracking on simulated muon sample: (a) fraction of muon hits tagged as inliers as a function of maximal number of iterations N max , for different sizes of initial sample s; (b) composition of the XY hits formed by muons µ, and secondary electromagnetic particles: electrons e and photons γ, resulting from the interaction of muons with the detector. The inner circle represents the recorded hit composition from GEANT4 particle identification. The outer circle represents the proportion of hits tagged by the tracking algorithm as inliers and outliers. The RANSAC parameters have been set to t = 50 mm, N max = 20, and s = 2.

Figure 5 . 4 :

 54 Figure 5.4: (a) and (b) are GEANT4 event displays respectively showing a 10 GeV/c muon (in solid blue line) and 10 GeV/c electron (in solid red line), both emitted at the same position, 50 cm above the detector with an incident angle θ = 10 • to the z-axis. In (b) the incident electron triggers an electromagnetic shower (secondary gammas in yellow and electrons in red) after crossing the lead shielding panel at the center of the detector. The scintillator bars (sensitive parts) are depicted in green, and the passive shielding (both steel and lead materials) appears in red. (c) and (d) below show the same events after the RANSAC track reconstruction. We show in color scale the energy deposit dE recorded for each XY hit, around 4 MeV in ionization in each detection matrix (2cm thick) for the muon in (c), and a lot more widespread between secondary hits for the electromagnetic shower in (d). This information is however not used in the tracking algorithm. 109

Figure 5 . 5 :

 55 Figure 5.5: Mean reconstruction efficiency computed over all angular bins as a function of noise proportion, for each of the three 4-panel configurations.

Figure 5 . 6 :

 56 Figure 5.6: Angular residuals estimated with a 10 6 MC muons sample, with energy E 0 ∈ [10 -2 , 10 2 ] GeV for a proportion of 50% of outlier hits.

Figure 5 . 7 :

 57 Figure 5.7: RMSE of angular residuals histograms ∆θ (left column) and ∆φ (right column) as a function of noise proportion. Each line is a given telescope configuration of the 4-panel telescope.

Figure 5 . 8 :

 58 Figure 5.8: Processing time for the reconstruction of the 10 5 MC muon sample for both RANSAC and the Hough transform.

Figure 5 . 9 :

 59 Figure 5.9: Mean reconstruction efficiency ⟨ε reco ⟩ as a function of scintillator efficiency parameter ε scint , obtained from the muon MC simulation in GEANT4 (colored bands). The efficiency is derived for each of the three 4-panel telescope configurations: 3p1, 3p2, and 4p. The figure features calibration open-sky data points for each of the configurations. The hatch regions correspond to the intersection of the MC efficiency bands with the data uncertainty bands of the same configuration.

Figure 5 .

 5 Figure 5.10: RANSAC reconstruction efficiency as a function of muon kinetic energy, computed over a 10 6 MC muons sample, with energy E 0 ∈ [10 -2 , 102 ] GeV. We compare histograms for both telescope configurations with (green) and without (blue) a shielding panel (5 cm lead + 2 cm steel). The latter histogram is fitted with an erf function (solid green line), which gives a cut-off energy estimate of E c = (1.7 ± 0.9).10 -2 GeV.

  up min ]. These can further propagate to the computation of 1σ bounds on integrated flux according to Eq.1.6: the E low min is used to compute the upper bound of integrated flux I up , i.e integrating the muon spectrum on [E low min ; +∞[ and similarly to obtain the lower bound on flux I low , integrating on [E up min ; +∞[. This transmitted flux uncertainty is quantified here by the σ model term, shown in Fig.6.1.

Figure 6 . 1 :

 61 Figure 6.1: Mean integral flux as a function of opacity, linearized around opacity ϱ = 5.10 2 mwe. The bandwidth displays the uncertainty related to the transmitted flux computation in the standard rock σ model . The model of flux here is Guan's parametrization [90] of the Gaisser formula.

Figure 6 . 2 :

 62 Figure 6.2: Relative linear interpolation error on opacity from the modeled integral flux in the plane opacity/vertical incident angle (ϱ, θ).

Figure 6 . 3 :

 63 Figure 6.3: Fractional transmitted muon flux through (a) a pure water medium over standard rock and (b) a mixed medium composed of half water -half standard rock over standard rock, for three zenith angle values θ = {0, 60, 80} • . The color band covers the 1σ uncertainty range.

Figure 6 . 4 :

 64 Figure 6.4: (a) Histogram of the number of transmitted muons in a standard rock medium in the vertical angle deviation ∆θ and kinetic energy E plane; (b)∆θ distribution at energy bin E = 150 GeV, with its estimated standard deviation.

Figure 6 . 5 :

 65 Figure 6.5: Sketches of the main physical background sources that enter the muon telescope. The upper left and the bottom right sketch represent a low-energy muon scattered respectively on the object flanks and on the rear side of the detector. The upper right and bottom left sketches show the electron background flux that arises from the electromagnetic component of atmospheric showers. Taken from [165].

Figure 6 . 6 :

 66 Figure 6.6: GEANT4 event display of two "soft" muons of energy E µ = 200 MeV (straight solid blue lines starting in the figure upper left) that are entering the standard rock volume (here of planar surface in brown) and propagate over a few cm in the rock before being scattered back towards the muon telescope volume (in yellow). Along the muon tracks are produced secondary electrons (in red) and photons (in yellow). The muons have a very large incident angle with respect to the surface normal (θ > 85 • ).

Figure 6 . 7 :

 67 Figure 6.7: The four telescopes with their respective cones of sight. The SNJ telescope (upgraded NJ telescope) features two acceptance cones corresponding to its 3-panel (the lightest and widest yellow-shade cone) and its 4-panel configuration encompassing a narrower region of the dome.

Figure 6 . 8 :

 68 Figure 6.8: Photos of (a) SB, (b) SNJ telescopes, both located in the South-West region of the dome, with respective acceptance cones in blue and yellow in Fig.6.7; (c) BR telescope (red acceptance cone), (c) OM telescope, respectively located in the East region and in the north fault (Fente du Nord). The latter appears in the bottom right of the picture, very close to the rock.

Figure 6 . 9 :

 69 Figure 6.9: Event rate as a function of time in the four muon telescopes. SNJ event rates are shown for the first 3-panel configuration (3p1) and its full 4-panel configuration.

Fig. 6 .

 6 7). The muographies of the two South-West detectors, SB (Fig.6.10-a) and SNJ (Figs.6.10-b, 6.11), feature a high-density region in the East-North-East direction for the highest azimuth values, spanning over 10 • . The SNJ 4-panel muography (with an increased angular resolution by 49% compared to the 3-panel) has its acceptance cone targeting the most active region of the dome (cf. Fig.6.7). It also captures the extremity of the high-density region around φ = 40 • seen in the 3-panel muography. On top of this high density, the same superficial low-density region ranging from zenith θ ∈[START_REF] De | L'enigma dei raggi cosmici: Le più grandi energie dell'universo[END_REF][START_REF] Dempsey | Automatic precursor recognition and real-time forecasting of sudden explosive volcanic eruptions at Whakaari, New Zealand[END_REF] • is present. Here we do not show the second 3panel muography obtained with SNJ as it does not provide any additional information on the dome structure but allows us to increase the size of the data set and reduce statistical uncertainty. The acceptance cone of BR oriented towards the West seems also to intersect this same high-density region in its lowest azimuth region (cf. Fig.6.7) (ϕ ∈ [264, 280] • ).

Figure 6 .

 6 Figure 6.10: Two-dimensional mean density estimates in each 3-panel telescope. The blue markers on the topography (black solid line) represent angular coordinates of the main craters and pits angular, shown in Fig.6.7.

Figure 6 .

 6 Figure 6.11: Two-dimensional mean density estimate of the 4-panel configuration of the SNJ telescope, with increased resolution on the inner angular region of the 3-panel muography shown in Fig.6.10 (b).

  telescopes

2 :

 2 Mean statistical and systematic relative uncertainties with their standard deviation, associated with density estimates of Fig.6.10. The total systematic relative uncertainty, plotted in Fig.6.13, is defined bin per bin as ∆ tot sys = (∆ µ sys ) 2 + (∆ θ sys ) 2 .

Figure 6 . 12 :

 612 Figure 6.12: Statistical uncertainty associated with mean density estimates shown in Fig.6.10.

Figure 6 . 13 :

 613 Figure 6.13: Systematic uncertainty associated with mean density estimates shown in Fig.6.10.

  6.3: i.e in the i-th pixel, d syn i = N (d true i , σ i ), where d true i = F i,j ρ true j . All the data point uncertainties are set to σ i = ∆ tot i d true i

Figure 6 .

 6 Figure 6.14: Three-dimensional toy density model of the La Soufrière lava dome where density voxel values are set to ρ b = 2.0 g cm -3 excepted along the cylindrical anomaly in its center at density ρ a = 1.0 g cm -3 . The lines of sight of the SNJ telescope are displayed as yellow continuous lines. Both SB (blue square marker) and SNJ (yellow square marker) telescope positions are also displayed.

Fig. 6 .

 6 Fig. 6.15, each point corresponds to a given set of regularization parameters (σ textM , l c ), whose values are respectively coded by the marker color and size. The obtained L-curve allows to read optimal regularization parameters as (σ M , l c ) = (0.1 g.cm -3 , 200 m).

Figure 6 . 15 :

 615 Figure 6.15: Posterior standard deviation σ M as a function of the data misfit ϕ d . The color coding represents the prior density error σ M , and the marker size represents the correlation length l c as indicated in the legend. The star marker points out the optimal set of regularization parameters whose values are displayed on the graph.

Figure 6 . 16 :

 616 Figure 6.16: Density estimates obtained from the inversion of synthetic data generated in the four detectors. The model is shown for dome horizontal slices at four altitudes z={1250, 1300, 1350, 1400} m a.s.l. The voxel density values correspond to the posterior threedimensional density distribution ρ post . The telescope positions appeared in magenta. The figure insets show the input density toy model ρ true .

Figure 6 . 17 :

 617 Figure 6.17: Density estimates obtained from the inversion of synthetic data generated in one detector: the SNJ telescope. The model is shown for dome horizontal slices at four altitudes z={1250, 1300, 1350, 1400} m a.s.l. The voxel density values correspond to the posterior three-dimensional density distribution ρ post . The telescope positions appeared in magenta. The figure insets show the input density toy model ρ true

Figure 6 . 18 :

 618 Figure 6.18: Density estimates obtained from the inversion of synthetic data generated in three detectors: SB, SNJ and BR telescopes. The model is shown for dome horizontal slices at four altitudes z={1250, 1300, 1350, 1400} m a.s.l. The voxel density values correspond to the posterior three-dimensional density distribution ρ post . The telescope positions appeared in magenta. The figure insets show the input density toy model ρ true .

Figure 6 . 19 :

 619 Figure 6.19: Posterior synthetic data, shown in each of the four telescopes' sight planes. The color scale is showing the posterior mean density computed over all the crossed voxel posterior density values for each line of sight. The inset plot in each subplot is displaying the detector position and a slice (at altitude z=1300 m) of the true dome density model.

Figure 6 .

 6 Figure 6.20: Density estimates ρ post from real muography data inversion from the four telescopes. Four dome horizontal slices are shown at altitudes z = {1250, 1300, 1350, 1400} m a.s.l. The telescope positions appeared in squared markers with their associated color.

Figure 6 . 22 :

 622 Figure 6.22: Horizontal slice of Fig.6.20 (b), superimposed with the main structural features (in blue lozenge markers) and the main fumaroles (purple cone-shaped markers), already shown in Fig. 6.7.

Figure 6 . 23 :

 623 Figure 6.23: Vertical slices of the density model in the (Northing, Altitude) plane. The slice in (a) is taken for the Easting coordinate of the dome center.

6. 6

 6 Model interpretation 6.6.1 Insight on dome inner structure

Figure 6 . 24 :

 624 Figure 6.24: Uniaxial rock compressive strength shown for dome horizontal slices at four different altitudes z = {1250, 1300, 1350, 1400} m a.s.l, inferred from ρ post obtained after inversion of the four telescopes real data sets. The telescope positions appeared in magenta.

Figure 6 . 25 :

 625 Figure 6.25: Same density model horizontal slice as in Fig. 6.22, superimposed with electrical conductive anomalies A2 and A1 presented in [176].

Figure 6 . 26 :

 626 Figure 6.26: Vertical slice of density model at (cf. Fig. 6.23-b), superimposed with the isocontour points of electrical conductive anomalies A2 and A1 presented in [176]. N.B The northern part of the A1 region shown here is not represented in Fig. 6.25.

Figure 1 .

 1 Figure 1. (a) Map of La Soufrière lava dome featuring locations of the two telescopes SNJ and BR and their respective angular acceptance. The SNJ telescope has a 4-panel configuration and covers two regions at the same time: a broader region in light yellow shade with its 3-panel sub-configuration, and a tightened region using the 4-panel configuration. The main structural features, notably the Tarissan (TAR) and South (CS) craters, and active fumarolic vents on the dome summit are encompassed by both telescope angular apertures. (b) Photo of the SNJ telescope during its assembly, located at the South-West of the dome, composed of 4 detection matrices and an additional lead shielding panel at the center.

  al., 2003; Marteau et al., 2012; Saracino et al., 2016; Lo Presti et al., 2020; Peña-Rodríguez et al., 2020). With their robust, modular, and relatively simplified design compared to high-energy physics detectors, they are appropriate to deal with the main challenge of rough field operations (Lesparre et al., 2012a). Since 2017 two muon telescopes, respectively equipped with four and three scintillator panels, are installed at two different locations around the La Soufrière lava dome: at the South-West, and East of the volcano whose positions are shown in Fig. 1. The novel 4-panel configuration (4p), in addition to decreasing the low-energy particles background, allows increasing spatial resolution on the 2D density muography of the central part of the dome scanned region and offers two more 3-panel sub-configurations (3p1 and 3p2), which lead finally to three different muon images. As represented in Fig. 1, the 4-panel telescope named SNJ allows scanning at the same time a broad angular region using signal recorded in three panels with respective azimuth and zenith angles (ϕ 3p , θ 3p ) ∈ ([-17; 58] • , [43; 107] • ), as well as a narrower angular region centered on the dome bulk and its two main craters, using all 4 panels: (ϕ 4p , θ 4p ) ∈ ([-5; 46] • , [52; 98] • ). The two geometrical configurations have thus different angular acceptances. SNJ includes 4 detection matrices (front, middle1, middle2, and rear) interspaced by 60 cm; each matrix is an array of 16x16 bars, which is 80 cm long, 1 cm thick, and 5 cm wide. The 3-panel telescope, named BR, is located on the Eastern flank. It has 3 matrices (front, middle and rear) also interspaced by 60 cm: the front and rear matrices are composed of 32x32 bars, 80

  triggers electromagnetic showers, composed of secondary electrons and photons, characterized by a high hit multiplicity on the rear panel. Detailed descriptions of the muon telescope assembly used in this study can be found in Lesparre et al. (2012a) and Jourde et al. (2014); and description of the light and electronic readout as well as the timing system in Marteau (2009), Marteau et al. (2012), and Jourde et al. (2014).

  )where N sky (r i,j ) is the number of open-sky muons detected in r i,j , ∆T calib is the calibration run duration (around two days). The open-sky muon integral flux I sky (r i,j ) is estimated with CORSIKA[START_REF] Heck | CORSIKA: A Monte Carlo Code to Simulate Extensive Air Showers[END_REF], a Monte-Carlo simulation tool that allows accurate cosmic muons flux estimation.CORSIKA propagates extensive atmospheric showers, from their generation by primary cosmic-ray, e.g. an extra-galactic proton, interactions on upper atmosphere nuclei to their ground-level detection. Furthermore, this detailed MC framework allows us to consider geomagnetic effects alongside atmospheric parameters (e.g. density variations through a layered atmosphere model) that affect the muon flux, as studied in[START_REF] Cohu | Atmospheric and Geodesic Controls of Muon Rates: A Numerical Study for Muography Applications[END_REF].

  g. (Gaisser et al., 1990; Guan et al., 2015)) based on the muon production from mesons in the high atmosphere. A computation for each opacity of the muon minimal crossing energy E min is required to obtain the transmitted muon flux for a known density distribution. For this purpose, the work of the Particle Data Group (Workman & Others, 2022) is widely used. It provides an exhaustive review of the different interaction processes at stake, and their associated energy deposit contributions, during muon propagation in a dense medium. Finally, the muon energy spectrum Φ(E, θ) can be integrated over [E min ( ); +∞[:

Figure 2 .

 2 Figure 2. Data processing algorithm flowchart based on the RANSAC method.

Figure 3 .

 3 Figure 3. Three-dimensional representations of the hits formed in the telescope scintillator panels by: (a) an electromagnetic shower, triggered by the interaction of the primary electron with the shielding panel located between the Middle1 and Middle2 panels; (b) a simulated muon reconstructed with the RANSAC tracking algorithm, artificially polluted with synthetic outlier hits. The color scale represents the sum of energy deposits in X and Y bars, noted dE. The size of the XY hit is also proportional to dE. This information is however not used in the tracking algorithm.

Figure 4 .

 4 Figure 4. Results of RANSAC tracking on simulated muon sample: (a) fraction of muon hits tagged as inliers as a function of the maximal number of iterations N max , for different sizes of initial sample s; (b) composition of the XY hits formed by muons µ, and secondary particles: electrons e and photons γ, resulting from the interaction of muons with the detector. The inner circle represents the recorded hit composition from GEANT4 particle identification. The outer circle represents the proportion of hits tagged by the tracking algorithm as inliers and outliers. The RANSAC parameters have been set to t = 50 mm, N max = 20, and s = 2.

Figure 5 .

 5 Figure 5. Reconstruction efficiency maps for a 4-panel detector estimated from (a) Monte-Carlo simulated muons with GEANT4 reconstructed with RANSAC considering a scintillator efficiency of ε scint = 0.87, and (b) a real 4-panel detector data.

Figure 6 .

 6 Figure 6. Mean reconstruction efficiency ε reco as a function of scintillator efficiency parameter ε scint , obtained from the muon MC simulation in GEANT4 (colored bands). The efficiency is derived for each of the three 4panel telescope configurations: 3p1, 3p2, and 4p. The figure features calibration open-sky data points for each of the configurations. The hatch regions correspond to the intersection of the MC efficiency bands with the data uncertainty bands of the same configuration.

Figure 7 .

 7 Figure 7. χ 2 distribution (cf. Eq. 7) between estimated and calculated fluxes as a function of (θ, ∆ρ) on a 5 • -wide azimuth range around telescope central azimuth, for (a) SNJ and (b) BR datasets.

  Fig. 9 b)). We also retrieve on this East muography an overlaying low-density region above zenith 70 • . The lowdensity anomaly also seems to rise up to the surface below the South-Crater region, around azimuth 290 ± 4 • . On this muography, though, the shallow low-density anomaly appears broader and spans over the whole azimuth range above zenith 75 • , reaching the lowest values of -70% above azimuth 310 • , and zenith 65 • . This low-density region close to the surface corresponds to a superficial part of the dome on the north-west flank, which crosses highly-altered and clay-rich material, revealed during the 2009 landslide (Rosas-Carbajal et al., 2016; Heap et al., 2021). The main pits and craters, whose positions are projected on the BR muography, are part of a wide network of cavities spanning from the north fault to the Tarissan crater and expanding on the East flank. These structural features are expected to have a significant impact on the bulk density estimates.

Figure 8 .Figure 9 .

 89 Figure 8. Relative density variation two-dimensional maps estimated with the SNJ telescope (South-West) for (a) the three-panel configuration of SNJ (b) the four-panel configuration of SNJ. The map in (b) is projected onto the terrain model in (c): each line of sight (black line) features its density value, plotted at the entry point in the dome, and corresponds to a given dome thickness represented in (d). The main dome craters and pits positions of Fig. 1 are represented in blue lozenge markers.

  9 seems particularly impacted by the presence of such a network of cavities. In addition to the Tarissan and South craters, two additional pits of large size are located on the East-South-East flank: Breislack (BLK) and Gouffre 56 (G56), roughly 40 m deep (10% of the dome height), according to the description made by Kuster & Slive (1997), around azimuth 305 • and 312 • . Another important cavity, the Spallanzani cave of volume 55x35x11m 3 (Kuster & Slive, 1997), whose entrance is located in the north fault (FNO), is also likely to have an impact on the mean density measurement. The effect on the density of such volume was already estimated between 10 to 15% by Lesparre et al. (2012b). As these cavities are currently not taken into account when computing the total dome thickness traversed by the muons, shown in Fig. 8d 9c, these void spaces add to an average bulk density significantly lower than the average value of actual rocks. The western superficial low-density

  Previous studies[START_REF] Nishiyama | Monte carlo simulation for background study of geophysical inspection with cosmic-ray muons[END_REF][START_REF] Gómez | Forward scattering effects on muon imaging[END_REF][START_REF] Chen | Monte Carlo simulation for background noise study of the Wudalianchi volcanic density imaging[END_REF] have indeed enlightened that for such large targets as volcanoes, flux measurements are strongly impacted by lowenergy muons that are forward-scattered on the volcano flank facing the detector. These "soft" muons typically in the sub-GeV energy range can mimic dome-outgoing muon tracks and be reconstructed as such. This results in a measured flux that is over-estimated, which in turn leads to a lower opacity and thus lower average density estimate along the impacted line-of-sight. Indeed, this can be clearly observed in Fig.8a and bfor zenith angles higher than 85 degrees. Rock thicknesses in this area are so large that no muon events should be detected, therefore the low-density estimates arise from a large amount of forward scattered muons. Another source of contamination lies in the muon flux produced in the atmosphere below the telescope altitude. This backward muon flux can enter the telescope by its rear matrix, which especially impacts the most horizontal lines of sight above θ > 80 • and can significantly blur muon radiography close to the horizon, accounting for up to 50% of the detected events[START_REF] Jourde | Experimental detection of upward-going cosmic particles and consequences for correction of density radiography of volcanoes[END_REF]. The ideal rejection for this upward flux requires an ns clocking system for a precise time-of-flight measurement.[START_REF] Gómez | Forward scattering effects on muon imaging[END_REF] have shown for a simplified cone-shaped dome geometry that the forward scattering of muons increases with the zenith angle θ, exceeding 50% of the total number of recon-

FIGURE 1 :

 1 FIGURE 1: (a) photo of the 4-panel detector Super Nain Jaune, located near La Savane à mulets on the South flank of La Soufrière volcano; (b) a scheme of a scintillator matrix, adapted from[2]; (c) a scheme of a plastic scintillator bar, with its embedded WLS fiber, adapted from[12] .

FIGURE 4 :

 4 FIGURE 4: Angular resolution in Zenith (upper line) and Azimuth (bottom line) angles for the estimated muon trajectories, for the (a) RANSAC and (b) and straightness check processing.

3. 3 . 3 .

 33 Reconstruction efficiencyIn figure5, we have measured the muon reconstruction efficiency ϵ µ for each 4-panel detector line-of-sight. For each pixel, representing a given line-of-sight, we show the ratio of the number of reconstructed tracks and the total number of generated muon straight track. We observe for the RANSAC processing an increase by 10% of the mean efficiency ⟨ϵ µ ⟩ with respect to the straightness check-based processing.

FIGURE 5 :

 5 FIGURE 5: Muon reconstruction efficiency ϵ µ maps for (a) RANSAC with a mean efficiency ⟨ϵ µ ⟩ = 0.71 ± 0.11 and (b) straightness check processing with ⟨ϵ µ ⟩ = 0.61 ± 0.10. Each pixel is a given telescope line-of-sight.

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 A first misty footprint . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Origin of the discovery . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Muons in the current research world . . . . . . . . . . . . . . . . . . . . . 1.2.1 A door to new physics? . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Muons as a calibration tool . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Towards muography . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Muon physics: propagation in matter . . . . . . . . . . . . . . . . . . . . . 1.4 Modeling the muon flux: from the source to the detection . . . . . . . . . . 1.4.1 Birth and development of atmospheric showers . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 LAr experiments common framework: LArSoft . . . . . . . . . . . 2.3.2 Track reconstruction main steps . . . . . . . . . . . . . . . . . . . 2.3.3 Pandora . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cosmic muon analysis results . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Electron lifetime measurement with cosmics . . . . . . . . . . . . . 2.4.2 Gain measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Charge collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 dE/dx measurement for a MIP . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of expected muons . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Detector acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Number of detected muons . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Towards 3-D density modeling . . . . . . . . . . . . . . . . . . . . 3.3 What limits muography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Detector position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Acquisition time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Deflection from straight trajectory . . . . . . . . . . . . . . . . . . 3.4 The different technologies of muographers . . . . . . . . . . . . . . . . . . 3.4.1 Nuclear emulsion detectors . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Gas-filled detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Scintillator-based detectors . . . . . . . . . . . . . . . . . . . . . . 3.5 Early days of muography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 First density measurement in Australia . . . . . . . . . . . . . . . . 3.5.2 Exploring the pyramids . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Image Earth's subsurface and first probing of mountains . . . . . . 3.6 Application to volcanology . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Achievements and current projects . . . . . . . . . . . . . . . . . . 3.6.2 Adapting muography to volcanic monitoring . . . . . . . . . . . . 1.1 A cloud chamber picture taken by Anderson and Neddermeyer and featuring in their article of 1937

	Table des matières	xi
	Liste des figures	xxii
	1 Muons: from fundamental physics to imaging tools
	1.1

1.4.2 Muon flux on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Muon flux parametrizations . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Corrections to the muon flux . . . . . . . . . . . . . . . . . . . . . 1.4.6 Modeling the muon flux at La Soufrière de Guadeloupe . . . . . . . 1.5

2 Preliminary cosmic muons analysis in ProtoDUNE-Dual Phase 2.1 Introduction to DUNE and its scientific goals . . . . . . . . . . . . . . . . . 2.2 The ProtoDUNE Dual Phase detector . . . . . . . . . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Dual-Phase design . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Commissioning of ProtoDUNE-DP . . . . . . . . . . . . . . . . . . 2.2.4 ProtoDUNE-DP data . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Cosmic muon analysis . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 3 The era of muography: application to volcanology 3.1 Scattering muography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Absorption muography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Goal and principle . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 3.2.2

  4.6 Uniaxial rock strength (σ p ) as a function of wet bulk sample density (ρ W ).Taken from[START_REF] Michael | Alteration-Induced Volcano Instability at La Soufrière de Guadeloupe (Eastern Caribbean)[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.7Current network of extensometers, GNSS, and seismic stations deployed on La Soufrière de Guadeloupe lava dome, monitored by the OVSG. . . . . . . Three-dimensional representation of the lava dome electrical conductivity expressed in S/m as a blue-to-red color scale. The slices are oriented southnorth and east-west when viewed from the southeast. Reddish areas indicate more conductive material, which translates to more liquid and higher temperatures. Taken from[START_REF] Rosas-Carbajal | Volcano electrical tomography unveils edifice collapse hazard linked to hydrothermal system structure and dynamics[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Map featuring sight cones of the four detectors whose datasets were analyzed during the 2015-2019 period, targeting the main structural features of the dome. The "MT" detector is located in a fault (F30) shaped on the 30 th of August 1976, during the phreatic eruption. . . . . . . . . . . . . . . . . . 4.11 Layout of the muon telescope featuring three detection panels (in yellow).Their assembly relies on a circular basis to easily set the telescope azimuth.The horizontality is ensured thanks to adjustable scaffolding feet. The me-

4.8 Map of the GNSS horizontal velocities from 1995 to 2018 with respect to the Guadeloupe archipelago. Error ellipses are 95% of confidence. The same scale has been used for south Basse-Terre Island (a) and La Soufrière volcano (b). Permanent stations are located in orange markers. Taken from [147] . xvii 4.9 chanical structure is designed to additionally host a shielding panel on its central plane (not depicted here). The control box (in blue) that hosts the onboard computer is positioned between the middle and rear matrices. Taken from [142]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  2 + P 2 1 + P 2 cos(θ) P 3 + P 4 cos(θ) P 5 1 + P 2 1 + P 2 + P 4

  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Introduction to DUNE and its scientific goals . . . . . . . . . . . . 2.2 The ProtoDUNE Dual Phase detector . . . . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Dual-Phase design . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Commissioning of ProtoDUNE-DP . . . . . . . . . . . . . 2.2.4 ProtoDUNE-DP data . . . . . . . . . . . . . . . . . . . . . 2.2.5 Cosmic muon analysis . . . . . . . . . . . . . . . . . . . . 2.3 Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 LAr experiments common framework: LArSoft . . . . . . 2.3.2 Track reconstruction main steps . . . . . . . . . . . . . . 2.3.3 Pandora . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Cosmic muon analysis results . . . . . . . . . . . . . . . . . . . . 2.4.1 Electron lifetime measurement with cosmics . . . . . . . . 2.4.2 Gain measurements . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Charge collection . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 dE/dx measurement for a MIP . . . . . . . . . . . . . . . 2.5

  3.1 Scattering muography . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Absorption muography . . . . . . . . . . . . . . . . . . . . . . . . Early days of muography . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 First density measurement in Australia . . . . . . . . . . . 3.5.2 Exploring the pyramids . . . . . . . . . . . . . . . . . . . 3.5.3 Image Earth's subsurface and first probing of mountains . 3.6 Application to volcanology . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Achievements and current projects . . . . . . . . . . . . . 3.6.2 Adapting muography to volcanic monitoring . . . . . . .

3.2.1 Goal and principle . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Number of expected muons . . . . . . . . . . . . . . . . . 3.2.3 Detector acceptance . . . . . . . . . . . . . . . . . . . . . 3.2.4 Number of detected muons . . . . . . . . . . . . . . . . . 3.2.5 Towards 3-D density modeling . . . . . . . . . . . . . . . 3.3 What limits muography . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Detector position . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Acquisition time . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Deflection from straight trajectory . . . . . . . . . . . . . 3.4 The different technologies of muographers . . . . . . . . . . . . . 3.4.1 Nuclear emulsion detectors . . . . . . . . . . . . . . . . . 3.4.2 Gas-filled detectors . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Scintillator-based detectors . . . . . . . . . . . . . . . . . 3.5

  ; [211]; Tamburello et al., 2019; [147]; Jessop et al., 2021; Metcalfe et al., 2021).

  2

  .7. To estimate the volume encom-

						telescopes
		characteristics		Sacre bleu	Super	Baron Rouge	Orange
					(SB)	Nain Jaune	(BR)	Mécanique
						(SNJ)		(OM)
			number of panels 3	4	3	3
	configuration	length L [cm]	120	180	120	120
			scint. bars X × Y (front and rear	32 × 32	16 × 16	32 × 32	32 × 32
			matrices)
	position &	site		Savane	Parking lot	Roche	Fente	du
	orientation			South-West		Fendue	Nord
			X (UTM zone 20)	642611	642782	643346	642955
			Y (UTM zone 20)	1773798	1773683	1774030	1774561
			Z (WSG84)	1185	1145	1268	1345
			elevation [°]	10.0	15.1	19.5	13.9
			azimuth [°]	44.9	20.0	295.0	192.0
			start		24/08/2016	24/02/2019	09/03/2017	11/03/2017
	run		stop		01/04/2017	11/08/2019	19/08/2019	01/05/2018
			total	duration	202.9	145.0	640.3	191.2
			[days]
			number of events	1.38.10 7	2.23.10 7	1.1.10 8	2.81.10 6
			(raw)
	geometrical	max [cm 2 .sr]	2.78	11.11 (3p) /	2.78	2.78
	acceptance				6.25 (4p)
			min [m]		6	14 (3p) / 5	1	1
	rock thickness				(4p)
			center [m]	748	797 (3p) / 797	607	394
						(4p)
			max [m]		1272	1301 (3p) /	1164	1209
						1301 (4p)
	scanned volume	estimate [m 3 ]	1.33.10 7	1.41.10 7	9.89.10 6	3.83.10 6
	(V dome tot	∼ 5.10 7 m 3 )

Table 6 .

 6

	mean uncertainties	SB	SNJ (3p)	SNJ (4p)	BR	OM
	stat.⟨∆ stat ⟩	all θ θ > 70 •	0.24±0.17 0.07±0.06 0.10±0.09 0.10±0.10 0.11±0.11 0.30±0.17 0.09±0.07 0.15±0.10 0.12±0.10 0.17±0.11
	µ physics sys. ⟨∆ µ sys ⟩	flux absorption	0.10±0.07 0.10±0.02 0.11±0.03 0.11±0.04 0.16±0.31 0.07±0.09 0.07±0.10 0.07±0.10 0.07±0.08 0.07±0.07
	elevation sys. ⟨∆ θ sys ⟩	0.08±0.12 0.07±0.13 0.07±0.12 0.07±0.13 0.06±0.12
	total systematic ⟨∆ tot sys ⟩	0.17±0.14 0.16±0.14 0.17±0.13 0.17±0.13 0.20±0.31

https://pdg.lbl.gov/2011/AtomicNuclearProperties/)

From the German, "bremsen": to brake and "Strahlung": radiation.

A hadron is a composite particle made of quarks, and thus sensitive to the strong interaction.

A meson is a hadron composed of a quark q -anti-quark q pair.

The Volcanic Explosivity Index, or VEI, was proposed in 1982 as a way to describe the relative size or magnitude of explosive volcanic eruptions[START_REF] Newhall | The Volcanic Explosivity Index (VEI): An Estimate of Explosive Magnitude for Historical Volcanism[END_REF]

http://Diaphane-muons.com/

Quantum efficiency is defined as the number of photo-electrons emitted from the photocathode divided by the number of incident photons.

The latter was realized by LiDAR by the French Marine hydrographic and oceanographic service (SHOM) in the framework of the Litto3D © project[START_REF] Shom | Service hydrographique et océanographique de la marine)[END_REF]. The model has a spatial resolution of 1 m.

Rank deficiency = difference between the lesser number between the number of rows and columns and the rank, i.e the maximal number of linearly independent columns of the matrix.

L 2 functions ρ : R 3 → R

This appendix is an article submitted to Geophysical Journal International, currently in its first round of review (February 2023).

(a) (b) (c)

(a) RANSAC (b) Straightness check

Remerciements

Je souhaite adresser mes profonds et sincères remerciements :

• Aux membres de mon jury de thèse d'avoir lu, commenté et évalué mon travail,

ACKNOWLEDGMENTS

This work and the data shown were acquired in the context of two projects supported by the French Agence Nationale de la Recherche, projects DIAPHANE (ANR-14-ce04-0001) and MEGaMu (ANR-19-CE05-0033). We thank the editor C. Sens-Schönfelder, reviewer R. Nishiyama, and an anonymous reviewer, whose comments have contributed to improving the quality of our manuscript. The simulation and data analysis were performed on the S-CAPAD/DANTE platform of IPGP, France.

ACKNOWLEDGEMENTS

This work and the data shown were acquired in the context of the ANR MEGaMu. We also wish to thank Axel Le Berrigaud, for his contribution to the first applications of the RANSAC-based reconstruction during his master thesis at IPGP, in 2020.

DATA AVAILABILITY

The data presented in this work are not publicly available.

Hence, from Eq.5.3, the minimal number of iterations needed to reach a final free-outlier consensus set with the probability p (typically set to 99%) can be estimated with the following expression:

We show in Fig. 5.1, the relation between RANSAC parameters N as a function of the outlier ratio ϵ. As the size of the initial data set k grows, the number of iterations increases faster with ϵ. For our study, the size of the initial sample s and the maximal number of iterations N max were tuned to maximize the number of muon hits tagged as inliers, using the simulation of an open-sky muon run with the framework described in Sec. 5.2. As previously said, random sampling methods are notably designed to deal with the presence of outlier points polluting data sets. Nevertheless, the current version of RANSAC implemented in our processing algorithm significantly relies on the assumption that the proportion of outliers within muon events does not exceed 50% [START_REF] Roters | FestGPU: a framework for fast robust estimation on GPU[END_REF]. As shown in Fig. 5.1, in case of very polluted data sets (e.g ϵ ≤ 0.1), the number of iterations N dramatically increases, along with the computing time, which would therefore put in question the efficiency of such tracking method. Nevertheless, in practice, we see that for simulated muon tracking without any instrumental noise contamination, this number of iterations remains as expected significantly low. Indeed, in Fig. 5.3a for initial sample size s = 2 and N max ≥ 20 the true muon hit inlier fraction reaches a plateau at 98.4 ± 0.2%. Besides, as shown in 5.3b, the muon tracking also shows quite low contamination from electrons and photons: these secondary hits represent 14.3 ± 0.7% of the total hits, and nearly half of this fraction (6.7 ± 0.6%) have been tagged as outliers. In Sec. 5.3, we will see how RANSAC behaves in case of an artificially polluted data set.

SUMMARY

Muography is increasingly used to image the density distribution of volcanic edifices, complementing traditional geophysical tomographies. Here we present a new muon data processing algorithm, and apply it to a new generation of scintillator-based muon detectors, to image the relative density distribution in La Soufrière de Guadeloupe volcano (Lesser Antilles, France). Our processing method iteratively searches for the best fit of each muon trajectory, accounting for all the hits registered by the detector related to the particular muon event. We test the performance of our algorithm numerically, simulating the interaction of muons with our detector and accounting for its exact assemblage including the scintillator bars and lead shielding. We find that our new data processing mitigates the impact of spurious signals coming from secondary particles, and improves the amount of successfully reconstructed events. The resulting two-dimensional muon images at La Soufrière have higher angular resolution than previous ones and capture the heterogeneous structure of the dome. They show density anomalies located on the summit southern region, which includes a boiling acid lake and degassing fractures, where the rock is the most porous and fumarolic activity is ongoing. This work shows the importance of combining numerical simulations of muon propagation with precise raw data processing to obtain high-quality results. It is also a first step towards fully as-sessing the noise contamination sources when performing muon tomography, and their correction, prior to geophysical interpretations. Key words: Tomography -Volcano monitoring -Hydrothermal systems -Instrumental noise -Numerical modeling -Muography.

INTRODUCTION

Volcanoes are complex systems that vary significantly in terms of size, internal structure, and the dynamics of heat and mass transport from the magma chamber to the surface. These properties thus require dedicated studies for each particular volcano in order to understand their functioning and mitigate their potential hazards. Both active and dormant volcanoes carry potential edifice collapse hazards associated with their geometry and past activity, which usually leads to mechanically weak rocks and collapse-prone flanks (Heap & Violay, 2021).

Geophysical imaging is used to study the volcano's internal structure and is performed with different methods such as seismic, gravity, magnetotellurics, electrical resistivity tomography, and muography. Each of these methods is sensitive to the spatial distribution of a particular physical property, but they all provide complementary information useful to 1) determine regions of different rock types, in particular places where the rock has been altered or is naturally mechanically weak;

2) understand the magmatic and meteoric fluid pathways, the potential chemical reactions, and the heat transport dynamics inside the volcano; 3) relate processes detected in the surface to the main geological structures; and 4) calibrate numerical models meant to explain the volcano functioning.

When information about possible unstable and/or altered rock is available, failure surfaces can be suggested to test numerically the velocity and final emplacement of debris avalanches [START_REF] Peruzzetto | Modeling of partial dome collapse of la soufrière of guadeloupe volcano: implications for hazard assessment and monitoring[END_REF].

Muography is a relatively novel method used to image the density distribution inside volcanic edifices (Tanaka, Originally used to characterize the natural rock shielding overlying underground particle physics laboratories in the 1950s, muon absorption imaging was then transported towards archaeological JAIS-ID, 2021 FIGURE 2: An event display of cosmic muons emitted from a plane surface above the front panel of the 4-panel detector using the CRY MC generator. The scintillator matrices are colored in green, the shielding parts are in red, and a 100-mm-thick lead panel is located between the third and the fourth panel.

Muon vs electron signal

We compare in figure 3 the muon and electron charge signals dQ in the rear panel of our detector (bottom panel in figure 2), located behind the lead shielding panel. We observe that 96% of the muon signals on figure 3b, fitted with a Landau function, are tagged at as inlier. We also observe a suppression of noisy hits at lower ADC values. As for electrons, we observe a large proportion of outlier hits that populates the low ADC values, with a ratio outlier/inlier around 85%.

Angular resolution

In figure 4, we show the angular deviation in zenith (∆θ) and azimuth (∆ϕ) between the generated muon straight track and the reconstructed track. We observe for the RANSAC processing an improvement of the angular resolutions in both zenith and azimuth of the order of 30%, compared to the straightness check-based processing.

JAIS-ID, 2021

APPLICATION TO AN OPEN-SKY ACQUISITION

We present in this section the application of the RANSAC processing to real data taken with a 4-panels telescope, as described in 2. The data covers 36 hours and has been recorded in open-sky mode, measuring the vertical cosmic muon flux centered around zenith θ = 0 • . This run serves as a calibration step for measuring the experimental detector acceptance, which allows to take into account detection matrices flaws due notably to some imperfect optical couplings [10] in the dome-outgoing flux computation. The integrated acceptance analytical function, presented in [11], on the other hand, is purely geometrical and depends only on the dimensions of our detector: distance between front and rear matrices, number, length and width of scintillator bars in each matrices. We observe in figure 7 that the mean ratio between experimental and theoretical acceptances is 0.38 ± 0.02 using the RANSAC processing. This reduction factor is consistent with previous detection efficiency estimation obtained in [2]. As for the former processing algorithm, this ratio is equal to 0.19 ± 0.01, that is, reduced by a factor 2. This means that with the new RANSAC method we manage to reconstruct twice more events for real open-sky data.

SUMMARY AND OUTLOOK

In this paper, we have presented a new reconstruction algorithm, based on the RANSAC procedure, for scintillator-based muon telescope data and compared it with our former processing algorithm that relied on a track straightness check criterion without any trajectory fit for the incident particles. The main drawback of the RANSAC-based algorithm is in an increased processing time, multiplied by a factor 2 for 10 5 MC muon events on a standard 8Go CPU. In spite of this, we observe that a RANSAC-based reconstruction of the muon tracks in the detector offers significant advantages. From the RANSAC processing of a pure muon MC sample, we highlight :

• a 30% gain in muon angular resolution • a 10% increase in muon reconstruction efficiency • a 13% increase in the muon count Although in a real world data acquisition, we would not be able to perform precise particle identification, we have already observed for open-sky real data that the RANSAC-reconstructed event count is doubled compared to the one obtained with the former processing algorithm, leading to a telescope effective acceptance estimation closer to the theoretical integrated acceptance expected for the 4-panel telescope configuration. This increase in reconstructed event rate will need to be further confirmed with a realistic CORSIKA [13] cosmic flux generation, taking into account all the cosmic sources of physical noise, that were not present in the current GEANT4 simulation study.

FUTURE STUDIES

The RANSAC processing will be used to reconstruct the available tomography datasets taken both with new generation 4-panels telescopes presented in sec.2, and with former 3-panels telescopes. The 2D density radiographies obtained for different scanned regions of the lava dome will then be combined to serve as input for numerical modelling, conjointly with other geophysics data sets (e.g, from gravimetry surveys), in order to obtain a 3D bulk density distribution model of the lava dome at an expected unprecedentedly high resolution thanks to the uniqueness of the different data sets available and the quality of the event reconstruction. This will lead to a better characterization of the hydrothermal fluid circulation impact on rock alteration and the resulting partial flank collapse hazard.