Zoé

Léa

Jean Céleste

Gabi Lou

Manon Angélique

Jorge Jara
email: jara@geologie.ens.fr

Keywords: tectonics, faults, surface ruptures, distributed deformation, diffuse deformation, optical image correlation, Baluchistan, Ridgecrest Supershear earthquakes, Off-fault coseismic damage, Earthquake mechanics and observations

Robin whom I spent a lot of time with, discussing about science or not.

Apart from friends from work, I thank a lot my

RÉSUMÉ

Les grands séismes décrochants se produisent le plus souvent sur des systèmes de faille complexes, avec une partie de la déformation de surface distribuée entre la zone de rupture principale et le milieu environnant, également appelé off-fault. La déformation de surface totale associée à un séisme est donc difficile à observer et à quantifier sur le terrain. Dans cette thèse, nous utilisons la corrélation sub-pixellique d'images optiques de résolution 0,50 m pour mesurer le détail du champ de déplacement 3D, au plus proche de la zone de faille, le long des ruptures de Ridgecrest (2019, Mw6.4 et Mw7.1, Californie) et sur une portion de 30 km le long de la rupture du Baloutchistan (2013, Mw7.7, Pakistan). Pour ces deux cas d'études, nous cherchons à comprendre les mécanismes de distribution de la déformation cosismique à la surface, et les paramètrent qui influent sur cette distribution. Dans ce but, nous effectuons une analyse détaillée de la distribution de la déformation de surface. Nous quantifions avec une précision de 0,10 à 0,20 m le déplacement total au travers de la zone de faille, le glissement sur chacune des failles activées, ainsi que la déformation off-fault. Nos résultats montrent que la déformation de surface peut être localisée dans une zone de faille inférieur à 30-40 m, mais qu'elle se distribue souvent à travers des zones de plusieurs centaines de mètres à quelques kilomètres de large. Dans ces zones de déformation plus larges, la déformation off-fault apparaît soit par des failles et fractures secondaires visibles à la surface, soit par des gradients de déplacement continus de pente 10 -3 . Ces gradients sont interprétés comme étant de la déformation diffuse et sont largement observés le long de la rupture de Ridgecrest où ils représentent en moyenne 30% de la déformation de surface totale. Le long de la zone d'étude sur la rupture du Baloutchistan, nous mesurons en moyenne 16% de déformation off-faut, incluant la déformation diffuse et les zones de fractures secondaires. Cette différence s'explique par une plus grande proportion de zones de failles linéaires dans le cas de la rupture du Baloutchistan. A l'inverse, les ruptures de Ridgecrest sont bien plus complexes et présentent une segmentation du glissement en surface et en profondeur, corrélée aux variations de géométrie de la faille. Nous discutons de l'impact du contexte géologique sur la géométrie des failles et montrons que le système de faille principal qui a rompu lors du séisme du Baloutchistan, à l'exception des quelques grandes complexités géométriques, correspond à une interface majeure entre les couches sédimentaires du socle plissé du Makran. Inversement, le séisme de Ridgecrest s'est produit dans une zone à lithologie et fabrique géologique hétérogènes, ce qui favorise une géométrie de faille complexe et donc une plus grande distribution de la déformation à la surface. Nous comparons nos observations à une série de mesures sismiques qui mettent en évidence des zones de répliques kilométriques autour des failles, interprétées comme de la déformation inélastique. Cela a pour conséquence une modification des propriétés élastiques de la roche sur une large épaisseur, et a un fort impact sur l'accumulation des contraintes intersismiques et sur la dynamique de rupture cosismique. Dans une dernière étape, nous utilisons les données optiques, combinées à des données InSAR et GPS, pour mieux contraindre les modèles d'inversions cinématiques et améliorer notre compréhension du mécanisme de la rupture en profondeur. La caractérisation du processus de rupture et de la mécanique de la zone de faille, associée aux observations tectoniques sur la distribution de la déformation à la surface, sont essentielles pour comprendre le cycle sismique sur une faille et pour estimer les probabilités de rupture de surface et de mouvements du sol.

Mots clefs : tectonique, failles, ruptures de surface, déformation distribuée, déformation diffuse, corrélation d'images optiques, Baloutchistan, Ridgecrest

RÉSUMÉ DÉTAILLÉ

La tectonique des plaques met en mouvement les masses rocheuses superficielles de la Terre les unes par rapport aux autres. Ces masses rigides correspondent à des plaques lithosphériques, comprenant des domaines océaniques composés d'un croûte basaltique fine (<10 km), et des domaines continentaux constitués en majorité de roches granitiques, sur des épaisseurs de plusieurs dizaines de kilomètres [START_REF] Bertrand | La chaine des Alpes et la formation du continent europeen[END_REF][START_REF] Le Pichon | Models and structure of the oceanic crust[END_REF].

Les plaques tectoniques évoluent à la surface de la Terre selon des déplacements convergents, à l'origine des zones de subduction et des massifs montagneux, ou divergents, responsables des zones d'effondrement et d'ouverture de grands bassins [START_REF] Demets | Geological current plate motions[END_REF][START_REF] Wegener | Die Entstehung der Kontinente und Ozeane[END_REF]. Les plaques tectonique coulissent également horizontalement les unes par rapport aux autres, sous l'effet de mouvements cisaillants, sans compression ni ouverture entre les deux domaines [START_REF] Demets | Geological current plate motions[END_REF][START_REF] Wegener | Die Entstehung der Kontinente und Ozeane[END_REF]. Dans cette thèse, nous nous intéressons uniquement à cette dernière catégorie de mouvements cisaillants, qui affectent tout particulièrement les zones continentales.

La croûte continentale est soumise à de fortes contraintes provenant des limites de la plaque lithosphérique à laquelle elle appartient. Étant donné qu'elle n'est pas recyclée au cours du temps comme c'est le cas pour la croûte océanique, la croûte continentale accumule de la déformation au cours du temps [START_REF] Roquemore | Structure, tectonics, and stress field of the Coso Range, Inyo County, California[END_REF][START_REF] Woollard | Gravity anomalies, crustal structure, and geology in Alaska[END_REF].

Il en résulte un milieu structuralement hétérogène, qui présente de nombreuses zones de faiblesse concentrant les déformations tectoniques qui ne sont pas accommodées en limites de plaque. La croûte continentale ne peut donc pas être considérée comme un milieu purement rigide, mais bien comme une zone qui se déforme en réponse aux contraintes tectoniques. En raison de l'hétérogénéité de sa structure, la déformation de la croûte continentale se traduit par l'activation de failles multiples aux géométries complexes [START_REF] Petersen | Fault slip rates and earthquake histories for active faults in southern California[END_REF][START_REF] Tapponnier | Active faulting and cenozoic tectonics of the Tien Shan, Mongolia, and Baykal Regions[END_REF]. Dans ce cas, les contraintes tectoniques se distribuent entre les nombreuses failles préexistantes et on parle alors de déformation distribuée. Cette complexité de la déformation continentale est largement visible sur le terrain et via l'imagerie satellite. Elle est observée à toute les échelles, depuis les grandes failles intracontinentales (>100 km; e.g., [START_REF] Tapponnier | Active faulting and cenozoic tectonics of the Tien Shan, Mongolia, and Baykal Regions[END_REF], aux microfailles observées en laboratoire (<1 cm; e.g., [START_REF] Faulkner | Scaling of fault damage zones with displacement and the implications for fault growth processes[END_REF].

Une faille est une interface rocheuse qui permet d'accommoder la déformation qui se produit entre deux blocs de roche qui se déplacent dans des directions différents, ou dans la même direction mais avec des vitesses différentes [START_REF] Anderson | The dynamics of faulting and dyke formation with applications to Britain[END_REF]. Cette déformation est soit compressive, soit extensive, ou encore cisaillante en fonction du contexte tectonique. La déformation cisaillante correspond à celle étudiée dans cette thèse, dans le contexte des failles continentales décrochantes. En réaction à la géométrie complexe des failles, la déformation cisaillante est également complexe. Elle se distribue à la fois entre les différentes failles activées mais également dans le milieu environnant, appelé off-fault (Mitchell and Faulkner, 2009;Rockwell et al., 2002). Mais, contrairement à la déformation localisée sur les failles, la déformation off-fault est difficile à mesurer sur le terrain car elle s'étend sur des grandes zones (> quelques centaines de mètres) et est caractérisée par des faibles gradients de déplacement du sol (10 -2 à 10 -4 ). Par conséquent, les mesures de déplacement de surface effectuées à la suite d'un séisme se concentrent généralement sur la partie visible de la déformation qui se produit sur les failles. Avec l'évolution de l'imagerie satellite et des méthodes de traitement et de corrélation d'images, il a été montré que la déformation off-fault constituait une part importante du déplacement du sol lors d'un séisme, pouvant représenter jusqu'à la moitié du déplacement total mesuré (Barnhart et al., 2020;[START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF][START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF].

Face à cette constatation, de nombreuses études ont été menées afin de mieux caractériser cette déformation off-fault et les paramètrent qui la contrôlent. Les résultats des recherches menées jusqu'ici montrent une corrélation entre la déformation off-fault et les paramètrent de rupture tels que la géométrie de la faille [START_REF] King | Role of Fault Bends in the Initiation and Termination of Earthquake Rupture[END_REF]Klinger et al., 2018), la structure et lithologie de la croûte continentale [START_REF] Teran | Geologic and structural controls on rupture zone fabric: A field-based study of the 2010 Mw 7.2 El Mayor-Cucapah earthquake surface rupture[END_REF][START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF], la déformation cumulée sur le long-terme [START_REF] Brodsky | Faults smooth gradually as a function of slip[END_REF]Mitchell and Faulkner, 2009), la profondeur de la croûte sismogénique [START_REF] Ampuero | Upper Limit on Damage Zone Thickness Controlled by Seismogenic Depth[END_REF][START_REF] Lefevre | Experimental evidence for crustal control over seismic fault segmentation[END_REF], ou encore la magnitude du séisme [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF]. Pour améliorer notre compréhension des mécanismes de la déformation off-fault, il donc est nécessaire d'améliorer les observations et la caractérisation des déformations de surface dans les régions sismiquement actives. Il est également important de comprendre les paramètres qui influencent la distribution de la déformation entre les failles et le milieu off-fault, et l'impact de ces processus sur le mécanisme de rupture. Pour étudier cette déformation, nous sommes focalisés sur des évènements ayant rompu la croûte superficielle et pour lesquels la déformation est visible directement sur le terrain et sur les images satellites.

Les problématiques soulevées dans cette thèse sont : Comment la déformation de surface se distribue lors d'un séisme et quels sont les mécanismes en jeu ? quel est le lien entre cette déformation de surface et le mécanisme du séisme en profondeur ? Dans quelle mesures la déformation de surface est influencée par la géométrie du système de failles et le contexte géologique ? Pour répondre à ces questions, nous utilisons des méthodes de corrélation d'images satellites optiques à très haute résolution (0,5 m) pour caractériser le champ de déplacement horizontal et vertical au plus proche de la zone de faille. Après une introduction générale sur la tectonique des plaques et la déformation de surface associée aux failles actives, suivie d'une présentation de la méthodologie utilisée, cette thèse est divisée en quatre grands chapitres.

Le premier chapitre présente une l'étude d'une section (~30 km) de la zone de faille qui a rompue lors du séisme du Baloutchistan (2013, Pakistan, Mw7.8). Dans ce chapitre, nous mettons en évidence les processus de déformation de surface qui affectent la croûte jusqu'à deux kilomètres en dehors du système de faille principal. Nous montrons une étroite relation entre la géométrie du système de faille et la déformation off-fault.

Finalement, nous mesurons la déformation post-sismique entre 6 mois et 3 ans après le séisme, et montrons que la déformation off-fault s'est produite avant cette période.

Dans le deuxième chapitre, nous mesurons le champ de déplacement associé à la séquence sismique de Ridgecrest (2019, Californie, Mw6.4 et Mw7.1). Nous y observons une zone de rupture particulièrement complexe composée d'une cinquantaine de failles qui s'entrecoupent. Les mesures de déplacement de surface montrent qu'une grande partie de celui-ci se produit en dehors du système de faille, à l'origine de gradients de déplacement d'échelle kilométrique. La zone totale affectée par la déformation de surface lors du séisme est donc plus beaucoup plus large (>1 km) que celle où se concentre les structures cassantes (de l'ordre de quelques centaines de mètres tout au plus). Nous comparons ces observations aux nombreuses mesures effectuées sur cette rupture via des méthodes géodésique, sismiques, numériques et de terrain, et discutons des liens entre la géométrie du système de faille, le processus de rupture, et la déformation totale observée à la surface.

Enfin, nous utilisons nos mesures haute-résolution combinées à des données GPS et InSAR pour contraindre les modèles de rupture en profondeur. Les résultats montrent que le glissement sur les failles en surface est largement inférieur au glissement qui se produit sur les failles en profondeur. La différence entre le déplacement superficiel et celui qui se produit sur le système de faille en profondeur est appelé shallow slip deficit (SSD; [START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF]. Dans le cas du séisme de Ridgecrest, le SSD moyen évalué dans les modèles est équivalent à la déformation diffuse moyenne mesurée à la surface. Cela implique que la déformation diffuse permet de compenser le déficit de glissement qui se produit sur les failles à faible profondeur.

Le troisième chapitre regroupe les résultats d'une étude multidisciplinaire à l'interface entre la sismologie, la géodésie, et les méthodes de modélisation numériques, et menée dans le cadre d'une étude collaborative sur les séismes supershear. Cette étude met en lumière une réduction de la zone de déformation totale au niveau de la transition de la rupture vers une vitesse supershear, c'est-à-dire supérieure à celle des ondes de cisaillement. Le travail effectué dans le cadre de cette thèse a consisté à mesurer le champ de déplacement associé à la zone de transissions supershear du séisme de Kokoxili (2001, Mw7.8, Tibet). Les résultats obtenus corroborent les résultats des calculs théoriques et des modèles numériques. Étant donné la faible résolution des images disponibles dans la zone du séisme de Kokoxili en 2001 (10 m), les observations effectuées dans cette étude ne permettent pas de caractériser les processus de déformation au coeur de la zone de faille.

Par conséquent, les résultats et interprétations développés ci-après se réfèrent principalement aux études sur les séismes du Baloutchistan et de Ridgecrest.

Mise en évidence de réseaux de failles complexes

Grâce aux méthodes de corrélation d'images optiques à haute résolution, nous mesurons le détail du champ de déplacement au coeur des zones de failles des séismes de Ridgecrest et du Baloutchistan. A partir de ces mesures, nous mettons en lumière des géométries de failles particulièrement complexes. Afin de mieux caractériser la géométrie de ces failles et la déformation qu'elles accommodent, nous dérivons les valeurs de la dilatation, du cisaillement et de la rotation à partir des cartes de déplacement horizontal. Cela permet une mise en évidence de la géométrie du système de faille, une caractérisation des processus de déformation, ainsi qu'une mesure de l'amplitude des déformations sur chacune des failles. Dans le cas du Baloutchistan, nous observons deux complexités géométriques majeures. La première consiste en un relais compressif, appelé C2, qui constitue la plus grande complexité de la rupture, longue de 200 km au total. La deuxième correspond à un changement d'azimut de la faille principale, auquel est associé une large zone densément fracturée. Cette deuxième zone est appelée F1. Nous montrons que la rupture s'est propagée à la fois sur des sections continues et bien définies dans le paysage, et sur des failles plus discontinues et dont la trace à la surface est plus ténue. Nous observons également de nombreuses failles qui se croisent, certaines presque à angle droit. Dans le cas de Ridgecrest, de nombreuses failles orthogonales sont également observées, avec la particularité que les deux systèmes orthogonaux participent au mécanisme principal de la rupture. En effet, cette séquence sismique a activé deux jeux de failles, l'un dextre et orienté nord-ouest et l'autre sénestre et orienté nord-est. Le système de faille total activé à la surface est alors constitué d'une cinquantaine de failles différentes.

Des profils de déplacement recoupant les différentes failles observées permettent de mesurer le déplacement accommodé par chacune d'entre elle avec une précision de 5 à 20 cm. La comparaison de ces mesures avec les mesures faites par interprétation d'image optique et de terrain permet de valider la méthode utilisée pour mesurer la déformation de surface. Cela montre que la corrélation d'image satellite permet une mesure détaillée, précise et fiable des déplacements de surface lors d'un séisme. Elle est donc particulièrement adaptée pour étudier les phénomènes sismiques dans des zones où le travail de terrain est impossible. La quantification du déplacement via les méthodes de corrélation optique permet tout particulièrement de mettre en évidence les petites failles secondaires (< 1 km de long) qui accommodent des déplacements de plusieurs mètres.

Les résultats pour les deux zones d'étude, Baloutchistan et Ridgecrest, montrent que le déplacement mesuré sur les failles principales n'est pas toujours représentatif du déplacement total qui s'effectue en surface, surtout dans les zones où la géométrie des failles et complexe.

Mesure des gradients de déformation qui se produisent en dehors des failles

En plus de permettre une caractérisation détaillée des failles, la corrélation d'images optiques à haute résolution permet de détecter des gradients de déplacement du sol qui se produisent sur des zones plus larges (>1 km) autour du système de faille. Ces gradients ne sont pas facilement mesurables sur le terrain en raison de leur grande extension horizontale et au fait qu'ils ne présentent pas de marqueurs visibles, comme des fractures.

Ces gradients sont continus et affectent de manière homogène le milieu autour des failles.

La déformation du sol s'effectue donc de manière diffuse dans le milieu et sur de grandes zones. On appelle alors ce processus de la déformation diffuse, en opposition à la déformation localisée qui se produit sur les failles. Selon le contexte, cette déformation peut se produire soit off-fault, c'est à dire le long d'un système de faille ayant rompu la surface, soit représenter l'intégralité de la déformation à la surface si le système de faille est resté enfoui en profondeur. Afin de quantifier l'amplitude de la déformation qui se produit en dehors des failles, nous utilisons des profils de plusieurs kilomètres de large (4-8 km) placés au travers de la zone de rupture, et mesurons le déplacement total qui a affecté la zone. La déformation dans le milieu correspond alors à la différence entre le déplacement total et la somme du déplacement sur les failles.

Dans le cas de la rupture du Baloutchistan, la déformation diffuse est mesurée uniquement au travers de la zone de relais C2, à la fois au centre du relai compressif et dans le toit de la rupture. Elle représente en moyenne 4% du budget total de déplacement sur la zone étudiée (30 km de long). Par contre, au niveau de la complexité F1, nous mettons en évidence des gradients de déformation off-fault plus élevés et affectant une zone inférieure à 500 m dans le toit de la rupture. Cette déformation est associée à des fractures et est différente de la déformation diffuse décrite précédemment. Elle accommode en moyenne 12% du budget total de déplacement sur la zone d'étude. Dans le cas de Ridgecrest, il est mis en évidence une déformation diffuse comparable à celle mesurée autour de la complexité C2, par contre, elle est présente sur la quasi-totalité de la zone de rupture. Par ailleurs, la déformation diffuse n'est pas toujours associée à une faille visible à la surface. Elle est alors le résultat d'une déformation profonde. Le long des ruptures de Ridgecrest, il est mesuré en moyenne 30% de déformation diffuse par rapport à la déformation totale mesurée en surface, dans la composant sénestre pour le foreshock, et dans la composante dextre pour le mainshock. Par contre, aucune déformation off-fault par fracturation n'est observée dans la zone d'étude de Ridgecrest.

La déformation off-fault peut donc représenter une grande partie de la déformation totale se produisant à la surface lors d'un séisme. Cette conclusion pose la question de la représentativité des mesures de déplacement de surface généralement effectuées sur les failles par rapport au déplacement total associé au séisme. La déformation qui se produit de manière diffuse pose notamment la question de la mesure de la déformation sur le terrain, étant donné que celle-ci s'étend sur de grandes zones (> 1km) et n'est pas détectable à l'oeil nu. De plus, nous montrons qu'elle représente jusqu'à 30% du budget total de déformation lors du séisme de Ridgecrest. La suite de ce travail se concentre donc sur la caractérisation de cette déformation diffuse et de son rôle dans le processus de rupture. Dans ce but, nous utilisons les données optiques haute résolution pour contraindre un modèle d'inversion cinématique qui permet de représenter le glissement cosismique qui a eu lieu sur le plan de faille en profondeur.

Relation entre la déformation diffuse, la magnitude du séisme, et la géométrie du système de failles L'inversion conjointe des données de déplacement de surface pour la modélisation du processus de rupture du séisme de Ridgecrest 2019 permet de contraindre le glissement sur la faille en profondeur. En général, les modèles soufrent généralement d'un manque de données à proximité de la faille afin de contraindre le glissement dans la partie superficielle du modèle. Dans ce travail, utilisons les données optiques qui offrent des mesures au plus proches des failles, ce qui permet une bonne représentation du glissement sur la faille à des faibles profondeurs (0-4 km).

Les résultats de ce travail montrent que la quantité de glissement se produisant en profondeur sur le plan de faille est supérieur au déplacement sur les failles en surface. La différence entre ces deux valeurs représente le SSD. On montre alors que le SSD est équivalent à la déformation diffuse mesurée en surface. Cela implique que la déformation diffuse compense le SSD et permet d'accommoder une quantité de déplacement similaire en surface et en profondeur. La déformation diffuse est donc corrélée au SSD qui se produit le long de la faille. Ce résultat a des implications majeures pour la compréhension du processus sismique, et montre qu'aucune déformation de surface n'est nécessaire pendant les phases inter-ou post-sismiques pour compenser le SSD. Cela montre également la nécessité de considérer la déformation de surface sur une largeur en moyenne supérieure 2 km autour des failles afin de prendre en considération l'ensemble des mécanismes mis en jeu.

Ces résultats sont comparés avec les observations faites le long de la rupture du Baloutchistan, ainsi qu'avec des observations et modèles publiés sur d'autres séismes. A partir de cette comparaison, nous montrons que le SSD est directement corrélé à la magnitude du séisme, avec un SSD de 0% le long des ruptures de magnitude supérieur à 7.5. Par conséquent, la déformation diffuse est également corrélée à la magnitude du séisme et se produit dans de faibles proportions le long des ruptures Mw>7.5. Cela s'explique par le fait que les séismes de forte magnitude rompent l'intégralité de la croute sismogénique jusqu'à la surface avec un glissement maximal propageant généralement à la surface.

Enfin, nous montrons que la distribution spatiale de la déformation diffuse, le long du système de failles, dépend de la géométrie des failles. C'est le cas par exemple de la rupture de Ridgecrest qui apparait segmentée, et met en jeu 3 segments de faille séparés par des variations d'azimut et des complexités géométriques. Chaque segment mesure entre 15 et 20 km de long, et présente des proportions de déformations diffuse différentes.

A l'inverse, le long de la rupture du Baloutchistan, la déformation diffuse se concentre dans les zones de complexité, à la transition entre des sections de faille linéaires. En comparant ces observations à celles effectuées sur d'autres séismes, il est montré que malgré la forte magnitude du séisme, les complexités géométriques agissent comme des barrières à la propagation du glissement sur le plan de faille et génèrent de la déformation diffuse dans le milieu. Par contre, le budget total de déplacement reste inchangé au travers des différentes complexités.

Conséquences de nos observations pour la compréhension de la mécanique des zones de faille

La déformation diffuse correspond à une zone de déformation large, de l'ordre de 2 km, qui se produit autour du système de faille principal. Dans l'hypothèse où cette déformation serait de type cassante, elle devrait affecter les propriétés mécaniques des roches. Cela influencerait alors sur le comportement de la zone de faille au cours des phases d'accumulation et de relâchement des contraintes tectoniques. Nous discutons alors des différents mécanismes pouvant être responsable de la déformation diffuse. Cette déformation étant largement observée dans des zones de socle granitique, nous proposons que les mécanismes principaux sont de type cassant, mais dont les traces (fractures) ne sont détectables sur le terrain ou par corrélation d'images satellite en raison de leur taille (inférieure à quelques centimètres). Cette interprétation est soutenue par le fait que la large zone de déformation diffuse le long des ruptures de Ridgecrest est corrélée à la distribution des répliques en profondeur. Par conséquent, la déformation diffuse serait un processus cassant, se produisant de manière cosismique ou dans les premières heures et les premiers jours suivant la rupture. Ceci implique que le bilan énergétique de la rupture ainsi que les mouvements du sol générés lors du séisme sont modifiés par la contribution de la déformation diffuse. Cependant, il est nécessaire de vérifier quels sont les mécanismes précis qui entre en jeu dans le processus de déformation diffuse afin de pouvoir estimer ses conséquences sur le comportement de la zone de faille et les caractéristiques dynamiques du séisme. Ces questions font l'objet de propositions de recherches exposées dans les perspectives de cette thèse (section 8.2).

Implications pour l'évaluation de l'aléa sismique

La caractérisation de la déformation diffuse qui se produit dans le milieu autour des failles est essentielle pour la compréhension et la prévision du comportement des failles, de leur potentiel sismogénique, et de leur potentiel de destruction en cas de rupture. Or, bons nombres d'estimations des magnitudes et temps de récurrence des séismes sont basées sur des mesures qui prennent en compte uniquement la zone de faille principale. Le biais sur le déplacement estimé du séisme, induit par la non prise en compte de la déformation diffuse, peut atteindre plusieurs mètres. Cela remet en question les estimations de taux de déformation sur les failles utilisées pour calculer les récurrences des séismes. En effet, celle-ci se basent essentiellement sur des mesures paléosismologique effectuées sur le plan de faille principal. Étant donné que la déformation diffuse est corrélée au SSD et à la magnitude du séisme, ce biais est plus important pour les séismes de faible magnitude.

L'incertitude sur le budget de déplacement total associé au séisme s'ajoute à une incertitude quant au rôle de la déformation diffuse dans le comportement sismique des failles. Notamment, il n'a pas encore été montré si la déformation diffuse se produisait de manière cosismique lors de la rupture ou de manière asismique dans les jours et semaines qui suivent la rupture. Si la déformation diffuse se produit de manière cosismique, elle influe sur la propagation dynamique de la rupture, et potentiellement sur le budget énergétique et le rayonnement des ondes sismiques. Finalement, nos données permettent de mieux comprendre le rôle de la géométrie des failles faille et du contexte géologique dans le contrôle du processus de rupture et de la distribution de la déformation. Cela permet de mieux contraindre les scénarios de rupture basés sur une caractérisation des géométries des réseaux de failles qui tentent de prédire la capacité de la rupture à se propager au travers des complexités géométriques. L'analyse de l'évolution des complexités géométriques avec le glissement cumulé et donc des scénarios de rupture représente une piste de recherche importante pour le futur.

INTRODUCTION AND STATE OF THE ART

Our knowledge of natural processes evolves along with an increasing collection of natural observations and models that attempt to reproduce it. The more scientific technics improve, the more we observe complexity in the natural processes, raising the issue of the degree of complexity that is necessary to better understand and predict natural processes. We address this issue in this thesis since we image and analyze details of earthquake ruptures of amplitudes and scales that are generally below consideration. To integrate this work into the broad scientific state of the art, and show the interest of going into the details of the mechanisms, I start this introduction with the origin of the global plate tectonics theory, then go down at the scale of faults and their earthquake cycle, and finally describe the current knowledge on fault structure at scales of kilometers to meters.

From Plate tectonics to earthquake cycle

Plate tectonics

In the late 19ths', the presence of mountains and basins at the surface of the Earth was explained by the "thermic contraction" model. This model proposes that the Earth is a mass in fusion that cools down through times, with oceans and continents as part of a unique granitic system (Suess, 1885). This theory was refuted because the distribution of elevations at the surface of the Earth is bimodal, while thermic contraction would generate a gaussian or normal distribution. Indeed, the bimodal distribution of the altitudes reveals that oceans and continent are part of two different systems with different composition and density [START_REF] Dana | On some results of the earth's contraction from cooling including a discussion of the origin of mountains and the nature of the earth's interior[END_REF][START_REF] Willis | Discoidal structure of the lithosphere[END_REF]. Arising from isostasy theories [START_REF] Hayford | Supplementary Investigation in 1909 of the Figure of Earth and Isostasy[END_REF], it was proposed that continents could move horizontally on a more viscous basement if a sufficient horizontal stress was applied [START_REF] Wegener | Die Entstehung der Kontinente und Ozeane[END_REF]. Numerous geological, geophysical, paleontological and paleoclimatic observations support the idea that continental masses have moved horizontally through time. For example, the similarity between mountain ranges between Europe and north America [START_REF] Bertrand | La chaine des Alpes et la formation du continent europeen[END_REF], or the similarity of continental margins east and west of the Atlantic ocean [START_REF] Haug | Les géosynclinaux et les aires continentales[END_REF], suggests that American and African continents were aside each other in past geological times. This gave birth to the "continental drift" theory in 1910-1920 [START_REF] Wegener | Die Entstehung der Kontinente und Ozeane[END_REF].

After World War II, the development of geophysical technics, especially seismology [START_REF] Gutenberg | Low-Velocity Layers in the Earth, Ocean, and Atmosphere[END_REF], gravity [START_REF] Talwani | Crustal structure of the mid-ocean ridges: 2. Computed model from gravity and seismic refraction data[END_REF] and paleomagnetism [START_REF] Irving | II. Analysis of the paleomagnetism of the Torridonian sandstone series of North-West Scotland[END_REF], revealed that oceanic and continental crusts are part of a larger system, the lithospheric plates. Lithospheric plates consist in rigid bodies that move on a more ductile asthenosphere [START_REF] Dietz | Continent and Ocean Basin Evolution by Spreading of the Sea Floor[END_REF]. In addition, investigation led on the nature and structure of the oceanic crust revealed the fact that oceanic crust is being renewed at medio-oceanic ridges, drifting continents away from each other (Le [START_REF] Le Pichon | Models and structure of the oceanic crust[END_REF]. This is referred to as the "oceans spreading" theory [START_REF] Wilson | A New Class of Faults and their Bearing on Continental Drift[END_REF].

Starting in the late-1967, the "ocean spreading" theory evolved to the "plate tectonics".

J. Morgan, D. McKenzie and X. Le Pichon proposed a model of rigid lithospheric plates that are separated by oceanic ridges, subduction zones or transform faults [START_REF] Mckenzie | Speculations on the Consequences and Causes of Plate Motions*[END_REF][START_REF] Morgan | Rises, trenches, great faults, and crustal blocks[END_REF][START_REF] Pichon | Sea-floor spreading and continental drift[END_REF]. Oceanic ridges are areas where oceanic lithosphere is created. Subduction zones are the boundaries where lithosphere deeps onto the asthenosphere and is recycled. Transform faults are boundaries where no material is either created or recycled, but where lithospheric plates slide aside each other. The exhaustive inventory of tectonic plates by Le Pichon leads to a model with 6 major plates: Eurasian, American, African, Indian, Pacific and Antarctic plates (Le [START_REF] Pichon | Sea-floor spreading and continental drift[END_REF]Dewey, 1972).

Following the years 1995-2000, and the development of Global Positioning Systems (GPS), geophysicists could measure millimetric displacement of the continental plates all over the world [START_REF] Herring | Geodetic applications of GPS[END_REF]. This leads to much more detailed models of the plate tectonics, comprising at least 25 rigid plates to more than 50 depending on the degree of detail in the model [START_REF] Bird | An updated digital model of plate boundaries[END_REF][START_REF] Demets | Geological current plate motions[END_REF][START_REF] Kreemer | A geodetic plate motion and Global Strain Rate Model[END_REF]. At each of these plate transitions, and because the plates are moving following different displacement vectors, areas of plate junctions are subjected to very high constraints, either diverging, converging or shearing. These constraints are accommodated through rock deformation that for most parts, is characterized by rock fracturing, and sliding of the rocks against each-other along interfaces that are called faults. This triggers an intense seismic activity [START_REF] Aki | Seismic displacements near a fault[END_REF], and explains the worldwide distribution of earthquake epicenters (Fig. 1.1B). But, part of the worldwide seismicity occurs away from plate boundaries, and within the lithospheric plate itself [START_REF] Sykes | Intraplate seismicity, reactivation of preexisting zones of weakness, alkaline magmatism, and other tectonism postdating continental fragmentation[END_REF]. This seismicity is triggered by distributed deformation that occurs over wide regions, located nearby or sometimes disconnected from the plate boundaries. This comes in contrast to the localized deformation that occurs at the plate interfaces, and is an evidence that lithospheric plates are not entirely rigid and can deform in their inner part. Intraplate distributed seismicity challenges the initial model of rigid lithospheric plates, and raises the issue of the nature of this seismicity.

Faults, earthquakes and earthquake cycle

A fault is an interface where differential displacements between distinct blocks or rocks are accommodated (Figs. 1.2 and 1.3). Faults occur at every scales, from the plate tectonic interfaces (e.g., [START_REF] Hubert-Ferrari | Morphology, displacement, and slip rates along the North Anatolian Fault, Turkey[END_REF][START_REF] Tapponnier | Active faulting and cenozoic tectonics of the Tien Shan, Mongolia, and Baykal Regions[END_REF], to small faults observed on the field (e.g., [START_REF] Chester | Implications for mechanical properties of brittle faults from observations of the Punchbowl fault zone, California[END_REF][START_REF] Shipton | Damage zone and slip-surface evolution over μm to km scales in high-porosity Navajo sandstone, Utah[END_REF] or generated in laboratory (e.g., [START_REF] Scholz | Microfracturing and the inelastic deformation of rock in compression[END_REF][START_REF] Tullis | Experimental deformation of dry westerly granite[END_REF]. In any of these cases, the deformation accommodated by faults are either diverging, converging or shearing, in response to the constraints that are applied on these interfaces (Figs. 1.2 and 1.3).

Along a fault plane, deformation is governed by frictional and rheological laws (Byerlee, 1978;[START_REF] Scholz | Microfracturing and the inelastic deformation of rock in compression[END_REF]. Fault friction prevents the fault from sliding continuously through time such that they accumulate the tectonic loading during what is called the inter-seismic period. This tectonic loading is accumulated through elastic deformation in the crust [START_REF] Byerly | Earthquake Mechanisms[END_REF]Scholz, 1998), for tens to thousands of years depending on the seismic behavior of the fault and of the concerned region [START_REF] Avouac | From Geodetic Imaging of Seismic and Aseismic Fault Slip to Dynamic Modeling of the Seismic Cycle[END_REF]. The seismic behavior of a fault varies from purely unstable, where the locked fault accumulates stress until it overcomes the static friction for which instability and seismic slip can be triggered, to entirely stable, where the fault is unlocked and slides continuously through time without any accumulation of stress, which make impossible unstable slip (Scholz, 1998). There exist some exceptions of faults which are known to accommodate displacement continuously and aseismically through time like, for example, the Central San Andreas fault, that exhibit creep rates up to 30 mm/year [START_REF] Burford | Slip on the San Andreas fault in central California from alinement array surveys[END_REF][START_REF] Khoshmanesh | Multiscale Dynamics of Aseismic Slip on Central San Andreas Fault[END_REF]. Similarly, the North Anatolian faults exhibit some creeping portions with rates varying from 8 to 50 mm/year over the past 70 years (Rousset et al., 2016). In contrast to stable creeping faults, locked faults that accumulate elastic stress through time are prone to unstability. When stresses become too high for the fault to remain locked, the fault unlocks and slip is triggered on the fault plane, and elastic rebound of the medium trigger seismic wave radiations [START_REF] Anderson | Parametric study of near-field ground motion for a strike-slip dislocation model[END_REF][START_REF] Byerly | Earthquake Mechanisms[END_REF]. The fault rupture propagates at speeds that generally approach the shear wave speed, about 3 km/s [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF][START_REF] Freund | Dynamic Fracture Mechanics[END_REF], but the rupture speed can sometime overcome the shear wave speed, and propagates with velocities up to 5-6 km/s (Das, 2007;Vallée et al., 2008). In this case, the seismic waves and associated ground motions can be far most destructive [START_REF] Mello | Identifying the unique ground motion signatures of supershear earthquakes: Theory and experiments[END_REF]. The unstable slip phase, also called the co-seismic phase, is the one representing major hazard to infrastructures and populations because of the ground acceleration and seismic waves [START_REF] Anderson | Parametric study of near-field ground motion for a strike-slip dislocation model[END_REF][START_REF] Bouchon | The motion of the ground during an earthquake: 1. The case of a strike slip fault[END_REF].

Eventually, after the co-seismic phase, after-slip and creep on the fault, as well as poroelastic and visco-elastic deformations of the lower-crust and mantle, occur to accommodate the remaining heterogeneities in the stress field after the onset of rupture.

This phase can last from months to years depending on the earthquake magnitude [START_REF] Barbot | A unified continuum representation of post-seismic relaxation mechanisms: semi-analytic models of afterslip, poroelastic rebound and viscoelastic flow: Semi-analytic models of postseismic transient[END_REF][START_REF] Diao | Overlapping post-seismic deformation processes: afterslip and viscoelastic relaxation following the 2011 Mw 9.0 Tohoku (Japan) earthquake[END_REF]. But, the fault behavior is not homogeneous in space nor it is constant through time. In fact, the stable vs unstable behavior (Scholz, 1998) of the fault is function of the strength of the fault, that is the capacity of a fault to accumulate stress and withstand before unstable slip (Fig. 1.4;[START_REF] Avouac | From Geodetic Imaging of Seismic and Aseismic Fault Slip to Dynamic Modeling of the Seismic Cycle[END_REF][START_REF] Mclaskey | Slow and fast ruptures on a laboratory fault controlled by loading characteristics[END_REF][START_REF] Thomas | Lithological control on the deformation mechanism and the mode of fault slip on the Longitudinal Valley Fault, Taiwan[END_REF]. It will vary following parameters like depth, lithology, pore pressure, stress heterogeneities, tectonic loading and seismic history of the fault [START_REF] Avouac | From Geodetic Imaging of Seismic and Aseismic Fault Slip to Dynamic Modeling of the Seismic Cycle[END_REF]. Hence, the seismic cycle will not reproduce similarly in time and space, neither will the seismic events (Fig. 1.4). A portion of a fault can host large earthquakes, while neighboring portions will host more repetitive and lower magnitudes events or eventually aseismic slip [START_REF] Avouac | From Geodetic Imaging of Seismic and Aseismic Fault Slip to Dynamic Modeling of the Seismic Cycle[END_REF][START_REF] Bakun | The Parkfield, California, Earthquake Prediction Experiment[END_REF]. Also, a portions of a fault that generated a large magnitude earthquake can also produce smaller size events or aseismic creep in the following cycles [START_REF] Chen | Creeping faults: Good news, bad news?[END_REF]Rousset et al., 2016). So, it is necessary to acquire data over a wide range of spatial and temporal scales to characterize the system variability and evolutions (Fig. 1.4). of the local tectonic strain, the seismic cycle can be constituted of infinite combinations and varieties of stable and unstable sliding processes (Di Toro et al., 2012).

Several models explain to a first order the time recurrence of seismic events on a defined portion of a fault [START_REF] Shimazaki | Time-predictable recurrence model for large earthquakes[END_REF], as well as the space distribution of coseismic slip during the different events along the strike of the fault [START_REF] Schwartz | Fault Behavior and Characteristic Earthquakes: Examples From the Wasatch and San Andreas Fault Zones[END_REF]. The example from Figure 1.4 shows a heterogeneous slip model for a portion of fault. In this model, the rate of shear stress accumulation, that is the slop of the grey curve, varies during the earthquake cycle, as well as the strength of the fault, that depends on the seismic history of the fault. Hence, because of the various parameters that are implied in the seismic behavior of a fault, the moment when shear stress overcomes fault strength occurs at various thresholds, and over various time intervals. These models are constrained by data on the fault slip rates, and on the recurrence and magnitude of past earthquakes. Current fault slip rates are general measured using geological observations (e.g., [START_REF] Hubert-Ferrari | Morphology, displacement, and slip rates along the North Anatolian Fault, Turkey[END_REF][START_REF] Schwartz | Fault Behavior and Characteristic Earthquakes: Examples From the Wasatch and San Andreas Fault Zones[END_REF], InSAR [START_REF] Peltzer | Transient strain accumulation and fault interaction in the Eastern California shear zone[END_REF][START_REF] Tong | High-resolution interseismic velocity data along the San Andreas Fault from GPS and InSAR[END_REF] and GNSS [START_REF] Gan | Strain accumulation across the Eastern California Shear Zone at latitude 36°30′N[END_REF][START_REF] Vernant | What can we learn from 20years of interseismic GPS measurements across strike-slip faults?[END_REF] data that report the displacement of the lithospheric plates. Magnitude of recent earthquakes are known from geodetic (e.g. [START_REF] Chen | Cascading and pulse-like ruptures during the 2019 Ridgecrest earthquakes in the Eastern California Shear Zone[END_REF][START_REF] Jolivet | The 2013 Mw7.7 Balochistan earthquake: Seismic potential of an accretionary wedge[END_REF][START_REF] Lasserre | Coseismic deformation of the 2001 Mw = 7.8 Kokoxili earthquake in Tibet, measured by synthetic aperture radar interferometry[END_REF] and seismic data [START_REF] Allen | The Potential for Earthquake Early Warning in Southern California[END_REF]. Magnitude and recurrence of earthquakes that occurred before the instrumental period, as well as long-term slip rates (>10 2 years), are generally retrieved using paleoseismology (e.g., [START_REF] Lefevre | Slip deficit and temporal clustering along the Dead Sea fault from paleoseismological investigations[END_REF][START_REF] Schwartz | Fault Behavior and Characteristic Earthquakes: Examples From the Wasatch and San Andreas Fault Zones[END_REF] and geomorphology (e.g., [START_REF] Haddon | Surface slip during large Owens Valley earthquakes[END_REF][START_REF] Hubert-Ferrari | Morphology, displacement, and slip rates along the North Anatolian Fault, Turkey[END_REF] methods. Our ability to predict the evolution of the earthquake cycle model for a fault, or a fault system, will then depend upon our ability to measure the current and past deformations, as well as our knowledge of the interactions between the parameters that govern the fault behavior and its variations with time and space.

However, most observations are made at the surface of the Earth, while the faults are large and deep structures that extend several tens of kilometers at depth. This raises the issue of our ability to infer fault properties that occur at depth from measurements at the surface. To do so, we use observations on exhumed faults, that inform on the fault structure at depth [START_REF] Lin | Spatial variations in damage zone width along strike-slip faults: An example from active faults in southwest Japan[END_REF]Mitchell and Faulkner, 2009), as well as laboratory experiments [START_REF] References Aben | High strain rate deformation of porous sandstone and the asymmetry of earthquake damage in shallow fault zones[END_REF][START_REF] Lockner | A generalized law for brittle deformation of Westerly granite[END_REF] and numerical modeling [START_REF] Allison | Earthquake cycle simulations with rate-and-state friction and power-law viscoelasticity[END_REF][START_REF] Kaneko | Shallow slip deficit due to large strike-slip earthquakes in dynamic rupture simulations with elasto-plastic off-fault response: Modelling shallow slip deficit[END_REF]) that attempt to reproduce fault behaviors.

1.1.3. From surface deformation to slip a on fault plane: the dislocation model Surface deformation observed after an earthquake corresponds to the response of the Earth's crust cover to slip on a fault at depth, which possibly ruptured the surface (Fig.

1 .5). It is a 3D problem for which we have information only at the surface using geodesy and field observations, and along the fault plane a depth using seismic data. Geodesy and field observations inform on the displacements that affect the ground surface. Seismology provides the location of the seismic event (foreshocks, mainshocks, aftershocks; [START_REF] Hauksson | State of stress from focal mechanisms before and after the 1992 landers earthquake sequence[END_REF], and the kinematics of the ruptures (source geometry and mechanism, frequency content, rupture displacement, timing and velocity ; [START_REF] Byrne | Great thrust earthquakes and aseismic slip along the plate boundary of the Makran Subduction Zone[END_REF][START_REF] Prejean | Observations of Earthquake Source Parameters at 2 km Depth in the Long Valley Caldera, Eastern California[END_REF][START_REF] Wald | Rupture model of the 1989 Loma Prieta earthquake from the inversion of strong-motion and broadband teleseismic data[END_REF]. Retrieving fault and fault slip characteristics from the combined analysis of geodetic and seismic data is called an inversion (Jolivet et al., 2014;[START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF]Wang et al., 2020). Inversion requires defining the mechanical behavior of the medium rocks that link the fault to the surface, through the calculation of Green's functions [START_REF] Hartzell | Earthquake aftershocks as Green's functions[END_REF] However, large uncertainties remain on the structure of the crust that is complex, heterogeneous, and largely variable from one region to another [START_REF] Burov | Rheology and strength of the lithosphere[END_REF][START_REF] Hauksson | Regional tectonics of the Coso geothermal area along the intracontinental plate boundary in central eastern California: Three-dimensional Vp and Vp/Vs models, spatial-temporal seismicity patterns, and seismogenic deformation[END_REF]. Rheological models are used, but they generally over-simplify the problem which can lead to bias in the inversion and so in the rupture mechanism we model at depth [START_REF] Barbot | Effect of a compliant fault zone on the inferred earthquake slip distribution[END_REF][START_REF] Marchandon | Origin of the shallow slip deficit on a strike slip fault: Influence of elastic structure, topography, data coverage, and noise[END_REF].

Rheology of the crust is inferred from rock mechanics experiments (e.g., [START_REF] Lockner | A generalized law for brittle deformation of Westerly granite[END_REF] and observations of crustal deformations along natural faults (e.g., [START_REF] Watts | Lithospheric strength and its relationship to the elastic and seismogenic layer thickness[END_REF]. The seismogenic crust is the part of the crust that is subjected to earthquakes (Scholz, 1998). It is inferred from the depth distribution of earthquakes, and varies from 15 to 25 km in different regions of the globe [START_REF] Maggi | Earthquake focal depths, effective elastic thickness, and the strength of the continental lithosphere 4[END_REF][START_REF] Nazareth | The Seismogenic Thickness of the Southern California Crust[END_REF]. The simplest rheological model for the seismogenic crust is the elastic-brittle model [START_REF] Byerly | Earthquake Mechanisms[END_REF]. Elastic strain, which is recoverable, is accumulated in the rocks until deviatoric stress is larger than the rock strength. At this stage, the rock fails and fractures propagate through the medium [START_REF] Lockner | A generalized law for brittle deformation of Westerly granite[END_REF]. This corresponds to brittle deformation. If a preexisting fault is present in the medium, the medium starts to deform when deviatoric stress overcomes the fault strength, which do not require to rupture new faults [START_REF] Ikari | On the relation between fault strength and frictional stability[END_REF]. At the surface, we observe two deformation patterns that are associated with this model. On one hand, there is the accumulation phase, where the medium is subjected to loading on both edges and that rocks accumulate elastic deformation (Fig. 1.5A). This elastic deformation is recoverable. On the other hand, when the stresses applied overcome the strength of the rocks, or of the pre-existing fault if any, brittle deformation is triggered and the strain accumulated in the medium is released through displacement on a fault.

The deformation that occurs on the fault is finite and cannot be recovered. Conversely, the medium retrieves its initial undeformed state (Fig. 1.5B). The first case, imaged by but the dislocation cuts through the surface [START_REF] Segall | Earthquake and Volcano Deformation[END_REF]. d1 is the downdip end of the dislocation (10 km).

(a) (b)

This model can explain the fault behavior over one earthquake cycle, and at a first order.

Indeed, we showed that the earthquake cycle of a fault varies in time and space. In addition, along natural faults, it was observed that the deformation does not occur only along the fault plane but, that the rocks surrounding the faults are affected by fractures and also accommodate part of the deformation (e.g., [START_REF] King | Role of Fault Bends in the Initiation and Termination of Earthquake Rupture[END_REF][START_REF] Scholz | Fault growth and fault scaling laws: Preliminary results[END_REF][START_REF] Shipton | Damage zone and slip-surface evolution over μm to km scales in high-porosity Navajo sandstone, Utah[END_REF]. Hence, faults have a certain width over which deformation is distributed, and this can reach tens to hundreds of meters (e.g., [START_REF] Faulkner | On the internal structure and mechanics of large strike-slip fault zones: field observations of the Carboneras fault in southeastern Spain[END_REF][START_REF] Mcgill | Surficial slip distribution on the central Emerson fault during the June 28, 1992, Landers earthquake, California[END_REF][START_REF] Tchalenko | Structural Analysis of the Dasht-e Bayaz (Iran) Earthquake Fractures[END_REF]. This raises the question of what is the structure of the fault zone? How does it impact on mechanical and seismic behavior of a fault, and on the surface displacement pattern associated with the fault rupture?

Distributed deformation: from the tectonic scale to the scale of faults

In the previous section, we explained the simple elastic-brittle model that is widely used to describe fault behavior, but we also showed that natural faults are more complex than a planar interface of infinitesimal thickness. Faults have a certain width, and the medium around the fault is subjective to non-elastic, also called inelastic, deformation. The total width of the deformation zone during an earthquake includes the fault zone, and the inelastic deformation zone around the fault. Numerous observations characterize this width of deformation and showing that rock deformation distributes over large regions, from the tectonic scale to the scale of faults. In the following section, we expose the current knowledge on distributed deformation processes and the issues that it raises in terms of seismic and mechanical behavior for a fault, as well as for seismic hazard assessment. This thesis focuses on the study of continental faults because the method we use, and that is described in section 2., is applicable to visible and emerged ground surfaces only. Moreover, continental faults can sometime be located in the direct vicinity of critical assets like cities or nuclear power plants, and so are direct threats to populations. In addition, faults sometime cross critical lifelines like railroads and pipelines. In this case, the long-term behavior of the faults must be characterized and taken into account in the constructions so that the structures can withstand before fault displacements and ground accelerations. strike-slip Kokoxili earthquake ruptured a 450-km section of the Kunlun fault in China (Klinger et al., 2006(Klinger et al., , 2005)). In 1905, the Mw≥8 strike-slip Tsetserleg-Bulnay earthquake sequence ruptured a 676-m long portion of the Bulnay fault system (Bulnay, Tsetserleg, Teregtiyn and Drugen faults) in Mongolia [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF]. In 1857, the Mw7.9 strikeslip Fort Tejon earthquake rupture a >300-km long portion of the San Andreas fault near Los Angeles city in California, USA [START_REF] Harris | In the shadow of 1857-the effect of the Great Ft. Tejon Earthquake on subsequent earthquakes in southern California[END_REF][START_REF] Zielke | High-Resolution Topography-Derived Offsets along the 1857 Fort Tejon Earthquake Rupture Trace, San Andreas Fault[END_REF].

Several areas in the world are known for accommodating crustal deformation over very wide (>10 2 km) areas that are cut out by numerous faults (Fig. 1.6). The example of the Mongolian active fault network (Fig. 1.6) is a typical example of distributed faulting, similarly to what is observed north of the India-Eurasia collision (Cheng et al., 2021;Klinger et al., 2005) or in the Basin and Range and Eastern California Shear Zone, located east of the San Andreas fault in California [START_REF] Gan | Strain accumulation across the Eastern California Shear Zone at latitude 36°30′N[END_REF][START_REF] Wesnousky | Active faulting in the Walker Lane[END_REF]. In these regions, tectonic loading is distributed between all the active faults of the area. As a consequence, each fault has a different slip rates, that is below that of the regional tectonic loading [START_REF] Petersen | Fault slip rates and earthquake histories for active faults in southern California[END_REF][START_REF] Walker | Active tectonics of an apparently aseismic region: Distributed active strike-slip faulting in the Hangay Mountains of central Mongolia[END_REF]. This makes difficult the characterization of the earthquake cycle of the faults, since the individual fault slip rates can be less than 1mm/yr in some regions, and with a few rare earthquakes that could be recorded [START_REF] Bollinger | 25,000 Years long seismic cycle in a slow deforming continental region of Mongolia[END_REF][START_REF] Petersen | Fault slip rates and earthquake histories for active faults in southern California[END_REF]. [START_REF] Walker | Active tectonics of an apparently aseismic region: Distributed active strike-slip faulting in the Hangay Mountains of central Mongolia[END_REF].

To be able to characterize the fault deformation and associated earthquake cycle in these regions, a good combination between measurements over a long-time scale, using paleoseismology, geomorphology and historical records, and data on the current active faults, acquired through field surveys, geodetic and seismic data, is necessary. Historical data inform on the previous large earthquake events with long-period recurrences (>10 3 years), while data on the current active faults, that are more detailed both in time and space, allow for a good characterization of the fault structures and of the parameters that control the ruptures. Also, paleoseismology and geomorphology are based on the analysis of surface fault deformation to infer past magnitudes of earthquakes. However, in the previous section, we described the uncertainty that remains on the relation between surface deformation and fault processes at depth. Hence, detailed data on recent fault deformations allow to better understand these relations, and so to better interpret the historical observations. From all these observations, several parameters are highlighted as having a strong influence, or even controlling, to some extent, the fault behavior and rupture process. These parameters are:

-The tectonic loading rate determines the amount of strain that is accumulated by the faults, and hence, the amount that is released during the seismic events [START_REF] Dolan | Extreme multimillennial slip rate variations on the Garlock fault, California: Strain super-cycles, potentially time-variable fault strength, and implications for system-level earthquake occurrence[END_REF]. It also influences the frictional properties of the fault zone [START_REF] Guérin-Marthe | Earthquake Nucleation Size: Evidence of Loading Rate Dependence in Laboratory Faults[END_REF][START_REF] Mclaskey | Slow and fast ruptures on a laboratory fault controlled by loading characteristics[END_REF].

-The rock lithology, stratigraphy and heterogeneity impact on the frictional behavior the fault [START_REF] Thomas | Lithological control on the deformation mechanism and the mode of fault slip on the Longitudinal Valley Fault, Taiwan[END_REF][START_REF] Verberne | Frictional Properties of Sedimentary Rocks and Natural Fault Gouge from the Longmen Shan Fault Zone, Sichuan, ChinaFrictional Properties of Sedimentary Rocks and Natural Fault Gouge from the Longmen Shan Fault Zone[END_REF] and on the distribution of the deformation at the surface [START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF][START_REF] Teran | Geologic and structural controls on rupture zone fabric: A field-based study of the 2010 Mw 7.2 El Mayor-Cucapah earthquake surface rupture[END_REF][START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF].

The pre-existing geological weaknesses from the crust also control the location of the ruptures [START_REF] Bullock | Lithological controls on the deformation mechanisms operating within carbonatehosted faults during the seismic cycle[END_REF][START_REF] Floyd | Spatial variations in fault friction related to lithology from rupture and afterslip of the 2014 South Napa, California, earthquake[END_REF][START_REF] Vallage | Multitechnology characterization of an unusual surface rupturing intraplate earthquake: the M L 5.4 2019 Le Teil event in France[END_REF].

-The segmentation of the fault controls the length of the rupture as well as the rupture pattern on the fault [START_REF] Klinger | Relation between continental strike-slip earthquake segmentation and thickness of the crust[END_REF][START_REF] Oglesby | What Can Surface-Slip Distributions Tell Us about Fault Connectivity at Depth[END_REF]. Fault segmentation refers to the fact that strike-slip faults exhibit along-strike variations in their geometry, and potentially in their slip rates [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF][START_REF] Duross | Fault segmentation: New concepts from the Wasatch Fault Zone, Utah, USA[END_REF]Klinger et al., 2006;[START_REF] Schwartz | Fault Behavior and Characteristic Earthquakes: Examples From the Wasatch and San Andreas Fault Zones[END_REF]. The length of a strike-slip rupture depends on the number of fault segments on which slip propagated (Cheng et al., 2021;[START_REF] Oglesby | What Can Surface-Slip Distributions Tell Us about Fault Connectivity at Depth[END_REF].

Individual length of the fault segments is about 10 to 25 km , and was shown to scale with the depth of the seismogenic zone [START_REF] Klinger | Relation between continental strike-slip earthquake segmentation and thickness of the crust[END_REF] -Eventually, neighboring faults can interact with each other [START_REF] King | Static stress changes and the triggering of earthquakes[END_REF] and so, the seismic behavior of one fault is not independent from that of the neighboring faults [START_REF] King | Static stress changes and the triggering of earthquakes[END_REF][START_REF] Mildon | Coulomb pre-stress and fault bends are ignored yet vital factors for earthquake triggering and hazard[END_REF].

Distributed deformation at the scale of faults, from depth to the surface

As explained in the previous parts, continental crust is heterogeneous and presents heterogeneous and complex weaknesses due to its tectonic history, which result in the activation of multiple faults with complex geometries (Fig. 1.6). This is observed at the scale of plate tectonics, at regional scales, and also at the scale of one fault. Indeed, after an earthquake rupture, several surface-rupturing fault scarps are generally observed in the field (Fig. 1.7). These faults scarps can be sub-parallel, oblique and eventually cross-cutting each other (Fig. 1.7). As a consequence, the total amount of deformation that occurred at the surface is sometimes difficult to estimate. Yellow arrows highlight the sub-parallel fault scarps [START_REF] Teran | Geologic and structural controls on rupture zone fabric: A field-based study of the 2010 Mw 7.2 El Mayor-Cucapah earthquake surface rupture[END_REF].

This complexity is also observed at depth with different characteristics. Observations on exhumed faults shows that fault deformation occurs over a large width of several meters at least (Fig. 1.8), which contradicts the simple brittle-elastic model presented previously.

Within basement or lithified sedimentary rocks, and most likely at large confining pressures under a few kilometers depth, deformation concentrates on a thin interface where rocks are metamorphized into fault gouge and cataclasite or ultracataclasite (Fig.

1 .9;[START_REF] Faulkner | On the internal structure and mechanics of large strike-slip fault zones: field observations of the Carboneras fault in southeastern Spain[END_REF][START_REF] Fossen | Deformation bands in sandstone: a review[END_REF][START_REF] Lin | Spatial variations in damage zone width along strike-slip faults: An example from active faults in southwest Japan[END_REF]Mitchell and Faulkner, 2009). This is referred to as the fault core. It can either manifest through a single fault core (Fig. 1.9A) or through a zone with multiple fault cores (Fig. 1.9B). In both cases, large deformation that occurs at the fault core also triggers damage in rocks from the surrounding medium. This damage is characterized by distributed fractures (Fig. 1.9;[START_REF] Faulkner | On the internal structure and mechanics of large strike-slip fault zones: field observations of the Carboneras fault in southeastern Spain[END_REF][START_REF] Kim | Fault damage zones[END_REF][START_REF] Lin | Spatial variations in damage zone width along strike-slip faults: An example from active faults in southwest Japan[END_REF]Mitchell and Faulkner, 2009;Okubo et al., 2019;Thomas and Bhat, 2018). Hence, faults appear to be 3D structures with a complex geometry. This structure is reproduced using numerical models (Fig. 1.10) but, similarly to inversion problems, uncertainty remains on the evolution of the 3D structure of the fault zone from depth to the surface since rock type and stress conditions vary with depth, and from one region to another.

Surface off-fault inelastic deformation

In this thesis, we focus on the observation of surface deformation patterns to better understand the 3D structure of the fault in the shallow crust, which can eventually enlighten characteristics of the fault system at larger depth. Current observations of surface deformation after an earthquake report complex networks of interacting small faults, similar to Figure 1.7, which are through to connect at depth with the fault structure described in Figures 1.8 and 1.9. No fault cataclastic deformation is observed at the surface since it requires high confining pressure. Sometimes, we observe one or several major fault structures that accommodate a large part of the deformation, and smaller fractures and deformation features that lie around this primary system. In some other cases, faults are not even visible at the surface and all the deformation is accommodated through small fractures [START_REF] Brooks | Buried shallow fault slip from the South Napa earthquake revealed by near-field geodesy[END_REF] and other inelastic deformation processes as warping or rotations [START_REF] Mcgill | Surficial slip distribution on the central Emerson fault during the June 28, 1992, Landers earthquake, California[END_REF]. This generates gradients of deformation at the surface that differ from the localized displacement offsets that are generally observed across major fault structures (Fig. 1.5b).

Following these observations, an effort was made to characterize the parameters that influence the distribution of deformation at the surface. This allows predicting the patterns of deformation (localized or distributed), and the extension of the area affected by this surface deformation for future surface-rupturing events. It was shown that low confining pressures acting at the surface favorize the distribution of the deformation over wide areas that are hundreds of meters-to kilometers-large [START_REF] Ampuero | Upper Limit on Damage Zone Thickness Controlled by Seismogenic Depth[END_REF][START_REF] Finzi | Structural Properties and Deformation Patterns of Evolving Strike-slip Faults: Numerical Simulations Incorporating Damage Rheology[END_REF]. Weak materials including damaged rocks and sedimentary deposits might also increases the distribution of the deformation at the surface (Roten et al., 2017;[START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF]. It was also shown how the geometrical complexity of the fault zone is associated with wide surface deformation zones (Klinger et al., 2018;[START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF]. Lithological contrasts across the fault zone were shown to influence the way inelastic deformation distributes around the faults [START_REF] References Aben | High strain rate deformation of porous sandstone and the asymmetry of earthquake damage in shallow fault zones[END_REF][START_REF] Teran | Geologic and structural controls on rupture zone fabric: A field-based study of the 2010 Mw 7.2 El Mayor-Cucapah earthquake surface rupture[END_REF].

The pre-existing geological fabric [START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF], and long-term cumulative deformation on the faults [START_REF] Brodsky | Faults smooth gradually as a function of slip[END_REF]Mitchell and Faulkner, 2009) influence on the geometry of the faults, which in turn affects the surface deformation. Finally, the magnitude of the earthquake [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF], and depth of the seismogenic crust [START_REF] Ampuero | Upper Limit on Damage Zone Thickness Controlled by Seismogenic Depth[END_REF][START_REF] Lefevre | Experimental evidence for crustal control over seismic fault segmentation[END_REF] were also shown to have an impact. There are not one leading parameter that drives the distribution of surface deformation, and interactions between the different parameters are complex and need to be characterized. From these observations, it was highlighted that the fault system that is primarily visible at the surface sometimes accommodate less than half of the total surface displacement over the entire fault zone [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF][START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF][START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF] 1999; [START_REF] Shelef | Deformation processes adjacent to active faults: Examples from Eastern California[END_REF]e.g. Fig. 1.11) and deformation bands [START_REF] Fossen | Deformation bands in sandstone: a review[END_REF][START_REF] Shipton | Damage zone and slip-surface evolution over μm to km scales in high-porosity Navajo sandstone, Utah[END_REF]. The ensemble of these deformation processes that act around the major fault planes, primary or secondary, are referred to as off-fault deformation (OFD). The proportions of OFD relatively to the amount of deformation that is accommodated through localized slip on the faults is function of the parameters listed before, that govern the distribution of the deformation at the surface (e.g., [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF][START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF][START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF]. The relative importance of each off-fault deformation mechanisms may also dependent on these parameters. Hence, there is a need to quantify the exact amount of OFD that occurs during a rupture, and how it relates to the characteristics of the fault zone. This should allow us to improve our knowledge on relations between surface deformation and the tectonic and geological controls.

Problematics and outline of the thesis

This study is motivated by the observed complexity of fault systems, for which many questions remain open. Technical developments allow for new observations of fault complexity, and open new possibilities to analyze the parameters that drive crustal deformation. In this thesis, we raise the following issue:

-How is surface deformation accommodated at during an earthquake?

-How does it relate to the earthquake process at depth? -What are the controlling parameters?

To answer these questions, we use high-resolution (0.5 m) optical image correlation methods to measure the horizontal and vertical displacement fields associated with two different surface-rupturing earthquakes. With this resolution, we are able to image the detail of the ruptures and to characterize the deformation patterns inside the fault zone.

This allows us to discuss the processes of distribution of the surface deformation.

Following an introduction on the methodology, this thesis is divided in four major chapters. The first chapter consists in a study of a complex fault zone that ruptured during the 2013 Baluchistan earthquake (Mw7.8, Pakistan) (S. Antoine et al., in prep.). In this chapter, we highlight surface deformation occurring up to several hundreds of meters or a few kilometers away from the visible fault plane. We also show a correlation between complex fault geometries and the distribution of the deformation over wide areas at the surface. Finally, in this same chapter, we provide a constrain on the timing of the deformation through the measurement of the long-term (from 6 months to 3 years) postseismic deformation. The second chapter of this thesis consists in a study of the 2019

Ridgecrest earthquake sequence (Mw6.4 and 7.1, California), for which very complex fault geometries and deformation patterns are observed (S. Antoine et al., BSSA, 2021).

Similar to the 2013 Baluchistan rupture, we highlight deformation that occurred over a width of ~2 km at the surface, including the visible fault structures. We highlight a relation between the distribution of the deformation at the surface and the fault geometry, rupture mechanism, and pre-existing structures from the basement. In this same chapter, we use our results to add further constrains to fault slip inversions and improve the modeling of the rupture process at depth (S. Antoine et al., in prep.). We compare the results with the surface displacement measurements, and propose interpretations on the role of wide surface deformation zones in the general rupture process. The third chapter is shorter and consists in a multidisciplinary study that uses theoretical and numerical modeling of dynamic rupture, along with observations on naturals fault made using seismic and geodetic technics, to highlight relations between the dynamic rupture characteristics and the surface deformation patterns (Jara et al., in review at Proceedings A). In the last chapter of this work, we discuss the previous results and interpretations, and show how they bring new elements for the general comprehension of earthquakes and surface ruptures. We also show the implications of this work for the assessment of earthquake hazard based on surface displacement observations. This thesis ends with a section of conclusions, followed by some perspectives that present possibilities for future research.

DATA AND METHODS: HIGH-RESOLUTION SATELLITE OPTICAL

IMAGE CORRELATION

Benefits of optical image correlation

So far, imaging of the coseismic surface displacement field is most often performed using interferometry of synthetic aperture radar (SAR) images, also referred to as InSAR (e.g., [START_REF] Fialko | The complete (3-D) surface displacement field in the epicentral area of the 1999 MW7.1 Hector Mine Earthquake, California, from space geodetic observations[END_REF][START_REF] Fielding | Surface Deformation Related to the 2019 Mw 7.1 and 6.4 Ridgecrest Earthquakes in California from GPS, SAR Interferometry, and SAR Pixel Offsets[END_REF][START_REF] Massonnet | The displacement field of the Landers earthquake mapped by radar interferometry[END_REF]. This method measures the phase difference between SAR images acquired at different periods, and it retrieves the ground displacement in the line of sight (LOS) of the satellite. It yields a centimeterscale precision in the LOS but is less sensitive in the horizontal component. In addition, InSAR saturates close to the fault zone due to phase decorrelation where large gradients of ground displacements occur. Correlation of the SAR amplitude image is sometimes used to avoid phase decorrelation close to the fault [START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF][START_REF] Fielding | Surface Deformation Related to the 2019 Mw 7.1 and 6.4 Ridgecrest Earthquakes in California from GPS, SAR Interferometry, and SAR Pixel Offsets[END_REF][START_REF] Michel | Measuring ground displacements from SAR amplitude images: Application to the Landers Earthquake[END_REF]; however, the precison of this method is limited by the image geometry and radiometry quality as well as its ground resolution (> 1m). To overcome this issue, satellite optical image correlation appears as a good solution to image the ground displacement field, since satellite optical images benefit from high radiometric and geometric quality [START_REF] Crippen | Measurement of subresolution terrain displacements using SPOT panchromatic imagery[END_REF][START_REF] Leprince | Automatic and Precise Orthorectification, Coregistration, and Subpixel Correlation of Satellite Images, Application to Ground Deformation Measurements[END_REF][START_REF] Puymbroeck | Measuring earthquakes from optical satellite images[END_REF].

Also, most recent optical satellite images have ground resolutions that overcome that of SAR images (<0.5 m).

State of the art of the satellite optical images

The wide use of optical images in Earth-Science first started with the collection of Landsat optical images, in 1972, by the NASA (USA), and with a ground resolution of 80 m. The images were used for example for mapping active faults from the Eastern California Shear Zone [START_REF] Ford | Faults in the Mojave Desert, California, as Revealed on Enhanced Landsat Images[END_REF] and central Asia [START_REF] Le Dain | Active faulting and tectonics of Burma and surrounding regions[END_REF][START_REF] Tapponnier | Active faulting and cenozoic tectonics of the Tien Shan, Mongolia, and Baykal Regions[END_REF]. More recent Landsat-8 images (2013) have a ground resolution of 15 m. In the years 1990 to 2000, SPOT-1 to SPOT-4 satellites from CNES were launched, and captured optical images with a ground resolution of 10 m. These images were extensively used in the development of image correlation technics [START_REF] Crippen | Measurement of subresolution terrain displacements using SPOT panchromatic imagery[END_REF][START_REF] Leprince | Automatic and Precise Orthorectification, Coregistration, and Subpixel Correlation of Satellite Images, Application to Ground Deformation Measurements[END_REF][START_REF] Puymbroeck | Measuring earthquakes from optical satellite images[END_REF]. Quickly after that, higherresolution images were delivered through the SPOT-5 program, starting in 2002, with a resolution of 5 to 2.5 meters [START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF](Vallage et al., , 2015;;[START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF]. Quite simultaneously, the Worldview Imaging Corporation, now called DigitalGlobe, launched series of satellites capable of capturing high-resolution images (Earlybird and QuickBird, respectively, with a 3 and 0.6 m image ground resolution). It was followed in 2007 by the first-generation Worldview satellites capable of capturing images at ground resolutions lower than 0.5 m (Barnhart et al., 2020[START_REF] Barnhart | Evidence for slip partitioning and bimodal slip behavior on a single fault: Surface slip characteristics of the 2013 Mw7.7 Balochistan, Pakistan earthquake[END_REF][START_REF] Zinke | Three-Dimensional Surface Deformation in the 2016 MW 7.8 Kaikōura, New Zealand, Earthquake From Optical Image Correlation: Implications for Strain Localization and Long-Term Evolution of the Pacific-Australian Plate Boundary[END_REF]. In 2011, the CNES was also able to acquire high-resolution images (0.5 m) through the launch of the Pleiades satellites (Delorme et al., 2020). Following year 2012, SPOT-6/7 images at 1.5 m also allowed a relatively high-resolution imaging of the Earth but with the advantage of a wider geographical coverage.

Taking advantage of the high-resolution of the images and their good radiometric and geometric quality, SPOT images, followed by the Worldview and Pleiades images, are now broadly used to capture the near-fault displacement field after earthquakes. The 30m Landsat-8 images, and 10-m Sentinel-2 images are also used to provide first order and large-scale maps of the displacement field (e.g., [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF][START_REF] Chen | Cascading and pulse-like ruptures during the 2019 Ridgecrest earthquakes in the Eastern California Shear Zone[END_REF].

Datasets used in this study

In this thesis, we benefit from an access to the high-resolution Pleiades and WorldView images, at 0.5 m ground resolution, though the Committee on Earth Observation Satellites Before the earthquake (Fig. 2.1 left), the image dataset is complex because as there is no a-priori on the zone of potential future earthquakes, there is no systematic strategy of image acquisition to build an archive pre-earthquake. So, we obtain a large and various dataset that includes various types of images (mono, stereo, and tri-stereo) from both the WorldvView and Pleiades satellites. The images are taken at various dates before the earthquake, sometimes with a few years difference. This increases the chances of having ground texture variations between the acquisitions, especially due to weathering, seasonal changes, and fluvial activity. Also, having images from different satellites makes challenging the characterization of the image geometry of acquisition when calculating the Digital Surface Models (DSM). The distribution of the viewing angles is also less ideal for images taken independently (Fig. 2.1), while it is an important characteristic of the image set when retrieving the 3D ground geometry [START_REF] Puymbroeck | Measuring earthquakes from optical satellite images[END_REF][START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF]. Finally, the images most-often only partially overlap, and so complex combinations with a large number of images are required to cover the entire rupture with minimum 2 images in any area of the ground before the rupture.

After the earthquake rupture (Fig. 2.1 right), because the image acquisition occurs upon request and with the aim of imaging the fault with high-quality images, we benefit from a tri-stereo Pleiades image acquisition (PL_1, PL_2, PL_3) of the rupture with, for each area of the ground, 3 images acquired at the same date, from the same satellite, and that have a good distribution of viewing angles, and with overlapping ground footprints.

Pleiades images are 20 to 25-km large. We obtained the tri-stereo Pleiades images for both the 2013 Baluchistan and 2019 Ridgecrest case of studies.

The third study on the 2001 Kokoxili rupture differs from the two previous because no high-resolution images are available for this rupture before the date of the earthquake.

So, we used pairs of mono-SPOT-4 images with a 10-m ground resolution. We also obtain the ground topography using a unique post-earthquake digital surface model at 90 m ground resolution (SRTM). Still, horizontal correlations between the pre-and postearthquake images are performed using the same method, and the same tool than the other studies. This method is described in the following sections.

General principle of optical image correlation

Optical image correlation can be applied to images of the ground taken at different periods of time to measure the ground surface displacements. It consists in correlating pairs of images with similar georeferencing and acquisition geometry, and measuring how groups of pixels that represent similar features of the ground have displaced between the two acquisitions. In panchromatic images, the ground texture is captured by variations of the grey amplitude of pixels in the image, from 0 to 255. Because in this study image correlation is applied to measure coseismic displacements, we will refer to the two images that are correlated as the pre-and post-earthquake images, respectively.

Optical images are never acquired with an exact similar geometry, i.e., viewing angle, satellite altitude and area acquired on the ground (Fig. 2.1). Hence, before performing the pixel correlation between the images, an important step of the processing consists in the orthorectification of the images. This step is even more important for stereoscopic and tri-stereoscopic acquisition, for which images are taken with a relatively high-angles (up to 20°) to the nadir, that is the vertical to the ground.

Orthorectification consists in correcting the images from their individual geometry of acquisition by projected them into a common ground geometry. To do so, one must calculate the digital surface models (DSMs) that provide the 3D geometry of the ground (elevation at each point of the image), or use available ones. Studies that apply optical image correlation use various strategies to orthorectify the images. A large-scale and lowresolution DSMs, like 90-m SRTM, can be used to correct both the pre-and postearthquakes images. This gives a good correction of the image geometry but topographic artefacts will remain (e.g., Klinger et al., 2006). This strategy is used in the case of the 2001 Kokoxili rupture study. Most accurate method is to use a pre-earthquake DSM to correct the pre-earthquake images, and a post-earthquake DSM to correct the postearthquake images, with in both cases a resolution that is consistent with that of the images. In general, the pre-and post-earthquake DSMs are directly derived from the optical images used for the correlation, if these are stereo-or tri-stereo datasets (Barnhart et al., 2020;Delorme et al., 2020). This method reduces the bias due to incorporation of the tectonic signal in the topographic corrections, and to georeferencing and resampling of datasets from different origins. This method was applied to the 2013 Baluchistan and 2019 Ridgecrest case of studies.

DSM calculation, orthorectification, and horizontal correlation are performed using MicMac (see 2.4.). Detailed orthorectification process is not described here but can be found in Rupnik et al. (2018Rupnik et al. ( , 2016) ) for that used in MicMac, and [START_REF] Puymbroeck | Measuring earthquakes from optical satellite images[END_REF] and [START_REF] Leprince | Automatic and Precise Orthorectification, Coregistration, and Subpixel Correlation of Satellite Images, Application to Ground Deformation Measurements[END_REF] for the general concepts applied in other software products, like COSI-Corr.

Optical image correlation using MicMac

Combined with the access to high-resolution images, this study benefits from recent developments in optical image correlation methods that enable to resolve ground displacements with an unprecedented resolution (Delorme et al., 2020;[START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF]. These developments were made in the framework of a collaboration between experts in photogrammetry from the Institut Geographique National (IGN) and the Tectonic team from Institut de Physique du Globe de Paris (IPGP). The aim was to exploit methods initially used to retrieve 3D geometries of objects from images taken with different viewing angles, like Digital Surface Models (DSMs) for example, to correlate images taken with a similar geometry but at a different period to measure displacements of the ground. It is now part of a software called MicMac, which regroups numerous tools for photogrammetric analysis of images and image correlation, managing both satellite images and camera photographs (Rupnik et al., 2018[START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF](Rupnik et al., , 2016)).

MicMac has proved to be efficient in retrieving complex surface displacements from subpixel optical image correlation (Delorme et al., 2020;[START_REF] Rosu | Measurement of ground displacement from optical satellite image correlation using the free open-source software MicMac[END_REF][START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF]Vallage et al., , 2015)). Major advance in this correlation software is that it allows for preserving the input resolution of the optical images through the correlation process. Hence, from the correlation of WorldView and Pleiades images, we obtain displacement map with ground resolutions of 0.5 m. MicMac retrieves the displacement of a group of pixel (5pixels large in this method) between the two acquisitions, and repeats the operation at every pixel location in the pre-earthquake image (every 0.5 m). Conversely, methods like COSI-Corr measure the displacement for independent groups of pixels (the term "correlation window" is also used in the literature) that do not overlap (e.g., Barnhart et al., 2020;[START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF][START_REF] Zinke | Three-Dimensional Surface Deformation in the 2016 MW 7.8 Kaikōura, New Zealand, Earthquake From Optical Image Correlation: Implications for Strain Localization and Long-Term Evolution of the Pacific-Australian Plate Boundary[END_REF]. So, it retrieves the displacement values every n pixels in the image and provides a less-resolved displacement map. Correlation window used in the COSI-Corr studies is generally fixed at 9 pixels, so the MicMac method resolves displacement patterns of spatial scales about 9 times greater.

MicMac also allows to calculate DSMs with a high spatial precision and resolution.

Indeed, MicMac applies a bundle block adjustment to each image to empirically refine the camera position and parameters of acquisition, that are provided only with a ~10 m precision by the spatial agencies (Rupnik et al., 2016). It also preserves the image resolution when calculating the DSMs. So, the ground geometry is retrieved with a 0.5 m ground resolution. Eventually, the vertical uncertainty associated with the DSMs processed with MicMac is close to the ground resolution, which is 0.5 m (Rupnik et al., 2018). Horizontal image correlation with MicMac consists in searching the position of similar groups of pixels (n=9 pixels in this thesis, so 4.5 m) in the pre-and post-earthquake images (Fig. 2.2). The group of pixel of central position (x1,y1) is selected in the preearthquake image, and searched for in the post-earthquake image by calculating correlations at various (x,y) positions in the image, within a defined search space. The search space defined in the post-earthquake image is used to limit the area of research to an a-priori displacement inferred from other geodetic or field studies (this will vary for each case of study and is detailed in the corresponding chapters). The size of the step while moving inside of the search window reduces until 1/10 th of pixel through the correlation process so it can detect displacements down to 0.05 m using input images at 0.5 m (Fig. Green box is the corresponding group of pixels on the post-earthquake images (bottom). In the left panel, the step in the correlation is one pixel (pixel correlation), so displacements lower than the pixel size cannot be detected. In the right panel, step in the correlation is 1/10 th of pixel, so displacements lower than the pixel size can be retrieved (in this example we measure a displacement of 2.5 dx, with dx=1/10 th of pixel, to the East of the image).

Measuring the vertical displacement field

Until recently, the vertical displacement field associated with an earthquake was only measured using InSAR. However, as explained in 2.2., InSAR decorrelates in areas of large displacements close to the fault and is limited by the SAR image resolution. Taking advantage of the high-resolution pre-and post-earthquake DSMs used for the image orthorectification (see section 2.3.), one can measure the vertical displacement field close to the fault. The vertical accuracy of such vertical displacement map (>10 cm) is lower than that of InSAR maps (a few centimeters), but it provides a detailed measurement of the vertical displacement field within the fault zone. The vertical offsets between the preand the post-earthquake DSMs are retrieved using either the Iterative Close Point method (ICP), or the vertical difference method (Barnhart et al., 2019a;Delorme et al., 2020).

On one hand, ICP method brings into alignment two sets of three-dimensional point clouds. It computes a rigid body transformation, including rotation and translation, of the pre-earthquake point cloud and minimizes the mean square error with the post-earthquake point cloud (Barnhart et al., 2019a;Besl and McKay, 1992;Nissen et al., 2012;[START_REF] Scott | The M7 2016 Kumamoto, Japan, Earthquake: 3-D Deformation Along the Fault and Within the Damage Zone Constrained From Differential Lidar Topography[END_REF]. But, this method smoothes the deformation signal by the size the kernel that is used for the body transformation (Barnhart et al., 2019a;Nissen et al., 2012). On the other hand, the vertical difference of DSMs simply measures the elevation difference at all (x,y) locations in the DSMs, considering that they are correctly co-registered like for the horizontal image correlation. When DSMs have a resolution lower than the coseismic displacement, artifact from topographic translation across the fault can occur (Barnhart et al., 2019a). In this case, we correct the pre-earthquake DSM using the coseismic horizontal displacement field measured previously, to re-align the topographic features (Fig. 2.4; Barnhart et al., 2019a;Delorme et al., 2020). Then, the only difference that remains between the two DSMs is the co-seismic vertical displacement field. Applying this method to our high-resolution DSMs, we can retrieve vertical displacements with a resolution of 0.5 m, without decorrelation close to the fault. The vertical difference method we use in this thesis was developed by Arthur Delorme at IPGP (Delorme et al., 2020). S. L. Antoine (1) , Y. Klinger (1) , A. Delorme (1) and R. D. Gold (2) (1) Université de Paris, Institut de physique du globe de Paris, CNRS, Paris, France, (

U.S. Geological Survey, Golden, Colorado, USA.

Peer Review DISCLAIMER: This draft manuscript is distributed solely for purposes of scientific peer review. Its content is deliberative and predecisional, so it must not be disclosed or released by reviewers. Because the manuscript has not yet been approved for publication by the U.S. Geological Survey (USGS), it does not represent any official USGS finding or policy.

Abstract

Imaging the near-fault displacement field for large earthquakes has long been a challenging task while it became a critical information for surface rupture hazard assessment. Methods such as InSAR, GPS, and field-work have a hard time grasping the fine-scale displacement pattern near the surface rupture. Optical correlation methods using high-resolution images appear to be a good solution to characterize the complexity of the displacement in the near-field. In this study, we use sub-pixel correlation of 0.5-m resolution optical images to measure the near-fault displacement field for a 30-km long area of the 2013, Mw7.7, Baluchistan rupture (Pakistan). This work aims at characterizing the distribution of on-versus off-fault displacements within the first kilometers around the primary fault zone. Our results reveal very heterogeneous displacement patterns and bring to light the geometrical complexity of the fault system. We perform a detailed analysis of the slip distribution across two major structures observed along the fault trace (a fault bend and a relay zone) and we show that the deformation can be either very localized in a 20-to 30-m wide fault zone, or conversely distributed over hundreds of meters to kilometers off the fault plane. We also quantify with a precision of 0.1 to 0.2 m the distribution of surface displacements along the fault system and propose a complete slip budget representing variations of on-and off-fault slip. We combine this analysis with existing surface rupture maps and propose different mechanisms to explain the OFD patterns: visible distributed fractures, continuous microscale deformations through microcracking and deformation bands, or both.

Introduction

The geometry of strike-slip faults has a strong control on the surface extent of the rupture [START_REF] King | Role of Fault Bends in the Initiation and Termination of Earthquake Rupture[END_REF][START_REF] Wesnousky | Seismological and structural evolution of strike-slip faults[END_REF] and on the distribution of slip on-and offfault (Klinger et al., 2018;Vallage et al., 2015). Geometrical variations in the fault trace, hereafter named "complexities", locally modify the distribution of stress accumulated during the interseismic period which, eventually, impact on the propagation of the rupture [START_REF] Aydin | Surface rupture at a fault bend: the 28 June 1992 Landers, California, earthquake[END_REF][START_REF] King | Static stress changes and the triggering of earthquakes[END_REF][START_REF] Mildon | Coulomb pre-stress and fault bends are ignored yet vital factors for earthquake triggering and hazard[END_REF][START_REF] Mildon | Evaluating models of Coulomb stress transfer: Is variable fault geometry important[END_REF]. In fact, complexities are favorable points for earthquakes to start if stress has accumulated in excess over time, but it can also act as a barrier and cause rupture arrests [START_REF] King | Role of Fault Bends in the Initiation and Termination of Earthquake Rupture[END_REF]. To a lesser extent, geometrical complexities cause variations in the velocity of the rupture, which enhance higher frequency and destructive seismic wave radiations where velocity transitions are occurring [START_REF] Duan | Multicycle dynamics of nonplanar strike-slip faults[END_REF][START_REF] Okuwaki | Role of geometric barriers in irregular-rupture evolution during the 2008 Wenchuan earthquake[END_REF].

Observations (Vallée et al., 2008) as well as numerical models (Hu et al., 2016) also suggest that complexities, because they impact the rupture velocity, can encourage transitions to supershear speed. The rupture is then far more destructive as it manifests a shear shock wave front [START_REF] Mello | Identifying the unique ground motion signatures of supershear earthquakes: Theory and experiments[END_REF]. In addition, co-seismic strain across these complexities is generally oblique to the directions of the faults so, deformation will tend to distribute outside of the primary fault plane and generates secondary cracking and faulting as well as deformation in the off-fault medium [START_REF] Bürgmann | Strain accommodation about strike-slip fault discontinuities in granitic rock under brittle-to-ductile conditions[END_REF][START_REF] Griffith | Rough faults, distributed weakening, and off-fault deformation[END_REF]Okubo et al., 2019). This deformation is referred as off-fault deformation. It is a difficult process to characterize and quantify on the field as long as it can spread hundreds of meters to kilometers off the primary fault plane [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF][START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF]Vallage et al., 2015). Also, decorrelation in InSAR data close to the fault for large-magnitude earthquakes prevent from observing such deformation patterns.

Advances in satellite imagery and correlation methods opened new possibilities for analyzing the details of the surface ruptures and the associated deformation processes, including ductile deformations, over widespread areas (e.g., Ajorlou et al., 2021;Barnhart et al., 2020;Delorme et al., 2020;[START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF]Vallage et al., 2015;[START_REF] Zinke | Three-Dimensional Surface Deformation in the 2016 MW 7.8 Kaikōura, New Zealand, Earthquake From Optical Image Correlation: Implications for Strain Localization and Long-Term Evolution of the Pacific-Australian Plate Boundary[END_REF]. This makes possible the characterization of parameters influencing surface deformation such as the basement fabric [START_REF] Fialko | Simple shear origin of the cross-faults ruptured in the 2019 Ridgecrest earthquake sequence[END_REF][START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF][START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF], the obliquity of the fault [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF]Vallage et al., 2015), the type of rocks [START_REF] Zhou | Characterizing Complex Surface Ruptures in the 2013 Mw 7.7 Balochistan Earthquake Using Three-Dimensional Displacements[END_REF][START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF], the surface topography [START_REF] Zinke | Three-Dimensional Surface Deformation in the 2016 MW 7.8 Kaikōura, New Zealand, Earthquake From Optical Image Correlation: Implications for Strain Localization and Long-Term Evolution of the Pacific-Australian Plate Boundary[END_REF], the magnitude of the displacement [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF][START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF][START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF], and the geometry of the fault system [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF]Klinger et al., 2018;[START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF][START_REF] Zhou | Characterizing Complex Surface Ruptures in the 2013 Mw 7.7 Balochistan Earthquake Using Three-Dimensional Displacements[END_REF][START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]. Detailed fault displacement data are crucial to characterize these relations between fault geometry, offfault deformation, displacement magnitude and geological context.

The main objective of this work is to characterize and quantify the impact of the fault geometry on the surface slip distribution and off-fault deformation processes. With this aim, we first describe the seismotectonic context of the study and the method used to measure the 3D surface displacement field. Then, we describe the geometry of the fault zone, including different geometrical complexities, and the distribution of surface displacement across these complexities. Finally, we build a complete slip budget in the fault parallel-component and we analyze the evolution of on-and off-fault displacements along strike in regards of the fault geometry.

Context of the study: the Baluchistan earthquake

The Mw 7.7 Baluchistan earthquake continental earthquake occurred the 24 September 2013 in an arid area south of the Chaman strike-slip fault system, in Pakistan, at the transition with the eastern Makran subduction range. The Makran range results from the collision of Arabia and Eurasia at 2.7 cm/year [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF]. Among the few earthquakes reported along the subduction, the largest event occurred in 1945 (Mw8.4, [START_REF] Byrne | Great thrust earthquakes and aseismic slip along the plate boundary of the Makran Subduction Zone[END_REF]. No significant event, however, was reported onshore before the 2013 Baluchistan earthquake. The Chaman fault system accommodates 2.9 cm/year [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF] of northwards motion of the Indian plate relative to the Eurasian plate, through sinistral shearing. Until 2013, the largest earthquake known in this area was the 1935 Mw7.7 Quetta earthquake [START_REF] Bilham | The 1892 Chaman, Pakistan, Earthquake[END_REF], which occurred on the Ghazaband fault located east of the Hoshab and Chaman faults (Fig. 3.1). The 2013 earthquake ruptured the Hoshab fault for ~200 km. This fault connects the Chaman strike-slip fault system to the Makran compressional range, two fault systems with azimuths about 90° apart (Fig. 3.1). At first order, the fault's geometry is controlled by folded Miocene-Oligocene sedimentary rocks associated with the Makran subduction accretionary prism.

The Hoshab fault outcrops at the toe of the topography, where basement tends to thrust over unconsolidated surface deposits [START_REF] Panagos | European digital archive on soil maps (EuDASM): preserving important soil data for public free access[END_REF].

Due to remoteness and access difficulties, this event has been studied almost exclusively using combinaison of seismic, Radar (correlation and InSAR) and optical data [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF][START_REF] Barnhart | Ball-andsocket tectonic rotation during the 2013 Mw7.7 Balochistan earthquake[END_REF][START_REF] Jolivet | The 2013 Mw7.7 Balochistan earthquake: Seismic potential of an accretionary wedge[END_REF][START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF], allowing for the characterization of the source parameters. A more detailed characterization of the fault geometry and of the near-fault displacement patterns was performed using highresolution optical images and image correlation (Barnhart et al., 2019[START_REF] Barnhart | Evidence for slip partitioning and bimodal slip behavior on a single fault: Surface slip characteristics of the 2013 Mw7.7 Balochistan, Pakistan earthquake[END_REF][START_REF] Cheng | Permanent Co-Seismic Deformation of the 2013 Mw7.7 Baluchistan, Pakistan Earthquake From High-Resolution Surface Strain Analysis[END_REF][START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF][START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF]Vallage et al., , 2015;;[START_REF] Zhou | Characterizing Complex Surface Ruptures in the 2013 Mw 7.7 Balochistan Earthquake Using Three-Dimensional Displacements[END_REF]Zhou et al., , 2015;;[START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF]. The rupture was predominantly strike-slip with a mean left-lateral slip of 8 m (Gold et al., 2015;Vallage et al., 2015), and a small thrusting components of 3 to 4 m (Avouac et al., 2014;[START_REF] Barnhart | Ball-andsocket tectonic rotation during the 2013 Mw7.7 Balochistan earthquake[END_REF][START_REF] Jolivet | The 2013 Mw7.7 Balochistan earthquake: Seismic potential of an accretionary wedge[END_REF]. It initiated on a 65°NW dipping fault [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF] and propagated bilaterally to the north and to the south-west. To the north, the rupture ended after 50 km when it reached a section of the Chaman fault, which is assumed to creep [START_REF] Jolivet | The 2013 Mw7.7 Balochistan earthquake: Seismic potential of an accretionary wedge[END_REF]. To the south-west, the rupture propagated for 150 km and generated up to 12 m of fault-parallel slip, 2 m of shortening, and 3-4 m of uplift [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF]Barnhart et al., 2019;[START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF][START_REF] Jolivet | The 2013 Mw7.7 Balochistan earthquake: Seismic potential of an accretionary wedge[END_REF]Vallage et al., 2015). Mean ratio of fault-parallel to fault-normal slip is of 6:1 (Barnhart et al., 2019). In the middle of the southern part of the rupture (N26.456°, E64.968°), where fault slip is close to its maximum, a kilometric-scale compressional relay zone links two major faults sections [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF]Vallage et al., 2015). This relay is hereafter referred to as the C2 complexity, following Vallage et al. (2015) and [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF].

It is also known as the Wahli Kaur step-over (WKS) in [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF] and [START_REF] Barnhart | Evidence for slip partitioning and bimodal slip behavior on a single fault: Surface slip characteristics of the 2013 Mw7.7 Balochistan, Pakistan earthquake[END_REF].

Using high-resolution optical imagery, we take the opportunity of such a large rupture in a dry and vegetation-free environment to document and discuss the relations between fault geometry and surface deformation patterns in the area of the C2 relay zone, together with the fault bend located 5 km north of C2 where complex fault geometries were also observed (F1; Vallage et al., 2015). In total, here we document near-fault surface displacements over a 30-km long fault stretch centered on the C2 and F1 complexities.

Other satellite-based studies that provide a comprehensive view of the 2013 rupture will be used as a reference frame to our near-fault study.

Data and methods

In this study, we used high-resolution (0.5 m) optical image correlation to measure the 3D near-fault surface displacement field related to the 2013 Baluchistan earthquake in the area of the C2 and F1 fault complexities (Figs. 3.1 & 3.2A). In order to orthorectify the images later used for correlation, we computed two different Digital Surface Models (DSMs) using images acquired respectively before and after the earthquake. These DSMs were also used to assess the vertical deformation due to the earthquake. The pre-earthquake dataset consists of 6 diachronique mono-images acquired by the WorldView satellite before the 2013 earthquake, from August 2011 to December 2012 (Supp. 3.1), with at least two images of each area of interest. The post-earthquake dataset consists of tri-stereo Pleiades images acquired in September 2016, 3 years after the earthquake (Supp. 3.1). Thus, our displacement maps include 3 years of post-seismic deformation in addition to the co-seismic deformation. Amplitudes of post-seismic deformation measured using InSAR range from 0.01 to 0.10 m in LOS [START_REF] Peterson | A viscous accretionary prism: InSAR observations following the 2013 Baluchistan, Pakistan earthquake[END_REF][START_REF] Wang | Observations and Modeling of Crustal Deformation due to Recent Large Earthquakes around the Tibetan Plateau[END_REF].

We used the MicMac photogrammetric tools [START_REF] Rosu | Measurement of ground displacement from optical satellite image correlation using the free open-source software MicMac[END_REF][START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF] to first, calculate the pre-and post-earthquake DSMs, second, to correct the images from viewing angle and topography distortions (orthorectification) and then, to measure the horizontal displacement field through correlation of the pre-and post-earthquake corrected images. We obtained 2 pre-earthquake orthoimages from the orthorectification of the WorldView images, and 3 post-earthquake orthoimages from the orthorectification of the Pleiades images. So, the 6 possible pairs of pre-and post-earthquake orthoimages were correlated and the results were stacked, allowing for noise and artefacts reduction.

The MicMac software allows preserving the input spatial resolution of the images through the correlation process, and thus it yields a horizontal displacement fields with a ground resolution of 0.5 m.

The horizontal displacement maps at 2.5 m resolution already available for the whole Baluchistan rupture from Vallage et al. (2015) were used as a far-field reference to correct our near-fault displacement maps, that are only available for a ~8 km wide area around the fault, from residual ramps. Ramps were corrected such that profiles along and across the fault zone in our results overlapped with similar profiles taken in Vallage et al.

(2015)'s results, which were corrected using far-field undeformed references. These profiles are individual profiles (not stacked profiles), crossing the entire high-resolution map both along and across the fault zone. Once the maps were corrected from these ramps, they were projected onto fault-parallel and fault normal-directions (Fig. 3.2 and Supp. 3.2).

Vertical deformation due to the 2013 rupture was assessed by calculating the vertical difference between the pre-and post-earthquake DSMs. Before that, the pre-event DSM was adjusted for the horizontal displacements according to horizontal offsets retrieved during the previous step to ensure that similar pixels from the pre-and the post-event DSM were aligned with each other before they got differentiated (Delorme et al., 2020).

This enables us to avoid spurious vertical signal in areas that combine steep topography and large horizontal displacement and to keep only the coseismic vertical displacement.

After corrections of the ramps, the resulting fault-parallel, fault-normal, and vertical displacement maps (Fig. 3.2A) still present some visible artifacts: vertical stripes due to the WorldView sensor, remaining topography after orthorectification (active fluvial network for the most part), and decorrelation due to presence of clouds. From profiles in the displacement maps, we estimated that the amplitude of the signal related to the linear stripes remains within 0.10 m. However, some topographic artefacts can reach few tens of centimeters to few meters in amplitude. Hence, in the following work we paid special attention to ensure that the profiles we used for data analysis and slip quantification were not affected by such artifacts. Thus, in summary, although the theoretical detection threshold using sub-pixel correlation should be about 0.005 m according to the resolution of our input images (Rosu et al., 2014), considering the residual level of noise in our data we conservatively assess that our actual precision in measuring the displacements is 0.10 m. Still, the vertical component of displacement is more affected by topographic noise and locally the noise can reach larger values. Vertical uncertainty of the DSMs processed with MicMac is of the order of the spatial resolution, so 0.50 m (Rupnik et al., 2018).

This means that post-seismic deformation after the 2013 rupture (0.01-0.10 m in LOS) occurs within the noise from this study Eventually, we derived the curl, dilatational strain, and norm of maximum shear strain components of deformation from the horizontal displacement maps. To do so, the maps were first down-sampled to a resolution of 2.5 m to reduce the correlation noise and avoid high-frequency spurious variations. It is equivalent to applying a low-pass filter as long as we average the displacement values over a 2.5-m window using a nearest neighbor interpolation. The derivations were performed using Matlab codes from [START_REF] Allmendinger | Structural Geology Algorithms: Vectors and Tensors[END_REF] that enable calculating the Lagrangian finite strain tensor from displacement gradient maps. These maps were used to characterize the deformation mechanisms that occurred at the two major complexities imaged in our study area, the F1 and C2 fault zones.

Results and interpretations

In the following section, our aim is to characterize the geometrical relations between faults, and the relations with the on-and off-fault displacement patterns. In our study area (Fig. 3.2A), the rupture propagated from north-east to south-west through multiple geometrical complexities. The two main complexities are called C2 and F1 (dashed ellipses in Fig. 3.2A), which are the focus of this work. We use the term "fault slip" to describe localized deformation on any fault which is larger than our detection threshold (~0.10 m). In the displacement maps, localized deformation arises as clear-cut transitions in the displacements. Conversely, the term "off-fault deformation (OFD)" is used to describe the deformation that spreads into large areas (~0.1-1 km) around the primary and secondary faults. It is recognized in the displacement maps as patches of gradual variations in the displacement amplitudes, and where no localized slip offsets can be identified.

The F1 complexity

The F1 complexity consists in a 7 km-long bend that accommodates a change in azimuth of 3° in the primary fault trace (P1, Fig. 3.2A & 3.3). It is characterized by a large area of distributed extension and shearing where numerous cracks and faults were mapped in the hanging-wall (Vallage et al., 2015). The main fault scarp, located to the south-east, has a zigzagging geometry with a bimodal distribution of azimuths (N216 and N259; [START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF]. The N216 orientation is similar to the basement fabric in this region and was interpreted as a re-activation of pre-existing faults during the 2013 rupture [START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF].

Amplitudes of fault-parallel displacement across F1 approach 11 m, however, it is distributed over a large area, from the trace of primary fault P1 to 500 m to the north-west (Fig. 3.2A & 3.3A). In the area of Figure 3.3A, and crossing profile AA', we observe a secondary fault located ~350 m north-west of the zigzagging fault P1. This secondary fault bounds the area of distributed fault-parallel displacement for ~500 m and accommodated ~4 m of left-lateral slip. This secondary fault appears clearly in the curl and norm of maximum shear strain maps, meaning that it accommodated left-lateral motion, similar to the primary fault plane. However, it did not accommodate significant shortening or extension as there is no signal at this location in the dilatation map. This is coherent with the fact that the fault does not appear on the fault-normal displacement map (Fig. 3.3B). ). This subsidence occurred within an area of a few kmwide, that lies along-side the trace of P1 for the entire length of the F1 complexity, and located few hundred meters north-east of P1 (Fig. 3.2A & 3.3C). We notice that this area of subsidence correlates with other cracks mapped by Vallage et al. (2015). Conversely, compression was measured on fault P1 that accommodates the shortening through localized deformation.

When comparing the displacement patterns across the F1 fault zone, to the linear and localized fault sections that bound the complexity to the north and to the south, we notice that fault-normal displacement amplitudes are two to three times greater across F1 (4-6 m) than across the linear areas (~2m). This was qualified by Vallage et al. (2015) as an excess of fault-normal displacement, caused by a decrease in the fault dip angle towards the surface. As a consequence to this excess of shortening, subsidence occurred within the hanging-wall.

South of the F1 complexity, the rupture propagated linearly for 4 km before crossing two small bends, each about 200 m in length (Fig. 3.2C), with the easternmost one that developed into a pull-apart basin. The surface displacement in the fault-parallel component is limited to a narrow region of about 100 m in width, corresponding to the light red-colored area in Figure 3.2C. Up to 2.5 m of subsidence in the hanging-wall are associated with these features. They also present a strong fault-normal displacement (Fig. 3.2A), similar to what was described in F1 (Fig. 3.3B-C). Again, in this case, the spatial extent of the fault-parallel displacements does not correlate with the fault-normal and vertical displacements. This speaks in favor of a low fault-dip angle close to the surface, that generated horizontal translation of the hanging-wall to the south-east and extension and subsidence within the hanging-wall.

The C2 complexity

The C2 complexity consists in a 5-km long and 2-km wide compressional relay zone that links primary faults P1 and P2 (Fig. 3.4B). It has been identified in previous studies as the main complexity along the 2013 Baluchistan rupture [START_REF] Barnhart | Evidence for slip partitioning and bimodal slip behavior on a single fault: Surface slip characteristics of the 2013 Mw7.7 Balochistan, Pakistan earthquake[END_REF]Gold et al., 2015;[START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF]Vallage et al., 2015). Fault P1 branches off from the pushup and dies out about 5-km south of the push-up. On the other hand, to the north of the push-up several faults connect to form the fault P2, which extends linearly westward. No significant complexity could be identified farther west along the rupture and thus fault P2 corresponds to the last continuous fault branch of the Baluchistan surface rupture after the C2 relay zone.

The C2 complexity starts, to the north-east, with a 400 m-long and 200 m-large extensional jog, located 2 km south-west of the two small bends from Fig. 3.2C. There, unlike the structures described in section 4.1.1., displacement patterns in the three components are correlated, and subsidence reaches 1.5 m (Fig. 3.2A and faults P1 and S2

in Fig. 3.4A). Hence, this area was not impacted by deeper variations of the fault geometry, unlike the previous complexities we described.

The body of the push-up is constituted of seven secondary faults that strike either eastwest or north-south, and link the two primary faults, P1 and P2, that strike ~N235 (faults S1 to S7; Fig. 3.4). Fault S1, S3, S6 and part of fault S4 had not been previously identified in the satellite images (Vallage et al., 2015), although they correspond to well defined slip discontinuities in our displacement and deformation maps (Figs. 3.2A, 3.4A and 3.5A). For faults S3 and S6, the presence of large drainages along the rupture trace makes difficult the identification of the faults using only photo-interpretation. In this case, the high-resolution displacement maps were necessary to fully resolve the surface ruptures.

Orientations of the secondary faults S1 to S7 can be classified into three categories: mountain range azimuth (red, Fig. Considering the definition of localized fault slip and off-fault deformation we stated earlier in section 4., fault S1 corresponds to an area of mixed deformation processes: localized deformation is visible on the eastern side of F1 (east of profile GG') while the western part corresponds to OFD gradients. South-west of F1, we also measured 4 m larger fault-normal displacement (Fig. 3.5B), as well as ~2 m lower uplift, relatively to the rest of the push-up (Fig. 3.5C). This observation, associated with the fact that P1

propagates further south-east in this area, reminds of the mechanism of flattening of the fault to the surface in the F1 complexity. In the case of F1, we saw that the fault propagated farther downhill and generated extension and subsidence within the hangingwall.

After the C2 relay zone, the rupture propagated almost linearly on fault P2, following the mountain range azimuth (Fig. 3.2A). 38°. An estimations of the dip angle at deeper depth using the distributed displacement gradients observed in the hanging-wall yields a dip angle of 56°, which is consistent with published values [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF]Zhou et al., 2015). According to Vallage et al. 3.6). Interestingly, vertical motion on fault S1 cannot be described as a well localized offset, but rather as a steep negative gradient of displacement. This supports the previous observation that this area deformed in part through OFD.

Displacement profiles in 3D view

To better understand how deformation distributed throughout the main geometrical complexities, we displayed cross-fault displacement profiles in 3D to visualize alongstrike variations of the slip distribution. Fault-parallel displacements, corresponding to the main component of displacement for this event, are presented for the C2 and F1 complexities (Fig. wall of C2. Also, these gradients are irregular and display changes of slope with, sometimes, small localized secondary offsets, highlighted in red. Thus, in this area the deformation mechanism was heterogeneous. Significant part of the OFD was taken by distributed cracks that accommodated each some localized deformation (~0.10-2 m; Fig.

3.7A

). This cracks have a limited length of 50-500 m and do not display clear-cut geometries in the displacement maps, thus they are not considered as secondary faults. The 3D plot across the C2 relay zone (Fig. 3.7B) covers the whole relay zone, from the small extensional relay zone to the north (S2-P1) to the connection with primary fault P2

to the south. To the north of the relay zone, slip profiles are simple with one offset of 9.5 m on primary fault P1 and some OFD for ~1 km in the hanging wall. At the center of the push-up area, we measured multiple fault offsets with the major offset on P1 and two offsets on S4 and S5 (profile BB ',Fig. 3.6). Secondary fault S1 is indicated in a zoom window (Fig. 7C) as long as it is not visible at the scale of the plot. Secondary fault S1 is very discontinuous and part of the displacement in this area was accommodated by OFD processes (Fig. 3.5), with the displacement profiles that resemble those observed in the F1 fault zone (Fig. 3.7A). Whether it is localized or not, S1 fault zone accommodated up to 4 m of slip that were initially taken by the primary fault P1. West of this area, we observe an inversion of the sense of motion between faults S4, S6 and S7. Offset on fault S6 is dextral while offsets on faults S4 and S7 are senestral. This can also be seen in the curl map from Figure 3.5E where the fault appears in yellow. This is due to its orientation, at ~90 degrees from fault S4 and S7, which made dextral slip possible. This area constitutes the transition point of the push-up where slip on primary fault P1 drops drastically to stabilize around 2 m and slip on faults S4 and S5 increases. Faults S4 and S5 accommodated 2/3 rd of the slip budget for 800 m before the junction with primary fault P2. This final connection constitutes the end of the push-up with slip from faults S4 and S5 that is transferred to primary fault P2, which becomes the major fault of the system with 7.5 m of slip. It is important to notice that, overall, the amount of localized fault slip after the relay zone (7.5 m on P2 plus 2 m on P1) meets again with the initial amount of localized slip on P1 before the relay zone (9.5 m).

In addition to localized slip that distributed between the different faults, we measured OFD gradients, essentially in the hanging-wall of the relay zone, but also between faults S4 and S5, and at some locations along S1 and S3. To the north of the complexity, displacement gradients in the hanging wall are positive while moving towards the fault, meaning that no OFD was accommodated in this area, likewise the northern tip of the F1 complexity (Fig. 3.7A). Conversely, to the center of C2, displacement gradients within the hanging-wall, as well as inside of the relay zone, are negative towards the primary fault, meaning that OFD occurred. However, these gradients differ greatly from those observed in the F1 area. In fact, OFD gradients within the hanging-wall (north of S5) and the pushup area (between S4 and S5) are of low angle, and do not mark a clear transition with neighboring areas where no OFD occurred. Also, no cracks were mapped in this region, neither localized slip offsets were detected in our displacement maps. So, we propose these gradients to be associated with large-scale and continuous modifications of the bulk volume characteristics, referred as to diffuse deformation. As the precision of our data is 0.10 m and our spatial resolution 0.50 m, we cannot exclude that cracks smaller than our resolution, and/or that accommodated amounts of displacement lower that our detection threshold, are included within the diffuse deformation gradients.

To go forward with the analyze of OFD processes, in the following we quantify these OFD gradients and we analyze the results in light of the rupture geometry.

3.1.5.3. Quantification of fault slip and off-fault deformation along the fault-system

In this section, using displacement profiles across the fault zone, that are placed every 100 m along our study area (315 profiles in total), we measured the total displacement offsets as well as localized fault slip on each fault of the system (Fig. 3.9). We have focused on the fault-parallel component of slip, since it corresponds to the main component of displacement for this earthquake. Also, because our measurements only address the near-fault deformation zone (4-5 km-wide profiles), we used results from Vallage et al. (2015), at 2.5 m resolution, to complement our data for the far-field area and trace ~9 km-long profiles (Fig. 3.8 and Supp. 3.2). Doing so, we ensured that farfield trend is well considered. The profiles correspond to stacks of the data along 100 mwide and ~8 km-long profiles, that frame the entire fault zone and catch the far-field trends (C2 is 2.5 km large for example; Fig. 3.8). Using the following relation: OFD = total slipsum of localized slip, we calculated the amount of OFD along the fault system and analyze the results in regards of the fault geometry. For each profile we estimated the total displacement by using the step between the linear regressions of the slip gradients in the footwall and hanging-wall, far from the fault (Fig. 3.8, black dashed lines). To quantify slip that localized on the different faults of the system, we used displacement profiles of a limited length of a few hundreds of meters, that were placed every 100 m apart along each fault of the system. In that case, linear regressions were taken very close to the slip offset to avoid incorporating OFD in our measurement, and only measure the amount of slip accommodated by the fault. ). In the complexities, fault slip can drop to only 30% of the total slip budget, with the remaining part of the displacement that was accommodated through OFD (grey area, Fig. 3.9B), so up to 70% of the total displacement budget. Mean OFD for this area of the Baluchistan earthquake is of 16%.

Apart from the F1 and C2 complexities, we observe two other patches of OFD: the tip of fault P1, located south-west of C2, that displays ~2 m of OFD which occurred in-between and around the small-faults oriented NE-SW; and a portion of fault P2, from 25 to 27.5

km, where cracks that lie along the primary fault accommodated deformation. In the area of the tip of fault P1, OFD was accommodated through both low-angle gradients (as in the hanging-wall of C2, Fig. 3.7B) and distributed faulting (as in the F1 fault zone, Fig.

3.7A

). In fact, many faults were mapped on the field in this area, but slip offsets lower than our data precision (0.10 m) are included within the OFD budget. In the area where OFD occurred along fault P2, from 25 to 27.5 km, the OFD gradients are alike those observed in the F1 area and, similar to F1 as well, many cracks were mapped within the first 200-300 m of the hanging-wall. Thus, in our study area, it is clearly observed that OFD occurred in areas of complex fault geometry, and was potentially associated with off-fault cracks, while simple fault zones localized ~100% of the co-seismic on a single fault structure.

We also observe that the OFD budget is highly variable along-strike. We measured abrupt lateral variations of OFD of few meters from one profile to the next. This variability of OFD is the reflect of the variability of fault slip, as OFD is directly derived from the quantification of localized slip on individual faults. This variability is strongly correlated with the presence of distributed cracks in the fault zone. This has been particularly observed at the F1 complexity, the S1 fault zone and the OFD zone in the hanging-wall of P2 (Fig. 3.9B). This reveals the mechanism of OFD in these regions, that was driven by the crack density and distribution. This type of OFD differs greatly from the OFD observed within the hanging-wall of C2 and between faults S4 and S5 (Fig. 3.8), that accommodated OFD over wider areas and without visible cracks to the surface (Fig.

3.7B).

These variations in OFD and fault slip are not encountered in the total slip budget that displays low-frequency and low-amplitude variations as described in the first paragraph of this section. Hence, OFD acted as a buffer to compensate fault slip variability, so that the total co-seismic slip budget remained stable and followed a large-scale pattern. It appears clearly that high-frequency fault slip variability was generated by the local-scale fault geometry (continuity and complexity of the fault zone and presence of cracks in the off-fault medium; Fig. 3.9) while low-frequency total slip variations were driven by the earthquake mechanism and deep fault geometry and asperities [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF].

Eventually, the spatial extent of the OFD zone was also measured along our study area.

As explained previously and based on the slip profile from Figure 3.8, the edge of the OFD zone corresponds to the points where the surface displacement profiles start to depart from the far-field slope (black dashed line, Fig. 3.8), that is itself estimated from the large-scale results from Vallage et al. (2015). For every profile analyzed in the study area, OFD was restricted to the hanging wall. This might be a direct consequence of the geometry of the fault plane, that dips ~50-55° to the north-west [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF], and which generated higher deformation within the hanging wall of the rupture. Moreover, the OFD zone was only been reported at the two complexities C2 and F1, where OFD distributed over areas wider than 200 m. Elsewhere, OFD remained close to the fault (30-200 m) and overlapped with the surface rupture map at the scale of the figure. So, large scale (>few hundred meters) gradients of OFD were measured only at the two major geometrical complexities.

But, the extend of the OFD zone differs subsequently between the C2 and F1 complexities. In F1, OFD is detected up to 500 m away from the primary fault trace (Figs.

3.3A, 3.6 profile AA' and 3.7A) while OFD in the C2 push-up was detected as far as 1.5

km north-west of the complexity (Figs. 3.6 profile BB',3.7A and 3.8). This comes as a direct consequence of the difference in the deformation mechanism between the F1 and C2 areas, with the large-scale diffuse deformation that occurred in the hanging-wall of C2 (Fig. (2014). Hence, there are large discrepancies between our estimation of mean OFD and what has been published so far. These discrepancies arise principally from a difference in the methodology used to quantify OFD, and the fact that we measured deformation features at different scales.

In fact, [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF] define OFD as the difference between far-field offsets (>350 m from the primary fault trace) taken across optical-based displacement maps at 5 m ground resolution, and near-field offsets (<10 m from the primary fault trace) remotely measured on 0.5 m ground resolution optical images. Similarly, in [START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF], the far-field displacement budget was obtained from the correlation of Landsat-8 images at 15-m ground resolution and it was compared to offsets in geomorphic features measured on 0.5-m ground resolution optical images. Estimation of the far-field budget may not be much different between these two studies and our work because all of them capture the large-scale displacement patterns, even so we took a special care in considering the verylarge scale gradients (> 1km) around the fault to include diffuse deformation within the total displacement budget. However, concerning the near-fault displacements, [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF] and [START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF] use offsets measured remotely on optical images while we considered offsets taken across displacement maps. This strongly impacts the estimation of the localized faut slip: using the displacement maps, we included the fault damage zone within the localized fault slip offsets (mean fault zone width is 50 m, min is <5 m and max is 380 m), while it might not be the case when considering offsets on visible features to the surface (e.g., <10 m in [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF]. We did not focus on separating the "very localized" slip offset (<10 m) from the "total localized" slip offset (10-200 m) within the displacement maps as we are more interested in deformation processes that occurred at scales larger than a few hundred meters and that are particularly difficult to measure on the field. OFD as defined by [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF] and [START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF], that includes nearfault damage zone, occurred all along the rupture with proportions that vary with the local geology and the local (< km) fault structural complexity. Thus, within the localized fault slip offsets we measured, there might be 28 to 45% of the deformation that spread over areas larger than 10 m. OFD, as we define it in this study, represents another 30% of the deformation budget that occurred around the major geometrical complexities (>km) (Fig.

3 .9). This comparison has to be taken with great care as long as [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF] and [START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF] did not measure offsets lower than ~1 m, and [START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF] include secondary faults within the OFD budget while we considered them as part of the localized slip budget. This adds supplementary uncertainty when comparing the different data sets.

Mechanisms of off-fault deformation

In addition to observations made in this study, we know from previous works, based on both surface measurements (field and geodetic) and dynamic rupture modeling, that OFD occurs through distributed cracking as well as diffuse deformation processes [START_REF] Bürgmann | Strain accommodation about strike-slip fault discontinuities in granitic rock under brittle-to-ductile conditions[END_REF][START_REF] Griffith | Rough faults, distributed weakening, and off-fault deformation[END_REF]Okubo et al., 2019). However, we do not know to what extent each mechanism acted in the OFD budget, when considering deformation processes that occurred at very large-scales (> 200 m), and what processes were exactly involved.

In the case of the F1 complexity, extensional deformation occurred within the hangingwall, due to the along-dip geometry to the fault (Vallage et al., 2015), favoring the development of large and dense networks of fractures (Okubo et al., 2019;Thomas and Bhat, 2018). These fractures extend for several hundred meters within the hanging-wall, which is coherent with the orders of magnitude that outcome from the model. This deformation mechanism is discontinuous and explains the strong variability in the fault slip curve in this area of the rupture (Fig. 3.9B), as long as the slip budget will depend on whether or not we cross cracks in the off-fault medium. We refer to this deformation mechanism as "distributed cracking".

No similar pattern was observed in the hanging-wall of C2, where continuous OFD gradients were measured. This characteristic, the continuity of the gradient, shows that the underlying deformation mechanisms was a continuous process, at least at our scale (spatial resolution of 0.5 m and detection threshold of 5-10 cm). There also are no visible fractures or deformation structures in the displacement maps nor in the optical images.

OFD in C2 occurred within the basement rocks of the Makran range, that for most part, consist in sandstones and shales [START_REF] Bakr | Geological Map of Pakistan. (East and West Pakistan)[END_REF]. These rocks can deform through plastic processes [START_REF] Fossen | Deformation bands in sandstone: a review[END_REF][START_REF] Hamiel | Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks[END_REF]) but, at high strain rates, brittle processes are still likely to occur [START_REF] References Aben | High strain rate deformation of porous sandstone and the asymmetry of earthquake damage in shallow fault zones[END_REF][START_REF] Fossen | Deformation bands in sandstone: a review[END_REF]. Hence, microfracturing and deformation bands (i.e., tabular zones of grain reorganization through grain sliding, rotation and/or fracturing) are the most probable processes responsible for the OFD gradients we observe. This deformation mechanism was, previously in this study, referred to as "diffuse OFD".

Relations with the fault geometry and the rock lithology

We illuminated in section 4.3. a relation between the complexity of the fault zone and the occurrence of OFD (Fig. 3.9B). This relation has also been highlighted by [START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF]. However, even if C2 and F1 concentrated the largest amounts of OFD in our study area, they do not present the same patterns of OFD. So, hereafter, we analyze the structure of C2 and F1 to highlight the relations between characteristics of the complexities and amounts and types of OFD.

The C2 complexity consists in a compressive relay zone that links the two major branches of the 2013 rupture. These faults cut through basement rocks and are well defined in the landscape in this area. Secondary faults constituting C2 are also, for most part, well defined in the landscape (Fig. 4A). Also, areas of C2 that were uplifted during the 2013 event (up to 4 m, Fig. 3.2A) are visible in the long-term topography. Overall, the topographic features seem to follow the geometry of the complexity. So, we propose C2 to be a long-term relay zone with faults deeply embedded into the basement rocks. Using the same methodology than in section 4.2., and considering that faults P1 and S5

propagate linearly at depth under C2, we propose at a very first order that the downdip end of the C2 complexity occurs around 700 m depth. Conversely, the connection between P1 and P2 remains unclear and estimations of fault dip angle based on inversion of geodetic data predict that P2 has a lower dip angle than P1, thus the two faults may never directly connect.

On the other hand, at the F1 complexity, the fault cuts through the surface farther downhill, into recent alluvial deposits. Topographic features follow the zigzagging fault trace in some areas but the fault scarp in this region is never higher than 2 to 5 m, so once to twice the 2013 rupture vertical uplift. We also estimated the downdip end of the F1 complexity to ~275 m, almost three times less than the C2 complexity. So, F1 appears to be a more recent, because less marked in the cumulative topography, and smaller complexity (both in width and in depth) than C2.

First conclusion based on these observations is that, even if C2 and F1 vary greatly in terms of size (both length and width) and relative age to the cumulative deformation of the fault, also referred as maturity of the fault zone [START_REF] Dolan | How well do surface slip measurements track slip at depth in large strike-slip earthquakes? The importance of fault structural maturity in controlling on-fault slip versus off-fault surface deformation[END_REF][START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF], they accommodated similar proportions of OFD (mean of 30%, and maximum of 70%). Hence, there appear to be no direct scaling between the size and "maturity" of the geometrical complexity, and the amount of OFD. From this, it comes out that the difference between C2 and F1 might not lie in the amount of OFD that was measured, but rather in the mechanisms that drive OFD.

In F1, where OFD occurred primarily through distributed cracking, fault complexity is observed at small-scale (~100 m) and is embedded into low-cohesion rocks.

Unconsolidated sediments are more likely to host distributed cracking than the consolidated rocks that have higher strength [START_REF] Kushnir | Characterizing the physical properties of rocks from the Paleozoic to Permo-Triassic transition in the Upper Rhine Graben[END_REF], and so to generate high-amplitude OFD gradients. Conversely, C2 is deeply embedded into basement rocks and so, OFD deformation in this region is more likely to affect a large volume of rocks and generate large-scale (~ km) gradients of OFD at the surface [START_REF] Segall | Earthquake and Volcano Deformation[END_REF]. In addition, basement rocks have a higher strength than sediments, and will most probably deform through diffuse deformation in a first stage, before that opening of large fractures (> 0.5 m), and localization of the off-fault deformation on these fractures, is required to release larger strain.

However, we notice that, across the S1 fault zone, which is part of the C2 complexity, OFD gradients are alike those observed in the F1 complexity while the rock type at S1 corresponds to basement. This area of the push-up is jammed between numerous orthogonal faults, and was affected by cumulative uplift through time. Thus, basement rocks must have been packed and intensively deformed through times such that they reached a very low strength that allow distributed cracking OFD, similar to F1. Another common point between S1 and F1 is the small-scale of the complexities in the fault trace (100-500 m), with the presence of discontinuous cracks. S1 is not easily distinguishable both in the landscape and in the displacement maps. In terms of geometrical relation with the other faults of the system, we also notice that S1 is located in between primary fault P1 and secondary faults S6 and S7, with S6 oriented perpendicularly (Fig. 3.4A). We hypothesize that S1 could have developed to cross-cut faults S6 and S7 in response to the right-angle azimuthal variations between faults P1, S6, S7 and S4 (Fig. 3.4A). S1 would then be a newly developing fault that display small-scale (~100 m) discontinuity and roughness, unlike other secondary faults of the relay zone that correlate with the longterm topography. This example, in addition to the F1 area, shows that OFD mechanisms are in part driven by the rock lithology, the scale at which fault complexity occurs and the long-term deformation on the fault. However, relation between those three parameters and the OFD pattern is not unambiguous. Supplementary observations on other faults systems are required to clarify these relations.

Eventually, the differences in OFD mechanisms we highlighted above might also explain why previous studies found out that larger amounts of OFD occur in areas of younger and/or thicker sediments [START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF][START_REF] Teran | Geologic and structural controls on rupture zone fabric: A field-based study of the 2010 Mw 7.2 El Mayor-Cucapah earthquake surface rupture[END_REF][START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF]. In fact, these studies did not take into account the diffuse OFD gradients since they were not detected before, either because of the resolution of the studies, or because the focus was not given to large-scale (>~300 m) diffuse deformation patterns.

Conclusions

In this paper, using 3D high-resolution (0.5 m) optical image correlation, we described the near-fault deformation patterns associated with a 30 km-long area of the 2013

Baluchistan rupture. This area includes two large geometrical complexities. We quantified the amount of on-and off-fault displacement with an accuracy of 0.20 m and evidenced a direct relation between the fault zone complexity and the occurrence of OFD at large-scale (>~300m) within the hanging-wall of the rupture. We draw the lector attention on the difference between OFD assessed in this study and OFD estimated in previous works. OFD in this study corresponds to deformation that occurs outside of the primary and secondary faults, and which does not include the fault damage that extends a few hundred meters (< 200-300 m) around the fault. We consider OFD mechanisms that occur at greater distances from the fault (>~300 m) and with smaller deformation gradients (0.1-1 m of slip/100 m).

We showed no difference in the amounts of OFD between the two fault complexities analysed in this study, even if the rock lithology and the scale of the geometrical complexities differs. Instead, we show that large-scale OFD occurs either through detectable (slip > 0.10 m and length >0.50 m) and distributed fractures, or through microscale and continuous modifications of the bulk properties (microfracturing and deformation bands), referred to as diffuse deformation. We propose that OFD mechanisms are driven by the type of rocks which are cut by the faults, and scale of the fault complexity (hundreds of meters or kilometers): OFD through visible and distributed fractures is more likely to occur in areas of small-scale (< km) fault complexity embedded into low-cohesion rocks, and eventually with an extensional context; diffuse OFD is more likely to affect high-cohesion rocks in areas of large-scale fault complexity. Both of these parameters are likely to be influenced by the long-term deformation accumulated by the fault zone, but this needs to be further investigated before drawing simple interdependency relation.

This work highlights new deformation processes that occur over wide areas around active faults, and which could not be detected on the field for most part. These need to be taken into account when estimating the deformation budget of an earthquake. This work opens new possibilities for taking advantage of optical images to focus on diffuse ground deformations that are not visible in the field, and push further the analysis of the relations with the fault zone complexity, rock lithology and long-term evolution of the fault system.

Acknowledgments

This work was partly supported by TOSCA CNES and by the ANR-18-CE31-0012

projects. This study contributed to the IdEx Université de Paris ANR-18-IDEX-0001. This is IPGP contribution XXXX. Numerical computations were partly performed on the S-CAPAD platform, IPGP, France. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. We proposed several possible mechanisms of deformation that could be responsible for the patterns we observed. However, the characterization of the deformation mechanisms relies on the assumption that it occurred coseismically at high strain rates. Recent studies show that part of the surface deformation associated with an earthquake may in fact occur during the early post-seismic phase, in the days and weeks following the rupture [START_REF] Brooks | Buried shallow fault slip from the South Napa earthquake revealed by near-field geodesy[END_REF]Milliner et al., 2020). In the case where diffuse deformation would occur at low strain rates, during the post-seismic phase, plastic deformations, and for example flowing of the materials [START_REF] Tullis | Experimental deformation of dry westerly granite[END_REF], could become possible processes of deformation. In addition, this implies that OFD would not participate to the energy budget of the coseismic rupture, and so to the ground motions. To assess the seismic hazard associated with a fault, both the total fault displacement and coseismic ground motions have to be characterized to design buildings and lifelines.

In the previous section, we measured deformation over a long period of time, from 2011-2012 to 2016, while the September 2013 rupture lasted only about one minute [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF]. As a consequence, there is large uncertainty on the part of post-seismic deformation that is included in the displacement maps calculated in section 3.1. Previous studies that measured post-seismic deformation using InSAR, from October-December 2014 to June 2017, report values of 0.01 to 0.10 m in the LOS [START_REF] Peterson | Viscous Accretionary Prisms: Viscoelastic Relaxation of the Makran Accretionary Prism Following the 2013 Baluchistan, Pakistan Earthquake[END_REF][START_REF] Wang | Observations and Modeling of Crustal Deformation due to Recent Large Earthquakes around the Tibetan Plateau[END_REF]. These measurements show that the overall post-seismic signal occurred over large-scales (>5 km) and did not localize on the fault. With the optical image correlation method, we investigate small-scale (< 1 km) and localized post-seismic deformations that could have occurred within the fault zone, as a response to the geometrical complexity of the coseismic rupture and displacement pattern.

Data and methods

To measure post-seismic deformation after the 2013 Baluchistan rupture, we used series of optical images acquired from 45 days to 3 years after the rupture. We first use stereo SPOT-6 images acquired in November 2013, in the area of the C2 and F1 complexities, with a ground resolution of 1.5 m (Tab.

Results

In this section, we investigate the post-seismic deformation after the 2013 Baluchistan rupture both at the center of the rupture, where maximum coseismic slip was measured and large complexities were observed, and at the tips of the rupture (Figure 3.10). Figure All the displacement maps calculated for the central area of the rupture, either spanning

2013 to 2016 or 2014 to 2016, show a large-scale linear ramp related to orthorectification bias. Topographic and correlation noise is also visible. However, there are no ground displacements that arise from the background noise and that are correlated with the geometry of the fault zone (in red in Figure 3.11A). Stacked displacement profiles, placed across the C2 relay zone, allow to draw a similar conclusion (Fig. 3.11E,F). In Figure 3.11F, a small signal 0.10 m of amplitude is visible at the location of fault strand S4. This signal does not exhibit the profile of a displacement profile, but rather looks like a noise peak. The fact that it is of the amplitude of the topographic noise present elsewhere in this mountainous region allow us to conclude that this signal corresponds to noise.

However, it is possible that the maps include small-amplitude (<0.10 m) offsets but that are below the detection threshold when considering the topographic noise. Large scale (>5 km) post-seismic deformation can also be present in the maps, but this type of signal cannot be distinguished from the low-frequency artefacts. Because optical displacement maps are subjected to substantial low-frequency noise sources (satellite position and orientation, camera geometry, and orthorectification errors), they are not adapted to measuring displacement patterns that are entirely decorrelated with the surface fault trace.

We also investigated post-seismic deformation at the tips of the ~200 km-long Baluchistan rupture. In these areas, post-seismic deformation is expected to occur along the fault plane because of stress homogenization between the tip of the rupture and the locked (or creeping) zones [START_REF] Jolivet | The 2013 Mw7.7 Balochistan earthquake: Seismic potential of an accretionary wedge[END_REF][START_REF] Peterson | Viscous Accretionary Prisms: Viscoelastic Relaxation of the Makran Accretionary Prism Following the 2013 Baluchistan, Pakistan Earthquake[END_REF][START_REF] Wang | Observations and Modeling of Crustal Deformation due to Recent Large Earthquakes around the Tibetan Plateau[END_REF]. SPOT- Solène L. Antoine (1) , Yann Klinger (1) , Arthur Delorme (1) , Kang Wang (2) , Roland Bürgmann (2) , Ryan D. Gold (3) (1) Université de Paris, Institut de physique du globe de Paris, CNRS, Paris, France, (

6
Department of Earth and Planetary Sciences, University of California Berkeley, California, USA, (3) U.S. Geological Survey, Geologic Hazards Science Center, Golden, Colorado, USA.

Ambraseys, 1970) and geodetic observations (e.g., Delorme et al., 2020;Milliner et al., 2015;[START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF]Vallage et al., , 2015)). Numerical approaches also enable us to establish a relation between observations of fault zone complexity and the mechanical and physical properties of the fault itself (e.g., [START_REF] Finzi | Predicting rupture arrests, rupture jumps and cascading earthquakes[END_REF]Klinger et al., 2018). The et al., 2016;[START_REF] Schwartz | Fault Behavior and Characteristic Earthquakes: Examples From the Wasatch and San Andreas Fault Zones[END_REF]. These variations also coincide with changes in the trend of the fault that often are separated by prominent fault structural complexities [START_REF] Schwartz | Fault Behavior and Characteristic Earthquakes: Examples From the Wasatch and San Andreas Fault Zones[END_REF]. Segmentation of fault systems is also Geometrical barriers are important features because they represent either stopping or favorable initiating points for earthquake ruptures, depending on local stress conditions [START_REF] Biasi | Bends and ends of surface ruptures[END_REF][START_REF] Finzi | Predicting rupture arrests, rupture jumps and cascading earthquakes[END_REF][START_REF] King | Role of Fault Bends in the Initiation and Termination of Earthquake Rupture[END_REF]Klinger et al., 2006;[START_REF] Lozos | Small intermediate fault segments can either aid or hinder rupture propagation at stepovers[END_REF]. At these locations, surface displacements can be particularly difficult to characterize and to quantify as part of the slip is distributed in the surrounding medium through secondary faulting and diffuse deformation [START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF][START_REF] Zhou | Characterizing Complex Surface Ruptures in the 2013 Mw 7.7 Balochistan Earthquake Using Three-Dimensional Displacements[END_REF], Klinger et al., 2006). This occurs principally around the major geometrical barriers, although it can also be observed to a lesser extent at every geometrical complexity along any surface rupture (Milliner et al., 2015(Milliner et al., , 2016a)).

In this study, the term "localized deformation" characterizes slip that occurs on an actual fault, of either primary or secondary strands. Conversely, "diffuse deformation" is used to describe any type of deformation, elastic and inelastic, characterized by large-scale displacement gradients in the bulk medium, without surface rupturing. Near-fault diffuse deformation generally arises from the bulk response to subsurface variations of slip on the fault, including shallow slip deficit [START_REF] Brooks | Buried shallow fault slip from the South Napa earthquake revealed by near-field geodesy[END_REF][START_REF] Nevitt | Mechanics of near-field deformation during co-and post-seismic shallow fault slip[END_REF]. It is encountered at all spatial scales and the contribution to the total surface displacement pattern remains poorly understood. Inelastic diffuse deformation regroups any deformation apart from elastic processes. The current understanding of processes that accommodate inelastic diffuse deformation include block rotations [START_REF] Mcgill | Surficial slip distribution on the central Emerson fault during the June 28, 1992, Landers earthquake, California[END_REF][START_REF] Shelef | Deformation processes adjacent to active faults: Examples from Eastern California[END_REF], micro-scale brittle deformations [START_REF] Hamiel | Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks[END_REF][START_REF] Mcgill | Surficial slip distribution on the central Emerson fault during the June 28, 1992, Landers earthquake, California[END_REF], and granular flow [START_REF] Fossen | Deformation bands in sandstone: a review[END_REF][START_REF] Hamiel | Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks[END_REF].

We note that the definition of diffuse deformation used in this study refers to large-scale (>500 m) gradients of displacement. It differs from the more commonly used definition of off-fault deformation (OFD), which is inferred from the difference between fault offsets measured using satellite imagery analysis over a limited aperture of a couple hundred meters, and that measured on discrete faults in the field [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF][START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF]Milliner et al., , 2015;;[START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF].

Earthquake sequences involving cross faults [START_REF] Hudnut | Cross-fault triggering in the November 1987 Superstition Hills earthquake sequence, southern California[END_REF][START_REF] Kilb | Implications of diverse fault orientations imaged in relocated aftershocks of the Mount Lewis, ML 5.7, California, earthquake[END_REF], such as the 2019 Ridgecrest sequence, generate particularly complex surface ruptures as two sets of faults at high angles to one another accommodate the earthquake deformation.

Thus, high-resolution data that describe the surface displacement distribution are crucial to characterize the slip on the different faults that ruptured, the geometrical relations between the faults, and the way faults (primary and secondary), as well as zones of diffuse deformation, contribute to the total earthquake displacement budget. Key questions about this earthquake sequence include the role of the two different sets of faults in the earthquake mechanisms, and how interactions between the primary fault system and the cross faults influenced the coseismic surface displacement budget and deformation patterns.

In the following, we document, analyze and discuss near-fault surface displacement patterns for the 2019 Ridgecrest earthquake sequence. We first describe the context of the earthquake sequence, including the Mw6.4 foreshock and the Mw7.1 mainshock, and we review the different observations and measurements already available. Next, we detail the optical image-correlation methodology used to measure the three-dimensional (3D)

surface displacement field associated with the earthquake rupture sequence, and we describe the results obtained. We then extract slip values from the displacement maps across the entire fault zone width and measure the slip distribution in the parallel, perpendicular, and vertical components of the deformation. We also discriminate between the contributions of localized slip on discrete faults and diffuse deformation occurring in the surrounding medium. Finally, we compare these results with previously published kinematic inversions and subsurface fault geometries inferred from relocated aftershocks to relate earthquake process at depth with patterns of surface displacement we have observed.

The Ridgecrest earthquake sequence

The 2019 Ridgecrest earthquakes (Mw6.4 and Mw7.1), referred to as the Ridgecrest sequence, occurred during 4-6 July 2019 in southern California. The two events are located at the transition between the Southern Walker Lane and the Eastern California shear zone (Fig. 4.1). The Southern Walker Lane and Eastern California shear zone are part of a system of distributed faulting that accommodates 20% of the relative dextral motion between the Pacific and North America plates [START_REF] Carlson | Kinematics of the westcentral Walker Lane: Spatially and temporally variable rotations evident in the late Miocene Stanislaus Group[END_REF]. Three large historical earthquakes that occurred within this region include the 1872 Mw7.7 Owens

Valley [START_REF] Haddon | Surface slip during large Owens Valley earthquakes[END_REF][START_REF] Crippen | Measurement of subresolution terrain displacements using SPOT panchromatic imagery[END_REF]Mw7.3 Landers (Massonnet et al., 1993;[START_REF] Sieh | Near-field investigations of the Landers earthquake sequence[END_REF], and 1999 Mw7.1 Hector Mine earthquakes [START_REF] Fialko | The complete (3-D) surface displacement field in the epicentral area of the 1999 MW7.1 Hector Mine Earthquake, California, from space geodetic observations[END_REF][START_REF] Treiman | Primary Surface Rupture Associated with the Mw 7.1 16 October 1999 Hector Mine Earthquake, San Bernardino County, California[END_REF]. The Ridgecrest sequence was unexpected since it occurred on a system of faults that was not identified as potential active faults. However, retrospective analysis of highresolution topography data has revealed structures predating the 2019 ruptures (Jobe et al., 2020). The Mw7.1 mainshock rupture southeastern end is located just a few kilometers north of the sinistral Garlock fault, which is a major active fault in southern California.

After the 2019 Ridgecrest sequence, surface creep (Barnhart et al., 2020;Ross et al., 2019) as well as clusters of seismicity [START_REF] Shelly | A High-Resolution Seismic Catalog for the Initial 2019 Ridgecrest Earthquake Sequence: Foreshocks, Aftershocks, and Faulting Complexity[END_REF] were triggered on a section of the Garlock fault, which has not ruptured in the last thousand years (with the exception of a Mw5.7 earthquake recorded in 1992, which was probably triggered by the Landers earthquake; [START_REF] Dolan | Extreme multimillennial slip rate variations on the Garlock fault, California: Strain super-cycles, potentially time-variable fault strength, and implications for system-level earthquake occurrence[END_REF][START_REF] Petersen | Fault slip rates and earthquake histories for active faults in southern California[END_REF]. Therefore, the occurrence of the 2019 Ridgecrest earthquake sequence and its triggering activities on the Garlock fault raise concerns about future seismic hazard in this area of southern California. Ridgecrest surface ruptures, are highlighted in red. Large historical earthquake ruptures are shown in orange [START_REF] Haddon | Surface slip during large Owens Valley earthquakes[END_REF][START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF]. The blue polygon around the Ridgecrest earthquake shows the area covered by the post-earthquake, high-resolution

Pleiades images (Tab. S4.1).

The Mw6.4 and Mw7.1 events of the Ridgecrest sequence ruptured consecutively a system of high-angle strike-slip cross faults (northeast-and northwest-trending) within 34 hours.

The first-order geometry of the ruptures was highlighted by seismological data (e.g. [START_REF] Huang | Spatio-temporal foreshock evolution of the 2019 M 6.4 and M 7.1 Ridgecrest, California earthquakes[END_REF]Milliner and Donnellan, 2020;Ross et al., 2019;Wang et al., 2020). This event is referred to as the foreshock of the sequence. The Mw7.1 mainshock occurred on 6 July 2019 and ruptured a northwest-trending set of right-lateral faults for 50 km (Milliner and Donnellan, 2020;Ross et al., 2019). Inversions of geodetic and seismological data (e.g., [START_REF] Chen | Cascading and pulse-like ruptures during the 2019 Ridgecrest earthquakes in the Eastern California Shear Zone[END_REF][START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF][START_REF] Qiu | Slip Complementarity and Triggering between the Foreshock, Mainshock, and Afterslip of the 2019 Ridgecrest Rupture Sequence[END_REF]Wang et al., 2020;Xu et al., 2020), optical imagery (Barnhart et al., 2020;[START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF]Milliner and Donnellan, 2020), and field-based measurements [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF][START_REF] Ponti | Documentation of Surface Fault Rupture and Ground-Deformation Features Produced by the 4 and 5 July 2019 Mw 6.4 and Mw 7.1 Ridgecrest Earthquake Sequence[END_REF] indicate a maximum of 1.5 to 2 m of left-lateral slip for the foreshock and 5 to 6 m of right-lateral slip for the mainshock. Coseismic slip and aftershocks are limited to the upper ~15 km of the seismogenic crust (Cheng and Ben-Zion, 2020;[START_REF] Huang | Spatio-temporal foreshock evolution of the 2019 M 6.4 and M 7.1 Ridgecrest, California earthquakes[END_REF]Ross et al., 2019;[START_REF] Shelly | A High-Resolution Seismic Catalog for the Initial 2019 Ridgecrest Earthquake Sequence: Foreshocks, Aftershocks, and Faulting Complexity[END_REF]. At the surface, the ruptures cut through young alluvium and Quaternary lake and playa deposits, as well as bedrock outcrops of Mesozoic granite [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF][START_REF] Jennings | Geologic map of California : Trona sheet[END_REF][START_REF] Zimmaro | Liquefaction and Related Ground Failure from July 2019 Ridgecrest Earthquake Sequence[END_REF].

Data and methods

To characterize and quantify the surface displacements associated with the Ridgecrest earthquakes, we used a series of high-resolution (0.5 m) optical images that were acquired before and after the earthquake sequence by the WorldView and Pleiades satellites (Tab.

S4.1). The pre-earthquake images are a combination of archived mono-images from the WorldView and Pleiades satellites acquired between May 2012 and June 2018 (Tab.

S4.1). No major earthquake (M>5) occurred in this area since the 1995 Ridgecrest sequence [START_REF] Hauksson | Preliminary report on the 1995 Ridgecrest earthquake sequence in eastern California[END_REF], and the interseismic strain rate in this area is relatively low as indicated by GNSS and InSAR measurements (e.g., [START_REF] Gan | Strain accumulation across the Eastern California Shear Zone at latitude 36°30′N[END_REF][START_REF] Peltzer | Transient strain accumulation and fault interaction in the Eastern California shear zone[END_REF]. Thus, images spanning 2012 to 2018 can be used safely as reference for the preearthquake period. The post-earthquake dataset includes tri-stereo Pleiades images with three different acquisitions covering the northwestern, the center, and the south-eastern parts of the rupture (Tab. S4.1). Those images were acquired ~1.5 month after the earthquakes, over a span of 20 days. Post-seismic deformation for the 2-month period after the earthquake series is estimated to be about 5 cm in the line of sight (LOS) of Sentinel-2 satellites [START_REF] Brooks | Rapid Geodetic Observations of Spatiotemporally Varying Postseismic Deformation Following the Ridgecrest Earthquake Sequence: The U.S. Geological Survey Response[END_REF][START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF]; therefore, the surface displacements derived in this study likely include limited post-seismic deformation of a few centimeters. Because there was no stereo (or tri-stereo) image acquisition with <0.5-m resolution between the two earthquakes of the 2019 sequence, our measurements combine surface displacements associated with both the mainshock and the foreshock earthquakes, as well as ~2 months of post-seismic deformation.

Milliner and Donnellan (2020) used 3-m resolution Planet Labs satellite images, collected on a daily basis, to separate surface faulting from the two events. This enables us to assume that the northeast-trending surface rupture is associated with the foreshock earthquake while the northwest-trending surface rupture occurred during the mainshock earthquake (Barnhart et al., 2020;Ross et al., 2019). To image the whole area of the Ridgecrest surface rupture, we used 14 combinations of pre-and post-earthquake images, referred to as tiles (Fig. S4.1 and Tab. S4.2). The image processing described below was performed independently on the 14 tiles, which were subsequently mosaicked to obtain a continuous displacement map.

To obtain the horizontal displacement field, we performed sub-pixel cross correlation between the sets of pre-and post-earthquake images using the MicMac package [START_REF] Rosu | Measurement of ground displacement from optical satellite image correlation using the free open-source software MicMac[END_REF][START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF]Rupnik et al., , 2016)). This method preserves the input image resolution of 0.5 m through the correlation process and allows us to measure surface displacements with a detection threshold down to 5 cm (~1/10 th of the image resolution) for the most favorable signal to noise ratios (Delorme et al., 2020;[START_REF] Rosu | Measurement of ground displacement from optical satellite image correlation using the free open-source software MicMac[END_REF]. Prior to correlation, to correct for distortions related to variable viewing angles and topography, each image was orthorectified using a digital surface model (DSM) with a resolution consistent with the images. Independent DSMs for the pre-and post-earthquake scenes were used to avoid orthorectification errors due to the coseismic displacement between the pre-and post-earthquake image acquisitions.

For the pre-earthquake dataset, we ran image orthorectification with two different preearthquake DSMs. We compared the correlation results obtained with the two methods for each image tile, and we kept the results showing lower orthorectification artefacts (Fig. S4.2). On one hand, using MicMac, we calculated a 0.5-m resolution pre-earthquake DSM from the combination of diachronic WorldView and Pleiades images (Tab. S4.1) .

On the other hand, we used a 2-m resolution pre-earthquake DSM from Willis et al. The sub-pixel correlation was performed between all possible pairs of ortho-images.

Because two to three pre-earthquake images as well as three post-earthquake images were orthorectified for each tile (Tab. S4.2), we obtained between six and nine correlation results for each displacement component (east-west and north-south). Thus, for each component, we calculated a mean displacement map from all the correlation results, with a weighting based on the correlation score map of each correlation result, to average out noise and to obtain the cleanest possible displacement maps.

Finally, we computed the difference between the pre-and post-earthquake DSMs to obtain the vertical displacement field. We used the north-south (N-S) and east-west (E-W) displacement maps computed previously to account for the horizontal displacement of the corresponding pixels between the pre-and post-earthquake DSMs. We used a bivariate spline approximation to sample the post-earthquake DSM. Provided that our horizontal displacement maps are correct, this approach ensures that we measure only elevation differences due to the vertical displacement of the earthquake and that we do not introduce biases due to the topographic mismatch associated with horizontal displacements (Delorme et al., 2020). Vertical uncertainty associated with the DSMs processed with MicMac is close to the ground resolution, which is 0.5 m (Rupnik et al., 2018). Vertical uncertainty for the 2-m pre-earthquake DSMs is not reported in [START_REF] Willis | CaliDEM: Ridgecrest, CA region 2m digital surface elevation model[END_REF]. Topographic noise in the vertical displacement maps arises from both vertical errors in the DSMs and, in some areas, an imperfect match between the 0.5-m DSMs processed with MicMac and the oversampled 2-m DSMs from [START_REF] Willis | CaliDEM: Ridgecrest, CA region 2m digital surface elevation model[END_REF].

The 14 tiles of ground displacements in the east-west, north-south and vertical components were computed independently; thus, they need to be brought together into a common reference frame and be denoised to obtain consistent displacement maps for the whole Ridgecrest rupture. For that purpose, first, the tiles were corrected one by one from spurious signals like linear stripes (CCD artefacts) and sinusoidal artefacts (residual signal from micro-vibrations of the satellite) by modeling this noise in areas that do not include coseismic signal and then by subtracting the artefacts model from the whole tile.

Next, we removed a linear ramp in each tile using neighboring tiles as a reference, assuming that the tiles must overlap in values in common areas. However, in the horizontal components of displacements, this procedure could introduce a long wavelength drift across the full mosaic, as the mosaic is built step by step by aggregation of neighboring tiles. Thus, to ensure that our correction method does not add supplementary artefacts to the final displacement maps, we compared our final horizontal maps to 10-m resolution displacement maps that we computed from correlation of Sentinel-2 optical images (Fig. S4.3). Only one Sentinel-2 optical scene is needed to cover the entire Ridgecrest area. Because the displacement maps derived from Sentinel-2 include zones with known zero deformation, they can be corrected for any large-scale drift to ensure that these zero-deformation zones are actually equal to zero. These lowerresolution data were then used as a reference to correct potential large-scale trends in our high-resolution results and to ensure that long wavelength signals in our displacement maps represent coseismic deformation and not artefacts. For the vertical component, the coseismic signal is mostly limited to the near-fault domain, so we extracted a linear ramp in each tile to remove any large-scale offset to a zero reference. Then, the tiles were merged.

We checked the low-frequency corrections applied to our displacement maps against particularly at bends and in relay zones. This type of deformation is difficult to measure in the field because it is spread across large distances from the fault (> 500 m), and as it is difficult to find specific markers at the surface that allow for measuring this part of the deformation (e.g., Rockwell et al., 2002). In the optical-based displacement maps, diffuse deformation shows up as large-scale displacement gradients and can be measured. In the example of profile BB' (Fig. 4.2B), we observe a diffuse deformation gradient at a stepover located south of the intersection between the foreshock and the mainshock rupture traces. The zone of diffuse deformation extends ~1 km to the northeast of the mainshock surface rupture (Fig. 4.2A) and accommodates up to 1 m of displacement (Fig.

4.2B).

Using the east-west and north-south displacement maps (Figs. 4.2A and 4.3A), we derived in two-dimensional (2D) the curl (anti-symmetric part of the displacement gradient, Fig. 4.3C), as well as the dilatational and maximum shear components of strain (Fig. 4.3D,E) to better image the deformation processes associated with the foreshock and mainshock ruptures. The equation for the maximum shear strain is 𝛾 = ±(𝜆 1 -𝜆 2 )/ 2(𝜆 1 𝜆 2 ) 1/2 , with 𝜆 1 as the maximum elongation and 𝜆 2 as the minimum elongation (Ramsay, 1967). The magnitude of this value does not depend upon the orientation of the shear. These calculations were performed using MATLAB codes by [START_REF] Allmendinger | Structural Geology Algorithms: Vectors and Tensors[END_REF] to calculate the Lagrangian finite strain tensor from horizontal displacement gradients. Before performing the strain calculations, we downsampled the displacement maps to consider only deformation patterns coherent over at least a few meters, and to reduce correlation noise [START_REF] Cheng | Permanent Co-Seismic Deformation of the 2013 Mw7.7 Baluchistan, Pakistan Earthquake From High-Resolution Surface Strain Analysis[END_REF]. Downsampling the displacement maps is equivalent to applying a low-pass filter because we consider an "averaged" displacement value over a 2-m or 5-m window using the nearest neighbor epicentral area. This value is 2 orders of magnitude greater than the maximum possible strain, 0.5%, which can be accommodated by elastic deformation [START_REF] Lockner | A generalized law for brittle deformation of Westerly granite[END_REF]. This 0.5% threshold corresponds to the elastic limit of rocks, calculated as the mean ratio of yield stress to Young's modulus. So, any strain larger than this 0.5% threshold must correspond to inelastic strain. In our case most of this inelastic deformation actually corresponds to brittle deformation along faults. This 10% value is in agreement with results from Barnhart et al. (2020) who also measured surface deformation using image correlation for the Ridgecrest earthquakes. However, unlike Barnhart et al. (2020), we do not measure significant dilatational strain across the fault zone (Fig. 4.3E). Instead, similar to what was found by [START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF], our deformation signal is dominated by rotation and shear strain (Fig. 4.3C,D). This difference in the dilatational strain might be related to the regularization while inverting the surface displacement fields in Barnhart et al. (2020). Conversely, our method directly derives 2D strains from the surface displacement fields, without performing any inversion.

To the north of the mainshock rupture, we identify a set of faults that exhibit changes in strike from N42° to N15° when moving northward (Figs. 4.2A and 4.3A). These faults form a right-stepping connection with the Coso Basin located to the north of the Ridgecrest surface rupture [START_REF] Jobe | Evidence of Previous Faulting along the 2019 Ridgecrest, California, Earthquake Ruptures[END_REF]. They have a similar orientation to the pre-2019 earthquake structures [START_REF] Jobe | Evidence of Previous Faulting along the 2019 Ridgecrest, California, Earthquake Ruptures[END_REF] that relate to the ENE-WSW (N65°-N75°) extension in the Coso Basin [START_REF] Roquemore | Structure, tectonics, and stress field of the Coso Range, Inyo County, California[END_REF]. Just south of this set of faults, near the northern border of China Lake, we observe a 5-km large pull-apart basin bounded by two faults that accommodate slip both in the horizontal and vertical components. This basin is located at a bend in the rupture trace, with the main fault azimuth changing from N140°

to N152° in the southward direction. This basin shows evidence of long-term subsidence [START_REF] Jobe | Evidence of Previous Faulting along the 2019 Ridgecrest, California, Earthquake Ruptures[END_REF]. It also represents the largest subsidence feature along the Ridgecrest rupture, with a maximum vertical displacement of -2 m at its center (Fig. ). This is also consistent with the subsidence that we measured in the area (Fig. 4.4C). Compressional strain is conversely measured in the small compressive jog between fault strands 10 and 11 (Fig. 4.4F). Figure 4.4F shows, however, that for sections with an azimuth close to N152 (mean azimuth in this area of the rupture), the dilatation pattern is mostly close to zero, alternating frequently from dilation to compression. [START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF] suggested that this pattern could be a consequence of variations in the curvature of the fault at length scales of hundreds of meters, which represent small alternate transtensional and transpressional bends. We propose, at a much smaller scale, that pixel-by-pixel discrepancies in the E-W and N-S displacement maps in the fault zone can result in such high-frequency variations in the dilatational strain.

In addition to complex localized deformation, the epicentral area is also characterized by some diffuse deformation. One example is the potential blind fault identified to the northeast of the area imaged in Figure 4.4A. At this location, a fault was neither mapped in the field nor could a localized offset be detected in the displacement map. However, we could measure about 0.10 m of right-lateral diffuse deformation across that feature. Diffuse deformation pattern is also visible in the displacement maps between fault strands 5, 7, and 8 (Fig. 4.4A), and it appears as negative rotation (anticlockwise) in the curl map (Fig.

4.4D,G

). This pattern occurs in response to right-lateral slip on subparallel fault strands 5, 7, and 8.

South of the mainshock epicentral area, a large secondary subparallel fault, located east of the primary fault trace (Fig. 4.5B), corresponds to a right-lateral fault that ruptured twice at depth during the foreshock and the mainshock earthquakes [START_REF] Huang | Spatio-temporal foreshock evolution of the 2019 M 6.4 and M 7.1 Ridgecrest, California earthquakes[END_REF]Ross et al., 2019;[START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF]. Even if there is no unambiguous evidence that slip reached the surface on this NW-striking fault during the foreshock (DuRoss et al., 2020;Milliner and Donnellan, 2020), this rupture is located above one of the seismic sub-events forming the foreshock (Ross et al., 2019). Between this fault strand and the primary fault strand, we observed a small area that moved west while the rest of the block moved east (Fig. with the right-lateral sense of motion of the mainshock fault and suggests that the foreshock rupture propagated on two distinct structures on either side of the mainshock causative fault. One possible explanation is that the fault that was activated during the mainshock is likely more active and accumulates larger deformation at the fault zone, than the one activated during the foreshock, since the main fault is more closely aligned with the current regional deformation pattern. Thus, the main active fault zone may represent a barrier for the propagation of the foreshock rupture.

In this area, the displacement associated with the mainshock rupture is, at a first order, accommodated by one primary fault (Fig. Profile DD' (Fig. 4.7B), which goes across a small compressive bend that is located along the foreshock rupture, shows deformation localized on three fault strands for the faultparallel component. These fault strands are separated from each other only by ~120 m, and no significant diffuse deformation is accommodated between the localized ruptures.

However, shortening and vertical displacements occur only on the two fault strands located to the south, while the northern strand accommodates only strike-slip motion.

This suggests that some partitioning occurred between the fault-parallel and fault-normal displacements throughout the fault system.

To quantify along-strike variations of surface displacements, we extracted 123 and 460 swath profiles across the surface ruptures associated with the Mw6.4 foreshock and the Mw7.1 mainshock, respectively. Swath profiles are similar to those from Figure 7: 100m wide and 8-km long (Fig. 4.7), perpendicular to the local strike of the rupture (Fig.

4.2)

. Successive profiles are spaced every 120 to 150 m. For each profile, the total surface displacement was estimated by measuring the offset at the fault location between linear regressions fitting respectively the displacements on either side of the fault zone (Figs.

4.7 and S4.5). The parameters of each regression were tuned to fit the greatest arctangentshape or linear-shape gradients and accommodate the variable width of the fault zone that ranges from highly localized (30-40 m) to widely distributed (~2.5 km). In addition, each profile was manually inspected and adjusted to make sure that each fit is not biased by spurious data related to noise in the correlation maps. The total displacement measured on the profile includes either localized slip on faults or diffuse deformation, or both, as

shown in profile CC' (Fig. 2020) reveals that our measurements are generally larger than prior work. This result also applies to previously published higher resolution optical image-based studies [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF][START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF]Fig. S4.6). A first explanation, regarding results from Milliner and Donnellan (2020), is that they use lower-resolution images combined with large correlation windows (input image resolution is 3-5 m and output resolution of the displacement maps is 87 m), which might result in averaging out the displacement signal over large areas, leading to an underestimation of the peak signal. A second explanation, regarding all optical-based results, relates to the method used to measure the total displacement. In this study, we include diffuse deformation gradients taken over a wide aperture (>500 m) that might arise from variations of slip on the fault plane at shallow depth. As described previously in this study, diffuse deformation might include the elastic response of the medium to subsurface slip, in addition to surface inelastic deformation. Thus, our data represent the maximum displacement envelop for this rupture. It includes all the measurable surface displacements in the rupture area, up to a few kilometers away from the primary fault, without distinction on the type of deformation. Other studies do not incorporate these large-scale gradients into their surface slip estimations. They focus on fault displacement and local off-fault deformation over apertures of a few hundred meters [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF][START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF]Milliner and Donnellan, 2020), and provide estimations of the surface inelastic deformation along major faults. Finally, regarding the field observation, as already observed for previous events (Klinger et al., 2005;Milliner et al., 2015;[START_REF] Zinke | Three-Dimensional Surface Deformation in the 2016 MW 7.8 Kaikōura, New Zealand, Earthquake From Optical Image Correlation: Implications for Strain Localization and Long-Term Evolution of the Pacific-Australian Plate Boundary[END_REF], the field data usually only include offsets along localized deformation structures. In fact, except in unusual cases where long piercing lines are available (Rockwell et al., 2002;[START_REF] Rockwell | Surface rupture and slip distribution of the 1940 Imperial Valley earthquake, Imperial fault, southern California: Implications for rupture segmentation and dynamics[END_REF], it is difficult to incorporate diffuse deformation into field measurements. Thus, the field data generally report a minimum measurement of fault displacements compared to a more integrated measurement performed by image correlation (Delorme et al., 2020).

4.9C

). This local increase in horizontal slip is also visible the displacement maps from Barnhart et al. (2020). We could not rule out the effect of the prominent bedrock ridge east of fault strand 5 on the orthorectification process due to potential inaccurate topographic determination during the DSM calculation. Our measurements of the vertical motion in this area through the DSM differencing, however, are consistent with field observations (30-50 cm;[START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF], making the case of spurious topography unlikely. The fact that this excess slip is located along the section of fault that ruptured during the foreshock earthquake may not be a coincidence. We could hypothesize the right-lateral pre-rupture of fault strand 16 at depth during the foreshock earthquake [START_REF] Huang | Spatio-temporal foreshock evolution of the 2019 M 6.4 and M 7.1 Ridgecrest, California earthquakes[END_REF]Ross et al., 2019;[START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF] despite the fact that their orientation would be well suited for normal motion associated with the extensive Coso area. Instead, the curl map (Fig. 4.10C) shows that the displacement on these northeast-trending faults corresponds to left-lateral strike slip. This is consistent with their orientation, roughly parallel to the orientation of the foreshock.

Measurement of the displacement in the direction parallel to these faults (Fig. 4.10D) reveals that each fault has accommodated between 0.1 and 0.5 m of left-lateral motion.

However, taken together, the cumulative left-lateral displacement across the whole fault set is nearly zero (Fig. 4.10D), suggesting that this left-lateral motion is mostly related to individual block rotation inside the dextral shear zone corresponding to the northernmost extent of the mainshock rupture, following a bookshelf faulting pattern [START_REF] Tapponnier | Bookshelf faulting and horizontal block rotations between overlapping rifts in southern Afar[END_REF][START_REF] Wesnousky | Active faulting in the Walker Lane[END_REF].

Similar conclusions have been reached by [START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF] who used 6-m resolution horizontal displacement and strain maps to characterize the bookshelf faulting behavior.

They propose bookshelf faulting to occur beyond fault tips to allow transitions between distinct right-lateral faults, as long as clockwise rotation and left-lateral shear are kinematically equivalent to regional dextral shear [START_REF] Platt | Kinematics of rotating panels of E-W faults in the San Andreas system: What can we tell from geodesy?[END_REF]. We propose, in addition, that bookshelf faulting can arise from the interaction between diffuse shear and preexisting structures in the crust, as it has also been suggested for the 1986 Mw5.7

Mount Lewis earthquake [START_REF] Kilb | Implications of diverse fault orientations imaged in relocated aftershocks of the Mount Lewis, ML 5.7, California, earthquake[END_REF]. Indeed, the similarity in direction between the foreshock rupture (Fig. 4.2A) and the strike of relocated aftershocks clusters between 3 and 9 km depth (Ross et al., 2019;[START_REF] Shelly | A High-Resolution Seismic Catalog for the Initial 2019 Ridgecrest Earthquake Sequence: Foreshocks, Aftershocks, and Faulting Complexity[END_REF], suggests that this northeast direction might correspond to some geological fabric affecting the basement at a regional scale. Although we do not have geological evidence of a predominant northeastdirected crustal fabric in the area, clusters of background seismicity with bimodal orientation (north-east and north-west) were evidenced in this region before the rupture [START_REF] Fialko | Simple shear origin of the cross-faults ruptured in the 2019 Ridgecrest earthquake sequence[END_REF]. However, the locations of these northeast-trending faults identified at the surface, with the exception of the foreshock rupture, do not directly correlate with the northeast-trending clusters of aftershocks at depth. Northeast-trending faults identified in this study are short (0.1 to 3 km) and located close by within specific deformation zones (Figs. 4.2A,4.6 and 4.10). In contrast, the clusters of aftershocks are scattered all along the rupture.

At the northern end of the mainshock, the specific geometry of the northeast-trending cross faults might be indicative of the rupture process itself. Some of the faults have a curved geometry, while others have a sigmoidal shape with the northeastern part of the fault strands bent toward the south (northeast of the HH' profile in Fig. 4.10A), as illustrated by fault strands 2, 6, and 7 in Figure 4.10A. This is consistent with the deformation expected from dextral shear along the azimuth of the main rupture (N140°).

It also implies that these cross faults were possibly activated either before or during the rupture period when the area went under dextral shear deformation. A possible scenario is, although the main rupture did not reach to the surface, that the dynamic stress pulse located ahead of the rupture propagating northward triggered motion along this preexisting fabric (Okubo et al., 2019;[START_REF] Poliakov | Dynamic shear rupture interactions with fault bends and off-axis secondary faulting[END_REF][START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF] and led to the left-lateral displacement visible at the surface. In a second stage, these left-lateral fault strands were sheared right laterally, producing the sigmoid shape fractures visible in the displacement maps, as it has already been suggested for other evens (Klinger et al., 2005). Because the dextral fault never reached the surface, the sigmoid shape is still well preserved.

In the displacement maps, we can see that these faults are asymmetric with respect to the location of the right-lateral motion (Fig. 4.10A-C). The faults extend farther away on the northeastern side of the zone of dextral shear. The difference in length might be interpreted as a direct mechanical response of the medium to the location of the tensional lobe of the rupture on the eastern side of the northeast-trending fault system (Fig.

4.10A),
where opening of faults is facilitated [START_REF] Kim | Fault damage zones[END_REF]Okubo et al., 2019). This asymmetric pattern may be enhanced by the different mechanical response of the surficial sediments themselves, formed mainly by lacustrine/alluvium deposits to the east of the rupture, while the western side is formed by more indurated alluvial fan sediments and lake deposits overlying uplifted basement seating at shallow depth [START_REF] Jennings | Geologic map of California : Trona sheet[END_REF]. The area located between 10 and 20 km south of the epicenter, between sub-events 2 and 3, shows limited slip at depth (Fig. 4.11B). At the surface, we measure lower fault slip and larger diffuse deformation, up to 50% of the total slip measured (Fig. 4.11A). This area of decreased slip is located just south of the Mw6.4 left-lateral foreshock rupture.

Coulomb stress models show that this area experienced a negative stress change of 1 MPa after the foreshock rupture (Wang et al., 2020). The negative Coulomb stress change generated by the foreshock could have inhibited slip on the mainshock rupture but likely favored diffuse deformation in the surrounding medium as rocks are weaker under tension (Okubo et al., 2019;Sammis et al., 2009). The influence of the foreshock rupture on the mainshock displacement pattern could explain the sharp decrease in total surface displacement measured at the transition between segments 2 and 3 (Fig. 4.8B).

The fact that the kinematic models fit our data supports our strategy to measure surface deformation. Numerical models consider the bulk medium as elastic, so part of the diffuse deformation we measured using large-scale (>500m) gradients might include some elastic deformation generated by subsurface fault slip variations. However, without a modeling strategy that includes our near-fault data in the inversion process, we cannot quantify the amount of elastic vs inelastic deformation that occurred at the surface.

In attempt to qualitatively assess the importance of possible inelastic contribution, we predicted the elastic surface displacement associated to a blind rupture (Fig. 4.12). On one hand, we used the results of the kinematic inversions as an input for our elastic model and compare the result to our displacement data. The fault extends from 1.5 km below the surface to 7 km deep with a uniform slip of 3 m, and dips either at 90° or at 70° SW (red and orange curves, Fig. 4.12). With this strategy, we could not fit the data in the near field, although amplitudes of the displacements away from the fault are similar. On the other hand, we directly fitted the data by assigning the fault at a depth of 500 m (blue curve, Fig. 4.12). In this case, although amplitudes of the displacements away from the fault do not correspond, we could fit the displacement gradient within the fault zone. [START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF] used this same strategy to fit the surface displacements 5 km to the south-east, still in the area of bookshelf faulting and diffuse deformation, and obtained results for a fault located 350 m below the surface. However, elastic strain of magnitudes as high as 10 -3 and 10 -4 (0.5-0.05%) are unlikely to be accumulated over months within the last hundred meters of the crust (post-earthquake acquisitions occurred 1 to 2 months after the rupture, Tab. S4.1) and would be released within the early post-seismic phase [START_REF] Brooks | Rapid Geodetic Observations of Spatiotemporally Varying Postseismic Deformation Following the Ridgecrest Earthquake Sequence: The U.S. Geological Survey Response[END_REF][START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF]. So, it is likely that diffuse deformation represents inelastic permanent deformation. ). We showed that diffuse deformation is principally located along the northern and southern segments of the rupture (81% along S1 and 39% along S3). In contrast, in the central part of the rupture, which includes the epicentral region, we measured lower diffuse deformation (18%). The northern and southern areas are characterized by a significant shallow slip deficit (SSD; [START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF] according to the geophysical models (S1 and S3, Fig. 4.11B). Maximum slip at depth along S1 and S3 ranges between 4 and 6 m, while surface displacements reach a maximum of 2 m. Along these segments, strain distributes within the medium to accommodate the variations of fault slip toward the surface, generating diffuse deformation gradients at the surface. Using our observations only, we cannot quantify the part of deformation that is elastic and inelastic.

However, we can measure the spatial extent of the surface deformation zone, referred to as fault zone width (FZW; Figs. 4.13A and S4.9). This enables us to infer the volume of rock that is affected by diffuse deformation and determine the depth to which the SSD extends along the rupture.

The FZW is defined as the central area where deformation departs from the far-field gradients taken as references for the linear regressions on each side of the fault (Figs.

4.7A and 4.12 ). Along the mainshock surface rupture, the maximum fault zone width measured in our study reaches 4 km, and the mean FZW averages 2 km (Fig. 4.13A); thus, we measure deformation processes occurring within the first 2 to 4 km of the crust [START_REF] Segall | Earthquake and Volcano Deformation[END_REF]. Along the foreshock rupture, the FZW averages 680 m (Fig. S4.9). Hence, processes impacting the foreshock near-fault surface deformation patterns might occur at very shallow depths, likely a few hundred meters. Measurements of FZW from this study are compared to regional FZW reported in [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF], in parallel with the total slip budget from this study (Fig. S4.10).

We use elastic modeling to infer the potential depth of the source of the displacements we image at the surface. We model surface displacements associated with a 4-m rightlateral strike-slip dislocation in an elastic medium (Fig. 4.13B). The bottom of the fault is fixed at 15 km depth, and the top of the fault is allowed to vary from 4 to 0 km, 0 km corresponding to a surface rupture. Using these simulations, we show that the wavelength of the surface elastic deformation associated with slip ending at a depth greater than a few kilometers is too long to significantly impact the amount of total displacement measured using 8-km long profiles. This confirms our suggestion that the deformation signal discussed here mostly relates to surficial deformation and does not result from elastic deformation associated with slip at large depths (>2 km). New measurements presented in this study fill the gap between InSAR data, which image the regional displacement field and mostly bring information on fault slip at depth (>1-2 km depth), and field and optical data that focus on fault offsets and off-fault deformation at local scale (<500 m). In this study, we measured both a total displacement curve, which reflects slip at a shallow depth (<~2 km), and a fault slip curve that reflects localized fault deformation that reached the surface along the 20 fault strands that constitute the mainshock surface rupture. Diffuse deformation reflects the discrepancies between slip at shallow depth and surface slip. Hence, the fact that the kinematic slip predictions fit our data supports the existence of a shallow slip deficit along segments 1 and 3 of the mainshock rupture. The presence of a SSD is consistent with the magnitude of the mainshock earthquake. [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF] proposed that slip deficit on strike-slip faults most often occurred for earthquake of magnitudes lower than Mw7.5, based on comparisons with slip models on various earthquakes and on the observation that no SSD was modeled for very large events such as the 2001 Mw7.8 Kokoxili earthquake, the 2002 Mw7.9 Denali earthquake, or the 2013 Mw7.7 Baluchistan earthquake.

Similar to the mainshock surface rupture, we measured a mean diffuse deformation of 31% along the foreshock surface rupture (Fig. S4.7). This value of 31% of diffuse deformation differs greatly from the ~59 and 56% of off-fault deformation (OFD) measured, respectively, by [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF] and [START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF]. This difference is mainly due to the fact that fault damage occurring in the 10 to 100 meters around the fault is included within the localized fault slip budget in our study. Hence, diffuse deformation measured in this study refers only to large-scale (>100 m) displacement gradients. Also, using our displacement maps, we could measure slip on a larger number of fault strands (Fig. S4.8) than those observed in the field. This leads to a smaller estimation of diffuse deformation for the foreshock rupture.

Apart from these considerations, we also observe that the FZW from the mainshock earthquake is not impacted by the transition from segment 2 to 3 and by the junction with the foreshock rupture (Fig. 4.13A). Continuity is observed in the FZW along the northeastern side of the fault zone, from fault strand 16 to 19, which have a similar azimuth but are ~10 km apart from each other. A similar observation is made along the southwestern side of the fault zone, between fault strand 18 and 20 that are ~7 km apart.

Thus, it seems that the fault zone at depth is not impacted by the junction with the foreshock rupture. This supports our previous observation that the foreshock rupture is discontinuous across the mainshock and that the mainshock rupture constitutes the major structure of the 2019 sequence and a barrier to the propagation of the foreshock rupture.

Conclusions

In this paper, we presented a dataset of surface displacement measurements for the 2019 Ridgecrest (California) earthquake sequence, derived from correlation of 0.5-m resolution optical satellite images. We measured details of the surface displacements associated with both the foreshock and the mainshock earthquakes. This work highlights the respective importance of fault complexity and diffuse deformations in surface rupture processes. We show that slip measured in the field along primary surface ruptures underestimates the total amount of displacement. Diffuse deformations, which include both elastic and inelastic processes, can account for a significant part, up to 100%, of the total surface deformation in areas where the earthquake rupture barely reaches the surface. When considered along the entire length of the foreshock and mainshock ruptures, the diffuse deformation represents 31% of the total surface displacement budget.

Localized deformation on small secondary faults, when considered together, also accommodate a significant part of the surface displacement budget, up to 50% in some areas of the rupture. The spatial distribution of the diffuse deformation and of the localized deformation on faults emphasizes lateral variations of the rupture process reflecting structural fault segmentation. These variations are primarily related to the geometry of the fault, which in turn depends on a fault's seismic history, the long-term evolution of the major geometrical complexities with cumulative slip, and interactions with preexisting structures from the geological fabric [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF][START_REF] Lefevre | Slip deficit and temporal clustering along the Dead Sea fault from paleoseismological investigations[END_REF]. Comparisons with other optical-based results and kinematic slip inversions enable us to corroborate the fact that SSD did occur along the foreshock and mainshock ruptures of the Ridgecrest sequence. This SSD affects the first ~2 km of the crust and is responsible for part of the diffuse deformation we measured to the surface. However, it is not yet possible to discriminate with certainty between the elastic and inelastic contributions of the diffuse deformation using the currently available numerical models. Inelastic deformation participates to the budget of finite deformation for this earthquake and it must be taken into account when estimating the recurrence and future magnitudes of earthquakes in the area. It also indicates that deformation distributes over a wider area than what is directly visible to the surface. If the deformation is elastic, it means that elastic strain has accumulated during the rupture, which should be released during the next seismic cycle.

Data and Resources:

The Reitman, an anonymous reviewer, and Rui Chen associate Editor for insightful comments that improved this manuscript. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

Supplemental materials

The supplementary materials present, at first, elements on methodology we used to generate the horizontal and vertical displacement maps. We provide the references for the [START_REF] Thomas | Effect of Brittle Off-Fault Damage on Earthquake Rupture Dynamics[END_REF]. We also present the map of the fault zone width for the foreshock rupture (Fig. S4.9). Finally, we plot the fault zone width measured in this study against the total displacement budget from this study and the regional fault zone width measured in Gold et al., 2021 (Fig. S4.10).

Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government. S4.1. Pre-earthquake DSM used for the orthorectification of pre-earthquake images is indicated for each tile. "0.5 m" refers to DSMs calculated in this study using the pre-earthquake dataset from this study and the MicMac correlation method. "2 m" refers to the 2-m ground resolution DSM from [START_REF] Willis | CaliDEM: Ridgecrest, CA region 2m digital surface elevation model[END_REF].

Supplemental Figures: Diffuse deformation varies substantially following the complexity of the fault system.

Average value of diffuse deformation along the foreshock rupture is 31%. Along the mainshock rupture, from 10 to 25 km, regional FZW from [START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF] is below 300-400 m while we a FZW averaging 2 km. In fact, in this area, and as shown in Figure 13A, diffuse deformation extends around 2 km within the northeastern block of the rupture; even so, no secondary rupture is visible at the surface. Both along the foreshock and mainshock ruptures, there is no evident correlation between the FZW and the total surface displacement budget, although lowest values of FZW along the mainshock rupture occur just north of the mainshock epicenter.

Joint inversion for the rupture kinematics of the 2019 Ridgecrest earthquake sequence, with GNSS, InSAR and high-resolution optical imagery offset data: rupture complexity and diffuse deformation

This work was done in close collaboration with K. Wang (UC Berkeley, California, USA), who provided numerical codes which allow to jointly invert GNSS, InSAR and optical data, and try to resolve fault slip at depth. These codes were initially used to invert GNSS and InSAR data only (Wang et al., 2020;[START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF] So far, coseismic fault slip models are principally constrained using InSAR, GNSS, and seismic data [START_REF] Avouac | The 2013, Mw 7.7 Balochistan earthquake, energetic strike-slip reactivation of a thrust fault[END_REF][START_REF] Grandin | Rupture process of the Mw = 7.9 2015 Gorkha earthquake (Nepal): Insights into Himalayan megathrust segmentation[END_REF][START_REF] Jolivet | The 2013 Mw7.7 Balochistan earthquake: Seismic potential of an accretionary wedge[END_REF]. However, these inversions lack data to constrain the displacements within the first kilometers around the faults. It leads to large bias in the estimation of the fault rupture patterns close to the surface, which also impacts on the slip distribution at larger depths [START_REF] Marchandon | Origin of the shallow slip deficit on a strike slip fault: Influence of elastic structure, topography, data coverage, and noise[END_REF][START_REF] Xu | Refining the shallow slip deficit[END_REF]. Our understanding of rupture process is subjected to the variability in the modelling strategies, and so are the ground motions predictions [START_REF] Beresnev | Uncertainties in Finite-Fault Slip Inversions: To What Extent to Believe? (A Critical Review)[END_REF][START_REF] Causse | Constraining the roughness degree of slip heterogeneity[END_REF]. Following these observations, more recent models started to incorporate near-faut optical data to better resolve the rupture process at shallow depths (e.g., Delorme et al., 2020;[START_REF] Grandin | September 2005 Manda Hararo-Dabbahu rifting event, Afar (Ethiopia): Constraints provided by geodetic data[END_REF][START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF][START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF][START_REF] Xu | Refining the shallow slip deficit[END_REF]. Near-fault and dense optical data allow for assessing the detail of the fault ruptures at the surface, the distribution of slip along these faults (e.g., [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF][START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF][START_REF] Milliner | Bookshelf Kinematics and the Effect of Dilatation on Fault Zone Inelastic Deformation: Examples From Optical Image Correlation Measurements of the 2019 Ridgecrest Earthquake Sequence[END_REF][START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF][START_REF] Zinke | Three-Dimensional Surface Deformation in the 2016 MW 7.8 Kaikōura, New Zealand, Earthquake From Optical Image Correlation: Implications for Strain Localization and Long-Term Evolution of the Pacific-Australian Plate Boundary[END_REF], and the diffuse deformation in the medium [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]Figs. 3.9 and 4.9). However, the opticaldata used so far to constrain near-faut displacement inversions have ground resolutions of several meters, and present large noise compared to InSAR data (e.g. Barnhart et al., 2019;[START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF][START_REF] Xu | Refining the shallow slip deficit[END_REF]. In this study, we take advantage of the high-resolution optical (0.5 m) displacement maps from Antoine et al. (2021) (Figs. 4.2a and 4.3a,b), that offer a greater level of detail than the data used so far (see Fig. 3.6, profile BB' for an example).

In the case of the 2019 Ridgecrest earthquake sequence, there exist a wide range of coseismic slip models, and which are constrained by combinations of InSAR, GNSS, optical and seismic data (Barnhart et al., 2019;[START_REF] Chen | Cascading and pulse-like ruptures during the 2019 Ridgecrest earthquakes in the Eastern California Shear Zone[END_REF][START_REF] Gao | Coseismic faultslip distribution of the 2019 Ridgecrest Mw6.4 and Mw7.1 earthquakes[END_REF][START_REF] Goldberg | Complex Rupture of an Immature Fault Zone: A Simultaneous Kinematic Model of the 2019 Ridgecrest, CA Earthquakes[END_REF][START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF][START_REF] Li | The 2019 Mw 6.4 and Mw 7.1 Ridgecrest earthquake sequence in Eastern California: rupture on a conjugate fault structure revealed by GPS and InSAR measurements[END_REF][START_REF] Liu | Coseismic Rupture Process of the Large 2019 Ridgecrest Earthquakes From Joint Inversion of Geodetic and Seismological Observations[END_REF][START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF][START_REF] Pollitz | Kinematics of Fault Slip Associated with the 4-6 July 2019 Ridgecrest, California, Earthquake Sequence[END_REF][START_REF] Qiu | Slip Complementarity and Triggering between the Foreshock, Mainshock, and Afterslip of the 2019 Ridgecrest Rupture Sequence[END_REF]Ross et al., 2019;[START_REF] Tung | Structural Controls Over the 2019 Ridgecrest Earthquake Sequence Investigated by High-Fidelity Elastic Models of 3D Velocity Structures[END_REF]Xu et al., 2020b;[START_REF] Yue | The 2019 Ridgecrest, California earthquake sequence: Evolution of seismic and aseismic slip on an orthogonal fault system[END_REF]. At a first order, the models report a similar rupture process, with most of the slip occurring from ~3 to 10 km depth, and the maximum slip occuring in the region of the mainshock epicenter (e.g. Fig. 4.14; see Fig. 4.15a for epicenter locations).

A comparative study of several of the coseismic slip models is available in Wang et al. (2020). Slip for the mainshock rupture, that has occurred on the NW-striking fault system, is right-lateral. Slip for the foreshock rupture, that has occurred on the SW-striking fault system, to the exception of a small contribution along the NW-striking fault system [START_REF] Qiu | Slip Complementarity and Triggering between the Foreshock, Mainshock, and Afterslip of the 2019 Ridgecrest Rupture Sequence[END_REF]Wang et al., 2020), is left-lateral. In detail, the slip models propose different rupture patterns with varying distributions and amplitudes of maximum fault slip at depth. For example, the amplitude of maximum slip ranges from 4 to 10 m in the models. This variability arises from the various inversion strategies that are used, and whether the models seek to reproduce the rupture complexity close to the surface or not. A first strategy consists in using model segments that follow a first order fault geometry, extending uniformly either vertically [START_REF] Gao | Coseismic faultslip distribution of the 2019 Ridgecrest Mw6.4 and Mw7.1 earthquakes[END_REF][START_REF] Goldberg | Complex Rupture of an Immature Fault Zone: A Simultaneous Kinematic Model of the 2019 Ridgecrest, CA Earthquakes[END_REF][START_REF] Liu | Coseismic Rupture Process of the Large 2019 Ridgecrest Earthquakes From Joint Inversion of Geodetic and Seismological Observations[END_REF][START_REF] Pollitz | Kinematics of Fault Slip Associated with the 4-6 July 2019 Ridgecrest, California, Earthquake Sequence[END_REF][START_REF] Qiu | Slip Complementarity and Triggering between the Foreshock, Mainshock, and Afterslip of the 2019 Ridgecrest Rupture Sequence[END_REF]Wang et al., 2020;Xu et al., 2020b), or with a ~85°NW fault dip for the foreshock rupture (Chen et al., 2020;Wang et al., 2020), and a ~75-85°NE fault dip for the mainshock ruptures (Barnhart et al., 2019;[START_REF] Chen | Cascading and pulse-like ruptures during the 2019 Ridgecrest earthquakes in the Eastern California Shear Zone[END_REF]. These models most often propose a simple fault slip distribution similar to that described in the previous paragraphs (e.g., Fig. 4.14a).

Other models consider a more complex fault geometry at the surface, with a large number of fault segments, including smaller secondary faults, and with different dip angles [START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF][START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF]Ross et al., 2019;[START_REF] Yue | The 2019 Ridgecrest, California earthquake sequence: Evolution of seismic and aseismic slip on an orthogonal fault system[END_REF]. Some faults join at shallow depth, while some other propagate to the downdip end of the model. In this case, the rupture pattern is more complex, with a larger heterogeneity in the slip distribution both at depth and at the surface (e.g., Fig. 4.14b).

As shown in section 4. 1.6.2., all the models, whatever the inversion strategy, present a surface slip distribution (slip on the first patch of the fault model at the surface) that is similar at first order to the total surface displacement measured on optical displacement maps [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]black curve in Fig. 4.11). But, this means that diffuse deformation is incorporated to the amount of slip that is modeled on the shallow part of the fault plane. To be able to separate the contributions of faults and diffuse deformation in the inversion, near-fault and dense data are required. This will allow to model shallow slip without incorporating the diffuse part of deformation, and to analyze the complementarity between slip along the fault plane, and diffuse deformation at the surface. In this aim, in this study, we perform a joint inversion of near-fault highresolution optical maps from Antoine et al. (2021) (Figs. 4.2a and 4.3a,b) along with InSAR data from Wang et al. (2020), and GPS data.

Strategies of previous studies that included optical data

There exist several studies that used optical data to invert for the shallow part of rupture.

For example, [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF] combined 2.5 m-resolution optical displacement maps to InSAR data to constrain fault slip and fault geometry in the sub-surface of the 2013 Baluchistan rupture. But, to avoid incorporating small-scale complex deformations while inverting the large-scale rupture process, optical data were masked in the first 2 km around the faults. Using the same method, Delorme et al. (2020) inverted the surface displacements of the 2016 Mw6.5 Norcia earthquake using 3D optical displacement maps to fill the gap in InSAR data. Because the rupture is much smaller for this earthquake, they kept the data in the first 1-2 km around the faults. However, these two studied did not perform a joint inversion of optical and InSAR data with different weights. Rather, they incorporated the optical displacement maps into a unique map constitutes of InSAR and optical data. This does not allow for an independent re-sampling, and relative weighting of the data that would moderate the datasets relatively to their noise and uncertainty. Joint inversion of InSAR, GNSS and optical data was performed by [START_REF] Xu | Refining the shallow slip deficit[END_REF] for the 1992 Mw7. 3 Landers, 1999 Mw7.1 Hector Mine, and2010 Mw7.2 El Mayor-Cucapah earthquakes. Still, they considered surface fault offsets taken along 1-2 km-long across-fault profiles, rather than the entire displacement maps, which does not allow for considering the diffuse deformation patterns over wide fault zone width (FZW).

Studies that performed a joint inversion of the full optical-derived displacement field near the faults include, for example, [START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF] and Barnhart et al. (2019) for the 2019 Ridgecrest ruptures, and [START_REF] Grandin | September 2005 Manda Hararo-Dabbahu rifting event, Afar (Ethiopia): Constraints provided by geodetic data[END_REF] for the 2005 Manda Hararo-Dabbahu rifting event (Ethiopia). But, these studies use ground displacement maps of relatively low resolution, of > 5 m in [START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF], 87 m in Barnhart et al. (2019), and 128 m in [START_REF] Grandin | September 2005 Manda Hararo-Dabbahu rifting event, Afar (Ethiopia): Constraints provided by geodetic data[END_REF], and a simple fault geometry in Barnhart et al. (2019), which do not allow to retrieve details of the rupture pattern. In our work, we take advantage of the high-resolution (0.5 m) 3D optical displacement maps available for the foreshock and mainshock ruptures of the 2019 Ridgecrest earthquake [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]Figs. 4.2a and 4.3a,b), to perform a joint inversion of geodetic data on a realistic fault geometry.

Data and model geometry

We use static GNSS observations from the UNAVCO continuous stations (96 stations with an average spacing of 30 km; stations CCC, P594, P595, and P580 are located within 20 km around the ruptures and used to compare with the model predictions). Ground displacements due to both the Mw6.4 and Mw7.1 ruptures were computed by the Scripps Orbit and Permanent Array Center (SOPAC). We use the Sentinel-1 InSAR data in LOS from [START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF], downsampled with an iterative quadtree-based scheme [START_REF] Wang | Slip model of the 2015 Mw 7.8 Gorkha (Nepal) earthquake from inversions of ALOS-2 and GPS data[END_REF]. Quadtree sampling method provides a high-spatial sampling of the data where displacement gradients are high (this includes the areas close to the faults, and some regions of noise), and downsamples to lower spatial resolutions where gradients of displacements are lower [START_REF] Marchandon | Origin of the shallow slip deficit on a strike slip fault: Influence of elastic structure, topography, data coverage, and noise[END_REF][START_REF] Wang | Slip model of the 2015 Mw 7.8 Gorkha (Nepal) earthquake from inversions of ALOS-2 and GPS data[END_REF]. As a consequence, near-fault data have more weight in the inversion than those away from the faults, which are largely downsampled [START_REF] Marchandon | Origin of the shallow slip deficit on a strike slip fault: Influence of elastic structure, topography, data coverage, and noise[END_REF]. To remove the highsampling in of data in areas of noise, the iterative method uses surface data predictions from an initial slip model, to refine the data resampling in a following iteration. However, technical issues are encountered when using the iterative method for the optical data, so they are sampled only one time with the quadtree method in this version of the model (Fig. 4.15). Iterative-resampling issues will be addressed in further works. Before the resampling, the optical data are masked at the edges, in the regions >2 km away from the fault zone, because of the lower accuracy of optical data compared to InSAR in these regions. We also mask the fault trace over a 100-m large area, to avoid a potential shift between the exact location of the fault trace on the ground and our model geometry.

All the datasets considered, GNSS, optical, and InSAR, include displacements from both the Mw6.4 foreshock and the Mw7.1 mainshock earthquake; hence, the fault displacement for the two earthquakes is inverted simultaneously. GNSS data are from 4 July from 0:00 to 17:00 UTC, to 6 July from 04:00 to 24:00 UTC [START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF]. InSAR data, from the C-band of Sentinel-1 satellite, were acquired on the 3 and 4 July before the rupture (3 July ∼18:51 PDT for the ascending track A064, 4 July ∼06:52 PDT for descending track D071), and on the 9 and 16 July after the rupture (9 July ∼18:51 PDT for ascending track T064, 16 July ∼06:52 PDT for descending track T071) [START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF]. Optical images used to retrieve near-fault ground displacement were acquired from 2012 to 2018 for the pre-earthquake ones, and from 26 August to 8

September 2021 for the post-earthquake ones. Detailed information on the images are provided in the Supplementary of [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF], in section 4.1. from this thesis (Tab. S4.1). As discussed in section 4.1.4., interseismic deformation across the Ridgecrest faults does not overcome the detection threshold of our image correlation method (5 cm); however, post-seismic deformation as large as ±6 cm in LOS [START_REF] Brooks | Rapid Geodetic Observations of Spatiotemporally Varying Postseismic Deformation Following the Ridgecrest Earthquake Sequence: The U.S. Geological Survey Response[END_REF][START_REF] Wang | Co-and Early Postseismic Deformation Due to the 2019 Ridgecrest Earthquake Sequence Constrained by Sentinel-1 and COSMO-SkyMed SAR Data[END_REF]) might be included in the displacement maps. This level of accuracy, however, is beyond the level of detail we seek to reproduce in the slip model.

In the case of the Ridgecrest earthquakes, there is no need to non-linearly invert for the fault geometry since it is already documented in detail from field observations [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF][START_REF] Ponti | Documentation of Surface Fault Rupture and Ground-Deformation Features Produced by the 4 and 5 July 2019 Mw 6.4 and Mw 7.1 Ridgecrest Earthquake Sequence[END_REF] and aftershock analysis [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF][START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF][START_REF] Plesch | Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence[END_REF]. We chose a model that fits the surface trace of the 5 major rupture faults strands visible in optical displacement maps (Figs. 4.2 and 4.3).

These faults are considered as major faults since they accommodate the largest amounts of displacement and they represent the most continuous displacement features on the ground. The fault model is traced as close as possible from the field-based fault map [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF], and is constituted of 23 linear model segments. Each segment is discretized in rectangular patches with 15 patches along-dip, with an along-dip width that ranges from ~1 km at the surface to ~3.5 km at 30 km depth. Along-strike, fault patches vary in length from ~500 m at the surface to ~2-3 km at depth. This fault model considers a vertical fault geometry, even so seismic data report variations in shallow fault dip from 70 to 90 degrees. However, at depth, all the faults are sub-vertical, with a dip close to 85-90° [START_REF] Plesch | Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence[END_REF]. was made using a quadtree-algorithm [START_REF] Wang | Slip model of the 2015 Mw 7.8 Gorkha (Nepal) earthquake from inversions of ALOS-2 and GPS data[END_REF]. The data are represented by 1342 patches of size ranging from 200 m close to the fault, and in noisy areas like to the south-west of the mainshock epicenter, and to 2 km far from the fault.

Inversion method and parametrization

We use MATLAB codes from Wang et al. (2020) that we adapted to include the nearfault optical displacement maps. We calculate the Green's functions, that relate the fault slip on a single fault patch at depth to a displacement of the ground surface, for the three datasets. Before we invert the Green's functions, they are moderated using the weight matrices of the datasets. Weight matrices are calculated using the uncertainty on the measurements. Green's function are computed using a 3D Okada solution in a homogenous elastic half space [START_REF] Okada | Surface deformation due to shear and tensile faults in a half-space[END_REF]. Weight matrices allow for considering the effect of the uncertainty and noise in the data. Uncertainty on the GNSS dataset is different for each station, average uncertainty on GNSS data is 0.37, 0.23, and 0.46 cm in the east, north, and vertical components, respectively. Uncertainty on the InSAR and optical data is considered uniform across the maps, and is of 2 and 15 cm, respectively. GPS and InSAR data were both fixed with a weight of 1, and optical data with a weight of 0.5. Inverting optical data with a similar weight than InSAR and GPS gives rise to large residuals between the model and the InSAR data. On the other hand, lowering the weight of the optical data at 0.5 does not impact the surface slip distribution in the model over amplitudes larger than 10-20 cm. Large residuals with the InSAR arising when using a weight of 1 for the optical data come from the fact that optical data include inelastic deformation close to the fault. These cannot be accounted for when using elastic modeling, and so, forcing the model to reproduce the optical data leads to a bias in the larger-scale and mostly elastic deformation signal measured by the InSAR. Still, InSAR data are more sensitive to vertical deformations. By lowering the weight of the optical data, we give more credit to the vertical component of displacement, compared to the horizontal component, in the area where InSAR and optical overlap (from 0.5 to 2-4 km around the faults). This should be the focus of further work and reflection on the inversion strategy.

In addition, because of the heterogenous spatial distribution and noise of the data, regularization must be applied to reduce model instabilities. Regularization strategies between the different studies vary a lot and it results in different fault slip distributions among the models. It has been shown that regularization impacts significantly on the slip distribution [START_REF] Causse | Constraining the roughness degree of slip heterogeneity[END_REF][START_REF] Ortega-Culaciati | An EPIC Tikhonov Regularization: Application to Quasi-Static Fault Slip Inversion[END_REF]; hence, one must find the optimal regularization parameter that allows a good trade-off between a heterogenous slip model that fits best the surface displacement data, and a smooth slip model that has larger misfit to the data (e.g., [START_REF] Goldberg | Complex Rupture of an Immature Fault Zone: A Simultaneous Kinematic Model of the 2019 Ridgecrest, CA Earthquakes[END_REF]. In this study, regularization is applied through a slip positivity constrain (the fault cannot slip backward) and a second-order Laplacian spatial smoothing kernel. To assess the best regularization parameter (R), we tested its effect on the fault slip distribution at depth and at the surface (Fig. 4.16a). We also used a L-curve that shows the evolution of InSAR residuals (average of ascending and descending median residuals, see Tab. X) with increasing R (Fig. Eventually, GNSS data are sparse and present a good fit with most of the different models.

We tested R values around that used in Wang et al. (2020), which is 1x10 -4 .

Results show that, for a regularization larger 1x10 -4 , maximum slip (M) and maximum surface slip (S) converge, and remain constant around 5 m after R=2x10 -4 . The two curves converge when maximum slip occurs along the surface of the (Fig. 4.16a). This is the result of the smoothing effect of regularization, that flattens and widens the large slip patches. Since the large slip occurs at shallow depth, widening of the patch results in an increase of slip at the surface (Fig. 4.16c). When regularization is of the order of 1x10 -5 and 1x10 -6 , the maximum slip patch remains at depth, between 2 and 8 km, depending on the model. The amplitudes reach 10 m, and remains constant for a regularization below 2x10 -5 even if the spatial distribution varies (Fig. 4.16d,e). Under this same value of R=2x10 -5 , maximum surface slip starts to increase and overcome the maximum surface fault slip from field and optical measurements (horizontal black line, Fig. 4.16a). Also, regularization below 2x10 -5 leads to higher heterogeneity in the fault slip distribution, which is largely subjected to the choice of the inversion scheme [START_REF] Beresnev | Uncertainties in Finite-Fault Slip Inversions: To What Extent to Believe? (A Critical Review)[END_REF]. Hence, R=2x10 -5 offers the best representation of slip both at the surface, and at depth. The use of a L-curve, representing the InSAR residuals versus the model regularization (Fig. 4.16b), supports our choice of R=2x10 -5 , which is very close from the "knee" of L-curve, at R=1x10 -5 . After analysis of the L-curve, the value of R=2x10 -5 is preserved because it allows for a better fit with the surface displacement data than R=1x10 -5 (Fig. 4.16a), and results in an increases in the InSAR misfit of only 0.06 cm compared to R=1x10 -5 . One of them, P5 is located along F2, at the transition with F4. The other one, P6, is located in the shallow part of F1, above the location of P1 along F2. The fact that P1 stops about 2 km from the surface on F2, while P6 starts at this same depth and extends up to the surface along the sub-parallel F1 fault, is coherent with seismic data that suggest that those two fault joint at shallow depth [START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF][START_REF] Plesch | Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence[END_REF].

Interpretation of the aftershock distribution highlights a dip angle of 55° for F1, and of >80 for F2 [START_REF] Plesch | Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence[END_REF]. Considering that, in the central part of F2, the two faults are separated of 1.8 km, and using trigonometry relationships, we find that F1 and F2 connect at a depth of 2.6 km if F2 is vertical, and 3.4 km F2 dip at 80°NE (we make the strong assumption that the faults propagate linearly at depth). These depth values are coherent with the depth of P3, from 0 to 3 km. respectively (Fig. 4.18c,d). We also obtain a good fit with the GNSS data (Fig. Some areas also present large residuals because of low-frequency noise that remain in the optical data. This is the case of the south-western end of F4 in the E-W displacement map, for example. Also, a ~5 km-long area of positive residual as large as 1 m is observed north-east of F2 in the N-S optical component. This area is disconnected from the faults, and occurs until the edge of the map. These arguments, in addition that no such misfit is observed with the InSAR data (circled area,Fig. 4.18), suggest that it is noise from the data. But, the optical data were corrected with a large-scale Sentinel-2 optical displacement map, and the misfit between the optical data and the Sentinel-2 map is -0.3 m (see Fig. S3 in Antoine et al., 2021, section 4.1). Hence, the source of this noise is still unclear and needs to be further investigated.

Discussion

We described in Introduction of this works the variety of slip models that exist for the 2019 Ridgecrest earthquake sequence. We described how amplitudes and spatial distribution of slip on the fault varies depending on the model geometry and resolution, and of the data that are used. Within this variety of models, those which have a slip distribution coherent at first order with the slip distribution modeled in this work, represented by slip patches P1 to P6 (Fig. 4.17 4.14). Major differences between these two models and that from this study (Fig. 4.17 We could set up a dip of 75°NE to the faults from the mainshock in the shallow part, following observations made on the southeast portion of F1, to F2 and F3 [START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF][START_REF] Plesch | Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence[END_REF]Ross et al., 2019). This value increases linearly until an angle of 90° is reached at 5 km depth (Figure 4.19). Because of technical limitations that still need to be overcome, we could not yet input an opposite fault dip angle for the northwest portion of F1, similar to what is inferred in the seismology. Foreshock fault dip angle is fixed at 80° NW at the surface, and vertical after 5 km depth. Our tests (Figure 4. [START_REF] Freund | The mechanics of dynamic shear crack propagation[END_REF] show that shallow variations in the fault dip angle do not impact on the slip distribution and amplitude. This new model does not allow for a better fit with the surface data either, and display similar residuals than the previous model (Fig . 4.18). Variations of the dip angles from 75°NE to 85°NE on the different portions, as told by the seismology, were also tested, but this did not impact the slip model, nor or the residuals.

This shows that the slip distribution inferred using the vertical fault model (Fig. 4.17)

does not primarily depend upon the sub-surface fault geometry. Still, variations of shallow fault dip angles, and of the impact on the surface displacement predictions and misfit with the data, will be more extensively analyzed in further work. Comparison of the mainshock slip model in its shallowest part (0-1 km) with optical surface displacement data (Fig. 4.20), shows that the model allows for a good representation of the surface displacement accommodated on faults. To have a consistent comparison, we cumulated the slip curves of the sub-parallel fault strands that overlap with the same trace in the model. F1 corresponds, on the ground, to fault strands 1,3,[START_REF] Dunham | Attenuation of radiated ground motion and stresses from threedimensional supershear ruptures[END_REF][START_REF] Bhat | Off-fault damage patterns due to supershear ruptures with application to the 2001 M w 8.1 Kokoxili (Kunlun) Tibet earthquake[END_REF][START_REF] Okubo | Dynamics, Radiation, and Overall Energy Budget of Earthquake Rupture With Coseismic Off-Fault Damage[END_REF][START_REF] Okubo | Modeling earthquakes with off-fault damage using the combined finite-discrete element method[END_REF][START_REF] Das | A numerical study of two-dimensional spontaneous rupture propagation[END_REF][START_REF] Burridge | The stability of a rapid mode II shear crack with finite cohesive traction[END_REF]and 19 (Fig. 4.9a). F2 follows the trace of fault strands [START_REF] Das | GEOPHYSICS: The Need to Study Speed[END_REF][START_REF] Burridge | Admissible Speeds for Plane-Strain Self-Similar Shear Cracks with Friction but Lacking Cohesion[END_REF]and 16 (Fig. 4.9a). F3 corresponds to fault strand 20 (Fig. Total surface displacement curve inferred from optical imagery is also reported with a grey solid line.

Comparison is also shown with the geodetic model from Wang et al. (2020). The geodetic model from Wang et al. (2020) fits better with the total surface displacement curve (grey;

Fig . 4.20), meaning that it incorporates part of the diffuse deformation in the slip that is modeled on the fault plane. Differences between models in this study (Fig. 4.17) and that of Wang et al. (2020) lie in the fact that we included optical displacement maps, adjusted the model geometry to the trace of the faults in the optical maps, increased resolution of the mesh, and used a lower regularization. These characteristics allow for a more detailed representation of the slip patterns, and for a better fit with the surface displacement data.

Evolution of average fault slip with depth, including slip on both the mainshock and foreshock ruptures, is presented against models from Wang et al. (2020), and other published models reviewed in Wang et al. (2020) (Fig. 4.21). We also present the effect of regularization on the along-dip average slip distribution by plotting slip curves of our model with R=2x10 -5 , corresponding to the model from Figure 4.17, and models obtained with larger regularization (6x10 -5 and1x10 -4 ). Models obtained with lower regularization do not fit the surface displacement data (Fig. 4.16a) and are considered as unlikely. Figure 4.21 shows that larger slip magnitudes occur at depths ranging from 3 to 6 km. This is similar to other models presented in 2021). The fact that maximum slip occurs between 3 and 6 km is often observed for strike slip ruptures, with slip in the shallow portions of the faults that is smaller than slip modeled on the deeper parts (e.g., [START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF][START_REF] Gombert | Revisiting the 1992 Landers earthquake: a Bayesian exploration of co-seismic slip and off-fault damage[END_REF]Wang et al., 2020;[START_REF] Xu | Refining the shallow slip deficit[END_REF]. This is referred to as a shallow slip deficit (SSD; [START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF].

The origin of the shallow slip is widely discussed. Some propose that shallow slip deficit is accommodated through plastic strain in the off-fault medium [START_REF] Kaneko | Shallow slip deficit due to large strike-slip earthquakes in dynamic rupture simulations with elasto-plastic off-fault response: Modelling shallow slip deficit[END_REF]Roten et al., 2017). Some other showed that shallow slip is accommodated during the post-seismic phase and could account for this SSD [START_REF] Brooks | Buried shallow fault slip from the South Napa earthquake revealed by near-field geodesy[END_REF]Milliner et al., 2020). Eventually, it has been shown that the modeling assumptions on the crustal structure [START_REF] Barbot | Effect of a compliant fault zone on the inferred earthquake slip distribution[END_REF][START_REF] Marchandon | Origin of the shallow slip deficit on a strike slip fault: Influence of elastic structure, topography, data coverage, and noise[END_REF][START_REF] Ragon | Accounting for uncertain 3-D elastic structure in fault slip estimates[END_REF], the smoothing and regularization effects on the slip distribution [START_REF] Causse | Constraining the roughness degree of slip heterogeneity[END_REF][START_REF] Ortega-Culaciati | An EPIC Tikhonov Regularization: Application to Quasi-Static Fault Slip Inversion[END_REF]also Figs. 4.16 and 4.10 in this study), or the lack of near-fault data to constrain shallow slip in the model [START_REF] Marchandon | Origin of the shallow slip deficit on a strike slip fault: Influence of elastic structure, topography, data coverage, and noise[END_REF][START_REF] Xu | Refining the shallow slip deficit[END_REF], could arise in a bias in the representation of slip on the shallow parts of the crust. Using our inversion strategy, we exclude some of the possibilities above. Because we included a high-resolution optical dataset near the fault, and used a low smoothing constrain, we reduce the bias associated with data scarcity and regularization. We also reduce the bias associated with the fault geometry by using a complex fault model similar to the fault geometry observed in the optical data. We showed that the distribution and amplitudes of slip in the shallow part of the model it robust because it fits the surface displacement data However, in the model, 1.12 m of average strike-slip is obtained along the 0-1 km fault patches of fault F1 to F5 (Fig. 4.22). Differences between the two values can be explained by slip on secondary faults that are not included in the model. This concerns fault strands [START_REF] Dunham | Near-source ground motion from steady state dynamic rupture pulses[END_REF][START_REF] Thomas | Effect of Brittle Off-Fault Damage on Earthquake Rupture Dynamics[END_REF][START_REF] Templeton | Off-fault plasticity and earthquake rupture dynamics: 1. Dry materials or neglect of fluid pressure changes[END_REF]9 and 14 (Fig. 4.9). Also, under-estimation of slip along fault F2 in the model compared to optical data (Fig. 4.9) speaks in favor of a larger fault slip in the optical data compared to the model. In addition, average total surface displacement, measured from optical imagery, in the fault-parallel component (Figure 4.9b, black curve), is 1,8±0.07 m. This value includes the 1.26 m of localized slip on faults, but also 0.54 m of diffuse deformation. It is also equivalent to the amount of slip that is modeled at depth, on the fault plane. This shows that the combinations of complex faulting and diffuse deformation at the surface, allows for accommodating an equivalent amount of displacement at the surface, compared to what is modeled on the fault plane at depth. Hence, when considering surface displacements over a wide zone (~2 km), there is no SSD associated with the 2019 Ridgecrest ruptures. Finally, InSAR data used in our inversion were acquired 3 to 10 days after the rupture.

These data cover part of the near-fault domain and fit well with our model (Fig. 4. 18a,b).

Even InSAR is more sensitive to vertical displacements, it yields a centimetric precision in LOS. Hence, diffuse deformation of metric amplitudes, as measured along the northern and southern sections of the mainshock rupture, must have been captured by InSAR. This means that diffuse deformation occurred co-seismically, or within the first 3 days of the rupture.

Conclusions at this stage of the study and future improvements

In this study, we showed that the joint inversion of GNSS, InSAR and high-resolution near-fault optical data allow for a good characterization of the rupture process in the shallow crust. By testing various values of the regularization parameter in the inversion process, we obtain a slip distribution that fits all the datasets and preserves slip homogeneity along-strike and along-dip. From this model, we highlight a heterogeneous slip distribution with 6 patches of slip distributed along the major rupture plane, striking to the north-west. We highlighted a 43% decrease of slip along the fault plane in the shallow crust compared to what is modeled at depth, but which is entirely compensated by secondary faulting and diffuse deformation measured at the surface. This means that co-seismic slip occurs with the same magnitudes at depth and at the surface, even if uncertainty remains on the magnitude and distribution of slip at depth because of modeling assumption. This work shows that model parameterization and resolution, and data completeness close to the fault, highly affect our ability to retrieve the rupture pattern from depth to the surface. Detailed near-fault data, complex fault geometry, refined meshing, and low regularization were used in this study to refine the fault slip model ant depth and in the shallow crust. The model was compared to a detailed characterization of the surface rupture pattern, including localized slip on faults and diffuse deformation occurring at larger scales. Results show that, along the 2019 Ridgecrest ruptures, total surface displacement measured over a wide deformation zone (2 km on average), is equivalent to the slip modeled at depth. Hence, there is not SSD along the ruptures. We

show that this result is coherent with numerical modeling of the elasto-plastic response of the fault zone during a dynamic rupture, and that the data we use constrain the surface diffuse deformation the first three days after the rupture. Eventually, this model needs to be tested with more complex fault geometries, that is a realistic fault geometry with variations of fault slip angles with depth, and, a realistic crustal model that takes into account variations in the elastic properties of the crust. de déformation mis en jeu, mais permet tout de même une analyse de la variation de la taille de la zone de faille le long de la rupture, qui évolue entre ~100 m et ~2 km. Par ailleurs, cette étude s'insère dans un contexte scientifique différent des chapitres précédents de cette thèse, se rapprochant des domaines de la sismologie et de la mécanique des roches. Dans ces deux domaines de spécialité, ce que nous appelons de la déformation off-fault ou diffuse est appelée de manière générale endommagement. Ce terme englobe toutes les formes de déformation qui se produisent en dehors du plan de la rupture, et sera utilisé dans la suite de ce chapitre.

English introduction:

Previously in this thesis, we demonstrated the benefits of high-resolution optical image correlation to characterize static characteristics of surface ruptures after an earthquake like the fault slip and the diffuse deformation patterns. In this chapter, based on theoretical and numerical modelling, and combined with observations on natural fault ruptures, we

show that high-resolution optical image correlation opens possibilities for analyzing dynamic characteristics of ruptures. Particulary, we show how variations in width of the damage zone around the fault, that can be inferred from geodetic and seismic observations, could correspond to variations in the rupture speed from sub-shear to supershear. This work is part of a multi-disciplinary and collaborative study realized with with J. Jara, L. Bruhat, M. Y. Thomas, K. Okubo, E. Rougier, A. J. Rosakis, C. G.

Sammis, Y. Klinger, R. Jolivet and H. S. Bhat. The manuscript is currently in review at

Proceedings and is named "Signature of transition to supershear rupture speed in coseismic off-fault damage zone" (full manuscript in the supplements from this thesis).

In the following, I present my work on optical image correlation and summarize the results from the other parts. In this work, the resolution of the optical images, as well as the limitations in the numerical and theoretical calculations do not allow for differencing near-fault damage from more diffuse deformation zones. For simplicity and consistency regarding the manuscript submitted at Proceedings A, we will refer to any deformation occurring off the primary fault plane as damage.

Relation between rupture speed and off-fault damage

The development of powerful theoretical and numerical tools allows for incorporating off-fault damage in rupture models (Klinger et al., 2018;Okubo et al., 2019;Thomas and Bhat, 2018). As described in the introduction of this thesis, in section 1.2.2., off-fault damage consists in an area of plastic deformation within the volume that directly surrounds the fault [START_REF] Faulkner | On the internal structure and mechanics of large strike-slip fault zones: field observations of the Carboneras fault in southeastern Spain[END_REF]Mitchell and Faulkner, 2009). Damage accommodates a large part of the surface deformation in addition to the localized deformation that is accommodated through fault slip. The numerical models used in this study allow for assessing the rupture damage along a propagating rupture, and for assessing variations in the damage patterns in relation with the evolutions in the rupture characteristics (Okubo et al., 2019;Thomas and Bhat, 2018). From these models, it was observed a drastic reduction in damage around the propagating rupture tip when the rupture transitions from a subshear to a supershear speeds. This is explained by the fact that stresses around the propagating rupture tip depend upon the rupture speed 𝑣:

with 𝜎 𝛼𝛽 0 is the initial stress state and 𝐾 𝐼𝐼 𝑑𝑦𝑛 is the stress intensity factor which can be approximated by with 𝑐 𝑝 is the P-wave speed, 𝑐 𝑅 the Rayleigh wave speed, 𝐾 𝐼𝐼 𝑠𝑡𝑎 is the static stress intensity factor of the cracks, that is function of the current crack length 𝐿 ̂ [START_REF] Freund | Dynamic Fracture Mechanics[END_REF].

When the rupture speed increases, the dynamic stress intensity 𝐾 𝐼𝐼 𝑑𝑦𝑛 factor monotonically decreases toward zero, and so the does the stress concentration around the fault tip. 𝐾 𝐼𝐼 𝑑𝑦𝑛 is responsible for opening cracks in the off-fault domain (Thomas and Bhat, 2018). As a consequence, if 𝐾 𝐼𝐼 𝑑𝑦𝑛 decreases, less damage is triggered in the off-fault domain compared. For strong velocity changes, like a transition from subshear to supershear speeds, it was observed that the area of damage around the fault drastically reduces (Okubo et al., 2019;Thomas and Bhat, 2018). This pattern is observed all along the transition zone, which is of the order of 5 km in the models.

In the following we show that optical image correlation allows for detecting such changes in the damage zone width, similarly to the work that was done on OFD and diffuse deformation for the 2013 Baluchistan and 2019 Ridgecrest ruptures. We apply the optical image correlation method to measure the displacement field associated with the 2001

Mw7.8 Kokoxili rupture (China), which allows for an accurate localization of supershear transition zone for this rupture. The 2001 Mw~7.8 Kokoxili (China) earthquake is a strike-slip event that ruptures ~450 km of the Kunlun fault. Mean rupture speed of this earthquake ranges between 3.3-3.9 km/s, which is larger than the shear wave speed, and hence reported as a supershear rupture (Bouchon and Vallée, 2003;Robinson et al., 2006). This earthquake was well recorded by satellite images, allowing us to investigate the spatial distribution of coseismic off-fault damage by focusing on the fault zone width. We use SPOT-1 to SPOT-4 images, covering the 2001 Kokoxili rupture area from 1988 to 2004 with a ground resolution of 10 m, allowing for change detection of 1 m [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]Delorme et al., 2020;Vallage et al., 2015). The maximum strike-slip displacement measured at the surface is around 8 m (Klinger et al., 2006(Klinger et al., , 2005)). Fault zone widths observed on the field are at least few tens to a few hundreds of meters (Klinger et al., 2006(Klinger et al., , 2005)). Thus, these observations allow us to detect fault zone width variations from satellite image correlation for this earthquake.

Prior to correlation, the SPOT images were corrected from viewing angle and topography using a unique 90 m SRTM digital surface model. The pre-and post-earthquake corrected paired images were correlated using the MicMac package [START_REF] Rosu | Measurement of ground displacement from optical satellite image correlation using the free open-source software MicMac[END_REF][START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF], which measures the horizontal surface displacement between the acquisitions (Fig. 5.1). The MicMac method enables preserving the resolution of the input images, that is 10 m, since it measures the displacement at every pixel location in the preearthquake image. Size of the pixel pattern that is considered in the pre-earthquake images and looked for in the post-earthquake image is 5 pixels, so 50 m. A regularization of 0.3 is applied to oblige MicMac to consider realistic values of displacement relatively to the neighboring pixels, even if this means considering pixels with a lower correlation score (with a regularization of 0, MicMac takes the result with the best correlation score without regards on the coherence with the surrounding pixels). We place 460 stacked profiles perpendicular to the fault trace in the displacement map every 500 m to analyze variations in the surface displacements and fault zone width, with the same method as in sections 3.1.4.2. Stacked profiles are rectangles of 40~km-long and 1~km-wide in which individual slip profiles are stacked, reducing noise and artifacts (Fig. 5.2). On each profile, displacement is translated in the direction parallel to the fault, and we observe the coseismic offset produced by the earthquake. This allow us to evaluate the width of the fault zone in the strike-slip direction over 396 clear-enough profiles (Fig.

5.2)

. Fault zone widths observed on the field are at least a few tens to a few hundred meters (Klinger et al., 2006(Klinger et al., , 2005)), hence, using maps of displacement with a ground resolution 10 m, we infer we can detect fault zone width variations from satellite image correlation for this earthquake. The analysis of the profiles highlights three main domains in fault zone width. The first domain, which represents the majority of the rupture, exhibits a mean value of 238±80m, and it corresponds to the strike-slip section of the rupture where only one fault zone is observed at the surface (Figs. 5.3B, 5.4A and 5.5). The second domain presents a width larger than 1km lasting for about 15km. This section shows a complex rupture pattern, where the rupture splits into two fault zones evolving into slip-partitioning between pure strike-slip along the southern fault strand and pure normal faulting along the northern strand. The slip-partitioning lasts for about 70 km eastward (Fig. 5.3A,B; Klinger et al., 2005). The third domain is located right before the onset of the slip-partitioning region, where the average fault zone width drops to 127±39 m over ~11 km long segment (Figs.

5.3B, 5.4B and 5.5). This value is about half of the mean fault zone width of 238 m calculated for the whole fault zone (ignoring the slip-partitioning area). This area colocates with the region where the rupture is inferred, from teleseismic data, to have transitioned to the supershear regime (Vallée et al., 2008). With these observations, and considering the theoretical analysis and numerical modeling conducted in the previous sections, we interpret this localized fault zone width reduction as a shrinkage of the offfault damage zone associated with the rupture transitioning to supershear speeds. Hence, this characterized region represents a natural observation of the supershear transition zone. start to observe a localized fault zone relatively to the area more to the west. The location of this profile correlates with the ending point of the push-up identified as a potential trigger for supershear transition (Vallée et al., 2008). East of profile 168, the localized fault zone width is indicated by light-red dashed lines (Fig. 5.4B). West of profile 168, the fault zone width is not well defined, as it can be seen in the optical images (Fig. 5.4A).

Moreover, the beginning of the supershear transition zone inferred from optical image correlation (Fig. 5.3B) correlates with the location of a push-up in the rupture zone (Figs.

5.4B and 5.5). Observations (Robinson et al., 2006;Vallée et al., 2008) and numerical models (Bruhat et al., 2016;Hu et al., 2016) suggest that geometrical complexities impact on the rupture velocity, and can trigger transitions of the rupture from subshear to supershear velocities, and vice versa. The example of this push-up shows how fault geometry can influence the rupture propagation and the coseismic deformation pattern.

The optical image correlation results (Fig. 5.3) are coupled with observations of aftershock distributions for the 2013 Mw7.5 Craig (Alaska), the 2002 Mw7.8 Denali (Alaska), and 1999 Mw7.4 Izmit (Turkey) earthquakes, which are known to be partially supershear. The analysis of the aftershock distribution was produced by other authors from this study, and is not part of the work made in this thesis. In fact, early aftershocks after a rupture are mainly governed by the state of stress left in the wake of the rupture [START_REF] Hardebeck | The static stress change triggering model: Constraints from two southern California aftershock sequences[END_REF]. Since damage is directly related to the state of stress around the rupture tip, as explained previously in section 5.1., it is investigated whether the aftershock distribution can highlight variations in the width of the damage zone around the fault. As a consequence, it could allow to refine the location of the supershear transitions along the fault rupture. Results shows that kilometric gaps in the aftershock distribution of these ruptures correlate with the regions of supershear transitions inferred from seismology. Similarly to the optical image correlation results (Fig. 5.3), observations on the aftershock distributions supports the theory and models on the damage evolution of supershear ruptures (section 5.1). Details on the aftershock observations can be found in the complete version of the study (in review at Proceedings A, in supplements from this thesis). This work provides hints on the damage patterns that can be observed along a supershear rupture transition zone, and comes as a support to location technics from the seismology. It opens new opportunities for analyzing the earthquake supershear transition mechanisms, like for example the relation with the fault geometry.

DISCUSSION

According to numerous published observations on surface fault ruptures (chapter 1; Figs [START_REF] Okubo | Modeling earthquakes with off-fault damage using the combined finite-discrete element method[END_REF], it is now admitted that the surface deformation associated with an earthquake generally distributes over wide areas, rather localizing in a narrow fault zone. This makes challenging the quantification of the surface deformation after an earthquake, whereas it constitutes a fundamental information for the understanding of the rupture process, and for the assessment of the earthquake hazard. It raises the following issues:

1.7-1.
-How is surface deformation accommodated at during an earthquake?

-How does it relate to the earthquake process at depth? -What are the controlling parameters?

To answer these questions, we used high-resolution optical image correlation to measure the detail of the fault ruptures and off-fault deformation associated with two recent surface-rupturing events: the 2019 Ridgecrest earthquakes, and a 30-km long area of the 2013 Baluchistan earthquake. Optical image correlation, as applied in this thesis (chapter 2), allows measuring the 3D ground displacement field with a resolution of 0.5 m, over wide regions. We used large-aperture profiles (> 4 km) across the displacement maps to measure the ground displacements occurring away from the faults initially detected on the field.

Results obtained for the 2013 Baluchistan earthquake show that gradients of displacement affect the off-fault medium up to 2 km away from the primary fault rupture. This off-fault deformation represents, in the strike-slip component, 3 m for a total of 9 m of displacement across the fault, in a region where the fault geometry is particularly complex. Because the gradients are continuous and not correlated with cracks at the ground surface, we refer to this pattern as diffuse deformation.

Imaging the 2019 Ridgecrest earthquake displacement field revealed similar diffuse deformation patterns, although to a larger extent [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]. It represents on average 30% of the total surface displacement budget in the strike-slip component. This diffuse deformation affect, on average, an area wide of 2 km, including the visible fault ruptures. Because the 2019 Ridgecrest rupture is widely investigated using field, geodetic, seismological, and numerical methods, it allows crosschecking our correlation results with other existing observations. This comparison first allows to validate our measurements, and to address the issue of the nature of the diffuse deformation (elastic or inelastic), and of its role in the rupture process.

In the following discussion, we first expose field and seismic observations that support the existence of a wide (> 1 km) deformation zone around faults. We also show examples of diffuse deformation that was detected in other studies, although it was not quantified.

Second, since we showed that diffuse deformation compensates for the shallow slip deficit (SSD; section 4.2.), we show the impact of considering diffuse deformation in the estimation of earthquake magnitude and fault slip rates. Then, comparing the observations made along the 2013 Baluchistan and 2019 Ridgecrest ruptures, we highlight relations between diffuse deformation, fault geometry, earthquake magnitude and geological context. Using these two cases of study, and complementary observations on other strikeslip ruptures, we bring to light general characteristics of earthquake ruptures. Finally, we discuss the mechanisms that could be responsible for diffuse deformation and how it would impact on the fault mechanical and seismic behavior.

Observations of wide deformation zones around faults

Observations of the deformation around faults are generally limited to the area of fractured rocks over width of a few hundred meters (Figs. 1.7-1.11). But, in this work, we highlighted kilometric-wide deformation zones around fault ruptures. This type of deformation was never quantified in previous studies, nor it was considered in the displacement budget of earthquakes, while we showed that it represented 30% of the surface displacement budget along the 2019 Ridgecrest rupture (Fig. 4.9). However, there exist observation of wide deformation zones that support the fact that diffuse deformation as we observed along the 2019 Ridgecrest and 2013 Baluchistan ruptures is a common feature for earthquakes. Indeed, several field observations report fractures kilometers away from fault systems (e.g., [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF], while seismological data evidence wide aftershocks clusters around faults (e.g., [START_REF] Liu | A Structural Interpretation of the Aftershock "Cloud" of the 1992 Mw 7.3 Landers Earthquake[END_REF]. Finally, some optical image correlations for other earthquake ruptures captured deformation patterns similar to those highlighted in this thesis (e.g., [START_REF] Scott | The M7 2016 Kumamoto, Japan, Earthquake: 3-D Deformation Along the Fault and Within the Damage Zone Constrained From Differential Lidar Topography[END_REF]. In the following part, we describe these observations and how they compare to the diffuse deformation pattern measured in this thesis. 2013) also reported 2 km-wide areas of damaged rocks at the junction between the Arimae-Takatsuki Tectonic Line (ATTL) and the Rokkoe-Awaji Fault Zone (RAFZ), in southern Japan. However, these observations inform on the total number of fractures present in the medium, that is inherited from the long-term fault activity, and the regional tectonic history. It does not inform on the deformation that affected the area during the last earthquake.

Observations on recent surface ruptures allow to characterize the deformation zone associated with the last earthquake rupture, and co-seismic deformation zones of a few kilometers-wide could be highlighted. For example, 2 km-wide fault zones constituted of sub-parallel and cross-cutting fault strands were evidenced along the Loma Prieta, San Andreas, California, earthquake rupture (1989, Ms7.1;[START_REF] Aydin | Right-lateral-reverse surface rupture along the San Andreas and Sargent faults associated with the October 17,1989, Loma Prieta, California, earthquake[END_REF]. Similar patterns were observed in the trace of the 1992 Landers [START_REF] Johnson | Shear zones formed along long, straight traces of fault zones during the 28 June 1992 Landers, California, earthquake[END_REF][START_REF] Mcgill | Surficial slip distribution on the central Emerson fault during the June 28, 1992, Landers earthquake, California[END_REF], and 1999 Hector Mine earthquakes [START_REF] Treiman | Primary Surface Rupture Associated with the Mw 7.1 16 October 1999 Hector Mine Earthquake, San Bernardino County, California[END_REF], in Eastern California Shear Zone. Other examples include the 2001 Kokoxili (China) rupture, where a >2 km-large fault zone is observed along a 100-km long partitioning area (Klinger et al., 2005), and the 1905 Tsetserleg-Bulnay (Mongolia) rupture, where secondary cracks were observed more than 2 km away from the primary fault system [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF].

Along these ruptures, a wide deformation zone was evidenced using visible deformation features on the ground that mark the deformation, such as fractures (e.g., [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF], mole tracks [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF], or offset features (e.g., Rockwell et al., 2002).

However, results from the present work show that not only secondary fractures accommodate the deformation around the faults, but that the medium also deforms through processes producing continuous gradients of displacement at the ground surface over kilometric scales. These gradients were not detected in the field because of the absence of visible markers; however, we show how several seismic and geodetic observations support our measurement of a wide diffuse deformation zone associated with earthquake ruptures.

Seismic observations of a wide deformation zones

Seismic studies use the relocation of epicenters associated with background seismicity, mainshocks, and aftershocks, to analyze the spatial distribution of seismicity before, during, and after an earthquake. Some studies revealed that seismicity associated with an earthquake rupture distributes over a wide area, of scales comparable with the width of the diffuse deformation zone highlighted in this work (>1 km).

For example, kilometric-wide areas of mainshock seismicity and aftershocks were highlighted along the 1992 Landers rupture [START_REF] Liu | A Structural Interpretation of the Aftershock "Cloud" of the 1992 Mw 7.3 Landers Earthquake[END_REF]. It was shown that only 0.1 to 2% of the aftershock population occurred within 400 m around the fault, 40% within 0.5 km, and 60% to 65% within 1 km. Relocation error in the horizontal component for the relocated aftershocks is 0.31 to 0.35 km. This pattern was interpreted as triggered seismicity on fault planes that are well away from the primary fault trace [START_REF] Liu | A Structural Interpretation of the Aftershock "Cloud" of the 1992 Mw 7.3 Landers Earthquake[END_REF].

Similarly, [START_REF] Kato | Source fault model of the 2018 Mw 5.6 northern Osaka earthquake, Japan, inferred from the aftershock sequence[END_REF] showed that aftershocks following the 2018 Mw5.6

Osaka, Japan, earthquake were segregated within clusters located from 0.5 to 2 km on either side of the fault zone. This result was obtained using relocated seismicity from the day of the rupture to 12 days after. Relocation error for this study is 0.24 km, in the horizontal component. Distribution of the epicenters (Fig. 6.2) shows that seismicity, on the day of the rupture, affects the entire fault zone, from the fault core to 2 km on either side. But, after one day, seismicity on the fault zone starts to decay and aftershocks cluster from ~1 to 2.2 km west, and 0.5 to 1.5 km east of the fault. This highlights that deformation during an earthquake affects a wide volume of rocks around the faults, of width similar to that observed in the optical images (>1 km). In addition, aftershocks following the 2019 Ridgecrest mainshock were showed to occur over kilometric wide areas (Fig. 6.3;Ross et al., 2019;[START_REF] Shelly | A High-Resolution Seismic Catalog for the Initial 2019 Ridgecrest Earthquake Sequence: Foreshocks, Aftershocks, and Faulting Complexity[END_REF]. Interestingly, this correlates with the width of the deformation zone we inferred from surface measurements that includes diffuse deformation (Fig. 4.13). Particularly, south-east of the epicenter, aftershocks align along two parallel planes (dashed blue lines; Fig. 6.3) that correlate with the edges of the fault zone we observed at the surface (Fig. 4.13). Also, to the north, a >5km large cluster of aftershocks correlates with the location of the bookshelf faulting area, where we showed that diffuse deformation represented the totality of the surface displacement (Fig. 4.10). From these observations, there seems to be a correlation between diffuse deformation at the surface and distributed seismicity at depth. et al., 2018).

Diffuse deformation patterns are also visible in the vertical displacement map of the 2010 Mw7.2 El-Mayor Cucapah, Mexico, earthquake (Zhou et al., 2015). Vertical displacements were calculated using a pre-earthquake LiDAR digital elevation model, and a post-earthquake DSM derived of Pleiades tri-stereo images. Figure 6.5 presents the vertical displacement field in the area of the Paso Inferior Accommodation Zone (PIAZ).

In this area, a NE-SW gradient of displacement of 1 m slip over 1 km width is visible from the Paso superior fault to the Laguna Salada fault. In the field, ~1 m of surface slip was measured on the fault from the PIAZ, compared to ~2 m and ~3-4 m along the southwest and north-east neighboring faults, respectively [START_REF] Fletcher | Assembly of a large earthquake from a complex fault system: Surface rupture kinematics of the 4 April 2010 El[END_REF]. Hence, by adding the diffuse deformation to the offsets measured on the faults, surface displacement across the PIAZ yields values similar to that of the neighboring faults. Also, the fact that diffuse deformation links two localized fault zones, that are the Paso superior fault to the Laguna Salada fault, suggests that these two faults are connected at depth. This pattern is similar to that highlighted at the north of the 2019 Ridgecrest rupture (Fig. 4.10), where surface diffuse deformation reveals the deformation of a fault structure that remains buried at depth. In the case of the 2019 Ridgecrest earthquake, this type of diffuse deformation pattern is correlated with diffuse seismicity at depth (Fig. 6.3). showed that the diffuse deformation pattern observed at the surface could be reproduced by dynamically activated off-fault fractures, and that this pattern is widely observed at geometrical complexity like the junction between these three faults. Finally, we found evidence of diffuse deformation patterns in the displacement field of the 2008 M w 7.9

Sichuan earthquake, China [START_REF] Elliott | Earth Observation for Crustal Tectonics and Earthquake Hazards[END_REF]. Using SAR image correlation at ~100 m ground resolution, [START_REF] Elliott | Earth Observation for Crustal Tectonics and Earthquake Hazards[END_REF] highlighted 5-km large patches of displacements of higher amplitude (+3 m) in the off-fault. They associated this feature associated with the presence of a blind fault at depth.

Hence, diffuse deformation was observed in several other studies, and was associated with areas of fault complexity (Klinger et al., 2018), or uncomplete rupture of a fault (Figs. 6.4 and 6.5;[START_REF] Elliott | Earth Observation for Crustal Tectonics and Earthquake Hazards[END_REF]. In this work, we showed that diffuse deformation could represent up to 30% of the total coseismic slip budget for an earthquake. Hence, it is necessary to understand the origin of this deformation and how it is complementary to the localized deformation on the fault, at the surface and at depth. With this aim, we perform a joint inversion of GPS, InSAR and high-resolution optical data to model the rupture pattern at depth. This work was fulfilled for the 2019 Ridgecrest earthquake.

Because this earthquake was observed using various methods, it allows for a good constraint of the slip model, and for a comparison between our results and other models and observations.

Comparison with slip models: relation between surface diffuse deformation and the rupture process at depth

After using the high-resolution optical displacement maps to characterize and quantify diffuse deformation patterns, we use these data to perform a joint inversion of the 2019

Ridgecrest displacement pattern at depth, with GNSS and InSAR data (section 4.2). The results show that, on average, the larger slip is located between 3 and 6 km at depth. This pattern is widely observed for strike-slip ruptures, and is referred to as shallow slip deficit (SSD; [START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF]. It is widely discussed whether this SSD is an artefact from modeling technics [START_REF] Marchandon | Origin of the shallow slip deficit on a strike slip fault: Influence of elastic structure, topography, data coverage, and noise[END_REF][START_REF] Xu | Refining the shallow slip deficit[END_REF], or if it corresponds to an actual deficit of slip in the shallow part of faults, that would be compensated by interseismic and/or post-seismic shallow fault creep [START_REF] Fialko | Evidence of fluid-filled upper crust from observations of postseismic deformation due to the 1992 Mw7.3 Landers earthquake[END_REF][START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF]Milliner et al., 2020;[START_REF] Yue | The 2016 Kumamoto Mw = 7.0 Earthquake: A Significant Event in a Fault-Volcano System[END_REF], or inelastic deformation of the off-fault medium [START_REF] Fialko | Deformation on Nearby Faults Induced by the 1999 Hector Mine Earthquake[END_REF][START_REF] Marone | On the Mechanics of Earthquake Afterslip[END_REF]Roten et al., 2017). Current measurements of inelastic fault zone width are of the order of 10 2 m (e.g., Barnhart et al., 2020;[START_REF] Scott | The M7 2016 Kumamoto, Japan, Earthquake: 3-D Deformation Along the Fault and Within the Damage Zone Constrained From Differential Lidar Topography[END_REF][START_REF] Zhou | Characterizing Complex Surface Ruptures in the 2013 Mw 7.7 Balochistan Earthquake Using Three-Dimensional Displacements[END_REF]. However, this values does not take into consideration the deformations that affects the crust at larger distances from the faults, as highlighted in this work using optical image correlation. In fact, we showed how the surface diffuse deformation measured along the 2019 Ridgecrest rupture (section 4.1) is equivalent to the SSD estimated in the models (section 4.2). This means that surface deformation associated with an earthquake is equivalent the deformation that occurs at depth. Figure 6.6

summarizes the results from the comparison between the surface displacement data (Fig.

6.6a

) and the fault slip model that reports the slip distribution at depth (Fig. 6.6c). Figure 6.6b presents a simplistic model of a fault zone cross section, based on the previous results. 4.22). SSD is reported at a 3-km depth, where maximum displacement on the fault starts decreasing toward the surface.

As a consequence to the fact that part of the surface coseismic displacement occurs through diffuse deformation in the medium, there is a large bias (30%) in the estimation of the earthquake displacement, and magnitude, from field measurements that report the displacement on the faults. In the following, we show the impact of considering diffuse deformation in the calculation of the earthquake magnitude and seismic energy, and in the estimation of the fault geologic slip rates, and earthquake recurrence. These calculations are based on the example of the 2019 Ridgecrest ruptures, for which field data are available.

Using large-aperture profiles across the ruptures, we measured an average surface displacement, including diffuse deformation, of 0.73±0.09 m and 2.13±0.06 m for the foreshock and mainshock events, respectively [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]. As a comparison, field measurements report on average (from the cumulative moving maximum curve) 0.5 m and 1.7 m for the foreshock and mainshock ruptures, respectively [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF].

These measurements are made on the faults and do not include diffuse deformation. To assess the impact of diffuse deformation on the characterization of the earthquake event,

we calculate the seismic moment, moment magnitude, and seismic energy [START_REF] Bormann | Seismic Sources and Source Parameters, New Manual of Seismological Observatory Practice (NMSOP)[END_REF][START_REF] Hanks | A moment magnitude scale[END_REF] using displacement values from both field data [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF], and optical image correlation results [START_REF] Antoine | Diffuse Deformation and Surface Faulting Distribution from Submetric Image Correlation along the 2019 Ridgecrest, California, Ruptures[END_REF]. Details of our calculation, and of the comparison with field data from [START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF], are presented in Table 6.1. The seismic moment 𝑀 0 of an earthquake is given by the equation: we estimate an increase of +0.1 𝑀 𝑤 when including the diffuse deformation in the displacement budget (Tab. 6.1). Finally, we estimate the impact on the seismic energy 𝐸 (Joules):

𝑀 0 =
𝐸 = ∆𝜎 2𝜇 𝑀 0
with a stress drop (∆𝜎) of 2.6 MPa for the foreshock, and 8.5 MPa for the mainshock [START_REF] Sheng | Stress Field Variation During the 2019 Ridgecrest Earthquake Sequence[END_REF]. This gives a seismic energy of 1.71x10 15 J for the foreshock earthquake, and of 4.98 x10 16 J for the mainshock earthquake, when considering diffuse deformation (Tab. 6.1). Compared to field results from DuRoss et al. ( 2020) (Tab. 6.1), it yields an increase of 0.54 x10 15 J for the foreshock, and 1.37 x10 16 J for the mainshock.

These results have a large uncertainty from the fact that we project diffuse deformation on the fault plane, while more accurate calculations should use estimations of the fractured area [START_REF] Ben-Zion | Spatial variations of rock damage production by earthquakes in southern California[END_REF]. Still, our results are in the range of what is found for earthquakes of similar magnitude. For the 1992 Landers and 1999 Hector Mine, seismic energy calculated by [START_REF] Boatwright | Radiated Seismic Energy and Mean Square S-Wave Spectral Comparisons for Large Earthquakes[END_REF], from regional surface waves, is 3.7x10 15 J and 3.4x10 15 J, respectively. As a comparison, the seismic energy of a Mw5.5 earthquake is of the order of 10 13 [START_REF] Bormann | Seismic Sources and Source Parameters, New Manual of Seismological Observatory Practice (NMSOP)[END_REF], and that of a Mw>8 of the order of 10 17 [START_REF] Bormann | Seismic Sources and Source Parameters, New Manual of Seismological Observatory Practice (NMSOP)[END_REF][START_REF] Ye | Energy Release of the 2013 Mw 8.3 Sea of Okhotsk Earthquake and Deep Slab Stress Heterogeneity[END_REF] et al., 2021). For the calculation of 𝑀 0 , w used a rigidity (𝜇) of 30 GPa and a fault width (W) of 10 km. For the calculation of 𝐸 𝑠 , we used a stress drop (∆𝜎) of 2.6 MPa for the foreshock, and 8.5 MPa for the mainshock [START_REF] Sheng | Stress Field Variation During the 2019 Ridgecrest Earthquake Sequence[END_REF]. Moment magnitudes estimated from the seismology are 6.4 for the foreshock and 7.1 for the mainshock (Ross et al., 2019;[START_REF] Shelly | A High-Resolution Seismic Catalog for the Initial 2019 Ridgecrest Earthquake Sequence: Foreshocks, Aftershocks, and Faulting Complexity[END_REF].

In addition, using simplistic calculations, we show the impact of the diffuse deformation in estimating fault slip rates and earthquake recurrence. First, considering two theoretical events dated with ages of 1000 years difference, a characteristic coseismic displacement for two earthquakes of 1.7 m would give a slip rate of 1.7mm/yr. Considering a coseimsic displacement of 2.13 m, that includes diffuse deformation, would give a slip rate of 2.13 mm/yr. Hence, including diffuse deformation in the earthquake displacement budget increases of 20% the estimated fault slip rate. On the other hand, considering a slip rate of ~1 mm/yr, that is an average slip rate for faults from the Southern Walkner Lane [START_REF] Wesnousky | Active faulting in the Walker Lane[END_REF], and a coseismic displacement of 1.17 m based on field data, yields a recurrence time of 1170 years for the earthquake. Considering a coseismic displacement of 2.13 m, which includes diffuse deformation, we obtain a recurrence time of 2130 years, which is almost twice larger than when using the field data. These calculations only illustrate at first order the impact of the under-estimation of the surface displacement on the estimation of earthquake recurrence, since the ~1 mm/yr slip rate might be itself under-estimated from the fact that it does not include the diffuse deformation. However, it shows how significant is the impact of diffuse deformation on the estimation of the earthquake and fault parameters.

Hence, ignoring diffuse deformation in the estimation of the displacement budget of an earthquake results in an underestimation of the earthquake characteristics, of the fault slip rates, and of the recurrence time of earthquakes. However, this bias in earthquakes and faults characteristics are likely not similar for every earthquake rupture. Indeed, we showed that the 2013 Baluchistan rupture, along our study area, exhibits low proportions of diffuse deformation. We measured 16% of off-fault deformation, which includes both diffuse deformation and smaller scale (<500 m) brittle off-fault deformation. Hence, there are major differences in the proportions of diffuse deformation between the 2013

Baluchistan and 2019 Ridgecrest ruptures. To understand what controls the proportions and spatial distribution of diffuse deformation, in the following, we explore the relation between diffuse deformation and several earthquake parameters like SSD, fault geometry, and geological context.

Primary control of earthquake magnitude on SSD and diffuse deformation

Along the 30-km long area of the Baluchistan rupture, to the exception of the C2 relay zone that is a the main kilometric-scale complexity along the 200-km long rupture trace, we did not measure diffuse deformation. When comparing with other displacement maps that report the entire displacement field for the 200-km long 2013 Baluchistan rupture (e.g., [START_REF] Barnhart | Evidence for slip partitioning and bimodal slip behavior on a single fault: Surface slip characteristics of the 2013 Mw7.7 Balochistan, Pakistan earthquake[END_REF][START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF]Vallage et al., 2015), we draw a similar conclusion based on the fact that the rupture is well-marked in the displacement field and seems to accommodate the entire displacement associated with the earthquake.

Measurements of the inelastic fault zone widths from these displacement maps corroborate our observations, with an average width of 100m-200m [START_REF] Cheng | Permanent Co-Seismic Deformation of the 2013 Mw7.7 Baluchistan, Pakistan Earthquake From High-Resolution Surface Strain Analysis[END_REF][START_REF] Zhou | Characterizing Complex Surface Ruptures in the 2013 Mw 7.7 Balochistan Earthquake Using Three-Dimensional Displacements[END_REF]. Hence, we propose that there was no SSD along the 2013 Baluchistan rupture. This conclusion is consistent with results from [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF], who used InSAR and near-fault optical data to constrain the rupture process, and found no SSD for this earthquake. [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF] justify the general absence of SSD for this earthquake by the fact that, to fulfill such a large magnitude, the earthquake requires to rupture the entire width of the crust, from depth to the surface. Using a comparison with other strike slip ruptures, they propose that earthquakes with Mw>7.5 do not exhibit SSD, compared to Mw~7 that show variable proportions of SSD. Since we observed a correlation between SSD and diffuse deformation, this would imply that more diffuse deformation occurs from Mw<7.5 earthquake ruptures. (2017) were obtained using optical data combined with other datasets in the inversion process.

To test this hypothesis, we plot the percentage of SSD of eleven 6.5<Mw<7.8 strike-slip earthquakes against their magnitude (Fig. 6.7). The results show a decrease of the SSD, from ~65-80% for Mw6.5 earthquakes, to 0% for Mw>7.5 earthquakes. Earthquakes of magnitudes 7.0<Mw<7.3 display variables amounts of SSD, from 3 to 43%. Hence, it appears that SSD correlates with the magnitude of the earthquake rupture. As a consequence, diffuse deformation correlates with the magnitude of the earthquake.

Indeed, measurements made along the surface rupture of the 2001 Kokoxili rupture using correlation of SPOT images at 10 m ground resolution (chapter 5 of this thesis), show an average deformation zone width of 230 m (with the exception of the area of partitioning), suggesting that all the deformation is localized on the primary fault structure. We similarly showed that the 2013 Baluchistan rupture was well localized, with likely no diffuse deformation to the exception of the 4-km long C2 area. Conversely, we showed examples of diffuse deformation along the 2016 Kumamoto (Fig. 6.2), 2010 El-Mayor Cucapah (Fig. 6.3), and 2019 Ridgecrest earthquakes (section 4.1), that have a magnitude lower than 7.5, and a SSD of 11% to 43%. Finally, for the Mw6.5 2003 Bam earthquake [START_REF] Binet | Horizontal coseismic deformation of the 2003 Bam (Iran) earthquake measured from SPOT-5 THR satellite imagery[END_REF][START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF], measurements from [START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF] show that ground deformation across the rupture is larger 1.5 km away from the fault (0.5-0.6 m), than at the fault itself where only a few centimeters could be measured. This highlights a large proportion of diffuse deformation for this rupture, comparable to the amount of SSD that occurred.

For the 2018 Mw7.5 Palu earthquake, that is of intermediate magnitude compared to the examples cited before, no SSD was measured on average (Socquet et al., 2019) but, diffuse deformation patterns are visible at the surface and correlated with sections where SSD occurred along the rupture (Fig. 6.8). Indeed, the maximum slip of 5 m propagated at the surface from 0.8°S to 1.1°S (Fig. 6.8a) along a localized section of the rupture (Fig.

6.8b

). A profile across this area shows that all the slip occurred on a narrow fault zone (Fig. 6.8c). Field measurements report a fault zone width of ~40 m in this area [START_REF] Jaya | Distribution and morphology of the surface ruptures of the 2018 Donggala-Palu earthquake, Central Sulawesi, Indonesia[END_REF]. But, along some sections of the rupture, we observe SSD in the model (SSD1-3), correlated with diffuse deformation patterns at the surface. Despite the fact that some noise is visible in the images in these areas of the rupture, due to vegetation in the region of Palu, it is clear that the rupture is less localized along these sections. It shows that, even if on average there is no SSD along this rupture, some parts display small SSD and more diffuse deformation patterns at the surface. In these areas field measurements report a fault zone width of 320 m [START_REF] Jaya | Distribution and morphology of the surface ruptures of the 2018 Donggala-Palu earthquake, Central Sulawesi, Indonesia[END_REF]. On the optical displacement map, the diffuse deformation pattern is ~1 km wide (Fig. 6.8b). From the model, we estimate a SSD of ~1.5 m along these sections, and so similar amplitudes of diffuse deformation at the surface. occurs over a narrow fault zone, ~40 wide from the field observations [START_REF] Jaya | Distribution and morphology of the surface ruptures of the 2018 Donggala-Palu earthquake, Central Sulawesi, Indonesia[END_REF].

Conversely, along the 2013 Mw7.7 Baluchistan earthquake, we showed that no such sections of diffuse deformation are observed, and that the rupture is well-localized along its entire length. Similarly, in the model of [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF], no SSD is modeled. As an exception, one patch of SSD is visible in the modeled, beneath the C2 complexity. It corresponds to the area where we measured diffuse deformation over a 2-km large and 4km long area, in the hanging-wall of the rupture. Similarly, along the 2002 Denali rupture, models from [START_REF] Elliott | Coseismic deformation of the 2002 Denali fault earthquake: Contributions from synthetic aperture radar range offsets[END_REF] display no SSD on average. But, the slip distribution is heterogenous, with a few patches of slip that sometimes remain limited to depths larger than 3 km depending on the model parameters that are tested. Still, they are associated with changes in the strike of the earthquake, which suggest a geometrical control on the SSD and surface displacement pattern. This is similar to what was observed along the C2 relay zone of the 2013 Baluchistan rupture. This shows that, on one hand, for earthquakes of Mw>7.5, no SSD occurs along the rupture to the exception of a few major complexities that locally trigger SSD and diffuse deformation. On the other hand, it shows that the fault geometry controls the persistency of a few patches of SSD and diffuse deformation along Mw>7.5 ruptures, that overall display a localized rupture pattern.

Secondary control of the fault geometry on the SSD and diffuse deformation

In chapter 3.1 and 4.1 from this thesis, we showed that diffuse deformation is correlated with variations in the fault geometry, both for the 2013 Baluchistan and 2019 Ridgecrest ruptures. In the previous section of this discussion, we also showed that, despite the control of the earthquake magnitude on the average proportion of diffuse deformation, the rupture geometry seems to control its distribution along the strike of the fault.

First, in the case of the 2019 Ridgecrest earthquakes, complexity is present along the entire fault system. We particularly highlighted a complex geometry in the epicenter area, at the. northwestern tip of the rupture, and at the junction between the foreshock and mainshock ruptures. However, lowest proportions of diffuse deformation were measured in the epicentral area. This shows that the complexity of the rupture traces do not explain the distribution of diffuse deformation for this rupture. But, we also highlighted a segmentation of the rupture, with three domains that display different amounts of total slip and diffuse deformation (Fig. 4.9b), correlated with variations in the strike of the rupture (Fig. 4.9a). In fact, the rupture changes from a N152 strike at the center of the rupture, to a strike of N140 in its north-western part, and of N134 in its south-eastern part.

Segments that strike N140 and N134 display the largest proportions of diffuse deformation, of 80% and 40%, respectively. Conversely, the segment that strike N152, along which the rupture initiated and maximum slip was measured both at depth and at the surface, displays 18% of diffuse deformation, with most of it that occurs at the transition with a major extensional relay-zone north-west of the segment (Fig. 4.9a,b).

Comparison of the segment orientations with the regional stress field shows that the central part of the rupture, that strike N152, is oriented 33 degrees to the maximum regional compression axis (N5°; [START_REF] Fialko | Simple shear origin of the cross-faults ruptured in the 2019 Ridgecrest earthquake sequence[END_REF]Fig. 6.9). Conversely, the N140 and N134-striking sections are oriented 45 and 51 degrees, respectively, from the maximum compression axis (Fig. 6.9). From the Mohr-Coulomb-Anderson theory [START_REF] Anderson | The dynamics of faulting and dyke formation with applications to Britain[END_REF]Scholz, 2019), it was predicted that the optimal angle for a fault to rupture is ±25-30° to the maximum compression axis (Scholz, 2019;[START_REF] Sibson | Rupture nucleation on unfavorably oriented faults[END_REF].

Hence, the N152 segment is more favorably oriented for rupture than the N140 and N134

segments. As a consequence, larger slip occurs along this segment, and no diffuse deformation is measured. It shows that the rupture segmentation impacts on the rupture process and along-strike distribution diffuse deformation.

Then, for Mw>7.5 earthquakes, we showed that diffuse deformation was unlikely to occur because, during the rupture, the coseismic slip propagated all the way to the surface on a narrow fault zone. Still, we highlighted the persistency of patches of diffuse deformation along geometrical complexities, like the C2 relay zone of the Baluchistan earthquake, also associated with local SSD. These geometrical complexities generally mark the transition between successive narrow localized fault ruptures, and highlight the segmentation of the rupture [START_REF] Duross | Fault segmentation: New concepts from the Wasatch Fault Zone, Utah, USA[END_REF][START_REF] Klinger | Relation between continental strike-slip earthquake segmentation and thickness of the crust[END_REF][START_REF] Schwartz | Fault Behavior and Characteristic Earthquakes: Examples From the Wasatch and San Andreas Fault Zones[END_REF]. From field and satellite image-based measurements, it was observed that fault displacement across these complexities was decreasing compared to the slip measured along fault segments [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF]Klinger et al., 2006). This is different what is found for other 7.1<Mw<7.3 events earthquakes (3-17%; Fig. 6.4), we discuss the influence of the tectonic and geological contexts on this rupture.

Geological control on the earthquake rupture

As explained in the previous section, the geometry of a fault rupture exerts a control on the pattern of displacements along the rupture. It was shown for several ruptures that the geological structure (e.g., [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF][START_REF] Teran | Geologic and structural controls on rupture zone fabric: A field-based study of the 2010 Mw 7.2 El Mayor-Cucapah earthquake surface rupture[END_REF] and pre-existing faults and fabric from the crust (e.g., [START_REF] Barchi | The influence of subsurface geology on the distribution of earthquakes during the 2016--2017 Central Italy seismic sequence[END_REF][START_REF] Fialko | Simple shear origin of the cross-faults ruptured in the 2019 Ridgecrest earthquake sequence[END_REF][START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF] can influence the rupture pattern. Hence, in the following, we analyze how the relation between the geological and tectonic context, and the rupture patterns of the 2013 Baluchistan and 2019 Ridgecrest earthquakes.

On one side, the 2019 Ridgecrest rupture occurred within a distributed right-lateral deformation zone located 120 km north-east of the San Andreas fault within the North America plate, and called the Southern Walker Lane (Fig. 3.1). In this region, it exists numerous faults that have different orientations. Also, a basement fabric was highlighted in the geology [START_REF] Jobe | Evidence of Previous Faulting along the 2019 Ridgecrest, California, Earthquake Ruptures[END_REF] and in the background seismicity [START_REF] Fialko | Simple shear origin of the cross-faults ruptured in the 2019 Ridgecrest earthquake sequence[END_REF] and aftershock (Ross et al., 2019) clusters. The geological map also indicated that [START_REF] Jennings | Geologic map of California : Trona sheet[END_REF]Fig. 6.9) faults oriented NW-SE and NE-SW, of length similar to the faults activated during the 2019 ruptures, are present all over the region. Hence, the 2019 Ridgecrest event propagated within a network of orthogonal faults, which explain its particularly complex and discontinuous geometry of the ruptures. For example, the foreshock rupture developed on two faults that are orthogonal to the mainshock average rupture azimuth (Figs. 4.2 and 4.6), and the junction area is associated with large amount of diffuse deformation, around 1 m (Fig. 4.9). Also, to the north-west of the rupture, we showed how the basement fabric was reactivated through bookshelf faulting to accommodate right-lateral shear at the surface (Fig. 4.10). This results in high proportions of diffuse deformation along this segment (80%). Quit differently, the Baluchistan earthquake ruptured a pre-existing interface between layers of exhumed folded Miocene-Oligocene sedimentary rocks of the compressive Makran range (Fig. 6.10). Hence, the rupture benefits of a pre-existing and almost plane interface to propagate, which reduces the degree of complexity of the fault system at depth. Even if a basement fabric was highlighted in this region, with orientations orthogonal to the fault azimuth [START_REF] Vallage | Geological structures control on earthquake ruptures: The M w 7.7, 2013, Balochistan earthquake, Pakistan: Geological Control on Earthquake[END_REF], it has a subsidiary role in controlling the rupture process compared to the most prominent curved faults (>200 km) oriented NNE-WSW. Inversion of the fault geometry using optical and radar satellite data in [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF] shows that the fault plane has a relatively planar geometry, following a curved fault trace at the surface. Dip angles vary from 50-60°NW at the center of the rupture, to 75°NW along the western brand, and 90°NW along the northern branch. The fact that the rupture benefitted from a pre-existing and continuous geological interface to propagate may have favored a localized of the deformation with a narrow deformation zone of 100-200 wide on average. However, we highlighted the presence of a major geometrical complexity along the trace of this fault, which is associated with diffuse deformation in the hanging-wall of the rupture. From the results of Vallage et al. (2015) and [START_REF] Zhou | Characterizing Complex Surface Ruptures in the 2013 Mw 7.7 Balochistan Earthquake Using Three-Dimensional Displacements[END_REF], we see that this complexity is the only one that allows a junction between two disconnected faults forming the rupture. Other complexities from the rupture include the C1 flower structure, but which occurs within the trace of a same fault, and the B1, B2, F1 and C3 complexities, that consist in azimuthal variations within a fault plane [START_REF] Vallage | Distribution de la déformation en contexte tectonique oblique : applications aux séismes de Denali (2002) et du Balochistan[END_REF]. Interestingly, [START_REF] Lauer | Fault Geometry and Slip Distribution of the 2013 Mw 7.7 Balochistan Earthquake From Inversions of SAR and Optical Data[END_REF] This analysis brings to light a control of the geological context on the fault geometry, that partly explains the rupture pattern of these two faults. This controls is however secondary to the control of the earthquake magnitude on the rupture pattern, and hence on the diffuse deformation that is observed at the surface.

Possible mechanisms responsible for diffuse deformation

In this thesis, we showed that diffuse deformation occurs primarily along-side geometrical complexities like fault junctions, relay zones, and bends, and in areas where part of the rupture did not reach the surface. Hence, diffuse deformation seems to be partly related to the termination of faults, either in complexities, or along blind faults. We know from observations [START_REF] Kim | Fault damage zones[END_REF] and theoretical calculations (section 5. 1.1. and Scholz et al., 1993) that stresses at the tip of fault ruptures trigger inelastic deformation of the intact rocks in the surrounding medium. It is referred to as a process zone. When accumulated over the entire length of the rupture, it creates a damage zone around the fault (Mitchell and Faulkner, 2009). The width of the damage zone depends on the fault length [START_REF] Scholz | Fault growth and fault scaling laws: Preliminary results[END_REF] and geometry [START_REF] Kim | Fault damage zones[END_REF][START_REF] King | Role of Fault Bends in the Initiation and Termination of Earthquake Rupture[END_REF], the nature of the rocks [START_REF] References Aben | High strain rate deformation of porous sandstone and the asymmetry of earthquake damage in shallow fault zones[END_REF][START_REF] Faulkner | On the internal structure and mechanics of large strike-slip fault zones: field observations of the Carboneras fault in southeastern Spain[END_REF], and dynamic rupture parameters like the rupture velocity (section 5.1.1.). It generally reaches a few hundred meters in width [START_REF] Faulkner | Scaling of fault damage zones with displacement and the implications for fault growth processes[END_REF]Fig. 1.9). In complex rupture zones, the damage zone can even affect several kilometers of rocks around the faults (Klinger et al., 2018).

The major difference between the field observations, that generally report faults and offset features, is that diffuse deformation is not associated with any visible deformation First, by measuring the acoustic emissions in experimental rock deformation studies, it was established that modifications of stress below 95% of the failure stress can trigger diffuse microfracturing of the rocks [START_REF] Lockner | Quasistatic fault growth and shear fracture energy in granite[END_REF][START_REF] Scholz | Microfracturing and the inelastic deformation of rock in compression[END_REF]. This mechanism could generate micro-displacements on series of microfractures, and thus generate displacement gradients similar to those observed in the diffuse deformation zones. The fact that seismicity was evidenced well away from the primary rupture planes, during and after earthquakes (Figs. 6.2 and 3), is coherent with such a micro-fracturing of the rock volume. Moreover, because this type of deformation requires stresses much lower than the complete failure of the rocks, it can act at larger distances than the generally observed damage zone.

A second possibility involves the triggering of millimeter-size to centimeter-size fractures that result from the complete failure of the rocks. This mechanism requires that stresses in the medium reach the rock failure stress. The ground displacement associated with the opening of a large number of these fractures could also result in continuous gradients of displacements, and diffuse seismicity. When distributed over kilometers, these fractures could be hard to detect on the field. Still, in proportions as large as what was observed along the 2019 Ridgecrest ruptures, this type of fracturing should have been detected.

Another possible mechanism could be the reactivation of pre-existing faults (1-10 m in length) from the medium, with amplitudes which are below what can be detected in the field or using satellite data (±5 mm). This type of deformation was detected by Xu et al. (2020) in the region of the 2019 Ridgecrest ruptures, using decomposed high-pass filtered stacked unwrapped phase of Sentinel-1 interferograms. However, this process alone cannot account for the metric amplitude of diffuse deformation we measured around the faults, nor it could generate continuous gradients of displacement over wide areas around the fault zone.

Other possibilities to deform the rocks around the ruptures involve deformation bands, clay smears [START_REF] Fossen | Deformation bands in sandstone: a review[END_REF], and viscous deformations [START_REF] Owen | Deformation processes in unconsolidated sands[END_REF], that can occur in the sedimentary rocks and deposits. Because both the 2019 Ridgecrest and the 2013 Baluchistan events ruptured through sedimentary deposits (Figs. 6.9 and 6.10), these deformation processes might have occurred. Deformation bands include dilatation, compaction, or shear bands, and they correspond to areas of modified rock porosity with reorganization of the grains [START_REF] Fossen | Deformation bands in sandstone: a review[END_REF]. Clay smears form slip surfaces in rocks that have >40% of phyllosilicate content [START_REF] Fossen | Deformation bands in sandstone: a review[END_REF]. Eventually, viscous deformation can occur in saturated unconsolidated sediments [START_REF] Owen | Deformation processes in unconsolidated sands[END_REF], but it generally form turbulent features, like what is observed in liquefied areas [START_REF] Zimmaro | Liquefaction and Related Ground Failure from July 2019 Ridgecrest Earthquake Sequence[END_REF]. Deformation bands and clay smears could generate ground displacement patterns similar to what was measured by Xu et al. (2020), with offsets of a few millimeters that distribute over width of a few hundred meters. However, it might not generate continuous displacement gradients. Considering the fact that diffuse deformation was also measured across areas where the rupture is embedded into granitic rocks, without sedimentary coverage (Fig. 6.9), deformation bands and clay smear may likely not be the primary mechanism responsible for diffuse deformation. Moreover, such plastic deformation mechanisms would not trigger seismicity as it was observed around faults (Figs. Understanding these mechanisms are of prime importance for the characterization of the impact of diffuse deformation on the mechanical behavior of the rocks in the fault zone.

In the following, we develop several observations and models that characterize the impact that inelastic deformation has on the mechanical and seismic behavior of the fault zone.

Considering that diffuse deformation is an inelastic process, this provides insights on the impact that diffuse deformation can on the earthquake process.

Impact of diffuse deformation for the fault mechanical and seismic behavior

Along this work, we showed that in addition to rock damage occurring in the first hundreds of meters around a fault, diffuse deformation was also affecting the rock volume at kilometric scale. This diffuse deformation is most likely inelastic, as discussed in sections 4. 1.6.2 and 4.1.6.3,and associated with seismic activity (Figs. 6.2 and 3). As a consequence, the mechanical properties of the rocks around the faults are affected in a permanent manner, and so is the mechanical behavior of the fault [START_REF] Vidale | Damage to the shallow Landers fault from the nearby Hector Mine earthquake[END_REF].

A major consequence of inelastic deformation zones, according to the literature, is a decrease in elastic stiffness (Faulkner et al., 2006;[START_REF] Lockner | Quasistatic fault growth and shear fracture energy in granite[END_REF] and seismic wave velocity [START_REF] Passelègue | Development and Recovery of Stress-Induced Elastic Anisotropy During Cyclic Loading Experiment on Westerly Granite[END_REF][START_REF] Vidale | Damage to the shallow Landers fault from the nearby Hector Mine earthquake[END_REF] of the rocks. Observations report about 10% reduction in the P and S wave velocities in first 1-2 km around the faults [START_REF] Vidale | Damage to the shallow Landers fault from the nearby Hector Mine earthquake[END_REF]. One on hand, reduction in elastic stiffness (decrease in Young elastic moduli and increase of Poisson ratio; Faulkner et al., 2006) impacts on the mechanism of stress accumulation within the damage zone around the fault. Particularly, it generates rotations in regional stress orientations which allows for the activation of faults that are initially unfavorable to slip (Faulkner et al., 2006). Reduction in elastic stiffness of the rocks also decreases the seismic coupling (amount of elastic energy that can be released seismically) of the fault zone which, in turn, reduces the stress drop of future seismic events [START_REF] Lyakhovsky | Scaling relations of earthquakes and aseismic deformation in a damage rheology model[END_REF]. This can result in part in the interseismic strain that is released aseismically during the interseismic or post-seismic phases, or through recurrent small seismic events, rather than during major rupture [START_REF] Lyakhovsky | Scaling relations of earthquakes and aseismic deformation in a damage rheology model[END_REF]. However, the decrease in elastic stiffness of the host rock is partly recovered after the onset of rupture, up to a state that is close from the initial conditions [START_REF] Passelègue | Development and Recovery of Stress-Induced Elastic Anisotropy During Cyclic Loading Experiment on Westerly Granite[END_REF][START_REF] Vidale | Damage to the shallow Landers fault from the nearby Hector Mine earthquake[END_REF]. Hence, it is the accumulation of the off-fault deformation through the seismic cycles that controls the evolution of the fault zone mechanical behavior.

On the other hand, reduction in elastic stiffness of the rocks also impacts the dynamic characteristics and propagation of the rupture (Ben-Zion and [START_REF] Ben-Zion | Seismic radiation from regions sustaining material damage[END_REF]Templeton and Rice, 2008). Numerical modeling of dynamic rupture shows that fault ruptures associated with the opening of off-fault cracks display smaller propagation velocities than ruptures embedded in a purely elastic medium (Okubo et al., 2019). For example, this can prevent a rupture from transitioning to supershear rupture speeds (Okubo et al., 2019).

Seismic efficiency of the dynamic rupture, that is the ratio between radiated seismic energy and dissipated energy because of on-and off-fault fracturing, also decreases because part of the energy associated with the primary fault rupture is captured by the inelastic damage zone (Okubo et al., 2019). However, the propagation of new fractures in the off-fault medium is also responsible for new sources of seismic moment and wave radiations that superimpose to the contribution of the primary fault rupture. Seismic moment arising from fracturing in the off-fault medium can be close from the primary fault seismic moment (Ben-Zion and Ampuero, 2009). In the case of the Ridgecrest ruptures, we estimated at a first order that diffuse deformation contributed to an increase of ~40% of the seismic moment (Tab. 6.1). So, there is a balance between the dissipation of seismic energy that is captured by the off-fault cracks, and the rupture energy generated by the opening of off-fault cracks. Yet, the exact balance between these two mechanisms is unclear to us. Finally, seismic radiations generated by the off-fault cracks enhance the high-frequency content of the overall seismic radiated energy (Okubo et al., 2019;Shi and Day, 2013), as well as the shear wave scattering [START_REF] Eberhart-Phillips | Surface seismic and electrical methods to detect fluids related to faulting[END_REF]Okubo et al., 2019). As a consequence, a rupture sustaining large off-fault damage has a more complex seismic signature with enhanced high frequency content, especially in the nearfault region.

The processes listed before are described for crystalline rocks that deform through brittle processes, while ruptures sometimes propagate through sedimentary rocks. Plastic deformation within the sedimentary layers would have the opposite effect of brittle deformation, and increases the elastic stiffness of the host rock (Ben-Zion and [START_REF] Ben-Zion | Seismic radiation from regions sustaining material damage[END_REF]. Even if we showed that diffuse deformation is more likely represented by brittle deformation because it is associated with seismic activity, we cannot exclude that part of the diffuse deformation is represented by plastic deformations that would have almost opposite effects on the rocks' mechanical properties. This shows the necessity to better characterize the nature of diffuse deformation, for example using geological and geophysical methods to investigate the properties of the rock in areas where diffuse deformation was measured (see section 8.2).

CONCLUSIONS

In this thesis, we used high-resolution (0.5 m) optical image correlation to measure the 3D displacement field associated with both a part of the 2013 Baluchistan earthquake (30km long; section 3.1), and the 2019 Ridgecrest earthquake sequence (section 4.1), and.

The results allow to analyze the ground displacements within the fault zone with an unprecedented level of detail. We also imaged the horizontal displacement field associated with the 2001 Kokoxili earthquake using images at a 10-m resolution (section 5.). This work is included in a multi-disciplinary study focusing on supershear earthquakes, representing only a minor part of this thesis compared to the high-resolution studies. The major conclusions of this thesis are developed hereafter.

Imaging complex fault geometries and measuring localized deformation on faults

Image correlation at 0.5 m along the 2019 Ridgecrest and 2013 Baluchistan ruptures revealed areas of complex fault geometries, where numerous faults accommodate displacement, either in a direction sub-parallel or cross-cutting the main rupture. A majority of these faults was mapped in the field in the case of Ridgecrest, or through photo-interpretation of optical images in the case of the Baluchistan. However, using the high-resolution correlation method, we were able to detect supplementary faults that were not mapped. Moreover, we derived the rotation, dilatational strain, and shear strain from the horizontal displacement maps. This allowed us to characterize the faults' mechanism and detailed geometry, as well as the amount of rotation and strain accommodated by each fault. The results reveal that the faults accommodate strain of the order of 10 The results show that small (~1 km) faults located along the primary fault ruptures could accommodate up to several meters of displacement. This highlights the necessity of measuring the displacement on all the faults of the system, and not only on the major faults that are not always representative of the amount of displacement accommodated across the system. In the case of the 2019 Ridgecrest earthquakes, these results are consistent with field measurements yielding similar values. Although our study provides a larger number of measurements on a larger number of faults, and with a smaller uncertainty compared to field observations. This shows that this method is reliable to measure the fault displacement along ruptures where field investigations cannot be undertaken.

Measuring large-scale displacement gradients: the diffuse deformation

It was shown by numerous previous studies (section 6.1) that surface deformation during an earthquake can affect an area of several kilometers wide, in addition to the displacement that occurs on the surface ruptures. However, there are no measurements of the coseismic displacement accommodated across these areas. In the present work, using large-aperture profiles (4-8 km) across the rupture zones of the 2019 Ridgecrest and 2013

Baluchistan earthquakes, we detected ground displacement gradients, in the strike-slip component, affecting the medium up to 2 km away from the faults. These gradients are homogeneous in space and have a slope of the order of 10 -3 -10 -4 . They are referred to as diffuse deformation. This diffuse deformation was not detected on the field because its typical patterns are not characterized by surface deformation markers like faulted offset features. It was not either measured in other optical image correlation studies that generally measure only the deformation on the faults using ~500-m long profiles. Along our study area of the 2013 Baluchistan earthquake, diffuse deformation represents only 4% of the total surface displacement budget of the earthquake, in the strike-slip component. The total displacement budget corresponds to the total displacement measured across the fault zone, including diffuse deformation and displacement on the faults. Along the 2013 Baluchistan ruptures, we also measured deformation away from the faults that was accommodated in a heterogeneous manner, and associated with cracks mapped on the field. This deformation represents another 12% of the total displacement budget of the earthquake that is not accommodated by the faults. However, it occurs in a more restricted area around the faults (<500 m) and cannot explain the larger scale (>1 km) deformation around faults. Along the 2019 Ridgecrest ruptures, diffuse deformation represents 30% of the total surface displacement budget of the earthquake, in the strikeslip component. Hence, diffuse deformation represents an important part of the total surface displacement that occurs during an earthquake. Following this observation, we raised the question of what is the role of diffuse deformation in the rupture process? And how does it compare to the slip distribution at depth? We answered to these questions using a joint inversion of surface displacement data for the 2019 Ridgecrest earthquakes, summerized hereafter in section 7.3.

Relation between diffuse deformation, earthquake process at depth, and fault geometry

The joint inversion of surface displacement data for the modeling of the 2019 Ridgecrest earthquake rupture process (section 4.2) integrates the near-fault high-resolution optical displacement maps calculated in this thesis (section 4.1). This allows to constrain the rupture process in the shallow part of the fault model. The results of this work show that the amount of slip occurring at depth on the fault plane is equivalent to the amount of displacement measured at the surface over wide-apertures profiles, which includes diffuse deformation in the medium and localized deformation on the faults. This has major implications for the understanding of the earthquake process, and shows that no surface fault creep is necessary during the inter-or post-seismic phases to reconcile the displacement occurring at depth along the fault, and what is occurring at the surface.

However, it requires to consider surface deformation over width larger than 1-2 km around the faults. In addition, it has major implications on the estimation of earthquake magnitude and fault slip rates from surface observations, since part of the displacement budget of the earthquake occurs away from the visible fault system. From this observation, we came to the conclusion that diffuse deformation is directly correlated with the shallow slip deficit (SSD) that occurs at depth on the fault plane.

We compared these results with the observations made along the 2013 Baluchistan rupture, and published observations and models for other strike-slip ruptures. From this comparison, we showed that the SSD is directly correlated with the magnitude of the earthquake, with larger SSD along Mw<7.5 ruptures, and no SSD along Mw>7.5 ruptures.

As a consequence, diffuse deformation is also correlated with the earthquake magnitude, with low SSD proportions occurring along Mw>7.5 ruptures. Because large magnitude (Mw>7.5) earthquake require to rupture the entire width of the fault plane to achieve such a large seismic moment release, maximum slip generally propagates to the surface on the fault ruptures, and no diffuse deformation is measured. As a consequence, fault slip displacements measured along the fault of such large magnitude earthquake ruptures should yield values equivalent to the total amount of surface deformation. Conversely, it means that fault slip displacements values measured along the faults of smaller magnitude (Mw<7.5) likely underestimate the total coseismic slip budget, up to 30% according to our study on the 2019 Ridgecrest earthquake.

Finally, we showed that the distribution of the diffuse deformation, along the strike of the fault system, was dependent on the geometry of the faults. We showed that the 2019

Ridgecrest ruptures were segmented with different proportions of diffuse deformation along each segment, and that these proportions were primarily controlled by the orientation of the fault segment relatively to the maximum regional compression axis.

Along the 2013 Baluchistan ruptures, we showed that the diffuse deformation concentrates in areas of fault complexity, at the transition between more linear fault sections. We found similar evidence, for other Mw>7.5 earthquakes, of diffuse deformation concentrated around major geometrical complexities in the fault trace. This shows that, despite the large magnitude of the earthquake, geometrical complexities act as barriers to the propagation of slip on the fault plane. As a consequence, part of the fault displacement is accommodated in the surrounding medium through diffuse deformation.

Still, the total budget of the earthquake remains unchanged across the complexities.

Nature of diffuse deformation and impacts for earthquake hazard

In the last chapter of this work (sections 6.6), we discussed the possible mechanisms that could be responsible for the diffuse deformation patterns we observed. Because diffuse deformation was measured both in regions of sedimentary and crystalline rocks, we proposed that the predominant deformation processes would be brittle deformation of the rocks, implying fractures at a scale below what is detectable on the field or using optical image correlation. This proposition is supported by the fact that the wide diffuse deformation zone along the 2019 Ridgecrest ruptures is correlated with the distribution of aftershocks. Hence, diffuse deformation would be a brittle process, occurring coseismically or within the first hours and days following the rupture, and responsible for the radiation of seismic waves. This implies that the energy budget of the rupture, the seismic radiations, and the ground motions, include a contribution from the diffuse deformation (sections 6.7

). Yet, we do not know what exact mechanisms are brought into play during diffuse deformation, while it is a needed information to characterize the impact of diffuse deformation on the dynamic characteristics of the earthquake. These questions remain unanswered, and propositions for future research on this topic are proposed in the following chapter.

PERSPECTIVES: TOWARD A BETTER UNDERSTANDING OF DIFFUSE DEFORMATION AND FAULT COMPLEXITY

Three complementary research perspectives can be proposed to improve the characterization of diffuse deformation processes, and of its relation with the rupture pattern and fault geometry.

Optical image correlation

The first perspective is to apply the high-resolution optical image correlation method to every surface-rupturing earthquake for which high-resolution optical images will be available. Satellite or airborne images are preferable to image large rupture although drone images can also be used at a smaller scale. In addition to using the high-resolution correlation method to characterize fault slip complexity, one has to pay attention to the large-scale deformation gradients occurring around the primary fault ruptures to assess the diffuse deformation that was so far overlooked. However, imaging the earthquake displacement field using high-resolution optical image correlation requires to have highresolution optical images acquired before the earthquake ruptures. Moreover, to be able to orthorectify the images and measure the vertical displacement field, either stereo or tristereo acquisitions are required to reconstruct the 3D ground geometry before the rupture.

Possibly, this can be done using an external DSM if any is available. However, this DSM must have a resolution close to that of the images. This raises the issue of what images and topography data need to be acquired on active faults, such that the post-earthquake images can be compared to pre-earthquake dataset ones?

Moreover, we showed that all the coseismic displacement associated with the 2019

Ridgecrest and 2013 Baluchistan earthquakes reached the surface during the rupture or within the few days following the rupture. To better constrain the mechanisms that are brought into play in the diffuse deformation, one has to characterize the kinematics of diffuse deformation, and the timing relatively to the onset of rupture. This requires the acquisition of optical images dataset at high-resolution in the days and weeks after the rupture.

Benefitting from a recurrent high-resolution optical image acquisition on a fault zone could also provide information on the post-seismic, or inter-seismic surface deformations.

For example, creeping deformation is often measured using InSAR, GPS or creepmeters.

Although these methods are more sensitive that optical methods, optical image correlation could provide detailed maps (<0.5 m ground resolution) of the creeping fault displacement patterns within the fault zone. Still, this approach requires to detect centimeter-scale ground displacement, which could be achieved using centimeterresolution drone images for example.

Field investigations

The second perspective is an in-situ investigation of the rock properties and ground deformation features in the regions where diffuse deformation was measured from optical image correlation. Since the deformation likely occurred below what is detectable with naked eyes, subsurface geophysical technics should be used to investigate the mechanical properties of the rocks at the scale of the fault zone (~km). Seismic wave velocity [START_REF] Eberhart-Phillips | Surface seismic and electrical methods to detect fluids related to faulting[END_REF][START_REF] Vidale | Damage to the shallow Landers fault from the nearby Hector Mine earthquake[END_REF] and anisotropy [START_REF] Shi | A Study of Seismic Anisotropy of Wenchuan Earthquake Sequence[END_REF][START_REF] Zhang | Seismic anisotropy in the shallow crust of the Loma Prieta segment of the San Andreas Fault System[END_REF] are generally used to this aim. Radar-and electrical-based methods are also adapted to study the ground properties at a local scale [START_REF] Carrière | Combining Electrical Resistivity Tomography and Ground Penetrating Radar to study geological structuring of karst Unsaturated Zone[END_REF][START_REF] Eberhart-Phillips | Surface seismic and electrical methods to detect fluids related to faulting[END_REF]. In addition, rock strength and porosity could be measured in the laboratory [START_REF] References Aben | High strain rate deformation of porous sandstone and the asymmetry of earthquake damage in shallow fault zones[END_REF][START_REF] Hamiel | Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks[END_REF], and small-scale (< 1cm) deformations and fractures observed on rock thin sections under microscope (Mitchell and Faulkner, 2009).

Analogue modeling

In this thesis, we showed that the fault geometry controls the spatial distribution of diffuse deformation. We also proposed that the fault geometry is influenced by the geological structures, and more particularly by the pre-existing faults affecting the basement. Beside numerical modeling that allows to study fault complexity (e.g., [START_REF] Fattaruso | Response of deformation patterns to reorganization of the southern San Andreas fault system since ca. 1.5 Ma[END_REF][START_REF] Jiao | Fault Segmentation Pattern Controlled by Thickness of Brittle Crust[END_REF] and inelastic deformation around faults (e.g., [START_REF] Finzi | Structural Properties and Deformation Patterns of Evolving Strike-slip Faults: Numerical Simulations Incorporating Damage Rheology[END_REF]Roten et al., 2017), one way to analyze the relation between diffuse deformation, fault geometry, and pre-existing structures, is the modeling of faults and associated deformation using clay box [START_REF] Cooke | Evolution of fault efficiency at restraining bends within wet kaolin analog experiments[END_REF] or sand box models [START_REF] Dooley | Analogue modelling of intraplate strike-slip tectonics: A review and new experimental results[END_REF]. Using these This type of modeling is widely use to study the construction of complexities, like relay zones [START_REF] Cooke | Evolution of fault efficiency at restraining bends within wet kaolin analog experiments[END_REF][START_REF] Dooley | Geometries and kinematics of strike-slip fault systems : insights from physical modelling and field studies[END_REF]. Applied with larger amounts of cumulative deformation on the fault system, this approach could provide insights on fault evolution over long time periods, and on the co-existence between different types of complexities in the fault zone.

Signature of transition to supershear rupture speed in coseismic off-fault damage zone

Jorge Jara ,8 and Harsha S.

Bhat 1

Most earthquake ruptures propagate at speeds below the shear wave velocity within the crust, but in some rare cases, ruptures reach supershear speeds.

The physics underlying the transition of natural subshear earthquakes to supershear ones is currently not fully understood. Most observational studies of supershear earthquakes have focused on determining which fault segments sustain fully-grown supershear ruptures. Experimentally cross-validated numerical models have identified some of the key ingredients required to trigger a transition to supershear speed. However, the conditions for such a transition in nature are still unclear, including the precise location of this transition. In this work, we provide theoretical and numerical insights to identify the precise location of such a transition in nature. We use fracture mechanics arguments with multiple numerical models to identify the signature of supershear transition in coseismic offfault damage. We then cross-validate this signature with high-resolution observations of fault zone width and early aftershock distributions. We confirm that the location of the transition from subshear to supershear speed is characterized by a decrease in the width of the coseismic off-fault damage zone. We thus help refine the precise location of such a transition for natural supershear earthquakes.

Introduction

While most earthquake ruptures propagate at speeds below the shear wave speed (sub-Rayleigh regime), some earthquakes can occasionally accelerate above the shear wave speed. Such events are known as supershear earthquakes. The earthquake rupture speed, and specifically its abrupt changes, control the high-frequency radiation [1,2]. As an earthquake goes to supershear speeds, it manifests Mach fronts that produce unusually large ground motion at distances far from the fault [3][START_REF] Dunham | Near-source ground motion from steady state dynamic rupture pulses[END_REF][START_REF] Dunham | Attenuation of radiated ground motion and stresses from threedimensional supershear ruptures[END_REF]. Rupture speed also governs the spatial extent of the off-fault coseismic damage zone, i.e., the volume within the crust directly surrounding the rupture that experienced mechanical damage [START_REF] Bhat | Off-fault damage patterns due to supershear ruptures with application to the 2001 M w 8.1 Kokoxili (Kunlun) Tibet earthquake[END_REF][START_REF] Templeton | Off-fault plasticity and earthquake rupture dynamics: 1. Dry materials or neglect of fluid pressure changes[END_REF][START_REF] Thomas | Effect of Brittle Off-Fault Damage on Earthquake Rupture Dynamics[END_REF][9][START_REF] Okubo | Dynamics, Radiation, and Overall Energy Budget of Earthquake Rupture With Coseismic Off-Fault Damage[END_REF][START_REF] Okubo | Modeling earthquakes with off-fault damage using the combined finite-discrete element method[END_REF]. Coseismic off-fault damage refers to fractures created or reactivated in the off-fault volume due to the dynamic rupture. As variations in rupture speed affect the seismic radiation and the off-fault coseismic damage, understanding the conditions for a rupture to transition from the sub-Rayleigh to the supershear regime would greatly help constrain fault properties (geometry, friction, lithology, etc.) and traction conditions that promote supershear ruptures. In addition, knowing how, why, and where earthquakes attain supershear speeds would help in the reliable estimation of earthquake hazard assessment [1,[START_REF] Das | GEOPHYSICS: The Need to Study Speed[END_REF]. Whether supershear ruptures occur in nature was a matter of debate for a long time. While theoretical and numerical models demonstrated in the early 1970s that earthquakes could propagate at supershear speeds [START_REF] Burridge | Admissible Speeds for Plane-Strain Self-Similar Shear Cracks with Friction but Lacking Cohesion[END_REF][START_REF] Hamano | Dependence of rupture time history on the heterogeneous distribution of stress and strength on the fault plane[END_REF][START_REF] Andrews | Rupture velocity of plane strain shear cracks[END_REF][START_REF] Das | A Numerical Study Of Rupture Propagation And Earthquake Source Mechanism[END_REF][START_REF] Das | A numerical study of two-dimensional spontaneous rupture propagation[END_REF][START_REF] Burridge | The stability of a rapid mode II shear crack with finite cohesive traction[END_REF][START_REF] Freund | The mechanics of dynamic shear crack propagation[END_REF], the absence of field observation and their extreme rarity in laboratory experiments [START_REF] Wu | Stick-slip propagation velocity and seismic source mechanism[END_REF] first suggested that supershear earthquakes could not exist in nature. It was not until the Mw 6.5 Imperial Valley earthquake (California, 1979) that a supershear rupture was inferred for the first time [START_REF] Olson | Finite faults and inverse theory with applications to the 1979 Imperial Valley earthquake[END_REF][START_REF] Archuleta | A faulting model for the 1979 Imperial Valley earthquake[END_REF][START_REF] Spudich | Imperial Valley Earthquake Using a Short Baseline[END_REF]. Pioneering laboratory experiments [START_REF] Rosakis | Cracks Faster than the Shear Wave Speed[END_REF][START_REF] Xia | Laboratory Earthquakes: The Sub-Rayleigh-to-Supershear Rupture Transition[END_REF][START_REF] Passelègue | From Sub-Rayleigh to Supershear Ruptures During Stick-Slip Experiments on Crustal Rocks[END_REF] together with observations from the 1999 Mw 7.4 Izmit and the 1999 Mw 7.2 Düzce earthquakes in Turkey [START_REF] Bouchon | Seismic imaging of the 1999 Izmit (Turkey) Rupture inferred from the near-fault recordings[END_REF][START_REF] Bouchon | How fast is rupture during an earthquake? New insights from the 1999 Turkey Earthquakes[END_REF], then conclusively confirmed that supershear ruptures are more common than previously expected. Supershear ruptures have now been inferred for several, albeit rare, events: the 2001 Mw 7.8 Kunlun (China) earthquake [START_REF] Bouchon | Observation of long supershear rupture during the magnitude 8.1 Kunlunshan earthquake[END_REF][START_REF] Robinson | The M w 7.8, 2001 Kunlunshan earthquake: Extreme rupture speed variability and effect of fault geometry[END_REF][START_REF] Vallée | The 14 November 2001 Kokoxili (Tibet) earthquake: High-frequency seismic radiation originating from the transitions between sub-Rayleigh and supershear rupture velocity regimes[END_REF], the 2002 Mw 7.8 Denali (Alaska) earthquake [START_REF] Ellsworth | Near-Field Ground Motion of the 2002 Denali Fault, Alaska, Earthquake Recorded at Pump Station 10[END_REF][START_REF] Dunham | Evidence for a Supershear Transient during the 2002 Denali Fault Earthquake[END_REF], the 2010 Mw 6.9 Qinghai (China) earthquake [START_REF] Wang | The 2010 Qinghai, China, Earthquake: A Moderate Earthquake with Supershear Rupture[END_REF], the 2012 Mw 8.6 off-Sumatra (Indonesia) earthquake [START_REF] Wang | Supershear rupture on multiple faults for the Mw8.6 Off Northern Sumatra, Indonesia earthquake of[END_REF], the 2013 Mw 7.5 Craig (Alaska) earthquake [START_REF] Yue | Supershear rupture of the 5 January 2013 Craig, Alaska ( M w 7.5) earthquake[END_REF], the 2013 Mw 6.7 Okhotsk (Kamtchatka) earthquake [START_REF] Zhan | Supershear rupture in a Mw 6.7 aftershock of the 2013 Sea of Okhotsk earthquake[END_REF][START_REF] Zhan | Supershear rupture in the 24 May 2013 M w 6.7 Okhotsk deep earthquake: Additional evidence from regional seismic stations[END_REF], the 2013 Mw 7.7 Balochistan (Pakistan) earthquake [START_REF] Wang | Backprojection analyses from four regional arrays for rupture over a curved dipping fault: TheMw 7.7 24 September 2013 Pakistan earthquake[END_REF], and most recently the 2018 Mw 7.5 Palu (Indonesia) earthquake [START_REF] Bao | Early and persistent supershear rupture of the 2018 magnitude 7.5 Palu earthquake[END_REF][START_REF] Socquet | Evidence of supershear during the 2018 magnitude 7.5 Palu earthquake from space geodesy[END_REF].

Several efforts have been made to identify and characterize these supershear earthquakes using various methods [START_REF] Bouchon | How fast is rupture during an earthquake? New insights from the 1999 Turkey Earthquakes[END_REF][START_REF] Vallée | The 14 November 2001 Kokoxili (Tibet) earthquake: High-frequency seismic radiation originating from the transitions between sub-Rayleigh and supershear rupture velocity regimes[END_REF][START_REF] Bao | Early and persistent supershear rupture of the 2018 magnitude 7.5 Palu earthquake[END_REF][START_REF] Bouchon | The Aftershock Signature of Supershear Earthquakes[END_REF][START_REF] Walker | Illuminating the near-sonic rupture velocities of the intracontinental Kokoxili M w 7.8 and Denali fault M w 7.9 strike-slip earthquakes with global P wave back projection imaging[END_REF]. The methods are, in most cases, designed to reveal along which segment the rupture propagated at supershear velocities. The location of the transition from sub to supershear speeds is usually inferred as follows. The fault is subdivided into segments for which an average rupture speed is determined, using kinematic inversion methods. If the average rupture speed of a segment is deduced to be supershear, supershear transition is presumed to have occurred in between subshear and supershear segments. However, due to the lack of dense near-fault records, rupture speed estimates from seismological inversions and backprojection techniques rely on teleseismic records, leading to rough estimates of the transition's location (i.e., tens of kilometers precision). Even in a situation where near-field data exists, such as during the 2002 Denali earthquake [START_REF] Ellsworth | Near-Field Ground Motion of the 2002 Denali Fault, Alaska, Earthquake Recorded at Pump Station 10[END_REF][START_REF] Dunham | Evidence for a Supershear Transient during the 2002 Denali Fault Earthquake[END_REF], records are still too sparse to precisely locate the transition.

Numerous theoretical and numerical models have been developed to characterize the mechanics of supershear transition. On planar faults with homogeneous stress and strength, the occurrence of supershear ruptures depends on the S ratio: S =(τpτ 0 )/(τ 0 -τr), where τ 0 is the background shear stress and τp and τr are the peak and residual frictional strengths [START_REF] Andrews | Rupture velocity of plane strain shear cracks[END_REF][START_REF] Das | A Numerical Study Of Rupture Propagation And Earthquake Source Mechanism[END_REF][START_REF] Das | A numerical study of two-dimensional spontaneous rupture propagation[END_REF]. Supershear ruptures occur if S<1.77 in 2D [START_REF] Andrews | Rupture velocity of plane strain shear cracks[END_REF][START_REF] Das | A numerical study of two-dimensional spontaneous rupture propagation[END_REF], or S<1.19 in 3D [START_REF] Dunham | Conditions governing the occurrence of supershear ruptures under slipweakening friction[END_REF], and the fault length is long enough for transition to occur [START_REF] Andrews | Rupture velocity of plane strain shear cracks[END_REF]. The time and location of the transition are determined by how far S is from the Burridge-Andrews threshold of 1.77. However, natural faults do not have homogeneous stress and strength conditions before an earthquake. Numerical studies of supershear transition have also focused on spatial heterogeneity in stress and/or strength distribution, on a planar fault [START_REF] Dunham | Conditions governing the occurrence of supershear ruptures under slipweakening friction[END_REF][START_REF] Dunham | A supershear transition mechanism for cracks[END_REF][START_REF] Liu | Transition of mode II cracks from sub-Rayleigh to intersonic speeds in the presence of favorable heterogeneity[END_REF] and on a non-planar fault [START_REF] Shi | Rupture dynamics and ground motion from 3-D rough-fault simulations[END_REF][START_REF] Bruhat | Rupture complexity and the supershear transition on rough faults[END_REF][START_REF] Hu | Supershear transition mechanism induced by step over geometry[END_REF], while Kaneko & Lapusta [50] and Hu et al. [51] looked at the effect of the free surface on supershear transition.

Despite these efforts, the conditions for supershear transition in nature are still poorly understood because the imprecise location of the transition does not allow us to properly characterize the conditions on, or off, the fault that led to it. In addition, it also appears difficult to precisely capture this transition in both laboratory earthquakes (e.g., Mello [START_REF] Mello | Identifying the Unique Ground Motion Signatures of Supershear Earthquakes: Theory and Experiments[END_REF]) and numerical models. This problem is mainly because the transition happens over very small length scales (of millimeters in the lab and over a few elements in numerical methods). A new approach that focuses specifically on the characterization of the region of transition is thus required to better understand why earthquakes accelerate to supershear speeds.

Although the modeling efforts helped better understand the physics of supershear ruptures, it is not straightforward to apply that knowledge in the field, especially since we do not directly measure the stress and strength along the fault. Therefore, we propose a new approach based on the physical characterization of parameters that may promote supershear ruptures. This approach focuses on the detailed analysis of coseismic off-fault damage as a proxy to characterize the transition to supershear speeds. In this work, we cross-validate this idea by writing a theoretical analysis and comparing it with several numerical approaches. Then, we look at the observations available for natural supershear ruptures to corroborate this approach.

Coseismic Off-Fault Damage Around The Rupture Tip While

Approaching Supershear Transition: Theoretical Analysis Coseismic off-fault damage depends on the evolution of stress during the propagation of a rupture in the crust. We use Linear Elastic Fracture Mechanics (LEFM) to provide closed-form solutions to quantify the state of stress around the rupture tip.

Consider a semi-infinite plain-strain crack that moves at speed v ≤ c R , where c R is the Rayleigh wave speed. Let the origin of the polar coordinate system (r, θ) coincide with the crack tip. The near-tip stress, which also depends on the rupture speed v, is given as [START_REF] Freund | Dynamic Fracture Mechanics[END_REF],

σ αβ (r, θ,v)= K dyn II (v) √ 2πr f II αβ (θ,v)+σ 0 αβ α, β =1, 2 (2.1
)

σ 33 = ν(σ 11 + σ 22 ) (2.2)
where σ 0 αβ is the initial stress state and K dyn II is the dynamic stress intensity factor which can be approximated as

K dyn II ≈ 1 -v/c R p 1 -v/cp K sta II ( L) (2.3)
where cp, the P-wave speed, is the limiting speed for a mode II crack and L = R t 0 v(t)dt is the current crack length and c R is the Rayleigh wave speed [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]. Here K sta II (L)=∆τ √ πL is the static stress intensity factor of the crack and ∆τ is the stress drop defined as σ 0 yx -τr. Here σ 0 yx is the initial shear stress on the fault and τr is the residual frictional strength. This solution by Freund [START_REF] Freund | Crack propagation in an elastic solid subjected to general loading-II. Nonuniform rate of extension[END_REF] allows us to transform a static solution into a dynamic one. The rupture velocity dependence of the stress field makes it undergo a Lorentz-like contraction affecting both the stress field and its angular distribution as the rupture speed approaches the Rayleigh wave speed, c R [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]. This contraction has already been observed and verified experimentally for ruptures approaching the Rayleigh wave speed [START_REF] Svetlizky | Classical shear cracks drive the onset of dry frictional motion[END_REF]. For a crack-like rupture, there is a competing effect of the increase in stress intensity factor with increasing crack length. If it is a stable slip pulse of fixed length, as most earthquake ruptures [START_REF] Heaton | Evidence for and implications of self-healing pulses of slip in earthquake rupture[END_REF], then the effect of the crack length is invariant with rupture speed.

When an earthquake rupture transitions to the supershear regime, the rupture first accelerates to the Rayleigh wave speed. Within the framework of LEFM, as the rupture approaches the Rayleigh wave speed, K dyn II monotonically decreases to zero, strongly reducing the stress concentration at the rupture tip. Thus the off-fault domain affected by this stress concentration also shrinks. We illustrate this effect by calculating the extent of the region where the stress state exceeds the Drucker-Prager failure criterion, i. e. F DP ≥ 0, where pδ ij ) and tan φ = f is the static coefficient of friction. This yield criterion, effectively a smooth approximation to the Mohr-Coulomb law using invariants of the stress tensor [START_REF] Drucker | Soil Mechanics and Plastic Analysis or Limit Design[END_REF], takes into account all possible planes of slip, as opposed to Mohr-Coulomb where the potential slip planes need to be defined a priori or are optimally oriented.

F DP = p J 2 + p tan φ. ( 2
Using the LEFM solutions for a mode II crack, we can compute F DP . The closed-form expressions are quite long, but the key variables that affect the solution can be written as

F DP = F " 1 √ r , p L 1 -v/c R p 1 -v/cp ! , ∆τ, θ, ν, tan φ, σ 0 αβ # (2.5)
Thus, given a rupture velocity history, v(t), one can then calculate the region around the fault where the Drucker-Prager yield criterion is violated.

We illustrate the solution in Figure 1. Consider a fault in an elastic medium subject to an initial stress state (see Table 1 for all the parameters used). Assume that the rupture is accelerating to the Rayleigh wave speed (Figure 1a). The instantaneous static stress intensity factor increases with the increasing length of the crack (Figure 1b). However, Freund's universal function, F II (v), monotonically decreases to 0 (Figure 1c). Consequently, the dynamic stress intensity factor first increases with increasing crack length but soon starts decreasing as it is dominated by F II (v) (Figure 1d). Thus, we should expect to see an initial increase in the spatial extent of the damage followed by a reduction as the rupture approaches the limiting speed. We illustrate this effect at key moments in the rupture's history (marked by points numbered 1 through 9 in Figure 1d). First, we compute, in Figure 1e, the 'instantaneous' damage, i. e., not accounting for the damage accumulated from the past history of the rupture around the rupture tip. We observe that the spatial extent of the domain where Drucker-Prager failure criteria is violated first increases (time points 1 through 3) and then decreases (time points 4 through 9) as the rupture accelerates. We can now superimpose all the snapshots of instantaneous damage to visualize the 'cumulative' damage accumulated by the rupture throughout its history (Figure 1f). Note that this last exercise is purely for illustrative purposes as the LEFM solution does not really account for the history of the rupture. These limitations will be overcome in the following sections. Nevertheless, to the first order, this approach illustrates that, as the rupture approaches the Rayleigh wave speed, the width of the coseismic off-fault damage zone should decrease significantly.

It is also worth noting that in the calculations above, the off-fault medium remains elastic. Thus, the rupture is insensitive to changes in the constitutive response of the medium due to coseismic off-fault damage. To resolve these issues, we need to take advantage of numerical simulations that allow for proper constitutive descriptions of the off-fault medium, including feedback from changes in elastic properties on the rupture behavior and that also allow for transition from subshear to supershear speeds.

Coseismic Off-Fault Damage Around The Rupture Tip During

Supershear Transition: Numerical Analysis While many efforts have been made to reveal the dynamics of supershear ruptures [START_REF] Bhat | Off-fault damage patterns due to supershear ruptures with application to the 2001 M w 8.1 Kokoxili (Kunlun) Tibet earthquake[END_REF][START_REF] Dunham | Conditions governing the occurrence of supershear ruptures under slipweakening friction[END_REF][START_REF] Dunham | A supershear transition mechanism for cracks[END_REF][START_REF] Liu | Transition of mode II cracks from sub-Rayleigh to intersonic speeds in the presence of favorable heterogeneity[END_REF][START_REF] Shi | Rupture dynamics and ground motion from 3-D rough-fault simulations[END_REF][START_REF] Kaneko | Supershear transition due to a free surface in 3-D simulations of spontaneous dynamic rupture on vertical strike-slip faults[END_REF][START_REF] Gabriel | Source properties of dynamic rupture pulses with off-fault plasticity[END_REF], the importance, during such event, of the complex feedback between the dynamic rupture and the mechanically evolving medium was not fully considered. However, new modeling approaches have recently emerged, allowing for dynamic activation of coseismic off-fault damage around faults and its feedback on rupture dynamics. In this section, we discuss two modeling strategies that allow for off-fault damage associated with brittle fracture. In the following, we show that the observed damage pattern does not depend on any specific constitutive law used to account for damage. Instead, crustal damage is a universal feature that can aid in identifying the region of supershear transition in the field.

(a) Modeling strategies

The two numerical models presented in this study account for fracture damage-dominated brittle rheology. Earthquake ruptures induce large dynamic strain rates ( ≥ 1s -1 ) around the rupture tip that cannot be fully accounted for by classical Mohr-Coulomb or Drucker-Prager plasticity, typically used low strain rates ( ≤ 10 -5 s -1 ) and the two models presented below allow to account for such large dynamic strain. Okubo et al. [START_REF] Okubo | Dynamics, Radiation, and Overall Energy Budget of Earthquake Rupture With Coseismic Off-Fault Damage[END_REF][START_REF] Okubo | Modeling earthquakes with off-fault damage using the combined finite-discrete element method[END_REF] explicitly model the sub-kilometric off-fault fractures using a continuum-discontinuum numerical framework. The micromechanical model [START_REF] Thomas | Effect of Brittle Off-Fault Damage on Earthquake Rupture Dynamics[END_REF]9] accounts for coseismically activated "microfractures" (∼10's of meters) by computing the dynamic changes in the constitutive response due to these fractures. The difference between the two models is essentially in the modeled off-fault fracture networks scale, leading to two distinct modeling strategies. In essence, the two brittle failure models are complementary, and they should, ideally, be combined. Nevertheless, they are good proxies for modeling off-fault damage at various scales, and we further show that they produce similar results.

To reproduce coseismic damage at relatively large scales (i.e., 50m to several km), Okubo et al. [START_REF] Okubo | Dynamics, Radiation, and Overall Energy Budget of Earthquake Rupture With Coseismic Off-Fault Damage[END_REF][START_REF] Okubo | Modeling earthquakes with off-fault damage using the combined finite-discrete element method[END_REF] use the software suite HOSSedu, developed by the Los Alamos National Laboratory (LANL) [START_REF] Rougier | OSS.edu2.x (Hybrid Optimization Software Suite -257Educational Version[END_REF][START_REF] Knight | HOSS: an implementation of the combined finite-discrete element method[END_REF]. The numerical algorithms behind this tool are based on the combined Finite-Discrete Element Method (FDEM) proposed by Munjiza [START_REF] Munjiza | The combined finite-discrete element method[END_REF], to produce dynamically activated off-fault fracture networks. The key feature here is that FDEM allows each interface between the finite elements, describing the off-fault medium, to have its own tensile and shear failure criterion. Thus, these interfaces can rupture under appropriate traction conditions. When the earthquake rupture propagates, the dynamic stress field around its tip will increase to violate the tensile or shear failure criteria leading to off-fault fracture damage. The lower limit of fracture resolution is around 50m, which is defined by the minimum size of the discretized mesh. For a further detailed description of the method, see Okubo et al. [START_REF] Okubo | Dynamics, Radiation, and Overall Energy Budget of Earthquake Rupture With Coseismic Off-Fault Damage[END_REF][START_REF] Okubo | Modeling earthquakes with off-fault damage using the combined finite-discrete element method[END_REF].

The second modeling strategy relies on laboratory experiments [START_REF] Walsh | a The effect of cracks in rocks on Poisson's ratio[END_REF][START_REF] Walsh | The effect of cracks on the compressibility of rock[END_REF][START_REF] Faulkner | Slip on 'weak' faults by the rotation of regional stress in the fracture damage zone[END_REF] and field observations [START_REF] Hiramatsu | Seismological evidence on characteristic time of crack healing in the shallow crust[END_REF][START_REF] Froment | Imaging along-strike variations in mechanical properties of the Gofar transform fault, East Pacific Rise[END_REF] that show significant changes in elastic properties related to fracturing. Observations in the field show up to 40% coseismic reduction in P-and S-wave velocities on spatial scales of hundreds of meters normal to the fault and few kilometers in depth. Following the damage constitutive laws proposed by Ashby & Sammis [67], Deshpande and Evans [START_REF] Deshpande | Inelastic deformation and energy dissipation in ceramics: A mechanism-based constitutive model[END_REF], Bhat et al. [69], Thomas et al. [8] implemented an energy-based micromechanical brittle rheology, as developed by Rice [START_REF] Rice | Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms[END_REF], to account for such dynamic change of bulk rheological properties during earthquakes. In short, at each equilibrium state, the Gibbs free energy density Ψ of the damaged solid is defined as the sum of (1) the free energy Ψ e of a solid, without flaws, deforming purely elastically and (2) the free energy Ψ i corresponding to the contribution of the current set of microcracks. Using thermodynamic arguments, Ψ = Ψ e + Ψ i is used to derive the new stress-strain constitutive law and the changes of elastic properties in the medium at the equilibrium stage. In the model, following laboratory experiments, the evolution of Ψ i is determined by taking into account the effect of loading rate and crack-tip velocities on crack growth (see Zhang & Zhao [71] for an overview). A complete description of the model can be found in Bhat et al. [69] and Thomas et al. [START_REF] Thomas | Effect of Brittle Off-Fault Damage on Earthquake Rupture Dynamics[END_REF].

(b) Model set up

For comparison purposes, we set up the two brittle rheology models to be as close to each other as possible. In both cases, we consider a 2-D plane strain medium, with a 1-D right lateral fault embedded in it and loaded by uniform background stresses. The maximum compressive stress σ 1 , and the minimum compressive stress σ 3 are in the xy plane, whereas the intermediate principal stress σ 2 coincides with σzz. The fault plane makes an angle of 60 o with σ 1 with normal uniform traction (σ 0 yy ) and shear traction (σ 0 xy ) everywhere except in the nucleation zone of the micromechanical model where the shear traction is slightly above the nominal static strength. In the FDEM model, the dynamic rupture is initiated by locally decreasing the static friction, in the nucleation zone, instead of the local change of σ 0 xy . These different nucleation strategies do not affect the results since this study is focused on damage occurring when the rupture is dynamic (sub and supershear). For each model, the initial shear stress and the S-ratio were chosen so that the rupture transitions to supershear speed reasonably early on.

Rupture propagation along the main fault plane is governed by a slip-weakening friction law [START_REF] Palmer | The growth of slip surfaces in the progressive failure of overconsolidated clay[END_REF]. Static friction is set at 0.6, which corresponds to a value measured in laboratory experiments for a large range of rocks [START_REF] Byerlee | Friction of rocks[END_REF], and dynamic friction at 0.1 as observed in high slip-rate experiments [START_REF] Wibberley | Recent advances in the understanding of fault zone internal structure: a review[END_REF].

The fault length is 64km for the micromechanical model and 115km for the FDEM model. The domain width is determined based on the numerical method and on the scale of the fracture networks accounted for in the calculations. For the micromechanical model, the width around the fault is 6km with absorbing boundary conditions to avoid reflections interfering with the propagating dynamic rupture. For the FDEM model, we simply set the domain large enough (86km) so that the reflections do not arrive on the fault over the computation duration. We non- Process zone size (m) 1057 SS -ratio 1.2 dimensionalize all length scales by the static size of the process zone (R 0 ) so that the models could be compared to natural or laboratory earthquakes. R 0 is defined as

R 0 = 9π 16(1 -ν) µG (τp -τr) 2 . (3.1)
Here ν is Poisson's ratio, µ is shear modulus, G is the fracture energy associated with the slipweakening law, τp and τr are the static and dynamic frictional strengths, respectively. Reference values for the common parameters between the two models are summarized in Table 1, whereas parameters specific to different modeling strategies are provided in the Supplementary Information (Tables S1 &S2). Schematics and parameters for the simulations can also be found in Figure S1.

(c) Influence of rupture dynamics on off-fault damage

To understand the complex interaction between the main fault and the surrounding medium undergoing coseismic damage, it is necessary to simultaneously look at the rupture dynamics, the associated stress field around the fault, and the triggered damage in bulk. Here we use the micromechanical model to underline the key features that arise from numerical studies. We note that the same conclusions can be drawn from the FDEM model. In the micromechanical model, under a compressive regime, damage occurs by the growth of pre-existing "flaws". They represent the faults, joints, cracks, mineral twins, defects in the crystal structure, grain boundaries, etc., we observe in nature. Frictional sliding occurs on these preexisting fractures when the shear stress overcomes the frictional resistance acting on the fracture interface. As the faces slide in opposing directions, it creates a tensile wedging force that opens wing cracks at the tips of the shear fracture. The wing cracks grow when their stress intensity factors overcome the fracture toughness. In Figure 2 emphasize the regions where shear sliding is likely to occur, i.e., F DP > 0 (equation 2.4), and therefore where we expect wing cracks to grow. Fault slip rate (white curves) and cumulative damage (greyscale) are superimposed on each snapshot of F DP . New coseismic damage induced by the displayed stress field is highlighted in red. As the rupture propagates below the Rayleigh wave speed (Figure 2a), the damage is essentially generated behind the rupture front, in the tensional quadrant. This feature has been seen by Okubo et al. [10] as well, using the FDEM model. Damage also occurs ahead of the rupture front, where the S-wave field concentrates, although the resulting damage density is much smaller than behind the rupture front. The location of newly formed cracks corresponds to the region where F DP is positive. The transition to supershear velocity (Figure 2b) directly impacts the generation of coseismic damage. Around t =6.5s a new pulse is generated ahead of the rupture (Figure 3b). The transition to supershear velocity coincides with a decrease in the width of the damage zone (Figures 2b and3b-d). As we argued in section 2, this is likely related to the decrease in the stress intensity factor as the rupture speed increases. When approaching the Rayleigh wave speed, K dyn II becomes small, even if the total length of the fault that has ruptured ( L in equation 2.3), has increased significantly. Moreover, a snapshot of F DP at t =7.2s (Figure 2b) shows that the first pulse is not strong enough to change the state of stress optimally, thus reducing even further K dyn II . These two factors likely explain why damage mostly occurs behind the front of the pulse propagating at sub-Rayleigh rupture speed. After t =7.2s, the second pulse ahead of the rupture front becomes stronger and the first pulse weaker (Figure 3d-e). Consequently, damage now essentially occurs behind the rupture front propagating at supershear speed and in the area behind the Mach cone (Figure 2c). Therefore, even if the sub-Rayleigh pulse propagates now inside the "transition zone", it induces little to no further damage. Therefore, this local reduction in the damage zone width is expected to remain unchanged for the earthquake duration and could, in principle, be observed in the field.

Finally, it is worth noticing that while the surrounding medium records the transition to supershear rupture (via the damage zone width), the final displacement along the fault, that geodesy could provide, carries no such information. As one can see in Figure 3a, the slip continues to accumulate in the "transition zone" after the passage of the rupture, while the damage zone width remains unchanged (Figure 2c).

(d) Comparison between the different modeling strategies

We now perform a similar exercise using the FDEM model and choose parameters such that the micromechanical and FDEM models are as close to each other as possible. We adopt a slightly different yield criterion to reflect on the way damage is accounted for in this numerical method (see section (a) and Okubo et al. [10]). We evaluate the potential regions of failure, in shear, by calculating the invariant form of the Mohr-Coulomb yield function, F MC (see equation 4.142 of Chen & Han [START_REF] Chen | Plasticity for structural engineers[END_REF]):

F MC = R MC p J 2 + p tan φ -c (3.2)
where c is the cohesion and R MC is given by:

R MC =sin ⇣ Θ + π 3 ⌘ sec φ + 1 √ 3 cos ⇣ Θ + π 3 ⌘ tan φ (3.3)
where

cos 3Θ = 3 √ 3 2 J 3 J 3/2 2 ; J 3 = 1 3
s ij s jk s ki (3.4) where J 3 is the third invariant of the deviatoric stress tensor. Failure occurs when F MC ≥ 0.

Note that when R MC =1 and c =0, F MC becomes the Drucker-Prager criterion as described in equation 2.4. Figure 4 displays the F MC criteria when the rupture propagates at supershear velocities and the cumulative damage pattern resulting from the entire coseismic rupture. On top of the area undergoing F MC > 0 in relation to the rupture propagation along the main fault plane, we observe positive values at the tip of the discretized off-fault fractures, far away from the rupture front. This feature is not so evident within the micromechanical model because the fractures are homogenized in the constitutive law (section (a)). We also observe a shrinkage in the spatial extent of damage when the rupture transitions to supershear, as seen earlier with the micromechanical model.

In addition to the above described numerical experiments using brittle rheology, we acknowledge that Templeton & Rice et al. [START_REF] Templeton | Off-fault plasticity and earthquake rupture dynamics: 1. Dry materials or neglect of fluid pressure changes[END_REF] first explored the extent and distribution of off-fault plasticity during supershear rupture (see Figure 13 of Templeton & Rice et al. [START_REF] Templeton | Off-fault plasticity and earthquake rupture dynamics: 1. Dry materials or neglect of fluid pressure changes[END_REF]). They conducted a 2D in-plane dynamic rupture modeling with Drucker-Prager elasto-plastic constitutive law. In their model, the degree of damage is inferred from the accumulation of plastic strain. Although plasticity, as a proxy for damage, essentially holds only at low-strain rates [START_REF] Zhang | Effect of loading rate on fracture toughness and failure micromechanisms in marble[END_REF][START_REF] Davis | Plasticity and geomechanic[END_REF], they nevertheless pointed out a "remarkable contraction" in the damage zone width associated with supershear transition.

We can thus conclude, quite confidently, that this reduction in damage zone width (a gap in off-fault fractures) is a universal characteristic of supershear transition and is insensitive to the constitutive law used to model damage.

Natural observations of coseismic off-fault damage

Theoretical and numerical models, with three different rheological descriptions of off-fault damage [START_REF] Templeton | Off-fault plasticity and earthquake rupture dynamics: 1. Dry materials or neglect of fluid pressure changes[END_REF][START_REF] Thomas | Effect of Brittle Off-Fault Damage on Earthquake Rupture Dynamics[END_REF][START_REF] Okubo | Dynamics, Radiation, and Overall Energy Budget of Earthquake Rupture With Coseismic Off-Fault Damage[END_REF], all suggest that the region affected by the stress field around the rupture tip shrinks during the supershear transition. This shrinkage of the stress field results in a narrow off-fault damage zone. We now look for a direct (or indirect) signature of this reduced damage zone in the case of natural supershear earthquakes, verifying their location with corresponding kinematic models that invert for rupture speeds.

Vallage et al. [START_REF] Vallage | Inelastic surface deformation during the 2013 Mw 7.7 Balochistan, Pakistan, earthquake[END_REF] and Klinger et al. [78] showed that there is a one-to-one relationship between the features of the displacement field, around the fault, of an earthquake and off-fault fracture damage. Using the FDEM method described earlier, Klinger et al. [START_REF] Klinger | Earthquake Damage Patterns Resolve Complex Rupture Processes[END_REF] showed that the observed, spatially diffuse, displacement field around a fault is due to displacement accommodated by offfault fractures (see Figure 3 in [78]). This feature is in stark contrast with the sharp displacement field obtained, around a fault, when displacement is only accommodated by the main fault. Thus deviations from this sharp displacement field would allow us to characterize the width of the damage zone around a fault.

We can also make a reasonable assumption that the newly created/reactivated off-fault fractures are likely to host early off-fault aftershocks (based on the observed stress field). We should then expect a reduction in the spatial extent of the early off-fault aftershocks at the location of the supershear transition. A more thorough analysis would involve continuing the simulations described earlier for up to a week, or so, after the earthquake. However, computational limitations make this beyond the scope of the present study.

(a) Optical image correlation to observe off-fault coseismic damage

Recent developments in satellite optical image analysis and sub-pixel correlation methods allow for detecting displacement variations due to an earthquake down to sub-metric resolutions (e.g., Vallage et al. [77], Barnhart et al. [79], Delorme et al. [80] and Ajorlou et al. [81]). These methods enable characterizing the surface rupture geometry, the amount of surface displacement, and the width of the zone affected by this displacement after an earthquake, also referred to as the "fault zone width" [START_REF] Milliner | Quantifying near-field and off-fault deformation patterns of the 1992 M w 7.3 Landers earthquake[END_REF]. The fault zone width reflects the lateral extent of rock damage, and secondary deformation features around the fault [START_REF] Rockwell | Lateral Offsets on Surveyed Cultural Features Resulting from the 1999 Izmit and Duzce Earthquakes, Turkey[END_REF][START_REF] Mitchell | The nature and origin of off-fault damage surrounding strikeslip fault zones with a wide range of displacements: A field study from the Atacama fault system, northern Chile[END_REF]. In the field, the fault zone width corresponds to the last deformation features observed when moving away from the fault core [START_REF] Rockwell | Lateral Offsets on Surveyed Cultural Features Resulting from the 1999 Izmit and Duzce Earthquakes, Turkey[END_REF]. The same definition is employed in geodetic studies providing a more detailed characterization of this region due to the density and resolution of the observations [START_REF] Klinger | Earthquake Damage Patterns Resolve Complex Rupture Processes[END_REF].

The 2001 Mw 7.8 Kunlun (China) earthquake is a strike-slip event with ∼ 400km long surface rupture and a mean rupture speed between 3.3-3.9 km/s, which is larger than the shear wave speed and hence reported as a supershear rupture [START_REF] Bouchon | Observation of long supershear rupture during the magnitude 8.1 Kunlunshan earthquake[END_REF][START_REF] Robinson | The M w 7.8, 2001 Kunlunshan earthquake: Extreme rupture speed variability and effect of fault geometry[END_REF]. This earthquake was well recorded by satellite images, allowing us to investigate the spatial distribution of coseismic off-fault damage by focusing on the fault zone width. We use SPOT-1 to SPOT-4 images, covering the 2001 Kunlun fault area from 1988 to 2004 with a ground resolution of 10m, allowing for change detection of less than 1m [START_REF] Vallage | Inelastic surface deformation during the 2013 Mw 7.7 Balochistan, Pakistan, earthquake[END_REF][START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF]. We focus on the central part of the rupture dominated by fault parallel motion [START_REF] Klinger | High-Resolution Satellite Imagery Mapping of the Surface Rupture and Slip Distribution of the Mw 7.8, 14 November 2001 Kokoxili Earthquake, Kunlun Fault[END_REF][START_REF] Klinger | Evidence for an earthquake barrier model from Mw 7.8 Kokoxili (Tibet) earthquake slip-distribution[END_REF], obtaining through optical image correlation the horizontal displacement field for the earthquake east-west and north-south components (Figure S2). The maximum strike-slip displacement for this area of the rupture is around 8 m [START_REF] Klinger | High-Resolution Satellite Imagery Mapping of the Surface Rupture and Slip Distribution of the Mw 7.8, 14 November 2001 Kokoxili Earthquake, Kunlun Fault[END_REF][START_REF] Klinger | Evidence for an earthquake barrier model from Mw 7.8 Kokoxili (Tibet) earthquake slip-distribution[END_REF]. Prior to correlation, SPOT images are corrected from viewing angle and topography using a unique 90m SRTM digital surface model. The pre-and post-earthquake corrected paired images are correlated using the MicMac package [START_REF] Rupnik | MicMac -a free, open-source solution for photogrammetry[END_REF][START_REF] Rosu | Measurement of ground displacement from optical satellite image correlation using the free open-source software MicMac[END_REF], which measures the horizontal surface displacement between the acquisitions (see Supplementary Information, Figure S2 to see the data employed in this study). The MicMac method enables preserving the resolution of the input images, that is 10m, since it measures the displacement at every pixel location in the preearthquake image. The pixel size pattern considered in the pre-earthquake images searched in the post-earthquake ones is 5 pixels, so 50 m. We applied. A regularization of 0.3 is used to oblige MicMac to consider realistic displacement values relative to neighboring pixels, even if this procedure assumes pixels with a lower correlation score (regularization 0 in Micmac means taking the best correlation score without regarding on the coherence of surrounding pixels). Output displacement maps show displacement at every pixel location ( every 10m) in the eastwest and north-south directions. We place 460 stacked profiles perpendicular to the fault trace in the displacement map (Figure S2) every 500 m to analyze variations in the surface displacements and fault zone width. Individual slip profiles are stacked on rectangles of 40km-long and 1kmwide, which reduces noise and artifacts (Stacking profiles from ENVI version 5.5.1; Exelis Visual Information Solutions, Boulder, Colorado). On each profile, displacement is estimated in the faultparallel direction, observing the coseismic offset produced by the earthquake. (see Supplementary Information, Figure S3 for a profile example). This procedure allows us to evaluate the fault zone width in the strike-slip direction, over 396 profiles (Fig. 5b). Field measurements of fault zone widths are at least a few tens to a few hundred meters in the case of this earthquake [START_REF] Klinger | High-Resolution Satellite Imagery Mapping of the Surface Rupture and Slip Distribution of the Mw 7.8, 14 November 2001 Kokoxili Earthquake, Kunlun Fault[END_REF][START_REF] Klinger | Evidence for an earthquake barrier model from Mw 7.8 Kokoxili (Tibet) earthquake slip-distribution[END_REF]. Thus, we infer from our results that using displacement maps with a ground resolution of 10m, a reliable estimation of the fault zone width can be detected employing satellite image correlation.

The analysis of the profiles highlights three main domains in fault zone width. The first domain, which represents the majority of the rupture, exhibits a mean value of 238 ± 80m, and it corresponds to the strike-slip section of the rupture where only one fault zone is observed at the surface (Figures 5, S4A , S5). The second domain presents a width larger than 1km lasting for about 15km. This section shows a complex rupture pattern, where the rupture splits into two fault zones evolving into slip-partitioning between pure strike-slip along the southern fault strand and pure normal faulting along the northern strand. The slip-partitioning lasts for about 70 km eastward [START_REF] Klinger | High-Resolution Satellite Imagery Mapping of the Surface Rupture and Slip Distribution of the Mw 7.8, 14 November 2001 Kokoxili Earthquake, Kunlun Fault[END_REF][START_REF] King | Slip-Partitioned Surface Breaks for the Mw 7.8 2001 Kokoxili Earthquake, China[END_REF] ( Figures 5,S4B,S5). The third domain is located right before the onset of the slip-partitioning region, where the average fault zone width drops to 127 ± 39m over 11km long segment. This value is about half of the mean fault zone width of 238m calculated for the whole fault zone (ignoring the slip-partitioning area). This area co-locates with the region where the rupture is inferred, from teleseismic data, to have transitioned to the supershear regime [START_REF] Bouchon | Observation of long supershear rupture during the magnitude 8.1 Kunlunshan earthquake[END_REF][START_REF] Robinson | The M w 7.8, 2001 Kunlunshan earthquake: Extreme rupture speed variability and effect of fault geometry[END_REF][START_REF] Vallée | The 14 November 2001 Kokoxili (Tibet) earthquake: High-frequency seismic radiation originating from the transitions between sub-Rayleigh and supershear rupture velocity regimes[END_REF]. The onset of slip-partitioning cannot cause a reduction in the damage zone width because the normal faulting motion was quite likely triggered by the dynamic stress field of the main supershear segment [START_REF] Bhat | Off-fault damage patterns due to supershear ruptures with application to the 2001 M w 8.1 Kokoxili (Kunlun) Tibet earthquake[END_REF]. This means that the overall dynamics in that region was still dominated by the rupture on the strike-slip segment. With these observations, and considering the theoretical analysis and numerical modeling conducted in the previous sections, we interpret this localized fault zone width reduction as a shrinkage of the off-fault damage zone associated with the rupture transitioning to supershear speeds (e.g., Figure 2). Hence, this characterized region represents a natural observation of the supershear transition zone.

(b) Distribution of early aftershocks as a proxy for off-fault coseismic damage Assuming that the nucleation of early aftershocks is mainly governed by the stress state left in the wake of the earthquake, we test whether the extent of coseismic off-fault damage in nature manifests in the spatial distribution of early off-fault aftershocks. The logic is that considering the state of stress in the wake of the earthquake, weakened regions will preferentially host early aftershocks, and thus we should observe a region with less events where the rupture transitioned to supershear. When re-analyzing the regions where established supershear ruptures transitioned from sub-Rayleigh to supershear speeds, we might find local minima in the spatial extent of early aftershocks. If this region coincides with kinematic observations of supershear transition, we would have an independent and more precise location of supershear transition. We analyze three reported supershear ruptures for which high-resolution aftershock catalogs are available, at least up to one month after the main event. The 1999 Mw 7.4 Izmit earthquake (Turkey) had a mean rupture velocity of about 4.8km/s [START_REF] Bouchon | Seismic imaging of the 1999 Izmit (Turkey) Rupture inferred from the near-fault recordings[END_REF][START_REF] Bouchon | How fast is rupture during an earthquake? New insights from the 1999 Turkey Earthquakes[END_REF], and a well-recorded 3-month aftershock catalog [START_REF] Bouchon | The Aftershock Signature of Supershear Earthquakes[END_REF] (Figure 6a for 1-month aftershock locations). In the case of the 2002 Mw 7.9 Denali earthquake (Alaska), rupture velocities of about 5.5km/s were reported [START_REF] Ellsworth | Near-Field Ground Motion of the 2002 Denali Fault, Alaska, Earthquake Recorded at Pump Station 10[END_REF], and 1year long catalog of aftershocks is available [START_REF] Ratchkovski | Aftershock Sequence of the Mw 7.9 Denali Fault, Alaska, Earthquake of 3 November 2002 from Regional Seismic Network Data[END_REF] (Figure 6b for 1-month aftershock locations). The 2013 Mw 7.5 Craig earthquake (Alaska) was also inferred as a supershear event, with rupture velocities between 5.5 to 6.0km/s [START_REF] Yue | Supershear rupture of the 5 January 2013 Craig, Alaska ( M w 7.5) earthquake[END_REF] and a 5-month aftershock catalog is available [START_REF] Holtkamp | A High Resolution Aftershock Catalog of the Magnitude 7[END_REF] (Figure 6c for 1-month aftershock locations).

We observe a paucity in aftershocks where the ruptures have transitioned to supershear, as highlighted by the pink boxes in Figures 7b,d computing the cumulative moment density released by aftershocks, considering those located within la =5km of the fault trace (see the Supplementary Information for the same analysis considering la =2.5km, Figure S6). We assume that each aftershock, having a seismic moment M 0 , is a circular crack and compute the slip distribution δ(r) using [START_REF] Udias | Source Mechanisms of Earthquakes[END_REF]:

δ(r)= 24∆τ 7πµ s ✓ 7M 0 16∆τ ◆ 2/3 -r 2 (4.1)
where r is the crack radius, µ =30GPa is the shear modulus, and ∆τ is the stress drop, assumed to be 3MPa. We bin the aftershock crack radius along-strike, based on the minimum aftershock magnitude for each case (Izmit Mw=2.4,Denali Mw=1.5,and Craig Mw=1.7). Thus the binning sizes employed to discretize the main fault are 90m for Izmit, 32m for Denali, and 40m for Craig earthquake. With the slip distribution, the cumulative seismic moment density of the i th bin containing N aftershocks is given by

m (i) c ≈ µ N X j=1 δ (i) j (4.2)
Note that δ (i) j is part of the slip distribution of the j th aftershock projected on the i th fault segment. We also note that we do not compute the full moment density tensor as we do not have focal mechanisms for all the aftershocks. As we are interested in density relative spatial variation (along the strike of the main fault) of this quantity, the above approximation is adequate. We compute the seismic moment density over two periods following the mainshock: 1 and 3 weeks (Figures 7a,c, and e). Focusing on the region where the rupture is expected to transition to the supershear regime (based on kinematic models), we systematically observe a small area characterized by a reduced seismic moment density and lack of aftershocks (Figures 7b,d,and f,pink boxes). The extent of these regions is different for each earthquake c. f., 3.6km for Izmit, 6km for Denali, and 3.2km for Craig. The difference in the smoothness of the cumulative moment curves for different earthquakes is due to the number of aftershocks available for the calculation. In the Craig earthquake case, we have 68 [START_REF] Rockwell | Lateral Offsets on Surveyed Cultural Features Resulting from the 1999 Izmit and Duzce Earthquakes, Turkey[END_REF] aftershocks at a distance of 5km from the main fault in 1 week (3-week), 2396 (4265) for Denali, and 641 (1118) for the Izmit earthquake.

In order to evaluate the robustness of our findings, we perform an uncertainty analysis of the seismic moment density estimation by accounting for errors in the location of aftershocks in each catalog. We employ a multivariate normal distribution {x} = N 2 (µa, σa), assuming that the location uncertainty of aftershocks is normally distributed. Here, µa corresponds to the aftershock epicenter, while σa is the covariance matrix of the location. This procedure allows us to generate, randomly, 10000 synthetic catalogs and repeat the along-strike moment density evaluation with time for each of the 10000 catalogs, deriving the mean and the standard deviation (1-σ and 10-σ) of the spatio-temporal evolution of seismic moment density (red areas in Figure 8 for 1-week, denoting the mean of the seismic moment density on the main fault ± the standard deviation). Such analysis suggests that despite potential changes in the spatio-temporal distribution of seismic moment density, our conclusions are robust (see the Supplementary Information for a comparison between 1 and 3-week using 1-σ and 10-σ, Figures S7, S8, andS9).

We consider 3-week as the maximum reasonable period to investigate the evolution of the seismic moment for these earthquakes to avoid potential effects of postseismic deformation. Afterslip is proposed as a mechanism driving the aftershock triggering (e.g., Hsu et al. [93] and Ross et al. [94]). For each studied event, the inferred transitional region co-locates with areas where some authors report afterslip occurrence [START_REF] Freed | Implications of deformation following the 2002 Denali, Alaska, earthquake for postseismic relaxation processes and lithospheric rheology[END_REF][START_REF] Hearn | Izmit earthquake postseismic deformation and dynamics of the North Anatolian Fault Zone[END_REF][START_REF] Ding | Coseismic and Early Postseismic Deformation of the 5 January 2013 Mw 7.5 Craig Earthquake from Static and Kinematic GPS Solutions[END_REF]. Therefore, the observed gap in the early aftershocks productivity (less than 3-week after the mainshock) is mainly related to the mainshock rupture. It is worth noting that results for the Mw 7.8 Kunlun earthquake reported by Robinson et al. [START_REF] Robinson | The M w 7.8, 2001 Kunlunshan earthquake: Extreme rupture speed variability and effect of fault geometry[END_REF], employing Harvard CMT solutions, also allude to the same conclusion. However, due to the lack of high spatio-temporal density of aftershocks in their catalog, we are more confident in our results using optical correlation techniques to characterize the transition zone. each event (green for sub-Rayleigh and orange for supershear) [START_REF] Ellsworth | Near-Field Ground Motion of the 2002 Denali Fault, Alaska, Earthquake Recorded at Pump Station 10[END_REF][START_REF] Yue | Supershear rupture of the 5 January 2013 Craig, Alaska ( M w 7.5) earthquake[END_REF][START_REF] Bouchon | The Aftershock Signature of Supershear Earthquakes[END_REF], while the starts denote the epicenter of each We note that in the distribution of moment plotted for the entire fault profile (Figure 7), local gaps in aftershock density exist and do not necessarily indicate a supershear transition. We hypothesize that such gaps could also be related to abrupt local changes in rupture speeds, off-fault medium strength and fault geometrical complexities which warrants further investigation.

To conclude, we show that the reduction in the spatial extent of early aftershocks (≤ 3-week after the mainshock), and the cumulative seismic moment density around the fault (≤ 5km on either side of the fault) is associated with the transition to supershear speed. This type of analysis also helps us identify, more precisely than kinematic models, the region where supershear transition occurs.

Conclusion and Discussion

Using theoretical arguments and numerical models, that account for coseismic off-fault damage, we have shown that supershear transition is characterized by a significant reduction in the width of the off-fault damage zone. This feature is due to a Lorentz-like contraction of the spatial extent of the stress field around a rupture tip. We then cross-validated this phenomenon with natural observations of coseismic off-fault damage zone width using image correlation and analysis of aftershock catalogs. We confirm that supershear transition is indeed characterized by a significant reduction in the width of the off-fault damage zone.

Our results are in general agreement with the published kinematic models for the Izmit [START_REF] Bouchon | How fast is rupture during an earthquake? New insights from the 1999 Turkey Earthquakes[END_REF][START_REF] Bouchon | Space and Time Evolution of Rupture and Faulting during the 1999 Izmit (Turkey) Earthquake[END_REF], Denali [START_REF] Ellsworth | Near-Field Ground Motion of the 2002 Denali Fault, Alaska, Earthquake Recorded at Pump Station 10[END_REF][START_REF] Dunham | Evidence for a Supershear Transient during the 2002 Denali Fault Earthquake[END_REF], Craig [START_REF] Yue | Supershear rupture of the 5 January 2013 Craig, Alaska ( M w 7.5) earthquake[END_REF], and Kunlun [START_REF] Bouchon | Observation of long supershear rupture during the magnitude 8.1 Kunlunshan earthquake[END_REF][START_REF] Robinson | The M w 7.8, 2001 Kunlunshan earthquake: Extreme rupture speed variability and effect of fault geometry[END_REF][START_REF] Vallée | The 14 November 2001 Kokoxili (Tibet) earthquake: High-frequency seismic radiation originating from the transitions between sub-Rayleigh and supershear rupture velocity regimes[END_REF] earthquakes, relative to the location where the rupture accelerates to supershear speeds. Our inferred location of the transition zone can sometimes be different from those found in kinematic inversions but only by a few kilometers.

While the theoretical and numerical models are still idealized (modeled on a 1D planar fault with uniform traction and frictional strength), the reduction in the width of the coseismic offfault damage zone, related to supershear transition, is observed for natural earthquakes (Figures 5 and7). This approach could thus be used to refine and precisely define the zone of supershear transition inferred from coarser kinematic models. Moreover, as the zone of supershear transition is now narrowed down to a few kilometers, it could also be used to guide future fieldwork to study the segment of the fault where such a transition occurred. A better understanding of the physical conditions for supershear transition might help us foresee the location of future supershear transition, although, as of now, this remains a difficult task.

The lack of aftershocks on the segment experiencing supershear rupture has been debated in the past. Bouchon & Karabulut [42], for instance, already concluded that the entire supershear segment collocated with a region of aftershock quiescence. To explain this, they claimed that the stress and strength conditions were more homogeneous on the supershear segments [START_REF] Bouchon | The Aftershock Signature of Supershear Earthquakes[END_REF], impeding large ground accelerations near the fault [START_REF] Bouchon | How fast is rupture during an earthquake? New insights from the 1999 Turkey Earthquakes[END_REF][START_REF] Ellsworth | Near-Field Ground Motion of the 2002 Denali Fault, Alaska, Earthquake Recorded at Pump Station 10[END_REF]. Our work, in contrast, focuses on characterizing the region where the rupture transitions to supershear speed, before the said supershear segment.

The results of this study are valid for a well-developed sub-Rayleigh rupture that transitions to supershear speeds. However, the 2018 Mw 7.5 Palu (Indonesia) earthquake might have either nucleated directly at a supershear speed, or transitioned very early [START_REF] Bao | Early and persistent supershear rupture of the 2018 magnitude 7.5 Palu earthquake[END_REF][START_REF] Socquet | Evidence of supershear during the 2018 magnitude 7.5 Palu earthquake from space geodesy[END_REF][START_REF] Zhang | Complex multiple-segment ruptures of the 28 September 2018, Sulawesi, Indonesia, earthquake[END_REF]100]. The model we develop here might provide insights to understand whether an earthquake can nucleate and propagate directly at supershear speeds, and if so, why that would be the case. Data Accessibility. All the catalogs and numerical simulation results employed in this work have been obtained from published works, cited in the main text and Supplementary Information. each event (green for sub-Rayleigh and orange for supershear) (Bouchon and Karabulut, 2008, Ellsworth et al., 2004, Yue et al., 2013), while the starts denote the epicenter of each earthquake and the arrows indicate the ruptures' direction. The pink boxes point out our proposed transition zone, also observed in a map view in b for Izmit, d for Denali, and f for Craig earthquakes.

3. 1 . 3 . 4 . 4 . 1 . 4 . 1 . 3 . 4 . 1 . 4 . 4 . 1 . 5 . 4 . 1 . 5 . 1 . 4 . 1 . 5 . 2 . 4 . 1 . 6 . 2 . 4 . 1 . 6 . 3 . 4 . 1 . 8 . 4 . 2 . 4 . 2 . 1 . 1 . 4 . 2 . 1 . 2 . 4 . 2 . 2 . 4 . 2 . 3 . 4 . 2 . 6 . 5 . 1 . 5 . 2 . 1 . 6 . 2 . 6 . 3 . 6 . 4 . 6 . 5 . 6 . 6 . 6 . 7 . 1 . 7 . 2 . 7 . 3 . 7 . 4 . 1 .

 13441413414415415141524162416341842421142124224234265152162636465666717273741 Context of the study: the Baluchistan earthquake . . . . . . . . . . . . . 3.1.4. Data and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. Results and interpretations . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5.1. Description of the geometrical complexities and of the associated displacements patterns . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5.1.1. The F1 complexity . . . . . . . . . . . . . . . . . . . . . . 3.1.5.1.2. The C2 complexity . . . . . . . . . . . . . . . . . . . . . 3.1.5.2. Displacement profiles across the C2 and F1 complexities . . . . . . 3.1.5.2.1. Individual displacement profiles . . . . . . . . . . . . . . . 3.1.5.2.2. Displacement profiles in 3D view . . . . . . . . . . . . . . 3.1.5.3. Quantification of fault slip and off-fault deformation along the fault-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6.1. Fault zone, damage zone and OFD zone: comparison with other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6.2. Mechanisms of off-fault deformation . . . . . . . . . . . . . . . . 3.1.6.3. Relations with the fault geometry and the basement lithology . . . 3.1.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.9. Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Post-seismic deformation after the 2013 Baluchistan rupture in areas of fault complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Data and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed deformation at the scale of a whole rupture: characterization, quantification, and relation with the rupture process at depth 107 Diffuse deformation and surface faulting distribution from submetric image correlation along the 2019 Ridgecrest, California, ruptures (Antoine et al., 2021) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ridgecrest earthquake sequence . . . . . . . . . . . . . . . . . . . Data and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the results and comparison with independent data . . . . Description of the rupture geometry and of the near-fault displacement patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantification of fault-parallel, fault-normal, and vertical displacements along the ruptures . . . . . . . . . . . . . . . . . . 129 4.1.5.3. Localized slip vs diffuse deformation for the Mw7.1 mainshock earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.1.6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6.1. Two sets of orthogonal faults and their impacts on the surface displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison with kinematic slip inversions and implications for the rupture segmentation and the processes of surface deformation . . . Fault zone width, spatial extent of the diffuse deformation, and shallow slip deficit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data and Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.10. Supplemental materials . . . . . . . . . . . . . . . . . . . . . . . . . . Joint inversion for the rupture kinematics of the 2019 Ridgecrest earthquake sequence, with GNSS, InSAR and high-resolution optical imagery offset data: rupture complexity and diffuse deformation . . . . . . . . . . . . . . . . . . . 4.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling fault rupture complexity in the shallow crust . . . . . . Strategies of previous studies that included optical data . . . . . . Data and model geometry . . . . . . . . . . . . . . . . . . . . . . . . . Inversion method and parametrization . . . . . . . . . . . . . . . . . 4.2.4. Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions at this stage of the study and future improvements . . . . 192 Relation between fault geometry, diffuse deformation, Relations between rupture speed and off-fault damage . . . . . . . . . . . . . 196 Application of optical images correlation to the 2001 Mw7.8 Kokoxili supershear rupture (Kunlun, China) and measurement of the fault zone width (FZW) . Observations of wide deformation zones around faults . . . . . . . . . . . . 206 6.1.1. Field observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.1.2. Seismic observations of a wide deformation zones . . . . . . . . . . . 208 6.1.3. Satellite observations of surface ruptures and diffuse deformation . . . 211 Comparison with slip models: relation between surface diffuse deformation and the rupture process at depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Primary control of earthquake magnitude on SSD and diffuse deformation . . 220 Secondary control of the fault geometry on the SSD and diffuse deformation . 224 Geological control on the earthquake rupture . . . . . . . . . . . . . . . . . . 226 Possible mechanisms responsible for diffuse deformation . . . . . . . . . . . 230 Impact of diffuse deformation for the fault mechanical and seismic behavior . 232 Imaging complex fault geometries and measuring localized deformation on faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Measuring large-scale displacement gradients: the diffuse deformation . . . . 236 Relation between diffuse deformation, earthquake process at depth, and fault geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Nature of diffuse deformation and impacts for earthquake hazard . . . . . Optical image correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.2. Field investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Figure 1 . 1 :

 11 Figure 1.1: (a) MORVEL plate tectonics model from DeMets et al. (2010) showing 25 assumed rigid lithospheric plates. Patterned red areas show diffuse plate boundaries. (b) Worldwide distribution of earthquake epicenters at depth shallower than 40 km from 1967 to 2007 (DeMets et al., 2010). Magnitudes larger than 3.5 in black and larger than 5.5 in red. Clusters of large magnitude epicenters occur at the plate boundaries highlighted in the MORVEL model in (a).

Figure 1 . 2 :

 12 Figure 1.2: Photography of a fault mirror in Greece. Limestone sedimentary rocks (whitish rocks to the right) are put in contact with slope deposits (brown and tilted deposits to the left) through the fault interface (polish surface of the limestone rocks). The fault accommodates vertical displacement between the two blocks, with the block at the left going down. Photography from J.-B. De Chabalier during the field training in Greece (1 st year of PhD), 2019.

Figure 1 . 3 :

 13 Figure 1.3: Field (a,c,d) and drone (b) images of the 2019, Mw7.1, Ridgecrest, California, mainshock ground ruptures. The faults accommodate (a,d) vertical, (b) left-lateral and (c) right-lateral deformation (DuRoss et al., 2020).

Figure 1 . 4 :

 14 Figure 1.4: Model of a heterogeneous seismic cycle on a fault. Failure on the fault plane occurs when shear stress overcomes faut strength. As a result, unstable co-seismic slip occurs on the fault and, over time, this mechanism allows to accommodate the tectonic loading. Fault strength increases following healing processes and decreases during the co-seismic phase. Other processes like pre-seismic and post-seismic deformation, either seismic or aseismic, can be observed. Depending on the characteristics of each fault and

Figure 1 .

 1 Figure 1.5A, can also correspond to the case of a rupture that did not reach the surface (or blind rupture; Segall, 2010) and for which the surficial part of fault remains locked and the medium subjected to elastic strain.

Figure 1 . 5 :

 15 Figure 1.5: (a) Left. Elastic half-space with a 2D dislocation at depth. Horizontal and vertical axis are in kilometers. Right. Surface displacement and strain profiles across the fault. d2 is the top edge of the dislocation (3 km). s is the slip vector. (b) Similar to (a)

Figure 1 . 6 :

 16 Figure 1.6: Geodynamics of Mongolia with surrounding parts of China, Kazakhstan and Russia. Active faults are represented by black lines. Background image is a hill-shade of the region[START_REF] Walker | Active tectonics of an apparently aseismic region: Distributed active strike-slip faulting in the Hangay Mountains of central Mongolia[END_REF].

Figure 1 . 7 :

 17 Figure 1.7:Aerial photographs of the 2010 Mw7.2 El Mayor-Cucapah earthquake rupture (Baja California, northwest Mexico). On average, 2 m of oblique-dextral slip were measured for this rupture. Blue arrows report the width of the rupture zone. Red arrows point on the principal fault scarp that accommodates most of the vertical deformation.

Fig. 1 . 8 :

 18 Fig. 1.8: Photograph of the Rokko fault within the Arimae-Takatsuki tectonic line, southwest Japan. The fault puts in contact granitic rock (left) with rhyolitic tuff (right). (b) Sketch of the fault zone with rock textures indicated. F1 and F2 indicate the location of the two sub-parallel fault cores. Inner fault zone is constituted of fault breccia. Rocks that surround the fault zone are foliated cataclasite. The fault zone (c) Fault striations indicate strike-slip shear[START_REF] Lin | Spatial variations in damage zone width along strike-slip faults: An example from active faults in southwest Japan[END_REF].

Figure 1 . 9 :

 19 Figure 1.9: Sketch of a typical strike-slip fault zone structure in a quartzofeldspathic country rock showing a (a) single fault core and a (b) multiple fault core, with associated damage zone (Faulkner et al., 2003). In (a), the damage zone is uniformly distributed around the single fault core, with a decay of the crack density when moving away from the fault core. In (b), fracture lenses separate the numerous fault gouge bands (equivalent to fault cores), and damage zone corresponds to the area located around the last fault gouge bands on both sides of the fault zone. Horizontal scale (width) of the fault zone is ~10 2 m in (a) and ~10 3 m in (b).

Figure 1 . 10 :

 110 Figure 1.10: Distribution of plastic strain (𝜀 𝑝 ) for a rupture of variable width (W), embedded in a uniform visco-elasto-plastic material (Ampuero and Mao, 2017). "C" and "T" indicate the compressive and tensional side of the rupture, respectively. Green dashed line indicates the location of horizontal cross sections of the fault zone not presented in this thesis but available in the original publication. Larger plastic deformation occurs on the tensional side of the rupture, where opening of fractures is favored.

Figure 1 .

 1 Figure 1.11: (a) Field photography of the surface deformation associated with the Mw6.0 2014 South Napa earthquake. En-echelon fractures are highlighted in yellow. (b) Trench log across the rupture zone. Fractures are drawn in orange. The black sample was collected for mechanical tests and has a composition of 50% of clay with a coefficient of friction of ~0.40[START_REF] Brooks | Buried shallow fault slip from the South Napa earthquake revealed by near-field geodesy[END_REF].

(

  CEOS) Seismic Hazards Pilot from the Centre National des Etudes Spatiales (CNES), and European Spatial Agency (ESA). The Worldview images are shared by the U.S.Geological Survey in the NextView licensing agreement, which we access through our collaboration with Ryan D. Gold (USGS Golden, Colorado). For each case of study presented in this thesis, the details of the image references, dates, viewing angles and resolutions are described in the corresponding chapter. One common characteristic of the image datasets in the 2013 Baluchistan and 2019 Ridgecrest cases of study, in addition to their 0.5 m ground resolution, is that for each scene we have at least two to three images acquired with different viewing angles, before and after each earthquake rupture. This allows us to have a stereoscopic (2 images) or tri-stereoscopic (3 images) view of the ground and capture its 3D geometry before and after the earthquakes, which is necessary to measure the 3D displacement field.

Figure 2 . 1 :

 21 Figure 2.1: Example of geometry of acquisition of the optical images. WV indicates the WorldView satellite and PL indicates the Pleiades satellite. Image scenario in the two cases of study from this study (2013 Baluchistan and 2019 Ridgecrest) is similar.

Figure 2 . 2 :

 22 Figure 2.2: Sketch of optical image correlation using MicMac. Images were orthorectified prior to correlation so that they have the exact same geometry of acquisition and ground footprint. Group of pixel of n pixels in width (n=9 in this thesis) is in blue in the preearthquake images (left). The similar group of pixel (in red) is retrieved at the position (x+Δx, y+Δy) in the post-earthquake image (right). Search space is in green. Sub-pixel step used to measure displacement lower than the pixel size is represented in blue. The position of the group of pixels in the post-earthquake image is retrieved by calculating the correlation between the different groups of pixels at all possible locations in the search space, and by taking the greatest correlation score (S19 in the example of this figure).Modified from[START_REF] Vallage | Distribution de la déformation en contexte tectonique oblique : applications aux séismes de Denali (2002) et du Balochistan[END_REF]

  2.3). The best correlation score corresponds to the position (x2,y2) of the group of pixels in the post-earthquake image. With the positions (x1,y1) and (x2,y2) of the same group of pixels in the pre-and post-earthquake image, MicMac measures the displacement of the ground between the two acquisitions in the x and y directions of the image. These displacements are sometimes projected onto directions parallel or perpendicular to the fault azimuth. Exact parameters as well as corrections applied to the output displacement maps are detailed in the different chapters.

Figure 2 . 3 :

 23 Figure 2.3: Simple sketch of the principle of sub-pixel image correlation. Red box is the group of pixel selected in the pre-earthquake images (top). Green box is the corresponding

Figure 2 . 4 :

 24 Figure 2.4: Vertical difference of DSMs. Pre-earthquake DSMs is translated using the Δx and Δy ground displacements measured previously (see 2.4.) to re-align the topographic features with those of the post-earthquake DSM. Difference in elevations that remains between the translated pre-earthquake DSM and the post-earthquake DSM (Δz2) arise from the coseismic vertical displacement field (Delorme et al., 2020).

3. 1 .

 1 Detailed characterization of off-fault deformation patterns across complex fault zones and relations with the fault zone structure: zoom in the 2013 Baluchistan surface rupture (Antoine et al., in preparation)

Figure 3 . 1 :

 31 Figure 3.1: Regional context of the Baluchistan earthquake (modified from Avouac et al., 2014). The earthquake ruptured bilaterally 200 km of the Hoshab fault, at the intersection between the Makran subduction prism and the Chaman left-lateral fault system. No previous earthquake was reported on the Hoshab fault, but few large earthquakes occurred on the subduction thrust fault and on the Ghazaband fault. The focal mechanism (https://earthquake.usgs.gov/earthquakes/ eventpage/usb000jyiv/executive) revealed a predominantly strike slip (left-lateral) earthquake with a small thrusting component on a north-west dipping fault. The C2 compressional relay is the main geometrical complexity along this rupture. Dashed polygon around the rupture represents the area of Figure 3.2B.

Figure 3 . 2 :

 32 Figure 3.2: (A) Fault-parallel, fault-normal, and vertical displacement maps at 0.5 m ground resolution for a 30 km-long area of the Baluchistan earthquake centered on the C2 complexity (Fig. 3.1). P1 and P2 indicate the two main fault branches of the rupture in this area, that are separated by the C2 complexity. AA' and BB' displacement profiles are presented in Figure 3.6. (B) Optical image correlation of SPOT-5 images at 2.5 m ground resolution (modified from Vallage et al., 2015) for the whole Baluchistan rupture. Study area (red polygon) shown in (A) is located roughly in the middle of the rupture. The 2.5-m data are used as a far-field reference for correcting ramps and artifacts. (C)

Figure 3 . 3 :

 33 Figure 3.3: Displacement maps for a 3-km long section of the F1 complexity (Fig. 3.2A) in (A) the fault-parallel, (B) the fault-normal and (C) the vertical components. Surface rupture map from Vallage et al. (2015) is overlaid in (C). (D) Norm of maximum shear strain, (E) curl and (F) dilatation maps at 2.5-m ground resolution. Profiles are 1-km long and cross a 500-m long secondary fault that developed into the hanging wall and that accommodated mainly fault parallel displacement.

  3.4A), NNE-SSW orientation (dark blue, Fig. 3.4A) and ENE-WSW orientation (light blue, Fig. 3.4A). These orientations resemble the ones measured in Vallage et al. (2016) in the F1 complexity, with the NNE-SSW orientation (dark blue, Fig. 3.4A) being close from the ~N216 basement fabric orientation they have identified. In fact, even the westward termination of P1 seems to be influenced by the basement fabric: the tip of fault P1 (Fig. 3.2D) veer to strike NNE-SSW after the C2 relay zone.

Figure 3 .Fig. 3 .

 33 Figure 3.4: (A) Map of fault orientations in the C2 complexity: red when following the primary fault orientation (mountain range azimuth), light blue for ENE-WSW orientations, and dark blue for NNE-SSW orientations. Background surface rupture is from Vallage et al. (2015). Faults S1, S3, part of fault S4 and fault S6 were not reported in the rupture map, but were observed in our displacement maps (Fig. 3.2A). Background picture corresponds to a post-earthquake orthoimage. (B) Sketch of the surface rupture overlapped on a 0.5 m resolution post-earthquake DSM. Primary faults are in red. (C) Sketch of fault orientations.

Figure 3 . 5 :

 35 Figure 3.5: Displacement maps for a 3-km long section of the C2 complexity (Fig. 3.2A) in (A) the fault-parallel, (B) the fault-normal and (C) the vertical components. Surface rupture map from Vallage et al. (2015) is overlaid in (C). Faults S1 and S6 were inferred from the displacement maps but was not reported in the surface rupture map. In (A): red right-angles between S5, S6 and S7 highlight ~90° azimuthal variations ; dashed arrows overlapped on P1, S1 and S4 highlight gradual azimuthal variations that shift an east-west trend. (D) Norm of maximum shear strain, (E) curl and (F) dilatation maps at 2.5-m ground resolution. Profiles across the fault zone are 1-km long. Peak of strain and curl

Figure 3 . 6 :

 36 Figure 3.6: AA' and BB' profiles in the fault-parallel (red), fault-normal (green) and vertical (blue) components of slip, respectively across the F1 and C2 complexities (Fig. 3.2A). Positive fault normal offsets correspond to extension while negative offsets correspond to shortening. Fault locations are reported with the light grey areas. Gradients of displacement are highlighted with arrows in BB' and correspond to OFD. Black and grey BB' profiles were taken respectively from 2.5 m displacement maps of Zhou et al. (2018) and Vallage et al. (2015).

(

  2015), extension and subsidence occurred as a consequence of the change in fault dip and thus, the subsiding area visible at the surface and in the displacement profile (blue) should be located above the change in fault dip at depth. Hence, we estimated the change in fault dip to occur around 275 m depth.Displacement profiles across the C2 relay zone (BB', Fig.3.6) show four slip offsets and zones of OFD gradients both in the hanging-wall and in the push-up area. In the faultparallel component, left-lateral slip was mainly accommodated by faults P1 and S1 that run alongside each other, for a total slip accommodated within this area of 6 m. Another ~2 m was accommodated by faults S4 and S5 (~1 m each), in addition to some OFD within the relay, and north of the relay(red arrows, Fig. 3.6). While some limited shortening was accommodated by the fault S1, most of it was accommodated on primary fault P1. Extrados-extension occurred across fault S5, and through distributed OFD in the hanging-wall and in the push-up area (green arrows, Fig.3.6). In the vertical component, uplift localized on faults P1 and S4, while some distributed vertical deformation was also measured throughout the push-up structure and in the hanging-wall (blue arrows, Fig.

  3.7). Complementary plot for the fault-normal component in C2 can be found in Supplementary 3.3.The 3D plot across the F1 fault zone (Fig.3.7A) illustrates the along-strike distribution of surface slip at the northern tip of the complexity, before that the secondary fault observed in Figure3.3 quicks in: surface displacements distributed over a maximum width of ~300 m within the hanging wall. The three northernmost profiles in the plot correspond to reference profiles for a linear section of the fault, before the rupture enters F1. The red area reports slip on primary fault P1. It decreases from 12 m to the north-east to 7.5 m to the south-west. As slip on the fault decreases, the distributed deformation zone gets wider and catches a larger part of the displacement budget. OFD gradients, highlighted in blue (Fig.3.7A), are steep and mark a clear transition with the far-field slope (dashed line, Fig.3.7A), unlike previous OFD gradients observed in the hanging-

Figure 3 . 7 :

 37 Figure 3.7: Fault-parallel displacement profiles arranged in a 3D view on (A) the northern tip of the F1 complexity, (B) the C2 relay zone and, (C) zoom on the secondary fault S1 inside of C2 (Fig. 3.2B). Each profile corresponds to the stack of individual slip profiles over a width of 250 m for (A) and (B), and 100 m for (C). Slip on each fault is represented by the colored area, either in dark pink for the two primary faults (P1 and P2) or in light pink for the secondary faults (S1-7). Arrows highlighting offsets along fault S4, S6, and S7 indicate the change between sinistral and dextral slip.

Figure 3 . 8 :

 38 Figure 3.8: BB' fault parallel stacked slip profile from Figure 3.2A (inside the vertical dashed lines) merged with far-field data (> 2 km) from Vallage et al. (2015). Stack is 100m wide. Dashed black lines correspond to linear regressions in the hanging wall and footwall outside of the fault zone used to measure the total displacement offset associated with the entire fault zone width. Red dashed lines are projections of individual offsets for each fault of the push-up. Blue curves highlight OFD gradients. Blue and red lines along the x-axis respectively report the width of OFD gradients and fault slip offsets.

Figure 3 . 9 :

 39 Figure 3.9: (A) Surface rupture map from Vallage et al. (2015). Colors correspond to the classification of the faults used for the slip budget in (B). Dashed grey lines above the two complexities correspond to the extent of OFD gradients in the hanging wall and footwall (e.g., blue dashed line in Fig. 3.7A). (B) Fault parallel displacement budget along our study area. Black curve corresponds to the total displacement budget. Dashed pink

1 .

 1 3.7B). Mean OFD gradients for F1 approach 10 m/km (~1 m of slip accommodated over a width of 100 m) while it is only 1 m/km in the C2 area (~0.10 m of slip accommodated over a width of 100 m), without considering the gradients Fault zone, damage zone and OFD zone: comparison with other studiesIn this study, we estimated a mean OFD of 16% over a 30 km-long area of the 2013 Baluchistan rupture. Maximum values of OFD of 70% were encountered in areas where fault geometry is complex (C2 and F1). Mean OFD for the total Baluchistan rupture (~200 km) estimated respectively by[START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF] and[START_REF] Zinke | Surface slip and off-fault deformation patterns in the 2013 MW 7.7 Balochistan, Pakistan earthquake: Implications for controls on the distribution of near-surface coseismic slip[END_REF] are 45% and 28%. The area we study here gathers two of the largest complexities of the Baluchistan rupture and so, as long as OFD is larger in fault complexities (Fig.3.9B;[START_REF] Milliner | Comparison of coseismic near-field and off-fault surface deformation patterns of the 1992 Mw 7.3 Landers and 1999 Mw 7.1 Hector Mine earthquakes: Implications for controls on the distribution of surface strain[END_REF], we should have measured larger OFD than[START_REF] Gold | On-and off-fault deformation associated with the September 2013 Mw 7.7 Balochistan earthquake: Implications for geologic slip rate measurements[END_REF] andZinke et al.

3. 2 )

 2 . The pre-earthquake mono-images overlap each other such that at least 2 preearthquake images were available in the different location of the area of study(30-km long). WorldView images are roughly 15-km large. The two post-earthquake acquisitions are both constituted of three sub-images acquired with different viewing angles (tristereo). Pleiades images are roughly 20 to 25-km large.3.2. Near-fault and long-term post-seismic deformation after the 2013 Baluchistan rupture3.2.1. IntroductionIn the study of the 2013 Baluchistan rupture, we showed that large-scale off-fault deformation (OFD) occurred in regions where fault geometries are particularly complex.

  3.1). These images allow us to constrain postseismic deformation within the first month and a half after the rupture. However, the relatively low resolution of the SPOT-6 images compared to Pleiades images, does not allow for detecting displacement smaller than ~0.20 m. We then use stereo Pleiades images acquired in March and May 2014, a year and a half after the rupture (Tab. 3.1), and along the entire length of the fault rupture (~200 km), to investigate post-seismic displacement of the order of ~0.10 m. This second phase of correlation is performed along both the C2 and F1 complexities, and the terminaisons of the rupture. Both the SPOT-6 images from 2013, and the Pleiades images from 2014 were correlated with the tri-stereo Pleiades images dataset from 2016 (Tabs. 3.1 and 3.2), which include the images used to measure the coseismic displacements in the area of the C2 and F1 complexities (section 3.1.). Prior to correlation, all the images were orthorectified using DSMs calculated from the same image dataset, similar to the method presented in2.4. and 3.1.4. Correlation method is also similar to what was described in sections 2.4. and 3.1.4. Table 3.1: Detailed information and references of the satellites images used to investigate post-seismic deformation along the trace of the 2013 Baluchistan rupture.

2 :

 2 List of the correlations that were performed along the 2013 Baluchistan rupture trace, in the months and years following the coseimsic phase. Images correspond to those listed in Table3.1 and are referred using the same names. Correlation results at the scale of the ~200 km-long rupture are displayed in Figure3.10.Tests 2,[START_REF] Bhat | Off-fault damage patterns due to supershear ruptures with application to the 2001 M w 8.1 Kokoxili (Kunlun) Tibet earthquake[END_REF] 8 and 9 are shown inFigure 3.11, tests 5 and 10 in Figure 3.12, and test 11 in Figure 3.13.

3 .

 3 11 shows the results of the post-seismic deformation measurements made after the 2013 Baluchistan rupture along the area of the C2 and F1 complexities. The maps present the east-west component. In this area, we performed two types of correlation. The first one uses the images acquired in 2013 to 2016 (Tabs. 3.1 and 3.2), and benefits from a ground resolution of 1.5 m. The second one is performed between the images acquired in 2014 and 2016 (Tabs. 3.1 and 3.2), at a resolution of 0.5 m. One of the two correlation results in each area is shown in Figure 3.11A. Comparison between the two series of results is shown in Figure 3.11B,D in the area of the C2 complexity.

  images are not available for these areas, and only the Pleiades images acquired in 2014 and 2016 were used. The results show that, similarly to the central area of the rupture along C2 and F1, no post-seismic deformation larger than ~0.10 m occurred after December 2014 along the tips of the 2013 rupture (Figs.3.12 and 3.13). Only low frequency artefacts and topographic noise arise in the displacement maps.

Figure 3 . 10 :

 310 Figure 3.10: Location of the correlations along the trace of the 2013 Baluchistan rupture. Rupture map from Vallage et al. (2015) is overlaid in red.

Figure 3 .

 3 Figure 3.11: (A) East-west displacement maps from 2013 and 2014 to 2016 in the area of the C2 and F1 complexities of the 2013 Baluchistan rupture. Rupture map from Vallage et al. (2015) is overlaid in red. (B) Zoom in the east-west correlation map calculated from Pleiades images at 0.5 m resolution (test 6, Table 3.2). (C) Correlation score map for (B). (D) Zoom in the east-west correlation map calculated from SPOT-6 and Pleiades images at 1.6 m resolution (test 7, Table3.2). (E) East-west stacked displacement profile across

Figure 3 .

 3 Figure 3.12: (A) East-west displacement maps from March 2014 to October 2016 at the northern tip of the 2013 Baluchistan rupture (test 11, Table3.2). Rupture map fromVallage et al. (2015) is overlaid in red.

Figure 3 .

 3 Figure 3.13: (A) East-west displacement maps from March 2014 to September 2016 at the southern tip of the 2013 Baluchistan rupture (tests 5 and 10, Table3.2). Rupture map fromVallage et al. (2015) is overlaid in red.

4. 1 .

 1 Diffuse deformation and surface faulting distribution from sub-metric image correlation along the 2019 Ridgecrest ruptures (California, USA) Published in Bulletin of the Seismological Society of America: https://pubs.geoscienceworld.org/ssa/bssa/article-abstract/111/5/2275/607260/Diffuse-Deformation-and-Surface-Faulting

  combination of these different observations suggests a relationship between the fault geometry and the lateral extent of a coseismic rupture. The term "fault segmentation" is used to describe the laterally discontinuous fault geometry and the control these discontinuities exert on the propagation of the rupture. For example, changes in the fault scarp morphology and fault geometry combined with lateral variations in the recurrence intervals of past earthquakes along the Wasatch normal fault (Utah, USA, 370 km long) have enabled identification of individual fault segments that behave differently (DuRoss

  widely observed in strike-slip contexts, for example, the 2001 Kokoxili earthquake in central Tibet or the 1905 Mw>8 Tsetserleg-Bunlay earthquake in Mongolia. Using satellite optical imagery-based measurements,Klinger et al. (2006) and[START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF] described the segmentation of these two ruptures, respectively: they identified a decrease in surface fault slip every ~20 km along the rupture, which corresponds to the location of fault bends and stepovers. They conclude that large earthquakes are generated by the rupture of multiple fault segments, separated from each other by geometrical barriers.

Figure 4 . 1 :

 41 Figure 4.1: Tectonic context of the Ridgecrest sequence. Pacific plate relative motion to North America plate is 49 mm/yr (ten Brink et al., 2018). ECSZ stands for Eastern California shear zone. Quaternary faults (USGS database, see Data and Resources) are shown in black. Major faults, including the Garlock and San Andreas Faults, and the 2019

  (2019) that we resampled to a 0.5-m resolution. The DSM from[START_REF] Willis | CaliDEM: Ridgecrest, CA region 2m digital surface elevation model[END_REF] was computed from WorldView and GeoEye images acquired in 2008 and 2016 and processed using COSI-Corr[START_REF] Leprince | Automatic and Precise Orthorectification, Coregistration, and Subpixel Correlation of Satellite Images, Application to Ground Deformation Measurements[END_REF]. For 7 tiles out of 14 (Tab. S4.2), we chose to use the DSM from Willis et al. (2019) because the charge-coupled device (CCD) artefact from the WorldView sensor was too large in our own DSM and impacted our correlation results (Fig. S4.2). The external DSM from Willis et al. (2019) is oversampled and introduced into the MicMac chain to generate the ortho-images. It is co-registered with the Pleiades data based on the georeferencing information. For the post-earthquake dataset, post-earthquake DSMs were computed directly from the tri-stereo Pleiades acquisitions (Tab. S4.1). Images and DSMs processed with MicMac are automatically co-registered during the processing. Eventually, all images were orthorectified using the MicMac package.

  GNSS data from[START_REF] Floyd | Survey and continuous GNSS in the vicinity of the July 2019 Ridgecrest earthquakes[END_REF]. Only three point are located within our study area, which is limited to a width of 4-5 km on each side of the fault. For these three points, the difference between our data and the GNSS data, in the east and north components, ranges from 0.04 to 0.44 m. The mean difference is 0.15 m for the east component and 0.19 m for the north component. However, these comparisons are of limited utility because they are based only on three points. Also, the time period used to account for the earthquake deformation is much smaller in the GNSS dataset (July 2 nd to July 9 th , 2019) than in the optical-image dataset (2012-2018 to September 2019), leading to supplementary discrepancies between the two datasets. 4.1.5. Description of the results and comparison with independent data 4.1.5.1. Description of the rupture geometry and of the near-fault displacement patterns East-west, north-south and vertical displacement maps (Figs. 4.2 and 4.3A,B) reveal a complex surface rupture composed of several tens of fault strands of various lengths (<1 to 20 km). The results are almost in perfect agreement with the ruptures mapped by DuRoss et al. (2020) and Ponti et al. (2020), based on field observations crosschecked with geodetic data (e.g., optical, lidar). In a few cases, however, we identified additional subparallel, or longer, fault strands compared to what was mapped in the field. For example, some cross faults striking northeast, located to the north of the mainshock rupture and along the foreshock surface rupture, were not reported in the field maps but are visible in our maps (examples are indicated in Fig.4.2A with white arrows). These additional fault strands have a length of the order of 1-2 km and accommodate up to a few tens of centimeters of displacement. Thus, they need to be taken into account when considering the total displacement budget. The displacement maps also indicate zones where significant diffuse deformation occurred around the different faults and

  interpolation. Downsampling was either to a 5-m ground resolution to analyze the rotation and strain patterns at the scale of the whole rupture or to a 2-m ground resolution to analyze details of the rupture in some specific areas.

Figure 4 .

 4 Figure 4.2: (A) East-west surface displacement map for the 2019 Ridgecrest earthquake sequence. Ground resolution is 0.5 m. USGS epicenter locations (see Data and Resources) are reported with a red star for the foreshock (35.705°N and -117.505°W) and a blue star for the mainshock (35.77°N and -117.60°W). Surface rupture map from DuRoss et al. (2020) is overlaid in black. Examples of fault offsets measured in our displacement maps but not reported in the field are indicated with white arrows. Zones of cross faults striking north-east, to the north and south of the mainshock rupture, are indicated by dashed circles. Azimuths of the mainshock rupture primary fault trace are indicated at three locations with black arrows. Lateral displacement profiles AA' and BB', in the faultparallel component of displacement, are presented in (B). Lateral profiles CC' and DD'

Figure 4 .

 4 Figure 4.3: (A) and (B) Same as in Figure 4.2A for the north-south and vertical components of displacement. Field rupture map is overlaid in (B). (C) Curl, (D) norm of

  4.3B). No ground surface rupture could be identified in the field or in the displacement maps that bound the basin to the south. However, there we measure a sharp gradient of displacement in the east-west and vertical directions(Figs. 4.2A and 4.3B), suggesting that a blind fault ruptured at depth. The interior of the basin is characterized by strong decorrelation, which is associated with liquefaction that was recognized in the field by[START_REF] Zimmaro | Liquefaction and Related Ground Failure from July 2019 Ridgecrest Earthquake Sequence[END_REF]. The Mw7.1 epicenter is located just south of this pull-apart basin. At least eight distinct fault strands accommodated deformation in this area (Fig.4.4A). Altogether, they define an extensional jog with up to 2 m of subsidence (Fig.4.4C). Most of these structures were reported in the field map[START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF][START_REF] Ponti | Documentation of Surface Fault Rupture and Ground-Deformation Features Produced by the 4 and 5 July 2019 Mw 6.4 and Mw 7.1 Ridgecrest Earthquake Sequence[END_REF] although the detail of the structures and associated diffuse deformation zones could not be resolved using discrete measurements. In this area of the rupture, fault strand 5 is considered to be the primary fault because it represents the longest continuous fault strand and accommodates the most prominent right-lateral strike slip along the mainshock rupture. In the subsequent discussion, we refer to other faults observed in this region as secondary fault strands.

Figure 4 . 4 :

 44 Figure 4.4: (A-F) Displacement, rotation, and strain maps for the epicentral area of the Mw7.1 Ridgecrest earthquake. (A) East-west, (B) north-south, and (C) vertical displacement maps. (D) curl, (E) norm of maximum shear strain, and (F) dilatational strain maps at 5-m ground resolution. Inset (i) in (D) shows an area of diffuse rotation with adjusted color scale. Profiles across the fault zone in the curl, norm of maximum shear strain, and dilatation maps are presented respectively in (G), (H), and (I). Profiles in (G) and (I) are 50-m wide. Profile in (H) is 25-m wide. High-frequency noise results from ground texture variations. Fault labels as in Figure 4.9A.

  4.5A). Optical satellite images at this site show that this area is characterized by very fine sediments, likely clay, with multiple subparallel shorelines associated with successive high stands of the China Lake. These shorelines are outlined by whitish evaporite salt. Deposits of fine clay sediments are visible at the foot of the Argus Range (Fig.4.5B). Correlation can be observed between the shape of the displacement pattern on its southeastern extremity and the shape of the contour lines, suggesting some slope-instability process. Thus, we propose that this zone is located in between the two ruptures strands, whose motion seems inconsistent with the general deformation pattern, and is associated with local decoupling of the upper sedimentary layers from deeper units during lateral spreading of the clay-rich sediments in response to the earthquake shaking. This interpretation is consistent with documentation of shaking-induced structures in the field by[START_REF] Ponti | Documentation of Surface Fault Rupture and Ground-Deformation Features Produced by the 4 and 5 July 2019 Mw 6.4 and Mw 7.1 Ridgecrest Earthquake Sequence[END_REF].

Figure 4 . 5 :

 45 Figure 4.5: (A) East-west displacement map along the mainshock surface rupture where decoupling between the sediments and the base rocks is observed. Contour lines are extracted from a pre-earthquake DSM at 2 m ground resolution (Willis et al., 2019). (B) Landsat image of the surface rupture captured on 14 July 2019. Surface rupture map is from DuRoss et al. (2020) and shaking-induced structures from Ponti et al. (2020). The southeastern limit of the decoupling area seems to correspond to the limit between fine sediments from Indian Wells Valley and coarser deposits from the Argus Range (black

Figure 4 . 6 :Fig. 4 .

 464 Figure 4.6: (A) East-west and (B) north-south displacement maps at the junction between the foreshock and mainshock surface ruptures. The small cross faults indicated in (A) are examples of very small faults, roughly aligned with the foreshock azimuth, that were for most part not mapped in the field (Fig. 4.2A). In (B), en echelon faults in the trace of the mainshock surface rupture are indicated by black arrows. Only the major one, labelled "en echelon", was mapped in the field (Fig. 4.2A). (C) Map of the curl at 2-m ground resolution for a zoom in the foreshock/mainshock ruptures junction (see location in A and B). Long cross faults belong to the foreshock surface rupture. Small cross faults located between the en-echelon faults belong to the mainshock surface rupture. Diffuse rotation is visible between the small cross faults, indicated by denser yellowish color between faults. (D) Interpreted curl map with the visible cross faults belonging to the mainshock rupture mapped in red (0.1 to 0.4 m of N-S slip).

  4.6). However, when zooming on the junction, we can see that this primary fault is in fact formed by a series of en-echelon, left-stepping faults oriented in the north-south direction (Fig 4.6B). These en-echelon faults are connected by sub-parallel sinistral faults (Fig. 4.6C,D). These sinistral faults have similar orientations to the faults associated with the foreshock. Diffuse clockwise rotation is also evident between those faults (Fig. 4.6D). South of the junction with the foreshock rupture, the mainshock surface displacement is accommodated by two primary faults separated by a stepover (Fig. 4.2A). Several secondary faults located southwest of these primary faults are visible in the field rupture map (Fig. 4.2A) and slipped ~0.20 m. The mainshock surface rupture terminates to the southeast into a set of several small left-lateral faults with azimuths like those of the foreshock, the faults mapped to the northern end of the rupture (Fig. 4.2A), and at the junction between the mainshock and the foreshock (Fig. 4.6). All these structures accommodate left-lateral horizontal motion (Fig. 4.3C). 4.1.5.2. Quantification of fault-parallel, fault-normal, and vertical displacements along the ruptures In the following section, we document along-strike variations of the horizontal displacements, decomposed into fault-parallel and fault-normal directions, as well as of the vertical component, and we analyze the results with respect to the fault geometry described previously. Swath profiles 100-m wide and 8-km long placed perpendicular to the local strike of the rupture are used to characterize the processes accommodating deformation in the fault zone. Here, we focus on quantifying the number of fault strands activated with their respective amount of slip, and the diffuse deformation for the foreshock (Fig. 4.7A) and mainshock surface ruptures (Fig. 4.7B). The criteria used to differentiate localized fault slip from diffuse deformation are the angle and the shape of the displacement gradients.Localized fault slip shows up in profiles as subvertical linear gradients will well-defined extremities. Conversely, diffuse deformation shows up as long-wavelength gradients with smooth transitions towards the far-field trends. Width of the deformation zone cannot be used as a criterion to differentiate localized slip on a surface fault strand and diffuse deformation in the bulk medium. In fact, large-amplitude localized fault slip can generate a deformation zone several hundred meters wide[START_REF] Faulkner | Scaling of fault damage zones with displacement and the implications for fault growth processes[END_REF] Fig. S4.4).Conversely, a rupture zone with a low amplitude total offset accommodated through diffuse deformation can occur over a limited width of a hundred meters (e.g., blind fault in Fig.4.4A). Diffuse deformation is essentially described in the fault-parallel component of displacement because fault-normal and vertical displacement profiles are more affected by noise: thus, the difference between diffuse gradients and slip offsets can be ambiguous.

Figure 4 . 7 :

 47 Figure 4.7: (A) Fault-parallel, fault-normal and vertical displacements along profile CC' (Fig. 4.2A) across the pull-apart basin in the mainshock surface rupture. Faults 4 and 5 are antithetic and dip toward the center of the basin with an angle of 40° (fault dip angle is the arctan of vertical displacement over fault-normal displacement). (B) Fault-parallel, fault-normal, and vertical displacements along profile DD' across a compressional jog along the foreshock surface rupture (Fig. 4.2A). For each profile, the linear regression

  4.7A). Using this method for each profile, we measure the maximum surface displacement. Theoretical precision of the measurement is 0.05 m, but here we consider a conservative uncertainty of 0.10 m to account for the noise that is present in the displacement maps (orthorectification bias and correlation noise). Mean standard deviation for individual offset measurement, referred to as std1, is almost constant along the slip curve. This corresponds to the aleatoric uncertainty arising from the noise in the data. Respectively, for the parallel, normal, and vertical component of displacement, std1 is 0.06, 0.06, and 0.08 m for the mainshock, and 0.09, 0.06, and 0.14 m for the foreshock slip curves. Impact of the positioning of the linear regressions on each side of the fault, which corresponds to the epistemic source of error (std2), was assessed by analyzing independently several times the displacement profiles. These profiles were systematically taken across displacement maps at different resolutions(2, 3,[START_REF] Dunham | Near-source ground motion from steady state dynamic rupture pulses[END_REF] and 5 m, Fig. S4.5). This allows us to assess the influence of the map downsampling on the measured offsets. We then took the standard deviation over the four displacement values measured at various map resolutions at each point along the ruptures (Fig.S4.4).Std2 represents the reproducibility and reliability of the measure and was plotted as an error bar in Figure4.8. We separately estimated the mean std2 for the foreshock and the mainshock ruptures, since the amplitude of the displacement and number of profiles are different. In the case of the mainshock, we estimate a mean std2 of 0.07, 0.14, and 0.06 m, respectively, in the fault-parallel, fault-normal, and vertical components. In the case of the foreshock, we estimate a mean std2 of 0.09, 0.06, and 0.04 m, respectively, in the fault-parallel, fault-normal and vertical components.We also calculated the coefficient of variation (CV) on our measurements. CV corresponds to the ratio between std2 and the mean value of the displacements, measured in the maps at various resolutions. The mean CV in the fault-parallel component is 0.13 for the foreshock rupture and 0.04 for the mainshock rupture. This highlights that largemagnitude displacements as for the mainshock rupture are less impacted by correlation noise and present a lower variability than lower-magnitude displacements, as observed for the foreshock, which displays a larger CV. In the fault-normal component of displacement, CV is 5.09 and 1.46, respectively, for the foreshock and mainshock ruptures. Because CV is the ratio between std2 and the mean offset value at a point, it tends toward large values when the offset values are close to zero. Hence, the mean CV for the fault-normal component of displacement is likely to be less meaningful because of the recurrent changes from positive to negative sign in the displacements(Fig. 4.8).Furthermore, CV for the vertical displacement offsets could not be assessed because a large number of measurements are close to zero which renders CV essentially infinite.

Figure 4 . 8 :

 48 Figure 4.8: Fault-parallel (red), fault-normal (green), and vertical (blue) slip distributions for the Mw6.4 foreshock earthquake in (A) and for the Mw7.1 mainshock earthquake in (B). Epicenter locations (red stars) and junctions between the foreshock and mainshock rupture (green triangles) are projected on the x-axis. Positive fault-normal displacement corresponds to extension. Positive vertical displacement corresponds, for the foreshock, to uplift on the northern wall of the rupture, and for the mainshock, to uplift on the eastern wall of the rupture. Cumulative moving mean of field data from DuRoss et al. (2020) are in orange. Optical results from Milliner and Donnellan (2020) in black, cumulated for the mainshock over three of the subparallel fault strands considered in their study. Both datasets (DuRoss et al., 2020; Milliner and Donnellan, 2020) are in the fault-parallel

4. 1 . 5 . 3 .

 153 Localized slip vs diffuse deformation for the Mw7.1 mainshock earthquake While in the previous sections we had systematically integrated the deformation across the entire rupture zone (Fig.4.8B), in the following we examine the detail of the individual slip distribution for 20 fault strands along the mainshock rupture that can be recognized in our horizontal displacement maps (Fig.4.9A,B). Hence, in many places, this involves looking at the individual slip distribution for several subparallel rupture strands. Still, faults shorter than 1 km were aggregated with the largest structure nearby. Also, individual faults have widths ranging from a few meters to a few hundred meters (e.g., red profile in Fig.S4.4) and include some of the rock damage that occurs in the direct vicinity of the fault structure. This is different from what is generally done in other studies, that estimate localized deformation on visible fractures of widths of a few meters only, and fault damage over scales of a few hundred of meters[START_REF] Cheng | Permanent Co-Seismic Deformation of the 2013 Mw7.7 Baluchistan, Pakistan Earthquake From High-Resolution Surface Strain Analysis[END_REF][START_REF] Gold | Coseismic Surface Displacement in the 2019 Ridgecrest Earthquakes: Comparison of Field Measurements and Optical Image Correlation Results[END_REF]. Hence, by subtracting the sum of the localized deformation along each different rupture strand from the deformation integrated over the whole fault zone (Fig.4.8B), we estimate the amount of diffuse deformation, corresponding to the longwavelength and low-amplitude gradients of deformation that occurs away from the primarily visible deformation areas (e.g., Fig.4.2B, profile BB'), at every location along the rupture (Fig.4.9B).

Figure 4 . 9 :

 49 Figure 4.9: (A) Map of the 2019 Ridgecrest mainshock surface rupture from DuRoss et al. (2020). Colors and corresponding labels indicate the 20 continuous fault strands for which we have collected individual slip profiles. (B) Total displacement (black, from Fig. 4.8B) vs localized fault slip (colored lines). Field datapoints for each fault strand are shown indicated by similarly colored crosses. For sake of clarity, error bars for field data are shown only for data above our total displacement budget. The sum of the localized slip is shown in white and the diffuse deformation (total displacement minus localized slip) in grey. (C) GG' fault-parallel displacement profile taken across the area where an excess of fault slip is measured relative to the total displacement budget in (B). Dashed grey lines correspond to the linear regressions (slopes 1 and 2) used to measure the total surface displacement in blue. Green profile represents a possible slip profile using uniform slip gradients inside the fault zone (slope 2). A component of apparent left-lateral shearing near fault strand 5 leads to an excess or overshoot of fault slip over total displacement.

Figure 4 .Fig. 4 .

 44 Fig. 4.9B) dominates the total displacement budget (black curve, Fig. 4.9B), while localized deformation remains limited along fault strands 1, 2 and 3 (colored curves, Fig. 4.9B). When moving toward the epicenter, the amount of diffuse deformation increases in parallel with the total displacement budget. At its maximum along segment S1, diffuse deformation reaches 2 m in places where no localized deformation in the right-lateral direction is observed at the ground surface. Along fault strand 3, where localized deformation is found, the diffuse deformation represents 25% of the total displacement budget for the fault-parallel component. In total, diffuse deformation in this region of the rupture represents 81% for a mean right-lateral slip of 1.21 m.

1 .

 1 locally increased stress within the upper few kilometers of fault strand 5, making fault strand 5 more prone to slip during the mainshock rupture. Unfortunately, current models of Coulomb stress variations based on slip inversion do not have the spatial resolution to further address this issue. Segment S3 corresponds to an area where the slip distribution for the fault-parallel component remains almost constant, around 1.5 m, for a distance of 22 km from the junction with the foreshock rupture to the southern tip of the mainshock rupture. In this area, displacement is primarily localized on fault strands 18 and 19 that accommodate slip up to a maximum of, respectively, 1 and 1.3 m. These two faults are connected by a 1-km long relay zone where up to 1 m of fault-parallel shear is accommodated through diffuse deformation (profile BB' in Fig. 4.2B and Fig. 4.9B). There, the proportion of diffuse deformation approaches 80%. Conversely, toward the southern end of the rupture, where fault strands 19 and 20 are parallel, the amount of diffuse deformation is close to zero. Fault strand 20 is sub-parallel to fault strand 19, and it accommodates only a small amount of slip (~0.20 m). Additional faults were mapped by Ponti et al. (2020) in this area, but they do not correspond to visible slip offsets in our surface displacement maps.Diffuse deformation in S3 represents 39% of the total surface displacement budget for a mean right-lateral slip of 1.42 m. For the entire length of the mainshock rupture, diffuse deformation represents 31% of the surface displacement budget. Similar amounts of diffuse deformation were also inferred along the foreshock rupture, where 31% of the total left-lateral displacement is accommodated through diffuse deformation (Figs. Two sets of orthogonal faults and their impacts on the surface displacements Although the main rupture follows a general trend oriented northwest, the Ridgecrest earthquakes displacement field is also characterized by the activation of a second set of cross faults trending N42°, perpendicular to the mainshock surface rupture, including the ground ruptures associated with the foreshock. In addition to the foreshock rupture already documented in previous studies (e.g.,Barnhart et al., 2020;[START_REF] Chen | Cascading and pulse-like ruptures during the 2019 Ridgecrest earthquakes in the Eastern California Shear Zone[END_REF][START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF] Milliner and Donnellan, 2020), a large number of secondary faults associated with the mainshock rupture also align along the N42° trend. One several kilometer-long fault located south of the foreshock rupture, well mapped in the field[START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF], crosses the mainshock ruptures between fault strands 18 and 19 (Fig.4.9A) and can be followed for several kilometers. This oblique-slip cross fault accommodates 0.15 m of left-lateral slip and 0.15 m of shortening in the area south of the connection with the mainshock rupture. We identified smaller scale northeast-trending faults in the relay zone between faults 18 and 19 (Fig.S4.2B) alongside this long northeast-trending cross fault. We also mapped smaller northeast-trending faults at the junction between the foreshock and the mainshock ruptures (Fig.4.6). Finally, the northern and southern tips of the mainshock rupture are characterized by faults with similar azimuth(Figs. 4.2A and 4.10). Many of these faults, with the exception of those observed at the mainshock/foreshock junction (Fig.4.6), were identified in the field[START_REF] Duross | Surface Displacement Distributions for the July 2019 Ridgecrest, California, Earthquake Ruptures[END_REF]. Their displacement, however, could generally not be quantified in the field due to the lack of unambiguous markers.

Figure 4 .

 4 Figure 4.10: (A) East-west and (B) north-south displacement maps to the north of the mainshock rupture, where northeast-trending faults ruptured the surface (see Fig. 4.2A for location). Rupture map from DuRoss et al. (2020) is overlaid in black in (B). (C) Curl map calculated from horizontal displacement maps downsampled at 2 m. Blue

4. 1 . 6 . 2 .

 162 Comparison with kinematic slip inversions and implications for the rupture segmentation and the processes of surface deformationWe compare the distribution of slip at depth derived from several kinematic inversions compiled byWang et al. (2020) to our surface displacements for the mainshock of the 2019 Ridgecrest sequence. In our study, we subdivided the rupture into three segments, bounded by variations in the rupture azimuth and geometrical complexities (Figs. 4.2A, 4.8B, and 4.9B). At depth, the estimations of fault dip angle based on the relocated seismicity yield values ranging between 70° and 85° toward the SW for the northern segment S1(Ross et al., 2019) and between 70° and 75° toward the NE under the southern segment S3[START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF][START_REF] Plesch | Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence[END_REF]Ross et al., 2019). For the central segment of the rupture (S2),[START_REF] Plesch | Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence[END_REF] inferred a 55°NE fault dip angle for the primary fault plane (fault strand 5, Fig.4.7A).

Figure 4 .

 4 Figure 4.11: (A) Surface fault-parallel slip distribution along the mainshock rupture from this study in black (total displacement curve from Fig. 8B) and from five published kinematic inversions in color. For the models, predictions of slip on subparallel fault strands were added to obtain a total displacement prediction that is comparable to the displacement curve from this study (in black). Joint inversion from Wang et al. (2020) includes both seismic and geodetic data. Predictions of surface slip from the five kinematic inversions are projected onto a common line that follows the mean rupture azimuth (N140°), and values on subparallel fault strands are summed to obtain total surface displacement curves that are comparable with optical and field results from Figures 8B and 9B. Diffuse deformation calculated in this study (Fig. 4.9B) is shown in grey at the bottom. (B) Combined geodetic and seismological data inversion for the mainshock slip distribution on the primary fault plane from Wang et al. (2020). Slip vectors show relative motion of the east side of the kinematic fault model. Circular rupture front at a velocity of 2.2 km/s is in black. Blue circles are relocated aftershocks from Shelly (2020).

Figure 4 . 12 :

 412 Figure 4.12: Displacement profile (black), in the fault-parallel component of displacement, taken across the diffuse deformation area to the north of the mainshock rupture (ii' profile from Fig. 4.10E). Predictions of surface displacements from 2D elastic models are presented in blue, red and orange. Blue model corresponds to a direct fit of the data without data constraining the input parameters. Red and orange models use the kinematic slip models(Wang et al., 2020) as input parameter. The fault is vertical in the red profile, and dips 70° S-W, similar to the geometry inferred from the distribution of aftershocks(Ross et al., 2019), in the orange profile.

Figure 4 .

 4 Figure 4.13: (A) Map of the FZW along the mainshock surface rupture. Positions of the colored lines represent the western and eastern ends of the deformation zone in each of the 460 displacement profiles used to quantify slip along the mainshock rupture (e.g., Figs. 4.7A, vertical profile and Fig. S4.4). Colors represent the total displacement measured at each profile (Fig. 4.8B). Background map from DuRoss et al. (2020). FZW along the foreshock rupture is presented in Figure S4.9. (B) Surface displacement profiles for strike-slip motion on a vertical dislocation in an elastic medium. The upper limit of the dislocation varies from 0 to 4 km depth, and right-lateral slip is 4 m.

  Pleiades and WorldView images (Tab. S4.1), the map of the combination of pre-and post-earthquake images (Fig. S4.1) with the details of the image combinations and the digital surface models (DSMs) we used (Tab. S4.2), a comparison of an east-west displacement map generated with different orthorectification methods (Fig. S4.2), and a presentation of the low-frequency corrections we made on the east-west component displacement map for the entire study area (a similar method was applied to the northsouth component; Fig. S4.3).Then, we present a series of profiles across our displacement maps that allow for characterizing localized fault slip vs diffuse deformation (Fig. S4.4), a series of profiles across our displacement maps downsampled at various resolutions (Fig. S4.5), and a comparison of the displacement values we measured in this study against values measured in the field and in other optical-based studies (Fig. S4.6). Next, we present the detailed displacement budget (localized fault slip vs diffuse deformation) for the foreshock earthquake (Fig. S4.7) and the detailed map of the ruptures we used to measure localized fault slip along the foreshock rupture (Fig. S4.

. 1 :

 1 Figure S4.1: Map location of the 14 correlation tiles used to measure surface displacements after the 2019 Ridgecrest earthquakes. References of the images used for each tile are reported in TableS4.2.

Figure S4. 2 :

 2 Figure S4.2: (A and B) East-west displacement maps in the area of the jog between fault strands 18 and 19 (Fig. 4.2A). Pre-earthquake DSM used in (A) was calculated using a combination of WorldView and Pleiades images and the MicMac correlation method (tile 8; Fig. S4.1). Pre-earthquake DSM used in (B) corresponds to the 2-m DSM from Willis et al. (2019) over-sampled to a 0.5-m resolution. Artefacts in (A) consist in a series of vertical bands that arise from the shape of the CCD sensor of the WorldView satellite.This artefact is not visible in (B). Some of the faults aligned with azimuths ~N42, similar to the faults identified at the mainshock rupture tips(Figs. 4.2A and 4.10) as well as the foreshock rupture(Figs. 4.2A and 4.6), are indicated with black arrows. (C) JJ' profile across (A) in black and across (B) in red. Profile is taken east-west, parallel to the rows of the image so that there is no interpolation between the pixels (interpolation is required when profiles are oblique to the lines and rows of the image). Profiles show that although

Figure S4. 3 :

 3 Figure S4.3: (A) East-west displacement map of the 2019 Ridgecrest earthquake from Sentinel-2 optical images correlation at 10 m ground resolution. High-frequency noise comes from the shape of the CCD sensor. Only a first-degree polynomial correction using the far-field undeformed area as a reference was applied. No correction was made for other high-frequency artefacts. (B) Difference between east-west displacement maps

Figure S4. 4 :

 4 Figure S4.4: Displacement profiles across the northwestern tip of the mainshock rupture (locations in Figs. 4.2B and 4.10A). Swath profiles (grey) are 100 m-wide and ~10 km long. Filtered profiles are presented in colors. Stravinsky-Golay filter was used with a window of 300 m and a first-degree polynomial. Blue and green profiles are taken across areas where no rupture was observed to the surface, neither in the field observations nor in our optical-based displacement maps. Red profile crosses fault strand 3. This figureshows that the FZW cannot be used as a parameter to distinguish the modes of surface deformation (localized or diffuse). In fact, diffuse deformation can occur over varying scales: 2074 m for the blue profile and 754 m for the green profile. Also, the green and red profiles exhibit similar values of FZW, respectively 754 m and 730 m, while only the red profile display a subvertical slip offset corresponding to slip on a surface fault strand

Figure S4. 5 :

 5 Figure S4.5: Fault-parallel displacement profile across the epicentral area of the mainshock rupture taken across displacement maps at 2 m (black), 3 m (green), 4 m (red), and 5 m (blue). The profiles cross fault strands 5, 10, 12, and 13 (Fig. 4.4A,B). Profiles are shifted slightly to the top left corner of the figure from the 2-m to the 5-m profiles so that similarities between the profiles can be observed. Axis of the plot are aligned with the 2-m profile (black). Error (±) associated with each measurement of total displacement corresponds to the standard deviation on the linear regression measured directly by the COSI-Corr software, corresponding to std1 (linear regressions are represented with the black dashed lines for the 2-m resolution profile). Std2 corresponds to the standard deviation taken over the series of four offsets measurements and CV to sdt2 divided by the mean displacement.

Figure S4. 6 :

 6 Figure S4.6: Surface displacement curves for the Mw7.1 mainshock earthquake. Black crosses display the total displacement budget measured in this study (Figs. 4.8B and 4.9B). Blue and green crosses, respectively, from Milliner et al. (2021) and Gold et al. (2021), report values of fault slip for the major fault strands of the surface rupture. These three datasets are based on high-resolution optical image correlation measurements. Red points are field measurements from DuRoss et al. (2020). Error bars are in grey for all four datasets. The total displacements we measure exceed the values in the other studies because we integrate slip on the sub-parallel fault strands as well as diffuse deformation into the total displacement budget (Fig. 4.9B).

Figure S4. 7 :

 7 Figure S4.7: Surface displacement budget along the foreshock rupture. (A) Total displacement curve (in black, similar to the red profile in Fig. 4.8A) and fault slip curves in color. The fault slip curves are obtained using 200-to 400-m long profiles across the 15 fault strands that define the foreshock surface rupture, similar to what has been done for the mainshock rupture (Fig. 4.9A). Map of the 16 fault strands used to characterize surface slip for the foreshock rupture is presented in Figure S4.8. Red star along the xaxis is the location of the foreshock epicenter. Green triangle along the x-axis reports the junction point with the mainshock rupture. (B) Diffuse deformation budget, in grey, with the total displacement curve reported in black. Diffuse deformation can be negative in a few areas where resampling of the data on a common grid leads to some discrepancies where fault strands are very small (<1 km). Diffuse deformation accommodates 100% of the left-lateral displacement at the south-western tip of the rupture (Figs. 4.2 and 4.3).

Figure S4. 8 :

 8 Figure S4.8: Map of the 15 fault strands constituting the foreshock rupture, based on the displacement maps from this study. Colors and numbering correspond to those used in Figure S4.7. Background map (black) from DuRoss et al. (2020). The two maps overlap in most areas but several strands mapped in the field did not exhibit significant slip offsets (>10 cm). Conversely, fault strands4, 6, 7, 8, 14, and 15 were not mapped in the field while slip offsets were measured in the displacement maps.

Figure S4. 9 :

 9 Figure S4.9: Map of the FZW along the foreshock surface rupture. Positions of the colored lines represent the northern and southern ends of the deformation zone in each of the 123 displacement profiles we used to quantify slip along the foreshock rupture. Colors represent the total displacement measured at each profile (Fig. 4.8A). Background map is the rupture map from DuRoss et al. (2020). To the northeastern and southwestern tips of the foreshock rupture and where the surface displacements are of small amplitudes relatively to the noise, a fixed FZW of 400 m is used to retrieve the displacements. This width of 400 m is sufficient to capture the surface displacement patterns in these areas: no attempt was made to capture the details of the FZW at a local scale.

Figure S4. 10 :

 10 Figure S4.10: Plot of the FZW against the total surface displacement for (A) the foreshock surface rupture and (B) the mainshock surface rupture. Green triangles report the areas of junction between the mainshock and foreshock ruptures. Black curve is the total surface displacement budget estimated in this study (Fig. 4.8). Red curve is the FZW (including the diffuse deformation zones) from this study (Figs. 4.13A and 4.A9). Orange crosses report values of regional FZW from Gold et al. (2021). Regional FZW from Gold et al. (2021) is generally in agreement with our data, except the northwestern end of the mainshock rupture, that is not covered by the Gold et al. (2021) dataset. Along the foreshock rupture, many points from Gold et al. (2021) lie above our data. This corresponds to areas where we used a fixed FZW of 400 m (see explanations in Fig. S4.9).

Figure 4 . 14 :

 414 Figure 4.14: Coseismic slip model of the Mw6.4 foreshock and Mw7.1 mainshock earthquakes of the 2019 Ridgecrest sequence. (a) Model from Chen et al. (2020) that arises from the inversion of GNSS, InSAR, Sentinel-2 and PlanetLab optical data, and seismic data. (b) Model from Magen et al. (2020) that was derived from the joint inversion of GNSS, InSAR, and PlanetLab optical data. The figure is modified from Wang et al. (2020), which present a review of the existing coseismic slip models for the 2019 Ridgecrest earthquake sequence.

Figure 4 .

 4 Figure 4.15: (a) Model geometry overlapped with the field-based fault map from Duross et al. (2020). Colors and numbering refer to the 5 faults used in the model. (b,c,d) Sampled optical data in the vertical (b), east-west (c) and north-south (d) components. Sampling

  4.16b), to determine the R value for the best trade-off between model complexity and fit to the data. InSAR residuals are used as a reference for the model misfit to the data because InSAR covers most of the study area and present a small uncertainty, of a few centimeters. On the other hand, optical data cover only the fault zone, have an uncertainty about 10-20 cm, and present large misfit where the model cannot reproduce the rupture complexity.

Figure 4 .

 4 Figure 4.16: (a-b) Results of the regularization tests for the entire length of the fault model (50 km). (a) Evolution of the maximum surface slip (S) taken on the most superficial patches from the fault slip model (~0-1 km), in red, and the maximum slip (M) along the model, in blue, with increasing regularization (R). Slip is plotted against log(R) for the sake of clarity. Horizontal line plotted at S=5 m indicates the maximum surface slip

Figure 4 .Figure 4 .

 44 Figure 4.17 shows a 3D view of the model obtained from the joint inversion of GNSS, InSAR and optical data, with a weight of 1, 1, and 0.5, respectively, and a regularization of 2x10-5 . Results display 4 patches of large slip at depth, referred to as P1 to P4. The largest, P1, is located to the south-east of the mainshock epicenter, along F2, below the epicenter that is located at 2 km depth[START_REF] Huang | Spatio-temporal foreshock evolution of the 2019 M 6.4 and M 7.1 Ridgecrest, California earthquakes[END_REF]. Maximum of slip in P1 is reached at the junction between F1 and F2. P1 extends ~9 km along-dip and ~5 km along-strike with a maximum amplitude of 10 m from 4.4 to 7.3 km depth. P2 occurs along F1, just north-west of the epicenter. It extends 6 km downward from the surface, with a maximum slip around 9 m. P3 is located on the north-westernmost segment of the rupture (segment 1 in Fig.4.9), and it extends ~8 km along-dip and along-strike. P3 slip has a lower amplitude than slip in P1 and P2, from 5 to 8 m. P4 occurred along F3, south-east of the junction with F2, and it reaches a maximum slip around 6 m. Two other patches of large slip can be distinguished, but with smaller amplitude of than the others, from 3 to 5 m.

Figure 4 . 17 :

 417 Figure 4.17: Coseismic slip model of the 2019 Ridgecrest earthquake sequence from the kinematic slip joint inversion of InSAR, GNSS and optical data. Model segments are highlighted using the colors and numbering from Figure 4.14. Mainshock epicenter from Huang (2020) is indicated by a red star. Foreshock epicenter from Huang (2020) is reported by a blurred blue star because it is located behind F4 and F5, at the junction with F2 (see Fig. 4.14 for epicenter map locations). Slip model is presented in a north-east view in (a), and a south-west view in (b).

Figure 4 . 18 :

 418 Figure 4.18 : Residuals between the model and data in the ascending (a) and descending (b) components of InSAR, and in the east-west (c) and north-south (d) components of optical data. (e) Data (red) and modeled (blue) GNSS displacement for the closest stations to the ruptures. (f) Frequency of occurrence of the optical and InSAR residuals for amplitudes ranging between ±50 cm. Quantile distribution of the residuals is presented in

  4.18e).Maximum and minimum residuals for the InSAR data reach ± 50 cm for both the ascending and descending tracks. It occurs mainly around the pull-apart basin to the north-west of the mainshock epicenter, and between F1 and F2, where the ground ruptures are more complex than the fault model. Largest residuals with the optical data reach ± 2 m. It occurs most often along the fault traces, because the ground fault geometry does not perfectly fit with that of the fault model (these are masked in Figure4.18), and because of the complexity of the near-fault displacement field that is not entirely reproduced by the model. For example, we observe residuals of ± 2 m in the bookshelf faulting area, at to the north-west of the mainshock rupture, where surface displacements are accommodated only through diffuse deformation. In this region of the rupture, because the gradients are of large-wavelength, the resampling of the data averages the signal over the fault zone, which results in a lack of constrain close to the fault. Hence, the models puts all the displacement on the fault.

  ), include models from[START_REF] Gao | Coseismic faultslip distribution of the 2019 Ridgecrest Mw6.4 and Mw7.1 earthquakes[END_REF],[START_REF] Jin | Finite Slip Models of the 2019 Ridgecrest Earthquake Sequence Constrained by Space Geodetic Data and Aftershock Locations[END_REF],[START_REF] Li | The 2019 Mw 6.4 and Mw 7.1 Ridgecrest earthquake sequence in Eastern California: rupture on a conjugate fault structure revealed by GPS and InSAR measurements[END_REF],[START_REF] Liu | Coseismic Rupture Process of the Large 2019 Ridgecrest Earthquakes From Joint Inversion of Geodetic and Seismological Observations[END_REF],[START_REF] Tung | Structural Controls Over the 2019 Ridgecrest Earthquake Sequence Investigated by High-Fidelity Elastic Models of 3D Velocity Structures[END_REF],Wang et al. (2020), and Xu et al. (2020b). These models use various combinations of SAR, GNSS and seismic data to constrain the slip distribution.Xu et al. (2020b) is the only model from this selection that uses optical images to constrain the near field domain. Common point of all these models is that they use a model geometry constituted of several subparallel and cross-cutting faults, accounting for the rupture geometry observed at the surface (field and optical imagery) and in the seismology. The only exception is the model from[START_REF] Tung | Structural Controls Over the 2019 Ridgecrest Earthquake Sequence Investigated by High-Fidelity Elastic Models of 3D Velocity Structures[END_REF] that considers only two orthogonal curvilinear rough fault planes to describe the rupture. However, this models differs from all the other models by the fact that it uses the finite element method to account for a complex crustal structure. Other models that used optical data in the inversion process are not included in this selection. This is the case of the model fromBarnhart et al. (2019) and of[START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF] (Fig.

  ) lie in the model geometry and resolution.Barnhart et al. (2019) use only two linearly dipping planes to represent the foreshock and mainshock, and[START_REF] Magen | Fault Rerupture during the July 2019 Ridgecrest Earthquake Pair from Joint Slip Inversion of InSAR, Optical Imagery, and GPS[END_REF] use a complex model but with a resolution and meshing that do not allow for a gradual evolutions of slip along the fault plane (Fig.4.14). This shows that, despite the use of high-resolution near-fault data, the choice of the model geometry, resolution and meshing primarily impacts on the slip distribution that is inferred at depth. Our model geometry is fixed at the surface by the trace of the faults in the field and optical data (Fig. 4.15a), and at depth by the 3D fault model from Plesch et al. (2020) based on relocated hypocenters of the 2019 Ridgecrest sequence from Hauksson et al. (2020) and Ross et al. (2019), and selected focal mechanism from Hauksson et al. (2020). The results from Plesch et al. (2020) show that the faults dip vertically after ~5 km depth, hence, our vertical fault model has a realistic geometry below 5 km. Because we are interested in the shallow rupture processes, we attempted to model variations of fault dip angles at shallow depths.

Figure 4 . 19 :Figure 4 . 17 .

 419417 Figure 4.19: Coseismic slip model of the 2019 Ridgecrest earthquake sequence on a complex fault model that includes variations of the fault dip in the shallow part. Model parameters are similar to those used for the vertical model (Fig. 4.17). Fault dip angles at the surface is 75°NE along the mainshock model, and 80°NW along the surface foreshock model. Fault dip angle is 90° after 5 km depth for all the faults. Numbering of the slip patches (P1-P6) refers to similar slip patches observed using the vertical fault model fromFigure 4.17. Slip model is presented in a north-east view in (a), and a south-west view in (b).

  4.9a). General trend of the displacement curves (blurred lines in background in red, green, and blue for the field equivalents of F1, F2, and F3, respectively) is well-represented by our model, and highamplitude and small-scale variations from the model match well with those from the optical data. This shows the ability of the model to incorporate complexity in the slip distribution along the fault plane. This is allowed by the realistic fault geometry and the fine meshing (~500 m at the surface) of the fault model, as well as the detailed data near the fault.

Figure 4 .

 4 Figure 4.20 : (a) Comparison along the NW-fault system between surface fault slip from the coseismic slip model from Figure 4.17 (dashed lines, red for F1, green for F2, blue for F2, similarly to Fig. 4.15a), from Wang et al., 2020 (solid lines of the corresponding color), and from optical imagery data (blurred solid lines of the corresponding color).

Figure 4 .

 4 21, which include models from Chen et al. (2020), Jin and Fialko (2020), Liu et al. (2019), Ross et al. (2019), Wang et al. (2020), Xu et al. (2020a) and Yue et al. (

(Fig. 4 .

 4 [START_REF] Wu | Stick-slip propagation velocity and seismic source mechanism[END_REF]. Eventually, uncertainty remains on the effect of the homogeneous crustal model assumption. But, the model from[START_REF] Tung | Structural Controls Over the 2019 Ridgecrest Earthquake Sequence Investigated by High-Fidelity Elastic Models of 3D Velocity Structures[END_REF], that considers a realistic crustal model, shows results comparable to this model. Hence, we consider the effect of the crustal model as minor when compared to that of the model geometry, regularization, and data completeness.

Figure 4 .

 4 Figure 4.21: (a) Average and (b) normalized average slip along the depth of different coseismic slip models, including the mainshock and foreshock ruptures. Normalized average slip is obtained by dividing the average slip profile (left) of each model by the magnitude of its maximum slip. Model from this study with R=2x10 -5 (corresponding to the model from Figure 4.17) is in red, with R=1x10 -4 is in orange, and R=6x10 -5 in brown. Models from Wang et al. (2020) are in blue, with dark blue for the geodetic model, and light blue for the joint inversion-based model. Other published models overlapped in grey include those reviewed in Wang et al. (2020), hat is models from Chen et al. (2020), Jin and Fialko (2020), Liu et al. (2019), Ross et al. (2019), Xu et al. (2020b), and Yue et al. (2021). Non-zero slip at large depth (>15 km) in our model arise from a dip-slip component of displacement, but right-lateral slip drops to zero after 15 km depth.

Figure 4 . 22 :

 422 Figure 4.22: Along-depth distribution of co-seismic fault-parallel displacement between the faults and the medium diffuse deformation, including the foreshock and mainshock events. Light red area represents the average right-lateral slip inferred from the joint inversion of geodetic data (Fig. 4.17). Dark red line extrapolates the slip distribution from the maximum average at depth to the value of 1.26 m at the surface, that corresponds to the measured displacement on faults in the optical data. Blue red line extrapolates the slip distribution from the maximum average at depth to the value of 1,8±0.07 m at the surface, that corresponds to the total measured displacement in the optical data, including the diffuse deformation.

  Précédemment dans cette thèse, nous avons démontré les apports de la corrélation d'image optique à haute résolution pour caractériser le champ de déplacement en surface associé à des grandes séismes continentaux. Dans ce chapitre, basé sur des modèles théoriques et numériques combinés avec des observations sur des failles naturelles, nous montrons l'apport de la corrélation d'image optique pour l'analyse des caractéristiques dynamiques de la rupture. Tout particulièrement, nous montrons le lien entre la variation de la taille de la zone de faille en surface, et des changement brutaux dans la vitesse de la rupture. Ce travail fait partie d'une collaboration multidisciplinaire avec J. Jara, L. Bruhat, M. Y. Thomas, K. Okubo, E. Rougier, A. J. Rosakis, C. G. Sammis, Y. Klinger, R. Jolivet et H. S. Bhat. Le manuscrit intitulé « Signature of transition to supershear rupture speed in coseismic off-fault damage zone » est actuellement à l'étude à Proceedings A (manuscrit entier présenté dans les suppléments de cette thèse). Dans ce chapitre est présenté, tout d'abord, un résumé des théories numériques qui permettent d'assumer un lien entre la réduction de la taille de la zone de faille à la surface, et la transition de la rupture depuis une vitesse subshear, c'est-à-dire inférieur à la vitesse des ondes de cisaillement 𝑣 𝑠 , à supershear, c'est-à-dire supérieur à 𝑣 𝑠 . Dans un deuxième temps, nous développons les résultats de la corrélation d'images optiques effectuée sur la zone du séisme de Kokoxili (2001, Mw7.9, Chine), qui s'est en partie propagé une vitesse supershear. Cependant, contrairement aux précédents chapitres où nous nous avions accès à des images haute résolution, la résolution des images optiques disponibles pour cette zone avant le séisme est de 10 m. Cela ne permet pas d'étudier le détail des mécanismes

5. 2 .

 2 Application of optical images correlation to the 2001 Mw7.8 Kokoxili supershear rupture (China) and measurement of the fault zone width (FZW)

Figure 5 . 1 :

 51 Figure 5.1: East-west, north-south and correlation score obtained from correlation of paired SPOT images at 10 m ground resolution. Output displacement maps report displacements at every pixel location, so every 10 m, in the east-west and north-south directions.

Figure 5 . 2 :

 52 Figure 5.2: Example of one of the 396 profiles obtained from the optical correlation image processing along the fault. The step in the figure represents the coseismic offset produced by the earthquake, while the red bow denotes the region defined during this work as the fault zone width, illuminating the width of the off-fault damage zone.

Figure 5 . 3 :

 53 Figure 5.3: (A) Map of the strike slip section of the 2001 Mw~7.8 Kokoxili earthquake(China), where P1 denotes the transition zone reported for the event from seismological far-field data(Vallée et al., 2008). (B) Along-Strike fault zone width (black) and its

Figure 5 . 4 :

 54 Figure 5.4: SPOT image of the 2001 Kokoxili rupture area: (A) before that the rupture transitioned to supershear speed and (B) after that the rupture transitioned to supershearspeed. The rupture is indicated in red at some locations. Location of the push-up that is mentioned as a potential trigger for the supershear rupture(Vallée et al., 2008) is circled in red dashed lines. The rupture trace is located in some places with red arrows. The rupture is very clean in the images in B), unlike in A) where the trace is more ambiguous.

Figure 5 . 5 :

 55 Figure 5.5: Across-fault displacement profiles showing the fault-parallel component of displacement (left-lateral displacement). Profiles are arranged in a three-dimensional view. Plot axis show displacement amplitude and distance across the fault for the first profile to the bottom of the plot. The rupture propagates from the west to the east, that is from the bottom to the top of the plot. Profile 168 corresponds to the profile where we

6. 1 . 1 .

 11 Field observations Kilometric-wide deformation zones are reported for a large number of faults, including exhumed systems and surface ruptures. Along exhumed faults, for example, Wallace and Morris (1986) reported 4 to 6 km-wide shear zones penetrated by multiple sub-parallel and interconnected faults, along the Osburn fault in northern Idaho. Similarly, Faulkner et al. (2003) reported 1 km-wide deformation zones constituted of multiple strands of fault gouge separated by lens of fractured rocks in the exhumed Carboneras fault zone, southeastern Spain. Lin and Yamashita (

Figure 6 . 2 :

 62 Figure 6.2: Time evolution of the relocated mainshock and aftershocks with distance to the fault for the 2018 Mw5.6 Osaka, Japan, earthquake. DTZ is a zone of delayed aftershocks. Red dashed lines highlight major areas of aftershock activity, located ~1 to 2.2 km to the west, and 0.5 to 1.5 km to the east of the fault (modified from Kato and Ueda, 2019).

Figure 6 . 3 :

 63 Figure 6.3: Seismicity of the 2019 Ridgecrest earthquake sequence, including foreshock, mainshock, and aftershock sequences. AA' profile, which highlights the seismicity on the orthogonal basement fabric, is not shown in this figure. Black lines correspond to the surface ruptures of the 2019 sequence (DuRoss et al., 2020). Purple lines are quaternary faults from the USGS fault map. Dashed blue lines highlight clusters of aftershocks along northwest-southeast parallel planes. North-east shift of ~1 km between the seismicity cluster highlighted in dashed blue, and the surface fault trace results from the 80°NE dipping angle of the fault (modified from Ross et al., 2019).

Figure 6 . 4 :

 64 Figure 6.4: (i) LiDAR differential topography map at 25 m ground resolution along the Dozon section of the 2016 Mw7.0 Kumamoto, Japan earthquake. AA' profile is in (ii). D and D' report the area of diffuse deformation observed south-east of the rupture in AA'.The unmapped displacement discontinuity in (i) is indicated in (iii), and it correlates with the trace of the 2002 Imaizumi rupture. (ii) AA' vertical displacement profile across the Futagawa fault in blue. Yellow, red and blue areas report the domains used to measure displacement offsets over ±35, ±100 and ±1000 m large areas[START_REF] Scott | The M7 2016 Kumamoto, Japan, Earthquake: 3-D Deformation Along the Fault and Within the Damage Zone Constrained From Differential Lidar Topography[END_REF]. Red

Figure 6 . 5 :

 65 Figure 6.5: Vertical displacement map along the Paso Inferior Accommodation Zone (PIAZ) of the 2010 Mw7.2 El-Mayor Cucapah, Mexico, earthquake. Vertical displacements are derived from topography differing of a post-earthquake tri-stereo Pleiades DSM, and a pre-earthquake LiDAR digital elevation model. Diffuse deformation gradient between the Paso Superior fault and the Laguna Salaga Fault is indicated in blue (modified fromZhou et al., 2015).

Figure 6 . 6 :

 66 Figure 6.6: Conceptual model of displacement and structure associated with a fault zone that sustains diffuse deformation. The model is based on the results from optical image correlation (section 4.1.) and joint inversion of geodetic data (section 4.2.) along the 2019 Ridgecrest earthquake sequence. It presents the case of a unique fault plane that displays a symmetric displacement pattern. (a) Normalized surface displacement profile across the fault zone. The profile shows the distribution of the surface displacement between the primary fault structure, corresponding to the area where fractures are visible on the field, and the diffuse deformation zone. The total fault zone width corresponds to the area over which the total surface displacement (between the horizontal red dashed lines) is

  𝜇 × 𝐷 × 𝑆 with 𝜇 the average rigity of the medium (30 GPa), 𝐷 (m) the displacement on the fault, and S (m 2 ) the area of the fault plane. 𝑆 = 𝑊 × 𝐿 with 𝑊 (m) the along dip width of the fault plane (10 km based on the curve from Figure 4.21), and 𝐿 (m) the horizontal length of the fault. We use 𝐿 =18 km for the foreshock rupture, and 𝐿 = 55 km for the mainshock rupture, based on our measurements of surface displacement (Fig. 4.9). From this equation, we obtain a seismic moment of 3.94 x10 18 N.m for the foreshock earthquake, and 3.51 x10 19 N.m for the mainshock earthquake (Tab. 6.1). This yields an increase of seismic moment of 1.24 x10 18 N.m (+45%) for the foreshock earthquake, and 0.96 x10 18 N.m (+37%) for the mainshock earthquake, compared to the values based on the fault displacements (DuRoss et al., 2020) (Tab. 6.1). Considering the following relation for moment magnitude:

Figure 6 . 7 :

 67 Figure 6.7: Average SSD measured from earthquake rupture models, plotted against earthquake magnitudes, for strike-slip events of 6.5<Mw<7.8. Results from this work are in red (Antoine et al., 2021) and are discussed in section 5.1.??. Models from Antoine et al. (2021), Lauer et al. (2020), Socquet et al. (2019), Xu et al. (2020, 2016) and Yue et al.

Figure 6 . 8 :

 68 Figure 6.8: (a) Rupture model for the 2018 Mw7.5 Palu earthquake from the joint inversion of optical (Sentinel-2, 10 m) and InSAR data (Socquet et al., 2019). Slip on the fault is left-lateral. Areas of SSD (1,2,3) are indicated above the model. (b) Horizontal displacement map of the 2018 Mw7.5 Palu earthquake from correlation of Sentinel-2 optical images (Elliott et al., 2020). Areas where SSD is visible in the model are highlighted with white arrows. AA' profile is presented in (c). Color scale is similar in (a) and (b), although (a) represents the fault displacement and (b) the ground displacement. (c) AA' profile across the maximum slip zone. All the surface displacement

from the 2019

 2019 Ridgecrest ruptures, for which we observed variations in the fault displacements between the different segments. Because a Mw>7.5 earthquake is more likely to propagate the maximum slip all the way to the surface, fault displacements measured along the different segments of the rupture should reflect the total coseismic displacement. However, at the segment transitions, geometrical complexities would disrupt the propagation of slip along the fault plane, and part of the deformation will be accommodated off the primary fault system. As shown along the C2 complexity of the 2013 Baluchistan rupture, diffuse deformation compensates this decrease in localized fault displacement, and the total coseismic displacement budget remains constant across the complexity (Fig.3.9).Finally, because the 2013 Baluchistan and 2019 Ridgecrest ruptures occurred in two different geological contexts, we discuss of the possible control exerted by the geological context on the fault geometry, and hence, on the distribution of diffuse deformation. Also, because the 2019 Ridgecrest exhibits exceptionally large (30-43%) SSD compared to

Figure 6 . 9 :

 69 Figure 6.9: Geological background of the 2019 Ridgecrest surface ruptures. Field rupture map from DuRoss et al. (2020) is in red. Azimuths of the fault segments are indicated with arrows. Maximum compression axis 𝜎 1 is from Fialko and Jin (2021). Geological map is from Jennings et al. (1962). Pink terrains correspond to Mesozoic granitic rocks. Dark orange terrains, located for most part South of the Garlock fault, are Cenozoic

  also highlighted a decrease of the strikeslip component, and an increase of the fault-normal component of displacement southwest of the C2 relay zone. It was interpreted as a transition from the Chaman strike-slip fault system to the north, to the compressive Makran area to the south-west. Hence, the C2 relay zone represents a major complexity in the trace of the 2013 Baluchistan rupture, at the transition between two discontinues faults that belong to two different tectonic systems. As a consequence, despite the fact that the 2013 rupture has a magnitude of 7.8, this complexity represented a major barrier to rupture propagation and hence, triggered diffuse deformation around the fault ruptures. Diffuse deformation represents 30% of the total displacement budget in this area.

Figure 6 . 10 :

 610 Figure 6.10: Geological context of the 2013 Baluchistan rupture. The Hoshab fault, that ruptured on the day of the 2013 event, is highlighted in red. Location of the C2 complexity along the fault is highlighted with a red circle. Yellow terrains, noted Tmo, correspond to Miocene and Oligocene sedimentary shales and sandstones. Beige dotted terrains, noted Q, correspond to unconsolidated Quaternary alluvial deposits (modified from[START_REF] Bakr | Geological Map of Pakistan. (East and West Pakistan)[END_REF]

  marker in the field. It means that the deformation processes that are brought into play in the diffuse deformation affect the rocks at scales below what is visible by human eyes, orwhat is detectable with our methods (0.5 m in length and 0.1 m in amplitude). Moreover, the fact that these gradients are continuous and have a curved profile (concave or convex depending on the side of the rupture), shows that the deformation homogeneously affects the volume of rocks. These characteristics justify the name of diffuse deformation. To better characterize diffuse deformation and how it affects the properties of the rocks in the off-fault medium, we propose a series of mechanisms that could generate homogeneous gradients of displacement.

  ranging from 30 m to 200 m. The rotation maps highlight the left-lateral mechanism of the faults activated during the 2013 Baluchistan ruptures, to the exception of a right-lateral fault that accommodates slip in an area of particularly complex fault geometry. Along the 2019 Ridgecrest ruptures, the rotation map shows that both leftlateral northeast-striking, and right-lateral north-west striking faults were activated, which is corroborated by the field observations. In addition, dilatational and compressional deformation were principally observed on the 2013 Baluchistan faults for which an important fault-normal component was measured during the rupture. Along the purely strike-slip 2019 Ridgecrest ruptures, only small areas of compression and dilatation were measured, and they are associated with areas of fault geometrical complexity. Finally, along both the 2019 Ridgecrest and the 2013 Baluchistan ruptures, we showed that the basement fabric participated to the geometrical complexity of the ruptures, with the reactivation of pre-existing orthogonal structures.Because the ruptures are complex, one has to measure the displacement accommodated by each fault to understand the way coseismic displacement is distributed across the entire system. With this aim, we used profiles across the individual faults to measure the amount of strike-slip displacement accommodated by each fault, with a precision of 5 to 20 cm.

  methods, one can attempt to reproduce fault geometrical complexity with off-fault diffuse deformation, and analyze the evolution of the system with increasing cumulative deformation. In addition, in our study of the 2013 Baluchistan surface rupture, we brought to light clues on potential fault complexity evolution with cumulative deformation. Indeed, we observed two complexities with one of them that displays the characteristics of a recent fault complexity relatively to the long-term accumulated displacement on the fault. Using a large sandbox model (> 3 m), eventually including pre-existing heterogeneities in the medium, we could reproduce the long-term evolution of a fault zone and analyze the phases of construction, and destruction of fault geometrical complexities.

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Figure 1 .

 1 Figure 1. LEFM solution illustrating the temporal evolution of damage around a rupture tip. All plots are normalized by their maximum values. (a) Prescribed rupture velocity history normalized by the Rayleigh wave speed. (b) Evolution of the static stress intensity factor which is proportional to the length of the crack, p L(t). (c) Freund's universal function. Here c R and cp are the Rayleigh and P-wave speeds respectively. (d) Evolution of the dynamic stress intensity factor. (e) Instantaneous snapshots of the contours of the domain where the Drucker-Prager failure criterion is violated, F DP ≥ 0, at key points labeled 1 through 9 in (d). (f) Superposition of the contours where F DP ≥ 0 throughout the history of the rupture.

. 4 )

 4 Here p ≡ I 1 /3=σ kk /3 is the hydrostatic stress derived from the first invariant of the stress tensor,I 1 , J 2 = s ij s ji /2corresponds to the second invariant of the deviatoric stress tensor (s ij = σ ij -rspa.royalsocietypublishing.org Proc R Soc A 0000000 ..........................................................

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Figure 2 .

 2 Figure 2. Damage and Mohr-Coulomb Failure with the micromechanical model. Temporal evolution of a dynamic rupture occurring on a right-lateral fault embedded in granite (solid grey line at y =0), using the micromechanical model. The rupture is nucleated around (0, 0) denoted by the yellow star. Sub-figures display snapshots of the Drucker-Prager yield criterion (equation 2.4) when the rupture propagates at subshear (a) and supershear velocities (c), and during the transition (b). Fault slip rate (white curves) and the cumulative damage density, D, (grey scale) are superimposed. Instant damage happening in relation to the displayed stress field is underlined in red. Yellow star denotes the nucleation patch and C and T the compressional and the tensional quadrants, respectively.

  we display the Drucker-Prager criterion to rspa.royalsocietypublishing.org Proc R Soc A 0000000 ..........................................................

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Figure 3 .

 3 Figure 3. Slip and slip rate evolution with the micromechanical model. Cumulative slip (a), and slip rate, δ , (b-e) on the fault are displayed at various time steps. Colored curves correspond to the dynamic simulation with the damage evolution law, dotted grey curves represent a simulation with the same parameters but within a pure elastic medium. Grey box gives the location of the area where less damage is observed.

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Figure 4 .

 4 Figure 4. Damage and Mohr-Coulomb Failure with FDEM model. Dynamic rupture occurring on a right-lateral fault embedded in granite, using the FDEM model. Figure displays the invariant form of the Mohr-Coulomb yield criterion (equation 3.2) when the rupture propagates at supershear velocities. Black lines gives the spatial distribution of off-fault fractures that occurs during the entire event. Signature of the transition from the sub-Rayleigh to the supershear regime is highlighted by the gap in off-fault damage.

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Figure 5 .

 5 Figure 5. Optical Image Correlation Analysis. (a) Map of the strike slip section of the 2001 Mw 7.8 Kunlun earthquake (China), where P1 denotes the transition zone reported for the event from seismological far-field data [31]. (b) Along-Strike fault zone width (black) and its associated uncertainty (grey), obtained from the analysis of 40km long profiles, sampling the fault zone every 500m, on the surface displacement maps. The latter is derived from correlating pre-and post-earthquake SPOT-1 to SPOT-4 images. The 11km long red area highlights a region with a mean fault width (red dashed line) of only 127m compared to 238m recorded for the rest of the rupture (red line). The latter excludes the area where two parallel fault strands are activated and for which the fault zone is exceptionally large (> 1000m). (c) Zoom of the Figure 5b.

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Figure 6 .

 6 Figure 6. Aftershock Catalogs. 1-month aftershock distribution for Izmit (a), Denali (b), and Craig (c) earthquakes, colorcoded by time. Pink stars indicate the mainshock epicenters with focal mechanism. The continuous black lines denote the surface rupture for each event, while the dashed ones indicate a distance of 5km from the fault for each event. Brown dashed boxes are the target regions to explore the supershear transition based on published kinematic models (Bouchon et al. [27] for a, Ellsworth et al. [32] for b, and Yue et al. [36] for c. The pink boxes indicate this work's proposed transition zones.

  , and f. We quantify such lack of aftershocks by rspa.royalsocietypublishing.org Proc R Soc A 0000000 ..........................................................

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Figure 7 .

 7 Figure 7. Spatio-Temporal Seismic Moment Density Evolution. Cumulative seismic moment density projected on the main fault at different temporal scales (1-3 weeks), for Izmit (a), Denali (c), and Craig (e) earthquakes. All the aftershocks within a distance of 5 km from the fault are considered in the calculation (area denoted by the black discontinuous lines in Figures 6a, a ¯nd c). Color-coded arrows (at the top in a, b, and c) indicate the different speed regimes reported for

  earthquake and the arrows indicate the direction of the rupture. The pink boxes highlight our proposed transition zone, also observed in a map view in (b) for Izmit, (d) for Denali, and (f) for Craig earthquakes.

  rspa.royalsocietypublishing.org Proc R Soc A 0000000 ..........................................................

Figure 8 .

 8 Figure 8. High-Resolution Aftershock Catalog Statistical Analysis. Aftershock seismic moment density projected on the main fault for 1-week, considering all the earthquakes at a distance of 5km from the fault for Izmit (a), Denali (b), and Craig (c) earthquakes. Black curve is the mean seismic moment density on the main fault ± the standard deviation (1-σ) computed from 10000 synthetic random catalogs. Pink boxes highlight our proposed transition zone.

  rspa.royalsocietypublishing.org Proc R Soc A 0000000 ..........................................................

Figure S1 :

 S1 Figure S1: Schematics and parameters used for the simulations of dynamic ruptures in a 2-D in-plane model. We consider a right-lateral planar fault, embedded in a brittle off-fault medium (FDEM or micromechanical model). Slip-weakening friction (grey box) acts along the main fault plane.The medium is loaded by uniform background stresses with the maximum compressive stress σ 1 making an angle of 60 o with the fault plane. The thick grey line corresponds to the nucleation zone where either the initial shear stress is set-up to be just above the fault strength (micromechanical model), or we apply a local decrease of the static friction (FDEM model). Figures adapted from Thomas et al. (2017) and Okuboet al. (2019) .

Figure S2 :

 S2 Figure S2: (a) North-South and (b) East-West surface displacement maps for the strike-slip part of the 2001 Kunlun surface rupture. Results benefit from a 10-meters ground resolution after the horizontal correlation of SPOT (1 to 4) pre-and post-earthquake images using MicMac. North-South displacements are close to zero at the fault while there is a clear left-lateral offset in the East-West displacement map. (c) Correlation score provided by MicMac after the horizontal correlation of the SPOT images. Dark areas correspond to decorrelation in lakes, drainages or snow. White areas, representing areas of good correlation, are present around the fault zone and particularly between 91.5 and 92.5 o E.

Figure S3 :

 S3 Figure S3: Example of one of the 396 profiles obtained from the optical correlation image processing along the fault. The step in the figure represents the coseismic offset produced by the earthquake, while the red box denotes the region defined during this work as the width of the off-fault damage zone.

Figure S4 :

 S4 Figure S4: SPOT image of the 2001 Kunlun rupture area: (A) before the rupture transitioned to supershear speed, and (B) after the rupture transitioned to supershear speed. The rupture is indicated in red at some locations. The rupture trace is located in some places with red arrows. The rupture is clean in image B), unlike in A) where the trace is more ambiguous.

Figure S5 :

 S5 Figure S5: Across-fault displacement profiles showing the fault-parallel component of displacement (left-lateral displacement). Profiles are arranged in a three-dimensional view. Plot axes show displacement amplitude and distance across the fault for the first profile to the bottom of the plot. The rupture propagates from the west to the east, from the bottom to the top of the plot. Profile 168 corresponds tothe profile where we observe a localized fault zone relative to the westward area. Eastward to profile 168, the localized fault zone width is indicated by light-red dashed lines. Westward to profile 168, the fault zone width is not well defined, as seen in the optical images (IMAGE B, FigureS4).

Figure S6 :

 S6 Figure S6: Spatio-Temporal Seismic Moment Density Evolution. Cumulative aftershock seismic moment density projected on the main fault at different temporal scales (1-3 weeks), for Izmit (a), Denali (c), and Craig (e) earthquakes. All the aftershocks within a distance of 2.5 km from the fault are considered in the calculation (area denoted by the black discontinuous lines in Figures 6a, b and c on the main text). Color-coded arrows (on top of a, b, and c) indicate the different speed regimes reported for

Figure S7 :

 S7 Figure S7: High-Resolution Aftershock Catalog Statistical Analysis, Izmit Earthquake, 1-and 10σ. a and b show the aftershock seismic moment density projected on the main fault at different temporal scales (1-3 weeks), considering all the aftershocks at a distance of 5 km from the fault. The red area in each panel denotes the mean seismic moment density projected on the main fault ± the standard deviation (1-σ, indicated in the legend) of the 10000 synthetics catalogs performed for the analysis. The pink box indicates this work's proposed transition zone. c and d are the same plots than before (a and b), but considering a 10-σ standard deviation on the calculation.

Figure S8 :

 S8 Figure S8: High-Resolution Aftershock Catalog Statistical Analysis, Denali Earthquake, 1-and 10σ. a and b show the aftershock seismic moment density projected on the main fault at different temporal scales (1-3 weeks), considering all the aftershocks at a distance of 5 km from the fault. The red area in each panel denotes the mean seismic moment density projected on the main fault ± the standard deviation (1-σ, indicated in the legend) of the 10000 synthetics catalogs performed for the analysis. The pink box indicates this work's proposed transition zone. c and d are the same plots than before (a and b), but considering a 10-σ standard deviation on the calculation.

Figure S9 :

 S9 Figure S9: High-Resolution Aftershock Catalog Statistical Analysis, Craig Earthquake, 1-and 10σ. a and b show the aftershock seismic moment density projected on the main fault at different temporal scales (1-3 weeks), considering all the aftershocks at a distance of 5 km from the fault. The red area in each panel denotes the mean seismic moment density projected on the main fault ± the standard deviation (1-σ, indicated in the legend) of the 10000 synthetics catalogs performed for the analysis. The pink box indicates this work's proposed transition zone. c and d are the same plots than before (a and b), but considering a 10-σ standard deviation on the calculation.

  1.2.1. Continental strike-slip faults and distributed tectonic deformation Continental strike-slip faults are faults that cut through the continental crust and accommodate horizontal shear deformation. Continental active faults are not the host for the largest earthquakes on Earth (Mw>9), which generally occur in subduction zones. But still, they are associated with a large number of Mw7-8 earthquakes, with some of them that caused large damage to structures and populations. For example, the 2001 Mw.7.8

3. PART 1: DISTRIBUTION OF THE SURFACE DISPLACEMENTS IN A COMPLEX FAULT ZONE ALONG THE 2013 BALUCHISTAN RUPTURE

	fracturation visible du milieu. Cette déformation est observée dans des complexités à
	grande échelle qui affectent les roche endurée du toit de la rupturedu tout de la rupture.
	Cette déformation continue se produit probablement à une échelle inférieure à notre
	Résumé français : résolution, via de la microfracturation et des bandes de compression, dilatation et
	Mesurer le champ de déplacement au plus proche des zones de failles lors des grands cisaillement. Dans une deuxième partie de ce chapitre sur le séisme du Baloutchistan,
	séismes continentaux est un enjeu majeur pour la compréhension du mécanisme de la nous mesurons la déformation post-sismique à partir de 45 jours jusqu'à 3 ans après la
	rupture co-sismique et pour l'évaluation des risques sismiques. Les méthodes de rupture afin d'étudier l'évolution temporelle de la déformation dans la zone de faille. Nos
	corrélation optique qui utilisent des images à haute résolution permettent de caractériser résultats montrent qu'aucune déformation proche de la zone de faille et d'amplitude
	le détail du champ de déplacement au plus proche de la faille. Dans cette étude, nous supérieure à ~0.10 m, qui est notre seuil de détection, ne se produit entre 45 jours et 3 ans
	utilisons la corrélation sub-pixellique d'images optiques à une résolution de 0,5 m pour après le séisme. Nous n'avons par contre aucune image optique pour des dates antérieurs
	mesurer le champ de déplacement dans les 5 premiers kilomètres autour de la zone de à 45 jours, et donc nous ne pouvons contraindre la cinématique de la déformation diffuse
	faille, sur une zone de 30 km de long de la rupture du Baloutchistan (2013, Mw7.7, dans les premières semaines après la rupture. La question de la nature sismique ou
	Pakistan). Dans un premier temps, nous montrons que la déformation cosismique de asismique de la déformation diffuse pour cette rupture reste donc ouverte.
	surface est très complexe et qu'elle met en jeu de nombreuses failles qui interagissent,
	avec des géométries complexes. Nous observons tout particulièrement deux complexités
	d'échelle kilométrique, une zone de relai compressif et un changement d'azimut, pour
	lesquelles une large proportion de déformation off-fault (OFD) est mesurée. Nous
	mesurons avec une précision de 0,10 à 0,20 m le budget de déformation total à travers la
	zone de faille, qui évolue entre ~50 et 2,5 km de large, et le détail du glissement sur les
	différentes failles identifiées. La différence entre ces deux courbes nous permet d'obtenir
	la quantité d'OFD qui se produit le long de notre zone d'étude. Nous observons des
	maximums d'OFD jusqu'à 70% dans les deux complexités majeures de la rupture, et des
	valeurs minimums autour de 0% dans les zones où la rupture est plus simple. Dans ces
	zones plus simples, la zone de faille totale mesure tout au plus 100m à 200 m de large.
	Nous croisons ces mesures avec l'analyse de la géométrie de la rupture et de la
	géomorphologie de la zone de faille, et caractérisons les différents mécanismes d'OFD.
	Nous observons premièrement que l'OFD peut se produire par une facturation dense du
	milieu, distribuée sur plusieurs centaines de mètres autour de la faille principale, et pour
	laquelle certaines fractures sont visibles sur les images satellites. Ce mécanisme se
	produit principalement dans les complexités à petite échelle, qui affectent seulement la
	couche sédimentaire. Nous observons deuxièmement une déformation continue du
	milieu, c'est-à-dire qui génère des gradients continus de déplacement à la surface, sans

Table 3 .

 3

4. PART 2: DISTRIBUTED DEFORMATION AT THE SCALE OF A WHOLE RUPTURE: CHARACTERIZATION, QUANTIFICATION, AND RELATION WITH THE RUPTURE PROCESS AT DEPTH

  Les données sismiques et géodésiques ont révélé le mécanisme complexe de cette rupture en profondeur, mais également sa complexité en surface, égalent observée sur le terrain. Cela soulève la question de l'interdépendance des deux systèmes de failles accommoder la partie du glissement qui ne se propage jusqu'à la partie la plus superficielle du plan de faille. L'origine inélastique ou élastique de cette déformation diffuse et de son rôle dans le budget total de déformation des séismes de 2019 est discutée. Cependant, l'hypothèse d'une déformation inélastique finie est plus probable étant donné l'intensité des gradients mesurés. Dans un deuxième chapitre, nous utilisons les cartes de déplacement de surface haute résolution pour contraindre le glissement de surface sur la faille dans les modèles d'inversion cinématique. L'inversion est réalisée en utilisant des codes de K. Wang (U. C. Berkeley) qui intègrent les données optiques avec des données InSAR et GPS. Nous obtenons une distribution du glissement avec un niveau de détail supérieur aux inversions précédentes. La distribution du glissement modélisé dans la

	Résumé français :
	La séquence sismique de Ridgecrest (Californie, juillet 2019, Mw6.4 et Mw7.1) a rompu
	consécutivement un système de failles orthogonales (nord-est et nord-ouest) à 34 heures
	d'intervalle.

à la fois en profondeur et à la surface, et de l'impact sur la distribution du déplacement cosismique en surface. Dans un premier chapitre, nous utilisons la corrélation d'image satellite optique à haute résolution (WorldView et Pléiades) pour mesurer le champ de déplacement horizontal et vertical avec une résolution de 0,5 m sur la zone de faille et ses alentours. Nos mesures incluent les deux séismes majeurs de la séquence : le foreshock de Mw6.4 et le mainshock de Mw7.1. Nos résultats montrent des différences significatives avec les mesures géodésiques et géologiques précédentes, ce qui s'explique principalement par le fait que nous mesurons la composante diffuse de la déformation, qui se produit en dehors du système de faille. Les cartes de déplacement horizontal sont utilisées pour calculer la rotation, la dilatation et le cisaillement dans la zone de faille. On observe que la rotation et la déformation cisaillante ont un rôle prédominant dans le processus de rupture observé ici, ce qui est cohérent avec le mécanisme du séisme, alors que la dilatation ne joue un rôle qu'à l'échelle locale, dans certaines complexités géométriques. La mesure du déplacement le long des failles et dans le milieu (déformation diffuse) met en lumière une segmentation de la rupture, qui est cohérente avec des variations dans la géométrie de la zone de faille. Nous observons également une série de failles nord-est, activées à la fois lors du mainshock et du foreshock, et dont l'orientation correspond à la fabrique profonde identifiées dans les aftershocks de la séquence. Nous comparons nos résultats aux prédictions de glissement de surface proposées par les modèles d'inversion cinématiques et observons une corrélation entre les zones de déformation diffuse à la surface et les régions de déficit de glissement en surface par rapport au glissement en profondeur. Il semblerait donc que la déformation diffuse serve à partie supérieure de la faille montre une bonne corrélation avec les données optiques sur les failles ainsi que les données InSAR. Notre modèle montre que le glissement en surface sur la faille ne représente que la moitié du déplacement total qui a lieu en profondeur, entre 3 et 6 km, mais que ce déficit est comblé entièrement par la déformation diffuse que nous avons mesuré autour des failles.

Table S4 .

 S4 Combinations of pre-and post-earthquake images, referred to as tiles and used to perform the horizontal and vertical correlations are detailed in FigureS4.1 and TableS4.2.

	Supplemental Tables:
	Image reference Pre-earthquake images (mono-image each) Date (month-Off-nadir angle (°) Image identification day-year) Pl-pr1 05-28-2012 -33.2 DS_PHR1A_201205281845034_F Tile number (Fig. S4.1) Pre-earthquake images (mono-image each) Pre-Post-earthquake images earthquake (tri-stereo each) DSM R1_PX_W118N36_0409_02217 Pl-pr2 06-23-2012 -8.63 2 Pl-pr1 ; Wv-pr5 ; Wv-pr8 Pl-po1 0.5 m DS_PHR1A_201206231845554_S E1_PX_W118N35_0515_01840 3 Pl-pr1 ; Wv-pr7 Pl-po1 0.5 m
	Pl-pr3 4 5	06-23-2012 Pl-pr1 ; Wv-pr8 12.33 Pl-pr1 ; Wv-pr8	DS_PHR1A_201206231846279_F Pl-po3 2 m R1_PX_W118N35_0720_03528 Pl-po3 2 m
	Pl-pr4 6 Wv-pr1 7 Wv-pr2 8 Wv-pr3 9 Wv-pr4 0 Wv-pr5 1	07-02-2012 Pl-pr1 ; Wv-pr8 20.95 03-28-2016 5.8 Pl-pr3 ; Wv-pr7 03-28-2016 23.3 Pl-pr3 ; Wv-pr7 06-23-2018 28.2 Pl-pr1 ; Pl-pr3 07-29-2016 22.8 Pl-pr2 ; Wv-pr2 ; Wv-pr8 06-16-2018 26.6 Pl-pr3 ; Wv-pr1 ; Wv-pr6	DS_PHR1A_201207021827552_S Pl-po3 2 m E1_PX_W118N35_0822_01319 Pl-po3 0.5 m 102001004B3FF400 Pl-po3 2 m 102001004B17FA00 Pl-po3 0.5 m 103001007FA3FC00 Pl-po3 2 m 1020010053CCF600 1020010073BC6300 Pl-po2 2 m
	Wv-pr6 12	07-23-2016 Pl-pr3 ; Pl-pr4 27.0	1030010059B13200 Pl-po3	0.5 m
	Wv-pr7 13	11-18-2016 Pl-pr4 ; Wv-pr9 11.6	1020010059614C00 Pl-po2	0.5 m
	Wv-pr8 14	02-08-2016 Pl-pr4 ; Wv-pr4 20.2	1030010052953900 Pl-po2	0.5 m
	Wv-pr9	07-12-2016	24.8	1030010059031600
	Post-earthquake images (tri-stereo each)
				DS_PHR1B_201908191845340_F
				R1_PX_W118N35_0322_01871 ;
	Pl-po1	08-19-2019	-11.6 ; 4.29 ; 11.02	DS_PHR1B_201908191846005_F R1_PX_W118N35_0322_01921 ;
				DS_PHR1B_201908191846119_F
				R1_PX_W118N35_0322_01878
				DS_PHR1B_201908261841519_F
				R1_PX_W118N35_0815_01864 ;
	Pl-po2	08-26-2019	-10.84 ; 1.52 ; 11.71	DS_PHR1B_201908261842125_F R1_PX_W118N35_0815_01903 ;
				DS_PHR1B_201908261842296_F
				R1_PX_W118N35_0815_01846
				DS_PHR1A_201909081842078_F
				R1_PX_W118N35_0 18_04115 ;
	Pl-po3	09-08-2019 -12.36 ; -2.92 ; 13.83	DS_PHR1A_201909081842235_F R1_PX_W118N35_0 18_04273 ;
				DS_PHR1A_201909081842518_F
				R1_PX_W118N35_0518_04069

1: References of the images used in this study. Pl: Pleiades images. Wv: WorldView images. pr: pre-earthquake images. po: post-earthquake images.

Table S4 .

 S4

2: References of images used for each correlation tiles (Fig. S4.1). Image references are detailed in Table

Table 4 .

 4 1: Results of the tests on the model regularization from

	0,001	6,76	9,99	4,38	0,2707	0,6793	0,4749
	0,002	6,30	9,99	1,98	0,3087	0,6949	0,5018
	0,004	5,56	9,99	1,98	0,3584	0,7640	0,5612
	0,006	5,26	9,99	1,98	0,3657	0,8223	0,5939
	0,008	5,15	9,99	1,98	0,3831	0,8678	0,6254
	0,009	5,11	9,99	1,98	0,3679	0,8726	0,6202
	0,01	5,08	9,99	1,98	0,3587	0,8747	0,6167
	0,02	4,99	9,99	4,38	0,3632	0,9614	0,6623
	0,04	5,05	9,45	4,38	0,3819	1,0437	0,7127
	0,06	4,98	8,33	4,38	0,4391	1,0278	0,7334
	0,08	4,91	7,41	4,38	0,4509	1,0317	0,7413
	0,1	4,82	6,64	4,38	0,4659	1,0286	0,7472
	0,2	4,77	4,77	0	0,6568	0,785	0,7209
	0,4	4,97	4,97	0	0,6215	0,8446	0,7330
	0,6	5,04	5,04	0	0,5062	1,0773	0,7917
	1	5,02	5,02	0	0,4332	1,2536	0,8434

Table 4 .

 4

	2.

Table 4 .

 4 2: Quantiles of the residuals between model and InSAR and optical data, at different cumulative probabilities P. Maps of the residuals, and frequency plots are presented in Figure 4.18.With this model, we obtain median residuals of 0.36 and 0.76 cm with the ascending and descending components of the InSAR, respectively (Fig.4.18a,b). Median residuals with the optical data are -3.49 and 10.87 cm in the east-west and north-south components,

Table 6 .

 6 1: Detail of the calculations of the seismic moment 𝑀 0 , moment magnitude 𝑀 𝑤 , and seismic energy 𝐸 for the 2019 Ridgecrest earthquakes. White lines report calculations made on the base on fault displacement measurements from DuRoss et al. (2020). Light orange lines report calculations made using results of the optical image analysis (Antoine

  6.2 and 6.3). As a consequence, microfracturing of rocks below the failure stress, and opening of fractures at scales below what is detectable by human eyes and satellite data are the most likely processes that could occur in the diffuse deformation zones. Propositions of research to investigate the nature of diffuse deformation, based on field investigations and laboratory methods, are developed in section 8.2. from this thesis.

  1, * , Lucile Bruhat 1 , Marion Y. Thomas 2 , Solène Antoine 3 , Kurama Okubo 4 , Esteban Rougier 5 , Ares J. Rosakis 6 , Charles G. Sammis 7 , Yann Klinger 3 , Romain Jolivet

Table 1 .

 1 Table of parameters for the LEFM, Micromechanical and FDEM models

	Parameters Description	Values
	State of stress -σ 0 xx -σ 0 yy σ 0 yx	Fault-parallel stress (MPa) Normal stress on the fault (MPa) Shear stress on the fault (MPa) Nucleation Zone * (MPa)	37.7 60.7 19.9 36.4
	ψ Bulk properties Angle between σ 1 and fault (deg.) 60 ρ density (kg/m 3 ) 2700 cs S-wave speed (km/s) 3.12 cp P-wave speed (km/s) 5.62 ν Poisson's ratio 0.276
	Fault parameters
	fs	Static friction coefficient	0.6
	f d Dc R 0	Nucleation Zone † Dynamic friction coefficient Characteristic slip distance (m)	0.3 0.1 1.0

Li Q, Zhao B, Tan K, Xu W. 2020 Two main rupture stages during the 2018 magnitude 7.5

Sulawesi earthquake. Geophysical Journal International 221, 1873-1882.

ACKNOWLEDGEMENTS

My most sincere acknowledgements are addressed to Yann Klinger, who gave me the opportunity of this thesis and supervised my work during this long adventure. I discovered the possibilities that image correlation open in the field of Earth science, and had the chance to work on cutting-edge methods. Yann also gave me the opportunity to collaborate with researchers from abroad (R. Gold, R. Burgmann, K. Wang), and to go on a field mission in Mongolia. I thank a lot Yann for giving me all these opportunities, and for being such a great thesis supervisor! I also sincerely thank Arthur Delorme, who taught me the basis of optical image correlation.

Arthur has always been of a great help to resolve technical problems. All along my thesis, we interacted on technical issues and ways to improve our calculations. I thank the thesis Jury, Cécile Lasserre, Tom Rockwell, Daniel Faulkner, Michele Cooke, Robin Lacassin, and Oona Scotti, for accepting to review my work. I am grateful for the discussions we had on the day of the defense. I thank my thesis committee, Harsha Bhat and Pascal Bernard, who followed my work and evaluated the scientific progress every year. Harsha and Pascal gave me precious advice that helped me include this thesis into a broader scientific landscape. I thank a lot the Laboratoire de tectonique et mécanique de la lithosphère, IPGP, who gave me a warm welcome and provided a stimulating work atmosphere. The Covid pandemic got the better of it to the end, but I keep very good memories of all the time spent in this lab. I also thank all the researchers from IPGP and ENS who I had inspiring discussions with, and sometimes help with my work. I particularly think of Jean-Matthieu Nocquet, Romain Jolivet, Raphaël Grandin, and again Harsha Bhat and Pascal Bernard. I also thank, Benjamin Lauer and Amaury Vallage, former PhDs of the Tectonic team, who worked on the Baluchistan earthquake before me. I could benefit from their data and knowledge on the study area as a starting point for my research. I thank a lot my foreign collaborators, Ryan Gold from U.S. Geological Survey, and Roland Burgmann and Kang Wang from U.C. Berkeley, whom I was honored to work with. It was the occasion for me to challenge my work against people who have a different background, experience and who use different technics. This led to very nice work and publications. A special thank is addressed to Ryan Gold who provided us the WorldView images used to perform the high-resolution image correlation for the 2019 Ridgecrest and 2013 Baluchistan earthquakes. I thank Marc Pierrot-Desseligny and Ewelina Rupnik, from the Institut Geographique National. Marc and Ewelina developed the software MicMac which I used for image correlation. This is an outstanding software that provides many functions, and allows for great measurements of earthquake surface displacements and deformations. I thank IPGP for the S-CAPAD cluster facility. It was a necessary infrastructure for my work since the high-resolution calculations are heavy and cannot be performed directly on a computer. I also thank Geneviève Mougliny for her help in managing the cluster and the technical issue. I thank Ecole Doctorale STEP'UP and Clément Narteau for their support during my thesis. I also thank Hekla association, supported by Ecole Doctorale, who organized various sorts of events to allow networking between PhD students. I thank a lot these people and organizations for their role in the nice atmosphere at IPGP. I thank Université de Paris for the PhD fellowship, and the administrative follow-up.

Finally, I thank all my friends and family for their support in my good and bad times during this PhD. I thank all the people from IPGP and ENS, PhDs and others, for the discussions around drinks and lunches. A special thank is addressed to Roxane, Manon, Matthieu, and

Declaration of Competing Interests :

The authors acknowledge there are no conflicts of interest recorded. Supplementary 3.2: (A) Fault-parallel displacement map in a 30-km long area of the Baluchistan earthquake. Data at 0.5-m ground resolution (Fig. 3.2A in the fault-parallel component) cover only the near-fault domain. Large-scale and lower resolution results (2.5 m;Vallage et al., 2015) obtained after correlation of SPOT-5 images were used as a reference to correct our results from ramps and differentiate large scale displacement gradients from near-field OFD gradients. These data are visible outside of the 0. Shortening is measured on primary fault 1 but increases towards the south of the area and reaches a maximum of 5 m at the S1 fault zone (Fig. 3.5B).

Declaration of Competing Interests:

The authors acknowledge there are no conflicts of interest recorded.

Abstract

The 2019 Mw6.4 and Mw7.1 Ridgecrest earthquake sequence (California, July 2019) ruptured consecutively a system of high-angle strike-slip cross faults (northeast-and northwest-trending) within 34 hours. The complex rupture mechanism was illuminated by seismological and geodetic data, bringing forward the issue of the interdependency of the two fault systems both at depth and at the surface and of its effect on the final surface displacement pattern. Here, we use high-resolution (WorldView and Pleiades) optical satellite image correlation to measure the near-fault 3D surface displacement field at 0.5 m ground resolution for the two earthquakes. We point out significant differences with previous geodetic and geologic-based measurements, and we document the essential role of distributed faulting and diffuse deformation in producing the observed surface displacement patterns. We also derive strain fields from the horizontal displacement maps, and we highlight the predominant role of rotation and shear strain in the surface rupture process. We discuss the segmentation of the rupture based on the fault geometry and along-strike slip variations. We also image several northeast-trending faults with similar orientation to the deeply embedded shear fabric identified in aftershock studies and we show that these cross faults are present all along the rupture, including at a scale <100 m. Finally, we compare our results to kinematic slip inversions and show that the surface diffuse deformation is primarily associated with areas of shallow slip deficit; however, this diffuse deformation cannot be explained using elastic modeling. We conclude that inelastic processes play an important role in contributing to the total surface deformation associated with the 2019 Ridgecrest sequence.

Introduction

Geometrical complexity of earthquake surface ruptures has been described for a wide range of fault systems over the last decades based on geological (e.g., [START_REF] Choi | Geologic Inheritance and Earthquake Rupture Processes: The 1905 M $\ge$ 8 Tsetserleg-Bulnay Strike-Slip Earthquake Sequence, Mongolia[END_REF][START_REF] Griffith | Rough faults, distributed weakening, and off-fault deformation[END_REF]Klinger et al., 2005;Mitchell and Faulkner, 2009;