Je Souhaite

René-Édouard Plessix

Daniela Donno

Alexandrine Gesret

Mark Noble

Pierre Dublanchet

Carlos Pérez

Vi Nhu Élise

Fang Ba

Charles-Antoine Wang

Sven Lameloise

Yves- Marie Schilke

Tiago Batany

Jihane Barros

Julien Belhadj

Yubing Cotton

Keurfon Li

Hao Luu

Alexandre Jiang

Michelle Kazantsev

Tianyou Almakari

Abed Zhou

Maxime Benaichouche

Shanker Godano

Fares Krishna

Mehouachi

Bénédicte Zufri

Joshua Loubamono

Lartey

J'ai

Seismic waves are widely used for the characterisation of the subsurface as they can travel over long distances and image structures of interest for the oil and gas industry. Signals recorded at the surface during active seismic experiments are analysed to recover parameters governing the wave propagation, such as the pressure wave velocity eld. Several techniques aiming at resolving this inverse problem rely on the assumption of a scale separation between rapidly varying components of the velocity model, the re ectivity model, and a smooth macro-model controlling the kinematics of wave propagation. Migration algorithms have been developed for the recovery of the re ectivity model but they require a good estimate of the macromodel to deliver accurate images. Analysis of re ectivity images may in turn give indications about the accuracy of the estimated macro-model: this is the principle of Migration Velocity Analysis (MVA) techniques, based on the analysis of primary re ection data. The formulation of MVA considered here is based on extended re ectivity images depending on an additional parameter called subsurface-o set, a spatial delay introduced during the construction of the re ectivity image. The velocity analysis is carried on panels called Common Image Gathers (CIGs), function of depth and of the subsurface-o set, an accurate macro-model corresponding to energy focused at zero subsurface-o set in CIGs. Using Di erential Semblance Optimisation (DSO), the inversion strategy consists of nding the macro-model best focusing events in CIGs. However the method is known to have di culties in the presence of coherent noise such as multiple re ections, whose removal from recorded data remains a challenging problem. Multiples falsely interpreted as primary re ections result in spurious events in CIGs which do not necessarily focus for the correct macro-model, misleading subsequent velocity analysis. Even with only primary re ections, CIGs are altered with migration artefacts, also not focused for the correct macro-model, due to the limited extension of any seismic survey. Recent studies suggest replacing standard migration, which consists of applying the adjoint of the modelling operator to observed data, by inversion. These techniques have been shown to attenuate migration artefacts, but inversion formulas do not necessarily exist for multiples.

In this thesis, I propose to use iterative migration to build CIGs free of migration and multiple artefacts. This makes MVA a nested optimisation procedure: in the inner loop, the re ectivity should minimise data mis t for a xed value of the macro-model, while the defocused energy in CIGs is minimised in the outer loop. The improvements but also instabilities brought by this strategy are shown and analysed in the case of primaries only and then in the case of rst-order surface related multiples modelled with a second-order Born approximation. To limit the additional computational cost of iterative migration, a preconditioner is de ned to reduce the number of inner iterations. The preconditioner is a newly developed pseudo-inverse of the modelling operator, derived for primary re ections only. An issue with iterative migration is the apparition of low energy events at large values of h in CIGs. These events have little impact on data mis t but large in uence on the subsequent velocity analysis and its stability. i

Dans cette étude, je propose d'atténuer les artefacts de migrations et ceux dus aux multiples en construisant les CIGs par migration itérative des données. La MVA peut alors être vue comme un problème d'optimisation imbriqué : dans la boucle interne, la ré ectivité est obtenue en minimisant la di érence entre données observées et données calculées pour une valeur xe du macro-modèle, tandis que le macro-modèle est remis à jour dans la boucle externe pour minimiser l'énergie défocalisée dans les CIGs. Les avantages mais aussi les instabilités de cette stratégie sont illustrés et analysés d'abord dans le cas de primaires seuls, puis avec ajout des multiples de surface du premier ordre, modélisés par l'approximation de Born du second-ordre. Pour réduire le coût de calcul supplémentaire apporté par la migration itérative, un préconditionneur est introduit pour limiter le nombre d'itérations nécessaires dans la boucle interne. Le préconditionneur utilisé ici est une nouvelle proposition de pseudo-inverse de l'opérateur de modélisation des ré exions primaires. Il permet également d'accélérer la convergence de la migration itérative dans le cas de multiples.

Une di culté liée à la migration itérative est l'apparition d'évènements peu énergétiques pour des grandes valeurs de l'o set en profondeur. Ceux-ci ont une faible incidence sur la boucle d'optimisation interne mais ont une in uence néfaste sur l'analyse de vitesse et compromettent sa stabilité. Cette di culté est atténuée en ajoutant une régularisation sur le modèle de réectivité. Je propose aussi une alternative plus robuste appliquée à la méthode d'optimisation imbriquée, elle consiste en une modi cation de la fonction coût habituelle de la MVA.

Cette stratégie est testée sur des jeux de données synthétiques obtenus sous l'approximation acoustique à densité constante de l'équation des ondes, d'abord avec des ré exions primaires seules, puis avec des multiples de surface du premier ordre. En n sa robustesse est évaluée en l'appliquant à des données obtenues en modi ant le code de modélisation des données observées, en introduisant par exemple un modèle de densité variable ou en modélisant les multiples avec une condition de surface libre.

Les principales contributions de cette étude sont (1) l'analyse des limitations actuelles des méthodes d'inversion dé nies dans le domaine image ; (2) des propositions de stratégies plus robustes vis-à-vis de ces di cultés, qui béné cient à la fois au cas classique (ré exions primaires seules) et au cas étendu aux multiples de surface. caractérisés uniquement par le temps de trajet source-récepteur en laissant de côté l'amplitude et la signature des signaux sismiques. Dé nies dans le domaine temps, les méthodes de tomographie minimisent l'écart entre des temps de trajet calculés avec une estimation du modèle de vitesse et des temps de trajet pointés dans les données enregistrées. Elles sont couramment utilisées dans l'industrie et de nombreuses évolutions ont été proposées telles les tomographies de pente et la stéréotomographie.

L'autre famille de méthodes visant à estimer le macro-modèle est connue sous le nom d'« analyse de vitesse par migration » (Migration Velocity Analysis, MVA). Ces méthodes exploitent la redondance d'information contenue dans les données sismiques. Ces techniques sont dé nies dans le domaine image et peuvent utiliser la théorie des rais ou considérer le champ d'onde dans son ensemble. L'analyse se fait sur les ondes ré échies primaires et repose sur l'approximation de Born, qui dé nit une relation linéaire entre données ré échies et modèle de ré ectivité. Les di érentes variantes de MVA peuvent être classées en deux catégories. Dans l'orientation surface, le jeu de données est divisé en sous-ensembles indexés par un paramètre de redondance tel que la position de la source en surface ou la distance entre source et récepteur. Une image de ré ectivité est obtenue par migration pour chaque sous-jeu de données : on obtient alors une collection d'images représentant une même région de la subsurface. Si le bon modèle de vitesse est utilisé au cours de l'étape de migration, ces images doivent être semblables. Si le modèle de vitesse est erroné, les ré ecteurs peuvent ne pas être positionnés à la même profondeur pour chaque valeur du paramètre de redondance. Ce critère de cohérence permet la dé nition d'une fonction coût qui mesure la similarité entre les images. Le macro-modèle est déterminé par minimisation de cette fonction coût. Dans l'orientation profondeur, étudiée dans cette thèse, un délai spatial et/ou temporel est introduit pendant l'étape de migration lors de la corrélation des champs source et récepteur. Dans cette étude je considérerai uniquement un délai spatial appelé « o set en profondeur » et supposé horizontal. Pour chaque valeur de l'o set en profondeur, une image de ré ectivité est construite avec l'ensemble des données. On parle alors de modèle (de ré ectivité) « étendu ». L'introduction de ce degré de liberté supplémentaire permet de compenser les erreurs du macro-modèle et conduit à un nouveau critère pour l'évaluation du macro-modèle. Celui-ci est dé ni sur des panneaux 2D, appelés Common Image Gathers (CIGs), représentant le modèle de ré ectivité fonction de la profondeur et de l'o set en profondeur pour une position latérale xe. Avec le bon modèle de vitesse, l'énergie doit se focaliser autour de la valeur nulle de l'o set en profondeur, qui correspond au modèle de ré ectivité qui a un sens physique. La présence d'énergie à des valeurs non nulles de l'o set en profondeur, correspondant à des modèles de ré ectivité non physiques, traduit la présence d'erreurs dans le modèle de vitesse. J'utiliserai dans cette étude la stratégie d'« optimisation par semblance di érentielle » (DSO). Elle construit le macro-modèle de vitesse par minimisation d'une fonction coût qui pénalise l'énergie défocalisée dans les CIGs. Il a été montré, au moins dans des cas simples, que cette fonction est lisse et ne présente pas de minima secondaires dans une large région autour du minimum global. Le macro-modèle peut alors être obtenu par minimisation de cette fonction coût avec des méthodes d'optimisation locale. Le calcul du gradient associé est une étape essentielle de l'inversion et fera l'objet d'une grande attention dans cette étude.

La formulation profondeur de la DSO est théoriquement attrayante mais peu d'applications sur données réelles ont été publiées jusqu'ici. En plus du coût de calcul numérique de la méthode, plusieurs di cultés l'empêchent d'être plus largement adoptée. Tout d'abord, on constate en pratique l'apparition dans les CIGs d'artefacts liés à l'extension nie de sources et de récepteurs en surface. Ces artefacts ne focalisent pas pour le bon modèle de vitesse, qui ne constitue alors plus un minimum de la fonction coût de la DSO. Une deuxième di culté, connue sous le nom d'« artefacts du gradient », est la présence de fortes oscillations dans le gradient de la fonction coût aux troncatures du modèle de ré ectivité et autour de la position des ré ecteurs. Ces oscillations ne sont pas nécessairement atténuées par un lissage du gradient. Pour s'en a ranchir, la méthode de « contraction horizontale » [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF]Shen et Symes, 2015) a été proposée, mais elle présente le désavantage de ne plus dé nir la remise à jour du modèle de vitesse en tant que gradient d'une fonction objective. Finalement, les méthodes de MVA restent limitées à l'analyse de ré exions primaires seules. Si un formalisme similaire a récemment été dé ni pour le cas des ondes transmises et des ondes plongeantes, les ré exions multiples sont toujours considérées comme du bruit cohérent. En e et, interprétées comme des primaires elles conduisent à de nouveaux artefacts dans les CIGs qui, de même que les artefacts de migration, ne focalisent pas nécessairement pour le bon modèle de vitesse.

L'atténuation des ré exions multiples dans les données observées a fait l'objet de nombreuses recherches et demeure un problème ouvert. Plusieurs méthodes de prédiction des multiples contenus dans un jeu de données ont été proposées, certaines supposant la connaissance d'un modèle de vitesse. La méthode SRME (Surface Related Multiple Elimination, Verschuur et Berkhout, 1997) est aujourd'hui parmi les plus populaires et ne requiert aucune connaissance sur le sous-sol. En revanche, elle suppose une couverture dense de sources et récepteurs en surface. Quelle que soit la méthode employée, la prédiction des multiples n'est en pratique jamais parfaite et une étape de « soustraction adaptative » est nécessaire pour faire correspondre au mieux la prédiction aux multiples présents dans les données. Le danger est alors d'atténuer également l'information contenue dans des ré exions primaires chevauchant des multiples. Depuis quelques années, les multiples sont cependant abordés sous un autre angle : ils peuvent être considérés non pas comme du bruit mais comme une source d'information supplémentaire, notamment en raison de l'illumination di érente du sous-sol qu'ils procurent. Des algorithmes de migration prenant en compte les ré exions multiples ont récemment été proposés. Ils utilisent des procédures itératives telles qu'une migration au sens des moindres carrés. L'utilisation de multiples pour la reconstruction du macro-modèle de vitesse a jusqu'à présent été moins explorée.

Dans cette étude, je propose d'itérer l'étape de migration avant de remettre à jour le macromodèle. La migration itérative (chapitre 2) vise à déterminer un modèle de ré ectivité étendu minimisant une fonction coût qui mesure l'écart entre données observées et données calculées. L'objectif est d'atténuer les artefacts de migration et ceux dus aux multiples qui apparaissent à la première itération, équivalente à la migration classique. L'étude est restreinte au cas de multiples de surface du premier ordre modélisés par une approximation de Born du second ordre. L'analyse de vitesse par migration itérative ainsi dé nie est un problème d'optimisation imbriqué à deux niveaux et une méthode de calcul du gradient de la fonction coût externe doit être dé nie. Deux stratégies sont étudiées dans le chapitre 3. Dans un premier temps la méthode est testée dans le cas de primaires seuls, puis étendue aux multiples. Dans le chapitre 4, elle est comparée, dans le cas de primaires seuls, à une stratégie d'inversion « directe » où l'image migrée est obtenue par application aux données inverse d'un pseudo-inverse de l'opérateur de modélisation dans le domaine étendu. Une nouvelle formule d'inversion est proposée à cette occasion. De même que celles proposées par ten Kroode (2012) et [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF], 2017), et contrairement au cas étudié dans Lameloise et al. (2014), elle ne fait pas intervenir de quantités liées au rais, même si les calculs intermédiaires reposent sur l'approximation hautes fréquences. Cette stratégie atténue e cacement les artefacts de migration et améliore la qualité de la remise à jour du macromodèle (section 4.2). Une légère modi cation de la fonction coût de la DSO permet également de fortement atténuer les oscillations résiduelles localisées au niveau des ré ecteurs. Par ailleurs l'introduction du pseudo-inverse comme préconditionneur de la migration itérative permet d'en diminuer le coût numérique. Comparé à l'inversion directe, l'approche itérative produit des résultats de ré ectivité similaires. En revanche, le calcul du gradient de la boucle externe se révèle instable : celui-ci est très sensible à de légères modi cations du modèle de ré ectivité apparaissant entre deux itérations successives et ayant un faible impact sur la fonction objective de la migration. Cette di culté est illustrée dans le chapitre 3 et des régularisations stabilisant le calcul du gradient sont étudiées dans les chapitres 3 et 5. Finalement la robustesse de cette stratégie est testée dans le chapitre 6 sur des jeux de données synthétiques obtenus avec un code de modélisation di érent de celui utilisé pour l'inversion, par exemple avec une ondelette de source di érente ou un modèle de densité variable.

In exploration geophysics, seismic imaging consists of characterising the subsurface model parameters such as pressure and shear wave velocity or rock density with the analysis of non-destructive measurements performed at the surface. Emitted seismic waves propagate in all the directions in the subsurface and are distorted during propagation. In simple models, they are re ected and di racted at interfaces characterised by rapid variations of physical properties of the rocks. Sensors located at the surface record the waves reaching the surface after having interacted with the subsurface and provide the data used by geophysicists. These time recordings usually cannot be used directly for geological interpretation. Surface recordings have to be converted into maps of the subsurface. This crucial step is formulated as the resolution of an inverse problem aiming at nding the value of physical parameters governing the propagation of waves in the subsurface. The resolution of this inverse problem is a di cult step and many methods addressing it have been proposed. Among others issues, data may depend in a nonlinear way on the model parameters. This is the case for example when the model corresponds to the large wavelengths of the velocity model or when seismic waves re ect several times in the subsurface. This thesis especially focuses on these two cases.

In this introduction, we rst brie y summarize the main steps of seismic imaging used in exploration geophysics and then pay attention to the resolution of the inverse problem. Finally we motivate our work and explain the current limitations of image-domain methods, in particular how multiples can bias traditional imaging techniques. For that, we recap the physics of multiple bouncing waves and explain why multiples may bring valuable information on the subsurface instead of being removed in a pre-processing step.

From seismic acquisition to interpretation

Seismic experiments are in common use in the oil and gas industry for the subsurface imaging and reservoir management of a reservoir. Seismic waves can travel over long distances and are thus well suited to the study of the geologic structures involved in oil production, located at a depth of a few kilometres in the subsurface. Seismic exploration uses active seismic experiments, meaning the source at the origin of wave propagation is arti cially triggered, contrary to passive seismic considered in seismology for the study of natural earthquakes. We review the main steps of active seismic experiments in this section. The reader is referred to [START_REF] Sheri | Exploration Seismology[END_REF] and [START_REF] Yilmaz | Seismic Data Analysis[END_REF] for more details.

Seismic data acquisition

Seismic surveys can take place in land or marine environments. In the usual land acquisition, the source is a vibrating truck and an array of receivers called geophones measures the motion of particles. They are evenly placed at the surface along one or several lines ( gure 1.1a). The experiment is run many times with di erent source and receiver positions. Under the approximation of single scattered energy, a point of the subsurface is in this way illuminated through many di erent angles. The presence of a drilled borehole can also provide a di erent acquisition geometry called Vertical Seismic Pro ling (VSP) with sources at the surface and receivers in the well ( gure 1.1b). In the case of cross-well acquisition, sources and receivers are located in two di erent wells. In marine acquisition, the source is an air-gun and receivers called hydrophones sensitive to the pressure are distributed along several streamers towed by a speci c vessel. Alternatively receivers can be located at the sea oor in a display called Ocean Bottom Cable (OBC) ( gure 1.1c).

(a) surface acquisition (taken from Danish Energy Agency).

(b) VSP acquisition.

(c) OBC acquisition. The data recorded by a receiver is a function of time and is usually called a trace. The collection of traces recorded during an experiment is represented on a panel called shot gather with the time on the vertical axis, and the distance between source and receiver (also called o set) on the horizontal axis ( gure 1.2).

A shot gather records the Earth's response to the source excitation. Many events corresponding to di erent kinds of waves can be observed. We can distinguish between pressure waves (P-waves), where particles move parallel to the wave propagation direction, and shear waves (S-waves), where the particle motion is orthogonal to the wave propagation [START_REF] Aki | Quantitative Seismology[END_REF]. Besides, energy travels as surface and body waves. Surface waves are more energetic and provide information about the near-surface [START_REF] Socco | Surface-Wave Method for near-Surface Characterization: A Tutorial[END_REF][START_REF] Solano | Imagerie Sismique de La Proche Sub-Surface : Modi cation de l'inversion Des Formes d'onde Pour l'analyse Des Ondes de Surface[END_REF]. They are used in seismology but commonly considered as noise in seismic exploration which uses body waves. The latter can be classi ed according to their path in the subsurface:

• transmitted waves, such as direct and diving waves travel between the source and the receiver without being re ected;

• (primary-)re ected and di racted waves are generated at discontinuities of the Earth with strong impedance contrast;

• refracted waves travel along these interfaces; • multiple re ected waves are caused by strong re ectors such as the sea surface or the water bottom. Upgoing energy is sent back to the subsurface and can re ect a few times on the subsurface structures before being recorded.

Physical modelling

For the determination of the subsurface parameters, we rst need to de ne a physical law representative for the wave propagation observed on data recordings. It is mathematically formalised through a partial di erential equation. The most general visco-elastic wave equation involves the density, the attenuation of P and S waves and the 21 elastic coe cient of the sti ness tensor relating the stress tensor to the strain tensor. Simpli ed physics is usually assumed. If the subsurface is considered isotropic, the sti ness tensor reduces to the Lamé parameters λ and µ. The simplest approximation is the isotropic acoustic case. The Earth is assimilated to a uid parametrised by the P-wave velocity V P and the density ρ. Then only P-waves are considered and re ected events are caused by rapid variations of the acoustic impedance I P = ρV P . Note that the model parameters are function of the spatial coordinates: even for the constant density acoustic case, a large number of parameters should be speci ed for solving the wave equation. The simulation of wave propagation is usually performed with a nite-di erence resolution of the wave-equation [START_REF] Virieux | P-SV Wave Propagation in Heterogeneous Media: Velocity-stress Finite-di erence Method[END_REF][START_REF] Levander | Fourth-order Finite-di erence P-SV Seismograms[END_REF][START_REF] Operto | 3D Finite-Di erence Frequency-Domain Modeling of Visco-Acoustic Wave Propagation Using a Massively Parallel Direct Solver: A Feasibility Study[END_REF], or with nite element schemes (see Virieux et al., 2011, for a review). Modelling can also rely on a high-frequency 1.1. From seismic acquisition to interpretation approximation such as ray theory (Červený, 2005). The result of the forward problem is synthetic data calculated for all subsurface points, in particular at the receiver positions.

Preprocessing

Earth's parameters cannot be directly inverted from seismic measurements. Usually, preprocessing is needed before subsequent imaging steps. The rst processing stage consists of selecting the wave eld in the data that will be used for the analysis and of attenuating the noise inherent to every practical experiment [START_REF] Yilmaz | Seismic Data Analysis[END_REF]. Many seismic imaging procedures rely on primary re ection data only and consider other events like transmitted waves or multiples as coherent noise. These should be removed from the data before further analysis. In particular, the removal of coherent noise such as multiples has been an intense research topic [START_REF] Verschuur | Seismic Multiple Removal Techniques: Past, Present and Future[END_REF] and will be reviewed in section 1.4.3.

However, with the improvement of seismic imaging algorithms and computation capabilities, recent developments attempt to use all the information available in the data, like multiples or transmitted events, and more generally the full wave eld. In particular, the purpose of this study is to investigate the use of multiple re ections as signal rather than noise.

Definition of an objective function

The accuracy of model parameters used during modelling can be assessed by comparing observed data and calculated data. For a quantitative estimation, a scalar objective or cost function is de ned on the space of admissible models. It is designed such that the best model is a global minimiser or maximiser of the function. It can be de ned directly in the data-domain, measuring for example the di erences between observed and calculated data in the least-squares sense [START_REF] Tarantola | Inversion of Seismic Re ection Data in the Acoustic Approximation[END_REF]. Alternative formulations in the image domain detailed in section 1.3 rely on focusing or coherency criteria de ned on reconstructed images of the Earth [START_REF] Al-Yahya | Velocity Analysis by Iterative Pro le Migration[END_REF][START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. A simple example of image-domain strategy will be given at the end of this section.

The purpose of the inverse problem is to nd a set of parameters minimising the objective function. In practice, it is an ill-posed problem, meaning that several models can explain perfectly the data (Tarantola, 2005), due to an imperfect illumination of the subsurface or to coupling between parameters. This issue can be mitigated by adding a regularisation term to the objective function, usually enforcing the smoothness of the recovered model, or its consistency with a priori information about the subsurface [START_REF] Asnaashari | Regularized Seismic Full Waveform Inversion with Prior Model Information[END_REF].

Solving the inverse problem Optimisation strategy

The objective of the inverse problem is to determine a set of model parameters which minimise the objective function through an optimisation procedure. The most general approach involves global optimisation methods [START_REF] Sen | Global Optimization Methods in Geophysical Inversion[END_REF] which requires only the ability to compute the value of the objective function. Examples of such methods are simulated annealing [START_REF] Kirkpatrick | Optimization by Simulated Annealing[END_REF][START_REF] Ingber | Very Fast Simulated Re-Annealing[END_REF], genetic algorithms [START_REF] Holland | Adaptation in Natural and Arti cial Systems: An Introductory Analysis with Applications to Biology[END_REF] or particle swarm optimisation [START_REF] Kennedy | Particle Swarm Optimization[END_REF]. The drawback of these strategies is their computational cost, as numerous evaluation of the objective function are required.

Local optimisation methods [START_REF] Nocedal | Numerical Optimization[END_REF]) are a less expensive alternative. The gradient of the objective function is used to iteratively update an initial guess of the model, such that the value of the objective function decreases with iterations. Local optimisation requires the ability of computing the gradient of the objective function with respect to the model parameters. The adjoint-state method [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF][START_REF] Chavent | Nonlinear Least Squares for Inverse Problems[END_REF] provides a computationally e cient way of performing this derivation.

In practice the objective function is not necessarily convex because of the non-linear relationship between data and model parameters: gradient-based method may converge to a local minima. Therefore an other requirement for local optimisation is the knowledge of an initial model su ciently close to the true model. Inversion strategies considering rst the low-frequency content of the data and progressively incorporating higher-frequencies help mitigating the non-linear behaviour of the objective function and relax the requirement of an accurate initial model [START_REF] Bunks | Multiscale Seismic Waveform Inversion[END_REF]. Strategies to de ne convex objective functions will be detailed in sections 1.2 and 1.3.

Multiparameter inversion

Most applications of inversion techniques consider a mono-parameter description of wave propagation with the pressure-wave velocity V P . Moving to a multi-parameter inversion is challenging [START_REF] Operto | A Guided Tour of Multiparameter Full-Waveform Inversion with Multicomponent Data: From Theory to Practice[END_REF]. As pointed out before, due to limited acquisition with sources and receivers at the surface, di erent classes of parameters are coupled, meaning that they have a similar impact on seismic data. This issue is known as cross-talk or trade-o between parameter. To overcome this di culty, a suitable parametrisation is needed. In the acoustic example, the parameter couples can be (V P , ρ) or (V P , I P ) depending of the acquisition [START_REF] Virieux | An Overview of Full-Waveform Inversion in Exploration Geophysics[END_REF][START_REF] Zhou | Full Waveform Inversion of Diving & Re ected Waves for Velocity Model Building with Impedance Inversion Based on Scale Separation[END_REF][START_REF] Zhou | Velocity Model Building by Full Waveform Inversion of Early Arrivals & Re ections and Case Study with Gas Cloud E ect[END_REF].

Scale separation

In the following we consider a monoparameter inversion with the P-wave velocity model, noted c(x) from now on, as unknown. x = (x, y, z) or (x, z) denotes the spatial coordinates. The resolution of the velocity model image that can be recovered is limited by the frequency band of the data (typically 5 to 70 Hz) and the data acquisition setup. The model resolution is better described with spatial frequencies. As illustrated by the well-known sketch of Claerbout (1985) ( gure 1.3), two separate ranges of spatial frequencies of the model can be reconstructed from seismic data [START_REF] Jannane | Wavelengths of Earth Structures That Can Be Resolved from Seismic Re ection Data[END_REF], leading to a scale separation of the velocity model ( gure 1.4):

• the smooth slowly-varying component of the model ( gure 1.4b), called the background velocity model, or macro-model, controlling the kinematics of wave-propagation;

• rapid changes in the components of the model, responsible for the re ections, referred to as re ectivity ( gure 1.4c). This part can be physically interpreted as the Earth discontinuities.

Note that since the publication of Claerbout's book, the frequency gap has been progressively lled by improvement of seismic source design allowing to record lower frequencies in the data and the progress of imaging techniques allowing to recover a more detailed background -Spatial frequencies that can be resolved from seismic data (from Claerbout, 1985). -Illustration of the usual scale separation for the Marmousi model [START_REF] Brougois | Marmousi, Model and Data[END_REF]. The complete velocity model c( x) is decomposed into a smooth background part c 0 (x) and a perturbation δc(x) of the background model c 0 (adapted from [START_REF] Billette | Estimation de Macro Modèles de Vitesse En Sismique Ré exion Par Stéréotomographie[END_REF].

velocity model [START_REF] Nichols | Resolution in Seismic Inversion-Spectral Gap or Spectral Overlap, Which Is Harder to Handle? 74[END_REF][START_REF] Lambaré | Recent Advances in Ray-Based Tomography[END_REF]. However, the traditional scale separation remains the theoretical basis of many seismic imaging methods.

The scale separation can be written explicitly

c(x) = c 0 (x) + δc(x), (1.1)
where c 0 is the background velocity model and δc is the re ectivity. If |δc| |c 0 |, the re ectivity can be seen as a perturbation of the background velocity model. Under the Born approximation, the relationship between primary re ection data and model perturbation δc is linear. Data still depend non-linearly on the background velocity model c 0 , though.

The large and short-scale structures of the velocity model can be inverted simultaneously without scale separation, as in Full Waveform Inversion (FWI) presented in section 1.2.1. Alternatively, they can be recovered separately. Migration algorithms are designed to convert events recorded in time into a re ectivity map of the model perturbation δc function of depth. Under the Born approximation, the determination of δc is a linear inverse problem. Performing this conversion requires the prior knowledge of the background velocity model. The determination of the long-scale structure of the velocity model is performed with tomographic approaches, either in the data-domain [START_REF] Bishop | Tomographic Determination of Velocity and Depth in Laterally Varying Media[END_REF] or in the image-domain [START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. Migration and tomography strategies will be reviewed in sections 1.2.2, 1.2.3 and 1.3.

Postprocessing

Finally the quantitative knowledge of physical properties provided by the inversion is used to determine a model of the subsurface at the reservoir scale. Empirical relationship between V P and V S may give indications about the presence of hydrocarbon [START_REF] Tatham | Vp/Vs -A Potential Hydrocarbon Indicator[END_REF] and the permeability and porosity of rock materials [START_REF] Domenico | Rock Lithology and Porosity Determination from Shear and Compressional Wave Velocity[END_REF]. This information are used to manage the oil production of the reservoir. This thesis focuses on the resolution of the inverse problem in the isotropic acoustic approximation of the wave equation. We study a method called Migration Velocity Analysis (MVA), an image-domain technique aiming at recovering the background velocity model. The purpose of the thesis is to investigate the possible inclusion of multiple re ections in this approach, which traditionally considers only primary re ections. The principle and issues of MVA will be detailed in section 1.3. We now review standard time-domain strategies for the resolution of the inverse problem.

Time-domain methods for the resolution of the inverse problem

We detail in this section time-domain methods addressing the resolution of the inverse problem in the isotropic acoustic approximation of the wave equation, meaning that the unknown is the pressure wave velocity eld c(x). Image-domain methods will be described in section 1.3. We rst present the FWI strategy, which consider the full recorded traces to recover both the large and short scales of c(x). Then alternative strategies relying on the scale separation assumption ( gure 1.4) are detailed. We shall distinguish between methods dedicated to the recovery of the short-scale structures δc(x) (section 1.2.2) and those aiming at retrieving the background velocity model c 0 (x) (section 1.2.3).

Full Waveform Inversion

A data-fi ing procedure FWI [START_REF] Virieux | An Overview of Full-Waveform Inversion in Exploration Geophysics[END_REF][START_REF] Fichtner | Full Seismic Waveform Modelling and Inversion[END_REF] is an iterative procedure considering the complete recorded seismic traces. The associated objective function measures the least-squares mis t between recorded data and simulated data.

J FWI [c] = 1 2 P calc [c] -P obs 2 2 .
(1.2)

All type of waves are included in the modelling of the data (direct, diving, re ected, multiply scattered waves). First introduced by [START_REF] Tarantola | Inversion of Seismic Re ection Data in the Acoustic Approximation[END_REF], the method has gained in popularity with the increase of computer power. In theory di erent model parameters (velocity, density, attenuation, anisotropy) can be resolved, requiring a modelling engine able to reproduce the physics of wave-propagation as accurately as possible [START_REF] Warner | Which Physics for Full-Wave eld Seismic Inversion[END_REF]. However most applications use the acoustic approximation of the wave-equation and are interested in retrieving only the pressure-wave velocity model of the Earth because of the computational cost of elastic modelling and of the challenges of multi-parameter estimation [START_REF] Operto | A Guided Tour of Multiparameter Full-Waveform Inversion with Multicomponent Data: From Theory to Practice[END_REF].

Resolution analysis

As FWI tries to explain the complete data set, it should provide high-resolution images of the subsurface, with both the long and short-scale structures of the velocity model. The contribution of di erent kinds of waves in the construction of the velocity model can be analysed with the following relationship [START_REF] Devaney | A Filtered Backpropagation Algorithm for Di raction Tomography[END_REF][START_REF] Miller | A New Slant on Seismic Imaging: Migration and Integral Geometry[END_REF],

k = 2ω c 0 cos θ 2 n, (1.3)
linking the spatial frequency vector or wavenumber k at point x to the di raction angle θ associated to a source-receiver pair, n being the normalisation of vector k ( gure 1.5). In the shallow part of the subsurface, large di raction angles obtained for diving waves allow to recover the small wavenumbers, i.e. the large-scale structure of the velocity model. Because of the limited surface o set range, the deeper part of the subsurface is investigated mainly by re ected waves with a small di raction angle, and only the high-frequency part of the velocity model is recovered. In the deeper part of the subsurface, FWI thus behaves like a non-linear least-squares migration algorithm [START_REF] Mora | Elastic Wave-eld Inversion of Re ection and Transmission Data[END_REF](Mora, , 1989)). Methods extracting the information about the long-scale structure contained in re ected events will be presented in section 1.2.3. The cycle-skipping issue It is known that the FWI objective function (1.2) su ers from local minima because of the non-linear relationship between model and data [START_REF] Gauthier | Two-dimensional Nonlinear Inversion of Seismic Waveforms: Numerical Results[END_REF]. This problem is called cycle-skipping and imposes in theory the use of global optimisation procedures [START_REF] Tognarelli | Two-Grid Stochastic Full Waveform Inversion of 2D Marine Seismic Data[END_REF][START_REF] Galuzzi | Stochastic FWI on Wide-Angle Land Data with Di erent Order of Approximation of the 2D Acoustic Wave Equation[END_REF]. These methods are computationally expensive as they require many evaluations of the objective function. Gradient-based methods are less expensive, but they need a starting model close to the solution to avoid converging to a local minimum. More precisely the phase mismatch between data computed with the initial and true model should be less than half the shortest wavelength contained in the data.

To overcome this di culty, a hierarchical strategy consists of inverting rst the low-frequency content of the data and then progressively incorporating higher frequencies, as the basin of attraction of the mis t function is inversely proportional to the central frequency of the data [START_REF] Bunks | Multiscale Seismic Waveform Inversion[END_REF][START_REF] Pratt | Two-Dimensional Velocity Models from Wide-Angle Seismic Data by Wave eld Inversion[END_REF][START_REF] Sirgue | E cient Waveform Inversion and Imaging: A Strategy for Selecting Temporal Frequencies[END_REF]. The presence of large o sets recording in the surveys allows also to better constrain the background model with transmitted waves [START_REF] Shipp | Two-Dimensional Full Wave eld Inversion of Wide-Aperture Marine Seismic Streamer Data[END_REF]. This hierarchical strategy has been successfully applied on real data sets in marine [START_REF] Sirgue | 3D Waveform Inversion on Valhall Wide-Azimuth OBC[END_REF][START_REF] Vigh | 3D Prestack Time Domain Full Waveform Inversion[END_REF] and land [START_REF] Plessix | Application of Acoustic Full Waveform Inversion to a Low-frequency Large-o set Land Data Set[END_REF] environments. In typical acquisition geometries, for a target at a few kilometres depth the frequency content of the data should start around 1 Hz to avoid cycle skipping. If low-frequency data or large o sets are not available, other strategies have to be designed to avoid falling in a local minimum.

Alternative techniques modify the de nition of the objective function to enlarge the basin of attraction around the correct velocity model. The usual 2 -norm can be replaced by a more convenient distance. [START_REF] Métivier | Measuring the Mis t between Seismograms Using an Optimal Transport Distance: Application to Full Waveform Inversion[END_REF] use an optimal transport distance to measure the mis t between seismograms; the basin of attraction of the objective function is extended at the cost of a new optimisation problem dedicated to the calculation of the distance. Other strategies transform the signals of both observed and calculated data to a more convenient domain before subtraction. [START_REF] Shin | Waveform Inversion in the Laplace-Fourier Domain[END_REF]Cha (2008, 2009) have studied the use of the Laplace and Laplace-Fourier domain to perform the inversion and showed that an accurate smooth model could be built in these domains. In the Normalised Integration Method [START_REF] Donno | Estimating the Background Velocity Model with the Normalized Integration Method[END_REF], the transformation consists of integrating the square of the signals and can be interpreted as a measure of the accumulation of energy along the trace. The objective function compares monotone increasing signals and is more convex; however the processing of noise for this method should be investigated. Other methods consider the envelope of seismic data [START_REF] Wu | Seismic Envelope Inversion and Modulation Signal Model[END_REF][START_REF] Chi | Full Waveform Inversion Method Using Envelope Objective Function without Low Frequency Data[END_REF] to work on less oscillatory signals. All these transforms allow to mitigate the cycle-skipping issue but produce velocity models with a lower resolution than conventional FWI. However they can be considered as a rst inversion step dedicated to the building of an accurate starting model for FWI [START_REF] Tejero | Comparative Study of Objective Functions to Overcome Noise and Bandwidth Limitations in Full Waveform Inversion[END_REF], or for migration algorithms used to recover the short-scale structure of the velocity model.

Linearised waveform inversion

The purpose of migration techniques is to recover a map of Earth discontinuities corresponding to a perturbation of the velocity model. These methods are based on the classic scale separation and assume the knowledge of an estimate of the macromodel. They were historically designed for primary re ections only. Extension to multiple re ections will be presented in section 1. 4.4. The rst migration techniques used geometrical construction [START_REF] Bleistein | Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion[END_REF]. In a simple geologic context, an event appearing at time t in a trace corresponding to a source at position s and a receiver at position r is due to a di racting point situated at a location x such that the sum of traveltimes from s to x and from x to r is equal to t. All the subsurface points satisfying this criterion de ne an isochrone curve; in the case of an homogeneous velocity model, it is an ellipse with focus points at s and r. Repeating the process for all sources and all receivers, the summation of ellipses interferes constructively along the re ectors and destructively anywhere else [START_REF] Bleistein | Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion[END_REF].

Later the construction of a re ectivity model was reformulated by considering the propagation of complete wave elds and [START_REF] Claerbout | Toward a Uni ed Theory of Re ector Mapping[END_REF] introduced the concept of Imaging Condition. Interfaces are de ned as the location where a downgoing incident wave eld coincides in time with an upgoing re ected wave eld. In the case of primary re ections in surface acquisition, the downgoing wave eld is the source wave eld and the upgoing wave eld is the re ected wave eld recorded by the receivers. This leads to a three-step procedure:

• propagation of an estimation of the source wavelet to determine the source wave eld S(s, x, t) in the medium at all subsurface points and for all times;

• backpropagation of the data recorded at receiver positions at all subsurface points and for all times to determine the receiver wave eld R(s, x, t);

• application of an imaging condition to these wave elds S and R to determine an image of the subsurface. Many formulations exist for this imaging condition, the most commonly used is the cross-correlation at zero-lag of the source and receiver wave elds.

The procedure is repeated for each shot point, yielding as many migrated sections as shot points in the survey. The obtained images can be stacked to increase the signal to noise ratio. However, the similarities and discrepancies between these di erent images can also be used as information. This is the basic principle of MVA techniques presented in Section 1.3. This migration algorithm has been recognised as the rst iteration of the least-squares datatting inverse problem [START_REF] Lailly | The Seismic Inverse Problem as a Sequence of before Stack Migrations[END_REF][START_REF] Tarantola | Inversion of Seismic Re ection Data in the Acoustic Approximation[END_REF]. Under the Born approximation, this inverse problem is linearised. Assuming a xed estimate of the background velocity model c 0 , it consists of determining the model perturbation δc which best reproduces recorded data. The associated scalar mis t function is .4) where P obs stands for recorded data and P[δc] represents the Born-modelling of data with the model perturbation δc. As in the case of FWI (equation 1.2), this objective function measures the mis t between observed data and calculated data, except that here only the short-scale part of the velocity model is updated, and the associated inverse problem is linear.

J 0 [c 0 , δc] = 1 2 P[δc] -P obs 2 , ( 1
The migration procedure detailed above actually computes the gradient of the objective function (1.4) in δc = 0,

∂ J 0 ∂ δc c 0 , δc = 0 (x) = 2 c 3 0 (x) s t ∂ 2 ∂ t 2 S[c 0 ](s, x, t)R[c 0 ](s, x, t) dt ds, (1.5)
where S and R are the source and receiver wave eld, whose values depend on the background velocity c 0 (x). Note that the de nition of the mis t function (1.4) contains an implicit summation over the sources. De ning the re ectivity as the gradient of this function produces a post-stack image.

The migration operator is actually the adjoint of the Born modelling operator and provides only a qualitative image of the subsurface: the position of the re ectors is kinematically consistent with the assumed velocity model and the recorded traveltimes. However the method does not provide a correct estimation of their amplitude. To obtain a quantitative estimate of the re ectivity model, the minimisation problem presented above should be solved completely. This true-amplitude migration can be achieved with an approximate inverse of the Hessian of the objective function [START_REF] Beylkin | Imaging of Discontinuities in the Inverse Scattering Problem by Inversion of a Causal Generalized Radon Transform[END_REF][START_REF] Lambaré | Iterative Asymptotic Inversion in the Acoustic Approximation[END_REF][START_REF] Plessix | Frequency-Domain Finite-Di erence Amplitude-Preserving Migration[END_REF][START_REF] Kiyashchenko | An Unsplit Convolutional Perfectly Matched Layer Improved at Grazing Incidence for the Seismic Wave Equation[END_REF]. Alternatively an iterative procedure using gradient-based methods can be de ned to minimise the mis t function [START_REF] Nemeth | Least-squares Migration of Incomplete Re ection Data[END_REF][START_REF] Østmo | Finite-Di erence Iterative Migration By Linearized Waveform Inversion In the Frequency Domain[END_REF]. In chapter 2, an iterative migration procedure will be presented in the context of second-order Born modelling.

The result of the migration process depends strongly on the velocity model used to compute the source and receiver wave elds. With an incorrect velocity model, events may be positioned at the wrong depth and may not be well focused. In the following section, time-domain methods to construct the background velocity model are reviewed.

1.2.3. Tomographic methods: retrieving the large-scale structure of the velocity model

Traveltime tomography

Traveltime tomography techniques are developed with the ray theory, based on the highfrequency asymptotic approximation of the wave equation (Červený, 2005). Wave propagation in the subsurface is described by rays with propagation laws similar to those used in optics. In this formalism, the Green's functions describing the wave propagation can be decomposed into three terms, one accounting for the traveltime, one for the amplitude and one for the source signature. In tomographic methods, events in the data are characterised by their traveltime only.

Their amplitude and the nite-frequency nature of the data are not considered, contrary to FWI. The objective function of the associated inverse problem measures the di erence between traveltimes picked on a selection of events in the data with traveltimes computed with an estimation of the velocity model c:

J TT [c 0 ] = 1 2 τ[c 0 ] -τ obs 2 .
(1.6)

The velocity model is updated iteratively to minimise the traveltime di erences. The method requires a rst processing stage to pick events, as well as a modelling engine relying on the resolution of the Eikonal equation to compute traveltime maps corresponding to a velocity model [START_REF] Vidale | Finite-Di erence Calculation of Travel Times[END_REF][START_REF] Podvin | Finite Di erence Computation of Traveltimes in Very Contrasted Velocity Models: A Massively Parallel Approach and Its Associated Tools[END_REF][START_REF] Noble | Accurate 3-D Finite Di erence Computation of Traveltimes in Strongly Heterogeneous Media[END_REF]. Depending on the acquisition, di erent events can be picked.

The rst arrivals are mostly used in seismology. They consists of direct and diving waves and are easier to pick than re ections. In active seismic experiments, they are well suited when the acquisition geometry emphasises transmission e ects, for example in surface acquisition with large o sets, cross-well acquisition or with buried receivers and sources at the surface [START_REF] Nhu Ba | Détection Des Zones de Failles Par Tomographie En Transmission : Application à La Station Expérimentale de Tournemire[END_REF]. In the case of surface acquisition without large o sets, the depth penetration of these waves restricts the area where the background velocity can be recovered. Information about the deeper part of the model is contained in re ected waves.

Re ected events can be picked as well. In this case the model is parametrised both by a velocity model and a geometric description of the re ectors. Re ection tomography was historically developed in the time domain [START_REF] Bishop | Tomographic Determination of Velocity and Depth in Laterally Varying Media[END_REF][START_REF] Farra | Non-Linear Re ection Tomography[END_REF]. Each re ection in the data is associated to a re ector. An issue is that picking re ected events along a wide range of o set is a di cult task, especially with complex geology and noisy data [START_REF] Lailly | Smooth Velocity Models in Re ection Tomography for Imaging Complex Geological Structures[END_REF]. To improve the signal to noise ratio, the data can be stacked if the model is not too complex. Alternatively the analysis can be transposed to the migrated domain [START_REF] Stork | Re ection Tomography in the Postmigrated Domain[END_REF], where interfaces are easier to pick. Re ectors are picked on migrated sections, then the corresponding migrated events are modelled (or demigrated) to be compared to the observed traveltimes [START_REF] Jacobs | Sequential Migration-Aided Re ection Tomography: A Tool for Imaging Complex Structures[END_REF][START_REF] Grau | Geophysical Exploration in Areas of Complex Geology, II Sequential Migration-Aided Re ection Tomography: An Approach to Imaging Complex Structures[END_REF]. Note that with shorto sets only, there is an ambiguity between the velocity and the depth of re ectors [START_REF] Williamson | Tomographic Inversion In Re ection Seismology[END_REF]. Re ection tomography is more generally known to be an ill-posed inverse problem requiring external constraints. These constraints can be existing well data [START_REF] Le Stun | Taking into Account a Priori Information in 3D Tomography[END_REF], or information about the subsurface provided by other imaging methods, gravimetric or electromagnetic for example [START_REF] Lines | Cooperative Inversion of Geophysical Data[END_REF].

Other alternatives to the picking of continuous re ected events are slope tomography methods, and in particular stereotomography [START_REF] Billette | Velocity Macro-Model Estimation from Seismic Re ection Data by Stereotomography[END_REF][START_REF] Lambaré | Stereotomography[END_REF][START_REF] Guillaume | Dip-Constrained Non-Linear Slope Tomography -an Application to Shallow Channel Characterization[END_REF]. In addition to traveltimes, the slope of locally coherent events are picked on common shot and common receiver panels. The inverted velocity model should then explain both traveltimes and slopes of picked events. In this method there is no need to describe the model with a set of continuous layer interfaces, only the dip of local events is needed. Then the density of picked events is higher. As for traveltime tomography, local picking can be performed in the image domain [START_REF] Chauris | Migration Velocity Analysis from Locally Coherent Events in 2-D Laterally Heterogeneous Media, Part I: Theoretical Aspects[END_REF]. Then kinematic invariants such as traveltimes and slopes can be recomputed in the time domain.

Traveltime tomography has become a standard in the oil and gas industry for velocity model building since the late nineties [START_REF] Woodward | A Decade of Tomography[END_REF]. Although recent publications have focused mainly about FWI, progresses have been made to better constrain the inversion, include structural information like sharp velocity contrasts and make the method more and more automatic [START_REF] Lambaré | Recent Advances in Ray-Based Tomography[END_REF]. Improvement of both the method and the recording devices have increased the resolution of velocity model constructed by traveltime tomography [START_REF] Nichols | Resolution in Seismic Inversion-Spectral Gap or Spectral Overlap, Which Is Harder to Handle? 74[END_REF]. We now review extension of traveltime tomography that go beyond the high-frequency approximation.

Wave-Equation Traveltime Tomography

Wave-Equation Traveltime Tomography (WETT) is an extension of traveltime tomography which takes into account the nite-frequency nature of seismic data. [START_REF] Luo | Wave-equation Traveltime Inversion[END_REF] proposed to cross-correlate the traces of rst-arrival events in observed and calculated data instead of subtracting their traveltime. Then they minimise the time-lag maximising the crosscorrelation. van Leeuwen and Mulder (2010b) indicate that the traveltime error might not coincide with the time-lag maximising the cross-correlation because of errors in the source wavelet used for calculated data. They propose to increase the robustness of the approach with an objective function penalising non-zero time-lag. Compared to FWI, WETT yields velocity models with lower resolution but is less prone to cycle-skipping and can be used to build a starting model for FWI [START_REF] Wang | Integrated Inversion Using Combined Wave-Equation Tomography and Full Waveform Inversion[END_REF]. The resolution of the method can be increased by replacing correlation by deconvolution [START_REF] Luo | A Deconvolution-Based Objective Function For Wave-Equation Inversion[END_REF]. A related strategy is Adaptive Waveform Inversion (AWI) (Warner and[START_REF] Warner | Adaptive Waveform Inversion -FWI Without Cycle Skipping -Theory[END_REF] in which the coe cients of a Wiener lter applied to calculated data are determined at each iteration to match observed data. The velocity model is then updated so that the coe cients of the lter amount to a simple zero time lag, which corresponds to the case where calculated data perfectly reproduce observed data.

Reflection Waveform Inversion

Re ection Waveform Inversion (RWI) [START_REF] Xu | Inversion on Re ected Seismic Wave[END_REF][START_REF] Zhou | Fundamental Issues in Full Waveform Inversion[END_REF][START_REF] Brossier | Velocity Model Building from Seismic Re ection Data by Full-Waveform Inversion[END_REF] is an approach similar to FWI developed to extract information about the macromodel from re ected events. It is inspired by the Migration-Based Traveltime Tomography (MBTT) procedure [START_REF] Chavent | Determination of Background Velocities by Multiple Migration Fitting[END_REF][START_REF] Plessix | Automatic and Simultaneous Migration Velocity Analysis and Waveform Inversion of Real Data Using a MBTT/WKB J Formulation[END_REF]. It either assumes a scale separation between low and high frequency content of the velocity model [START_REF] Xu | Inversion on Re ected Seismic Wave[END_REF] or relies on a parametrisation with both the velocity and the impedance to naturally facilitate the scale separation [START_REF] Zhou | Velocity Model Building by Full Waveform Inversion of Early Arrivals & Re ections and Case Study with Gas Cloud E ect[END_REF]. An initial estimate of the macromodel is used to construct a migrated section. Then re ected events are modelled from the migrated section and the background velocity model, and compared to observed re ections in an objective function similar to FWI. With this strategy, transmission wavepaths are constructed between the re ectors and the surface as well as between the re ectors and the receivers, allowing to recover the long-scale part of the velocity model in areas of the subsurface not reached by diving waves. [START_REF] Zhou | Full Waveform Inversion of Diving & Re ected Waves for Velocity Model Building with Impedance Inversion Based on Scale Separation[END_REF] and Alkhalifah and Wu (2016b) proposed strategies to combine the information extracted from both FWI and RWI about the macromodel.

An alternative strategy using the scale separation is the Di erential Waveform Inversion (DWI) approach [START_REF] Chauris | Di erential Waveform Inversion -A Way to Cope with Multiples? 75th[END_REF]. A migrated section is computed from one shot gather and used to calculate synthetic data for the next shot. The di erence with the corresponding observed data is used as information about the errors contained in the macromodel.

Finally van Leeuwen and Mulder (2008a) propose to correlate traces of observed and simulated data separated by a spatial shift and with a temporal lag in order to build coherency panels function of the space and time shifts. The correct velocity model corresponds to focusing of energy at zero spatial and temporal delay. The criteria assessing the quality of the velocity model in this approach as well as in DWI are inspired by image-domain methods presented in the next section.

Migration Velocity Analysis

We present in this section image-domain methods for the resolution of the inverse problem. This is the class of methods which will be studied in the thesis.

The basic principle of image-domain methods is that seismic data are redundant. Using an initial velocity model, a collection of migrated images can be created with di erent subsets of the data, for example one image for each shot gather [START_REF] Al-Yahya | Velocity Analysis by Iterative Pro le Migration[END_REF]. Migrating with the correct velocity model should result in kinematically coherent images, meaning that re ectors in images produced with di erent shot-point experiments should be positioned at the same depth [START_REF] Al-Yahya | Velocity Analysis by Iterative Pro le Migration[END_REF]. Conversely, discrepancies between di erent images of the same Earth carry information about the errors in the estimated velocity model. The family of methods using this principle is called Migration Velocity Analysis (MVA).

Historically MVA emerged as an extension of the usual Normal Move Out (NMO) correction procedure. In this method the data are sorted into panels called Common Mid Point Gather, function of the surface o set and the time. With the assumption of horizontal structure of the Earth, each trace represents the re ection produced by the same subsurface point with di erent source-receiver spacings. Re ections then appear as hyperbolas with a traveltime increasing with o set. The NMO correction consists of nding the (1D-)velocity model that best explains these hyperbolas. The better the velocity model, the better traces stack constructively after correction. Although dipping events can be treated with the Dip Move Out correction, this method is limited to the case of simple velocity model structures. It has been transposed to the image domain by [START_REF] Al-Yahya | Velocity Analysis by Iterative Pro le Migration[END_REF] to handle more complex media.

Many di erent MVA approaches have been proposed. They all analyse primary re ections and rely on the separation of scale and the Born approximation. We will review them using the formalism introduced by [START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. MVA methods can be described in four steps:

• migration of the data to an image-domain parameterised with an additional parameter representing the redundancy of seismic data. For example in 2D, the migrated volume δc(x, z, h) is parameterised by lateral position x, depth z, and the redundant parameter h. Then the subsurface image and the data have the same dimension. The latter indeed depends on the source horizontal coordinate, the receiver horizontal coordinate and time. This is a key point of MVA strategies: even in an incorrect background velocity model, no information is lost during migration of the data, in the sense that data can be re-modelled in the same background velocity model using the migrated volume;

• de nition of a coherency or focusing criterion. It depends on the choice of the redundant parameter and states that the value of the re ectivity along the additional parameter carries the information about the velocity model. Note that stretching e ects and limited acquisition geometries are not necessarily taken into account in this principle. The analysis is performed on panels called Common Image Gathers (CIGs) representing a section of the re ectivity volume as a function of depth and of the extra-parameter for a xed lateral position;

• de nition of an objective function. Contrary to the data domain where observed data provide a natural reference for assessing the quality of an estimated velocity model, there is no obvious reference in the image domain. Instead, the objective function evaluates if the focusing criterion is satis ed. It is de ned by the general expression

J MVA [c 0 ] = 1 2 A δc[c 0 ](x, z, h) 2 , (1.7)
where operator A is called annihilator, and is chosen such that J MVA is extremal for the correct background velocity model. To allow minimisation with gradient-based methods, the objective function should be smooth and free of local minima over a large range of velocities. The partial derivative of the function with respect to the image, A T Aδc, is commonly called image residual.

• computation of the gradient of J MVA with respect to c 0 . This is usually done with the adjoint-state method [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF] for e ciency reasons. The background velocity model is then updated using non-linear local optimisation methods [START_REF] Nocedal | Numerical Optimization[END_REF].

In the following, we present two formulations of MVA: in the surface-oriented approach, the redundant parameter is related to the acquisition, contrary to the depth-oriented approach where it is introduced during the construction of the migrated image. Then we will focus on the depth-oriented formulation. We de ne the objective function in the image-domain and detail the resolution of the associated inverse problem. The general di culties and limitations of the method, appearing even in the case of primary re ections only, are presented. Finally we illustrate the principle of MVA on a simple 1D case and show that the presence of multiples interpreted as primary leads to an incorrect background velocity update. This motivates the need to deal with multiples in the MVA framework.

Surface-oriented MVA

A natural choice for the redundant parameter is the shot number or shot position [START_REF] Symes | Inversion of Re ection Seismograms by Di erential Semblance Analysis: Algorithm Structure and Synthetic Examples[END_REF][START_REF] Huang | Born Waveform Inversion via Variable Projection and Shot Record Model Extension[END_REF]. The idea is that images obtained from several experiments should be identical and independent of the position of the source. With this formulation, CIGs represent the image of the subsurface obtained with di erent subsets of the data. On these CIGs, events should be horizontal for the correct velocity model. Several authors suggest using the surface o set (distance between source and receiver) instead of the shot number [START_REF] Zhang | Angle Gathers from Reverse Time Migration[END_REF] as common o set sections provide large illumination, although it is less natural for wave-equation based modelling. An example of an image volume indexed by the surface o set is represented in gure 1.6. If the velocity model is incorrect, events in CIGs are not horizontal any more. In simple models, they curve upward for a too slow velocity model and downward for a too low velocity model. Then two strategies can be de ned. In the case of the Semblance criterion [START_REF] Chavent | Determination of Background Velocities by Multiple Migration Fitting[END_REF], the similarity between images produced with di erent subsets of the data is measured by stacking over the extension parameter and the objective function measures the energy of the stack. For the correct velocity model, the images sums up constructively and the energy of the stacked image is maximal. The objective function is free of local minima over a large range of velocity but may exhibit oscillations away from the correct model [START_REF] Chauris | Two-Dimensional Velocity Macro Model Estimation from Seismic Re ection Data by Local Di erential Semblance Optimization: Applications to Synthetic and Real Data Sets[END_REF].

With the Di erential Semblance Optimisation (DSO) [START_REF] Symes | Velocity Inversion by Di erential Semblance Optimization[END_REF] strategy, the objective function computes the derivative of the image with respect to the redundant parameter to measure the atness of events in CIGs [START_REF] Chauris | Two-Dimensional Velocity Macro Model Estimation from Seismic Re ection Data by Local Di erential Semblance Optimization: Applications to Synthetic and Real Data Sets[END_REF]. It has been shown, at least for simple 1D models, that the DSO functional has better convexity properties than the one de ned by the semblance principle [START_REF] Stolk | Smooth Objective Functionals for Seismic Velocity Inversion[END_REF]van Leeuwen and Mulder, 2010a). Note that the DSO functional was initially de ned as a regularisation term for FWI [START_REF] Symes | Inversion of Re ection Seismograms by Di erential Semblance Analysis: Algorithm Structure and Synthetic Examples[END_REF].

The estimation of the derivative is sensible to coherent noise and thus requires ltering before evaluation of the objective function [START_REF] Chauris | Two-Dimensional Velocity Macro Model Estimation from Seismic Re ection Data by Local Di erential Semblance Optimization: Applications to Synthetic and Real Data Sets[END_REF]. The recognition of continuous events across the o set range might be tedious as well. Then [START_REF] Chauris | Migration Velocity Analysis from Locally Coherent Events in 2-D Laterally Heterogeneous Media, Part I: Theoretical Aspects[END_REF] proposed to pick the slope of locally coherent events in CIGs and common o set sections to assess the quality of the velocity model. They show that this strategy is an equivalent in the image domain of the stereotomography procedure described before.

However as each individual image is obtained with only a subset of the data, kinematic artefacts may appear when complex wavepaths are considered [START_REF] Xu | Common-angle Migration: A Strategy for Imaging Complex Media[END_REF]Stolk and Symes, 2004;[START_REF] Zhang | Angle Gathers from Reverse Time Migration[END_REF]. In the depth-o set extended model presented in the next paragraph, each component of the migrated volume is constructed from the whole data set, the image is thus better constrained and is less prone to the apparition of artefacts [START_REF] Stolk | Modeling of Seismic Data in the Downward Continuation Approach[END_REF].

Depth-oriented MVA

In the depth formulation of MVA, the redundant parameter is independent of the acquisition and is introduced during the construction of the migrated image. The image domain is said to be extended and the extension parameter can be a spatial [START_REF] Rickett | O set and Angle-domain Common Image-point Gathers for Shot-pro le Migration[END_REF][START_REF] Shen | Di erential Semblance Velocity Analysis by Wave-Equation Migration[END_REF] or temporal (Sava and Fomel, 2006;[START_REF] Yang | Image-Domain Wave eld Tomography with Extended Common-Image-Point Gathers[END_REF] delay. Alternatively a scattering angle can be considered [START_REF] Sava | Angle-domain Common-image Gathers by Wave eld Continuation Methods[END_REF][START_REF] Biondi | Angle-domain Common-image Gathers for Migration Velocity Analysis by Wave eld-continuation Imaging[END_REF]. The spatial delay ( gure 1.7) is commonly referred to as depth-o set or subsurface-o set. In 2D, the general extended cross-correlation formula with the subsurface o set h = (h x , h z ) and the time-lag ∆t [START_REF] Sava | Extended Imaging Conditions for Wave-Equation Migration[END_REF] is

δc[c 0 ](x, h, ∆t) = 2 c 3 0 (x) s t ∂ 2 ∂ t 2 S[c 0 ](s, x -h, t -∆t)R[c 0 ](s, x + h, t + ∆t) dt ds. (1.8)
The section of this image at h = 0 and ∆t = 0 corresponds to the image of the physical re ectivity obtained with the classical migration formula (1.5) after summation over all sources and receivers. The introduction of the delays compensates for errors contained in the velocity model and allows to capture information that is not present in the usual migrated section. Energy focusing at non-zero values of the extension parameter carries information about the background velocity model. In the depth-extended domain, the coherency criterion used to assess the quality of the velocity model is the focusing of energy at zero-space lag and zero time-shift.

In practice, after computation of the source and receiver wave elds, a correlation is performed for each value of the subsurface-o set and the time-lag. To keep the migrated image size and the computational time within reasonable limits, the extension is usually made along a single extension parameter only. The subsurface-o set is usually chosen [START_REF] Shen | Automatic Velocity Analysis via Shot Pro le Migration[END_REF]. In the case of surface acquisition and horizontal structures, most of the information about the velocity model is contained along the horizontal component of the depth-o set. Therefore in the following, the subsurface-o set will be considered horizontal only, h = (h, 0), as represented in gure 1.7.

CIGs then represent the migrated section at xed lateral position and depend on the depth z and on the subsurface-o set h. Energy focuses around zero-o set for the correct background velocity model and spreads over non-zero o set in an inaccurate model. Events have a downward (upward) curvature for a too low (too high) velocity model [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF] (see gure 1.9a for an example). Note that this is the opposite for the surface-o set case.

The Semblance and DSO strategies used in surface-oriented MVA have equivalences in the depth-oriented approach. The Semblance principle consists of maximising the energy around zero-o set. Note that maximising the energy of the image at h = 0 corresponds to the surfaceoriented Semblance criterion. In the depth-oriented DSO formulation, the objective function should penalise defocused energy. This is the strategy used in this study.

Choice of a misfit function and inversion of the velocity model

Usually, the objective function of DSO is constructed by multiplying the image by the value of the subsurface o set [START_REF] Shen | Automatic Velocity Analysis via Shot Pro le Migration[END_REF][START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF]:

J DSO [c 0 ] = 1 2 hδc[c 0 ](x, h) 2 .
(1.9) This is the de nition used in this study. Alternatively, a mixed formulation involving both the semblance and the di erential semblance can also be formulated [START_REF] Shen | Automatic Velocity Analysis via Shot Pro le Migration[END_REF][START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF])

J[c 0 ] = 1 2 h δc[c 0 ](x, h) 2 - α 2 δ(h)δc[c 0 ](x, h) 2 .
(1.10) with a positive scalar coe cient α to be determined. This formulation allows to use the total information contained in the migrated image, as the energy at zero-o set discarded by the DSO term is used by the semblance term. In this formulation, away from the minimum, the optimisation is driven by the DSO term while the robustness of the semblance close to the solution is used in the last iterations.

The MVA objective function is non-linear and minimisation is performed with local optimisation schemes. The calculation of the gradient is the main computational e ort of the method. It is therefore performed with the adjoint-state method (Lameloise et al., 2014;Yang and Sava, 2015). As will be explained in the following of this section, this gradient is not always smooth although we want to update the background part of the velocity model. Therefore a smoothing is commonly applied to the gradient, at least in the rst iterations [START_REF] Lameloise | Quantitative Migration for a More Robust Migration Velocity Analysis[END_REF]. In theory the inversion should account for the Hessian of the MVA objective function, but its computation is not a ordable in practice. So far only an estimation of its diagonal computed as the product of the Hessian matrix with a unit vector has been used (Liu et al., 2014b;Shen and Symes, 2015).

Limitations of MVA

Despite an attractive formulation, not so many applications on real data have been published [START_REF] Chauris | Two-Dimensional Velocity Macro Model Estimation from Seismic Re ection Data by Local Di erential Semblance Optimization: Applications to Synthetic and Real Data Sets[END_REF][START_REF] Mulder | Automatic Velocity Analysis by Di erential Semblance Optimization[END_REF][START_REF] Alkhalifah | Tau-Migration and Velocity Analysis: Application to Data from the Red Sea[END_REF][START_REF] Shen | Automatic Velocity Analysis via Shot Pro le Migration[END_REF][START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF]Weibull et al., 2012a,b;[START_REF] Lameloise | Analyse de Vitesse Par Migration Quantitative et Introduction Des Ondes Transmises[END_REF]. We review some of the di culties of MVA which prevent its use in realistic industrial contexts.

Computational cost

To limit the computational cost and memory requirements of the method, the extension of the model is made only along the horizontal component of the subsurface-o set, and its vertical component as well as the time-lag parameter are not considered. Even in this simpli ed setting, MVA remains computationally expensive. This is the main limitation for the extension of MVA techniques to 3D-applications, where two extra dimensions are in principle added to the model space to match the data size.

A migration step should be performed at each iteration on the background velocity model, involving the calculation of the source and receiver wave elds and their cross-correlation. This cross-correlation should be performed for each value of the extension parameter. To reduce the computation time necessary for cross-correlation, Yang and Sava (2015) propose to compute CIGs only at a selection of image points instead of the whole image. van [START_REF] Van Leeuwen | A ordable Full Subsurface Image Volume -An Application to WEMVA[END_REF] propose to construct CIGs with only a random choice of traces.

A related issue is the maximum value of the subsurface-o set h max that should be considered. In practice it is chosen empirically, depending on the maximum value of the surface o set and the depth of the re ectors. If a too low value is chosen, some information may be lost during migration. If it is too high, the cross-correlations for the large values of h increase the computational cost without adding any information. As the background velocity model improves with iterations, energy should be more and more focused in CIGs and the value of h max could decrease. Using this idea, [START_REF] Fu | Reducing the Cost of Extended Waveform Inversion by Multiscale Adaptive Methods[END_REF] propose a multiscale/multigrid strategy starting with low frequency data and a coarse sampling of the subsurface-o set axis. At each iteration the data mis t is computed in two cases, rst considering all the subsurface-o sets until h max , second using only half of the subsurface-o sets range. This criteria is used to update the value of h max at each iteration. In parallel, the sampling of the subsurface o set-range is increased as higher frequency are included in the data.

Note that if this techniques may decrease the computational cost of cross-correlation, the cost due to propagation remains expensive, especially on ne grids.

Although computational e ciency is an important aspect of the method, this issue will not be speci cally addressed in the study, CIGs being calculated with a xed range and sampling of the extension axis.

Accuracy of migrated images

The re ectivity model used in MVA techniques is the result of a classic migration algorithm. As already mentioned, this image is the rst gradient of the objective function minimising the mis t between observed and calculated data and is produced using the adjoint of the Born modelling operator. Several authors suggest that this is not a su ciently accurate solution of the minimisation problem. [START_REF] Yang | Illumination Compensation for Image-Domain Wave eld Tomography[END_REF] report that under complex structures like salt dome, uneven illumination results in defocused energy even for the correct velocity model. As a remedy, they propose to incorporate illumination as a weight in the MVA objective function, such that defocusing due to poor illumination does not in uence the velocity update.

Even for very simple models, CIGs are perturbed with migration artefacts ( gure 1.8a) related to the limited source and receiver coverage of the acquisition setting [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF]. These artefacts always have an upward curvature, independently of the velocity used during migration and do not focus for the correct velocity. As a result, the MVA objective function is not minimal for the correct velocity model but for a model with slightly higher values (Lameloise et al., 2014) and its gradient does not provide a satisfactory update ( gure 1.8b). As these events are steeper than the ones associated to the true re ector, [START_REF] Weibull | Automatic Migration Velocity Analysis Using Reverse Time Migration[END_REF] and [START_REF] Chauris | Inversion Velocity Analysis -The Importance of Regularisation[END_REF] propose to apply a z-derivative to the CIGs to attenuate migration artefacts. Alternatively, [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF] suggests the application of tapers to smooth the truncation in source and receiver coverage. A more sophisticated approach consists of replacing the classic migration by a quantitative migration (Lameloise and Chauris, 2014;[START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. Lameloise et al. (2014) extend the ray+Born inversion approach of [START_REF] Lambaré | Iterative Asymptotic Inversion in the Acoustic Approximation[END_REF] to the extended model and de ne migration weights such that the Hessian of the migration objective function is almost diagonal. In practice the weights allow to compensate for uneven illumination in the CIGs. Migration artefacts are correctly attenuated ( gure 1.9a) and the gradient of the objective function improved ( gure 1.9b). However, this approach is limited to the use of ray theory in the computation of the CIGs. In this thesis, it will be extended to wave-equation based operators [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]Symes, 2015, 2017). We will show that this strategy involves the application of a z-derivative to the migrated image, similar to what was proposed by [START_REF] Weibull | Automatic Migration Velocity Analysis Using Reverse Time Migration[END_REF] and [START_REF] Chauris | Inversion Velocity Analysis -The Importance of Regularisation[END_REF], thus giving a formal justi cation to their strategy.

Gradient artefacts

A well-known issue pointed out by [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF], [START_REF] Vyas | Gradients for Wave-equation Migration Velocity Analysis[END_REF] and [START_REF] Chauris | Removing Spurious Oscillations in the Gradient of the Di erential Semblance Optimization Functional[END_REF] is the presence of strong oscillations in the gradient of the MVA objective function with respect to the background velocity ( gure 1.10a). They are known as "gradient artefacts". They occur at discontinuities in the re ectivity model and along the re ectors and prevent the MVA gradient to be used as a velocity update without a prior smoothing stage. [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF] propose to add a phase shift of 90°to the image residual with the application of a h-derivative before the computation of the gradient of the MVA objective The "horizontal contraction" approach allows to remove these spurious oscillations (from [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF].

function. This trick greatly improves the quality of the velocity update ( gure 1.10b) but is not fully understood. Shen and Symes (2015) recognise in this modi cation a warping technique: the derivative with respect to h applied to the image residual can be seen as a contraction of CIGs in the h direction, this contracted CIGs having their energy more focused at zero subsurfaceo set. Therefore Shen and Symes (2015) rename this technique "horizontal contraction". A disadvantage of this method is that the modi ed velocity update is not the gradient of an objective function any more. Moreover such a technique does not properly work in the case of low velocity anomalies (Shen and Symes, 2015).

Stability of the MVA gradient

Another numerical di culty in the computation of the MVA gradient has been recently pointed out [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF]. To obtain an accurate re ectivity model free of migration artefacts, one could treat migration as an inverse problem and carry on the minimisation of the associated objective function (1.4) iteratively. The resulting re ectivity model is used as input of the MVA objective function (1.9). This nested optimisation strategy (see also gure 1.22) will be extensively analysed in this thesis.

As recently observed by [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF], an issue with this approach is the apparent instability of the gradient of the MVA objective obtained after each inner iteration on the re ectivity: "the theoretical relation between the error in solving the inner problem and the error in the gradient computation has not been established. The reason is that the small 2 error in the data mis t does not imply a small error in the WEMVA operator. " (Huang, 2016, p. 100).

We illustrate this issue with a numerical result ( gure 1.11) extracted from chapter 3. The model considered here is similar to those shown in gures 1.8 to 1.10. A single horizontal re ector is located in a homogeneous background velocity model and the MVA gradient is computed in a too low homogeneous medium. Thus we expect a homogeneous negative gradient above the re ector position ( gure 1.11b). A good solution of the inner problem is found after 5 iterations. The subsequent re ectivity models provide very similar data mis t ( gure 1.11a). However using these successive re ectivity models as input to the MVA objective function leads to very di erent MVA gradients ( gure 1.11c), whose similarity to the expected velocity update decreases exponentially with the number of iterations ( gure 1.11d). This issue will be further illustrated in chapters 3 and 4 and possible solutions will be proposed in chapters 3 and 5. n) δc (7) G (n) -G (7) (d) Distance between the gradient obtained at iterations 7, 8, 9, 10 and 11 and the gradient obtained at iteration 7, function of the same distance between the re ectivity models. The dahsed blue line represents the curve y = 0.23(e 1.25x -1).

Sensitivity to noise

We now illustrate the sensitivity of the MVA gradient to noise in observed data ( gure 1.12). We use an example similar to the one of the preceding paragraph. We rst compute the re ectivity model and the associated MVA gradient with noise-free observed data. The method used to obtained these images will be detailed in chapter 4. The MVA gradient is homogeneous and negative above the re ector. Increasing the noise on observed data leads to an increased noise level in the re ectivity image, but the re ector is still clearly distinguishable. The migration operator is linear in the noise nevel, and so is the impact on the re ectivity image ( gure 1.12b).

On the contrary, the background velocity update is more altered by noise in observed data: artefacts with an incorrect sign appear and the gradient is not homogeneous any more. It di ers from the reference model with a polynomial behaviour (here, a power 1.13 as illustrated in gure 1.12b) when the noise level increases, so this is not as extreme as the instability shown in gure 1.11.

Beyond primary reflected waves

All the theoretical aspects of the method are developed for primary re ections. However, the background velocity updates could bene t from the inclusion of other events with di erent illumination in the MVA procedure.

Recently, the extension to transmitted events has been investigated (Shen and Symes, 2013;Chauris et al., 2013;[START_REF] Lameloise | Automatic Migration Velocity Analysis Applied to Direct Waves in a Crosswell Con guration[END_REF][START_REF] Lameloise | Quantitative Migration for a More Robust Migration Velocity Analysis[END_REF]. [START_REF] Lameloise | Quantitative Migration for a More Robust Migration Velocity Analysis[END_REF] de ne image functions for transmitted events, equivalent to the extended migrated sections for re ected waves. They are constructed with the same extended cross-correlation imaging condition, except that the full recorded wave eld including re ected and transmitted waves is backpropagated. The corresponding CIGs are sensitive to the velocity and exhibit the same focusing behaviour: energy is focused at zero subsurface-o set for the correct velocity model and spreads over non-zero o sets otherwise. The conventional MVA objective function penalising defocused energy can be used and the corresponding velocity updates are smooth and do not su er from the "gradient artefacts" observed for re ected waves.

Multiple re ections are still an issue for MVA. If they are not removed from the data, they mislead the velocity analysis as the methodology developed for primary re ection does not explain the kinematics of multiple re ections. Compared to the primary it may be mistaken for, a multiple travels in the upper part of the model. With the hypothesis of increasing velocity with depth, multiples then tend to favour lower apparent velocities. Hence they need to be removed from the data in a preprocessing stage or in the migrated sections with a muting in the fk domain [START_REF] Mulder | Automatic Velocity Analysis by Di erential Semblance Optimization[END_REF][START_REF] Li | Interval Velocity Estimation via NMO-Based Di erential Semblance[END_REF], for example. However the removal of multiple re ection is challenging and residual multiple energy may survive these ltering processes. As a remedy, [START_REF] Mulder | Automatic Migration Velocity Analysis and Multiples[END_REF] introduce a bias in the MVA objective function to favour higher velocities, thus compensating for multiples. This formulation relies on the assumption that multiples favour lower velocity, which does not hold for models where the velocity does not increase with depth.

Illustration of the principle and di iculties of MVA on a simple example

To conclude this part, we illustrate in this paragraph the basic principle of MVA and some issues due to the presence of multiples. For the sake of simplicity, we consider a pure 1D example ( gure 1.13). Although it may not be representative for the complexity of a 2D acquisition, it shares many properties with the 2D and 3D cases that will be considered in the following chapters. This 1D example will be used again in chapters 4 and 5.

The input is re ected observed data ( gure 1.13b) pre-computed in an exact homogeneous velocity model with a single re ector located at 400 m depth. MVA techniques aim at recovering the background velocity model. Here we start with an initial too low homogeneous background velocity, such that we expect the method to provide a positive update. The method is decomposed in two steps. First observed data are converted into a re ectivity image function of the subsurface by migration. This image (e) is obtained as the zero-lag cross-correlation of two wave elds (c) and (d) computed by propagating the source wavelet (a) emitted at z = 0 and by backpropagating observed data (b) recorded at z = 0. As the initial velocity model is too low, the re ector is located above its true position.

The so-called "image residual" (g) (which can be seen as an equivalent of the more natural data residual in data-domain methods) is obtained by multiplying the re ectivity image by |z -z ex | 2 , square of the distance to the true re ector depth. Hence the norm of the image residual (z -z ex )δc 2 is minimum when the re ector is located at its correct position. The objective of MVA is to nd a background velocity model minimising the norm of the image residual. Note that this 1D case is somehow arti cial as it requires the prior knowledge of the exact re ector position, which is an unknown in practice. Moreover, the approach is valid only for a single re ector. The 2D approach detailed at the beginning of this section is more general and does not require prior knowledge of the subsurface, but its principle is very similar.

The derivation of the velocity updates involves computing two new wave elds obtained as the interaction of the source (c) and receiver wave elds (d) with the image residual (g) (Lameloise et al., 2014;Yang and Sava, 2015). The summation of the zero-lag cross-correlations of these new wave elds with the source and receiver wave elds gives the opposite of the velocity update ((h) and (k)). We obtain a constant value above the re ector altered with unwanted oscillations around the re ector which can not be easily attenuated by smoothing. This is one of the issues of MVA which has been discussed in section 1.3.4.

Finally, we repeat the same experiment with observed data containing both the primary re ection and a surface multiple ( gure 1.14) to illustrate the potential issues caused by multiples. Hence, the multiple is treated as a primary and is migrated to twice the depth of the primary in the image domain ( gures 1.14e and 1.14g). As a consequence, the image residual would not be minimal for the correct velocity. There are two di erences with the preceding case on the velocity update ( gures 1.14h and 1.14k): it has a non-zero value below the re ector, where no information can be recovered in theory, and the sign of the update is incorrect. This phenomena will be observed in the 2D case as well in chapter 4 and illustrates the need for a proper inclusion of multiples in MVA, as well as in other inversion strategies.

Conclusion

The objective of the thesis is to investigate the inclusion of multiple re ections in the usual depth-oriented MVA. A strategy is developed, rst in the case of primaries only, to make the method more robust against migration artefacts and spurious oscillations in the gradient. The method is then extended to the case of multiples.

To further motivate the inclusion of multiple re ection in MVA, we explain in the next section why their removal is challenging and what bene ts they may bring to imaging procedures. In particular we report on recent extensions of migration techniques to multiples, which may help to de ne a MVA procedure handling multiples.

Multiple reflections 1.4.1. Physics of multiples

As de ned by [START_REF] Verschuur | Seismic Multiple Removal Techniques: Past, Present and Future[END_REF], primaries are waves that undergo only one upward re ection, whereas multiple re ections are characterised by at least one downward re ection. Because of spherical divergence and of the loss of energy occurring at each re ection, multiples e ectively visible in the data are associated to strong re ectors. This is typically the case in marine acquisition where the interface between the air and the sea acts as a mirror. The water bottom is also a strong re ector. An other example is the chalk layer in the North Sea located between highvelocity layers, creating two high impedance contrasts which are strong multiples generators [START_REF] Reinier | Building a Robust Depth Imaging Velocity Model in North Sea[END_REF].

A rst classi cation of multiples refers to the location where the shallowest downward re ection occurs. If it is the surface, the multiples are called surface-related multiples ( gure 1.15a). Note that the upgoing multiple re ection can be re ected at the surface again, leading to higher-order surface multiples ( gure 1.15b). In the following, a rst-order surface-multiple refers to the multiple event undergoing a single downward re ection at the surface, and a second-order surface-multiple as the corresponding event for two downward re ections, etc. When the shallowest downward re ection occurs at an interface between two layers in the subsurface, the term internal multiple is used ( gure 1.15c). Similarly, several order of internal multiples can be de ned.

In the case of marine acquisition, the reverberations occurring in the water column are called water-layer multiples ( gure 1.15d). Multiples undergoing at least one re ection below the water bottom and a reverberation in the water-layer are referred to as peg-leg multiples ( gure 1.15e).

In practical marine acquisition, both source and receivers are located a few meters below the sea surface. Then one part of the energy emitted by the source goes directly to the subsurface, while an other part re ects at the sea surface, leading to a surface multiple called source ghost.

The same phenomenon occurs at the receiver side and is called receiver ghost. These kinds of multiples are also observed in the case of OBC acquisition ( gure 1.15f). Naturally more complex paths in the subsurface which do not t in the previous categories are possible ( gure 1.15g).

One can also distinguish between long-period and short-period multiples. Long-period multiples ( gure 1.16a) appear in the data record as separate events from their relative primaries. Visually the same succession of events is repeated at regular intervals of time ( gure 1.17). This is the case with water-layer reverberations recorded in a deep-water environment. Oppositely, -Di erent kinds of multiple re ections (similar to Verschuur, 2013, pp. 8-9). The black point and the white triangle represent source and receiver positions, respectively.

short-period multiples ( gure 1.16b) are produced by reverberation within small layers such as chalk. These reverberations overlap with one another and with the original primary, resulting in a single event with a di erent wavelet signature in the recorded data. Many imaging procedures designed for primary re ections require multiples to be removed from data. On the contrary, some methods presented in section 1.4.4 include them as complementary information. In any case, one needs the ability to model multiple re ections. Standard multiple modelling techniques are now reviewed.

Modelling of multiples

Surface-related multiples can be included in modelling by replacing the usual absorbing boundary condition at the surface by a free-surface. Alternatively, multiples can be predicted using primary re ections. First-order surface related-multiples can indeed be considered as the (primary-)response of the Earth to an areal source made of the primary re ections reaching the surface. Then rst-order multiples can be obtained by re-injecting the primaries into a primaries modelling code. If the model of the rst layer is known (for example the water layer in marine acquisition), multiples can be predicted by adding a ctive roundtrip to the data through this layer with wave eld extrapolation [START_REF] Wiggins | Attenuation of Complex Water-bottom Multiples by Wave-equation-based Prediction and Subtraction[END_REF][START_REF] Wiggins | Attenuation of Complex Water-bottom Multiples by Wave-equation-based Prediction and Subtraction[END_REF]. Then primaries are transformed in rst-order multiples, rst-order multiples in second-order multiples, and so on. The surface re ection coe cient should be taken into account (the coe cient -1 is the simplest approximation).

A data-driven alternative requiring no model of the subsurface is possible. In this approach, a rst-order surface-multiple is obtained by autoconvolution of the primary re ection. The prediction is exact except for the source wavelet which is present twice in the multiple due to the autocorrelation. A shape correction has then to be applied. Similarly the second-order surface-multiple can be obtained by convolution of the primary with the rst-order multiple, and so on. In the 1D case and with a Dirac source, the total response including all order of surface-multiples can be described as an in nite series:

d(t) = P 0 (t) + P 0 (t) * P 0 (t) + P 0 (t) * P 0 (t) * P 0 (t) + • • • (1.11)
where * represent the convolution product and P 0 (t) the primary impulsive response of the Earth. For the 2D and 3D case, a summation over the re ection point at the surface has to be added. More generally, the multiples contained in a data set can be generated by convolving the total data set with the impulse primary response [START_REF] Verschuur | Adaptive Surface-Related Multiple Elimination[END_REF][START_REF] Weglein | An Inverse-scattering Series Method for Attenuating Multiples in Seismic Re ection Data[END_REF][START_REF] Dragoset | A Perspective on 3D Surface-Related Multiple Elimination[END_REF])

M(x s , x r , t) = x k P 0 (x k , x r , t) * d(x s , x k , t), (1.12)
where x k is the re ection point at the surface.

The interaction of internal multiples with the subsurface can be described by the Lippmann series [START_REF] Lippmann | Rearrangement Collisions[END_REF][START_REF] Ten Kroode | Prediction of Internal Multiples[END_REF], originally used in quantum mechanics. The Born approximation used to model primaries is actually the rst-order approximation of the series. Alternatively the Bremmer series can be considered [START_REF] Bremmer | Approximation as the First Term of a Geometric-Optical Series[END_REF][START_REF] De Hoop | Generalization of the Bremmer Coupling Series[END_REF]. Berkhout (2014a) describes a Full Wave eld Modelling (FWMod) strategy which takes as input a re ectivity model and one-way propagation operators from one depth-level to the following based on the velocity model. The modelling is performed recursively: in the rst roundtrip, the downgoing wave eld emitted by the source and the primary re ections are computed. The latter interact with the re ectivity model to generate both internal and surface rst-order multiples in the second roundtrip. The process is repeated for higher-order multiples and stopped when their amplitude is too weak.

Removal of multiples

Multiples are sometimes easy to recognise in the data, for example long-period multiple appear as a repetition of the primaries pattern. The dip of primaries and multiples events arriving at the same time in the data can also be inconsistent. In more complicated cases, multiples may easily be mistaken for primaries. Moreover, usual migration algorithms are designed to account for primary re ections only. Hence multiples falsely interpreted as primaries can lead to spurious events in the nal migrated image and incorrect interpretation of the geology ( gure 1.18). Guitton, 2005).

Numerous strategies to remove multiple re ections from seismic data have been designed. We review brie y the main families here and explain why multiple attenuation remains a challenging issue. The reader is referred to [START_REF] Verschuur | Seismic Multiple Removal Techniques: Past, Present and Future[END_REF] for an extensive review.

Radon transform

The rst family of multiple removal takes advantage of the fact that a multiple and a primary arriving at the same time in the data have not "seen" the same velocities in the Earth. With the hypothesis of a velocity increasing with depth, a multiple travels in shallower and slower layers of the subsurface. Then primaries and multiples do not exhibit the same move-out on Common Mid Point (CMP) gathers. Using a NMO-correction with the correct velocity, primaries are attened but multiples are not. A strategy using these move-out discrepancies transform the CMP gathers to a space where primaries and multiples are easily separable. Multiples are muted and the inverse transform is performed to output CMP gathers without primaries. The double Fourier transform mapping CMP gathers to the f -k domain [START_REF] Ryu | Decomposition (DECOM) Approach Applied to Wave Field Analysis with Seismic Re ection Records[END_REF] is an example of such a transformation, but the most popular choice is the parabolic [START_REF] Hampson | Inverse Velocity Stacking for Multiple Elimination[END_REF][START_REF] Kabir | Toward True Amplitude Multiple Removal[END_REF] or hyperbolic [START_REF] Foster | Suppression of Multiple Re ections Using the Radon Transform[END_REF] Radon transform. The method assumes that primaries and multiples in CMP gathers (with or without NMO-correction) can be described by di erent parabolic or hyperbolic functions. Then in the Radon domain, they should appear as focused events. After muting of the multiples, the primaries are reconstructed with the inverse Radon transform. In practice, artefacts appear in the Radon domain and primaries and multiples are not well separated, leading to inaccurate multiple attenuation. A remedy consists of rede ning this strategy as an inversion aiming at nding the model in the Radon domain that best ts the original data after inverse Radon transform. A sparsity constraint in the Radon domain is added to ensure primaries and multiples are easily separable [START_REF] Sacchi | High-resolution Velocity Gathers and O set Space Reconstruction[END_REF][START_REF] Trad | Latest Views of the Sparse Radon Transform[END_REF]. However the method still has di culties in case of complex geology when primaries and multiples do not exhibit enough move-out di erence or when they cannot be described by parabolic or hyperbolic functions. To better account for the propagation e ect in complex media, the same analysis can be transposed to the image-domain where the move-out discrepancies are analysed in CIGs function of the surface o set [START_REF] Duquet | Filtering Coherent Noise during Prestack Depth Migration[END_REF] or the scattering angle [START_REF] Sava | Multiple Attenuation in the Image Space[END_REF]. The velocity model is assumed to be relatively accurate, so that primaries are at in the angle-domain CIGs, contrary to multiples. Multiples are then easily recognisable and a similar strategy using the Radon transform is used to discriminate between at and curved events. As in the data-domain the possible overlap of primaries and multiples at zero-angle may harm the attenuation of multiples. The main hypothesis in this approach is the knowledge of a reliable velocity model.

Adaptive subtraction

The second family of multiple removal techniques is a two-step procedure. First the data set is used to compute a prediction of the multiples following one of the method presented in section 1.4.2. Then the prediction is subtracted from the data to yield the primary estimation.

This second step is di cult because the prediction of multiples is in practice never perfect. In wave eld extrapolation methods, inaccurate prediction may be caused by an incorrect model of the layer in which the multiple re ection is simulated. Multiples predicted by convolution of the data with the primary response contain the source wavelet twice. Moreover, predictions based on a 2D-model assumption do not take into account the 3D-propagation e ects. Therefore there may be inaccuracies in the phase, amplitude, and wavelet of predicted events [START_REF] Abma | Comparisons of Adaptive Subtraction Methods for Multiple Attenuation[END_REF] and simple subtraction of the prediction to the original data does not yield a good estimation of the primaries. The prediction has to be accommodated to the recorded multiples in a process called adaptive subtraction, which consists of applying a lter to the multiples before subtraction. The lter is determined by minimising the energy of the di erences between the original data and the ltered multiples in a least-squares sense. If primaries and multiples are overlapping in the data domain, minimising the energy can result in distortion of the primaries and/or residual multiple in the nal result [START_REF] Nekut | Minimum Energy Adaptive Subtraction in Surface-Related Multiple Attenuation[END_REF]. Alternatives consider replacing the least-squares criterion by the 1 -norm [START_REF] Guitton | Adaptive Subtraction of Multiples Using the L1-Norm[END_REF], among other possibilities [START_REF] Batany | Adaptive Multiple Subtraction: Uni cation and Comparison of Matching Filters Based on the q-Norm and Statistical Independence[END_REF]. The subtraction can also be performed in a domain where primaries and multiples are less likely to interfere such as the curvelet domain, Radon domain, or frequency domain [START_REF] Sacchi | High-resolution Velocity Gathers and O set Space Reconstruction[END_REF][START_REF] Donno | Curvelet-Based Multiple Prediction[END_REF][START_REF] Batany | Signal Separation in Convolutive Mixtures: Contributions to Blind Separation of Sparse Sources and Adaptive Subtraction of Seismic Multiples[END_REF].

A very-well known strategy based on multiple prediction and adaptive subtraction is the Surface-Related Multiple Elimination (SRME) method [START_REF] Verschuur | Adaptive Surface-Related Multiple Elimination[END_REF]. It is based on the observation that a surface-multiple can be decomposed in several primary re ections connected by a re ection point at the surface. If the source and receiver coverage of the acquisition is dense enough, these primaries are contained in the data and can be used to predict the multiples. SRME is implemented as an iterative method aiming at improving an initial estimation of the primaries [START_REF] Berkhout | Estimation of Multiple Scattering by Iterative Inversion, Part I: Theoretical Considerations[END_REF]. This estimation is convolved with the data to produce a prediction of the multiples. Then an adaptive subtraction is performed to remove the estimated multiples from the total data, yielding an improved estimation of the primaries. The rst estimate of the primaries can be the output of another multiple elimination technique or simply the complete data. This process converges very fast in practice and only a few iterations are needed [START_REF] Berkhout | Estimation of Multiple Scattering by Iterative Inversion, Part I: Theoretical Considerations[END_REF]. One major advantage of this method is that it is fully data-driven and requires no model of the subsurface. Conversely, the limitations of the method are related to the data acquisition. A requirement is that all the primaries composing the multiples should be recorded in the data. This may not be the case in typical marine acquisition, where the rst o sets near the source are missing for practical reasons. This represents also an issue for the extension to 3D marine acquisition where the direction perpendicular to the streamers is sampled with only a few lines. In practice, these limitations require a method to reconstruct the missing primaries, for example by interpolation of the data [START_REF] Kabir | Restoration of Missing O sets by Parabolic Radon Transform[END_REF]. Estimation of Primaries by Sparse Inversion (EPSI) (van Groenestijn and Verschuur, 2009) addresses the issue of inaccurate multiple prediction issue by recasting the prediction and the adaptive subtraction into a single inversion procedure. An objective function compares observed data to predicted primaries and predicted multiples. The corresponding iterative process should update the estimation of the primaries as well as the source wavelet. EPSI provides enhancements compared to SRME but is computationally more expensive.

Note that SRME is designed to deal with surface-related multiples only. A direct adaptation to internal multiples would consist of wave eld extrapolation to calculate the data that would be recorded at ctive sources and receivers located on the boundary generating the internal multiples. The process should be repeated for each layer and has the disadvantage of requiring an accurate velocity model to process the wave eld extrapolation. [START_REF] Jakubowicz | Wave Equation Prediction and Removal of Interbed Multiples[END_REF] proposes an alternative formulation in which the redatuming uses the primary re ections contained in the data, discarding the need for a subsurface model. Alternatively an inverse scattering series can be used to predict all internal multiples in a fully data-driven manner [START_REF] Weglein | An Inverse-scattering Series Method for Attenuating Multiples in Seismic Re ection Data[END_REF][START_REF] Ten Kroode | Prediction of Internal Multiples[END_REF] 1. 4

.4. Using multiples as valuable information

A lot of e ort has been put to develop multiple removal techniques because migration and velocity analysis techniques were designed for primary re ections only. However in the last decade, a di erent approach considering multiples as signal rather than noise has gained popularity. Multiples have travelled at least twice through the subsurface and therefore contain a lot of valuable information that may not be present in primary re ections. For example they may travel in area of the subsurface not illuminated by primaries like beneath salt domes ( gure 1.19). Moreover a surface multiple may originate from a virtual source not present in the survey, for example when short-o sets are missing in the acquisition, thus providing illumination with smaller angle re ections, di erent wavenumbers for imaging and better vertical resolution than primaries ( gure 1.20). In particular the cross line of a 3D marine acquisition is coarsely sampled and the so-called acquisition footprint is still visible in the data. [START_REF] Long | Mitigation of the 3D Cross-Line Acquisition Footprint Using Separated Wave eld Imaging of Dual-Sensor Streamer Seismic[END_REF] show that this artefacts can be mitigated with the use of multiple re ections. A correct use of multiple re ections could then possibly relax the need of dense source and receiver coverage at the surface. The sensitivity of multiples to the velocity model has been studied by some authors to detect small velocity changes in time-lapse experiments [START_REF] Snieder | Elastic Coda Wave Interferometry, a New Tool for the Instrumented Oil eld[END_REF][START_REF] Verschuur | Using Primaries and Multiples in Time-Lapse Imaging and Velocity Estimation[END_REF].

s r 0 r 1 r 2 • • • r n Figure 1.20.
-Illumination of a at re ector by primary re ections (blue, solid) and rst-order surface-related multiples (red, dotted). Multiples allow a wider coverage of the re ector with smaller scattering angles.

Imaging with multiples

A simple way of using multiple re ections consists of transforming them into pseudo-primaries which can be imaged with classic migration algorithms. The transform can be achieved with seismic interferometry [START_REF] Schuster | Interferometric/Daylight Seismic Imaging[END_REF], which states that the response recorded at a point B to a ctive source located in A can be obtained by cross-correlating the passive measurements made in A and B. When A and B are two receivers located at the surface, the cross-correlation of the traces recorded in A and B results in the primary re ection linking the two points [START_REF] Sheng | Migrating Multiples and Primaries in CDP Data by Crosscorrelogram Migration[END_REF][START_REF] Shan | Source-Receiver Migration of Multiple Re ections[END_REF][START_REF] Schuster | Interferometric/Daylight Seismic Imaging[END_REF][START_REF] Jiang | Migration of Multiples[END_REF][START_REF] Jiang | Migration Methods for Imaging Di erent-Order Multiples[END_REF]. Alternatively [START_REF] Berkhout | Combining Full Wave eld Migration and Full Waveform Inversion, a Glance into the Future of Seismic Imaging[END_REF]Verschuur (2003, 2006) deconvolve the data with an estimation of the primaries (obtained with SRME for example) to transform rst-order surface multiple into pseudo-primaries, second-order surface multiple into pseudo-rst-order surface multiple, and so on... Then a new application of the SRME method isolates the pseudo-primaries from higher-order multiples and a classic migration algorithm can be applied.

Conversely the usual migration algorithms can be adapted to multiple re ections. A rst attempt was made by [START_REF] Reiter | Imaging with Deep-Water Multiples[END_REF] who used Kirchho migration to separately image primary re ections as well as ghost re ection against the surface in OBC data. Their approach assumes that multiples and primaries are separated in time in the data and is thus restricted to deep-water acquisition. More recently this constraint was released by the development of sensors able to separate up and downgoing energy, thus discriminating between upgoing primaries and downgoing receiver ghosts [START_REF] Muijs | Prestack Depth Migration of Primary and Surface-Related Multiple Re ections: Part I -Imaging[END_REF][START_REF] Whitmore | Imaging of Primaries and Multiples Using a Dual-Sensor Towed Streamer[END_REF].

The usual imaging condition for primary re ection can be transposed to multiple re ections. It still consists of correlating a source and a receiver wave eld, except that the source wave eld is the forward propagation of the entire data set and the receiver wave eld the backward propagation of the same data set without primary re ections [START_REF] Berkhout | Multiple Technology: Part 2, Migration of Multiple Re ections[END_REF][START_REF] Guitton | Shot-Pro le Migration of Multiple Re ections[END_REF]Liu et al., 2014a). Compared to the usual primary imaging condition, the source wave eld does not originate at a single point but is an areal shot with virtual sources located at every receiver position. An advantage of this method is that there is no need to estimate the shape of the source wavelet as the new pseudo-source is directly the data recording. With this methodology, two images can be constructed, one from the migration of primaries, and one from the migration of multiples.

However, a correct separation of the primaries and the multiples is required and the nal image is contaminated with cross-talk artefacts ( gure 1.21). These spurious events are caused by downgoing energy interfering with an unrelated upgoing energy, for example a primary with a second-order surface multiple. To avoid these artefacts, each order of multiple should be isolated to produce a separate migrated image. As a remedy to the cross-talk artefacts, the use of the deconvolution imaging condition instead of the cross-correlation has been studied [START_REF] Muijs | Prestack Depth Migration of Primary and Surface-Related Multiple Re ections: Part I -Imaging[END_REF][START_REF] Whitmore | Imaging of Primaries and Multiples Using a Dual-Sensor Towed Streamer[END_REF], however this method has limited success in complex media [START_REF] Poole | Deconvolution Imaging Conditions and Cross-Talk Suppression[END_REF][START_REF] Tu | Limitations of the Deconvolutional Imaging Condition for Two-Way Propagators[END_REF]. More recent studies use a least-squares inversion approach based on tting the data reconstructed with the estimated re ectivity model and observed data [START_REF] Brown | Least-Squares Joint Imaging of Multiples and Primaries[END_REF][START_REF] Verschuur | Estimation of Multiple Scattering by Iterative Inversion, Part II: Practical Aspects and Examples[END_REF][START_REF] Wong | Imaging with Multiples Using Least-Squares Reverse Time Migration[END_REF][START_REF] Zhang | Least-Squares Reverse Time Migration of Multiples[END_REF][START_REF] Tu | Fast Imaging with Surface-Related Multiples by Sparse Inversion[END_REF]. When modelling data with re ectivity image contaminated with cross-talk artefact, extra-re ections will appear in synthetic data. These will be back-projected in the re ectivity update to attenuate the cross-talk artefacts.

If internal multiples are properly modelled, they can be included in the inversion procedure. This is the approach of the Full Wave eld Migration (FWM) technique [START_REF] Berkhout | Combining Full Wave eld Migration and Full Waveform Inversion, a Glance into the Future of Seismic Imaging[END_REF](Berkhout, , 2014b;;[START_REF] Soni | Full-Wave eld Migration of Vertical Seismic Pro ling Data: Using All Multiples to Extend the Illumination Area: Full-Wave eld Migration of Vertical Seismic Pro ling Data[END_REF] where the modelling is performed with the Full Wave eld Modelling procedure presented in section 1.4.2. The amplitude di erences between modelled and observed data is converted into re ectivity update.

More recently, new imaging techniques have been derived from the resolution of the Marchenko equation [START_REF] Wapenaar | Marchenko Imaging[END_REF]. As in reverse-time migration, Marchenko imaging consists of correlating a downgoing and an upgoing wave eld at each point of the subsurface. However in reverse-time migration, the downgoing wave eld is approximated by direct propagation in a non-re ective media of the source wavelet to the image point, and the upgoing wave eld is obtained in the same way by backpropagation of the residuals. In Marchenko imaging, all the internal multiples re ections encountered by the source and receiver wave eld are accounted for, so that the wave elds correlated at each subsurface location correctly include internal multiples [START_REF] Behura | Autofocus Imaging: Image Reconstruction Based on Inverse Scattering Theory[END_REF]. As a consequence, Marchenko imaging does not generate cross-talk.

These wave elds are obtained from surface recordings and an estimation of the direct arrival recorded at each subsurface location by iteratively solving the Marchenko equation [START_REF] Rose | Single-Sided' Autofocusing of Sound in Layered Materials[END_REF]Wapenaar et al., 2011). Note that a velocity model is needed for the direct arrival estimation. [START_REF] Singh | Marchenko Imaging: Imaging with Primaries, Internal Multiples, and Free-Surface Multiples[END_REF] extend this procedure to use simultaneously primaries, internal multiples and surface multiples.

1.5. Motivations and thesis outline 1.5.1. Motivations: towards a more robust MVA and extension to multiple reflections

Real seismic data contain both primaries and multiple re ections. Much e ort has been devoted to the removal of multiple re ections to provide migration algorithms with data containing only single-scattered events. As a consequence, Migration Velocity Analysis techniques have been designed for single-scattered events only. Recent developments showed that transmitted waves could be incorporated in MVA [START_REF] Shen | Subsurface Focusing Measurement of Diving Waves and Its Application to Re ection Tomography[END_REF][START_REF] Lameloise | Quantitative Migration for a More Robust Migration Velocity Analysis[END_REF]. Extending MVA to multiple re ections would have two advantages. First this would remove the di cult pre-processing stage of multiple attenuation, second the information contained in multiple re ections may be used to update the background velocity model in areas not illuminated by primaries. So far multiples have been considered as noise in MVA techniques [START_REF] Mulder | Automatic Velocity Analysis by Di erential Semblance Optimization[END_REF][START_REF] Li | Interval Velocity Estimation via NMO-Based Di erential Semblance[END_REF], and only a few proposals have been made to include them as valuable signal. van Leeuwen and Mulder (2008b) study the behaviour of multiples in the data-domain correlation methodology proposed in van Leeuwen and Mulder (2008a). They propose to update alternatively the re ectivity and the velocity model to both maximise the correlation in the data domain and minimise the mis t between observed and calculated data. Another strategy in the data domain is Joint Migration Inversion (JMI) [START_REF] Staal | Velocity Estimation Using Internal Multiples[END_REF]Berkhout, 2014c;[START_REF] Staal | Combined Imaging and Velocity Estimation by Joint Migration Inversion[END_REF]. It is an extension of the Full Wave eld Migration technique presented in section 1.4.4 which includes internal and surface multiples in the usual data mis t objective function. The amplitude of data residuals is used to update the re ectivity model and the phase is used to update the velocity model. Both models are updated simultaneously at each iteration. Note that this approach does not use an extended-re ectivity. To our knowledge, only two proposals have been made to include multiples in usual image-domain methods. [START_REF] Nasyrov | Velocity Analysis for VSP Data Using Multiples[END_REF][START_REF] Nasyrov | Velocity Analysis for VSP Data Using Multiples[END_REF] propose an original strategy in which primaries and rst-order surface-related multiples are migrated separately. They consider an objective function measuring the similarity of the two images, so that the discrepancies between the image obtained with primaries and the image obtained with multiples are used to update the velocity model. Alternatively, the sensitivity of the Marchenko imaging procedure to the background velocity has been very recently investigated by [START_REF] Díaz | Extended Imaging, Deconvolution, and Two-Way Wave elds: A Comparison[END_REF] who compute extended re ectivity images in subsurface-o set and angle domain with internal and surface multiples. The dependence of the Marchenko wave elds to the background velocity is not as explicit as in the case of RTM, thus a method for the derivation of a velocity update should still be investigated.

Even in the case of primaries only, MVA still faces di culties as discussed in section 1.3. Therefore before introducing multiples, a robust method to deal with primary re ection has to be determined. Lameloise et al. (2014) have shown that the issue of migration artefacts can be mitigated by introducing weights in the migration procedure. As a consequence, the gradient of the DSO objective-function is greatly improved. However their approach rely on ray theory and does not extend easily to multiple re ections. The extension to wave-equation based operators presented in chapter 4 is inspired from ideas by ten Kroode (2012) and [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF] who have developed a pseudo-inverse of the extended Born modelling operator. In chapter 4, we propose a slightly modi ed version and more importantly a coupling with velocity analysis (Hou and Symes, 2016b).

In this study I rst investigate a modi cation of the standard MVA procedure consisting of performing the migration step iteratively to determine a better solution of the migration inverse problem than the one provided by standard migration. The method is not limited to ray theory and can handle multiple re ections. The analysis is restricted to rst-order surface-related multiples modelled with a second-order Born approximation. An extended re ectivity section is constructed to minimise the data mis t between calculated and observed data including both primaries and rst-order surface-multiples. At the convergence, the iterative procedure is expected to yield CIGs free of migration and cross-talk artefacts, thus allowing to use standard MVA, here subsurface-o set DSO.

This results in a nested optimisation procedure ( gure 1.22). In the inner loop, the extendedre ectivity is determined as a minimisation of the data mis t, while in the outer loop the velocity model is updated to minimise defocused energy in the optimal re ectivity. There are two main di erences with the JMI approach. First here an extended-re ectivity is considered. This allows to capture all the information contained in the data residual, even in the case of an inaccurate initial velocity model. Second the update of the velocity is based on defocusing in the extended-domain and not on data-mis t. Besides, the JMI approach aims at recovering both the re ectivity and the velocity which are updated simultaneously at each iteration whereas here a nested optimisation approach is studied, whose main objective is the determination of a correct background velocity model.

In the synthetic examples, we will rst test our method on primaries only data, and then on data containing both primaries and rst-order surface multiples. We will focus on the rst gradient of the outer loop. The objective is that the result obtained with primaries and multiples in observed data should be close to the one that would be obtained with observed data free of multiples, meaning that multiples are correctly interpreted by the iterative migration algorithm and do not hinder the velocity analysis procedure. To reach this objective, a number of essential elements have to be developed, successively detailed in chapters 2 to 5. Some of them are also useful for a stable MVA in the case of primaries only: in particular, we pay attention at the issue pointed out by [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF] regarding the stability of the MVA gradient computed with a xed background velocity model and successive values of the re ectivity model. Each chapter concludes with a limitation, justifying further developments. The nal formulation is obtained in chapter 5.

Thesis outline

The thesis is organised as follows:

• Chapter 2 is dedicated to the iterative migration procedure, corresponding to the inner

inner loop iterative migration chapters 2 & 4 Initial c 0 δc = 0 Update c 0 Model P[c 0 , δc] Update δc Evaluate J 0 [c 0 , δc] Compute ∂ J 0 ∂ δc [c 0 , δc]
Stopping criterion for J 0 satis ed?

Compute ∂ J 1 ∂ c 0 [c 0 ] Evaluate J 1 [c 0 ]
Stopping criterion for J 1 satis ed? -Sketch showing the algorithm of MVA with iterative migration. In the inner loop, we optimise the re ectivity δc for a given c 0 in order to minimise the objective function J 0 measuring the data mis t P[c 0 , δc] -P obs . In the outer loop, we update the velocity model to minimise the objective function J 1 measuring defocused energy in the CIGs of δc.

loop of gure 1.22. The modelling of multiples with a second-order Born approximation is introduced and compared to a nite di erence modelling with a free-surface condition. An e cient derivation of the gradient of the migration objective function using the adjoint state method is presented. We show that even in the presence of multiples, iterative migration yields clean CIGs free of migration and cross-talk artefacts and therefore improve the properties of the associated MVA objective function.

• In chapter 3, two methods for computing an approximate gradient of the MVA objective function after iterative migration are investigated. In the rst case, we compute an exact gradient of an approximate objective function. In a second approach, we suppose that iterative migration has converged and compute the approximate gradient of the ideal MVA objective function. This method is preferred as it has a simpler implementation. It consists of solving a so-called "adjoint problem", which is a linear system very similar to iterative migration. We highlight the importance of introducing regularisation during migration to obtain coherent velocity updates. We discuss also here instabilities observed in the gradient and a rst possibility to attenuate them. The second possibility is developed in chapter 5. The approach remains expensive, as two iterative schemes have to be solved.

Chapter 4 aims at obtaining a faster convergence.

• In chapter 4, an approximate inverse of the extended-Born modelling operator is presented. This direct inversion formula is designed for primary re ections only and is free of ray quantities. It is a variant to the method introduced by Hou and Symes (2015). The new aspect is the coupling with MVA. The weights are introduced as a preconditioner to accelerate the resolution of iterative migration and of the adjoint problem, even in the presence of multiples. In the primary-only case, we compare the results of direct and iterative inversions: we show that although they provide very close re ectivity images, the associated gradients are not similar. We also underline the di culty to solve the adjoint problem compared to iterative migration [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF]. This justi es additional developments discussed in chapter 5.

• In chapter 5, the behaviour of iterative migration velocity analysis is studied on a pure 1D primary-only case to better understand the issues presented in chapter 4. A major advantage of the 1D analysis is the possibility to have explicit operators and to run a large number of iterations. We propose a simple modi cation of the MVA objective function consisting of applying a " lter" to the nal CIGs before penalising defocused energy.

The lter is function of the background velocity model. In the primaries only case, this modi ed approach improves the convergence of the adjoint problem and yields gradients close to those obtained after direct inversion, both in 1D and 2D. In the case of multiples, we show that introducing classical regularisation on the migration during the migration step is still essential.

• In chapter 6, the approach is applied on 2D synthetic data sets. The robustness of the method is discussed with observed data computed with a di erent modelling engine from the one used for the inversion, for example with the introduction of a variable density, or when a di erent source wavelet is used for inversion and for observed data.

• In chapter 7, the main conclusions of the thesis are summarised and we discuss remaining issues in the perspective of real data applications.

Realisations and contributions

Starting from an existing nite-di erence acoustic forward propagation code, I have built an iterative migration Fortran90 code computing extended CIGs by iterative minimisation of the data mis t with non-linear optimisation techniques. I implemented the two methods for the computation of the background velocity update presented in chapter 3, as well as the inversion formula described in [START_REF] Cocher | Seismic Iterative Migration Velocity Analysis: Two Strategies to Update the Velocity Model[END_REF] used here as a preconditioner. The implementation of adjoint operators has been veri ed with the dot-product test (Claerbout, 2014, p. 28) and gradient derived with the adjoint-state method [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF] have been compared with nite-di erence computations. My main contributions are

• the construction of CIGs in the subsurface-o set domain with consistent inclusion of rst-order surface-related multiples;

• the extensive study of the construction of the gradient of the MVA objective function after iterative migration, as detailed in chapters 3 and 5;

• the comparison of direct and iterative inversion schemes for the construction of background velocity updates and the proposition to lter nal CIGs in the iterative case to stabilise the method.

These developments allow a better understanding of the behaviour of MVA after direct or iterative inversions. We have proposed strategies for a more robust scheme, applicable in the case of surface-related multiples but also of primaries only.

Part of the results have been presented in La minimisation de la fonction coût associée à la migration itérative est réalisée avec des méthodes d'optimisation locale [START_REF] Nocedal | Numerical Optimization[END_REF] utilisant le gradient de la fonction coût pour remettre à jour le modèle de ré ectivité. Le calcul du gradient, obtenu par la méthode de l'état adjoint [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF], est détaillé. Dans le cas linéaire de primaires seuls, la minimisation de la fonction objective est équivalente à la résolution d'un problème linéaire. Dans les exemples numériques, ce système est résolu itérativement avec l'algorithme du gradient conjugué linéaire. Lorsque des multiples sont considérés, des techniques d'optimisation non linéaire doivent être employées, ici le gradient conjugué non linéaire utilisant la formule de Polak-Ribière et la recherche linéaire de Moré et Thuente (1994).

La migration itérative est illustrée dans le cas simple d'un unique ré ecteur horizontal et d'un macro-modèle homogène. Dans un premier temps, les données observées (synthétiques) et calculées ne contiennent que des ré exions primaires. Un macro-modèle homogène mais plus lent que le modèle exact, est utilisé. À la première itération, les artefacts de migration décrits par [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF] et Lameloise et al. (2014) apparaissent dans les CIGs et le minimum de la fonction coût évaluée pour des macro-modèles homogènes n'est pas atteint pour la bonne valeur de vitesse. En poursuivant les itérations, les artefacts de migration sont progressivement atténués et le minimum décalé vers la bonne valeur de modèle de vitesse. Le test est répété dans le cas où les multiples de surface du premier ordre sont ajoutés aux données observées. À la première itération, les multiples interprétés comme des primaires conduisent à de nouveaux artefacts dans les CIGs à une profondeur double de la position du vrai ré ecteur. À l'itération suivante, ces artefacts génèrent dans les données calculées de nouveaux évènements (primaires et multiples) absents des données observées, qui conduisent à leur tour à une remise à jour du modèle de ré ectivité. Ainsi les artefacts dus aux multiples, tout comme les artefacts de migration sont progressivement atténués.

Introduction

This chapter introduces an iterative migration scheme [START_REF] Nemeth | Least-squares Migration of Incomplete Re ection Data[END_REF][START_REF] Østmo | Finite-Di erence Iterative Migration By Linearized Waveform Inversion In the Frequency Domain[END_REF] with a speci c focus on surface-related multiples. A scale separation between a smooth background velocity model and a rapidly-varying re ectivity model is assumed. The purpose of migration is to determine a re ectivity model which allows to faithfully reproduce observed data, assuming a xed initial background velocity model. The latter may be inaccurate, therefore the re ectivity model space is extended with an additional parameter, the subsurface o set. A re ectivity model explaining observed data can then be obtained even in an incorrect velocity model. Here we want to determine a re ectivity model able to explain correctly both primaries and rst-order surface multiples. The evaluation of the quality of the background velocity model (MVA part) is investigated in chapter 3.

Migration is formulated here as an inverse problem which consists of minimising an objective function measuring the mis t between observed and simulated data. This corresponds to the inner loop of the scheme described in gure 1.22. In the forward problem, data corresponding to a re ectivity model are modelled with a second-order Born approximation: primaries are linear in the re ectivity, but multiples are not. The inverse problem is solved iteratively with a gradient-based method, the re ectivity model being updated with the gradient of the leastsquares objective function.

Starting from a zero model, the rst gradient does not provide a reliable solution to the inverse problem. First, multiples incorrectly interpreted as primaries result in spurious events in the re ectivity model which do not correspond to a physical re ector. Second, even in the case of primaries only, migration artefacts appear on Common Image Gathers (CIGs), de ned as panels displaying the re ectivity along the depth and the subsurface-o set for xed lateral positions (Lameloise et al., 2014;[START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF]. CIGs are later used to evaluate the quality of the background velocity model. Migration artefacts are thus an issue for the update of the background velocity model. We show that iterative migration allows to attenuate both migration artefacts and spurious events caused by cross-talk.

We begin by describing the forward problem consisting of modelling primaries and rst-order surface-related multiples with a second-order Born-approximation. The results are compared with a classical nite-di erence modelling with a free-surface condition to investigate the ability of the Born approximation to deal with high velocity contrasts, which generate strong multiples. We then show how the modelling is performed in the extended-domain with the horizontal subsurface-o set as extension parameter. The inverse problem associated to iterative migration and the corresponding objective function are presented. It is solved with non-linear gradient-based optimisation techniques and an e cient way of computing the gradient of the objective function is introduced. Finally, we show simple synthetic examples. First we consider primary re ections only to show the in uence of migration artefacts (migration smiles) in CIGs on the shape of the MVA objective function. We show how iterative migration can mitigate the problems caused by migration artefacts. The same tests are run with surface multiples and the attenuation of cross-talk artefacts as well as migration artefacts with iterations is illustrated.

Notations

We consider a surface acquisition and a 2D Earth model where a subsurface point is noted x = (x, z). We de ne

• the model space of physical model parametrised by x. For example the background velocity model c 0 (x);

• the extended model space of variables function of the subsurface point x and the subsurface o set considered strictly horizontal h = (h, 0), for example the extended re ectivity ξ(x, h);

• the observed data space obs parametrised by the source horizontal coordinate s, the receiver horizontal coordinate r and the angular frequency ω (or time t) in the frequency domain formulation. Observed data are noted P obs (s, r, ω). Although we use the time domain for the implementation, equations will be written in the frequency domain in the following to simplify calculations;

• the data space . We note P[c 0 , ξ](s, x, ω) the wave eld calculated with the estimated re ectivity model ξ. Contrary to P obs , P is known everywhere in the subsurface. We de ne the operator M : → obs which, applied to a simulated wave eld P[c 0 , ξ](s, x, ω), selects its values at receivers' locations. Hence MP can be compared to P obs during iterative migration.

Table 2.1 summarises the notations used for the di erent spaces and their size. For more details on the choice of the data and model spaces, the reader is referred to [START_REF] Blazek | A Mathematical Framework for Inverse Wave Problems in Heterogeneous Media[END_REF] and [START_REF] Liu | Extended Re ection Waveform Inversion via Di erential Semblance Optimization[END_REF]. Note that with the introduction of the subsurface o set, the extended model and the observed data space obs have the same dimension, which allows to alternate between these two spaces without losing information. and n h are the number of grid points considered in the extended model space. n t is the number of time samples. n s and n r are the number of sources and receivers considered in the acquisition. In the data space, n x can be restricted to a zone around the shot gather including at least all the receivers.

The inner product of two vectors ξ and χ of is noted

〈ξ | χ〉 = x h ξ(x, h)χ(x, h) dh dx, (2.1)
and the associated norm is ξ = 〈ξ | ξ〉 . Similarly we de ne the inner-product 〈 • | • 〉 and the norm • in the data space. The norm of the data residuals R, de ned as the di erence between observed data and the value of simulated data at receivers' positions,

R[c 0 , ξ](s, r, ω) = MP[c 0 , ξ] (s, r, ω) -P obs (s, r, ω), (2.2)
is given by

R obs = s r ω R(s, r, ω)
2 dω dr ds.

(2.3)

Finally we call F * and F -1 the adjoint and inverse operators of an operator F.

Modelling under the second-order Born approximation

We detail in this section how primaries and multiples are modelled with a second-order Born approximation in the extended domain. We rst review the usual Born approximation for primary re ections in the non-extended case.

First-order Born approximation

In the constant-density acoustic approximation, wave eld propagation obeys the following wave-equation

(iω) 2 c(x) 2 P(s, x, ω) -∆P(s, x, ω) = Ω(ω)δ(x -s), (2.4)
for a source located at s = (s, 0) with a wavelet Ω(ω). The velocity model is noted c(x).

The Born approximation is a linearisation of this wave-equation and relies on the scale separation presented in the introduction. We write c(x) = c 0 (x) + δc(x), (2.5) where c 0 and δc stands for the background velocity model and the model perturbation respectively. Accordingly we separate the data P into P = P 0 + δP. P 0 corresponds to the data modelled in the smooth background c 0 and δP is the perturbation caused by the model perturbation δc.

We suppose that the model perturbation is small with respect to the background velocity model, that is δc c 0 . Neglecting second-order term δPδc, P 0 and δP are solution of

0 P 0 (s, x, ω) = Ω(ω)δ(x -s), 0 δP(s, x, ω) = (iω) 2 ξ(x)P 0 (s, x, ω), (2.6a) (2.6b)
where we have de ned

• the wave operator 0 as

0 = (iω) 2 c 2 0 (x) -∆. (2.7)
Note that 0 is de ned with the smooth background velocity c 0 (x);

• the re ectivity ξ(x) as

ξ(x) = 2δc(x) c 3 0 (x)
.

(2.8)

The coe cient 2/c 3 0 (x) is included in the de nition of ξ(x) to further simplify calculations, although the physical re ectivity property is described by the dimensionless quantity ξ(x) = δc(x)/(2c 0 (x)). Note that the re ectivity de ned in equation (2.8) is not extended yet. The case of the extended model will be presented in section 2.3.4.

The data perturbation δP corresponds to the re ected wave due to the re ectivity ξ. In all the following, it will be noted P 1 .

The solutions of equation (2.6) involve Green's functions G 0 de ned as the solution of the wave-equation with an impulsive source 0 G 0 (s, x, ω) = δ(x -s).

(2.9)

Then one can verify that

     P 0 (s, x, ω) = G 0 (s, x, ω)Ω(ω), P 1 (s, x, ω) = Ω(ω) y (iω) 2 G 0 (s, y, ω)ξ(y)G 0 (y, x, ω) dy, (2.10a)
(2.10b) are solutions of equations (2.6a) and (2.6b), respectively.

Reflection at the free surface

To consider rst-order surface-related multiples, we need to model the re ection of upgoing primaries at the surface. The latter is known to act as a mirror and a rst guess is to model multiples by applying the primary modelling work ow to a new areal source made of primaries recorded at the surface multiplied by a constant coe cient R = -1. Numerical comparison with a nite di erences modelling and a free-surface condition suggests that an additional coe cient 2iω/c 0 should be applied as well. We explain the physical meaning of this coe cient on a pure 1D case in appendix A and assume that it is valid in the general 2D case, too.

In the following, we note P 2 the wave eld resulting from the re ection of the primaries P 1 at the surface. The associated wave-equation is 0 P 2 (s, x, ω) = M s P 1 (s, x, ω), (2.11) where operator M s : → is de ned by 2.12) and includes the additional re ection coe cient discussed in appendix A.

M s P 1 (s, x, ω) = (-1)δ(x -x surf ) 2iω c 0 (x) P 1 (s, x, ω), (
We test the accuracy of this approximation, in particular at non-zero o set, with the following numerical example. We consider a Ricker with a maximum frequency of 40 Hz as the source wavelet ( gure 2.1). The source and the receivers are buried in the subsurface assumed homogeneous with a velocity of 3000 m/s ( gure 2.2a). We rst simulate the propagation of the source with a classical second-order accurate space and time nite-di erence scheme. Absorbing boundary conditions with Perfectly Matched Layer (PML) [START_REF] Bérenger | A Perfectly Matched Layer for the Absorption of Electromagnetic Waves[END_REF]Komatitsch and Martin, 2007) are implemented on each edge of the model except on the surface where a free-surface condition is implemented with the method of images [START_REF] Levander | Fourth-order Finite-di erence P-SV Seismograms[END_REF][START_REF] Robertsson | A Numerical Free-surface Condition for Elastic/Viscoelastic Finite-di erence Modeling in the Presence of Topography[END_REF] ( gure 2.2c, left). Then we use the same code with PMLs on each edge of the model to solve equations (2.10a) and(2.11) ( gure 2.2c, middle). The di erence between the two results increases with o set ( gure 2.2c, right). The amplitudes accuracy decays with o sets ( gure 2.2b) as the re ection coe cient (-1) at the surface is set independent of the incident angle, but more importantly the traveltime remains accurate. Thus equation (2.11) stands as a good approximation to model the surface re ection.

Second-order Born approximation

First-order surface multiples P 3 are modelled with the same equation as primary re ections P 1 (equation 2.10b), except that the incident wave eld is P 2 instead of P 0 . Then, given a background velocity model c 0 (x) and a (non-extended) re ectivity model ξ(x), data modelling is performed by solving four wave-equations

         0 P 0 (s, x, ω) = Ω(ω)δ(x -s), 0 P 1 (s, x, ω) = (iω) 2 ξ(x)P 0 (s, x, ω), 0 P 2 (s, x, ω) = M s P 1 (s, x, ω) 0 P 3 (s, x, ω) = (iω) 2 ξ(x)P 2 (s, x, ω), (2.13a) (2.13b) (2.13c) (2.13d)
where M s is de ned in equation ( 2.12).

Comparison with a finite-di erence modelling and a free-surface condition

We have tested the validity of the Born approximation with respect to the free surface. In theory, the Born approximation is valid for small velocity contrasts. We want to study its behaviour in the case of high contrasts which may cause strong multiples. We compare data modelling with the second-order Born approximation described by equations (2.13) with a nite-di erence modelling and a free-surface condition on four examples. We consider a background velocity of 1500 m/s and two values for the velocity perturbation: 1600 m/s and 2500 m/s, corresponding to a limited (100/(1500 + 1600) = 0.03) and a larger (1000/(1500 + 1600) = 0.25) velocity contrast. For each value, we consider two cases • a horizontal re ective layer with homogeneous velocity below the interface located at z 0 = 200 m ( gures 2.3 and 2.4). In this case, the background velocity model used for second-order Born modelling is obtained by smoothing the exact slowness model σ(x) de ned as σ(x) = 1/c 0 (x) and the velocity perturbation is calculated as ; • a horizontal "di ractive interface", with a velocity perturbation localised at depth z 0 and a homogeneous velocity (1500 m/s) below and above the interface ( gures 2.5 and 2.6). The source and the receivers are located near the surface. Ghosts at the source and the receivers sides are created by the free-surface in the nite-di erence simulation. This is accounted for in the Born modelling by a time shift in the source wavelet and in the computed data

     Ω ghost (t) = Ω(t) -Ω t + 2 z s c 0 , P ghost (s, r, t) = P(t) -P t + 2 z r c 0 . (2.14a) (2.14b)
Once more, this approximation is valid for vertical wave propagation only.

Both in the re ection and di raction cases, multiples are accurately modelled when the velocity contrast is small. In the di raction case, the kinematics of multiples is respected even with a high velocity contrast, although the amplitudes are overestimated ( gure 2.6). In the re ection case with a higher velocity contrast , the Born approximation does not model the kinematics of primaries as accurately, and it is worse in the case of multiples ( gure 2.4). This is due to the propagation in a smooth model that modi es the velocities compared to the propagation in a blocky velocity model. Note however that the time shift between the two data sets is very similar at zero and far o set. Besides, there is not a unique choice for the de nition of the smooth model. It is obtained in these examples by smoothing the blocky slowness model, but other de nitions may lead to better results, at least at zero o set. In practice, there is no need to smooth an exact model: the MVA strategy aims actually at determining both a background velocity model and a re ectivity model allowing to reproduce observed data with a second-order Born approximation.

Eventually, we note that re ection multiples ( gures 2.3 and 2.4) appear as a scaled version of the primaries with an opposite wavelet, contrary to di raction multiples ( gures 2.5 and 2.6) which have a di erent wavelet from the primaries.

Non-linearity of multiples with respect to the reflectivity model Finally, we illustrate the linear dependence of primaries and non-linear dependence of multiples to the re ectivity model in the second-order Born approximation with a simple example ( gure 2.7). We consider two di erent re ectivity models ξ A and ξ B consisting of a single re ector located at 200 m depth for model A and 300 m for model B. A single shot is red from the middle point in the surface. The source wavelet is a Ricker with a maximum frequency of 40 Hz ( gure 2.1). We compute the response recorded by receivers at the surface, respectively P 1 [ξ A ] and P 1 [ξ B ] ( gure 2.7, rst and second columns). Then we compare the response obtained with the linear combination of models ξ C = ξ A + 2ξ B ( gure 2.7, third column) with the linear combination of the individual data sets P 1 [ξ A ] + 2P 1 [ξ B ] ( gure 2.7, fourth column). In the -Same as gure 2.5, but with a stronger velocity contrast.

case of primaries only ( gure 2.7, middle row) these data sets are similar, which con rms the linear relationship between velocity perturbation and primary re ections obtained under the rst-order Born-approximation. In the case of multiples two remaining events are visible, the multiple which re ects twice on the deeper re ector and the one which re ects once at each re ector. As expected, the relationship between data and model perturbation is thus non linear under the second-order Born approximation.

P[ξ A ] -300 0 300 o set (m) P[ξ B ] -300 0 300 o set (m) P[ξ C ] -300 0 300 o set (m) P[ξ C ] -P[ξ A ] -2P[ξ B ]

Introduction of the horizontal subsurface o set

We now extend the second-order Born modelling procedure to the extended domain. The space of physical models is extended with the subsurface o set h, considered strictly horizontal h = (h, 0). The extended re ectivity ξ(x, h) now depends on this extra parameter and lives in , and is de ned as

ξ(x, h) = 2δc(x, h) c 3 0 (x)
.

(2.15)

The modelling of primary re ections now includes a spatial delay at depth as illustrated on gure 2.8. Equation (2.10b) de ning P 1 is then transformed into

P 1 (s, x, ω) = (iω) 2 Ω(ω) y h G 0 (s, y -h, ω)ξ(y, h)G 0 (y + h, x, ω) dh dy, (2.16)
with a new integral over h. In the case of rst-order surface-related multiples, several de nitions are possible ( gure 2.9). We consider the case where a spatial delay is introduced at each re ection point ( gure 2.9a), as it is a more general case and leads to similar formulations for the de nition of P 1 and P 3 . The modelling of rst-order surface-related multiples now reads

P 3 (s, x, ω) = (iω) 2 z k G 0 (m, z -k, ω)ξ(z, k) G 0 (z + k, x, ω) dk dz m∈∂ Ω (-1) 2iω c 0 (m) P 1 (s, m, ω) dm, (2.17)
where m is a point located at the surface ∂ Ω. As before, the coe cient 2iω/c 0 (x) is introduced to properly simulate the re ection at the free-surface. The implementation of equations (2.16) and (2.17) in their current formulation is not straightforward. Applying the wave-operator 0 to this formula allows to derive a more convenient modelling scheme:

                   0 P 0 (s, x, ω) = K P 0 (s, x, ω) = Ω(ω)δ(x -s), 0 P 1 (s, x, ω) = K P 1 (s, x, ω) = h (iω) 2 P 0 (s, x -2h, ω)ξ(x -h, h) dh, 0 P 2 (s, x, ω) = K P 2 (s, x, ω) = M s P 1 (s, x, ω) 0 P 3 (s, x, ω) = K P 3 (s, x, ω) = h (iω) 2 P 2 (s, x -2h, ω)ξ(x -h, h) dh, (2.18a) (2.18b) (2.18c) (2.18d)
which involves the usual wave equation with modi ed source terms K P 1 and K P 3 that include the integral over the subsurface o set. Thus the introduction of the subsurface o set does not change the wave-equation to be solved, but only the source terms K P 1 and K P 3 . Note that there is no additional loop over h, except in the source terms. The number of wave-equations to be solved remains the same.

To simplify further expressions, we introduce the vector S ∈ which is zero everywhere in space except at the location of the seismic source: S(s, x, ω) = Ω(ω)δ(x -s). We also de ne the bilinear operator K -[u, χ] : × → for a vector u of and a vector χ of as

K -[u, χ](s, x, ω) = h (iω) 2 u(s, x -2h, ω)χ(x -h, h) dh.
( 2.19) This allows the modelling scheme to be written in a more compact way, which will be used in following calculations,

         0 P 0 (s, x, ω) = S(s, x, ω), 0 P 1 (s, x, ω) = K -[P 0 , ξ](s, x, ω), 0 P 2 (s, x, ω) = M s P 1 (s, x, ω), 0 P 3 (s, x, ω) = K -[P 2 , ξ](s, x, ω). (2.20a) (2.20b) (2.20c) (2.20d)
We have described how to solve the forward problem of the inner inverse problem. We now introduce the corresponding objective function and the resolution of the inverse problem, consisting of determining ξ from observed data for a given background velocity model c 0 (x).

Optimisation strategy

Iterative migration is an inverse problem posed in the extended domain . This is the inverse problem we consider in this chapter. It is solved as an unconstrained optimisation problem. In this section we present the objective function to be minimised and the optimisation strategy which will be used in numerical examples. We eventually explain how the gradient of the objective function is computed.

Definition of an objective function for iterative migration

The objective function J 0 of iterative migration reads

J 0 [c 0 , ξ] = J Migr [c 0 , ξ] + a φ φ[ξ].
(2.21)

Iterative migration aims at determining an extended re ectivity model ξ minimising J 0 . The estimated background velocity c 0 , which might be inaccurate, is kept xed during the minimisation of J 0 . The objective function J 0 is made of two contributions,

• the rst term evaluates the mis t between data computed with the re ectivity model ξ and observed data. It is expressed as

J Migr [c 0 , ξ] = 1 2 M P 1 [c 0 , ξ] + P 3 [c 0 , ξ] -P obs 2 obs (2.22)
The projection operator M : → obs selects the value of the wave elds P 1 and P 3 at the position of receivers before comparison with observed data;

• the second term is a regularisation function whose weight is controlled by the scalar a φ . In this chapter, we will consider simple regularisation functions such as the 2 norm

φ[ξ] = 1 2 ξ 2
, but other terms may easily be introduced.

Linear case of primaries only

In the numerical example section, we begin by applying iterative migration in a simple case without multiples. The rst-order Born modelling can be represented by a linear operator F : → obs . To simplify the following equations, we assume that the projection operator M is included in the de nition of F. With the 2 norm as regularisation, the objective function (2.21) now reads

J 0 [c 0 , ξ] = 1 2 F[c 0 ]ξ -P obs 2 obs + a φ φ[ξ]. (2.23)
Its minimisation is equivalent to the resolution of the linear system

F T F + a φ I ξ = F T P obs , (2.24)
which is obtained by zeroing the gradient of the objective function (2.23). An e cient method to solve this system is the linear conjugate gradient algorithm (Nocedal and Wright, 2006, p. 112).

Note that in practice an F and F T will not be expliclty computed and an equivalent formulation will be used (section 2.5.1).

Non-linear optimisation in the non-linear case of multiple reflections

With multiples, the modelling operator is not linear any more and non-linear optimisation techniques have to be used. With gradient-based methods, the re ectivity model at iteration

(n + 1) is updated from the re ectivity model at iteration (n) following ξ (n+1) = ξ (n) + α (n) d (n) , (2.25)
where d ∈ is called descent direction and the positive scalar α is called step size or step length determined by a procedure called linesearch. The process is initialised with ξ (1) = 0 and ξ is updated until a convergence criterion is satis ed, for example when the value of the objective function or the norm of its gradient goes below a given threshold. In the numeric examples shown at the end of this chapter, we set a maximum number of iterations N, so that the nal result is ξ (N+1) . d and α are determined such that the value of J 0 decreases at each iteration, that is J 0 (ξ (n+1) ) < J 0 (ξ (n) ).

The most simple choice for the descent direction is given by the opposite of the gradient of the objective function, that is d (n) = -g (n) , with g (n) the gradient of J 0 computed at ξ (n) ,

g (n) = ∂ J 0 ∂ ξ [c 0 , ξ (n) ], (2.26)
but alternative strategies providing faster convergence exist. The optimal strategy is a trade-o between the number of iteration needed to reach a satisfactory minimisation of the objective function and the numerical cost for determining d and α. We brie y review standard non-linear optimisation strategies in appendix B. The reader is referred to [START_REF] Nocedal | Numerical Optimization[END_REF] for an extensive review. In all the following numerical examples, we use the linesearch procedure of Moré and Thuente (1994), which ensures that the strong Wolfe conditions (see section B.2 and gure B.1) are satis ed. The procedure requires the evaluation of the value and the gradient of the objective function at successive trial points ξ (n) + α i d (n) , until a satisfactory value for α i is found. This may be expensive if numerous step size have to be tested. However we observed in practice that this optimal value α (n) opt is usually found in two or three trial steps. Besides as we set n) , the linesearch procedure already provides the value of J 0 (ξ (n+1) ) and g (n+1) at the next iteration. So the additional cost of a linesearch procedure satisfying the strong Wolfe conditions remains a ordable.

ξ (n+1) = ξ (n) + α (n) opt d (
We use the non-linear conjugate-gradient technique to determine the descent direction,

d (n) = -g (n) + β (n) d (n-1) , (2.27)
where the scalar β is given by the formula Polak-Ribière formula

β (n) = g (n) g (n) -g (n-1)
g (n-1) g (n-1) (2.28)

A step length satisfying the strong Wolfe conditions ensures that d is e ectively a descent direction (Nocedal and Wright, 2006, p. 122).

Derivatives of J 0

One of the key ingredients of local optimisation methods is the gradient of the objective function.

Its computation is the most computationally expensive step of the optimisation procedure and we present here an e cient way of deriving the gradient of J 0 based on the adjoint state method. We illustrate on an example the di erent contributions of the gradient to the re ectivity update. Eventually, the shape of the Hessian matrix is studied on a simple 1D-case.

Computation of the gradient of J 0

In the objective function de ned in equation (2.21), the contribution to the gradient due to the regularisation is straightforward. When the regularisation term is de ned as the 2 -norm of ξ, we have

φ[ξ] = 1 2 ξ 2 ; (2.29) ∂ φ ∂ ξ [ξ] = ξ.
(2.30)

In this section, we focus on the gradient of J Migr . It can be expressed as

∂ J Migr ∂ ξ = ∂ P 1 ∂ ξ [c 0 , ξ (n) ] + ∂ P 3 ∂ ξ (c 0 , ξ (n) ) T M T M(P 1 [c 0 , ξ (n) ] + P 3 [c 0 , ξ (n) ]) -P obs . (2.31)
The quantities ∂ P 1 /∂ ξ and ∂ P 3 /∂ ξ are called Fréchet derivatives and correspond to the partial derivatives of speci c data with respect to model parameters. Their computation is expensive, and we prefer to use a more e cient gradient computation method, the adjoint state method [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF]. This method will directly compute the product of the Fréchet derivatives with the residuals rather than explicitly deriving the two terms.

An augmented functional called Lagrangian is de ned with constraints added to the objective function. Each state variable P 0 , P 1 , P 2 and P 3 is associated to an adjoint variable λ 0 , λ 1 , λ 2 and λ 3 . The λ i are vectors of and are used as Lagrange multipliers of the state equations (2.20)

J Migr (ξ (n) , P 0 , P 1 , P 2 , P 3 , λ 0 , λ 1 , λ 2 , λ 3 ) = J Migr (ξ) -λ 0 0 P 0 -S -λ 1 0 P 1 -K -[P 0 , ξ] -λ 2 0 P 2 -M s P 1 -λ 3 0 P 3 -K -[P 2 , ξ] , (2.32)
where we have dropped the dependencies to (s, x, ω) for simplicity. The derivative of the Lagrangian with respect to ξ can be written .33) where the overlined variable J Migr refers to the extended Lagrangian (2.32). We decide to set the derivatives ∂ J Migr /∂ λ i to zero, which ensures that the P i are solutions of the state equations (2.20). Similarly we set the derivatives ∂ J Migr /∂ P i to zero. This way, the computation of the Fréchet derivatives is not required and we obtain the adjoint equations de ning the adjoint variables

dJ Migr dξ = ∂ J Migr ∂ ξ + 3 i=0 ∂ P i ∂ ξ ∂ J Migr ∂ P i + 3 i=0 ∂ λ i ∂ ξ ∂ J Migr ∂ λ i , ( 2
λ i          * 0 λ 3 = M T M(P 1 + P 3 ) -P obs * 0 λ 2 = K + [λ 3 , ξ] * 0 λ 1 = M T M(P 1 + P 3 ) -P obs + M * s λ 2 * 0 λ 0 = K + [λ 1 , ξ],
(2.34a)

(2.34b) (2.34c) (2.34d)
where operator K + [u, χ] : × → is de ned for u ∈ and χ ∈ by

K + [u, χ](s, x, ω) = h (iω) 2 u(s, x + 2h, ω)χ(x + h, h) dh, (2.35)
and can is an adjoint operator of K -(equation 2.19):

u K -[v, χ] = K + [u, χ] v (2.36) for (u, v) ∈ and χ ∈ .
Eventually the gradient is given by the partial derivative of the Lagrangian with respect to ξ:

∂ J Migr ∂ ξ = dJ Migr dξ = ∂ J Migr ∂ ξ = Q[P 0 , λ 1 ] + Q[P 2 , λ 3 ] (2.37) where operator Q[u, v] : × → is an extended cross-correlation de ned for (u, v) ∈ by Q[u, v](x, h) = s ω (iω) 2 u * (s, x -h, ω)v(s, x + h, ω) dω ds. (2.38) It is related to K -and K + by Q[u, v] χ = K -[u, χ] v = u K + [v, χ] . (2.39)
The adjoint variables are solutions of equations similar to the state variables. They involve the adjoint of the wave operator, corresponding to a back-propagation in the time domain. Note that the computation of λ 0 is not necessary and the computation of the gradient is less then twice as costly as the computation of the value of the objective function (table 2.2).

primaries only primaries and multiples evaluate J 0 2 (P 0 , P 1 ) 4 (P 0 , P 1 , P 2 , P 3 ) evaluate J 0 and ∂ J 0 ∂ ξ 3 (P 0 , P 1 , λ 1 ) 7 (P 0 , P 1 , P 2 , P 3 , λ 3 , λ 2 , λ 1 ) Table 2.2. -Number of wave-equations to be solved to evaluate the value and the gradient of the objective function at a point ξ ∈ . The number indicated here should be multiplied by the number of sources position considered in the acquisition.

We now explain the physical meaning of the di erent parts of the gradient.

Interpretation of the gradient

Noting that the source terms of the equations de ning λ 1 and λ 3 (equations 2.34a and 2.34c) have a common term, we can split the adjoint variable λ 1 into two contributions

λ 1 = λ 3 + λ 4 , (2.40) with λ 4 solution of * 0 λ 4 = M * s λ 2 .
(2.41) so that the gradient of J Migr has three contributions

∂ J Migr ∂ ξ = Q[P 0 , λ 3 ] g 1 + Q[P 0 , λ 4 ] g 2 + Q[P 2 , λ 3 ] g 3 . (2.42)
To illustrate the contributions of each term to the gradient, we consider a simple example. A single horizontal re ector is located at 600 m in a homogeneous velocity model (3000 m/s) ( gure 2.10a). In the trial re ectivity ξ t , a re ector is located at depth 400 m ( gure 2.10b). The modelling of both observed data ( gure 2.10c) and calculated data is performed with the correct velocity model. Four events are visible in the residuals ( gure 2.10d) de ned as the di erence between data calculated with the trial re ectivity and observed data. Each part of the gradient convert each event to the model space.

• the rst contribution ( gure 2.11a) noted g 1 in equation ( 2.42) interprets the four events in the residuals as primaries, resulting in four events in the gradient. Among them, only one corresponds to the true re ector located at 600 m depth. The three other events are cross-talk artefacts.

• the second and third contributions ( gures 2.11b and 2.11c) look very similar. They both interpret the four events in the residuals as multiples. The second term g 2 images a multiple on the source side, while the third term g 3 images a multiple on the receiver side.

They can be understood as follows: among the two re ections de ning the multiples, one is produced by the trial re ectivity ξ t ( gure 2.10b) and the gradient tries to position the second re ector which allows to reconstruct the events in the data. In our example, the interpretation of the two primaries as multiples results in two other cross-talk artefacts.

Finally the interpretation of multiples as multiples leads to two other events, but none of them is positioned at the depth of the re ector, because they are deduced from assumption that a rst re ection occurred in ξ t which is an incorrect model.

This example actually considers an extreme case, where the second and third part of the gradient do not bring any useful information and only mislead the re ectivity update. In practice, the inversion starts at ξ = 0, where only the rst part of the gradient is non-zero. It results in two kinds of events: cross-talk artefacts and re ectors whose positions are kinematically consistent with the primaries in the data. The latter allow a correct interpretation of multiples at following iterations by the second and third term in the gradient.

We have presented optimisation strategies for iterative migration and an e cient way to compute the gradient of the associated objective function. We now present applications on a simple synthetic example. . P e and M e are the primary and multiple corresponding to the true re ector, P t and M t the primary and multiple corresponding to the trial re ectivity.

Synthetic examples

g 3 = Q[P 2 , λ 3 ]
are the interpretation of the same events as multiple re ections. The labels indicate the corresponding events in the data residual ( gure 2.10d). re ected event is visible on observed data ( gure 2.12a). For the optimisation process, we use the 2 -norm as regularisation function and the linear conjugate-gradient algorithm. 10 iterations are performed in a too slow background velocity model (2500 m/s) ( gure 2.13).

Two sections of the extended re ectivity volume are shown: one at h = 0 m corresponding to the physical re ectivity ( gure 2.13, left), and a CIG extracted at x = 810 m ( gure 2.13, middle). The most right panel displays the same CIG multiplied by the absolute value of the subsurface o set |h|, the actual input of the MVA objective function. After one iteration, two main events are visible with opposite curvatures, indicated by the dashed lines for the rst iteration ( gure 2.13, top). Because of the too low velocity, the re ector is shifted toward the surface in the zero-o set section and spreads over non-zero o sets with a downward curve, as predicted by the theory [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF]. The event curved upward above the re ector is a migration artefact due to the limited extension of sources and receivers in the acquisition. It is especially visible in the panels where |h|ξ (n) is displayed ( gure 2.13, right). The data reconstructed with this re ectivity model does not match observed data ( gures 2.12b and 2.12c), as only one iteration is performed.

The residuals are greatly reduced with iterations ( gure 2.14) and the nal re ectivity model obtained after 10 iterations perfectly explains observed data ( gures 2.12h and 2.12i). Two improvements are visible on the nal re ectivity image gure 2.13. Deconvolution of the wavelet source results in a more localised re ector, but the main e ect of iterations is the attenuation of migration artefacts and the strengthening of the event corresponding to the true re ector.

We run the same experiment for the correct velocity model (3000 m/s) and for a too high velocity model (3500 m/s). This test is similar to those presented in Lameloise et al. (2014) and Hou and Symes (2016b). The CIGs obtained after a single iteration are all a ected by migration artefacts with an upward curvature ( gure 2.15, top), although in the case of a too high migration velocity, these artefacts look similar to the event corresponding to the true re ector and are not distinguishable. After 10 iterations all CIGs look cleaner, and defocusing of energy is only due to errors in the velocity model.

As a result, attenuation of migration artefacts in the CIGs improves the properties of the MVA objective function J 1 . To illustrate this, we plot in gure 2.16 the value of J 1 calculated after several iterations of migration performed in homogeneous velocity models ranging from 2500 m/s to 3500 m/s every 100 m/s. To ease the comparison, we have plotted a normalised version of J 1 :

J 1 [c 0 ] = |h| ξ (N+1) [c 0 ] 2 ξ (N+10) [c 0 ] 2 . (2.43)
As already analysed by Lameloise et al. (2014), after one iteration the minimum of J 1 is not at the true velocity (3000 m/s) because of migration artefacts not focusing to zero-subsurface o set for the correct velocity. The minimum is reached for a too low velocity model as migration artefacts always have an upward curvature. The position of the minimum is a compromise between the defocusing of migration artefacts (they move away from the h = 0 axis as far as the velocity increases) and the defocusing of the event corresponding to the true re ector (decreasing when the velocity approaches the true velocity). As migration artefacts are attenuated when iterating, the minimum is progressively shifted to the true velocity. 1 2 3 4 5 6 7 8 9 10 11 10 -1 10 0 (a)

iterations 1 2 3 4 5 6 7 8 9 2600 2800 3000 3200 3400 velocity (m/s)

J 0 ξ (n)
|h| ξ (N+1) 2 ξ (N+1) 2 N = 1 N = 5 N = 2 N = 10

Primaries and multiples in a homogeneous medium

We now run the same example with rst-order surface-related multiples. There are now two events in observed data ( gure 2.17a). At the rst iteration, both are interpreted as primary re ections, resulting in two events ( gures 2.18a to 2.18c). The shallower one is the correct interpretation of the primary re ection and is similar to the primary-only case of the previous example. The misinterpretation of the surface multiple as a primary results in a cross-talk artefacts at twice the depth of the true event. As a consequence, reconstructed data do not match observed data. Iterations e ciently reduce the mis t ( gures 2.17f, 2.17g and 2.19) and the nal re ectivity section is free of migration artefacts and looks similar to the one obtained with primaries only ( gure 2.18). We also repeat the test of computing CIGs in three homogeneous velocity models (too low, correct and too high) ( gure 2.20). After one iteration, all CIGs are altered with cross-talk artefacts ( gure 2.20, top). Iterating allows to attenuate these artefacts and to obtain CIGs similar to those obtained in the previous example ( gures 2.15 and 2.20, bottom). Note that in the special case of homogeneous background velocity models, cross-talk artefacts exhibit the same focusing behaviour as the true events, hence we do not present the equivalent of gure 2.16 in the multiple case. In chapter 4, we run a similar test in the case where velocity increases with depth and we show that cross-talk artefacts due to multiples favour lower velocities (see section 4.4.2 and gures 4.24b and 4.27 for more details).

Conclusion

We have described in this chapter the iterative migration scheme used to determine a model perturbation explaining observed data containing both primaries and rst-order surface-related multiples. Data modelling is performed with a second-order Born approximation. Compared to a nite-di erence acoustic propagation with a free surface condition, this approximation is accurate, provided that the velocity contrasts in the models are not too large (less than 1000 m/s), otherwise the kinematics of multiples is not correctly reproduced.

We have described an e cient way of computing the gradient of the migration objective function based on the adjoint state method. Synthetic examples show that a re ectivity section correctly explaining the data can be retrieved after a few iterations, even in an incorrect velocity model thanks to the extension of the model space with the subsurface o set. At the rst iteration, migration artefacts due to the limited acquisition geometry appear on CIGs and the MVA objective function is not minimum for the correct velocity. Moreover spurious cross-talk events appear since multiple re ections are interpreted as primary events. Iterating allows attenuating both migration and cross-talk artefacts. As a consequence the minimum of the MVA objective function is at the correct velocity. It should also improve the shape of its gradient which de nes a velocity updates. The derivation of this gradient and its analysis is detailed in the next chapter. Chapter 3.

ξ(z, x, h = 0) (b) ξ(z, x = x m , h) (c) 1 iteration |h|ξ(z, x = x m , h) (d) depth (m) (e) (f)
Iterative Migration Velocity Analysis

Résumé du chapitre 3

Ce chapitre s'intéresse à la remise à jour des grandes longueurs d'onde du modèle de vitesse de propagation des ondes P, ou « macro-modèle », par la méthode d'optimisation par semblance di érentielle (DSO). La qualité du modèle de vitesse est évaluée sur le modèle de ré ectivité étendu obtenu par l'algorithme de migration itérative présenté au chapitre 2. La fonction coût de la DSO pénalise l'énergie défocalisée dans ce modèle de ré ectivité, et son gradient dé nit une remise à jour pour le macro-modèle. La construction du macro-modèle par minimisation de cette fonction sera illustrée dans le chapitre 6. Dans ce chapitre, je m'intéresse exclusivement au calcul du premier gradient.

L'algorithme d'analyse de vitesse par migration itérative étudié dans cette thèse se présente sous la forme d'un problème d'optimisation à deux niveaux ( gure 1.22) [START_REF] Colson | An Overview of Bilevel Optimization[END_REF]. Dans la boucle interne, le modèle de ré ectivité est remis à jour pour une valeur xe du macro-modèle. Dans la boucle externe, le macro-modèle est remis à jour en pénalisant l'énergie défocalisée dans le modèle de ré ectivité solution du problème interne. Idéalement la fonction coût externe devrait être dé nie avec la solution optimale du problème interne. En pratique un nombre limité d'itérations est e ectué dans la boucle interne, notamment pour limiter le coût numérique de la méthode. Ainsi une approximation de la fonction coût idéale et une approximation de son gradient sont utilisées en pratique. Il a été montré par Huang (2016) que le calcul de ce gradient était instable en pratique : de faibles di érences entre deux modèles de ré ectivité successifs, obtenus après un certain nombre d'itérations dans la boucle interne, ont un faible impact sur la valeur de la fonction coût interne, mais une grande in uence sur la fonction coût externe et sur son gradient.

A n d'illustrer et de mieux comprendre ce problème, je compare dans ce chapitre deux stratégies pour calculer une valeur approchée du gradient de la fonction coût externe vis-à-vis du macro-modèle. Toutes deux utilisent la méthode de l'état adjoint [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF]. La méthode A évalue le gradient exact de la fonction coût approchée en tenant compte de la dépendance de chaque itération (interne) vis-à-vis du macro-modèle. Dans la méthode B, on suppose que le problème interne, appelé aussi problème direct, est résolu jusqu'à convergence, et on écrit les équations donnant le gradient exact de la fonction coût « idéale » dé nie avec cette solution optimale. Il faut alors résoudre un problème adjoint, linéaire même lorsque des multiples sont considérés, et le calcul du gradient fait intervenir les solutions optimales des problèmes direct et adjoint. En pratique, les calculs sont faits à partir des solutions approchées des problèmes direct et adjoint, si bien que la remise à jour du macro-modèle dans la méthode B est aussi un gradient approché.

Les deux méthodes sont testées dans le cas de primaires seuls sur un cas simple où un unique ré ecteur horizontal est positionné dans un macro-modèle homogène. Le macro-modèle initial est homogène et trop lent, si bien que l'on s'attend à obtenir un gradient homogène et négatif audessus du ré ecteur et nul en dessous. Les résultats obtenus en pratique illustrent l'importance déterminante du poids donnée à la régularisation dans la boucle interne. Avec une valeur satisfaisante de ce poids, les deux méthodes convergent après quelques itérations vers le même gradient, homogène et d'un signe cohérent avec la valeur du macro-modèle initial. En revanche si le poids de la régularisation est trop faible, les gradients données par les deux méthodes sont di érents et ne convergent pas vers une valeur stable. Des oscillations subverticales apparaissent, en particulier pour le gradient calculé avec la méthode A. Une analyse détaillée de cet exemple permet de mieux comprendre l'origine de ces instabilités : si la fonction coût associée à la migration itérative semble converger rapidement, la valeur de la fonction coût externe calculé pour les modèles de ré ectivité successifs ne se stabilise pas et augmente graduellement. Des évènements sont progressivement créés aux grandes valeurs de l'o set en profondeur, dans le noyau de l'opérateur de modélisation mais avec une in uence déterminante sur la fonction coût externe. Donner un poids su sant à la régularisation contraint mieux la ré ectivité aux grands o sets en profondeur et stabilise le calcul du gradient qui fournit alors une remise à jour cohérente du macro-modèle. En revanche un poids trop fort empêche une atténuation correcte des artefacts de migration ; de plus de fortes oscillations dans le gradient sont observées en pratique autour du ré ecteur.

Ainsi l'atténuation des artefacts de migration au cours de la migration itérative conduit à des remises à jour du macro-modèle cohérentes, pourvu qu'une régularisation adéquate soit implémentée. Par la suite, seule la méthode B, plus simple à implémenter et moins sensible à l'apparition d'artefacts, sera considérée. En revanche, la migration itérative et la résolution du problème adjoint sont coûteuses numériquement et le choix d'un poids adéquat pour la régularisation reste di cile. Celui-ci peut être obtenu en traçant des courbes de Pareto, mais pour un coût de calcul prohibitif. Des solutions à ces di cultés sont proposées dans les chapitres suivants.

Finalement j'étudie le même exemple dans le cas de des ré exions multiples. Après une itération de migration itérative, les artefacts dus aux multiples dans le modèle de ré ectivité conduisent à des valeurs non nulles du gradient en dessous du ré ecteur. En itérant l'étape de migration, les artefacts dus aux multiples sont atténués et on obtient un gradient similaire à celui obtenu dans le cas de primaires seuls, pourvu qu'un poids adéquat soit donné à la régularisation.

Introduction

The aim of seismic imaging is the recovery of model's parameter such as the pressure wave velocity, allowing to numerically reproduce surface measurements. Determining the velocity model by minimising the least-squares mis t between observed data and calculated data may lead to an inaccurate estimation because the full waveform objective function has many local minima. The alternative method studied here consists of decomposing the velocity model into a smooth background model controlling the kinematics of wave propagation and a high-wavenumber part representing the re ectivity of the subsurface. Then for a given background velocity model, primary re ections are assumed to depend linearly on the re ectivity and multiple re ections quadratically. We have detailed in the preceding chapter the iterative determination of an extended re ectivity model free of cross-talk and migration artefacts minimising data mis t. The extension of the model allows to use the redundancy of observed data and to capture information about possible inaccuracies in the estimated background velocity model. Iterative migration corresponds to the inner loop of the nested optimisation strategy illustrated in gure 1.22. In this chapter, we are interested in the update of the background velocity model in the outer loop minimising defocused energy in the nal re ectivity.

In this chapter we do not loop over the macro-model but only consider the derivation of the rst update. Ideally, this update should be the exact gradient of the MVA objective function which measures defocused energy in a re ectivity model de ned as the optimal solution of the migration inverse problem. This is not achievable in practice as we perform a limited number of iterations to determine the nal re ectivity, as described in chapter 2.

The purpose of this chapter is the determination of an e cient way to compute an approximate gradient of the ideal MVA objective function. We would like the method to be robust against the gradient instability issue presented in section 1.3 [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF]. This issue will be illustrated in this chapter: we describe two di erent strategies to compute approximations of the ideal MVA gradient and show that they do not provide similar results. Furthermore, small di erences on the re ectivity model appearing by further iterating in the inner loop result in large modi cations to the gradient. We underline in this chapter the importance of a su ciently strong regularisation coe cient to mitigate this e ect. An alternative solution will be introduced in chapter 5.

In this chapter, we describe and compare two approximations of the ideal gradient. Both derivations use the adjoint-state method. In the rst approach, we compute the exact gradient of an approximation of the ideal MVA objective function by di erentiating all the iterations leading to the nal re ectivity model. In the second approach, we assume that iterative migration has reached convergence and compute an approximate gradient of the ideal objective function. This optimality criterion is used to derive the equations of the second approach. These two methods are compared in terms of implementation and stability of the gradient computed for successive values of ξ.

In the rst part of the chapter, the two methods are presented and compared on their computational merits. Then we present results of both methods on a simple synthetic example, rst in the linear case of primary re ections only, modelled under the rst-order Born linear approximation. Then the same example is run with rst-order surface-related multiples using a second-order quadratic Born approximation. These examples illustrate the importance of regularising the migration objective function.

Computing the gradient of J 1

We investigate the computation of two di erent approximations of the gradient with respect to the background velocity model c 0 of the ideal MVA objective function

J ∞ 1 [c 0 ] = 1 2 Aξ ∞ [c 0 ] 2 , (3.1)
where ξ ∞ obeys the rst order optimality condition of J 0 [c 0 , ξ],

∂ J 0 ∂ ξ [c 0 , ξ ∞ ] = 0. (3.2)
Operator A : → is called annihilator. For more detail on the meaning of this operator, we refer to section 1.3. In the numerical applications, the annihilator will consist of a multiplication by the absolute value of the subsurface o set: Aξ(x, h) = |h|ξ(x, h). The subsurface o set is the extra parameter on which the focusing criterion is tested. For a correct model, ξ(x, h) should be null for h = 0. The annihilator penalises defocused energy arising in an incorrect background velocity model. We keep the notation A in this chapter for the sake of generality.

In the following, we rst compute the exact gradient of an approximation

J (N)
1 of J ∞ 1 obtained when N iterations are performed to solve the migration problem. We call this approach method A. In the second approach, called method B, we assume that iterative migration reaches convergence after N iterations and use the optimality criterion (3.2) to determine an approximate gradient of J ∞ 1 .

Method A: Exact gradient of an approximate objective function

In method A, we compute the exact gradient of the following approximation of J ∞ 1 , de ned as

J (N) 1 [c 0 ] = 1 2 Aξ (N+1) [c 0 ] 2 (3.3)
where ξ (N+1) is obtained after N iterations of gradient-based minimisation of the migration objective function (equation 2.21) described in the preceding chapter.

A general way of performing this computation is the use of automatic di erentiation algorithms [START_REF] Bell | Algorithmic Di erentiation of Implicit Functions and Optimal Values[END_REF]. Here we employ the adjoint state method [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF] to obtain an explicit expression allowing the analysis of each contribution to the total gradient. To keep simple expressions, we consider the case where iterative migration is performed with steepest descent and a linesearch procedure f α . Then the state equations for the computations are given by

ξ (1) = 0 (3.4a) ∀n ∈ [1, N]              g (n) = ∂ J 0 ∂ ξ c 0 , ξ (n) d (n) = -g (n) α (n) = f α ξ (n) , d (n) ξ (n+1) = ξ (n) + α (n) d (n) , (3.4b) (3.4c) (3.4d) (3.4e)
where g (n) is the gradient of the objective function, d (n) is the descent direction and α (n) the step length at iteration n. We associate an adjoint variable to each equation of the iterative process (equations 3.4b to 3.4e). These Lagrange multipliers can be interpreted as constraints on the state equations. The associated Lagrangian reads:

J A 1 = J (N) 1 [c 0 ] - N n=1 η (n) ξ (n+1) -ξ (n) -α (n) d (n) - N n=1 β (n) α (n) -f α (ξ (n) , d (n) ) - N n=1 δ (n) d (n) + g (n) - N n=1 γ (n) g (n) - ∂ J 0 ∂ ξ c 0 , ξ (n) (3.5)
where the scalars β (n) and the vectors of η (n) , δ (n) and γ (n) are the adjoint variables associated to the state variables α (n) , ξ (n) , d (n) and γ (n) respectively . Similar to section 2.5.1 for the derivation of the gradient of J 0 , the value of the adjoint variables is obtained by zeroing the derivatives of the Lagrangian with respect to the state variables,

η (N) = A T Aξ (N+1) (3.6a) ∀n ∈ [N, 1]                  β (n) = η (n) d (n) δ (n) = α (n) η (n) + β (n) ∂ f α ∂ d ξ (n) , d (n) γ (n) = -δ (n) η (n-1) = η (n) + ∂ 2 J 0 ∂ ξ 2 (c 0 , ξ (n) )γ (n) + β (n) ∂ f α ∂ ξ ξ (n) , d (n) . (3.6b) (3.6c) (3.6d) (3.6e)
The gradient of J 1 with respect to c 0 then equals to

G (N) A = N n=1 ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (n) γ (n) + N n=1 β (n) ∂ f α ∂ c 0 ξ (n) , d (n) . (3.7)
Thus method A is an iterative process initiated at iteration N with the adjoint source term

η (N) = A T Aξ (N+1) named image residual.
Then, equations (3.6b) to (3.6e) have to be solved from n = N to n = 1. Method A requires the ability of computing the product of secondorder derivatives of J 0 with the successive values of γ. An e cient technique to perform this calculation will be presented in section 3.2.3. Note that the linesearch function may depend on c 0 , hence the second term in the gradient. If we consider a descent method with a xed step, that is α (n) = α 0 , we have β (n) = 0.

Method B: approximate gradient of the ideal objective function

In method B, we derive a simpler expression with the assumption that iterative migration is performed until convergence [START_REF] Chauris | Inversion Velocity Analysis -The Importance of Regularisation[END_REF]. That is, we suppose that the nal re ectivity section obeys the optimality condition (3.2) and we consider the ideal MVA objective function (3.1), built with ξ ∞ instead of ξ (N+1) . The gradient can again be evaluated via the adjoint state method. The associated Lagrangian is

J B 1 [c 0 , ξ ∞ , σ] = J ∞ 1 [c 0 ] -σ ∂ J 0 ∂ ξ [c 0 , ξ ∞ ] . (3.8)
with σ the adjoint state that satis es;

∂ 2 J 0 ∂ ξ 2 [c 0 , ξ ∞ ] σ = A T Aξ ∞ .
(3.9)

The ideal solution σ ∞ of this linear problem is the deconvolved version of the image residual A T Aξ ∞ . The gradient of J 1 with respect to c 0 then reads

G ∞ B = - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ ∞ σ ∞ . (3.10)
This is the optimal gradient obtained when both equations (3.2) and (3.9) are veri ed. In practice we approximate ξ ∞ by ξ (N+1) and we solve iteratively the linear system

∂ 2 J 0 ∂ ξ 2 [c 0 , ξ (N+1) ] σ = A T Aξ (N+1) (3.11)
with M iterations. This leads to the approximate gradient of J 1 M+1) .

G (N,M) B = - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) σ (N+1,
(3.12)

Even if the migration process is non-linear, the adjoint equation (3.9) is a linear problem of the type Hσ = θ where H is the Hessian of J 0 and θ is the image residual. If the convergence assumption is satis ed, the residuals are small and the Hessian operator is positive semi-de nite. Hence equation (3.9) is easily solved with a linear conjugate-gradient algorithm. Compared to method A, the adjoint equations of method B are solved independently of the optimisation strategy used for iterative migration. The adjoint problem may actually be solved with a di erent number of iterations, as the velocity update only involves the last value of ξ (n) and σ (m) regardless of how they are computed. Figures 3.1 and 3.2 illustrate how adjoint variables are computed in both methods and how they are combined with state variables to compute the velocity update. We detail the advantages of both methods regarding implementation and memory requirements in section 3.2.5. First we elaborate on the computation of second-order derivatives of J 0 .

ξ (1) ξ (2) • • • ξ (N) ξ (N+1) γ (1) γ (2) • • • γ (N) ∂ 2 J 0 ∂ ξ∂ c 0 ξ (1) γ (1) ∂ 2 J 0 ∂ ξ∂ c 0 ξ (2) γ (2) • • • ∂ 2 J 0 ∂ ξ∂ c 0 ξ (N) γ (N) + + + + N n=1 ∂ f α ∂ c 0 (ξ (n) , d (n) ) = G (N) A A T A Figure 3.1.
-Sketch of method A. Thin black arrows show the order of computation of the variables. Thick grey arrows show how they are combined to compute the velocity update

ξ (1) ξ (2) • • • ξ (N) ξ (N+1) σ (N+1,1) σ (N+1,2) • • • σ (N+1,M) σ (N+1,M+1) - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) σ (N+1,M+1) = G (N,M) B A T A Figure 3.2.
-Same as gure 3.1 for method B. Note that the numbers N and M of iterations performed to solve the direct and adjoint problems are not necessarily the same.

Computation of the second-order derivatives of J 0

Both methods involve the second-order derivatives of J 0 . In method A, the product of ∂ 2 J 0 /∂ ξ 2 and ∂ 2 J 0 /∂ ξ∂ c 0 with the adjoint variable γ is required N times. In method B, the resolution of the adjoint system (equation 3.11) with the linear conjugate gradient requires M products of the Hessian ∂ 2 J 0 /∂ ξ 2 with a vector of and the nal computation of the gradient (equation 3.12) requires a single product of ∂ 2 J 0 /∂ ξ∂ c 0 with the nal value σ (N,M) . The contribution of the regularisation is straightforward, hence we focus on the second derivatives of J Migr . Following [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF], an e cient procedure to compute their product with a vector of is de ned with a second-order adjoint-state technique.

We consider a new scalar function Γ and a vector χ of

Γ c 0 , ξ, χ = ∂ J Migr ∂ ξ [c 0 , ξ] χ (3.13)
such that the gradients of Γ with respect to ξ and c 0 equal the desired matrix-vector products

         ∂ Γ ∂ ξ = ∂ 2 J Migr ∂ ξ 2 [c 0 , ξ] χ, ∂ Γ ∂ c 0 = ∂ 2 J Migr ∂ ξ∂ c 0 (c 0 , ξ) χ. (3.14a) (3.14b)
We now apply the classic adjoint-state method to compute the gradients of Γ. We de ne a Lagrangian with the adjoint variables µ i and ν i associated to the state variables P i and λ i , respectively. The state equations are given by equations (2.20) and (2.34). The Lagrangian reads

Γ[c 0 , ξ, χ, P i , λ i , µ i , ν i ] = Q[P 0 , λ 1 ] + Q[P 2 , λ 3 ] χ -µ 0 0 P 0 -S -µ 1 0 P 1 -K -[P 0 , ξ] -µ 2 0 P 2 -M s P 1 -µ 3 0 P 3 -K -[P 2 , ξ] -ν 3 * 0 λ 3 -M T M(P 1 + P 3 ) -P obs -ν 2 * 0 λ 2 -K + [λ 3 , ξ] -ν 1 * 0 λ 1 -M T M(P 1 + P 3 ) -P obs -M * s λ 2 .
(3.15)

We derive Γ with respect to the state variables P i and λ i to nd the adjoint equations:

                       0 ν 1 = K -[P 0 , χ] 0 ν 2 = M s ν 1 0 ν 3 = K -[P 2 , χ] + K -[ν 2 , ξ] * 0 µ 3 = M T M(ν 1 + ν 3 ) * 0 µ 2 = K + [µ 3 , ξ] + K + [λ 3 , χ] * 0 µ 1 = M T M(ν 1 + ν 3 ) + M * s µ 2 * 0 µ 0 = K + [µ 1 , ξ] + K + [λ 1 , χ] (3.16a) (3.16b) (3.16c) (3.16d) (3.16e) (3.16f) (3.16g)
The desired matrix-vector products are obtained by derivating Γ with respect to ξ and c 0

∂ Γ ∂ ξ [c 0 , ξ, χ] = ∂ 2 J Migr ∂ ξ 2 [c 0 , ξ] χ = Q[P 0 , µ 1 ] + Q[P 2 , µ 3 ] + Q[λ 3 , ν 2 ] (3.17) ∂ Γ ∂ c 0 [c 0 , ξ, χ] = ∂ 2 J Migr ∂ ξ∂ c 0 [c 0 , ξ] χ = 3 i=0 C[P i , µ i ] + 3 i=1 C[λ i , ν i ], (3.18)
where we de ned the classical normalised cross-correlation operator

C[u, v] : × → for two vectors (u, v) of as C[u, v](x) = 2 c 3 0 (x) s ω (iω) 2 u(s, x, ω)v(s, x, ω) dω ds (3.19)
Note that both products can be computed simultaneously. Their calculations are twice as expensive as the computation of the gradient ∂ J Migr /∂ ξ (table 3.1). Note that if we calculate only ∂ 2 J 0 /∂ ξ 2 , some direct and adjoint variables are not needed, leading to less computations. In particular, in the case of primaries only, this operator does not depend on ξ and is computed in the same way as the gradient of J 0 .

Stability of the gradient

Ideally the MVA objective function should be evaluated at ξ ∞ and its gradient G ∞ used to update the background velocity model. In practice we perform a limite number of iterations and use

G (N) A or G (N,M) B
. One may want to estimate the decrease in convergence speed made by considering approximate gradients for the minimisation of J ∞ 1 [START_REF] Friedlander | Hybrid Deterministic-Stochastic Methods for Data Fitting[END_REF]) and bound the error de ned as the di erence between the macro-model recovered after a few outer iterations c f 0 and the exact macro-model c * 0 . Ideally it should be bound by the error made in the resolution of the inner problem by performing only N iterations. This analysis is not trivial as the gradient is not linear in

ξ (∂ J 1 /∂ c 0 = [∂ ξ/∂ c 0 ] T A T Aξ
). Some simple results can be stated, though. We have shown in chapter 2 that after an insu cient number of inner-iterations, the approximate objective function of method A is not minimum for the correct macro-model on one hand, and the gradient obtained with method B is very unlikely to lead to an accurate macro-model estimation on the other hand. An other issue is that the null space of the MVA primaries only primaries and multiples

J 0 [c 0 , ξ] 2 4 ∂ J 0 ∂ ξ [c 0 , ξ] 3 7 ∂ 2 J 0 ∂ ξ 2 [c 0 , ξ] χ 3 11 ∂ 2 J 0 ∂ ξ∂ c 0 [c 0 , ξ] χ 6 14
Table 3.1. -Number of wave-equations to be solved to evaluate the value, the gradient of the objective function at a point ξ, and the product of its Hessian with a vector χ. The number indicated here should be multiplied by the number of source positions considered in the acquisition.

objective function J 1 is not empty, meaning that di erent background velocity models can lead to relatively well-focused energy in CIGs. As an illustration, we present in gure 3.3 some results of iterations on the background velocity models. These results and the approach used to obtain them will be detailed in chapters 4 and 6. We display the exact macro-model used to compute observed data and two results obtained after twenty iterations on the macro-model starting with two di erent initial guesses. These three di erent macro-models result in CIGs with very similar focusing property. They di er in particular on the edges of the model where the acquisition setting does not constrain the model well. Larger surface-o set may better constrain these areas. This illustrates the practical di culty to bound the error between the exact background model and the one recovered after several outer iterations.

Instead, we propose here to study the stability of the MVA gradient computed for a xed background velocity model c 0 and successive values of the re ectivity model computed during iterative migration [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF]. Ideally we would like to be able to bound the error between the ideal gradient G ∞ and its approximations G by the error on the re ectivity model ξ (N+1)ξ ∞ . We begin by presenting three preliminary results for which we do not have formal proofs, but which allow to bound the error in the gradient by the error in the resolution of the inner problem. Then we use the numerical results presented in section 3.3.1 to determine if these hypotheses are reasonable. -Results of inversion (left, 2nd and 3rd row) obtained with observed data modelled in the exact model shown on the rst row (left). These results are presented in more details in chapter 6. For each background velocity model, three CIGs are displayed at the lateral positions 500 m, 1500 m and 2000 m as well as three traces of the velocity models at the same locations. The red curve corresponds to the exact model. The blue and green curves correspond to the nal models displayed in the 2nd and 3rd row, respectively. These results are obtained with two di erent initial models (dashed curves), a homogeneous macro-model and a model with values increasing with depth.

Lipschitz conditions

The stability of the MVA gradient across inner-iterations is based on the following Lipschitz conditions,

∂ ∂ c 0 ξ ∞ -ξ (N+1) × ≤ Ξ ξ ∞ -ξ (N+1) , (3.20a) ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ ∞ - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) × ≤ Ξ 1 ξ ∞ -ξ (N+1) , (3.20b) ∂ 2 J 0 ∂ ξ 2 c 0 , ξ ∞ - ∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) × ≤ Ξ 2 ξ ∞ -ξ (N+1) , (3.20c)
where Ξ, Ξ 1 and Ξ 2 are three positive constants.

We make two simplifying hypotheses, rst a linear modelling F : → obs of the data P[c 0 , ξ] = F[c 0 ]ξ, so that the derivatives of J 0 read

∂ J 0 ∂ ξ [c 0 , ξ] = Hξ -b, (3.21a) ∂ 2 J 0 ∂ ξ 2 [c 0 , ξ] = H, (3.21b)
with b = F T P obs and the Hessian H = F T F + a φ I. Second, we use a steepest descent direction and a constant step size α, then

Hξ (N+1) = (I -C N )b, (3.22)
with the operator C = I -αH supposed to ensure convergence to ξ ∞ , which satis es

Hξ ∞ = b. (3.23) Then ∂ ∂ c 0 (ξ (N+1) -ξ ∞ ) = - ∂ ∂ c 0 H -1 C N b (3.24a) = - ∂ H -1 ∂ c 0 C N b + NH -1 ∂ C ∂ c 0 C N-1 b + H -1 C N ∂ b ∂ c 0 (3.24b)
The convergence of this expression is determined by operator C N and so is the convergence of the migration process (equation 3.22), which indicates that the Lipschitz condition (3.20a) may be acceptable.

In the linear case the Hessian

H = ∂ 2 J 0 /∂ ξ 2 [c 0 , ξ] is independent of ξ, so that condi- tion (3.20c) is trivial. Eventually we use equation (3.21a) to write ∂ 2 J 0 ∂ ξ∂ c 0 [c 0 , ξ (N+1) ] - ∂ 2 J 0 ∂ ξ∂ c 0 [c 0 , ξ ∞ ] = ∂ H ∂ c 0 (ξ (N+1) -ξ ∞ ) + H ∂ ∂ c 0 (ξ (N+1) -ξ ∞ ), (3.25)
which guarantees condition (3.20b) provided the rst Lipschitz condition (3.20a) is veri ed.

Stability of method A

In method A we consider the exact gradient of J (N)

1 , then the error can be expressed as N+1) , so that

∆G (N) A = G (N) A -G ∞ = ∂ ∂ c 0 ξ (N+1) T A T Aξ (N+1) - ∂ ∂ c 0 ξ ∞ T A T Aξ ∞ . (3.26) We write ξ ∞ = ξ ∞ -ξ (N+1) + ξ (
∆G (N) A = ∂ ∂ c 0 ξ (N+1) T A T A ξ (N+1) -ξ ∞ - ∂ ∂ c 0 ξ ∞ -ξ (N+1) T A T Aξ ∞ . (3.27)
Using the Lipschitz condition (3.20a), it leads to

∆G (N) A ≤ k A ξ (N+1) -ξ ∞ , (3.28)
with k A a scalar constant. Hence, the stability of method A relies only on the hypothesis formulated in equation (3.20a).

Stability of method B

For method B, the optimal gradient is given by (3.10). In practice we use an approximate gradient: M+1) . The error associated to these approximations reads

equation (3.2) is solved in N iterations to nd ξ (N+1) and equation (3.11) is solved in M iterations to nd σ (N+1,
∆G (N,M) B = G (N,M) B -G ∞ = ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ ∞ σ (∞,∞) - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) σ (N+1,M+1) . (3.29)
Similarly to the previous case, we want to bound the error ∆G

(N,M) B
by the errors made in the resolution of the direct and adjoint problems, that is:

∆G (N,M) B ≤ k ξ ξ (N+1) -ξ ∞ + k σ σ (N+1,M+1) -σ (N+1,∞) , (3.30)
with k ξ and k σ two constants. Therefore we decompose σ (N+1,M+1) into

σ (N+1,M+1) = σ (N+1,M+1) -σ (N+1,∞) + σ (N+1,∞) -σ (∞,∞) + σ (∞,∞) , (3.31)
so that the error is the sum of three contributions

∆G (N,M) B = ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ ∞ - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) σ (∞,∞) - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) σ (N+1,M+1) -σ (N+1,∞) - ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) σ (N+1,∞) -σ (∞,∞) .
(3.32)

Using equation (3.20b), the rst contribution in equation (3.32) may be bounded by ξ ∞ξ (N+1) . The second contribution can be bounded by the error in the resolution of the adjoint problem σ (N+1,M+1) -σ (N+1,∞) . For the third term, we use the fact that σ (N+1,∞) and σ (∞,∞) are the exact solutions of problems (3.11) and (3.9) respectively, so that

∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) σ (N+1,∞) - ∂ 2 J 0 ∂ ξ 2 c 0 , ξ ∞ σ (∞,∞) = A T A ξ (N+1) -ξ ∞ . (3.33) We write again σ (N+1,∞) = σ (N+1,∞) -σ (∞,∞) + σ (∞,∞) to obtain ∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) σ (N+1,∞) -σ (∞,∞) = ∂ 2 J 0 ∂ ξ 2 c 0 , ξ ∞ - ∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) σ (∞,∞) + A T A ξ (N+1) -ξ ∞ . ( 3

Comparison in terms of implementation

Methods A and B di er regarding implementation of the adjoint equations. In method A, the adjoint equations depend on the optimisation method chosen for iterative migration, whereas in method B, the adjoint variables are solution of a linear system solved independently of the calculation made for the migration. Method B is thus easier to implement. In addition, the adjoint equations of method A (equations 3.6) involve the values of the state variables of migration ξ (n) , g (n) , and d (n) at each iteration. Hence method A requires storage of 3N -vectors. An alternative is to recompute their value when required during the computation of the adjoint variables. But they are needed in the reverse order to the one in which they are computed: we need rst ξ (N) , then ξ (N-1) , etc. Recomputation is then very expensive. In comparison, the adjoint variables in method B are computed independently of the successive values of the state variables. Only the last value of ξ (n) is needed to initiate the resolution, then we only store the variables needed to use the conjugate-gradient algorithm.

In both methods the main computational step is the product of the Hessian with a -vector. We have presented in section 3.2.3 an e cient way to perform this calculation based on secondorder adjoint-state approach [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF]. Note that the wave elds P i and λ i involved in the matrix-vector products have already been computed during the migration step, so that storing them would save a lot of computation time. However, these 4D arrays (n t × n z × n x × n s ) are too big to be stored and recomputation is necessary. We cannot process data source by source as for example the calculation of ξ implies a summation over all sources.

In method A, computing the derivative of the optimal step length with respect to the velocity model and the state variables adds extra computation of J 0 derivatives, especially in the nonlinear case where the optimal step α (n) is the result of a sophisticated procedure involving several trial steps; the formulas for this derivatives may not be straightforward as well. Hence, method A is roughly 1.5 times more expensive in computation time.

Synthetic examples

Examples with primaries only

We consider the linear case of primary re ections only and a model similar to the one studied in the numerical application of section 2.6. The model is 360 m deep and 2700 m large. The exact and initial macro-models are homogeneous (3000 m/s and 2500 m/s, respectively). A single horizontal re ector is located at 300 m depth. Sources are located at each grid point between x = 540 m and x = 2160 m. Receivers are located at each grid point within ±540 m around the source. As we are dealing with a very simple model, the 2 -norm is chosen as regularisation function. In case of a more complex geology with dipping re ectors for example, we may need a regularisation term function of depth to account for the poorer illumination of deeper structures. The importance of this regularisation term for the velocity update will be discussed later (section 3.3.2). After 10 iterations of linear conjugate gradient with optimal step length, both the value of J 0 and J 1 reached convergence ( gures 3.4a and 3.4c) and the gradient of J 0 is close to zero ( gure 3.4b). The gradient of J 1 is computed with both methods after each iteration ( gure 3.5). For method B, we choose in this example to use the same number of iterations to solve equations (3.2) and (3.11). The convergence speed of the resolution of the adjoint problem will be studied in more details and compared to the one of the direct problem in chapters 4 and 5 with the introduction of preconditioning. Here we set M = N for simplicity. At the rst iteration, positive values appear above 150 m depth for both methods. They are related to migration artefacts appearing in CIGs after a single iteration ( gure 3.6). With iterations these artefacts are progressively attenuated. After 10 iterations, the gradients obtained with both methods are similar and have the expected constant negative value. The central part is homogeneous, while the e ect of the acquisition geometry alters the edges of the gradient. Note also that the imprint of the re ector is still visible on the nal velocity update (around z = 250 m).

Both methods converge with approximately the same speed, although the gradient obtained after 5 iterations for method B is already very close to the nal result.

Importance of regularisation

To illustrate the importance of the regularisation term in the migration objective function, we run the same example with a 50 times smaller value of a φ . The corresponding gradients of J 1 ( gure 3.7) have shapes similar to the previous case ( gure 3.5) for the rst iterations but after 10 iterations they are altered with sub-vertical spurious oscillations, emphasised on gure 3.7 with dashed lines for method A. Moreover the gradients obtained for both methods do not look similar, although the value of J 0 seems to reach convergence ( gure 3.8a).

Actually, the norm of the penalised CIGs still increases steadily after 10 iterations ( gure 3.8c). This norm is not used in the inner loop iterative process, but we expect it to converge to a depth (m) stable value as does ξ. This result is not satisfactory and indicates that residual energy at large values of h in the CIGs ( gure 3.9, right) has a weak impact on the objective function J 0 , but is ampli ed by the multiplication by h and deeply in uences the value of J 1 and the velocity update. A su ciently strong regularisation term allows to better constrain the re ectivity model (2016), that no theoretical relation between the error on the gradient and the error in the inner iterations can be found. More precisely, referring to section 1.3, gure 1.11d suggests that the error increases exponentially.

ξ(z, x, h = 0) ξ(z, x = x m , h) 1 iteration |h|ξ(z, x = x m , h) depth (m)
We now identify the terms in the calculation of the gradient with method A that are responsible for the sub-vertical spurious oscillations. We use an example with smaller dimensions but with the same re ectivity and velocity value. Here, after 8 iterations of migration with steepest descent and optimal step size, we obtain with method A the gradient presented in gure 3.10a, which is altered with the same kind of artefacts. We control the correct computation of the gradient by also deriving the gradient with a nite di erence approach, requiring only to evaluate the objective function. As this is very expensive (proportional to the number of model parameters), we restrict the computation to a single line at depth z = 96 m and obtain a similar result ( gure 3.10b). These spurious oscillations are already visible in the adjoint variables (see for example γ (8) in gure 3.11a) and come from small oscillations located above the re ector in the nal migrated image, especially at large o sets. Their energy in ξ (N+1) is weak compared to the re ector, but they are strengthened in the computation of successive adjoint variables by two mechanisms. The rst is the iterative application of the Hessian ∂ 2 J Migr /∂ ξ 2 [c 0 , ξ (n) ] in the construction of the adjoint variables. In the linear case, it does not depend on ξ (n) and its application to a vector χ of is a sequence of a data modelling step using χ followed by a migration of this data back to the -space. Both steps involve two Green's functions from the source and the receiver position to a point of the subsurface. In a smooth model, the asymptotic amplitude term of the Green's function decays with the square-root of the distance, therefore the Hessian has a dynamic e ect of strengthening the shallow events and attenuating deeper ones. As the background velocity used for modelling and migration is the same, the diagonal term of the Hessian has no kinematic e ect and the position of events is thus not modi ed. We show the e ect of the Hessian applied to the adjoint variable γ (8) ( gure 3.11a), which contains both horizontal events (emphasised with blue dashed lines) and vertical events (red dashed lines) with the same shape as the artefacts of the velocity update. By nature, these artefacts are always above the interfaces and thus are ampli ed after application of the Hessian ( gure 3.11b), while the deeper horizontal events are attenuated. The regularisation term mitigates the attenuation of deeper events and prevents the apparition of artefacts.

The second origin of artefacts comes from the linesearch contributions (n) , d (n) ) ( gure 3.12a) to the construction of adjoint variables in method A (equation 3.6). They depend on the corresponding gradient of migration ∂ J 0 /∂ ξ[c 0 , ξ (n) ], which has non-negligible energy above the re ector in the last iterations ( gure 3.12b). The contributions of the linesearch also involve a sequence of modelling and migration steps applied to this gradient, hence, similarly to the application of the Hessian, the spurious events above the re ector are strengthened.

∂ f α /∂ ξ(ξ (n) , d (n) ) and ∂ f α /∂ d (ξ

Example with multiples

To investigate the ability of iterative migration velocity analysis to handle multiple re ections, we run the same example with rst-order surface-related multiples added to observed and calculated data. To take cross-talk events into account, the model is extended to 600 m depth. Two events are present in observed data: the primary re ection and its associated surface

∂ f α ∂ d (ξ (8) , d (8) )
to the equation of δ (8) . (b) Gradient of J 0 at the end of the iterations ξ (8) ] (section at h = 0). 100

g (8) = ∂ J 0 ∂ ξ [c 0 ,

Synthetic examples

multiple, recorded at approximately twice the time of the primary. To ensure the multiple event has enough in uence, the residuals between observed data and calculated data are multiplied by the recording time t in the de nition of J 0 . The data weight is taken into account in the derivation of the gradient of J 1 with respect to the background model. In the rst iteration, starting from ξ = 0, both events in observed data are interpreted as primary re ections. The primary re ection produces the same event as in the primaries-only case, while the multiple event adds an imprint at twice the depth of the true re ector. This artefact creates a new primary in the modelled data at the next iteration, which in turn will produce an update with a negative sign in the new re ectivity update. This way cross-talk artefacts are progressively attenuated.

We now compute the gradient of the MVA objective function after one and fteen iterations of migration. Compared to the primary only case, a more complex linesearch procedure f α is used during migration. Thus the derivation of the associated contributions to the exact gradient in method A become much more complex too. Moreover, method B seems less sensitive to vertical spurious oscillations illustrated in gure 3.7. Therefore we use method B for the computation of the gradient in this multiple example gure 3.14. At the rst iteration, the cross-talk artefact adds non-physical energy below the true re ector. After 15 iterations of migration, this artefact is largely attenuated and the gradient looks similar to the one obtained in the case where only primaries were considered. Choice of the regularisation coe icient Similar to the case of primaries only, a su ciently strong regularisation coe cient is required to avoid the apparition of spurious vertical oscillations. However if too much weight is given to the regularisation term, the attenuation of cross-talk artefacts is not as e cient and their imprint is still visible both on the nal re ectivity model and the MVA gradient. As an illustration, we compute the gradient obtained with method B and 15 iterations of iterative migration in three cases corresponding to three values of the regularisation coe cient ( gure 3.15). The choice of the regularisation is thus a trade-o between the smoothness of the gradient and the attenuation of artefacts and should in principle be the result of a Pareto curve analysis [START_REF] Hansen | The L-Curve and Its Use in the Numerical Treatment of Inverse Problems[END_REF]. However such an analysis is quite expensive, and a φ is chosen empirically in practice.

Conclusion

Re ectivity images produced after a single step of migration are perturbed with migration artefacts and cross-talk imprint of multiples. Minimising the migration mis t function iteratively attenuates these undesirable e ects and improves the properties of the MVA objective function.

In this chapter, we have introduced two approximations of the gradient of the ideal MVA objective function measuring defocused energy in a re ectivity model de ned as the optimal solution of the migration inverse problem. In the rst case, we compute the exact gradient of an approximate objective function de ned with the re ectivity model obtained after N iterations.

In the second one, we derive a simpler expression by assuming that the iterative process converged and compute an approximate gradient of the ideal objective function. Provided that su cient regularisation is applied, both methods yield consistent velocity updates, but the method assuming convergence is much easier to implement and requires less computational e ort and memory. We have provided a numerical example showing the instabilities of the gradient in a nested optimisation problem [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF] and have analysed some reasons for them. Regularisation is essential to produce a coherent velocity update; the 2 -norm is a simple but e cient choice.

Beyond the stability issues, the disadvantage of this iterative MVA process is its high computational cost: two problems have to be iteratively resolved to determine the gradient of the outer objective function updating the background velocity model. In chapter 4, the derivation of a preconditioner to accelerate the resolution of the direct and adjoint problems is investigated. The other di culty is the empirical choice of the regularisation weight as a trade o between the smoothness of the gradient and the attenuation of migration artefacts. In particular the fact that J 1 does not converge to a stable value in the inner iterations with small regularisation contrary to J 0 (see gure 3.8) is not satisfactory. This issue will be addressed and further analysed in chapter 5.

Résumé du chapitre 4

Ce chapitre décrit l'introduction d'un préconditionneur dans l'algorithme de migration itérative. L'objectif est d'accélérer la convergence de la migration itérative et de limiter le nombre d'itérations dans la boucle interne. Le préconditionneur utilisé ici est un pseudo-inverse de l'opérateur de modélisation de Born dans le domaine étendu, dé ni dans le cas de primaires seuls. Cet opérateur de modélisation est linéaire et noté F. Le pseudo-inverse est noté F † et est dé ni uniquement dans le cas de primaires seuls. L'obtention des CIGs par inversion « directe » à l'aide de cet opérateur est une alternative économique à la migration itérative pour l'atténuation des artefacts de migration. La fonction coût de DSO associée à l'inversion directe et son gradient sont étudiés et comparés à la stratégie itérative sur un exemple simple.

Le pseudo-inverse est présenté dans la section 4.2 qui est une transcription d'un article publié dans la revue Geophysics [START_REF] Cocher | Seismic Iterative Migration Velocity Analysis: Two Strategies to Update the Velocity Model[END_REF]. La formule présentée ici est inspirée de celle proposée par [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. Dans les deux cas, le pseudo-inverse est dé ni au sens asymptotique : la formule d'inversion est obtenue en exprimant l'opérateur de modélisation à l'aide de la théorie des rais, puis en dé nissant des poids tels que FF † soit proche de l'identité. Ces poids peuvent être interprétés à l'aide de la théorie des rais, mais la formule nale ne fait pas intervenir de quantités liées au rais. La formulation du pseudo-inverse est proche de celle de l'adjoint de l'opérateur de modélisation utilisé dans le cas de la migration classique. Des poids sont incorporés dans la formule de l'adjoint pour prendre en compte la décroissance géométrique et l'illumination inhomogène du sous-sol. En pratique les trois modi cations essentielles apportées à l'implémentation de l'opérateur adjoint sont : (1) l'application aux termes sources des équations dé nissant les champs sources et récepteurs de dérivées par rapport à la profondeur des sources et récepteurs ; ( 2 Les CIGs calculés en appliquant le pseudo-inverse aux données observées ne présentent pas d'artefacts de migration. De plus, le gradient de la fonction coût de DSO associée est plus homogène que celui obtenu dans le cas de la migration classique avec l'opérateur adjoint. En n une légère modi cation de la fonction coût de DSO consistant en une multiplication de la ré ectivité par une puissance spéci que du macro-modèle permet d'atténuer les artefacts du gradient localisés à la position des ré ecteurs. Ainsi remplacer l'opérateur adjoint par un pseudo-inverse pour le calcul des CIGs a des e ets béné ques sur l'analyse de vitesse pour un coût numérique semblable. En e et le temps de calcul supplémentaire demandé par l'application des poids listés ci-dessus est négligeable devant le coût de la propagation des champs d'onde. La stratégie d'inversion « directe » est donc une alternative e cace à la migration itérative dans le cas de primaires seuls et d'une acquisition dense, mais elle n'est pas aisément transposable au cas de multiples, où il faudrait dé nir un nouveau pseudo-inverse dont l'existence n'est pas garantie.

La section 4.3 détaille l'introduction du pseudo-inverse comme préconditionneur dans la migration itérative et dans la résolution du problème adjoint (cas de la méthode B dé nie au chapitre 3). Le préconditionnement peut être vu comme un changement de variable et ne demande que peu de modi cations de l'implémentation. La section 4.4 illustre sur un exemple simple l'accélération de la convergence de la migration itérative, dans le cas de primaires seuls, mais aussi dans le cas de multiples, bien que le pseudo-inverse utilisé comme préconditionneur ne soit dé ni que pour les ré exions primaires. Le préconditionnement est aussi testé dans le cas de primaires seuls pour le problème adjoint. S'il permet également une résolution accélérée, celle-ci demeure plus lente que pour le problème direct, même avec régularisation. Par ailleurs le gradient calculé dans le cas itératif avec la méthode B ne converge pas vers une valeur stable similaire au gradient obtenu par l'inversion « directe ». En particulier, introduire une puissance spéci que du macro-modèle dans la dé nition de la fonction coût de la DSO ne permet pas d'atténuer les oscillations résiduelles localisées au niveau du ré ecteur dans le gradient, contrairement au cas de l'inversion directe. Ces di cultés sont analysées dans le chapitre 5.

Introduction

We propose in this chapter a modi cation of the iterative migration process described in chapter 2. It consists of introducing a preconditioner in the minimisation of the migration objective function. Preconditioning can be interpreted as a change of variables allowing faster convergence. The preconditioner considered here is an approximate inverse of the extended Born modelling operator, de ned for primary re ections only. Its formulation is close to the usual adjoint's one, the di erence being the introduction of migration weights compensating for uneven illumination of the subsurface and for geometrical spreading. Preconditioning should result in faster attenuation of migration artefacts in CIGs. We investigate if this is also the case in the presence of multiples.

The derivation of the approximate inverse operator is presented in section 4.2, which is the transcription of an article published in Geophysics [START_REF] Cocher | Seismic Iterative Migration Velocity Analysis: Two Strategies to Update the Velocity Model[END_REF]. Only primary re ections are considered. The inverse operator considered here is close to the one presented in [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. Both derivations determine an approximate inverse F † of the extended Born modelling operator F such that their composition FF † is close to the identity operator. However [START_REF] Cocher | Seismic Iterative Migration Velocity Analysis: Two Strategies to Update the Velocity Model[END_REF] use a linearisation of the phase of FF † whereas Hou and Symes (2015) directly apply the stationary phase approximation, leading to di erent nal formulations for F † .

In terms of implementation, the formula of [START_REF] Cocher | Seismic Iterative Migration Velocity Analysis: Two Strategies to Update the Velocity Model[END_REF] is close to the standard migration algorithm, with three main modi cations: (1) vertical derivative with respect to source and receiver positions are applied to the source term in the equations de ning the source and receiver wave elds; (2) the source wave eld is constructed with the deconvolved version of the source wavelet; (3) a rst-order integration in time is applied before cross-correlation instead of a second-order derivative and a vertical derivative is applied to the result of the cross-correlation. These modi cations amount to applying the following weights, respectively:

(1) cosines of take-o angles at the sources and receivers positions; (2) deconvolution of the source wavelet; (3) cosines of the half-opening angle at the image point. A larger weight is given to small scattering angles and short surface o sets. The approach shares similarity with the quantitative extended migration presented in Lameloise et al. (2014). However no ray quantities are required in the nal expression, which makes this strategy amenable to wave-equation based MVA techniques.

This direct inversion formula is actually an alternative to iterative migration for the case of primaries. CIGs obtained with the inverse formula are free of migration artefacts and a new MVA objective function can be constructed with these CIGs. The derivation of the associated gradient is performed with the adjoint-state technique. Numerical examples show that it is smooth and homogeneous above the re ector. Moreover, a simple modi cation of the MVA objective function allows to remove the oscillations around the re ector described in section 3.3.1. The inversion strategy is then an alternative to the horizontal contraction [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF]Shen and Symes, 2015). The advantage of the direct inversion approach is that the velocity update is the gradient of an objective function, contrary to the horizontal contraction technique.

Inversion is an e cient alternative to iterative migration in the case of primaries. Computing the re ectivity image and the associated background velocity update with inversion is as costly as with standard migration. Iterative migration is much more expensive but keeps the ability of dealing with more complex cases such as multiples for which an inverse formula does not necessarily exists. In the second part of this chapter, we introduce migration weights in the iterative migration algorithm described in chapter 2 to improve its convergence rate. A strategy to properly include them in the optimisation process has to be de ned. The inversion formula of [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF] can be written as the composition of a weighting operator in the data space, the usual adjoint and a weighting operator in the model space. The iterative extension is thus natural and consists of changing the de nition of the norms in the data and extended-model space (Hou and Symes, 2016a). The inversion formula used here cannot be decomposed in this way. Instead, we choose to use the approximate inverse operator as a right preconditioner. In terms of implementation, this amounts to a change of variables in the de nition of the objective function of migration. Although the inverse formula does not take multiple re ections into account, it is expected to accelerate the convergence of iterative migration in the case of multiples as well as in the case of primaries only.

In the case of primaries only, direct and iterative inversions are expected to provide similar results for the re ectivity image, but not necessarily similar MVA gradients. We compare the corresponding background velocity updates obtained with both methods in the case of primaries only. Note that using method B presented in the preceding chapter, preconditioning does not change the way the gradient is computed, it only provides a more e cient way to nd solutions to the direct and adjoint problems. [START_REF] Cocher | Seismic Iterative Migration Velocity Analysis: Two Strategies to Update the Velocity Model[END_REF] This section is a transcription of the following article published in Geophysics, H. [START_REF] Cocher | Seismic Iterative Migration Velocity Analysis: Two Strategies to Update the Velocity Model[END_REF]. From Migration to Inversion Velocity Analysis. Geophysics, 82(3), S207-S223.

Definition of migration weights

Abstract

Migration Velocity Analysis is a technique de ned in the image domain to determine the background velocity model controlling the kinematics of wave propagation. In the presence of discontinuous interfaces, the velocity gradient used to iteratively update the velocity model exhibits spurious oscillations. For more stable results, we propose to replace the migration part by an inversion scheme. By de nition, migration is the adjoint of the Born modelling operator, whereas inversion is its asymptotic inverse. We propose new expressions in 1D and 2D cases, based on two-way wave-equation operators. The objective function measures the quality of images obtained by inversion in the extended domain depending on the subsurface o set. In terms of implementation, the new approach is very similar to classical migration velocity analysis. A 1D analysis shows that oscillatory terms around the interface positions can be removed by multiplying the inversion result with the velocity at a speci c power before evaluating the objective function. Several 2D synthetic data sets are discussed through the computation of the gradient needed to update the model parameters. Even for discontinuous re ectivity models, the new approach provides results without arti cial oscillations. The model update corresponds to a gradient of an existing objective function, which was not the case for the horizontal contraction approach proposed as an alternative to deal with gradient artefacts. It also correctly handles low velocity anomalies, contrary to the horizontal contraction approach. Inversion Velocity Analysis o ers new perspectives for the applicability of image domain velocity analysis.

Introduction

Migration Velocity Analysis (MVA) is a technique to determine the Earth's properties from seismic surface measurements. It is based on the assumption that the model parameters can be split into a background model containing the large scale structure of the model and a model perturbation characterising the details. The background model controls the kinematics of wave propagation. Under the Born approximation, the re ected data linearly depend on the model perturbations supposed to be small compared to the background model. In a given model, the re ectivity part is obtained through migration de ned as the crosscorrelation between the incident wave eld and the back-propagated residual wave elds [START_REF] Claerbout | Toward a Uni ed Theory of Re ector Mapping[END_REF]. Migration is by de nition the adjoint of the Born modelling operator and maps the data residuals to some model perturbations in the image domain. Once data have been migrated, the quality of the background model is evaluated by measuring the focusing of such images. Except for semblance measurements in the stacking power approach, the traditional principle consists of splitting the data into subsets, such as common shot or common o set gathers and to migrate them independently in the same background model [START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. If all images are consistent, then the model is said to be optimal for migration. Moveout residuals observed in Common Image Gathers (CIGs) indicate that the background model is not su cient and should be updated. The nal image indeed should not depend on the acquisition geometry, at least the part correctly illuminated. Di erential Semblance Optimisation (DSO) is a way to evaluate the quality of the model used for migration [START_REF] Symes | Velocity Inversion by Di erential Semblance Optimization[END_REF][START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. For surface-oriented shot or o set, residual energy is computed by comparing adjacent images. Such a measure is known to have a convex behaviour, at least for simple models.

More recently, an extension has been proposed. Instead of splitting the input data into di erent subsets, the full data set is migrated but a spatial or temporal shift is introduced in the imaging condition (Faye and [START_REF] Jeannot | Prestack Migration Velocities from Focusing Depth Analysis[END_REF][START_REF] De Bruin | Angle-dependent Re ectivity by Means of Prestack Migration[END_REF]Sava and Fomel, 2006;[START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. For a spatial shift, typically horizontal, we refer to the subsurface o set. We only consider this approach here. With the additional subsurface o set, the image domain becomes extended [START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. The main reason for this approach is the following: the dimension of the model space should be the same as the dimension of the data space (table 4.1). The investigated model is correct if energy focuses around the zero subsurface o set. As the model and data spaces have the same dimension, it is potentially possible to reconstruct the observed data from the re ectivity, even in an inaccurate model [START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF].

MVA in the extended domain still faces a number of challenges (Lameloise et al., 2014;[START_REF] Lameloise | Analyse de Vitesse Par Migration Quantitative et Introduction Des Ondes Transmises[END_REF]. (1) Due to limited acquisition, migration smiles are visible in CIGs. They always have an upward curvature [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF][START_REF] Li | Analysis of RTM Extended Images for VTI Media[END_REF]. For simple models, a downward (upward) curvature indicates a too low (high) velocity. Due to the presence of migration smiles, the optimal velocity is underestimated; in that model, there is a compromise between downward residuals and upward artefacts. ( 2 leading to local minima (Lameloise et al., 2014).

(3) As revealed by [START_REF] Vyas | Gradients for Wave-equation Migration Velocity Analysis[END_REF] and [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF], the gradient of the DSO objective function with respect to the background model contains unwanted oscillations around the re ector positions, especially when the re ectivity is discontinuous. It means that MVA does not fully behave in a tomographic algorithm for which we would expect a homogeneous update at least for simple models [START_REF] Sava | Wave-Equation Migration Velocity Analysis[END_REF][START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]Alkhalifah and Wu, 2016a). The straightforward solution is to spatially smooth the gradient, but in practice, a discontinuous re ectivity has an impact on a much larger zone than the zone containing the discontinuities. Moreover, the artefacts can have the same sign, especially around continuous interfaces: a smoothing approach would not destroy them. As a partial solution, [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF] have introduced a modi cation in the computation of the gradient such that spurious oscillations are removed. This spectacular e ect is not fully understood yet. It is however now clear that the modi ed expression is not the gradient of an objective function (Shen and Symes, 2015). The main objective of this paper is to discuss possible alternatives to overcome the artefacts. Another limitation of DSO is its sensitivity to coherent noise, such as linear noise or multiples [START_REF] Chauris | Two-Dimensional Velocity Macro Model Estimation from Seismic Re ection Data by Local Di erential Semblance Optimization: Applications to Synthetic and Real Data Sets[END_REF][START_REF] Mulder | Automatic Velocity Analysis by Di erential Semblance Optimization[END_REF]. In practice, the input data set for MVA should only contain re ected data. Finally, operational CPU cost and memory requirement are real di culties for the 3D implementation [START_REF] Duveneck | A Pragmatic Approach for Computing Full-Volume RTM Re ection Angle/Azimuth Gathers[END_REF][START_REF] Van Leeuwen | A ordable Full Subsurface Image Volume -An Application to WEMVA[END_REF]. The main reason is the increased size of the model space, in principle up to the size of the data space (table 4.1).

In practice, only few applications on real data have been published. Among others, we can cite [START_REF] Symes | Velocity Inversion by Di erential Semblance Optimization[END_REF], [START_REF] Chauris | Two-Dimensional Velocity Macro Model Estimation from Seismic Re ection Data by Local Di erential Semblance Optimization: Applications to Synthetic and Real Data Sets[END_REF], [START_REF] Mulder | Automatic Velocity Analysis by Di erential Semblance Optimization[END_REF], [START_REF] Alkhalifah | Tau-Migration and Velocity Analysis: Application to Data from the Red Sea[END_REF], [START_REF] Shen | Automatic Velocity Analysis via Shot Pro le Migration[END_REF], Weibull and Arntsen (2013) and [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF]. Recently, approaches have been proposed to overcome some of the di culties encountered by MVA. In particular, MVA can potentially be extended to transmitted waves (Chauris et al., 2013;[START_REF] Shen | Subsurface Focusing Measurement of Diving Waves and Its Application to Re ection Tomography[END_REF][START_REF] Biondi | Simultaneous Inversion of Full Data Bandwidth by Tomographic Full-Waveform Inversion[END_REF][START_REF] Lameloise | Automatic Migration Velocity Analysis Applied to Direct Waves in a Crosswell Con guration[END_REF][START_REF] Lameloise | Quantitative Migration for a More Robust Migration Velocity Analysis[END_REF]. By considering higher orders in the Born approximation, it is also possible to take into account multiples [START_REF] Staal | Velocity Estimation Using Internal Multiples[END_REF][START_REF] Cocher | Imaging with Surface-Related Multiples in the Subsurface-O set Domain[END_REF]. In these approaches, spurious oscillations related to truncated interfaces are not addressed.

To deal with this issue, we propose here to replace migration by inversion (ten Kroode, 2012; [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. Migration is the adjoint of the Born modelling operator. It is de ned as the rst gradient of the least-squares mis t between modelled and observed data. An alternative is to fully solve the quadratic minimisation. The rst attempts coupled to velocity analysis were obtained by iteratively determining the model perturbation (Liu et al., 2014b;[START_REF] Chauris | Removing Spurious Oscillations in the Gradient of the Di erential Semblance Optimization Functional[END_REF][START_REF] Cocher | Imaging with Surface-Related Multiples in the Subsurface-O set Domain[END_REF]. The shape of the objective function appears to be modi ed and more quadratic; some artefacts in the gradient were also attenuated. This motivates further investigations on iterative MVA. The main drawback with the iterative approach is the CPU-cost, proportional to the number of iterations. If we suppose that iterative migration has converged, the implementation simpli es but it is still very expensive (Cocher et al., 2017a), except if iterative migration can be accelerated with suitable preconditioners (Plessix, 2009). We propose here to investigate Inversion Velocity Analysis (IVA) by coupling inversion to velocity analysis (Liu et al., 2014b;[START_REF] Chauris | Inversion Velocity Analysis -The Importance of Regularisation[END_REF].

Inversion is a direct approach as an alternative to the iterative migration. It is remarkable that such an inversion formula exists (ten Kroode, 2012; [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. The derivation is performed under the high frequency approximation and depends on the absence of multiple ray paths. It appears that the result is accurate, even in the presence of triplicated wave elds, as the normal operator is elliptic, i.e. asymptotically invertible, whether caustics are present or not [START_REF] Stolk | Kinematics of Shot-Geophone Migration[END_REF]ten Kroode, 2012). The strongest limitation is the absence of turning waves, as the sub-surface o set has only a horizontal component [START_REF] Biondi | Angle-domain Common-image Gathers for Migration Velocity Analysis by Wave eld-continuation Imaging[END_REF]. The nal formula only consists of a combination of wave-equation based operators without ray quantities.

The main di erences between inversion and adjoint versions are additional derivative operators applied to the Green's functions or observed data at the source and receiver positions and to the re ectivity image. These operators compensate for geometrical spreading and for uneven illumination. For that, a change of variable is needed between the surface coordinates and the subsurface coordinates. It is thus essential to work in the extended image domain where precisely the model size equals the data size. The main objective of this work is to replace the adjoint result by the inverted result in MVA and to study the bene t of this modi cation. We rst review the derivation of the Born inversion formula in 2D. It is not exactly the same as the one proposed by [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. We provide the equivalent result in 1D. Then we explain how to compute the gradient of the new IVA objective function with respect to the background velocity model. With the 1D formulation, it is possible to analyse the impact of inversion on the shape of the gradient. We propose a slightly modi ed inversion (multiplication of the inversion result by the velocity at a speci c power) such that, at least in 1D, the gradient does not contain spurious oscillations. We then apply the same strategy in 2D, with applications on a series of synthetic data, in particular with discontinuous re ectivity functions for which artefacts have been observed [START_REF] Vyas | Gradients for Wave-equation Migration Velocity Analysis[END_REF][START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF]. We focus on the shape of the gradient as it is the main ingredient to update the background model.

Migration versus Inversion

In Migration Velocity Analysis, the model m(x) = m 0 (x) + ξ(x) is split into two parts, where m 0 denotes the background model containing the large scale component of the velocity structure and where ξ is a model perturbation providing the detailed part of the model [START_REF] Wu | Simultaneous Inversion of the Background Velocity and the Perturbation in Full-Waveform Inversion[END_REF]. m 0 controls the kinematics of wave propagation, whereas ξ generates re ections and di ractions. As a choice, we consider m 0 being the squared slowness model. The two components (m 0 , ξ) are inverted in a nested loop approach: in a given model m 0 , the inner loop updates ξ in a migration process, typically starting from ξ = 0. The nal ξ depends on m 0 . The outer loop updates m 0 according to some focusing criteria. Under the Born approximation, the modelled data d linearly depends on ξ, the model perturbation, supposed to be small in front of m 0 . The rst objective function to determine the optimal ξ parameter is de ned as the least-squares di erences between observed data d obs and computed data d in ξ

J 0 (ξ) = 1 2 d(ξ) -d obs 2 . (4.1)
Both observed and modelled data are evaluated at source s, receiver r and time t or alternatively angular frequency ω. We de ne the Born operator 0 in the extended domain [START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF]. With the choice of the squared slowness model, it reads

0 (ξ)(s, r, ω) = -(iω) 2 Ω(ω) dx dh G 0 (s, x -h, ω)ξ(x, h)G 0 (x + h, r, ω), (4.2)
where Ω is the input source wavelet, typically a Ricker function. The two Green's function G 0 satisfy the wave equation, here the constant density acoustic wave equation in the model m 0 . The associated source term is a Dirac distribution in space and time. Extended domain means that the model perturbation ξ depends on the spatial coordinates x = (x, z) in 2D and on an extra parameter, here a spatial horizontal shift h = (h, 0) in 2D (Sava and Fomel, 2006;[START_REF] Symes | Migration Velocity Analysis and Waveform Inversion[END_REF][START_REF] Sava | Extended Imaging Conditions for Wave-Equation Migration[END_REF]. Compared to classical Born modelling, an extra loop on h allows to consider non-physical models in the sense that the two Green's functions are evaluated at two distinct positions xh and x + h. The rst strategy to determine ξ consists of de ning it as the rst gradient of J 0 with respect to ξ and for ξ = 0. This is the classical migration approach, here formulated in the extended domain, yielding

( mig d obs )(x, h) = -ds dr dω (iω) 2 Ω * (ω)G * 0 (s, x -h, ω)d obs (s, r, ω)G * 0 (x + h, r, ω). (4.3)
This adjoint formulation implies the complex conjugate version G * 0 of the Green's functions and an integration over the acquisition parameters (s, r, ω). The second strategy consists of minimising J 0 and to de ne ξ through the inverse operator inv (ten Kroode, 2012; [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. Two aspects are essential in the derivation of such operator: rst, the extended domain plays a crucial role as the size of the data domain is the same as the size of the model domain. During the derivation, a change of variables is applied. The second element is the microlocal analysis, meaning that inv is the inverse of 0 under high frequency approximation and some additional approximations, mainly that there is no triplicated wave eld. The inverse operator should compensate for geometrical spreading and uneven illumination. A general form inspired from [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF] is

inv (δd)(x, h) = k(x, h) p ds dr dω(iω) ν Ω(ω) s z G * 0 (s, x -h, ω)δd(s, r, ω)
r z G * 0 (x + h, r, ω), (4.4) where p is the partial derivative with respect to variable p. For example, s z indicates the vertical derivative with respect to the source position. The values of p in p , ν in (iω) ν and k(x, h) in front of the integral need to be speci ed such that the composition of the two operators inv • 0 = , where is the identity operator. Let ξ be the integrand of the product of the two operators. It is linear in ξ and satis es

( inv • 0 )ξ(y) = dy (y, y )ξ(y ), (4.5)
with y = (x, h). inv is indeed an inverse if (y, y ) = δ(y -y ). After derivations reported in appendix 4.2.10, the nal expression for the 2D case reads

( inv δd)(x, h) = 32 m 0-m 0+ p Ω(ω) iω s z G * 0 (s, x -h, ω)δd(s, r, ω) r z G * 0 (x + h, r, ω) ds dr dω , (4.6) with p = 1 2 m 0+ m 0- + 1 z + 1 2 m 0- m 0+ -1 hz , (4.7)
where Ω is the inverse of Ω, m 0+ = m 0 (x -h), m 0-= m 0 (x + h). z is the vertical derivative at the image point, whereas hz is the derivative with respect to the vertical subsurface o set.

It is important to note that the nal expression does not contain ray quantities. In an invariant model, m 0-= m 0+ = m 0 and p = z . This result is not exactly the same as the one published in [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF], as explained in the Discussion section. The main di erences between the adjoint and inverse versions are the following: (1) instead of a second derivative of the seismic source, a rst-order integration in time has to be applied to the inverse of the source function;

(2) a normalisation factor 32 m 0-m 0+ has to be applied on the re ectivity to retrieve absolute amplitudes;

(3) the vertical derivatives with respect to the source and receiver positions are applied to the Green's function. The main dynamic e ect is a multiplication by the cosine weight at the source and receiver positions; (4) similarly, for laterally invariant model, an additional vertical derivative has to be applied at the image points. It acts as if the kernel would be multiplied by cos θ cos ϕ, where θ and ϕ are the half-opening and dip angles at the image point, measured from the normal to the dip and the horizontal, respectively (see gure C.1 p. 265 for the de nition of these angles). Such weights emphasise short o sets, small dips and vertical rays at source and receiver positions. Only wave-equation based operators are present in the de nition of the inverse operator inv . Following the same strategy, we propose equivalent and new expressions for the 1D case. This case will be useful in the next section to analyse the shape of the gradients with respect to m 0 .

( 0 ξ)(ω) = -(iω) 2 Ω(ω) z G 2 0 (z, ω)ξ(z) dz, (4.8a
)

( mig δd)(z) = - ω (iω) 2 Ω * (ω)G 2 * 0 (z, ω)δd(ω) dω, (4.8b
)

( inv δd)(z) = 16m(z) z ω Ω(ω) (iω) [ s z G * 0 ] 2 (z, ω)δd(ω) dω . (4.8c)
In 1D, there is no need to extend the model. We consider a single source and a single receiver, both at the surface. The source and receiver Green's functions are the same. We recognise the similar terms. The main di erence with the 2D case is the factor 16m(z) in front of the vertical derivative in equation (4.8c). In the case of a homogeneous model m 0 and an original re ectivity function de ned as R e δ(z -z e ), for a single re ector at depth z e , the expressions can be further simpli ed using analytic formulations for the Green's functions, yielding in 1D

ξ mig (z) = R e v 0 2 2 v e 2 2 g mig 2 z e v e - z v 0 , (4.9a
)

ξ inv (z) = 2 R e v 0 v e v 0 2 g inv 2 z e v e - z v 0 , (4.9b)
where g mig (t) and g inv (t) are obtained by crosscorrelating the seismic wavelet with itself or with its inverse, namely g mig (t) = Ω(t) ⊗ Ω(t) and g inv (t) = Ω(t) ⊗ Ω(t). The homogeneous velocities values are v e and v 0 , respectively in the exact and tested cases. In 1D homogeneous models, the adjoint and inverse expressions are very similar. For the inverse case, a deconvolution is applied; the same time shift appears in the two expressions. A di erent normalisation term appears. Note that there is no geometrical spreading and no need to compensate for uneven illumination.

We now have the expressions in 1D and 2D for the adjoint and inverse approaches. We introduce them in a velocity analysis approach.

MVA versus IVA

Two nested loops are de ned in image domain velocity analysis. The inner loop determines the re ectivity model ξ mig or ξ inv through the J 0 objective function (equation 4.1). The second objective function for the outer loop indicates how focused the energy is in the extended domain. For the adjoint and inverse formulation, we de ne

J α mig (m 0 ) = 1 2 m α 0 |h|ξ mig 2 , (4.10a) J β inv (m 0 ) = 1 2 m β 0 |h|ξ inv 2 . (4.10b)
The multiplication by the annihilator |h| aims at evaluating how energy is concentrated around h = 0 (Sava and Fomel, 2006). For the correct model m 0 , energy is expected to be localised in the physical domain. Extra terms m α 0 and m β 0 have been introduced inside the 2 norm. In the classical formulation, α = β = 0. They should be interpreted as an additional weight for more exibility in the shape of the gradient of J α mig and J β inv with respect to m 0 . As explained later, the additional term in the gradient for α or β if not equal to zero is localised around the re ectivity: this will be used to attenuate oscillations in the nal gradient. We use the adjoint state technique [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF] as an e cient tool for the gradient derivation. More details are provided in appendix 4.2.11. The adjoint state technique can be seen as a minimisation process under constraints. Instead of only considering m 0 as unknown, we enlarge the model space by incorporating other variables. We add Lagrangian constrains to force the wave eld to satisfy the wave equation. For an e cient implementation, we de ne the following variables S 0 , R 0 , µ s , µ r and λ (appendix 4.2.11): S 0 and R 0 are the forward and backward wave elds, whereas µ s and µ r are their adjoint variables, respectively. The nal gradient for α = 0 and β = 0 reads

∂ J 0 mig ∂ m 0 = -(iω) 2 S 0 (s, x, ω)µ * s (s, x, ω) ds dω -(iω) 2 R 0 (s, x, ω)µ * r (s, x, ω) ds dω, (4.11a) ∂ J 0 inv ∂ m 0 = -(iω) 2 S 0 (s, x, ω)µ * s (s, x, ω) ds dω -(iω) 2 R 0 (s, x, ω)µ * r (s, x, ω) ds dω + λ(x, h) ∂ p ∂ m 0 ξ 0 (x, h) dh, (4.11b)
with ξ inv (x, h) = p ξ 0 (x, h). p is de ned in equation (4.7). In laterally invariant models, p does not depend on m 0 . The adjoint variable λ 0 = a 2 ξ mig in the adjoint case and λ 0 (x, h) = * p a 2 p ξ 0 in the inversion case (appendix 4.2.11). µ s and µ r are solution of the wave equation for a source term excited by the product of λ 0 by the back-propagated or forward wave elds ( gure 4.1). In the gradient expression (equations 4.11a and 4.11b), S 0 and µ s contribute in a tomographic mode as they correlate from any interface to the surface. The same holds for R 0 and µ r . For non zero α and β values, the gradient expressions are simply obtained as a combination of the gradient for α or β = 0 and a positive term located around the position of the re ectors

∂ J α mig ∂ m 0 (x) = m 2α 0 (x) ∂ J 0 mig ∂ m 0 (x) + 2αm 2α-1 0 (x) dh |h| 2 ξ 2 mig (x, h), (4.12a) ∂ J β inv ∂ m 0 (x) = m 2β 0 (x) ∂ J 0 inv ∂ m 0 (x) + 2βm 2β-1 0 (x) dh |h| 2 ξ 2 inv (x, h). (4.12b)
We have seen that the gradient of the second objective function can thus be e ciently derived with the adjoint state technique. The adjoint approach contains 3 terms whereas the inverse 4 contributions. The two strategies have very similar implementation and the inverse is not really more di cult. In the next section, we analyse the di erent contributions to understand if they act in a tomographic mode and/or they have an oscillating contribution around the re ector position. This leads to the determination of an optimal β value. Then we present 2D applications with the same β.

Analysis of the gradient in 1D

The 1D case is a particular case. Instead of de ning a spatial shift h introduced in the imaging condition, we propose to de ne the annihilator as zz e , where z e is the exact depth of the re ector. This is only valid for a single re ector. However, the expressions are similar to the ones in 2D and help us understand the structure of the gradient. For homogeneous exact v e and initial v 0 models, the expressions for the 1D adjoint and inverse gradients are

G α mig (z) = - R 2 e v 4 e 2 8 v 2α-5 0 ( y -z e ) 2 g mig 2 z e v e - y v 0 g mig 2 z e v e - z + y + |z -y| 2v 0 d y + α R 2 e v 4 e 2 6 v 2α-4.5 0 (z -z e ) 2 g 2 mig 2 z e v e - z v 0 (4.13) G β inv (z) = 4R 2 e v 2β+1 0 v e v 0 4 ( y -z e ) 2 g inv 2 z e v e - y v 0 g inv 2 z e v e - z + y + |z -y| 2v 0 d y + 2 β + 2 R 2 e v 2β 0 v e v 0 4 (z -z e ) 2 g 2 inv 2 z e v e - z v 0 (4.14) + 4R 2 e v 2β 0 v e v 0 4 ( y -z e )g inv 2 z e v e - y v 0 g inv 2 z e v e - z + y + |z -y| 2v 0 d y.
The rst term in G α mig has a non zero contribution above the position of the re ector after migration, while the second term is positive and localised around the re ector. approach makes the signal more localised in space. From ξ inv , the modelled data nicely matches with the observed data ( gure 4.2, right). Both phase and amplitude are correctly retrieved.

For the gradient, we compare three approaches: (1) the inverse approach obtained with the adjoint state method (equation 4.11b), ( 2) a nite-di erence approach and (3) the analytic expression valid in homogeneous background models (equation 4.14). The nite-di erence approach consists of evaluating the gradient by computing J β inv for slightly perturbed models around the reference model. This is a much more expensive approach than the one provided by the adjoint state method but this is a ordable in 1D. We rst consider β = 0. The three approaches give very similar results for the migration approach ( gure 4.3), both if the initial model has a too low and too high value (2500 or 3000 m/s for a correct velocity at 3000 m/s or 2500 m/s, respectively). The gradient is homogeneous above the re ector, while oscillations are present around the re ector. In this case, the amplitudes of these oscillations are larger than the homogeneous part. We conclude that the DSO gradient in 1D has a tomographic mode together with a migration mode around the re ectivity. The additional contribution requires in practice to smooth the gradient before updating the model.

The same results hold for the inversion approach, still for β = 0 ( gure 4.4, top). It is interesting to note that the oscillations around the re ectivity are always positive, even if the initial model is lower or larger than the exact model. The three approaches (adjoint, nitedi erence and analytic) are also consistent. The α and β values introduced in the objective function o er additional possibilities. Appendix 4.2.12 indicates that there is an optimal β = -3/2 value for which the gradient becomes much smoother ( gure 4.4, middle). An alternative would be to optimise the shape of the gradient as in [START_REF] Wu | Simultaneous Inversion of the Background Velocity and the Perturbation in Full-Waveform Inversion[END_REF]. To further analyse this, we rst display the three contributions to the gradient in 1D ( gure 4.4, bottom). The red curve is always positive by de nition as it contains only squared terms (second term in equation 4.12b). It appears that the oscillations in the rst term are in phase, with an opposite sign, to the second contribution (blue line). The third contribution is smooth (black line), leading as a nal result to a smooth gradient (dashed line).

This e ect is con rmed by analysing more carefully the expression of the gradient in homogeneous models (equation 4.14). The integration over y is performed from 0 to the maximum depth z max . We split this integration between 0 to z and then between z to z max . We then integrate by parts. For β = -3/2, the oscillating contributions are exactly opposite. More details are provided in appendix 4.2.12. The same analysis with the gradient associated to migration does not lead to a smooth gradient. We conclude that in 1D, the gradient related to inversion can have a smooth behaviour for a particular normalisation (β = -3/2). We now investigate if these conclusions can be extended to 2D.

2D applications

We study the shape of the gradient of J α mig and J β inv with respect to the background model and for α = 0, β = 0 and β = -3/2. We investigate ve 2D synthetic cases. The rst one contains a single horizontal re ector to study (1) how the inversion impacts the shape of the gradient, and (2) if β = -3/2 is still a good candidate as in the 1D case. Then, we deal with discontinuous re ectivity sections, rst considering a single di raction point and then interfaces with abrupt truncations. They are typical cases for which the classical gradient exhibits spurious oscillations [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF][START_REF] Vyas | Gradients for Wave-equation Migration Velocity Analysis[END_REF]. The fourth case contains a complex rough interface between two homogeneous structures. Here, the observed data are not modelled under the Born approximation. A pre-processing is needed to remove the direct arrival. Finally, the last case contains a low velocity anomaly in which the horizontal contraction solution does not provide a proper solution as explained in Shen and Symes (2015). In this example, we perform non-linear iterations to update the velocity model; in other cases, we display the rst velocity update. In all cases, the data are generated with a nite-di erence time domain constant density acoustic code under the Born approximation [START_REF] Noble | Inversion Non Linéaire de Données de Prospection Pétroliere[END_REF]. Even if the velocity model is not invariant, we simplify p to z . The classical migrated section in a homogeneous model at v 0 = 2500 m/s exhibits a at interface around z = 290 m and low frequency variations above it ( gure 4.5a). This can be understood by looking at the CIGs ( gure 4.5c). For this speci c depth interface and maximum surface o sets, migration smiles are present o the zero-subsurface o set. The shape of such events are predicted by [START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF]. With the application of the inverse instead of the adjoint, events are more localised due to the deconvolution in the inverse formula, but more importantly migration smiles are removed: defocused energy corresponds to an incorrect velocity model ( gures 4.5b and 4.5d).

Single horizontal interface

The associated gradient is not homogeneous above the interface for the migration case ( gure 4.6a) due to the contribution of migration smiles. As their curvature is always towards the surface, a positive (red) contribution is added to the gradient. Note that the gradients are computed with respect to the squared slowness: we thus expect only negative (blue) values above the interface. This is partly the case for the inversion gradient and β = 0 ( gure 4.6b), but the impact of the re ectivity around z = 200 m is visible and adds a positive (red) contribution to the gradient. This e ect was already observed in the 1D case ( gure 4.4). A closer look shows that the gradient has also a negative contribution below the interface: this could appear to be strange, but can be explained as follows: energy is visible in CIGs for depths below z = 200 m and non-zero subsurface o sets as the velocity model is too slow ( gure 4.5d). The contribution of the gradient below that depth is due to the third term in equation (4.11b). This was not visible in 1D for which there is no need to rely on an extended model. With the introduction of β = -3/2, oscillations around the interface and the smooth update below disappear. As for the 1D case, the second term in equation ( 4.12b) cancels unwanted oscillations for β = -3/2, even if there is no formal proof in 2D. When the velocity v 0 is higher than the exact velocity v e , the same conclusions hold ( gures 4.7 and 4.8). CIGs could appear to be cleaner in the migration case but energy due to an incorrect model is partly superimposed with migration smiles ( gure 4.7a): the associated gradient indeed oscillates with positive and negative values above the interface ( gure 4.7c). With the inversion approach, CIGs and gradients are cleaner ( gures 4.7d and 4.8c). For β = 0, the gradient has a positive (red) contribution around the re ector as in 1D and as for the previous case. For β = -3/2, the gradient is homogeneous above the interface. Note that in gures 4.6c and 4.8c, edge e ects for x = 0 and x = 3500 m, are visible with lower energy due to limited acquisition.

We conclude from this rst test that the inversion scheme with β = -3/2 provides a smooth gradient as in a tomographic mode, at least for continuous interfaces.

Di raction model

The extreme discontinuous case is a model with a single di raction, here embedded in a homogeneous model v e = 3000 m/s. The maximum surface o set is ±600 m. The exact di raction anomaly is at (x, z) = (1800 m, 240 m). For a too slow velocity model v 0 = 2500 m/s, energy for h = 0 follows a di raction curve, but energy is visible above it in the migration case Both for migration and inversion, the gradient is oscillating ( gures 4.9c and 4.9d). We propose to smooth it as a regularisation before iterating over the background model. The Gaussian smoothing lter is expressed as

g σ (r 0 ) = 1 2πσ e - r 2 0 2σ 2 , (4.15)
where r 0 is the distance. The smoothing parameter is chosen as σ = λ data /2, with λ data being the mean wavelength of the data. Only negative values are obtained for the inversion case, but not for the migration case ( gures 4.9e and 4.9f).

Discontinuous interfaces

The third example consists of 4 truncated interfaces inspired from Shen and Symes (2015). The maximum surface o set and frequency content of the data are the same as for the di raction case. The smoothing parameter σ equals the half-wavelength. Once more, the migrated re ectivity contains energy above the interface positions, leading to oscillatory gradients, especially around the termination of interfaces ( gures 4.10a, 4.10c and 4.10e). Inversion results are much sharper ( gure 4.10b). The gradient is negative (blue) everywhere, even without smoothing ( gures 4.10d and 4.10f). Note that the re ectivity sections ( gure 4.10) can be reconstructed from the di raction case ( gure 4.9) as there is a linear relationship between ξ mig and the exact re ectivity. This is not true for the gradient as it is not linear in ξ mig . The same remark holds for the inversion case: gure 4.10f cannot be reconstructed as a linear combination of gradients associated to a single di raction point as in gure 4.9f.

Rough interface

In the next example, the model is extracted from the BP salt dome model [START_REF] Billette | The 2004 BP Velocity Benchmark[END_REF]. Here the velocity above the rough interface is set to 3000 m/s and below to 3800 m/s ( gure 4.11a). As before, the observed data is computed with a nite-di erence modelling code, but here without relying on the Born approximation. Absorbing boundaries are implemented via Perfect Matched Layers [START_REF] Bérenger | A Perfectly Matched Layer for the Absorption of Electromagnetic Waves[END_REF]Komatitsch and Martin, 2007), including for the top interface. As a pre-processing, the direct arrival is removed. The maximum surface o set is ±600 m and the maximum frequency content of the data is 40 Hz.

We compare the result obtained with the migration and inverse approaches. For the migration case obtained for v 0 = 2500 m/s, the interface is globally too shallow but also defocused and contains low frequency energy above the interface. As for the simple case ( gure 4.6a), the gradient is mainly positive at very shallow depths and then becomes negative as expected ( gure 4.11c). With the inversion approach, the re ectivity is sharper ( gure 4.12a) and the gradient has a mean negative (blue) value ( gure 4.12b). After smoothing with σ = λ data /2, the gradient is mainly negative, except in the left part where some oscillations are visible ( gure 4.12c). A stronger smoothing with σ = λ data would remove the positive values ( gure 4.12d). It is interesting to compare the observed data and the data after inversion and modelling ( gure 4.13). For the derivation of the gradient, we used β = -3/2 to remove the imprint of the interfaces in the gradient. Here we use β = 0 to check the quality of inversion. The rst events correspond to re ections, whereas later events are associated to di ractions. Events are correctly retrieved. The largest residuals are visible at large o sets. Traces extracted for o sets -300 m, 0 m and +300 m show that both phases and amplitudes are correctly retrieved, meaning that inversion provides indeed an inverse, even in an incorrect model, here a homogeneous model at v 0 = 2500 m/s ( gure 4.14). Note that the way to generate the data (without Born approximation) is not the same as the one used to de ne the modelling (with Born approximation). We would expect small di erences at least in the case of small velocity contrasts: it seems to be robust as the velocity contrast is here +800 m/s.

Low velocity anomaly

Finally, we investigate the case of a velocity model containing a low velocity zone of -500 m/s ( gure 4.15a), leading to triplicated wave elds. A similar example is discussed in Shen and Symes (2015) as this is a model for which the horizontal contraction [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF] does not provide a correct answer (Shen and Symes, 2015).

After migration in v 0 = 3000 m/s, the re ectivity gathers show de ected energy, supposed to be horizontal in the exact model ( gure 4.15b). The associated gradient is expected to display a red (positive) anomaly. This is not the case ( gure 4.15c). The zones in the dotted ellipses are positive (red) due to migration smiles in the CIGs. The structure of the central part (solid ellipse), with a negative (light blue) value below and between positive values, is similar to the one presented in Shen and Symes (2015), gure 2c in their paper. As explained in Shen and Symes (2015), the horizontal contraction in the case of triplicated wave elds does not provide a proper solution. With the inversion approach, the re ectivity section is similar ( gure 4.16a) but the gradient displays vertical stripes ( gure 4.16b). This is only the rst step in model building. After 10 iterations with a non-linear conjugate gradient, the inverted background model contains a main velocity anomaly ( gure 4.16c). The colorbar scale is the same as the one gure 4.15a. As the model only contains three re ectors, the shape of the anomaly is not perfectly constrained, but is localised around the correct position (dotted circle). Finally, the re ectivity in the inverted model displays relatively horizontal shapes. The focusing is largely improved after updating the background velocity model as indicated by the CIGs ( gure 4.17).

We conclude from these di erent investigations that the inversion formulas proposed for the 1D and 2D cases (equations 4.6 and 4.8c) are indeed inverse and not adjoint, as the data after inversion followed by modelling are nicely retrieved, even in an incorrect velocity model. The combination of inversion and velocity analysis provides a gradient that does not su er from artefacts as described by [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF][START_REF] Vyas | Gradients for Wave-equation Migration Velocity Analysis[END_REF]. We propose to incorporate m β 0 in the de nition of the second objective function to assess the quality of focusing in CIGs. With β = -3/2, the imprint of re ector is not visible in the gradient. Finally, the gradient may be smoothed with a characteristic length proportional to the wavelength of the data. In the last example, triplicated wave elds are visible. Even if the derivation of the

Discussion

The inversion formula 4.6 is not the same as the one provided by [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. In the derivation proposed here, we have supposed that the main contribution in equation (4.5) is obtained when y is close to y [START_REF] Beylkin | Imaging of Discontinuities in the Inverse Scattering Problem by Inversion of a Causal Generalized Radon Transform[END_REF]: we thus linearised the phase of the operator, whereas [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF] directly applied the stationary phase approximation. More importantly, the main advantage of their formulation is that they express the inverse operator as a combination of a model preconditioner W model , the classical adjoint operator and a datadomain preconditioner W data . Such an expression allows to easily adapt the iterative conjugate gradient by simply modifying the norms in the data and model domains (Hou and Symes, 2016a).

In our case, the inversion formula (equation 4.6) cannot be recast as a combination of di erent preconditioners, but the vertical source derivatives are applied to the Green's function and not to the observed data. Note that the integration over sources is performed along the horizontal component of the source position and not the vertical one.

In the case of laterally invariant model, the inversion expressions simplify in the two approaches. In [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF], the W model operator is obtained through two Lapacians in the (x, z) and (h, z) domains = -∇ 2

x,z -∇ 2 h,z (equation 30 in [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. In our approach, only a vertical derivative z is needed. The number of time integration also di ers. [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF] considered a linearisation of the velocity model, whereas we used a squared slowness perturbation. The weighting factor di erence 2/v 3 0 appears twice in the modelling and inverse operators. It means that the di erences between [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF] and the approach proposed here is a factor

= 2 v 0 ωk z k 2 z + k 2 x k 2 z + k 2 h 2 v 0 ω k z , (4.16)
where (k z , k x , k h ) are the wavenumbers associated to (z, x, h). The second equality is only valid for small k x and k h values. Operator is adimensional and it is similar to a quantity arising in two-way re ection travel time. For the optimal β, a multiplication by m -3/2 0 in front of the re ectivity function in equation (4.6) modi es the 32m 0 factor to 32m 0 • m -3/2 0 = 32v 0 . In 1D, it would lead to 16v 0 .

In terms of implementation, [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF] need to apply the W data operator to the data residuals. The derivative with respect to the vertical component of the source and receiver can be obtained with the free surface (ten Kroode, 2012; [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]. The operator can be obtained with two 2D Fourier transform, in (x, z) and (h, z) domains. In our approach, the implementation is very similar to the classical adjoint implementation. The vertical derivatives at the source and at the receiver positions are applied to the Green's functions and can be obtained by introducing a dipole instead of point sources. For the computation of the gradient with respect to the model perturbation, only two extra variables ξ and λ are needed, compared to classical MVA. ξ and λ are function of ξ 0 and λ 0 (appendix-4.2.11).

For the numerical illustrations, we assume that the background model was laterally invariant. In that case, the p operator (equation 4.7) simpli es to a vertical derivative z . The study of the correct operator is beyond the scope of this article. It would imply the derivative with respect to h z , taken at h z = 0. More research is needed to know how to compute such quantity, as CIGs depend on h x and are computed for h z = 0 only.

The additional weights introduced in the inversion formula are mainly cosines of takeo angles at the source and receiver positions, and cosine of opening and dip angles at the image point. The main e ects are visible for shallow depths and large o set contributions. As illustrated in gures 4.5 and 4.7, the migration smiles are largely attenuated. Weibull and Arntsen (2013) proposed to introduce a vertical derivative z to improve the robustness of MVA. Here, we give a justi cation for the introduction of this weight that is explicitly de ned in the inversion formula. It is interesting to note that the same derivative in z also appears in the inverse one-way Born formula [START_REF] Joncour | Preserved-Amplitude Angle Domain Migration by Shot-Receiver Wave eld Continuation[END_REF]. Here, we rely on two-way wave-equation operators. The vertical derivative is a high-pass lter, attenuating the most dipping events in CIGs, especially vertical events associated to lower wavenumbers. An important aspect is the introduction of the determinant (equation 4.23) due to a change of variables between the surface acquisition variables and the subsurface image variables. An alternative is to compensate for uneven illumination through ad-hoc weights usually derived in simple models [START_REF] Wu | Directional Illumination Analysis Using Beamlet Decomposition and Propagation[END_REF][START_REF] Yang | Illumination Compensation for Image-Domain Wave eld Tomography[END_REF][START_REF] Tang | Tomographically Enhanced Full Wave eld Inversion[END_REF]. The inversion formula provides an automatic and correct strategy.

Instead of considering inversion, one could solve the least-squares functional in an iterative manner (equation 4.1). The computational cost is proportional to the number of iterations. However, it may have a number of advantages. When the data do not linearly depend on the re ectivity, as in the case of surface multiples, inverse formula does not necessarily exist. For the derivation of the gradient with respect to the background model, one would need to take into account the iterations to derive the optimal re ectivity ξ. An alternative is to suppose that the convergence has been reached, leading to a simple linear problem to be solved (Cocher et al., 2017a). More investigations are needed to further compare MVA with iterative or direct inversion of the re ectivity.

Finally, the gradients only provide the rst iteration needed to update the background velocity model. For a complete minimisation, the estimation of the Hessian of the second objective function is essential (equation 4.10b). A rst possibility is to estimate the Hessian in a homogeneous model. An interesting alternative is proposed by Shen and Symes (2015) for the estimation of the diagonal term. The derivation was in the case of the horizontal contraction. More research, beyond the scope of this article, is needed to determine an equivalent approach when inversion is coupled to velocity analysis.

Conclusions

We have replaced migration with the inverse of the modelling operator in image domain velocity analysis. This is an automatic way to compensate for uneven illumination and to remove migration smiles especially visible for shallow depth and large o sets. As a consequence, the gradient associated to the quality of focusing has a smoother behaviour in simple models and does not su er from oscillations even in models with truncated re ectivities. Compared to an alternative solution based on horizontal contraction, the velocity update is the gradient of an objective function. To remove the imprint of the re ectivity, we have proposed to use the inverse of the modelling operator multiplied by the velocity at a speci c power determined in 1D and applicable in 2D. The next step consists of applying non-linear iterations to fully determine the velocity model.

Acknowledgements

The authors are grateful to the editor, to the associated editor T. Alkhalifah, to the reviewers W. Symes and B. Sun, and to an anonymous reviewer, as they all have provided very detailed comments to improve the initial manuscript. The authors would like to thank W. Symes for inspiring this work. They are grateful to him and to R.-É. Plessix for fruitful discussions. The research described in this article was carried out as a part of the Paris Exploration Geophysics Group project (GPX) funded by the French National Research Agency (ANR), CGG, Total and Schlumberger.

Inversion (appendix-A)

We develop the main steps to derive the inversion formula (equation 4.6). Under the high frequency approximation, the Green's function in the frequency domain is given by G 0 (s, x, ω) = K(ω)A(s, x)e iωT(s,x) , (4.17)

where T and A are the travel times and geometrical amplitudes solutions of the Eikonal and transport equations. In 1D, K(ω) = 1/iω and in 2D, K(ω) = 1/ iω, introducing an integration or half-integration in the time domain. Let de ne k x = iω∇T(s, xh) + iω∇T(r, x + h). The operator obtained as a combination of the forward and inverse operators in equation (4.5) reads

(y, y ) = k(x, h) dk (iω) 2+ν+β ∂ (s, r, ω) ∂ (k) A 2 (s, x -h)A 2 (r, x + h) e ik•(y -y) m 1/2 0s m 1/2 0r cos β s cos β r , (4.18)
where y = (x, h) is the vector in the extended space, and k = (k x , k h ) the associated wavenumber vector. m 0s and m 0r are model values at the source and receiver positions, whereas β s and β r are take-o set angles for the source and receiver positions. To obtain equation ( 4.18), we have rst used the following approximations

s z G 0 (s, x, ω) (iω) cos β s m 1/2 0s G 0 (s, x, ω) (4.19) r z G 0 (r, x, ω) (iω) cos β r m 1/2 0r G 0 (r, x, ω) (4.20) z G 0 (s, x -h, ω)G 0 (r, x + h, ω) (iω)(+m 1/2 0-cos θ s + m 1/2 0+ cos θ r )G 0 (s, x -h, ω) G 0 (r, x + h, ω) (4.21) hz G 0 (s, x -h, ω)G 0 (r, x + h, ω) (iω)(-m 1/2 0-cos θ s + m 1/2 0+ cos θ r )G 0 (s, x -h, ω) G 0 (r, x + h, ω), (4.22)
where θ s and θ r are the angles at the image points xh and x + h associated to the source and receiver positions. The model values m 0-and m 0+ are also evaluated at xh and x + h. Then we apply a change of variable from (s, r, ω) to k and replace the Green's functions by their high frequency approximation. The squared amplitudes are due to the application of the forward and inverse operators.

The remaining quantities to be evaluated are the determinant and the amplitude terms. Here we use the fact that the kernel e ik (y-y ) is oscillating and that the amplitude terms have a smoother behaviour (ten Kroode, 2012): we thus evaluate the amplitudes only at y. The 3 × 3 determinant in 2D can easily be obtained as k s and k r have explicit expressions, yielding

∂ (k) ∂ (s, r, ω) = -(iω) 2 m 1/2 0+ m 1/2 0- ∂ θ s ∂ s ∂ θ r ∂ r cos θ s cos θ r m 1/2 0-sin θ s + m 1/2 0+ sin θ r -sin θ s -sin θ r m 1/2 0-cos θ s + m 1/2 0+ cos θ r -cos θ s cos θ r -m 1/2 0-sin θ s + m 1/2 0+ sin θ r , = 2(iω) 2 m 1/2 0+ m 1/2 0- ∂ θ s ∂ s ∂ θ r ∂ r m 1/2 0+ cos θ s + m 1/2 0-cos θ r . (4.23)
The determinant expression contains terms evaluated at xh and x + h, and partial derivatives of the angles θ s and θ r at the image positions xh and x + h, with respect to the source and receiver positions. Note that the determinant also has the term m 2012) and [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF].

A 2 (s, x -h) = 1 8π 2 1 m 1/2 0s cos β s ∂ θ s ∂ s , (4.24a) A 2 (r, x + h) = 1 8π 2 1 m 1/2 0r cos β r ∂ θ r ∂ r . (4.24b)
Once more, it is remarkable that the two derivative terms |∂ θ s /∂ s| and |∂ θ r /∂ r| cancel out in the expression. The cos β s and cos β r terms are also removed in equation ( 4.18): this was the reason why vertical derivatives at the source and receiver positions were introduced in equation (4.4).

Gradient derivation for the inversion case (appendix-B)

We explain how to compute the gradient of the objective function J 0 inv with respect to the model m 0 (equation 4.11b). The expression for β di erent from zero is simply given in equation (4.12b). With the Lagrangian formalism, the principle consists of extending the objective function and of introducing the state equations are constrains [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF]. The strategy is the same for the classical DSO approach.

J = 1 2 aξ 2 -λ(x, h) ξ(x, h) -p ξ 0 (x, h) -λ 0 (x, h) ξ 0 (x, h) -ds dω S * 0 (s, x -h, ω)R 0 (s, x + h, ω)
µ s (s, x, ω) 0 S 0 -S s (s, x, ω)

-µ r (s, x, ω) * 0 R 0 -R s (s, x, ω) , (4.25)
where J now depends on (m, λ, ξ, λ 0 , ξ 0 , µ s , S 0 , µ r , R 0 ). The rst part in equation (4.25) is the original objective function. The other parts are respectively the de nition of the re ectivity variables ξ and ξ 0 , and of the forward S 0 and backward R 0 wave elds, solutions of the wave equation with source terms S s and R s . The associate adjoint variables are λ, λ 0 , µ s and µ r . The total gradient reads

dJ dm 0 = ∂ J ∂ m 0 + ∂ λ ∂ m 0 ∂ J ∂ λ + ∂ ξ ∂ m 0 ∂ J ∂ ξ + ∂ λ 0 ∂ m 0 ∂ J ∂ λ 0 + ∂ ξ 0 ∂ m 0 ∂ J ∂ ξ 0 + ∂ µ s ∂ m 0 ∂ J ∂ µ s + ∂ S 0 ∂ m 0 ∂ J ∂ S 0 + ∂ µ r ∂ m 0 ∂ J ∂ µ r + ∂ R 0 ∂ m 0 ∂ J ∂ R 0 . (4.26)
In order the avoid to compute the Fréchet derivatives (derivatives of λ, ξ, λ 0 , ξ 0 , µ s , S 0 , µ r and R 0 with respect to m 0 ), we set as a choice and as a de nition for the adjoint variables λ, λ 0 , µ s and µ r , the derivative of J with respect to λ, ξ, λ 0 , ξ 0 , µ s , S 0 , µ r and R 0 to zero, leading to eight equations λ = a 2 ξ, (4.27a) ξ(x, h) = p ξ 0 (x, h), (4.27b)

ξ 0 (x, h) = S * 0 (s, x -h, ω)R 0 (s, x + h, ω) ds dω, (4.27c) * 0 R 0 = R s (s, x, ω), (4.27d)
0 S 0 = S s (s, x, ω). (4.27e)

λ 0 (x, h) = * p λ = * p a 2 p ξ 0 , (4.27f)
= -32 2 m 2 a 2 2 z ξ 0 for laterally invariant models, (4.27g)

= a 2 ξ 0 for the adjoint version, (4.27h) *

0 µ s = λ 0 (x + h, h)R 0 (s, x + 2h, ω) dh, (4.27i
)

0 µ r = λ 0 (x -h, h)S 0 (s, x -2h, ω) dh. (4.27j)
equations (4.27a) to (4.27e) are solved in the reverse order. ξ 0 is obtained by crosscorrelating the forward wave eld S 0 with the back-propagated residual wave eld R 0 . Then, ξ is obtained by simply applying the di erential p operator onto ξ 0 . For the λ 0 variable, the adjoint of p should be applied to λ. In a laterally invariant model, m 0+ = m 0-= m 0 . As a(h) does not depend on x, λ 0 is obtained by applying the second derivative in z (equation 4.27g). As for S s and R s , µ s and µ r are solution of the same wave equation, but for di erent source terms for which the modi ed re ectivity λ 0 is activated by R 0 or S 0 (equations 4.27i and 4.27j). For the nal expression of the gradient, both 0 and p depend on m 0 , leading to three terms (equation 4.11b).

Optimal β parameter (appendix-C)

In the 1D case, we demonstrate that oscillatory terms in equation (4.14) cancel out. The strategy consists (1) of splitting the integrals over y from 0 to z max into two parts, and (2) of integrating by part. According to equation (4.14) Note that q(z, y) = p(z) if z ≥ y, and p( y) otherwise. We consider G 1 and rst split the integrals between 0 to z and between z to

, G β inv consists of three terms G 1 , G 2 and G 3 , namely G 1 (z) = γ v 0 z max 0 d y ( y -z e ) 2 g inv (p[ y])g inv (q[z, y]), (4.28a) G 2 (z) = γ 2 (β + 2)(z -z e ) 2 g 2 inv (p[z]), (4.28b) G 3 (z) = γ z max 0 d y ( y -z e )g inv (p[ y])g inv (q[z, y]), ( 4
z max G 1 (z) = γ v 0 z 0 ( y -z e ) 2 g inv (p[ y])g inv (q[z, y]) d y + γ v 0 z max z ( y -z e ) 2 g inv (p[ y])g inv (q[z, y]) d y, (4.30a) = γ v 0 g inv (p[z]) z 0 ( y -z e ) 2 g inv (p[ y]) d y + γ v 0 z max z ( y -z e ) 2 g inv (p[ y])g inv (p[ y]) d y.
(4.30b)

We have simpli ed q(z, y) depending if z ≥ y or not. The rst part of equation ( 4.30b) is integrated by parts, yielding

- γ 2 g inv (p[z]) ( y -z e ) 2 g inv (p[ y]) z 0 + γ g inv (p[z]) z 0 ( y -z e )g inv (p[ y]) d y = - γ 2 (z -z e ) 2 g 2 inv (p[z]) + γ z 0 d y ( y -z e )g inv (p[ y])g inv (q[z, y]) d y, (4.31)
In a similar way, we integrate the second part of equation (4.30b)

- γ 4 ( y -z e ) 2 g 2 inv (p[ y])) z max z + γ 2 z max z ( y -z e )g 2 inv (p[ y]) d y = γ 4 (z -z e ) 2 g 2 inv (p[z]) + γ 2 z max z ( y -z e )g inv (p[ y])g inv (q[z, y]) d y.
(4.32)

The integration by parts introduced a multiplication by -v 0 /2. We have supposed that

g inv (p[0]) = g inv (p[z max ]) = 0.
This is reasonable if z max if large enough. We combined the two parts and get

G 1 (z) = - γ 4 (z -z e ) 2 g 2 inv (p[z]) + γ z 0 ( y -z e )g inv (p[ y])g inv (q[z, y]) d y + γ 2 z max z ( y -z e )g inv (p[ y])g inv (q[z, y]) d y.
(4.33)

The total gradient reads

G β inv (z) =G 1 (z) + G 2 (z) + G 3 (z), (4.34) = γ 2 β + 3 2 G 2 (z) + 2γ z 0 ( y -z e )g inv (p[ y])g inv (q[z, y]) d y + 3γ 2 z max z ( y -z e )g inv (p[ y])g inv (q[z, y]) d y.
(4.35)

For β = -3/2, the rst oscillatory term is zero. By analogy with the horizontal contraction approach, the remaining terms have a smooth behaviour: for an objective function J 1 = 1/2 hξ(x, h) 2 , the adjoint source wave eld is ∂ J 1 /ξ = h 2 ξ. Modi ed versions have been proposed such as sign(h)h 2 ∂ ξ/∂ h and h∂ ξ/∂ h [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF]Shen and Symes, 2015). They are not associated any more to a gradient of an objective function, but usually lead to smooth gradients, except for example in the case of a low anomaly velocity (Shen and Symes, 2015). The equivalent in 1D would be (z -z e )∂ ξ/∂ z. This is exactly the term we nd in G β inv (z) when β = -3/2 (equation 4.35). As a conclusion, we do not prove that G β inv (z) is smooth, but has a similar behaviour as the terms in the horizontal contraction (Shen and Symes, 2015), while being a gradient of a speci c objective function.

This concludes the manuscript submitted to Geophysics.

Introduction of migration weights in the iterative migration process

In section 4.2, we have introduced an approximate inverse F † of the extended Born-modelling operator F. Applying this pseudo-inverse to observed data produces CIGs free of migration artefacts, leading to homogeneous gradients of the associated MVA objective function. In this section we explain how F † can be introduced in the iterative migration process. This modi cation is expected to accelerate the convergence speed of iterative migration, but not necessarily to stabilise the MVA gradient of the nested optimisation process. The weighted iterative scheme de ned here is di erent from the one proposed by Hou and Symes (2016a). Their strategy is based on the approximate inverse formula of [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF], which can be expressed as a modi cation of the usual adjoint with two weighting operators:

F † = W mod F T W data . (4.36a)
Then Hou and Symes (2016a) propose a modi ed iterative scheme using the standard linear conjugate gradient algorithm with new de nitions for the norms in the extended model and data spaces,

〈ξ | χ〉 W mod = 〈ξ | W mod χ〉 , (4.36b) 〈u | v〉 W data = 〈u | W data v〉 obs , (4.36c)
with the requirement that W mod and W data are positive de nite operators.

The inversion formula derived in section 4.2 cannot be decomposed in this way, so another strategy has to be de ned. Here we propose to use the approximate inverse F † as a right preconditioner. Note that the use of an amplitude-preserving migration operator as a preconditioner has already been studied in other contexts. [START_REF] Sevink | Three-Dimensional, Nonlinear, Asymptotic Seismic Inversion[END_REF] use it for iterative migration, but do not consider an extended model space. [START_REF] Métivier | Combining Asymptotic Linearized Inversion and Full Waveform Inversion[END_REF] use the Beylkin migration operator as a left preconditioner in FWI and show that it allows to accelerate the recovery of the short-scale structure of the velocity model.

In the following, we detail a preconditioned iterative scheme based on the inverse de ned in section 4.2, rst in the general non-linear case, then in the linear-case, corresponding to primaries modelled with the rst-order Born approximation.

Preconditioned iterative migration: non-linear case

In the general non-linear case, the migration objective function J 0 : → is de ned as (the implicit dependence of J 0 and P on c 0 is omitted in this section and in section 4.3.2 for the sake of clarity),

J 0 [ξ] = 1 2 MP[ξ] -P obs 2 obs + φ[ξ], (4.37)
where operator M : → obs selects the value of calculated data at receiver positions. Preconditioning may be seen as a change of variables. Let us de ne a new variable ξ ∈ obs such as

ξ = F † ξ, (4.38)
where F † is the pseudo-inverse of the extended Born modelling operator F, as de ned in section 4.2. Replacing ξ by F † ξ in equation (4.37) leads to a new objective function J 0 : obs → ,

J 0 ξ = 1 2 MP F † ξ -P obs 2 obs + φ F † ξ , (4.39) with        J 0 ξ = J 0 [ξ], ∂ J 0 ∂ ξ ξ = (F † ) T ∂ J 0 ∂ ξ [ξ] (4.40a) (4.40b)
The preconditioned iterative migration scheme now consists of nding the value of ξ in the data space minimising J 0 . The corresponding value in the image space is a minimiser of J 0 and can be computed as ξ = F † ξ. The inclusion of the preconditioner in the minimisation algorithm is easy. The algorithm is initiated with ξ (1) = 0 in , so that the initial value in obs is ξ (1) = 0. Then at each iteration, two steps are added to the classic algorithm, so that the value of ξ (n+1) is updated from ξ (n) in four steps:

(i) given ξ (n) ∈ obs , compute the corresponding value ξ (n) ∈ according to ξ (n) = F † ξ (n) ;

(ii) compute the value and gradient of the original objective function: J 0 [ξ (n) ] and g

(n) = ∂ J 0 /∂ ξ[ξ (n) ] ∈ ;
(iii) using equations (4.40b) and (4.40b), go back to the obs -space to determine the value of the new objective function J 0 , and its gradient g (n) ∈ obs with respect to ξ;

(iv) determine a descent direction d (n) ∈ obs and a step length α (n) ∈ to update ξ,

ξ (n+1) = ξ (n) + α (n) d (n) , (4.41)
and go back to step (i).

This optimisation algorithm is the same as the one described in chapter 2, except that it is performed in the observed data space instead of the extended model space. Only two additional calculations are needed: one to go from obs to (step (i)) where the usual gradient is computed, and one to go back to obs (step (iii)). This operation requires the ability to compute the adjoint (F † ) T of the approximate inverse F † . The calculation of this operator is detailed in appendix C. It can be interpreted as a weighted modelling favouring small re ection angles. Note that if the algorithm was initiated with a non-zero value ξ (1) , we would need to compute the inverse (F † ) -1 of F † to compute the corresponding value in the obs -space. Starting with ξ (1) = 0, this operator does not need to be computed.

The main computational cost remains the computation of the usual gradient in (resolution of seven wave-equations, as described in table 2.2). With preconditioning, four additional wave-equations have to be solved (table 4.2), two for the application of F † to ξ (n) and two for the application of (F † ) T to ∂ J 0 /∂ ξ[ξ (n) ] (appendix C).

F † obs → 2 (F † ) T → obs 2 (F † ) T F † → 3 Table 4.2.
-Number of wave-equations to be solved to compute the inverse map F † , its adjoint (F † ) T and the composition of the two operators. The number indicated here should be multiplied by the number of source positions considered in the acquisition.

In the non-linear case of multiples, the e ciency of the preconditioned strategy will be demonstrated on a numerical example (section 4.4.2). We now consider the special linear case of primaries only, where we explain theoretically why we expect the choice of F † as preconditioner to accelerate the convergence.

Preconditioned iterative migration: linear case

We now consider the linear case, where the modelling is described by the linear extended Born modelling operator F : → obs and the regularisation is the 2 -norm. In this speci c case, the implementation simpli es and the bene ts of the preconditioner are easier to understand.

The classic (equation 4.37) and preconditioned (equation 4.39) versions of the objective function read

     J 0 [ξ] = 1 2 Fξ -P obs 2 obs + a φ 1 2 ξ 2 , J 0 ξ = 1 2 FF † ξ -P obs 2 obs + a φ 1 2 F † ξ 2 . (4.42a) (4.42b)
Deriving these expressions with respect to ξ and ξ respectively lead to two linear systems

F T F + a φ I ξ = F T P obs , (F † ) T F T F + a φ I F † ξ = (F † ) T F T P obs . (4.43a) (4.43b)
The linear system (equation 4.43b) can be solved with a classic conjugate-gradient (CG) algorithm (Nocedal and Wright, 2006, p. 112). Note that this system is de ned in the obs -space, contrary to the classic linear system (equation 4.43a).

As F † is a pseudo inverse of F, the operator (F † ) T F T FF † = (FF † ) T (FF † ) should be close to the identity operator. Then, provided that the regularisation weight a φ is relatively small, the resolution of this new linear system should be much faster and should require less iterations.

In practice, the resolution of the preconditioned system (4.43b) with the classic CG-algorithm is equivalent to the resolution of the usual system (4.43a) with the preconditioned CG-algorithm (Nocedal and Wright, 2006, p. 119) and the preconditioner F † (F † ) T : → . This has the advantage of fewer modi cations in the implementation, in particular the resolution is still performed in the -space. Compared to the usual CG-algorithm, the additional computational cost is the product at each iteration of F † (F † ) T with a vector of , requiring three additional wave-equations to be solved (table 4.2).

Preconditioned resolution of the linear adjoint problem

We now consider the computation of the gradient of the MVA objective function studied in chapter 3. Using method A, introducing the preconditioner would change the equations to be solved and would add new contributions to the total gradient, as F † and (F † ) T depend on c 0 . In the following, we consider only method B where the nal re ectivity is de ned as the exact solution of the linear system

∂ J 0 ∂ ξ (ξ ∞ ) = 0.
Then introducing the preconditioner does not change the computation of the gradient, but only allows to nd an approximate solution of this linear system in a reduced number of iterations.

Moreover, the preconditioner can also be used to accelerate the resolution of the adjoint system, which is a linear problem

∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) η = A T Aξ (N+1) , (4.44)
where the adjoint variable is noted η from now on. In the linear case, this system is very similar to the direct problem (equation 4.43a),

F T F + a φ I η = A T Aξ (N+1) . (4.45)
Then the preconditioned CG-algorithm with F † (F † ) T as preconditioner can be used to solve the adjoint problem, both in the linear and non-linear cases. In the linear case at least, the preconditioner should in theory accelerate the resolution of the adjoint problem. This will be tested in section 4.4.3.

In summary, preconditioning does not change the computation of the gradient of the MVA objective function with method B (equation 3.12). The calculation only uses the last iterates ξ (N+1) and η (N+1,M+1) of the direct and adjoint problems. Preconditioning only provides a more e cient way to nd a solution to these problems in fewer iterations.

Synthetic examples

In this section, the behaviour of preconditioned iterative migration is illustrated on simple examples similar to the ones presented in chapters 2 and 3.

We rst consider the case of primary re ections only. Note that in this case, iterative migration is not necessarily required as direct inversion (section 4.2) already provides a satisfactory re ectivity image. The primary-only case is studied here to test the e ciency of the preconditioner to accelerate the convergence of iterative migration. Then we consider a case with rst-order surface-related multiples to determine if the preconditioner also improves the convergence rate of migration although it has not been designed for multiples. Finally, we investigate the e ciency of the preconditioner for the resolution of the adjoint problem with primaries only. We also compare the background velocity updates obtained with direct and iterative inversions.

Preconditioned iterative migration with primaries only

In this rst example, we consider primary re ections only and a simple model with a single re ector located at 300 m depth in a homogeneous velocity model (3000 m/s).

First iterate of the preconditioned iterative scheme

We rst compare the results obtained with the inverse formula derived in section 4.2 to the results of a single iteration of preconditioned iterative migration. The corresponding re ectivity images are de ned as F † P obs and F † (F † ) T F T P obs = F † (FF † ) T P obs , respectively. Note that the approximate inverse has been designed such that (FF † ) is close to the identity operator in the observed data space. Then we expect its transpose (FF † ) T to be close to identity as well, and thus the rst iterate should be close to the result of direct inversion.

Before looking at the re ectivity images, we compare the result of application of the operators (FF † ) and (FF † ) T to observed data in a too low, correct and too high velocity model ( gure 4.18). All results are expected to be very close to observed data. In practice, both operators correctly reproduce input data and residuals are mainly located at large surface o sets. However the transposed version (FF † ) T yields low-frequency artefacts before the primary event, especially for a too high velocity model. These artefacts might have undesirable e ects on CIGs.

We now consider the application of the approximate inverse to three data sets: (1) observed data, (2) result of application of (FF † ) to observed data and (3) result of application of (FF † ) T to observed data ( gure 4.19). The rst case corresponds to direct inversion and the third one to the rst iterate of preconditioned iterative migration, while the second case is considered only for comparison. The CIGs obtained in the second case are closer to the result of direct inversion than the rst iterate of the preconditioned scheme. In particular the low frequency artefacts observed on reconstructed data result in low frequency energy above the re ector in the case of a too high-velocity. This e ect is less visible for a correct and a too low initial velocity. Nonetheless, these CIGs show much improvement compared to the adjoint F T P obs , rst iterate of the non-preconditioned scheme. In particular, migration artefacts are greatly attenuated.

Eventually we plot the value of the normalised MVA objective function obtained in homogeneous background velocity models after one iteration with and without preconditioning, and the one obtained after direct inversion ( gure 4.20). With the inverse, the function is perfectly centred at the correct velocity model. Because of residual energy above the re ector, the minimum after one preconditioned iteration is not at the correct velocity model, but it is still more satisfactory than in the case of a single iteration without preconditioning.

This suggests that a single iteration of preconditioned iterative migration is not as e cient as direct inversion, and that additional iterations are required. This is actually not an issue as the preconditioned iterative scheme is not meant to be used in this primary-only case but in the more complex case of multiples for which iterations are required anyway to attenuate cross-talk artefacts.

Results a er a few iterations

We now perform ten preconditioned iterations on the same model ( gure 4.21). The main e ects of iterations on CIGs are additional deconvolution, strengthening of energy at large values of h (in the case of an incorrect velocity model) and attenuation of residual energy above the re ector. We plot the value of the objective function across iterations obtained with and without preconditioning ( gure 4.22). We consider the data mis t, as well as the norm of the residuals associated with the linear system (4.43a): F T F + a φ I ξ -F T P obs , which is actually the value of the gradient of J 0 with respect to ξ. As expected, the preconditioned strategy is faster at the rst iteration, but both versions reach the same level of data mis t after a few iterations. As in [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF], we consider a third possibility: iterative migration is performed without preconditioning but with the approximate inverse as initial guess F † P obs . In this case very few progress is made at the rst iteration because the approximate inverse is already a very good solution to the minimisation problem. Note that we cannot consider the preconditioned iterative minimisation with the approximate inverse as initial value because this would require the ability to compute (F † ) -1 .

Eventually we compute the value of the normalised MVA objective function for several homogeneous background velocity models ( gure 4.23) after several iterations of the preconditioned scheme. We saw that the minimum of the function does not correspond to the correct value of c 0 after a single iteration ( gure 4.20). This is progressively corrected and after ve iterations, the objective function of MVA is stable and minimum for the correct velocity.

In summary, we have shown that a single iteration of preconditioned iterative migration (FF † ) T (3rd column). We also consider the application of the adjoint operator F T to observed data (4th column). The CIGs of the rst and third columns correspond to the results obtained by direct inversion as in section 4.2, and to the rst iterate of the preconditioned iterative scheme described in section 4.3, respectively. The CIGs of the second and fourth columns are displayed for comparison. velocity (m/s)

|h| ξ (N+1) 2 ξ (N+1) 2
one iteration without preconditioning F T P obs one iteration with preconditioning F † (FF † ) T P obs approximate inverse F † P obs -Results of iterative migration in a too slow velocity model (2500 m/s) with primaries only. We plot the relative data mis t (left) and the relative normal residual associated with the linear system (4.43a). Red and blue curves correspond to the case of a zero initial guess (ξ (1) = 0) with classical and preconditioned iterative migration, respectively. The green curve corresponds to non-preconditioned migration initiated with the approximate inverse ξ (1) = F † P obs . The dashed line corresponds to the value obtained for the approximate inverse F † P obs .

2600 2800 3000 3200 3400 velocity (m/s) yields slightly degraded results compared to direct inversion because (FF † ) T is close but not equal to the identity. Its application to observed data yield small low-frequency artefacts. However the CIGs obtain after a single iteration with the preconditioner is free of migration artefacts, contrary to the non-preconditioned case. Besides the preconditioner also improves the convergence rate of iterative migration.

|h| ξ (N+1) 2 ξ (N+1) 2 N = 1 N = 5 N = 2 N = 10

Preconditioned iterative migration with multiples

We now discuss an example with rst-order surface-related multiples. We consider an exact model with a background velocity increasing with depth, from 2000 m/s at the surface to 3000 m/s at 1000 m depth ( gure 4.24a). The model is 2D but laterally invariant. The re ectivity model consists of a single re ector located at 475 m depth. Contrary to the case of homogeneous models considered in the preceding chapters, we expect events in CIGs corresponding to primaries and multiples (interpreted as primaries) to have di erent focusing behaviour. In particular, cross-talk artefacts should not be focused for the correct velocity model and should favour too low velocities. We consider several initial velocity models linearly increasing with depth. All start at the correct velocity at the surface (2000 m/s), but the velocity at 1000 m depth ranges from 2200 m/s to 3300 m/s ( gure 4.24a). We rst compute CIGs by applying the approximate inverse formula to observed data ( gure 4.24c) and plot the corresponding MVA objective function ( gure 4.24b). As observed data are computed with the Born approximation, primaries and multiples are available separately and we also plot the value of the objective function obtained for observed data containing primaries only or multiples only with the same range of initial velocity models ( gure 4.24a). In the case of primaries only, the objective function is minimum for the correct velocity, similarly to the results of section 4.4.1. Cross-talk artefacts linked to multiples focus for a too low velocity model ( gure 4.24c, 3rd column) and curve upward for the correct velocity model ( gure 4.24c, 4th column). As a consequence, the minimum of the MVA objective function considering both primaries and multiples is not obtained for the correct velocity model.

We now perform ten iteration of iterative migration without and with preconditioning using the highest initial velocity model considered in gure 4.24. As in the primaries-only example, we also consider the case of iterative migration without preconditioning starting with the approximate inverse F † P obs as initial guess. To help removing residual energy at large o set, we add regularisation with the Huber norm [START_REF] Guitton | Robust Inversion of Seismic Data Using the Huber Norm[END_REF]. In the rst iterations, the migration objective function decreases faster with preconditioning and with the strategy initialised with the approximate inverse ( gure 4.26), but the data mis t is similar for the three cases after a few iterations. The nal CIGs obtained with the three strategies are very similar ( gure 4.25), but the best attenuation of cross-talk artefacts is obtained in the preconditioned case. This is the optimisation strategy used in the following of the study.

Another practical advantage of the preconditioned strategy is related with the determination of an optimal step length. A general issue of linesearch techniques is the choice of an initial guess α 0 initiating the algorithm. Using the preconditioner, the linesearch strategy of Moré and Thuente (1994) returns step lengths in the interval [0.1, 1] at each iteration. Hence α 0 = 1 is always a good initial guess. Besides we can even directly set α = 1 for the rst iteration and save the computational cost of a linesearch. The rst gradient is very close to the approximate inverse, hence setting α = 1 yield a re ectivity model ξ (2) very well explaining primaries, so that residuals are mostly due to multiples. Finally we consider the same range of initial velocity models as in gure 4.24b and we compute the value of the MVA objective function obtained after several iterations of preconditioned iterative migration ( gure 4.27). After a single iteration, we obtain a result similar to the direct inversion case with a minimum obtained for a too low velocity model. With iterations cross-talk artefacts are attenuated and the minimum is progressively shifted to the correct velocity model.

Preconditioned resolution of the adjoint problem and associated gradient

Eventually we consider the computation of the gradient of J 1 with method B as described in section 3.2.2. We go back to the primaries-only example of section 4.4.1 and test if the preconditioner accelerates the convergence rate of the adjoint problem. We will show that the results are not fully satisfactory; hence we defer to chapter 5 the numerical applications in the case of multiples (section 5.3.2). We assume that the direct problem has been solved in seven iterations, meaning that the nal re ectivity is ξ (8) . The adjoint problem then consists of nding the solution η ∈ of Hη = b, (4.46) where H is the Hessian matrix of J 0 , which is independent of ξ in the linear case, and b = A T Aξ (8) is the image residual. A is the annihilator consisting of a multiplication by the subsurface o set and a power of the background velocity model,

[Aξ](x, h) = c β 0 (x)|h|ξ(x, h). ( 4

.47)

We choose β = 3/2, the value resulting in smooth gradients free of artefacts around the re ector in the case of direct inversion (section 4.2). The adjoint problem is solved with ten iterations of the conjugate gradient algorithm with and without preconditioner. We compare the convergence rate obtained with both strategies. Contrary to the direct problem, there is no norm in the observed data-space obs associated with the resolution of equation ( 4.46) which can be used as a convergence criteria. Here the conjugate gradient algorithm actually minimises

φ[η] = 1 2 〈η | Hη〉 -〈b | η〉 . ( 4

.48)

A di culty with this objective function is that we do not know the value obtained for the ideal solution of the linear problem. Therefore we will also look at the relative normal residuals of the linear system (4.46), de ned as

J Normal Res. [ξ] = Hξ -b b , (4.49)
which can be interpreted as the normalised norm of the gradient of φ (equation 4.48). Note that contrary to φ, there is no guarantee that the value of the normal residual decreases at each iteration.

Small regularisation

We rst consider a case with small regularisation, as in section 4.4.1. Using the same right-hand side term b, that is the same value for ξ (8) , we solve the adjoint problem in ten iterations, without and with preconditioning ( gure 4.28). We obtain a much lower convergence rate than for the direct problem, even with the preconditioner. The normal residual associated with the last value η (8,11) is just a little lower than for the initial value η (8,1) . At the last iteration, the adjoint variable η (8,11) is very oscillating ( gure 4.29), and residuals of the linear system are located mainly near the surface, that is away from the re ector position. This is related to the observations made in chapter 3: weak energy at large values of h, which does not bear relevant kinematic information, has large in uence on the resolution of the adjoint problem. Besides, note that the zero value at h = 0 in b is not recovered in Hη (8,11) . An undesirable consequence of the slow resolution of the adjoint problem is that the value of the associated gradient G (7,M) does not converge to a stable nal value ( gures 4.30 and 4.31). The sign of the gradient may even change across iterations (for example G (7,4) in the preconditioned case). Without preconditioning, the gradients exhibit vertical spurious oscillations very similar to those observed in the preceding chapter. When the adjoint problem is solved with the preconditioner we do not observe these artefacts. However the nal value is quite di erent from the one obtained after direct inversion, which is much smoother and homogeneous ( gure 4.32). More investigation is needed to understand these results (chapter 5). In the following we examine if a stronger regularisation helps to overcome these di culties.

Stronger regularisation

As in chapter 3, we consider increasing the value of the regularisation coe cient a φ . The convergence of the adjoint problem is improved but still much slower than for the direct problem ( gure 4.33). The preconditioner does not clearly improve the convergence rate. This can be explained by the fact that the operator (F † ) T (F T F + a φ I)F † is theoretically close to the identity only if the regularisation weight is not too high. The nal value η (8,11) is still very oscillating ( gure 4.34). Although the residuals of the linear system are weaker in the preconditioned case, the zero value at h = 0 in b is not recovered in Hη (8) and there are still strong residuals away from the re ector.

With stronger regularisation, the spurious oscillations appearing when no preconditioning is applied ( gure 4.30) are greatly attenuated ( gure 4.35), as already noticed in chapter 3. With η (8,11) Hη (8,11) b (8) without preconditioner

Hη (8,11) the preconditioner, the gradients are smoother and more homogeneous ( gures 4.35 and 4.36), but do not converge to a stable solution. Besides they are still quite di erent from the gradient obtained by direct inversion ( gure 4.37). In particular there are still residual oscillations around the re ector position.

As a conclusion, we note that although direct inversion and iterative migration yield very similar re ectivity images, the associated gradients are quite di erent. In particular, the use of the preconditioner does not relax the need for strong regularisation to obtain smooth velocity updates. An other issue associated with the iterative case is the di cult resolution of the adjoint problem, and the instability of the associated gradient. Further investigation is needed to understand this unwanted behaviour and design a more robust strategy. This is the purpose of chapter 5.

Conclusion

In this chapter, we have introduced an approximate inverse of the extended Born modelling operator. It is similar to the one proposed by [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF], the main di erence being that the derivation presented here uses a linearisation of the phase of FF † . The implementation of the approximate inverse resembles the adjoint's one with a similar computational cost. A direct inversion strategy consisting of applying this operator to observed data yields CIGs free of migration artefacts. Introducing a speci c power of the background velocity update in the associated MVA objective function results in a smooth and homogeneous gradient free of depth (m) wihout preconditioner G (7,1) with preconditioner depth (m) G (7,2) depth (m) G (7,3) depth (m) G (7,4) depth (m) G (7,5) depth (m) G (7,6) depth (m) G (7,7) depth (m) G (7,8) depth (m) G (7,9) 0 500 1000 1500

x-position (m) depth (m) 0 500 1000 1500 G (7,10) x-position (m) with preconditioner η (8,11) Hη (8,11) b (8) without preconditioner

M = 1 M = 3 M = 5 M = 7 M = 9 M = 2 M = 4 M = 6 M = 8 M = 10
Hη (8,11) The approximate inverse operator has been designed for primaries only. Iterative migration is not required any more in this case, at least for regularly sampled input data, but is still needed to handle multiple re ections. The approximate inverse has been introduced as a preconditioner to accelerate the resolution of both the direct and adjoint problems. Numerical examples show that the preconditioner allows a faster convergence of the direct problem. Despite the similarity between the direct and adjoint problems, the convergence rate of the latter is much slower, even with the preconditioner. As a consequence the gradient obtained for successive values of the adjoint problem is not stable and is quite di erent from the one obtained after direct inversion. Larger regularisation improves the smoothness of the gradient, but this is not a fully satisfactory solution as the adjoint problem still converges much slower than the direct problem. Besides, the preconditioner is not helpful when strong regularisation is applied.

These results may appear somehow surprising as direct inversion and iterative migration provide very similar re ectivity images. If we refer to chapter 3, the reason is that additional deconvolution in the case of iterative migration results in small oscillations in the nal re ectivity away from the re ector. These di erences allow to further reduce the data mis t but have large in uence on the MVA objective function and its gradient, leading to very di erent background velocity updates.

To better understand these issues and to improve the stability of the gradient computation in the iterative case, we conduct in the next chapter a pure 1D-analysis similar to the one presented in section 4.2.5 and propose a slightly modi ed MVA objective function leading to a more stable procedure. wihout preconditioner G (7,1) with preconditioner 0 200 400 depth (m) G (7,2) 0 200 400 depth (m) G (7,3) 0 200 400 depth (m) G (7,4) 0 200 400 depth (m) G (7,5) 0 200 400 depth (m) G (7,6) 0 200 400 depth (m) G (7,7) 0 200 400 depth (m) G (7,8) 0 200 400 depth (m) G (7,9) G (7,10) x-position (m) with preconditioner de calcul bien moindre. En outre, il permet une représentation des opérateurs de modélisation et de migration sous forme de matrices.

M = 1 M = 3 M = 5 M = 7 M = 9 M = 2 M = 4 M = 6 M = 8 M = 10
Comme dans le cas 2D, les instabilités dans le calcul du gradient sont liées à l'apparition d'énergie loin du ré ecteur et non porteuse d'information cinématique, mais qui in uence grandement la fonction coût de la DSO. Pour stabiliser ce calcul, je propose de faire porter la régularisation non pas sur la boucle interne comme étudié au chapitre 3, mais sur la boucle externe. La régularisation prend ici la forme d'un opérateur appliqué au résultat de la migration itérative avant pénalisation par l'annihilateur. Cet opérateur peut être interprété comme un ltre atténuant l'énergie non porteuse d'information cinématique dans le modèle de ré ectivité. Il est dé ni ici par F † F, composition du pseudo-inverse avec l'opérateur de modélisation. Il est important de noter que F † est construit de telle sorte que FF † soit proche de l'opérateur identité dans le domaine des données. En revanche F † F est di érent de l'identité dans le domaine image, en particulier il atténue les évènements dans les CIGs qui perturbent l'analyse de vitesse.

L'e et de cette régularisation est illustré dans les cas 1D et 2D. Il est possible de montrer qu'appliquer l'opérateur F † F au résultat de la migration itérative produit, dans le cas de primaires seuls, un modèle de ré ectivité étendu proche du résultat obtenu par application directe du pseudo-inverse aux données observées. Le calcul du gradient de la fonction coût externe est aussi beaucoup plus stable, sans condition particulière sur le poids donné à la régularisation dans la boucle interne, et le résultat également proche de celui obtenu dans le cas de l'inversion « directe ». L'opérateur F † F étant fonction du macro-modèle, des termes additionnels doivent être calculés pour obtenir le gradient. Cependant la résolution du problème adjoint étant accélérée par l'introduction de cet opérateur, le coût numérique total de la méthode est diminué avec l'ajout de la régularisation.

Dans le cas de multiples, la régularisation sur la boucle externe stabilise aussi le calcul du gradient, mais la régularisation sur la boucle interne demeure essentielle. Finalement, une stratégie alternative est étudiée dans le cas des multiples. Elle ne consiste à calculer qu'une seule contribution du gradient total de la boucle externe ; la remise à jour ainsi obtenue est dénommée « gradient tronqué » dans la suite du manuscrit. En particulier la résolution du problème adjoint n'est plus nécessaire, ce qui réduit grandement le coût de calcul de la remise à jour du macromodèle. Physiquement, cette simpli cation signi e que la migration itérative est utilisée pour obtenir un modèle de ré ectivité dans le domaine étendu sans artefacts de migration et sans artefacts dus aux multiples. Ce modèle de ré ectivité est utilisé pour régénérer un nouveau jeu de données constitué de ré exions primaires uniquement, cinématiquement cohérentes avec celles enregistrées dans les données observées. Finalement la stratégie d'inversion directe décrite dans la section 4.2 est appliquée à ce nouveau jeu de données.

Introduction

In this chapter, we analyse on a simple 1D case the behaviour of iterative migration velocity analysis. The advantage of this simple setting is the possibility to explicitly build modelling and migration operator and to run a large number of iterations. First we summarise the results obtained in 2D in the preceding chapters.

• In chapter 2, iterative migration has been introduced and we have shown that migration artefacts as well as cross-talk artefacts appearing on CIGs at the rst iteration are progressively attenuated ( gure 2.15);

• In chapter 3, we compared two methods yielding an approximate gradient of the MVA objective function after iterative migration. In the rst approach, we derived the exact gradient of an approximation of the ideal objective function. In the second one, we compute an approximate gradient of this ideal objective function. The second approach has a simpler implementation. The gradient is obtained in two steps. First an adjoint variable in the extended model space has to be computed as the solution of a linear problem, even in the presence of multiples. Then the gradient is computed using the last iterate of the direct and adjoint problems. Numerical evidence show that regularisation has to be introduced in iterative migration to ensure the smoothness of the gradient ( gures 3.5 and 3.7) and to prevent the apparition of spurious oscillations;

• In chapter 4, we have introduced an approximate inverse of the extended Born modelling operator. It is derived under the high frequency approximation, but involves waveequations operators only. For data containing primaries only, the application of this pseudo-inverse to observed data yields CIGs free of migration artefacts. The gradient of the associated MVA objective function is smooth and free of oscillations around the re ectors, provided that a speci c power of the background velocity is introduced in the annihilator. We showed how this approximate inverse can be used as a preconditioner to speed up the resolution of iterative migration ( gure 4.22), needed in the presence of multiples.

However several di culties remain, investigated in this chapter:

• If the modelling is linear, the direct and adjoint problems are very similar. They consist of the same system with two di erent source terms, Hξ = a for the direct problem, (5.1a) Hη = b for the adjoint problem, (5.1b) where H is the Hessian matrix of the migration objective function. However the resolution of the adjoint problem requires in practice more iterations than the direct problem. Using the approximate inverse operator as a preconditioner, a single iteration already provides a very good solution for the direct problem, but this is not the case for the adjoint problem ( gure 4.28);

• Although direct and iterative inversion result in very similar re ectivity images, they lead to quite di erent background velocity updates ( gure 4.32). In the iterative case, the gradient still has oscillations around the re ector and modifying the MVA objective function with a power of the background velocity does not allow to remove these oscillations.

In order to gain a better understanding of these issues and to propose solutions addressing them, we study here a 1D-case de ned in the same fashion as in section 4.2. Although the 1Dcase cannot reproduce all the features of 2D subsurface-o set extended migration, its reduced dimensionality allows to explicitly build and display modelling and inverse operators as matrices, and to considerably reduce computation time, so that performing tens or hundreds of iterations for solving both the direct and adjoint problems is a ordable. For example, it becomes possible to study the distribution of eigenvalues of the normal operator.

In this analysis we consider only the linear case of primaries. We have shown that direct inversion works already well in this case; the objective here is to understand why the iterative case is unstable and to propose a robust solution to this issue. At the end of the chapter, we discuss on a 2D numerical example how the conclusions can be extended to the case of multiples. We begin by presenting the 1D MVA approach. After a brief review of direct inversion with the approximate inverse operator, we detail the computation of the gradient in the case of iterative migration. A simple numerical example illustrates that the 1D and 2D cases have a similar behaviour. Eventually we propose a modi cation of the iterative MVA procedure to alleviate the di culties listed above.

In the second part of the chapter, this new strategy is applied on 2D examples, rst on primaries only, then on an example with primaries and rst-order surface multiples. The objective is to determine if the new approach de ned in 1D exhibits the same bene ts in the 2D case.

Analysis on a pure 1D case

Presentation of the 1D case

We consider a pure 1D case, similar to the one studied in section 4.2.5, with a single source and a single receiver located at the same position. In 2D, an extended-model space is de ned by adding an extension parameter to the model space . This is not necessary in 1D as the model space and the observed data space obs already have the same dimension, meaning that vectors of = are parametrised only by depth z. However we keep distinct notations and for consistency with the 2D case. In the following, the re ectivity is de ned as a perturbation of the background velocity model1 ,

ξ(z) = δc(z) 2c 0 (z) , (5.2)
and the re ectivity model consists of a single re ector located at z ex with a re ection coe cient R ex , ξ(z) = R ex δ(z -z ex ).

(5.

3)

The annihilator A : → is diagonal and de ned as a multiplication by the distance to the exact re ector position, (5.4) where z ex is the exact depth of the re ector. Note that this 1D-MVA approach is valid for re ectivity models with a single re ector only, but has a similar structure as the classical 2D DSO formulation. Then the MVA objective function is de ned in the general form as .5) where the diagonal operator C : → consists of a multiplication by c β 0 . As for the 2D case, using a speci c value for β may allow to remove unwanted oscillations in the gradient of J 1 (section 4.2).

[Aξ](z) = a(z)ξ(z) = |z -z ex |ξ(z),
J 1 [c 0 ] = 1 2 ACξ[c 0 ] 2 , ( 5
Here we consider the case where ξ is de ned as the result of iterative minimisation of

J 0 [c 0 , ξ] = 1 2 P[c 0 , ξ] -P obs 2 obs + a φ φ[ξ] = 1 2 F[c 0 ]ξ -P obs 2 obs + a φ φ[ξ],
( 5.6) for a xed background velocity model c 0 , with F : → obs the Born modelling operator and φ : → the regularisation function.

The source wavelet and its approximate inverse are noted Ω and Ω respectively. In the numerical example, Ω is a Ricker with maximal frequency 40 Hz ( gure 5.1). The exact and the initial velocity model are homogeneous (c exact 0 = 3000 m/s and c 0 = 2500 m/s). The re ectivity consists of a single re ector located at z ex = 400 m with R ex = 0.2. To be consistent with the 2D-implementation, we introduce a taper T (z) ( gure 5.2) in the -space by considering that J 0 is function of (T • ξ) instead of ξ. The taper will not be indicated in the following equations for the sake of clarity, but is taken into account in the implementation, including for the gradient expressions.

With the assumption of single scattering, data depend linearly on the re ectivity model and, for a xed velocity model c 0 , we can explicitly construct the (rectangular) matrices ( gure 5.3) corresponding to the following operators • F : → obs : modelling of primary re ections; • F T : obs → : migration of observed data; • F † : obs → : approximate inverse of F, de ned in section 4.2; • (F † ) T : → obs : adjoint of F † , which can be seen as a modi ed modelling operator.

Given ξ ∈ and P ∈ obs , these operators can be expressed with Green's functions, as in

                           Fξ (ω) = 4 c 2 0 (z) z (iω) 2 Ω(ω)G 2 0 (s, z, ω)ξ(z) dz, F T P (z) = 4 c 2 0 (z) ω (iω) 2 Ω * (ω)G * 2 0 (s, z, ω)P(ω) dω, F † P (z) = -4 ∂ ∂ z ω 1 (iω) Ω * (ω) ∂ ∂ z s G * 0 (s, z, ω) 2 P(ω) dω, (F † ) T ξ (ω) = 4 z 1 (iω) * Ω(ω) ∂ ∂ z s G 0 (s, z, ω) 2 ∂ ∂ z ξ(z) dz.
(5.7a)

(5.7b)

(5.7c) (5.7d) In essence, the adjoint and inverse expressions are very similar, except that Ω(ω) is replaced by Ω(ω) with a rst-order integration in time, and that two vertical derivatives are applied at the image point z and at the surface. The same observation holds for the forward modelling and the adjoint of the inverse operator.

In a homogeneous velocity model c 0 , the 1D Green's function reads

G 0 (s, z, ω) = c 0 2iω e -iωτ(s,z) , (5.8)
where τ(s, z) is the traveltime between the source s and the subsurface point z. The geometrical spreading is constant in 1D. Then we can easily derive explicit expressions for the four linear 2iωτ(s,z) ξ(z) dz,

operators                            Fξ (ω) = z Ω(ω)e -
F T P (z) =
ω Ω * (ω)e 2iωτ(s,z) P(ω) dω, 2iωτ(s,z) ξ(z) dz.

F † P (z) = 2 c 0 ω Ω * (ω)e 2iωτ(s,z) P(ω) dω, (F † ) T ξ (ω) = 2 c 0 z Ω(ω)e -
(5.9a)

(5.9b)

(5.9c) (5.9d) Compared to the adjoint F T , the approximate inverse F † introduces two modi cations: deconvolution of the source wavelet ( Ω instead of Ω) and correction of the amplitudes by a coe cient 2/c 0 . In 1D there is no need to correct for geometrical spreading nor for uneven illumination.

Direct inversion

Before analysing the iterative case, we brie y review the results obtained with direct inversion.

Here the re ectivity is de ned as the result of the application of the adjoint F T or inverse operator F † to observed data P obs : ξ mig = F T P obs , ξ inv = F † P obs .

(5.10a) (5.10b) We can control the accuracy of the approximate inverse F † by applying the modelling operator F to ξ inv ( gure 5.4). Recomputed data perfectly match observed data, which is not the case with the adjoint ξ mig . This indicates that F † is an accurate right inverse of F (FF † I). As already mentioned in chapter 4, F † is however not a left inverse of F (F † F = I) because the null space of the modelling operator F is not empty. It means that several di erent re ectivity models can lead to similar data set. For example, applying F † F to a spike re ectivity (as the red curve in to observed data (left). Observed data and data modelled from the adjoint and inverse re ectivity section (right).

We now derive analytic expressions for the gradient of the MVA objective function obtained when the re ectivity is de ned as F T P obs or F † P obs . The objective is to reproduce the results obtained with the procedure described in section 4.2.11 where wave-equations were solved with a nite di erence propagation code. To compute the background velocity update, we construct the following Lagrangians with the adjoint variables η mig and η inv in .

     J 1 [c 0 ] = 1 2 ACξ mig 2 -η mig ξ mig -F T P obs , J 1 [c 0 ] = 1 2 ACξ inv 2 -η inv ξ inv -F † P obs .
(5.11a)

(5.11b)

The adjoint variables η mig/inv are obtained by deriving the Lagrangians with respect to the state variables ξ mig/inv ,

η mig = C T A T ACξ mig , η inv = C T A T ACξ inv ,
(5.12a) (5.12b) and nally the gradient of J 1 with respect to c 0 is given by

         G mig ( y) = β -2 c 2β-1 0 ( y)a 2 ( y)ξ 2 mig ( y) + z Q mig ( y, z)η mig (z) dz, G inv ( y) = βc 2β-1 0 ( y)a 2 ( y)ξ 2 inv ( y) + z Q inv ( y, z)η inv (z) dz,
(5.13a) (5.13b) where the matrices Q mig : → and Q inv : → are de ned as

Q mig ( y, z) = ∂ F T P obs (z) ∂ c 0 ( y) , (5.14a) Q inv ( y, z) = ∂ F † P obs (z) ∂ c 0 ( y) . (5.14b)
The term βc 2β-1 0 a 2 ξ 2 in G mig and G inv originates from operator C. Note that there is an additional term -2c

2β-1 0 a 2 ξ 2 in G mig due to the coe cient 4/c 2 0 in the de nition of the adjoint operator (equation 5.7b).

After derivations reported in appendix D, we obtain the following expressions for Q mig and Q inv ,

Q mig ( y, z) = 16 c 3 0 ( y)c 2 0 (z) ω (iω) 4 Ω * (ω)G * 0 (s, z, ω)G * 0 (s, y, ω)G * 0 ( y, z, ω) dω (5.15a) = - 2 c 2 0 ω
(iω)Ω * (ω)e (iω) τ(s,z)+τ(s, y)+τ( y,z) P obs (ω) dω, (5.15b)

Q inv ( y, z) =    32 c 2 0 (s)c 3 0 ( y)c 0 (z) ω (iω) 4 Ω * (ω)G * 0 (s, z)G * 0 (s, y)G * 0 ( y, z) dω if y < z, 0 if y ≥ z, (5.15c) =    - 4 c 3 0 ω
(iω) Ω * (ω)e (iω) τ(s,z)+τ(s, y)+τ( y,z) P obs (ω) dω if y < z, (5.15d) where we have omitted the dependence of the Green's function to ω in equation (5.15c).

0 if y ≥ z,
In the same way as F T and F † , Q mig and Q inv di er by a coe cient 2/c 0 and by the source wavelet (Ω and Ω, respectively). However an additional di erence is that Q inv is upper triangular, contrary to Q mig ( gure 5.5). This may be surprising as the high-frequency expressions for the adjoint (5.9b) and pseudo-inverse (5.9c) are very similar. Actually the derivation of Q mig and Q inv starts from equations (5.7b) and (5.7c), which do not use asymptotic approximations. As explained in appendix D, the zero entries below the diagonal of Q inv are due to the derivative with respect to z in equation (5.7c). This illustrates that the high-frequency approximation of the gradient is not the gradient of the high-frequency approximation. The high-frequency approximation of the gradients G mig and G inv (equations (5.13)) derived here is di erent from the gradient that would be computed from the high-frequency approximation of the re ectivity images (equations 5.9b and 5.9c): in the rst case, we replace the Green's function by their expression after having deriving the gradient expression. In the second case, we would use the adjoint-state technique with equations (5.9) as state equations instead of equations (5.7). As equations (5.9b) and (5.9c) are very similar, this approach would lead to similar expressions for operators Q mig and Q inv , without the distinction between the case y < z and z > y (equation 5.15c), and Q inv ( gure 5.5b) would not be upper triangular any more. Compared to the adjoint case, the fact that Q inv is zero below the main diagonal changes the shape of the gradient around the re ector depth. In section 4.2, it has been shown that a speci c power2 of β allows to obtain a smooth gradient in the case of the inverse ( gure 5.6). Numerical tests (not shown here) indicate that this modi cation of the MVA objective function does not allow to remove oscillations in the case of the adjoint.

ξ mig = F T P obs ξ inv = F † P obs η mig = C T A T ACξ mig η inv = C T A T ACξ inv G mig G inv Figure 5.6.
-Computation of the gradient of J 1 with respect to c 0 (right panel), comparing the case where the nal re ectivity is taken equal to ξ mig = F T P obs and ξ inv = F † P obs . For this example, we took β = -1/2.

We have derived expression for the MVA gradients for the adjoint and pseudo-inverse operators. In the following section, we analyse the gradient in the iterative case.

Iterative inversion: expression of the gradient

Instead of using the approximate inverse F † , we determine the optimal re ectivity section by minimising iteratively the objective function (5.6). In the case of linear modelling (primaries only) and 2 regularisation, the minimisation of this objective function is equivalent to the resolution of the linear system F T F + a φ I ξ = F T P obs , (5.16) which involves the normal operator F T F : → . Note that iterative migration is an expensive technique for the case of primaries only as the approximate inverse already provides a good minimiser of J 0 . However iterative migration may deal with more complex cases like multiples, for which an inverse formula does not necessarily exist. Here we want to study the shape of the gradient of J 1 after iterative migration and restrict to this linear case for simplicity.

Computation of the gradient of J 1 Following method B de ned in chapter 3, we assume that the iterative migration process has reached convergence, meaning that the nal re ectivity ξ ∞ is de ned as

F T F + a φ I ξ ∞ = F T P obs .
(5.17)

To compute the gradient of J 1 , we de ne the following Lagrangian with equation ( 5.17) as constraint and the adjoint state η ∈ .18) Deriving this expression with respect to ξ ∞ leads to the following adjoint equation,

J 1 [c 0 , ξ ∞ , η] = 1 2 ACξ ∞ 2 -η F T F + a φ I ξ ∞ -F T P obs . ( 5
F T F + a φ I η = C T A T ACξ ∞ . (5.19)
ξ ∞ and η are solutions of the same linear problem for two di erent source terms. However the resolution of the adjoint problem is more di cult and requires more iterations as observed in section 4.4.3. Our interpretation of this di erent behaviour is that the source term F T P obs is in the image of the normal operator F T F, whereas there is no guarantee that the source term C T A T ACξ ∞ is in the image of F T F, meaning that a solution to equation (5.19) is less likely to be found, or at least more di cult. This point will be further illustrated in sections 5.2.4 and 5.2.7.

Assuming equation (5.19) is solved perfectly, the ideal gradient is made of three contributions

G ∞ (x) = β c 0 ACξ ∞ 2 (x) + ∂ ∂ c 0 (x) 〈η ∞ | F T P obs 〉 + ∂ ∂ c 0 (x) 〈η ∞ | F T Fξ ∞ 〉 . (5.20)
The rst term is due the power of c 0 that has been introduced in the de nition of J 1 . The second is due to the inner product 〈η | F T P obs 〉 and has already been analysed in section 5.2.2. Finally we compute the third contribution due to the normal operator,

∂ ∂ c 0 (x) η F T Fξ = ∂ ∂ c 0 (x) z ω y η(z)F T (z, ω)F(ω, y)ξ( y) d y dω dz (5.21a) = z ω y η(z) ∂ F T (z, ω) ∂ c 0 (x) F(ω, y)ξ( y) + η(z)F T (z, ω) ∂ F(ω, y) ∂ c 0 (x) ξ( y) d y dω dz (5.21b) = z ω y η(z) ∂ F T (z, ω) ∂ c 0 (x) F(ω, y)ξ( y) + ξ( y) ∂ F T (ω, y) ∂ c 0 (x) F(ω, z)η(z) d y dω dz (5.21c) = z Q 1 (x, z)η(z) dz + y Q 2 (x, y)ξ( y) d y - 2 c 0 F T Fξ (x)η(x) - 2 c 0 F T Fη (x)ξ(x), (5.21d)
with Q 1 and Q 2 two matrices similar to Q mig (equation 5.14a):

Q 1 (x, z) = ∂ F T P 1 (z) ∂ c 0 (x) and Q 2 (x, z) = ∂ F T P 2 (z) ∂ c 0 (x) , (5.22)
with calculated data P 1 = Fξ and P 2 = Fη.

In practice, the direct and adjoint problems are solved in a nite number of iterations (N and M iterations, respectively), leading to approximate solutions ξ (N+1) and η (N+1,M+1) and an approximate gradient G (N+1,M+1) made of seven contributions,

                                                 G (N,M) 1 (x) = - y Q (N+1) 1 (x, y)η (N+1,M+1) ( y) d y G (N,M) 2 (x) = - z Q (N+1,M+1) 2 (x, z)ξ (N+1) (z) dz G (N,M) 3 (x) = y Q mig (x, y)η (N+1,M+1) ( y) d y G (N,M) 4 (x) = - 2 c 0 F T P obs (x)η (N+1,M+1) (x) G (N,M) 5 (x) = 2 c 0 F T Fξ (N+1) (x)η (N+1,M+1) (x) G (N,M) 6 (x) = 2 c 0 ξ (N+1) (x) F T Fη (N+1,M+1) (x) G (N) β (x) = βc 2β-1 0 (x)a 2 (x)(ξ (N+1) ) 2 (x).
(5.23a)

(5.23b) (

(5.23f)

(5.23g)

If the direct (equation 5.17) and adjoint (equation 5.19) system are correctly solved, meaning that nal residuals are small, than for a small regularisation,

           P (N+1) P obs , Q (N+1) 1 Q mig , F T Fξ (N+1) F T P obs , F T Fη (N+1,M+1) C T A T ACξ (N+1) , (5.24a) (5.24b)
(5.24c) (5.24d) and most of the contributions in equation ( 5.23) cancel out,

     G 1 + G 3 0, G 4 + G 5 0, G 6 + G β 0 if β = -2.
(5.25a)

(5.25b)

(5.25c)

The only remaining contribution is G 2 . We expect that it is similar to the gradient obtained with direct inversion. We shall see on a numerical example that this is not the case for reasons explained after.

Preconditioning

To accelerate the resolution of the direct (equation 5.17) and adjoint (equation 5.19) problem, we use a preconditioner based on the approximate inverse F † . We brie y review how the modi ed preconditioned problem is de ned.

The original linear problem can be written in a compact way as

Hξ = b, (5.26)
with the Hessian H = F T F + a φ I and the vector b ∈ de ned as b = F T P obs for the direct problem and b = C T A T ACξ ∞ for the adjoint problem. This problem is solved with the linear conjugate gradient algorithm, which actually minimises the scalar objective function

ψ(ξ) = 1 2 ξ T Hξ -ξ T b.
(5.27)

We introduce the preconditioner F † in this system with the following change of variables

ξ = F † ξ (5.28)
with the new variable ξ ∈ obs . Replacing in equation ( 5.27) leads to (5.29) which corresponds to the linear system (5.30) where (F † ) T HF † (F † ) T F T FF † = (FF † ) T (FF † ) if the regularisation weight a φ is not too strong.

ψ ξ = 1 2 ξ T (F † ) T HF † ξ -ξ T (F † ) T b,
(F † ) T HF † ξ = (F † ) T b,
As F † has been designed to be an approximate inverse of F (FF † d d), this operator is expected to be close to the identity (note however that F † Fξ is not necessary close to ξ).

We recall that preconditioning does not change the way the gradient of J 1 is computed. It only provides a more e cient way to compute the solutions ξ (N+1) and η (N+1,M+1) of the direct and adjoint problems (equations 5.17 and 5.19).

Before comparing the gradients obtained by the direct (equation 5.23) and the iterative approach (equation 5.13b), we study the convergence rate of the direct and adjoint problems.

Convergence speed

We now consider the same example as in section 5.2.2 with a single re ector located at 400 m depth and a too low initial velocity. In this rst example we solve iteratively both the direct and adjoint problems with a weak regularisation weight a φ to compare the behaviour of the 1D and 2D cases. In this section, we focus on the convergence speed observed in the resolution of these problems.

We perform six iterations of iterative migration, both without and with preconditioning ( gure 5.7). The regularisation function is the 2 -norm and the regularisation weight a φ is supposed to be small. Similarly to the 2D case, preconditioning signi cantly accelerates the convergence. After one iteration, the data mis t has been reduced to 0.5 % of its initial value, compared to 13 % for the classical case without preconditioning. Additional iterations only result in further deconvolution of the source wavelet. The same comparison is performed for the adjoint problem ( gure 5.8). We use N = M = 100 iterations for both problems to obtain solutions ξ and η as accurate as possible. In this case, preconditioning does not really speed up convergence. The objective function associated with the linear conjugate gradient algorithm (equation 5.27, gure 5.8a, left) decreases slowly in the rst iterations, and the normal residual ( gure 5.8a, right), which is actually the norm of the gradient of the objective function, is not even monotonous. The nal solution η (101,101) is very oscillating ( gure 5.8b, left), similar to the observations made in the 2D case ( gure 4.29). We may have expected to obtain a nal value localised around 320 m, corresponding to the main peak in the nal re ectivity. However it is a relatively accurate solution to the adjoint problem as Hη (101,101) is close to the right-hand side term. (a) Norm minimised by the conjugate gradient algorithm (left, (equation 5.27) and relative normal residual (right) across iterations. The fact that preconditioning does not accelerate convergence (left) is not satisfactory. The approach proposed in section 5.2.7 will provide a more favourable behaviour. To understand the di erent behaviour of the direct and adjoint problems, we compute numerically the SVD decomposition of the matrices F T F and (F † ) T F T FF † ( gures 5.9a and 5.9b). To fairly compare the properties of the direct and adjoint problems, we consider normalised versions of their right-hand side terms ( gure 5.9c) and compute their projection on the eigenvectors of the matrices considered in the direct and adjoint problems ( gure 5.9d).

The convergence of the conjugate-gradient algorithm accelerates greatly when the eigenvalues are clustered (Nocedal and Wright, 2006, p. 118). In our case, the eigenvalues of the preconditioned system are more concentrated around 1 than the original one. This illustrates that FF † and (F † ) T F T are close to identity operators and is consistent with the observed improved convergence for the direct problem. Note that many eigenvalues are zero due to oversampling of the depth and time axes.

The di erent behaviour observed for the adjoint problem comes from the right-hand side term. For the direct problem in the preconditioned case, most of the energy of the projection is concentrated on eigenvectors associated with an eigenvalue close to 1, hence a single iteration provides a good solution to the direct problem. In the case of the adjoint problem, the projection of the right-hand side term spreads over a wider range of eigenvectors than for the direct problem, even with preconditioning. More importantly, non-negligible contributions come from eigenvectors associated with small eigenvalues, which explains that more iterations are required to solve the adjoint problem. These small eigenvalues are not all due to the oversampling of the time and depth axis, and using a coarser grid would not solve this issue.

Contrary to the direct problem, the right-hand side term of the adjoint problem cannot be expressed as the application of F T to a vector of obs . In section 5.2.7, we propose a modi cation of J 1 such that both problem have similar right-hand side terms with a more favourable projection on the eigenvectors of F T F and (F † ) T F T FF † . We rst discuss the shape of the MVA gradient and the role of regularisation.

Shape of the gradient

We compute the gradient with the values of ξ and η obtained after resolution of the direct and adjoint problems in 100 iterations each with preconditioning. This ensures that both problems are nearly perfectly solved ( gure 5.10). As a consequence, most of the contributions of the gradient cancel with one another as in equations (5.25) ( gure 5.11). However the remaining contribution G 2 has a strong negative peak at the re ector position and is strongly oscillating, even below the re ector.

The gradient obtained after iterative inversion is quite di erent from the one obtained after direct inversion. Contrary to iterative migration, the operator F † provides only an asymptotic inverse, as a consequence the nal re ectivity ξ (101) and the inverse ξ inv are not exactly similar, although they both result in very small relative data residuals (0.002 % and 0.25 %, respectively). The iterative result is more oscillating away from the re ector position. These oscillations allow to further reduce the data mis t, but do not carry useful kinematic information, which is contained in the main peak localised at the re ector position. After application of the annihilator, these oscillations are strengthened with respect to the main peak. Small perturbations of the re ectivity model in the null space of F away from the re ector leads to small modi cations on the value of J 0 , but have a large in uence on the gradient of J 1 and may mislead the velocity analysis. This is emphasised in this 1D case with the multiplication by zz ex , but the same conclusion holds in 2D. In the following, we investigate two ways of attenuating this undesirable e ect, rst by regularisation on ξ of the direct problem (section 5. - lter in the de nition of the MVA objective function (section 5.2.7).

6 • 10 -11 -4 • 10 -11 -2 • 10 -11 0 G total -6 • 10 -11 -4 • 10 -11 -2 • 10 -11 0 G 2 G 1 + G 3 G 4 + G 5 G 6 + G β -2 • 10 -8 0 2 • 10 -8 G 1 G 3 G 1 + G 3 0 100 200 300 400 500 600 -1 • 10 -7 -5 • 10 -8 0 5 • 10 -8 1 • 10 -7 depth (m) G 4 G 5 G 4 + G 5 0 100 200 300 400 500 600 -1 • 10 -10 0 1 • 10 -10 depth (m) G 6 G β G 6 + G β

Influence of regularisation

We consider regularisation of iterative migration as a rst remedy to the apparition of oscillations away from the re ector position. In the case of the 2 norm, we may increase the value of the regularisation weight a φ . This would yield a re ectivity image with energy more localised, but at the expense of a poorer data mis t. Consequently, the contributions G 1 + G 3 , G 4 + G 5 and G 6 + G β would not be negligible with respect to G 2 any more. Instead we consider using the Huber norm [START_REF] Guitton | Robust Inversion of Seismic Data Using the Huber Norm[END_REF], an intermediate between 1 and 2 norms, for the de nition of the regularisation function φ,

φ[ξ] = z hub ξ(z) dz, (5.31a)
with hub(x) de ned for any real number x as

hub(x) =      x 2 2s if |x| < s, |x| - s 2 if |x| ≥ s.
(5.31b)

The value for the threshold s needs to be de ned. In the following application we choose s = 0.01 (dimensionless quantity to be compared to the re ection coe cient de ned as ξ = δc/2c 0 ) to distinguish the main peak from the spurious oscillations. Note that the direct problem is not linear any more as the derivative of φ with respect to ξ cannot be written as a matrix-vector product. However the adjoint problem remains linear with a modi ed regularisation matrix,

F T F + a φ H φ [ξ ∞ ] η = C T A T ACξ ∞ , with H φ [ξ] = ∂ 2 φ ∂ ξ 2 [ξ].
(5.32)

The matrix H φ is diagonal, with 1/s on the diagonal where ξ ∞ is below the threshold s and zeros elsewhere.

We test the e ect of the Huber norm on the same example, with 10 iterations used to solve both the direct and adjoint problems. The nal re ectivity image is sparser, but the data mis t remains very low ( gure 5.12). The new right-hand side term has a more favourable projection on the eigenvectors of the linear system ( gure 5.13), consequently the convergence of the adjoint problem is easier and the preconditioner does bring some acceleration, as it does for the direct problem. The nal gradient ( gure 5.14) is slightly less oscillating than the one obtained in the previous case but is still a ected by a strong artefact at the re ector position. Besides, the determination of a φ and s is a tedious task. A Pareto analysis [START_REF] Hansen | The L-Curve and Its Use in the Numerical Treatment of Inverse Problems[END_REF] could indicate possible values as a compromise between data mis t and smoothness of the gradient. As this approach is relatively expensive, the value of a φ and s are in practice determined empirically.

New approach: application of a filter on the final reflectivity

As an alternative to regularisation, we consider applying a lter K : → to the result of iterative migration ξ ∞ before measuring defocused energy. The main objective is to remove F T Fη (11,11) (F T F + H φ )η (11,11) Figure 5.12. -Same as gure 5.10 with the Huber norm as regularisation, and after 10 iterations for both the direct and adjoint problems. -Normal residual across iterations for the adjoint problem (left). Eigenvalue distribution of (F † ) T (H + H φ )F † and projection on its eigenvectors of the right-hand side term of the adjoint problem in the preconditioned case. undesired energy in the re ectivity before evaluating the quality of ξ with respect to c 0 . This leads to the following objective function

0 -1 • 10 -11 -2 • 10 -11 -3 • 10 -11 G total 0 -1 • 10 -11 -2 • 10 -11 -3 • 10 -11 G 2 G 1 + G 3 G 4 + G 5 G 6 + G β -2 • 10 -10 -1 • 10 -10 0 1 • 10 -10 2 • 10 -10 G 1 G 3 G 1 + G 3
J 1 [c 0 ] = 1 2 ACKξ ∞ [c 0 ] 2 .
(5.33)

In the following we consider two lters K. The second one depend on c 0 , leading to additional terms in the velocity update. The gradient of the modi ed objective function is similar to the regular one. For simplicity, we note ζ = Kξ ∞ and build the following Lagrangian with the adjoint variables γ and η associated to ζ and ξ ∞ , respectively.

J 1 [c 0 , ξ, ζ, γ, η] = 1 2 ACζ 2 -γ ζ -Kξ ∞ -η (F T F + a φ I)ξ ∞ -F T P obs (5.34)
Deriving this expression with respect to the state variables ζ and ξ ∞ leads to the following adjoint equations,

γ = C T A T ACζ, F T F + a φ I η = K T γ = K T C T A T ACKξ ∞ .
(5.35a) (5.35b) The adjoint variable η is solution of a problem similar to equation (5.19), except that K T is added twice in the right-hand side term. The gradient eventually reads

∂ J 1 ∂ c 0 (x) =βc 2β-1 0 ζ 2 (x) - ∂ ∂ c 0 (x) η ∞ F T Fξ ∞ + ∂ ∂ c 0 (x) η ∞ F T P obs + ∂ ∂ c 0 (x) γ Kξ ∞ .
(5.36)

The three rst terms are similar to the previous case, except that K is now involved in the computation of ζ and η. If K depends on c 0 , the fourth contribution is non-zero and should be computed.

Frequency filter independent of c 0 We rst consider a lter K which does not depend on c 0 , so that the only di erence with the usual procedure is the modi ed right-hand side term of the adjoint problem. The lter is designed to remove the low and high-frequencies components of ξ (N+1) which do not bear relevant kinematic information. In this case, K can be seen as a taper in the wavenumber domain or as a convolution in the depth domain.

We perform 10 iterations without regularisation. The lter is empirically determined such that ζ is close to the result of the adjoint F T P obs in order to attenuate the spurious oscillations away from the re ector ( gure 5.15). The new right-hand side term has a projection on eigenvectors similar to the one obtained for the direct problem, and the convergence of the adjoint problem is almost as fast as for the direct problem ( gure 5.16). Ten iterations are su cient to obtain good solutions to both the direct and adjoint problems ( gure 5.17). The nal gradient is free of oscillations above and below the re ector, but is still oscillating around the re ector depth ( gure 5.18). So this is not yet a fully satisfactory solution. Moreover, in cases more complex than the homogeneous model considered here, the value of K should depend on the spatial positions and on the velocity values. In the next section, we propose a lter that ful ls these requirements.

Using F † F as a filter on the final reflectivity We now de ne a new lter using the approximate inverse F † ,

K = F † F. (5.37)
It consists rst of applying the modelling operator to the result of iterative migration, yielding new data which should be very close to observed data. Then applying the inverse operator should produce a new re ectivity image close to the inverse F † P obs , and the objective function should be close to the one studied in section 5.2.2. Note that F † has been designed such that FF † is close to the identity (F † F is not necessarily identity, though). The lter considered here should be close to the identity as well, but only for a speci c range of spatial frequencies. We would like the remaining frequency components to be attenuated. The source term of the adjoint problem now reads K T γ = F T (F † ) T γ and therefore lives in the image of operator F T , as does the source term of the direct problem.

With this new de nition, K now depends on c 0 and we compute the additional contributions F T Fη (11,11) (F T F + H φ )η (11,11) Figure 5.17. -Same as gure 5.12 without regularisation and with the introduction of a lter K in the MVA objective function. to the gradient (equation 5.36), noted G 7 , G 8 and G 9 ,

-2 • 10 -11 -1 • 10 -11 0 1 • 10 -11 G total -2 • 10 -11 -1 • 10 -11 0 1 • 10 -11 G 2 G 1 + G 3 G 4 + G 5 G 6 + G β -2 • 10 -11 0 2 • 10 -11 G 1 G 3 G 1 + G 3
∂ ∂ c 0 (x) γ F † Fξ (5.38a) = z ω y γ(z) ∂ F † (z, ω) ∂ c 0 (x) F(ω, y)ξ( y) + γ(z)F † (z, ω) ∂ F(ω, y) ∂ c 0 (x) ξ( y) d y dω dz (5.38b) = z ω y γ(z) ∂ F † (z, ω) ∂ c 0 (x) F(ω, y)ξ( y) + ξ( y) ∂ F T ( y, ω) ∂ c 0 (x) (F † ) T (ω, z)γ(z) d y dω dz (5.38c) = z Q 7 (x, z)γ(z) G 7 (x) + y Q 8 (x, y)ξ( y) G 8 (x) - 2 c 0 γ(x) F T (F † ) T ξ (x) G 9 (x) , (5.38d)
where Q 7 and Q 8 are matrices similar to Q inv and Q mig , respectively (equations 5.14a and 5.14b),

Q 7 (x, z) = ∂ F † P 7 (z) ∂ c 0 (x) and Q 8 (x, z) = ∂ F T P 8 (z) ∂ c 0 (x) , (5.39)
with calculated data P 7 = Fξ and P 8 = (F † ) T γ.

As in section 5.2.3, we now consider that the direct problem is solved in N iterations and the adjoint problem is solved in M iterations. Then the total gradient can be expressed as the sum update free of anomaly at the re ector position, which is preferable to additional iterations in the perspective of more expensive 2D applications. Finally, we investigate the stability of the gradient with respect to the number of iterations used to solve both the direct and adjoint problems. We compute the gradient obtained after each iteration on ξ (n) with and without the lter K = F † F. We also study the in uence of the preconditioner on the stability of the gradient computation. For the sake of simplicity, we consider that the same number of iterations is performed to solve the adjoint problem (M = N). We plot the mean value of the gradient obtained between z =0 m to 250 m, that is above the re ector ( gure 5.24). Without lter and without preconditioner, the value of the gradient is quite chaotic and its sign may change from an iteration to the next one. This is consistent with observations made by [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF]. Here we provide a numerical example where the sensitivity is so large that it depends on the precise number of iterations. The results are slightly better in the preconditioned case but still very oscillating. We obtain a much more stable behaviour when the lter is introduced in the objective function, the mean value being constant after 30 iterations. F T Fη (11,11) (F T F + H φ )η (11,11) Figure 5.21. -Same as gure 5.17 for the lter K = F † F.

-1 • 10 -10 -5 • 10 -11 0 G total G inv -2 • 10 -10 -1 • 10 -10 0 1 • 10 -10 2 • 10 -10 -2 • 10 -8 0 2 • 10 -8 G 1 G 3 G 1 + G 3 -5 • 10 -9 0 5 • 10 -9 G G G + G 5
G G G + G 8 G 7 G 6 + G 9 G 1 + G 3 G 2 + G 8 G 4 + G 5 G β Figure 5.22.
-Same as gure 5.18 for the lter K = F † F.

-1 • 10 -10 -Mean value of the gradient between z =0 m to 250 m (above the re ector where it is supposed to be constant with depth) obtained without and with the lter K = F † F after performing N =1 to 100 iterations of iterative migration. The adjoint problem is solved with the same number of iterations as the direct problem (M = N). Small 2 regularisation is applied. For each case we plot the result obtained without (left) and with (right) preconditioner introduced in the resolution of both direct and adjoint problems. Each plot is normalised individually by the mean value obtained between iterations 50 to 100.

-5 • 10 -11 0 G total G inv -2 • 10 -10 0 2 • 10 -10 -2 • 10 -8 0 2 • 10 -8 G 1 G 3 G 1 + G 3 -4 • 10 -9 -2 • 10 -9 0 2 • 10 -9 4 • 10 -9 G 4 G 5 G 4 + G 5 0 100 200 300 400 500 600 -1 • 10 -9 0 1 • 10 -9 depth (m) G 6 G 9 G 6 + G 9 0 100 200 300 400 500 600 -5 • 10 -10 0 5 • 10 -10 depth (m) G 2 G 8 G 2 + G 8 G 7 G 6 + G 9 G 1 + G 3 G 2 + G 8 G 4 + G 5 G β
The expected mean value is negative as this is the case of the blue curves. Locally the sign of the gradient is incorrect (positive values, red curves).

In summary, the introduction of the lter K = F † F in the de nition of the MVA objective function is essential to improve the stability of the gradient obtained after iterative migration and provides background velocity updates similar to those obtained after direct inversion, while keeping the possibility to perform iterations. We now want to test the applicability and e ciency of this new approach in the 2D case.

2D-Examples

In this section, we apply the new approach consisting of introducing a lter in the objective function of MVA:

J 1 (c 0 ) = 1 2 A[c 0 ]K[c 0 ]ξ (N+1) [c 0 ] 2 .
(5.42)

We will consider only the lter K = F † F based on the approximate inverse F † . From now on, the re ectivity is de ned as a squared slowness perturbation (ξ = 2δc/c 3 0 ). In equation ( 5.42), the annihilator consists of a multiplication by |h| and also includes a multiplication by a power of the background velocity model c β 0 , with β = 3/2. In section 5.3.1, we consider primaries only and an example similar to the one of chapter 4. The objective is to determine if the introduction of the lter K has the same bene ts as in the pure 1D case. We compare the results obtained with and without ltering nal CIGs and expect the lter to improve the convergence rate of the adjoint problem and to yield gradients similar to those obtained after direct inversion.

Then we move on to an example with primaries and rst-order surface-related multiples (section 5.3.2). This case was not considered in the 1D-analysis of the preceding section. We want to determine if the new approach extend to the case of multiples and yield coherent velocity updates.

Primaries only

We rst consider an example with primaries only similar to the one presented in section 4.4.3, to compare two approaches, without and with the lter K = F † F. A single at re ector is located at 300 m depth in a too low homogeneous velocity model (2500 m/s). The exact background velocity model is homogeneous too (3000 m/s). The model is 1620 m large and 450 m deep and is discretised on a 6 m × 6 m grid. Sources are located at each point of the surface with receiver also at every grid point within a maximum surface o set of ±540 m.

We perform seven iterations to solve the direct problem with preconditioning and small 2 regularisation, leading to ξ (8) . After application of K = F † F to this nal re ectivity, we obtain a CIG very close to the one obtained by direct inversion ( gure 5.25). In particular, spurious oscillations located above and below the re ector have been greatly attenuated. In ξ (8) , we observe around z = 350 m spurious events with an opposite curvature from the main event. We do not have a formal explanation for these events which may not be due to edge e ects as PML and tapers are implemented on each edge of the model. Note that they are greatly attenuated by the lter.

We now consider the adjoint problem. With the lter, the right-hand side term reads K T A T AKξ (8) instead of A T Aξ (8) . As in 1D, this new vector can be expressed as F T Q, result of application of the adjoint to a data set Q = (F † ) T A T AKξ (8) ∈ obs . As a consequence it is similar to the right-hand side term of the direct problem F † P obs with an event curved upward looking like usual migration artefacts in addition to the downward event corresponding to the true re ector ( gure 5.26, third column). The adjoint problem is solved in 10 iterations, with and without preconditioning for comparison. With the application of the lter, the adjoint problem is easier to solve and converge much faster, contrary to the original case ( gure 5.27). Moreover, the preconditioner accelerates the convergence rate, which is also satisfactory. Note however that the convergence rate is still slower than for the direct problem. The last iterates η (8,11) in the two cases are both very oscillating ( gure 5.26, rst column), but in the new approach, the residuals of the linear system are much smaller ( gure 5.26, fourth column). We now consider the sequence of gradients of J 1 with respect to c 0 associated to the successive values of adjoint variables obtained without and with the lter when the adjoint problem is solved with preconditioning. With the lter, this sequence converges to a stable gradient and the nal value is reached after ve iterations ( gures 5.28 and 5.29). Moreover, the nal gradient exhibit small oscillations, but is very close to the one obtained by direct inversion ( gure 5.30). This result is very similar to the one obtained in 1D ( gure 5.22).

As in the 1D case, the gradient can be decomposed into several parts. But contrary to the 1D case where we identi ed ten contributions (equation 5.40), the re ectivity has been de ned here as a square slowness perturbation ξ = 2δc/c 3 0 . With this parametrisation, the contributions G 4 , G 5 , G 6 and G 9 in equation (5.40) are zero. The remaining contributions are G 7 + G β , G 2 + G 8 and G 1 + G 3 ( gure 5.31, right column, top, middle and bottom plots respectively). As ξ (8) and η (8,11) are good solutions of the direct and adjoint problems respectively, the second and third η (8,11) Hη (8,11) b (8) = A T Aξ (8) without lter

Hη (8,11) b (8) -100 0 100 η (8,11) -100 0 100 h (m)

Hη (8,11) -100 0 100 h (m)

b (8) = K T A T AKξ (8)
-100 0 100 with lter

h (m)
Hη (8,11) b (8) 8,11) obtained in the resolution of the adjoint problem (1st column) and application of the Hessian H to η (8,11) (2nd column). We also display the right-hand side term b (8) of the linear system (3rd column) and the corresponding residual (4th column). We consider the case where no lter is introduced in the MVA objective function (top, similar to gure 4.29), and the case with a lter K = F † F based on the approximate inverse (bottom). The three most right plots of each line share the same colour scale.

contributions are small with respect to the rst one. The latter originates from the operator F † in the lter and has a physical meaning. It is the gradient that would be obtained by applying the direct inversion strategy to primary data re-computed with rst-order Born modelling from the nal re ectivity Fξ (8) . As this data set is very close to observed data, this contribution is very close to the gradient obtained with the direct inversion approach ( gure 5.30, top). Although this contribution is not the gradient of an objective function, we discuss in section 5.3.3 an alternative strategy using it as background velocity update.

Finally we comment on the edge e ects visible on the left and right side of the gradient ( gure 5.32, bottom left). Their extension (x =0 m to 550 m and x =1000 m to 1620 m) is approximately equal to the value of the maximum surface o set (540 m, here). Note that in the re ectivity section, the inhomogeneous part only is half of the maximum surface o set ( gure 5.32, top left). If we keep the same value for the maximum surface o set and extend the model laterally, the homogeneous central part is extended, but not the part altered by edge e ects ( gure 5.32, right). Therefore, in this example as well as in the following, we focus on the smoothness and coherency of the gradient in the central part. The attenuation of these artefacts is not speci cally addressed in this thesis. We observe that tapers on source and on receivers positions, properly included in the gradient computation, help mitigating this undesirable e ect. This issue deserves nonetheless further investigation. In summary, the introduction of the lter has a similar e ect in the pure 1D and the 2D cases. It improves the convergence rate of the adjoint problem and the associated gradient is close to the result obtained after direct inversion. Also, the introduction of the lter relax the requirement of su ciently strong regularisation illustrated in the examples of chapters 3 and 4 where no lter was considered.

Primaries and Multiples

We want to test this new approach on a case with rst-order surface-related multiples. We consider an example similar to the one studied in section 4.4.2. The exact model is laterally invariant with a single re ector located at 300 m depth and a background velocity model increasing with depth ( gure 5.33), leading to two events in observed data ( gure 5.34). The initial background velocity model is also increasing with depth, with the correct velocity at the surface but with a too low gradient. In all the following examples, we add a multiplication by z in the de nition of the annihilator to emphasise the impact of multiples. This weight is taken into account in the derivation of the gradient.

For reference, we rst apply the direct inversion strategy. If we remove multiples from observed data, we obtain a single event in CIGs, with a downward curve and the gradient is negative and homogeneous above the re ector, which is consistent with the too low initial velocity model ( gure 5.35a). With both primaries and multiples in observed data, a new event corresponding to the multiple interpreted as a primary appears around 600 m depth ( gure 5.35b). The actual multiple travelled twice in the exact model between the surface and depth (m) without lter G (7,1) with lter depth (m) G (7,2) depth (m) G (7,3) depth (m) G (7,4) depth (m) G (7,5) depth (m) G (7,6) depth (m) G (7,7) depth (m) G (7,8) depth (m) G (7,9) 0 500 1000 1500

x-position (m) depth (m) 0 500 1000 1500 G (7,10) x-position (m) with lter G (7,10) x-position (m) the exact re ector position. It is interpreted as a primary travelling in the initial velocity model between the surface and a depth roughly twice the one of the exact re ector. Hence, this "imaginary" primary has "seen" a higher velocity than the true multiple. As a consequence, part of the energy of the cross-talk artefact is defocused with an upward curvature and the gradient has positive contribution above the re ector We now consider iterative inversion with the lter F † F introduced in the de nition of the MVA objective function (equation 5.42). First, we study the case of observed data containing only the primary re ection ( gure 5.36). We obtain a gradient very similar to the case of direct inversion ( gure 5.35a), which indicates that the new approach works well in non-homogeneous models as well. -120 0 120 h (m)

M = 1 M = 3 M = 5 M = 7 M = 9 M = 2 M = 4 M = 6 M = 8 M = 10
Kξ (N+1)
-120 0 120 h (m)

A T AKξ (N+1)
-120 0 120 h (m) We now consider the full data set with the primary re ection and the rst-order surface multiple and use preconditioned non-linear optimisation as in section 4.4.2. The linear adjoint problem reads in the general case

K T A T AKξ (N+1)
∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) η = A T K T KA T ξ (N+1) , (5.43)
the di erence with the linear case being that the Hessian matrix depends on the nal value of ξ. This operator reads

∂ 2 J 0 ∂ ξ 2 =
∂ 2 P 3 ∂ ξ 2 (P 1 + P 3 -P obs ) + (5.44) where we have omitted the dependency to ξ (N+1) and operator M for readability. Because of the rst term in this expression, the Hessian may not be positive de nite in case of too strong residuals. This is an issue as the conjugate gradient algorithm is designed for positive de nite matrix only. This problem should not appear if iterative migration has converged, meaning that residuals are small, or if enough regularisation is added. An additional safeguard consists of exiting the algorithm as soon as a negative curvature is encountered [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF]. Using small 2 regularisation and ten iterations to solve the adjoint problem, we rst compare the gradient obtained after 1 and 10 iterations of migration ( gures 5.37a and 5.37b). In the rst case, residuals are still strong and a negative curvature has been encountered at the 10th iteration. Therefore the nal gradient is actually G (10,9) . Because of the cross-talk artefact in the CIG, the gradient is non-zero below the re ector. After ten iterations, the artefact is weaker and is further attenuated by the application of the lter. However, the gradient is not homogeneous and exhibits strong oscillations around the re ector. To obtain a more satisfactory result, we rst increase the number of iterations for the resolution of the direct and adjoint problems (30 and 20, respectively), but the gradient is still far from being homogeneous ( gure 5.37c). We notice that even after several iterations and application of the lter, the annihilator still strengthen residual energy at large values of h below the re ector ( gure 5.37c, 4th column). This suggests that these areas may not be well constrained. As a remedy, we add a Huber norm to the migration objective function J 0 with a bigger weight on these parts of the CIGs: (5.45) and we use again 10 iterations to solve the direct and adjoint problems ( gure 5.37d). Although the nal CIGs look very similar to the one obtained in the previous case, this modi cation greatly improves the gradient, which is homogeneous with the correct sign above the re ector and very weak below.

∂ P 1 ∂ ξ T ∂ P 1 ∂ ξ + 2 ∂ P 1 ∂ ξ T ∂ P 3 ∂ ξ + ∂ P 3 ∂ ξ T ∂ P 3 ∂ ξ + a φ ∂ 2 φ ∂ ξ 2 ,
φ[ξ] = x h hub h • z • ξ(x, h) dh dx,
As in the 1D case, we decompose the gradient into three parts ( gure 5.38). The rst one ( gure 5.38, 1st column) is due to the operator F † in the de nition of J 1 (equation 5.42). As already mentioned in the linear case, this is the gradient that would be obtained by applying the direct inversion approach to observed primaries re-computed from the nal re ectivity free of cross-talk artefacts. As a consequence this contribution is very similar to the one obtained by applying direct inversion to primary only ( gure 5.35a), provided that the nal re ectivity correctly explains observed data. The second contribution ( gure 5.38, 2nd column) is directly related to data residuals, and its share in the nal gradient decreases as far as the number of iterations increases ( gures 5.38a to 5.38c). With regularisation, the nal data residuals may be degraded, and this contribution may be not negligible any more (although it is still quite small in the example of gure 5.38d). Eventually we notice that the third part ( gure 5.38, 3rd column) is relatively strong in the four cases and does not necessarily add a coherent contribution to the gradient. Contrary to the linear case, it does not vanish when the direct and adjoint problems are perfectly solved. We conclude from this example that the new strategy consisting of adding a lter in the de nition of the MVA objective function is also e cient in the case of data containing rst-order surface multiples. The lter attenuates unwanted energy at large values of h as in the primaries only case and helps attenuating residual energy of cross-talk artefacts. Note however that contrary to the primaries only case, regularisation remains essential to iterative migration for a proper attenuation of cross-talk artefacts and consistent background velocity updates.

Alternative strategy

Both in the primaries only case and in the presence of multiples, we have isolated a contribution to the total gradient due to the operator F † in the lter K and showed that it is very close to the result of direct inversion applied on primaries only ( gure 5.31, top right and gure 5.38, 1st column). We propose as an alternative strategy to use this contribution only to update the background model. In this new approach, the MVA objective function to be minimised still reads

J 1 [c 0 ] = 1 2 A[c 0 ]F † [c 0 ]F[c 0 ]ξ (N+1) [c 0 ] 2 .
(5.46) -120 0 120 h (m)

Kξ (N+1)
-120 0 h (m)

A T AKξ (N+1)
-120 0 120 h (m) -120 0 120 h (m)

K T A T AKξ (N+1) (c)
Kξ (N+1)
-120 0 h (m)

A T AKξ (N+1)
-120 0 120 h (m) -120 0 120 h (m)

K T A T AKξ (N+1) (d)
Kξ (N+1)
-120 0 h (m)

A T AKξ (N+1)
-120 0 120 h (m) We brie y rederive an expression for the total gradient of the objective function (5.46) with respect to c 0 and identify the contribution which is considered in the new strategy. We de ne the following Lagrangian,

K T A T AKξ (N+1)
J 1 = 1 2 Aζ 2 -γ ζ -F † P calc -Q P calc -Fξ (N+1) -η ∂ J 0 ∂ ξ c 0 , ξ (N+1) (5.47)
The value of the adjoint variables γ, Q and η are obtained by zeroing the partial derivatives of the Lagrangian (5.47) with respect to the state variables ζ, P calc and ξ (N+1) ,

         γ = A T Aζ (N+1) , Q = (F † ) T γ, ∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) η = F T Q, (5.48a)
(5.48b) (5.48c) and the total gradient of the objective function (5.46) with respect to c 0 is obtained as the partial derivative of the Lagrangian (5.47) with respect to c 0 ,

∂ J 1 ∂ c 0 = "truncated gradient" ∂ ∂ c 0 Aζ 2 2 + ∂ ∂ c 0 γ F † P calc + ∂ ∂ c 0 Q Fξ (N+1) + ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) η. (5.49)
In the new approach, the background velocity update is de ned as the sum of the rst two contributions in (5.49), the two remaining contributions being dropped. Therefore this strategy will be referred to as "truncated gradient" in chapter 6.

We now explain the meaning of this new approach. We apply iterative migration to a data set containing both primaries and rst-order surface multiples to derive an extended re ectivity free of cross-talk artefacts. This extended image is used to re-compute primary re ections under the rst-order Born approximation. The value of this wave eld at receiver positions is kinematically consistent with the primaries contained in the original observed data set and de nes a new data set contains primaries only. The velocity analysis is performed on this data set as if it were observed data, using the direct inversion strategy presented in section 4.2.

The background velocity update used in the "truncated gradient" strategy is not the gradient of an objective function, but this de nition has several advantages. First, it requires less computational e ort as there is no need to solve the adjoint problem (5.48c) any more. Moreover the need to add proper regularisation and the di cult choice of parameters of the regularisation function is relaxed. Finally note that re-modelled primaries can be computed for source positions not present in the original acquisition, for example in the case of an irregular or incomplete acquisition. In the latter case, multiples may provide extra illumination to reconstruct a more detailed re ectivity model and compute primary re ections not recorded in the original acquisition. Also, the new strategy allows to provide the direct inversion strategy with data acquired with dense source and receiver coverage, which is one of the hypotheses made in the derivation of the approximate inverse F † .

Conclusion

We have studied the behaviour of iterative migration velocity analysis on a simple 1D case and observed the same properties as in the 2D case. Oscillations coming from the source deconvolution away from the re ector are ampli ed by the annihilator, leading to a very oscillating image residual. As a consequence, the resolution of the adjoint problem is slow and the background velocity update oscillating.

We have proposed two modi cations of the original procedure to alleviate these issues. First we have shown that adding a Huber norm to the migration objective function improves the properties of the adjoint problem and the associated gradient. A disadvantage of this solution is that two parameters have to be chosen empirically. Then, we have proposed to introduce a lter in the MVA objective function, so that spurious oscillations away from the re ector are attenuated. We showed that a lter based on the approximate inverse F † yields a gradient very similar to the one obtained after direct inversion. With this new objective function, the right-hand side terms of both the direct and adjoint problems can be expressed F T Q, and the adjoint problem converges much faster. As the lter depends on c 0 , additional terms have to be computed, as costly as two iterations of migration. However, the lter allows to solve the adjoint problem in a reduced number of iterations, so the lter actually reduces the computational expense of the method.

An application to a simple example suggests that this new approach works well in 2D as well. Compared to the examples of chapter 4, the adjoint problem converge faster ( gure 5.27), as well as the associated sequence of gradients ( gure 5.27). In this primary-only case, the need for su ciently strong regularisation is also relaxed. However, regularisation remains essential in the case of multiples to obtain coherent velocity updates.

In the case of multiples, we have proposed an alternative strategy consisting of using the nal result of iterative migration to generate a new data set made of primaries only. The direct inversion approach (section 4.2) is then applied on this new data set. Although the background velocity update de ned with this strategy is not the gradient of an objective function any more, this technique considerably reduces the computational expense of the method, and relaxes the di cult requirement to nd suitable regularisation parameters. Furthermore it may allow to regenerate reconstructed data with an acquisition di erent from observed data. For example, source positions missing in initial observed data can be added in the new data set and provide the dense source and receiver coverage assumed in the derivation of the approximate inverse. Chapter 6.

Application to synthetic data sets

Contents

Résumé du chapitre 6 L'objectif de ce chapitre est de tester la robustesse de l'approche dé nie au chapitre 5 lorsque le code de modélisation utilisé pour générer les données observées est di érent de celui utilisé lors de l'inversion. Dans l'optique d'applications sur données réelles (absentes de cette étude), il s'agit d'aller au-delà de la situation de « crime inverse » des tests numériques réalisés dans les chapitres précédents.

Dans la première partie de ce chapitre, je considère quatre modi cations du code de modélisation des données observées, tout en gardant le même code de modélisation pour réaliser l'inversion, et j'étudie l'in uence de ces modi cations sur les gradients calculés avec l'inversion directe et itérative, avec et sans multiples :

A -absence de basses fréquences dans l'ondelette de source ; B -modi cation de la forme de l'ondelette de source ; C -modélisation dans un modèle de vitesse lisse mais avec un modèle de densité variable dont les discontinuités sont responsables de ré exions primaires et multiples, modélisées avec une approximation de Born du second ordre ; D -modélisation des données avec densité et vitesse variables par di érences nies sans approximation de Born et avec une condition de surface libre pour la modélisation de multiples de surface.

L'absence de basses fréquences dans l'ondelette de source a une in uence négligeable sur le calcul du gradient (test A). En revanche, l'estimation de la forme de l'ondelette de source est essentielle pour la migration itérative dans le cas des multiples (en revanche, cela ne pose pas de di cultés dans le cas de primaires seuls) : si elle est trop éloignée de celle utilisée pour générer les données observées, la migration itérative ne parvient pas à construire un modèle de ré ectivité expliquant à la fois les ondes ré échies primaires et multiples (test B). L'utilisation de perturbations du modèle de densité au lieu de perturbations du modèle de vitesse pour générer les données observées (test C) n'a ecte pas la cohérence des gradients. La di érence principale est l'estimation de l'amplitude d'évènements correspondant à des grands angles de ré exions : elle se traduit dans le gradient par un poids di érent donné aux contributions des ré ecteurs peu profonds. En n l'utilisation d'une surface libre pour la modélisation des multiples de surface (test D) reste une di culté, en particulier la présence de « ghosts » aux positions des sources et des récepteurs. Ils sont ici pris en compte dans la modélisation des données calculées en utilisant une dérivée seconde de l'ondelette de source, mais cette approximation n'est valide que pour des angles d'incidence nuls à la surface et la migration itérative peine à trouver un modèle de ré ectivité expliquant correctement à la fois les primaires et les multiples.

Dans la deuxième partie du chapitre, des données synthétiques sont modélisées à partir d'un modèle par blocs (vitesse et densité variables) comprenant une anomalie lente de vitesse pour réaliser une inversion du modèle de vitesse. Les données observées sont modélisées avec des versions lissées de ces modèles et l'approximation de Born du second ordre (perturbations de vitesse et de densité). Lors de l'inversion, le modèle de densité est supposé constant. En pratique un lissage du gradient par un noyau gaussien est appliqué avant remise à jour du macro-modèle de vitesse. Dans un premier temps, seules les ré exions primaires sont considérées et la stratégie d'inversion directe est utilisée. Après quinze itérations, l'anomalie de vitesse est retrouvée, les CIGs sont aussi bien focalisés que dans le modèle exact, et la fonction coût atteint un niveau proche, légèrement inférieur, à celui obtenu dans le modèle exact. Dans un deuxième temps, les ré exions multiples sont ajoutées aux données observées et l'inversion est relancée à partir du même macro-modèle initial, les CIGs étant toujours calculés avec la stratégie d'inversion directe. Comparé au premier cas, de nouveaux évènements, non focalisés pour le bon modèle de vitesse, apparaissent dans les CIGs. Ainsi, au cours de l'inversion, la valeur de la fonction coût associée à l'analyse de vitesse atteint des valeurs bien plus faibles que celle obtenue dans le modèle exact. Le modèle de vitesse nal est di érent de celui obtenu dans le cas de primaires seuls et ne fait pas apparaître clairement l'anomalie de vitesse. Le modèle retrouvé réalise un compromis entre la focalisation dans les CIGs des évènements correspondant aux vrais ré ecteurs et la focalisation des artefacts dus aux multiples. Finalement un troisième cas est traité : la remise à jour du modèle de vitesse est calculée par la méthode du « gradient tronqué » après cinq itérations de migration itérative. La migration itérative permet d'atténuer les artefacts dus aux multiples et le modèle nal retrouvé est proche de celui obtenu dans le cas de primaires seuls avec l'inversion directe. Cela indique que les multiples ont été correctement interprétés au cours de l'inversion.

La dernière partie du chapitre est consacré au cas d'un trou d'acquisition, avec des sources manquantes au milieu du modèle. Les ré exions multiples peuvent potentiellement apporter une information complémentaire dans la partie centrale du modèle, non illuminée par les ré exions primaires. Le modèle de vitesse exact est latéralement invariant et un modèle homogène est utilisé comme modèle initial pour calculer le premier gradient (le macro-modèle n'est pas remis à jour dans cet exemple). On constate une absence d'énergie dans la partie centrale du modèle de ré ectivité obtenue par inversion directe dans le cas où les données observées ne comprennent que des ré exions primaires. Il en va de même pour le gradient associé. Il est intéressant de remarquer que, même en ne considérant que les ré exions primaires, la migration itérative permet de combler une partie du trou d'acquisition dans l'image de ré ectivité et dans la remise à jour du macro-modèle. En revanche l'introduction des ré exions multiples (à la fois dans les données observées et dans les données calculées) ne permet pas de combler d'avantage le trou d'acquisition, même après vingt itérations sur le modèle de ré ectivité.

Introduction

The aim of the thesis was to provide a more robust MVA procedure based on iterative migration, able to deal with multiple re ections. In chapters 2 to 5, we have progressively built a strategy, whose nal formulation can be summarised in four steps:

• preconditioned iterative migration to determine the re ectivity section ξ (N+1) best explaining observed data. The preconditioner is a pseudo-inverse F † of the extended modelling operator F. It has been derived for the case of primaries only (chapter 4) and is similar to the one proposed by [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF];

• application of the lter K = F † F to ξ (N+1) to remove unwanted energy at large values of h that does not have any meaning in terms of wave propagation kinematics. The ltered re ectivity ζ = Kξ (N+1) is used as input of the usual MVA objective function

J 1 [c 0 ] = 1 2 Aζ[c 0 ] 2 ; (6.1)
• computation of the image residual K T A T Aζ and resolution of a linear adjoint problem (even in the case of multiples) for the adjoint variable η

∂ 2 J 0 ∂ ξ 2 c 0 , ξ (N+1) η = K T A T AKξ (N+1) ; (6.2)
• computation of the approximate gradient G (N,M) , using the last iterate ξ (N+1) and the last iterate η (N+1,M+1) , as ,M+1) . ( 6.3)

G (N,M) = ∂ 2 J 0 ∂ ξ∂ c 0 c 0 , ξ (N+1) η (N+1
We have also introduced in section 5.3.3 an alternative strategy for the derivation of a background velocity update after iterative migration. It consists of using the extended re ectivity image resulting from iterative migration to recompute primary re ection data. Then the macromodel update is obtained with the direct inversion strategy applied to this new data set free of multiples, as if it was observed data. The background velocity update obtained with this method is only one of the contribution to the gradient of the objective function (6.1) with respect to c 0 , and is therefore referred to as "truncated gradient" in this chapter. The advantage of this strategy is that there is no need to solve the adjoint problem any more, so that the computational expense of the method is considerably reduced. Furthermore the direct inversion strategy has been shown to provide smooth consistent gradients in a wide variety of cases (section 4.2.6). Therefore we expect the truncated gradient to bene t of its favourable behaviour also in the case of multiples.

Currently the iterative migration strategy has been applied only on observed data modelled with a second-order Born approximation, that is with the same modelling tool as the one used in inversion. This is referred to as the "inverse crime" [START_REF] Wirgin | The Inverse Crime[END_REF] in inverse problems literature.

The purpose of this chapter is rst to evaluate the robustness of the approach when synthetic observed data are obtained with a di erent modelling code. This may allow us to determine possible limitations of our approach in the perspective of real data applications (not considered in this thesis). The second objective is to go beyond the computation of the rst gradient and run several non-linear iterations to update the background model. Finally we examine the ability of our MVA strategy to bene t from the additional information contained in rst-order surface-related multiples.

In the rst part of this chapter (section 6.2), we run four tests, named A, B, C and D, summarised in table 6.1. In tests A and B, we model observed data with the constant density acoustic waveequation and a modi ed source wavelet, the Ricker wavelet still being used for inversion. In tests C and D, observed data are modelled with a variable density acoustic propagation code, rst under the Born approximation, then with a full nite-di erence modelling and a free-surface condition. For each test, we compute six background velocity updates using di erent strategies (the letters in the list below correspond to the labels of sub gures in section 6.2):

(a) direct inversion on primary re ection data only; (b) direct inversion on data containing both primaries and rst-order surface-related multiples; (c) iterative inversion on primary re ection data only; (d) iterative inversion on data containing both primaries and rst-order surface-related multiples; (e) smoothed version of the gradient obtained in (c) after application of a gaussian blur; (f) "truncated gradient" obtained in the case of primaries, as described above; (g) same as (e) in the case of multiples (i.e. smoothed version of (d)); (h) same as (f) in the case of multiples.

The result of direct inversion (a) on primaries only is used as reference. The case (b) is considered to assess how multiples misinterpreted as primaries alter the result of (a). Then we compare the result of iterative inversion on multiples (d) with (a) to assess the robustness of our approach. To determine if potential failures of iterative inversion in the presence of multiples are due to multiples or to the iterative strategy, we also compute the gradient obtained by iterative inversion in the case of primaries only (c). Smoothing is commonly applied before updating the velocity model, therefore we show a smoothed version ((e) and (g)) of the gradient obtained after iterative inversion (d).

The same exact (in principle unknown) velocity model is used for tests A, B and C. Those tests are compared to a reference case where the same code is used for the modelling of observations and for inversion. The model is 1800 m large and 700 m deep, laterally invariant and discretised on a 4.8 m×4.8 m grid. The background velocity model is increasing with depth from 2000 m/s at the surface to approximately 2700 m/s at 700 m depth ( gure 6.1, left). The initial velocity model is similar to the exact model except that velocities are underestimated (from 2000 m/s at the surface to 2400 m/s at the maximum depth). Sources are located every 6 grid points at the surface with receivers at each grid point on the surface within ±700 m around the source. The model used in test D is di erent and will be presented in section 6.2.5. For all tests A, B, C and D, we de ne an annihilator similar to the one used at the end of

Aξ(z, x, h) = |h| • z • c β 0 (z, x)ξ(z, x, h). (6.4)
It consists of a multiplication by the absolute value of the subsurface o set to penalise defocused energy as well as a power β = -3/2 of the background velocity model which has been shown in chapter 4 to attenuate spurious oscillations appearing in the MVA gradient around re ectors' positions. To emphasise the in uence of deeper re ectors which may be hidden by cross-talk artefacts, we add a multiplication by the depth z in the de nition of the annihilator.

In the second part of this chapter (section 6.3), a blocky velocity model and a variable density model both with lateral variations are de ned to generate observed data under a second-order Born approximation. We run several outer iterations on the background velocity model and compare the results obtained by direct inversion on primary re ection data only and the "truncated gradient" strategy applied on data containing both primaries and rst-order surface multiples.

Finally we study in section section 6.4 an example with an incomplete acquisition where multiples may bring additional information compared to primaries. We want to test the ability of iterative MVA to use this information.

Robustness of Iterative Migration Velocity Analysis

This section presents and discusses results obtained when a di erent modelling tool is used for the modelling of synthetic observed data and for inversion. In tests A, B, and C (sections 6.2.2 to 6.2.4), re ections are due to a dense model perturbation (velocity model perturbation or density model perturbation). Before presenting the results of these tests, we show the velocity updates obtained in the reference case where the same modelling tool is used to compute observed data and calculated data.

Reference case

In the reference case, the re ectivity model is laterally invariant with rapidly varying values between 100 m and 500 m depth, and zero values above and below ( gure 6.1, right). Using a Ricker wavelet with maximum frequency 40 Hz, we obtain observed data with several events overlapping with one another ( gure 6.2). The rst gradient of the MVA objective function is expected to be negative above z = 500 m and zero below. We rst compute the gradients obtained after direct inversion. In the primaries only case, all events in CIGs have a downward curvature ( gure 6.3a). As expected, the gradient is negative above the deeper re ector and zero below. When direct inversion is applied to both primaries and rst-order surface multiples, additional cross-talk artefacts appear above and below the deeper re ector with an upward curvature ( gure 6.3b). As a consequence positive values appear on the gradient below the deeper re ector as well as strong edge e ects.

In the case of primaries only, the combination of iterative migration and ltering with F † F leads to CIGs very similar to those obtained by direct inversion ( gure 6.3c). The associated gradient has small non-zero values below the re ector but is quite close to the result shown in gure 6.3a. Finally in the case of multiples, iterative migration succeeds in removing cross-talks artefacts resulting in ltered CIGs very similar to the previous case ( gure 6.3d). The associated gradient has a consistent negative value above the deeper re ector and spurious positive values below are attenuated compared to gure 6.3b. These results are satisfactory, and smoothing the gradients yields consistent background velocity updates for subsequent inversion ( gures 6.4e and 6.4g). Note also that the "truncated gradients" ( gures 6.4e and 6.4g) are very similar to the gradient of gure 6.3a. -120 0 120 h (m)

Kξ (N+1)
-120 0 120 h (m)

A T AKξ (N+1)
-120 0 120 h (m) In summary we obtain results very similar to the case of a single re ector studied in section 5.3.2. However, the gradients presented here are less homogeneous. This can be interpreted as a superposition of the contributions due to the di erent events in the CIGs, although the gradient obtained with all the events in the CIGs is not formally a linear combination of the gradient obtained from CIGs containing single events. We also observe strong edge e ects for all gradients. As already mentioned in chapter 5 (see also gure 5.32), we are mainly interested in the central part of the gradient. With a larger model and the same acquisition parameters, the edge e ects would remain similar while the central part would be extended. In these examples we have considered relatively small models to limit the computational cost. We now begin the series of tests summarised in table 6.1 and focus on the central part of the gradients.

K T A T AKξ (N+1)

Test A -Sensibility to the lack of low frequencies in observed data

In this rst test, we use two di erent source wavelets for observations and inversion. Low frequencies up to 5 Hz are removed from the Ricker wavelet for observations ( gure 6.5). The original Ricker wavelet is used during inversion.

In the eight cases ( gures 6.6 and 6.7), we obtain results very similar to the reference case ( gures 6.3 and 6.4). The main di erence lies in the shape of events in CIGs modi ed in a similar way as the source wavelet is, but this has almost no in uence on the shape of the gradient. We conclude that the behaviour of the MVA techniques analysed in the preceding chapters with the standard Ricker wavelet should not be altered by the lack of low-frequencies inherent to real data acquisition. This is a di erent behaviour than for FWI [START_REF] Sirgue | E cient Waveform Inversion and Imaging: A Strategy for Selecting Temporal Frequencies[END_REF]Virieux and [START_REF] Virieux | An Overview of Full-Waveform Inversion in Exploration Geophysics[END_REF]. To our knowledge, the importance of source wavelet estimation has not been studied in the MVA literature yet.

Test B -Sensibility to inaccuracies of source estimation

In test B, observed data are generated with the same re ectivity model as in the reference, but with a di erent source wavelet, de ned as the derivative of the original Ricker wavelet ( gure 6.8). Inversion is still performed with the standard Ricker. The result of direct inversion is very similar to the reference case when applied to primaries only ( gures 6.3a and 6.9a). However multiples have a larger in uence than in the reference case ( gures 6.3b and 6.9b): cross-talk artefacts in CIGs have larger amplitudes compared to events due to primaries and the associated positive values in the gradient spread over a larger area. This may be related to the stretching of events in CIGs, decreasing with depth ( gure 6.10), so that deeper events have lower frequencies. Applying a time derivative to the source wavelet in observed data changes the frequency content of CIGs, and strengthens deeper events. Note that this is only a partial explanation as the approximate inverse should correct for the stretch.

Iterative inversion on primaries only results in CIGs very similar to the direct inversion case. The gradient is however altered by positive values around 100 m depth, which are not attenuated by smoothing ( gure 6.11e). Note that the "truncated gradient" ( gure 6.11f) is remarkably close to the result of direct inversion ( gure 6.9a). In the case of multiples, iterative migration fails to explain both primaries and multiples correctly ( gures 6.12 and 6.13). Some multiple re ections are still misinterpreted as primaries after ten iterations ( gure 6.12, bottom row, between 0.45 s and 0.6 s). Actually, the errors in the shape of the source are converted into a di erent re ector shape compared to the reference case. Primaries and multiples interact once and twice with re ectors respectively, hence changing the shape of the re ector can compensate for the source wavelet in the case of primaries, but this result in a wrong phase for predicted multiples. At the rst iteration, the re ectivity is optimised for primary re ection, then little improvement is made in the following iterations for both primaries and multiples ( gure 6.13, bottom). This results in poor nal data residuals and cross-talk artefacts not being attenuated in the nal CIGs ( gure 6.9d). This has unwanted consequences on the convergence of the adjoint problem ( gure 6.14, green curve), which may not be positive de nite in the case of multiples as -120 0 120 h (m)

Kξ (N+1)
-120 0 120 h (m)

A T AKξ (N+1)
-120 0 120 h (m) -120 0 120 h (m)

K T A T AKξ (N+1)
Kξ (N+1)
-120 0 120 h (m)

A T AKξ (N+1)
-120 0 120 h (m) -120 0 120 h (m)

K T A T AKξ (N+1) (d)
Kξ (N+1)
-120 0 120 h (m)

A T AKξ (N+1)
-120 0 120 h (m) discussed in section 5.3.2. A negative curvature is encountered after eight iterations and we stop the conjugate gradient iterations. Although the norm of the objective function associated with the CG-algorithm still decreases, the relative normal residuals increases from iteration 5, which is not satisfactory. A larger regularisation weight may mitigate this issue, but at the expense of a possible higher nal data mis t. The gradient computed after eight iterations of resolution of the adjoint problem looks consistent ( gure 6.9d) but its value and signs are actually not stable from one iteration to another. Note however that the "truncated gradient" ( gure 6.11h) which does not depend on the number of adjoint iterations, shows much improvement compared to gure 6.9b and therefore stands as a reliable alternative. This test indicates that the estimation of the source wavelet is a key point for the accuracy of iterative migration. A poor estimate prevents the attenuation of cross-talk artefacts and leads to inconsistent background velocity updates.

K T A T AKξ (N+1)

Test C -Density perturbation

We now consider a new exact model for the modelling of observed data. The exact (and initial) background velocity model are the same as those used in the reference case ( gure 6.1, left) but here with a zero velocity perturbation. Instead we assume that re ections originate only from a variable density model ρ(x), modifying the wave-equation (2.4) .5) For this test, we will decompose the density model ρ(x) into ρ(x) = ρ 0 (x) + δρ(x) and model data under a second-order Born approximation. Similar to the velocity perturbation case and in the case of primaries and multiples (right). The ve case shown here correspond to di erent linear systems and cannot be directly compared. Note that the relative normal residuals are actually the norm of the gradient of the CG-objective and may not decrease from one iteration to the following.

into (iω) 2 c 2 P -ρ∇ • 1 ρ ∇P = Ω(ω)δ(x -s). ( 6
(section 2.3.1), we decompose P into P = P 0 + δP and obtain

(iω) 2 c 2 (P 0 + δP) -(ρ 0 + δρ)∇ • 1 ρ 0 + δρ ∇(P 0 + δP) = Ω(ω)δ(x -s), (6.6) (iω) 2 c 2 (P 0 + δP) -(ρ 0 + δρ)∇ • 1 ρ 0 - δρ ρ 2 0 ∇(P 0 + δP) = Ω(ω)δ(x -s), (6.7) leading to (iω) 2 c 2 δP -ρ 0 ∇ • 1 ρ 0 ∇δP = -ρ 0 ∇ δρ ρ 2 0 • ∇P 0 + δρ∇ 1 ρ 0 • ∇P 0 , (6.8)
Assuming a homogeneous background density model ρ 0 (x) = ρ 0 , this expression simpli es into

(iω) 2 c 2 δP -∆δP = - 1 ρ 0 ∇δρ • ∇P 0 (6.9)
In test C, we set ρ 0 = 1, and we consider a laterally invariant density perturbation with the same shape as the velocity perturbation de ned in the reference case ( gure 6.1, right). The magnitude of the perturbation is determined such that the amplitude of re ection data at zero surface o set is the same as in the reference case ( gure 6.15). Moving from a velocity to a density perturbation does not change the kinematics of primary and multiples re ections. The main di erence is that amplitudes are lower for wide opening angles, corresponding to large surface o sets of early primary re ections due to the shallower re ectors ( gure 6.15, right). This di erence of amplitudes has a direct impact on CIGs obtained by direct inversion. Compared to the reference case ( gure 6.3a), the shallower re ectors have much lower amplitudes at large value of the subsurface-o set, so that only the defocusing of the deeper re ectors is interpreted in the velocity analysis ( gure 6.16a). The gradient looks more homogeneous but this is due only to the attenuation of shallower re ectors. The same e ect appears when multiples are added to observed data. Due to smaller opening angles than primaries, their amplitudes in observed data remain similar to the reference case. As a consequence they have higher in uence on velocity analysis: the amplitude of positive energy below the deeper re ector is relatively stronger than the above negative values due to primaries ( gure 6.16b). After iterative migration and ltering, we obtain CIGs very similar to the result shown in gure 6.16a, both in the case of primaries only and in the case of primaries and multiples ( gures 6.16c and 6.16d). The nal re ectivity correctly explains both primaries and multiples ( gure 6.17) although re ections are modelled with a velocity perturbation for the determination of the optimal model perturbation. Compared to the reference case ( gures 6.12 and 6.13, top), the nal mis t is slightly higher for multiples ( gure 6.18), but this does not prevent cross-talk artefacts from being attenuated. The associated gradients are very close to the one obtained in gure 6.16a, especially after smoothing ( gures 6.19e and 6.19g). Again, the truncated gradients ( gures 6.19f and 6.19h) are remarkably close to the velocity update obtained in the direct inversion case, meaning that the re ectivity image obtained after iterative migration allows to faithfully reproduce the kinematics of primary re ections. We conclude from test C that our approach correctly extracts the kinematic information contained in re ections originating from density perturbations. Iterative migration succeeds in nding a velocity perturbation model explaining correctly both primaries and multiples re ections. The amplitude di erence, especially for wide opening angles, between re ections originating from density and velocity perturbation is compensated by a modi cation of the value of the re ectivity model at large values of h. The MVA gradient is still consistent, the di erence being that a smaller weight is given in the background velocity update to re ections originating from shallow re ectors.

Test D -Observed data modelled without the Born approximation

Finally we test the ability of our approach to deal with observed data modelled without the Born approximation. We use an exact velocity model di erent from tests A, B and C but with the same dimensions and we keep the same acquisition parameters. The velocity increases with depth, similar to the previous case, but instead of a dense re ectivity, we add two layers with homogeneous velocity (2600 m/s) between 200 m and 400 m depth and 550 m and 600 m depth; the density model is variable with interfaces located at the same positions as the velocity discontinuities ( gure 6.20, red curves). Both models are laterally invariant and the velocity perturbations are not proportional to the density perturbations. Observed data are modelled with a nite-di erence variable density acoustic propagation code. In the case of primaries only, PMLs are implemented on each edge of the model while a free-surface condition is used to model multiples ( gure 6.21, 1st column). The main e ect compared to second-order Born approximation are the following:

• an angle dependent re ectivity coe cient at the free surface;

• ghost e ects at the source and receiver positions; • all orders of multiples are modelled, as well as internal multiples.

For comparison, we smooth the velocity and density models and de ne associated velocity and density perturbations ( gure 6.20, blue curves) to model data under a second-order Born approximation ( gure 6.21, 2nd column). To take ghosts e ects into account, we use a scaled second-order time derivative of the Ricker wavelet in the case of multiples ( gure 6.22). In the primaries only example, both data sets are very similar. Amplitudes are overestimated by the Born approximation, but with consistent kinematics and a similar amplitude ratio at zero and large o sets ( gure 6.21, top row). In the case of multiples, the kinematics is not as well reproduced by the Born approximation as in the previous case, but remains consistent. However the amplitudes are overestimated with much larger discrepancies at large o sets than at zero o sets. The origin of these errors is the use of a second-order time derivative to model ghost re ections, valid at zero-o set only.

We now compute the background velocity updates using the observed data sets obtained without the Born approximation ( gure 6.21, 1st column). For calculated data, the density model is assumed homogeneous and the initial velocity model is increasing with depth ( gure 6.20, dashed green curve). Direct inversion on primaries data yields CIGs with downward curved events only, which is consistent with the initial too low velocity model ( gure 6.23a). The central part of the associated gradient is smooth and homogeneous with negative values as expected. In the multiple case, we use the modi ed source wavelet introduced in the preceding paragraph to model ghost re ections for the inversion ( gure 6.22). Cross-talk artefacts with an upward curvature are superimposed on the events related to primaries, resulting in spurious positive values in the gradient ( gure 6.23b).

Iterative inversion in the primaries only case yields CIGs similar to direct inversion ( gure 6.23c). The gradient is negative but exhibits oscillations around re ector positions, which can be nonetheless attenuated by smoothing ( gure 6.24e). Again, the "truncated gradient" ( gure 6.24f) is very close to the result shown in gure 6.23a. This contrasts with the mixed results obtained with multiples. Cross-talk artefacts are still visible on CIGs ( gure 6.23d), although nal data residuals are very small ( gures 6.25 and 6.26). Our interpretation of this apparent contradiction is that the nal re ectivity contains events with downward curvature corresponding to the true re ectors which correctly explain primary re ections. However the amplitudes of multiples modelled from these re ectors do not match those in observed data, because of the inaccuracy of both the second-order Born approximation and the modelling of ghosts used here. The di erence is explained by residual cross-talk artefacts, which are nonetheless smaller than in gure 6.23b. As a consequence, spurious positive values are still visible in the gradient ( gure 6.23d). They are not attenuated after smoothing ( gure 6.24g). Note also that the convergence of the adjoint problem is slower than in the previous cases ( gure 6.14, orange curve). As in the previous example, we observe that the truncated gradient provides more consistent velocity updates and seems more insensitive to residual cross-talk artefacts in CIGs ( gure 6.24h).

We conclude from test D that iterative migration has di culty in presence of a free-surface, because source and receiver ghosts are not properly modelled. We used a second-order time derivative of the source wavelet to account for this e ect, but this approximation is valid for zero re ection angles only. A better modelling tool should be investigated to deal with this issue. As for the preceding examples, the best result is obtained here with the "truncated gradient" approach.

Iterations on the background velocity model

We now consider a 2D model with lateral variations and iterate on the background velocity model. The model is 2500 m large and is made of three 1D velocity pro les similar to the previous example with two sub-vertical faults and a low-velocity layer between 400 m and 550 m depth in the central part of the model ( gure 6.27, top). We also consider a variable density model with interfaces located at the same positions as in the velocity model, but with di erent contrast values.

We have seen in the preceding section that free-surface remains an issue as we cannot model properly ghost re ections at wide angles. Therefore we consider here a simpler case and model observed data under a second-order Born approximation. For that purpose we use smoothed versions of the velocity and density models ( gure 6.27, bottom). From now on, "exact velocity model" refers to this smoothed version. Although the forward modelling code for observations and inversion both use the second-order Born approximation, we are not in the position of an "inverse crime". Observed data are indeed modelled with variable density and both velocity and density perturbations, the density perturbations being stronger. For inversion, we assume a constant density model (ρ 0 = 1) and only model re ections due to velocity perturbations.

Starting with an homogeneous (2000 m/s) background velocity model as initial guess, the inversion aims at minimising the following MVA objective function .10) To prevent convergence towards background velocity models minimising the energy in CIGs instead of minimising only defocused energy, we consider an objective function normalised by the energy of the re ectivity image without multiplication by h [START_REF] Chauris | Two-Dimensional Velocity Macro Model Estimation from Seismic Re ection Data by Local Di erential Semblance Optimization: Applications to Synthetic and Real Data Sets[END_REF]. As the shallower interface is a strong re ector, a multiplication by z is introduced in the annihilator, so that deeper re ectors have enough in uence on the velocity model reconstruction. Besides, as velocities are underestimated in the initial velocity model, re ectors are migrated to shallower depths, and should be shifted toward deeper positions across iterations, which con icts with the z-multiplication in CIGs. Therefore a multiplication by z is also introduced in the normalisation term of the objective function (6.10). We rst use the direct inversion strategy on primaries only and perform 20 iterations of nonlinear conjugate gradient with the Polak-Ribière formula ( gures 6.28 and 6.29). To accelerate the convergence, we assume that the velocity in the rst layer is known and set to zero the gradient below 100 m depth. We also introduce a preconditioner consisting of a multiplication by the depth z α with α = 1 in the applications. Finally we smooth the resulting velocity update with a gaussian blur, with a stronger smoothing in the x-direction (σ x = 140 m and σ z = 90 m). These values are progressively decreased every four iterations to recover more detailed structure in the last iterations, as shown in table 6.2. depth ( gure 6.29, top row). The associated defocused energy drives the rst velocity updates and the shallower velocity contrast is recovered after a couple of iterations ( gure 6.28). Subsequent iterations address the focusing of events related to the low velocity anomaly at larger depths, progressively appearing between iterations 5 and 15. After 9 iterations, the value of the objective function has decreased to the value obtained in the exact model ( gure 6.30, left). Smaller values are reached by further attenuating defocused energy but this does not necessarily result in more relevant information on the background velocity model. In particular, velocity updates after iteration 15 are mostly located around the faults to the detriment of the low velocity layers on the left and right sides of the faults (around z = 400 m and z = 500 m, respectively) which are not recovered in the nal model. Finally note that CIGs are well focused after 5 iterations and similar to those obtained in the exact model ( gure 6.29, 4th and bottom row). Few modi cations are visible in subsequent iterations compared to the evolution of the background velocity model. Three di erent inversions strategies are shown: direct inversion on primaries only (left, corresponding to gure 6.28), direct inversion on primaries and multiples (centre, corresponding to gure 6.31), and iterative inversion on primaries and multiples with application of the lter F † F and the "truncated gradient" as background velocity update (right, nal result shown in gure 6.33b, bottom right).

J 1 [c 0 ] = h • z • c β 0 • ξ[c 0 ](z, x, h) 2 z • c β 0 • ξ[c 0 ](z, x, h) 2 . ( 6
The same example is run with rst-order surface multiples in observed data with identical smoothing parameters ( gures 6.31 and 6.32). We use direct inversion for the de nition of the extended image to evaluate the impact of multiples on the reconstruction of the velocity model. As already discussed in section 4.4.2, the associated MVA objective function is not minimal for the correct velocity model due to the presence of multiples. Here, a smaller value is reached after a single iteration ( gure 6.30, middle), indicating that inversion converges to a compromise model which should focus both cross-talks artefacts and events corresponding to true re ectors. The velocity models obtained in the rst iterations have a shape similar to the primaries only case ( gures 6.28 and 6.31), but after a few iterations, events with an upward curvature (around z = 400 m depth at iteration 5 for example, fourth row in gure 6.32) misguide background velocity updates and prevent a proper focusing of the event corresponding to the shallowest interface. Consequently, the value of the velocity at this interface is not correctly recovered, and neither is the low velocity anomaly. Finally we replace direct inversion by iterative migration, still with both primaries and rstorder surface multiples in observed data. We perform 5 inner iterations on the re ectivity model for each outer iteration and compute a background velocity update with the "truncated gradient" strategy: primaries are remodelled from the nal re ectivity model and the direct inversion strategy applied to this new data set to obtain a background velocity update. We begin by comparing the re ectivity sections obtained in the initial-velocity model ( gure 6.33a). The cross-talk artefacts appearing after direct inversion the complete data set (around z = 400 m) due to the multiple re ecting twice at the shallower interface are greatly attenuated after ve iterations. Although the gradient obtained with direct inversion on primaries and multiples is quite similar to the primaries only case, we obtain a much closer velocity update with the "truncated strategy" ( gure 6.33b, left column). Note that the largest values in the gradient are due to the faults, but consistent homogeneous updates are obtained on the left and right sides of the fault. The objective function decreases at the same pace as in the primary only case, and reaches a slightly lower value than the one obtained in the correct velocity model. Finally, the background velocity model recovered after fteen iterations is very similar to the result of inversion obtained in the primaries only case ( gure 6.33b, right column), showing that multiples have been correctly included in the process. We conclude from this example that the truncated gradient approach, consisting of applying the direct inversion to primaries only recomputed from the nal result of iterative migration, is an e cient strategy to deal with surface-related multiples, which otherwise prevent the inversion to converge to a reliable background velocity model. The inversion should in theory also be performed with the complete gradient, but this is more computationally expensive as the adjoint problem has to be solved and as more inner iterations are needed in practice to obtain consistent gradients. The truncated gradient is not formally the gradient of an objective function but e ciently deals with multiples at a reasonable computational cost.

Incomplete acquisition

Finally we consider an example with an incomplete acquisition to test the ability of our strategy to bene t from the additional information contained in multiples. For this example, only the rst MVA gradient is computed (we do not loop over the background velocity model). We consider a 700 m deep and 3750 m large model, discretised on a 4.8 × 4.8 m grid, with sources every six grid points from x = 0 m to x = 1500 m and from x = 2250 m to x = 3750 m ( gure 6.34). Receivers are located at each grid point at the surface within ±700 m around each source. Unlike sources, receivers are not removed from the central part of the model. Iterative migration is performed with this acquisition setting. After 20 iterations, data are re-computed for sources located every four grid points, including the central part. Using the truncated gradient strategy, velocity analysis is performed on this new data set with the direct inversion strategy (section 4.2).

We want to compare this background velocity update with the result obtained using the direct inversion strategy on the original data set. We consider a pure 1D velocity and density model ( gure 6.34) built by extending the central part of the exact model shown in gure 6.27. Hence the di erence between the results of direct and iterative inversion are not due to 2D e ects. The initial background velocity model is homogeneous (2000 m/s), so that we expect a negative gradient. 6) = F † Fξ (6) prim +mult -Results of 20 iterations on the background velocity model using the iterative migration strategy with observed data containing both primaries and rst-order surface multiples, and comparison with the results obtained with direct inversion. In the iterative case, the "truncated gradient" strategy is used to compute the background velocity update. direct inversion on primaries only (top), and truncated gradient computed after 20 iterations on J 0 with primaries only in both observed and calculated data (middle), and with primaries and multiples in both observed and calculated data (bottom). In the iterative case, the re ectivity section displayed here is the result of application of the lter F † F to the nal re ectivity model computed by iterative migration.

h (m) (
Twenty iterations are performed on the re ectivity section in the primary only case, as well as in the case of rst-order surface multiples ( gures 6.35 and 6.36). In the primary only case, iterative migration provides a re ectivity model with a smaller hole in the central part of the model compared to direct inversion ( gure 6.35, middle). The area with non-zero background velocity update is also reduced. In the case of multiples, we obtain a very similar re ectivity image ( gure 6.35, bottom). Although the multiples illuminate the central part of the model, we do not succeed in recovering a continuous re ector. Some cross-talk artefact are even still visible (around z = 400 m and x = 1600 m for example). As a consequence the MVA gradient does not exhibit improvement compared to the primary only case and there is even a stronger artefact with a wrong sign around x = 1600 m and x = 2200 m. A better result might be obtained by performing much more iterations, but the associated computational cost would not be a ordable. We conclude from this example, that iterative migration allows to extract more information on the edges of the model compared to direct inversion, even in the case of primaries only. However, the additional information contained in rst-order surface multiples is di cult to recover as it is located on the edges of the acquisition where strong edge e ects alter the gradient. Also the tapers applied to the sources and receivers on the edges of the acquisition reduce the weight given to multiples containing additional information.

Conclusion

In this chapter, we have tested the behaviour of iterative velocity analysis in the case of direct inversion and iterative migration when observed data are computed with a forward modelling code di erent from the one used during inversion. We have shown that our approach is robust against the lack of low frequencies in observed data and to mild errors in amplitude predictions. However, a reliable estimation of the source wavelet is required for iterative migration to obtain extended re ectivity images able to explain both primaries and surface-related multiples. In the perspective of real data applications, we have shown that ghost re ections are an issue and that they should be properly modelled or removed. Finally we have compared the background velocity update obtained as the complete gradient of J 1 with an alternative strategy introduced in chapter 5 and referred to as truncated gradient in this chapter. It consists of applying the direct inversion strategy to primary re ection data re-computed from the result of iterative migration. We have shown that this approach yields consistent background velocity updates and is more robust than the complete gradient computation with also a lower computational cost.

Then this strategy has been applied on a 2D synthetic example consisting of a model with lateral velocity and density variations, observed data being modelled under a second-order Born approximation with both velocity and density perturbations. After fteen iterations on the background velocity model, we obtained a result similar to the one obtained with the direct inversion strategy and observed data containing only primary re ections. This example shows the e ciency of our approach applied to data containing both primaries and rst-order surface multiples and yielding results similar to those obtained without multiples in observed data.

Finally, we have shown on an example with an incomplete acquisition that iterative migration allows to recover additional information about the re ectivity model and the associated background velocity update. However, it has not succeed in extracting the information contained in rst-order surface multiples. Further investigation is needed to achieve better results, in particular by extending the method to higher-order multiples.

Résumé du chapitre 7

Je me suis intéressé dans cette thèse à la détermination des grandes longueurs d'onde du modèle de vitesse de propagation des ondes P par les méthodes d'analyse de vitesse par migration. J'ai en particulier étudié l'orientation profondeur de l'optimisation par semblance di érentielle (DSO). La qualité d'un macro-modèle de vitesse est estimée en analysant la répartition d'énergie dans un modèle de ré ectivité, « étendu » avec l'o set en profondeur, obtenu par migration des données observées.

J'ai montré que remplacer l'étape de migration classique par une migration à amplitude préservée, ou inversion, pour le calcul des CIGs a une in uence déterminante sur le succès de la méthode. En particulier l'inversion permet de s'a ranchir des artefacts de migration qui ne satisfont pas le critère cinématique de la DSO. J'ai introduit et comparé dans ce manuscrit deux stratégies : une inversion « directe » où le modèle de ré ectivité est obtenu par application d'un pseudo-inverse aux données observées et une stratégie itérative où le modèle de ré ectivité est dé ni en tant que solution d'un nouveau problème inverse. Avec une légère modi cation de la fonction coût de la DSO, la stratégie utilisant l'inversion directe produit des remises à jour du macro-modèle cohérentes et lisses, et les « artefacts du gradient » sont fortement atténués. En revanche elle n'est dé nie que pour le cas de ré exions primaires. Pour pouvoir considérer également des ré exions multiples dans la DSO, j'ai étudié le cas où le modèle de ré ectivité est déterminé par migration itérative, d'abord dans le cas de primaires seuls où il peut être comparé à la stratégie utilisant l'inversion directe. J'ai montré sur des exemples numériques que la migration itérative atténue e cacement les artefacts de migration et fournit des modèles de ré ectivité proches de ceux obtenus par inversion directe. Par ailleurs, la convergence de la migration itérative peut être accélérée en utilisant un préconditionneur construit à partir du pseudo-inverse. La principale di culté du cas itératif est l'instabilité du calcul de la remise à jour du macro-modèle, qui est dé nie comme le gradient de la boucle externe d'un problème d'optimisation à deux niveaux. En pratique, cette instabilité signi e que de faibles di érences entre deux modèles de ré ectivité peuvent conduire à de grandes di érences sur le gradient. Elle est liée à l'apparition, au cours de la migration itérative, d'évènements aux grandes valeurs de l'o set en profondeur du modèle de ré ectivité. Cette énergie, non porteuse d'information cinématique, est ampli ée par l'application de l'annihilateur et perturbe la migration itérative. L'instabilité du calcul du gradient peut être atténuée en régularisant la boucle interne. La di culté est alors de déterminer le poids donné à la régularisation : ce choix est délicat, coûteux numériquement, et doit a priori être renouvelé à chaque remise à jour du macro-modèle. J'ai étudié au chapitre 5 une autre régularisation, portant sur la boucle externe et qui consiste à appliquer au résultat de la migration itérative un opérateur atténuant l'énergie non porteuse d'information cinématique et responsable des instabilités. Cette approche ne requiert pas de régularisation spéci que de la boucle interne et fournit des remises à jour du macro-modèle similaires à celles données par l'approche directe.

J'ai étudié l'extension de l'approche itérative aux cas des ré exions multiples de surface du premier ordre, modélisées ici avec une approximation de Born du second ordre. La migration itérative devient alors un problème d'optimisation non linéaire. J'ai illustré sur des exemples numériques l'atténuation au cours des itérations des artefacts dans les CIGs causés par l'interprétation des multiples comme ré exions primaires. Dans le cas de multiples, la régularisation de la boucle interne demeure essentielle pour obtenir des remises à jour du macro-modèle satisfaisantes, même avec la régularisation sur la boucle externe introduite dans le cas des primaires. Pour avoir un schéma plus exible et moins coûteux numériquement, j'ai nalement proposé une stratégie alternative où le modèle de ré ectivité étendu obtenu par migration itérative est utilisé pour générer un nouveau jeu de données ne comprenant que des ré exions primaires, cohérents cinématiquement avec ceux contenus dans les données observées. La remise à jour du macro-modèle est alors obtenue en appliquant la stratégie d'inversion directe à ce nouveau jeu de données. Cette remise à jour n'est pas le gradient d'une fonction objective mais elle demeure cohérente et robuste d'une part et moins coûteuse numériquement que le calcul du gradient complet d'autre part.

Dans cette étude, l'introduction des multiples dans la stratégie de DSO a été faite dans l'optique de s'a ranchir des artefacts liés aux multiples, sans réellement chercher à exploiter l'information supplémentaire qu'ils contiennent. Cela demanderait probablement de considérer tous les ordres de multiples de surface ainsi que les multiples internes. Des approximations de Born d'ordres plus élevés ne constituent probablement pas une solution satisfaisante. En particulier j'ai illustré les di cultés de cette approche lorsque les e ets de surface libre sont introduits dans la modélisation des données observées. Dans l'optique d'une prise en compte de tous les ordres de multiples et des multiples internes, la construction de CIGs fonction de l'o set en profondeur avec les méthodes d'imagerie de Marchenko a été étudié par [START_REF] Díaz | Extended Imaging, Deconvolution, and Two-Way Wave elds: A Comparison[END_REF][START_REF] Díaz | Extended Imaging and Tomography under Two-Way Operators[END_REF]. Cette approche semble prometteuse, mais il reste à dé nir une stratégie e cace pour le calcul de la remise à jour du macro-modèle.

En vue de possibles applications à des données réelles, j'ai testé dans le chapitre 6 la robustesse de la DSO dé nie avec la migration itérative sur des jeux de données obtenus avec une physique di érente de celle utilisée pour l'inversion. J'ai en particulier illustré la sensibilité de la migration itérative vis-à-vis de l'estimation de l'ondelette de source dans le cas de multiples. Il serait intéressant de chercher à introduire d'avantage de physique dans la méthodologie de la DSO. Par exemple, une méthodologie uni ée prenant en compte simultanément les ondes ré échies (primaires et multiples) et les ondes transmises suivant la stratégie de [START_REF] Lameloise | Quantitative Migration for a More Robust Migration Velocity Analysis[END_REF] reste à dé nir. Par ailleurs, la méthodologie conduisant à la formule d'inversion de perturbations du modèle de vitesse, proposée dans la section 4.2, pourrait être appliquée au cas où les ré exions sont causées par des perturbations du modèle de densité. Plus généralement, la physique de propagation des ondes utilisée pourrait également prendre en compte une densité variable ainsi que les e ets de propagation élastique, d'atténuation et d'anisotropie.

En n la stratégie d'optimisation utilisée pour l'inversion du macro-modèle n'a pas été étudiée en détail dans cette étude. [START_REF] Huang | Born Waveform Inversion via Variable Projection and Shot Record Model Extension[END_REF] utilisent la méthode de projection des variables (VPM) et une fonction objective di érente où le terme de DSO est assimilé à une régularisation de la fonction coût comparant données observées et données calculées. Par ailleurs, l'exploitation de l'information contenue dans le Hessien de la fonction coût de DSO n'a été que peu explorée dans la littérature, en particulier en raison du coût numérique élevé de son évaluation complète. C'est pourtant un aspect essentiel dans la remise à jour du macro-modèle. Le coût de calcul de la DSO et la taille mémoire qu'elle demande restent par ailleurs le principal obstacle à son extension au cas 3D.

Conclusions

In this thesis, I have investigated a Migration Velocity Analysis (MVA) technique for the resolution of the seismic inverse problem under the constant-density acoustic wave-equation approximation. This technique is de ned in the image domain and relies on the Born approximation and a separation of scales of the velocity model into a rapidly varying re ectivity model and a smooth macro-model. It aims at assessing the quality of an estimated background velocity model using the redundancy of seismic data. In the depth-oriented formulation, physical re ectivity images parametrised by spatial coordinates are extended with an additional variable h called subsurface-o set. After migration of seismic data to this extended domain, inaccuracies in the initial macro-model result in defocused energy at non-zero values of h. Using the Di erential Semblance Optimisation (DSO) strategy, the macro-model is iteratively corrected by minimising an objective function penalising defocused energy in Common Image Gathers (CIGs) until the extended re ectivity model becomes physical.

In this thesis, I have addressed the issue of spurious events appearing in CIGs which prevent the adoption of MVA techniques as standard seismic data processing tools. These artefacts are due to limited extension of acquisition geometries on one hand (Lameloise et al., 2014;[START_REF] Mulder | Subsurface O set Behaviour in Velocity Analysis with Extended Re ectivity Images[END_REF], and to multiple re ections misinterpreted as primary events on the other hand [START_REF] Mulder | Automatic Velocity Analysis by Di erential Semblance Optimization[END_REF][START_REF] Li | Interval Velocity Estimation via NMO-Based Di erential Semblance[END_REF], and do not properly focus for the correct velocity model. As a consequence, the MVA objective function is not minimum for the correct velocity model and its gradient does not provide a consistent background velocity update. Here I have investigated the use of inversion instead of migration to deal with both issues and obtain CIGs free of artefacts as well as consistent MVA gradients.

Inversion Velocity Analysis

The original formulation of MVA assumes primary re ection data only, and de nes a re ectivity image by application of the adjoint of the extended Born modelling operator to observed data. In the context of high-frequency approximation of the wave-equation, Lameloise et al. (2014) introduced an extended quantitative migration compensating for uneven illumination and geometrical spreading. Combined with the horizontal contraction technique [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF]Shen and Symes, 2015), this approach yields CIGs free of migration artefacts and smooth gradients. However ray theory is limited in the presence of complex geology, for which wave-equation based approaches are more appropriate. In this study, an approximate inverse using only wave-equation operators has been presented (section 4.2), with a formulation similar to other proposals [START_REF] Hou | An Approximate Inverse to the Extended Born Modeling Operator[END_REF]Symes, 2015, 2017). All these formulas are actually inverse operators in an asymptotic sense. Their derivation rely on ray theory and high-frequency approximations, but their nal expressions are free of ray quantities. This strategy results in improved background velocity updates compared to standard migration. Moreover the introduction of a power of the background velocity in the annihilator has been shown to remove oscillations located around the re ector position from the MVA gradient (section 4.2) yielding results similar to the horizontal contraction approach [START_REF] Fei | On the Gradient Artifacts in Migration Velocity Analysis Based on Di erential Semblance Optimization[END_REF]Shen and Symes, 2015). However the macro-model update derived with inversion is the gradient of an objective function, contrary to the case of horizontal contraction. Eventually, note that the advantage of this strategy is the recovery of true-amplitude images and CIGs free of migration artefacts.

As inversion formulas proposed so far are designed for primary re ections only, I have studied in this thesis iterative migration as a replacement for direct inversion. The nal formulation has been constructed progressively in chapters 2 to 5. The analysis has been performed in the case of primaries only, using the result of direct inversion as a reference, and then applied to the case of multiples. I also studied an equivalent 1D MVA problem in sections 4.2.5 and 5.2 which may not be representative for all the features of 2D subsurface-oriented DSO, but is a useful analysis tool as it exhibits a behaviour very similar to the 2D case with a much lower computational cost. The main conclusions of each chapter are summarised in table 7.1. I underline here the key aspects of the method.

In iterative MVA, the extended re ectivity image is not obtained via an inverse formula but as the solution of an inverse problem. For a given background velocity model, the model perturbation best explaining observed data in a least-squares sense is determined through an optimisation procedure. If calculated data are computed under the rst-order Born approximation (primaries only), the associated inverse problem is linear and can be solved with a standard conjugate gradient algorithm. I have shown that this technique e ciently attenuates migration artefacts in CIGs and yields results similar to direct inversion in the primary-only case. Compared to the result of direct inversion, we obtain after several iterations of migration a re ectivity image with a better deconvolution of the source wavelet, but this has little impact on the nal data mis t.

The derivation of the gradient of the outer objective function has been analysed in chapter 3. An approximate value of this gradient is computed using the nal result of two iterative problems: iterative migration on one hand, and a linear adjoint problem, similar to the direct problem but with a di erent source term on the other hand. One would expect the gradient to converge to a stable value at the same pace as iterative migration. Besides, as direct and iterative inversion provide similar re ectivity images, we may presume that the associated background velocity updates would be similar. However, numerical examples in chapters 3 to 5 have shown that this is not the case in practice. The value of the re ectivity for large h has been shown to have little impact on data residuals but a major in uence on the velocity analysis, after being ampli ed by the annihilator. Similar observations have been made by [START_REF] Huang | Born Waveform Inversion in Shot Coordinate Domain[END_REF]. This issue has been identi ed by comparing the value of both objective function J 0 and J 1 across inner iterations. The rst one reaches convergence after a few iterations, while the second does not stabilise because of small modi cation at large values of h ( gure 3.8). Another practical di culty is the slow convergence speed of the adjoint problem. As a consequence, the value of the gradient obtained with successive values of the adjoint variable is not stable. This is an undesirable behaviour as we would like the gradient to converge to a stable value after a given number of iterations performed for the resolution of both problems. I have shown that su cient regularisation on the re ectivity model helps mitigating these issues, but this solution is not fully satisfactory. In particular, the determination of regularisation parameters remains a tedious task and we still observe residual oscillations around re ector positions not present in the gradient obtained by direct inversion. For a more stable procedure, I have proposed to modify the usual MVA objective function by applying a " lter" depending on the background velocity model to CIGs before measuring defocused energy. This lter does not change the shape main realisations and conclusions limitations chapter 2

• an extended re ectivity model is de ned by iterative migration to minimise the mis t between observed data and calculated data • migration and cross-talk artefacts are greatly attenuated with iterations, improving the shape of the MVA cost function;

• a strategy should be de ned for the coupling with velocity analysis; chapter 3

• two methods for the computation of the MVA gradient after iterative migration are compared; • the selected method assumes that iterative migration reaches convergence. An adjoint variable is determined as the solution of a linear problem and the gradient is computed from the last iterates of the direct and adjoint problem;

• iterative MVA is computationally expensive as two iterative systems have to be resolved for the gradient computation; • su ciently strong regularisation is needed to obtain gradients free of spurious oscillations and stable across inner-iterations; chapter 4

• an approximate inverse of the extended Born modelling operator is de ned for primaries only, leading to a new IVA strategy; • introducing this approximate inverse as a preconditioner greatly accelerates the convergence speed of the direct problem;

• the adjoint problem converges much slower than the direct problem and the associated sequence of MVA gradients are not stable; • the MVA gradients obtained after direct and iterative migration are quite di erent despite the similarity of the associated re ectivity images; chapter 5

• iterative MVA is analysed on a pure 1D case. It exhibits a behaviour similar to the 2D case; • a lter F † F is introduced in the de nition of the MVA objective function to attenuate unwanted energy at large values of h; • with this modi ed MVA objective function, the adjoint problem converges much faster and the associated sequence of gradients is more stable with a shape similar to the gradient of direct inversion;

• the robustness of the approach regarding non-Born data should be investigated (beyond the inverse crime); chapter 6

• the direct and iterative inversion strategy are applied to a series of synthetic data computed with a modelling engine di erent from the one used during inversion; • direct inversion in the primaries only case is robust to inaccuracies of the source wavelet and to amplitude errors; • direct and iterative inversion strategies are robust with respect to density perturbations;

• in the multiple case, iterative migration fails to explain both primaries and multiples correctly if the shape of the source wavelet is inaccurately estimated; • application to real data. of defocused events in CIGs but attenuates spurious oscillations responsible for the instabilities described above. I have shown that this lter changes the source term of the adjoint problem which is now easier to solve. The value of the gradient is stable with iterations and similar to the one obtained by direct inversion. The additional cost represented by the application of the lter is largely compensated by the improved convergence of the adjoint problem, allowing to reduce the number of iterations.

Multiple reflections

I have studied the extension of this iterative procedure to the case of rst-order surface-related multiples. It consists of using a second-order Born approximation for forward modelling in the inner inverse problem. This changes the gradient formula in both the inner and outer minimisation problem. Iterative migration is not a linear inverse problem any more. Regarding the computation of MVA gradient, the adjoint problem is still linear but may not be positivede nite if nal data residuals are not small enough.

I have shown that cross-talk artefacts due to multiples are attenuated across iterations, leading to MVA gradients consistent with the primary-only case. However the computation of the migration objective function and of its derivatives is much more expensive compared to the linear case (table 3.1). Moreover iterative migration becomes a non-linear optimisation procedure. To reduce the computational expense of the method, we have proposed to use the approximate inverse as a preconditioner, which greatly improves the convergence of the direct and adjoint problem, even in the case of multiples. Finally we have proposed an alternative strategy where iterative migration is used to retrieve an extended re ectivity image free of cross-talk artefacts, allowing to re-compute primary re ections and to apply the direct inversion procedure which has been shown to be robust and computationally e cient. Although the background velocity update de ned by this strategy is not the gradient of an objective function, the procedure is lees computationally expensive as the iterative resolution of the adjoint problem is not necessary. Furthermore, we can reconstruct data for source positions missing in the original acquisition. Hence eventual additional information in the re ectivity image due to extrainformation contained in multiples may be incorporated in the re-computation of primaries.

In this study, multiples have been introduced with the objective of retrieving gradients similar to those obtained with multiple-free data. The models and acquisition settings used in the examples were not speci cally designed to use the additional information contained in multiples. We have shown that iterative migration succeeds in removing the imprint of multiples on the re ectivity, leading to gradients similar to those obtained in the case of primaries only. However, the example of section 6.4 shows that extracting additional information from multiples is a much more complex task, especially when considering only rst-order surface-related multiples. The areas investigated by this kind of multiples is actually small and located on the edges of the acquisition where edge e ects alter MVA gradient obtained in the primaries only case. Extending MVA techniques to all order of multiples and internal multiples may provide more favourable examples with wider illumination of the subsurface (see next section).

Perspectives 7.2.1. Using all orders of multiples

Extending MVA to all orders of multiples could be performed with the approach investigated by [START_REF] Díaz | Extended Imaging and Tomography under Two-Way Operators[END_REF], based on the resolution of the Marchenko equation. Using observed data and Green's function G 0 (s, x, ω) computed in an estimated macro-model c 0 (x), the iterative resolution of the Marchenko equations allows to retrieve the complete Green's functions G(s, x, ω) including all multiple re ections. This step is discussed in [START_REF] Wapenaar | Marchenko Imaging[END_REF] for the case of internal multiples and extended to surface multiples by [START_REF] Singh | Marchenko Imaging: Imaging with Primaries, Internal Multiples, and Free-Surface Multiples[END_REF]. The accuracy of the macromodel can then be evaluated in two di erent ways. First, one possibility is to decompose the complete Green's function into a downgoing G -and an upgoing wave eld G + to construct extended CIGs with a deconvolution imaging condition and measure defocused energy in a standard way [START_REF] Díaz | Extended Imaging, Deconvolution, and Two-Way Wave elds: A Comparison[END_REF]. An alternative consists of directly estimating the velocity model as G is reconstructed everywhere within the subsurface and is solution of the wave-equation

(iω) 2 c 2 (x)
G(s, x, ω) -∆G(s, x, ω) = δ(x -s), (7.1) the value of the velocity model c(x) can be recovered from the knowledge of the complete Green's function following

c 2 (x) = (iω) 2 G G ∆G G . (7.2)
This model can be decomposed into c(x) = c 0 (x) + δc(x). For the determination of G(s, x, ω), one needs to provide a smooth background model. If the kinematics is not correct, a standard MVA procedure can be de ned using the values of P obs and c 0 (x). Both the strategy of [START_REF] Díaz | Extended Imaging and Tomography under Two-Way Operators[END_REF] and our approach rely on an iterative procedure depending on the value of the macro-model c 0 (x). In [START_REF] Díaz | Extended Imaging and Tomography under Two-Way Operators[END_REF] the purpose is to retrieve the complete Green's function G, whereas here the unknown of the iterative process is an extended re ectivity model best explaining observed data. In both cases, the di culty is to compute the velocity update due to the dependence of the iterative process to the macro-model c 0 (x). Here we derive a strategy using the adjointstate method and the assumption that the output of the iterative migration is de ned as the minimiser of a cost function. Further investigations are required to de ne a similar strategy in the case of [START_REF] Díaz | Extended Imaging and Tomography under Two-Way Operators[END_REF]. It is possible that a regularisation term (equivalent of the lter K) has to be introduced with Marchenko-based MVA process. Note that the major advantage of the Marchenko approach is that both internal and surface multiples are consistently integrated in the imaging procedure, yielding CIGs free of cross-talk artefacts [START_REF] Díaz | Extended Imaging and Tomography under Two-Way Operators[END_REF].

Introducing more physics in MVA techniques

In section 4.2, an inversion formula allowing to retrieve an extended velocity perturbation explaining observed data has been presented. This formula was tested in section 6.2.4 on re ection data originating from a density perturbation and modelled under the Born approximation. Alternatively, one could also derive an inversion formula for density perturbation data, using a strategy similar to the one used in section 4.2. The di erence is that here, gradients of Green's function are involved instead of a their product with a second-order time derivative, which requires the derivation of new weighting operators di erent from the original inverse formula. This may enable an other inversion strategy for multi-component observed data. Using the surface recording of both horizontal and vertical displacements, one could try to invert for a velocity or a density perturbation, or for a velocity and impedance perturbation, as these two parameters exhibit less coupling [START_REF] Zhou | Full Waveform Inversion of Diving & Re ected Waves for Velocity Model Building with Impedance Inversion Based on Scale Separation[END_REF].

In this study we have used an explicit scale separation between a smooth background velocity model and a model perturbation. Following a strategy proposed by [START_REF] Zhou | Velocity Model Building by Full Waveform Inversion of Early Arrivals & Re ections and Case Study with Gas Cloud E ect[END_REF] in the framework of Full Waveform Inversion (FWI), one could also investigate a more natural scale separation, using velocity and variable density to parametrise the model, the velocity model controlling the kinematics of wave propagation and density accounting for the re ective property of the subsurface instead of the velocity model perturbation.

In the constant-density acoustic approximation, [START_REF] Lameloise | Quantitative Migration for a More Robust Migration Velocity Analysis[END_REF] show how transmitted waves can be included in MVA techniques by constructing extended images in a very similar manner to the usual procedure for re ections. They propose a strategy in which these two kind of events are used successively. Transmitted waves are used in a rst step to update the shallow part of the velocity model; then primary re ections are inverted to reach the deeper part of the model. In our approach, primaries and multiples are naturally inverted together. Inverting re ection and transmission data simultaneously could potentially better constrain the inversion. As pointed out by [START_REF] Lameloise | Analyse de Vitesse Par Migration Quantitative et Introduction Des Ondes Transmises[END_REF], an issue is that transmitted events are more energetic than re ections, requiring to introduce weights to balance the amplitudes of both kinds of events. Moreover, CIGs constructed with transmitted events do not have the same physical interpretation as CIGs built with re ection data; in particular they are not linked to a speci c interface. Hence, the inclusion of re ection and transmitted data in a uni ed framework needs further investigation.

Importance of the Hessian for iterations over the velocity model

A more thorough study should be led on the choice of optimisation strategies for the external loop aiming at determining the background velocity model. In particular the e ect of the Hessian of the objective function should be better taken into account. The gradients obtained by direct inversion in the case of primaries (section 4.2) are remarkably smooth, but their shape, and in particular the importance of side e ect vary greatly with the number of surface and subsurface-o sets, and the re ectors' depth. We may expect a proper introduction of the Hessian to attenuate these variations. Most applications on synthetic and real data "only" use the l-BFGS strategy to take second-order e ects in to account. An interesting strategy is proposed by Liu et al. (2014b) and Shen and Symes (2015) who compute an estimate of the diagonal of the Hessian matrix as the result of its application to a unit vector. An extension is the Truncated Newton strategy, already studied in the framework of FWI [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF][START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method: Quantitative Imaging of Complex Subsurface Structures[END_REF]. Contrary to the l-BFGS algorithm, these techniques require an e cient way of computing the product of the Hessian of the objective function with a vector of the space. One could use a second-order adjoint-state technique similar to the one presented in [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF][START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method: Quantitative Imaging of Complex Subsurface Structures[END_REF] and in section 3.2.3. An example of derivation of the matrix-vector product is presented in appendix E in the case of direct inversion. Note however that this approach is quite expensive. Moreover the positive-de nitiveness of the Hessian matrix should be investigated.

Inversion strategy

We have considered here a nested optimisation procedure with two separate objective functions, the inner inverse problem solving for the re ectivity for a given macro-model, and the outer objective function solving for the macro-model. Another possibility is to use a single objective function [START_REF] Fleury | Bi-Objective Optimization for the Inversion of Seismic Re ection Data: Combined FWI and MVA[END_REF][START_REF] Huang | Born Waveform Inversion via Variable Projection and Shot Record Model Extension[END_REF], where the rst term is related to data mis t and the second measures defocused energy in CIGs. Note that the DSO strategy was originally introduced as a regularisation for the FWI objective function [START_REF] Symes | Inversion of Re ection Seismograms by Di erential Semblance Analysis: Algorithm Structure and Synthetic Examples[END_REF]. An e cient procedure for the minimisation of this objective function is the Variable Projection Method (VPM) (van Leeuwen and Mulder, 2009;[START_REF] Rickett | The Variable Projection Method for Waveform Inversion with an Unknown Source Function[END_REF][START_REF] Huang | Born Waveform Inversion via Variable Projection and Shot Record Model Extension[END_REF]. It consists of inverting rst for the re ectivity ξ (following ξ = F † [c 0 ]P obs in the direct inversion case) and to replace in the de nition of the objective function. The rst step is similar to what have been proposed in this study, the second is di erent because we considered in this thesis only the second term in equation ( 7.3) for the derivation of a velocity model.

A di culty remains the choice of the weight λ. Furthermore, in the case of iterative migration with multiples, the expression replacing ξ in equation (7.3) should properly take cross-talk e ects into account, otherwise the second term of the objective function may mislead the background velocity update.

Extension to 3D

In 3D, the observed data space has ve dimensions (s x , s y , r x , r y , t), requiring two extension parameters for the model space, for example horizontal subsurface o sets h x and h y (table 4.1). A new inversion formula taking into account these new dimensions should be de ned with a strategy similar to the 2D case (section 4.2). The major issue of the extension is its computational cost, which disquali es the iterative determination of an extended re ectivity image with a large number of iterations. Preconditioning as proposed in this thesis or in Hou and Symes (2016a) is an essential element for the extension to 3D. In addition to 3D wave eld propagation, a cross-correlation should be performed for each couple (h x , h y ). New strategies such as those mentioned in section 1.3.4 should be further investigated for a more a ordable extension of MVA to 3D. Another di culty is that 3D acquisition do not provide dense source and receiver coverage in all directions, which may have undesirable e ects on re ectivity image and associated gradients.

Application to real data

As a prelude to real data applications, we have applied in chapter 6 the proposed MVA strategy to observed data computed with a forward modelling code di erent from the one used during inversion. We showed that an incorrect estimation of the source wavelet was an issue for the iterative case in the presence of multiples, but that direct inversion was insensitive to this kind of imperfection. More importantly, ghosts e ects due to the free-surface were shown to be an issue for iterative migration as they modify the amplitude and phase of re ection data, especially at wide angles. Ghost should be removed from observed data or properly included in the forward modelling to cope with this issue.

More generally, propagation e ects that are not accounted for in the forward propagation step may lead to inconsistent CIGs and velocity updates. For example the impact of attenuation and elastic e ects on the amplitude may be an issue for regular MVA strategies. Converted waves are also not predicted under the Born approximation.

Finally the irregularity of the acquisition geometry should be taken care of. Direct inversion formulas are indeed derived with the assumption of dense source and receiver coverage. Iterative migration may be an interesting alternative in this case as it deals more e ectively with irregular acquisition [START_REF] Nemeth | Least-squares Migration of Incomplete Re ection Data[END_REF].

• If condition (B.2) is veri ed, very small values of α ensure that the value of J 0 at iteration (n + 1) is smaller than its value at iteration (n). However to avoid performing numerous small steps, we would like to nd a value for α (n) close to the exact minimiser of J 0 along ξ (n) + α (n) d (n) . An optimal α value α (n) would lead to d (n) g (n+1) = 0.

(B.4)

Determining a good value for α (n) is actually a one-dimensional optimisation problem. We elaborate on linesearch strategies addressing this problem in the next section.

B.2. Linesearch

The linesearch procedure aims at nding the minimum of J 0 along the direction ξ (n) + α (n) d (n) .

In practice a scalar objective function ψ : → is de ned for xed values of ξ (n) and d (n) as ψ(α) = J 0 ξ (n) + αd (n) , (B.5a) and its gradient reads

ψ (α) = d (n) ∂ J 0 ∂ ξ ξ (n) + αd (n) . (B.5b)
For linear problems, ψ is quadratic and the exact minimiser can easily be determined. For non-linear problems, the shape of ψ is more complex and there is not necessarily an analytic formula for the minimiser of ψ. Then an inexact linesearch procedure is used to determine a "good" minimiser of ψ. The objective is to nd a value for α ensuring the following conditions called Wolfe conditions ( gure B.1):

• su cient decrease condition (also called Armijo condition) ψ(α) ≤ ψ(0) + c 1 αψ (0), (B.6a)

The scalar coe cient c 1 is positive, hence small values always satisfy this condition. The values of the coe cient c 1 and c 2 and the choice of the regular or strong curvature conditions depend on the descent direction strategy (detailed in the next section. For non-linear conjugate gradients methods, the strong Wolfe conditions are required and recommended parameters are c 1 = 1×10 -4 and c 2 = 0.1. For quasi-Newton methods, the regular curvature condition is su cient with c 2 = 0.9 [START_REF] Nocedal | Numerical Optimization[END_REF].

Algorithms ensuring the strong Wolfe conditions are relatively expensive because the value and the gradient of the objective function should be calculated at each trial step size. Therefore the choice of successive trial steps should be e cient. E cient strategies usually consist of two steps. First an interval containing an acceptable step size is determined. Then this interval is reduced iteratively until an acceptable step size is found. For our implementation, we have chosen the strategy of [START_REF] Mora | Elastic Wave-eld Inversion of Re ection and Transmission Data[END_REF] as recommended by Nocedal and Wright (2006, p. 162), using polynomial interpolations to compute the successive step size.

• the weighting operator W mod : → , which consists of a derivative with respect to z and a multiplication by a power α of c 0 ,

W mod ξ(x, h) (x, h) = - k c α 0 (x) ∂ ∂ z ξ(x, h) , (C.3)
If ξ is de ned as a velocity perturbation ξ = δc/(2c 0 ), then k = 8 and α = 0; if ξ is de ned as a squared slowness perturbation ξ = 2δc/c 3 0 , then k = 32 and α = 2. The vertical derivative in this weight can be interpreted under the high-frequency approximation as a multiplication by the cosine of the half-opening angle and the cosine of the dip angle at the image point ( gure C.1), iω c 0 (x) cos θ cos φ;

(C.4)

• the weighting operator W r : obs → , which in practice constructs a dipole source term around the depth z r of the receivers. Its practical implementation is

C.2. Equations for the adjoint map

We want to compute the adjoint (F † ) T of the operator F † . It is applied to a vector χ of and computes a vector Q = (F † ) T χ in obs . We use the adjoint-state method and de ne a scalar objective function and the derivatives of Γ with respect to P gives the application of (F † ) T to χ,

(F † ) T χ = ∂ Γ ∂ P = W * r ν 1 . (C.13)
These equations involve the adjoint of the weighting operators de ned in the previous section,

• the operators K -[P, ξ] and K + [P, ξ] : × → , de ned for P ∈ and ξ ∈ as Analytic expression for the MVA gradient a er direct inversion in 1D

This appendix is related to section 5.2.2 and details the derivation of the analytic expression of the background velocity update obtained after direct inversion in the 1D case (equations 5.13b, 5.15c and 5.15d) (we do not detail the case of the adjoint (equations 5.13a, 5.15a and 5.15b), which is very similar). The purpose is to obtain an analytic expression providing results similar to those obtained with the adjoint-state method and wave-equations operators (section 4.2.10). Note that the derivation does not assume a homogeneous background velocity model. We recall the expression of the objective function, The purpose of this appendix is the derivation of the expression of Q inv given in equation (5.15c).

J inv 1 [c 0 ] = 1 2 z a 2 (z)c

D.1. Derivation of Q inv

The derivation starts from equation (D.2), hence we need an expression for the derivatives of the Green's function G 0 (s, z, ω) with respect to s and z, and with respect to the value of c 0 at depth y. where the dependence of the Green's function to ω have been omitted.

D.2. High-frequency expression

For the numerical applications in section 5. We check the validity of equation (D.11) on a numerical example by comparing the gradient obtained with this expression with the gradient obtained by nite di erences and slightly perturbed model ( gure D.1). The latter consists of computing the value of J 1 (equation D.1) for velocity models with small velocity perturbations at position y c( y) = c 0 ( y) ± δc 0 . The computation is repeated for each depth y. Note that in this approach, ξ inv is not obtained through the asymptotic expression (D.13), but by solving wave-equations as described in appendix C.1. It is actually important to note that equation (D.14) has been obtained by replacing the value of the Green's functions by their high-frequency approximation (equation D.12) after the calculation of the derivative with respect to c 0 . One would obtain a di erent expression by applying the derivative with respect to c 0 to a high-frequency approximation of F † P obs such as equation (D.13). In particular, operator Q inv would not be upper triangular any more.

To illustrate this point, we consider the derivative of the Green's function G(s, z, ω) with respect to c 0 ( y). Applying the derivative rst leads to equation (D.6a) and then replacing the (D.17)

The main di erence is that the high-frequency approximation of the Green's function between s and z is insensitive to a perturbation below z, whereas the general Green's function may be a ected by a perturbation of the velocity model below z as a di racting point for example ( gure D.2, right). such that the derivative of Γ with respect to c 0 gives the desired matrix-vector product,

∂ Γ ∂ c 0 [c 0 , V] = ∂ 2 J 1 ∂ c 2 0 [c 0 ] V. (E.4)
Noting G the gradient of J 1 with respect to c 0 , we de ne the Lagrangian by de ning constraints on the state equations solved to determine the gradient with the adjoint variables γ ∈ , (Ξ 0 , Ξ, E, E 0 ) ∈ and (Π 0 , L 1 , N 1 , M 0 ) ∈ , with the value of k and α de ned in equation (C.3). Deriving the Lagrangian with respect to the state variable µ 0 , ν 1 , η 0 , η, ξ, ξ 0 , λ 1 , P 0 yields the following sets of adjoint equations

Γ = G V -γ G - ∂ f J 1 ∂ c 0 [c 0 , ξ] -C[µ 0 , P 0 ] -C[ν 1 , λ 1 ] -f mod [c 0 , η, ξ 0 ] -Π 0 0 P 0 -S -L 1 * 0 λ 1 -W r M T P obs -Ξ 0 ξ 0 -Q[P 0 , λ 1 ] -Ξ ξ -W mod ξ 0 -E η - ∂ f J 1 ∂ ξ [c 0 , ξ] -E 0 η 0 -W * mod η -N 1 0 ν 1 -K -[P 0 , η 0 ] -M 0 * 0 µ 0 -K + [λ 1 , η 0 ] , (E.
             0 M 0 = B[P 0 , V] * 0 N 1 = B[λ 1 , V] E 0 = Q[P 0 , N 1 ] + Q[M 0 , λ 1 ] E = W mod E 0 + ∂ f mod ∂ η [c 0 , η, ξ 0 ] V (E.7a) (E.7b) (E.7c) (E.7d)                    Ξ = ∂ 2 f J 1 ∂ ξ 2 [c 0 , ξ] E + ∂ 2 f J 1 ∂ c 0 ∂ ξ [c 0 , ξ] V Ξ 0 = W * mod E 0 + ∂ f mod ∂ ξ 0 [c 0 , η, ξ 0 ] V 0 L 1 = B[ν 1 , V] + K -[P 0 , Ξ 0 ] + K -[M 0 , η 0 ] * 0 Π 0 = B[µ 0 , V] + K + [λ 1 , Ξ 0 ] + K + [N 1 , η 0 ], (E.8a) (E.8b) (E.8c) (E.8d)
where operator B : × → is de ned for u ∈ and V ∈ as B[u, V](s, x, ω) = 2(iω) 2 V(x) c 3 0 (x) u(s, x, ω). (E.9)

Eventually the desired matrix-vector product is obtained as the gradient of Γ with respect to c 0 and reads

∂ 2 J 1 ∂ c 2 0 [c 0 ] V = ∂ Γ ∂ c 0 =C[Π 0 , P 0 ] + C[L 1 , λ 1 ] + C[N 1 , ν 1 ] + C[M 0 , µ 0 ] - 3V c 0 C[µ 0 , P 0 ] + C[ν 1 , λ 1 ] + ∂ 2 f J 1 ∂ c 2 0 [c 0 , ξ] V + ∂ f mod ∂ c 0 [c 0 , η, ξ 0 ] V + ∂ 2 f J 1 ∂ ξ∂ c 0 [c 0 , ξ] E.
(E.10)

We have divided the adjoint equations into two sets. In the rst group of equation, a vector E ∈ is determined through computations very similar to those used to determine ξ. The second set of equations looks like the one used to determine the gradient ∂ J 1 /∂ c 0 . Although only four "new" wave elds have to be computed (Π 0 , L 1 , M 0 , N 1 ), the approach is quite expensive because 4D wave elds arrays are too large to be kept in memory and need to be recomputed, rst to determine the value of E, then to compute the remaining contributions of the matrix-vector product.

Abstract

Active seismic experiments are commonly used to recover a model of the P-wave propagation velocity in the subsurface. "Migration Velocity Analysis" techniques aim at deriving a smooth background velocity model controlling the kinematics of wave propagation. First, a reflectivity image is obtained by "migration" of observed data using a first estimate of the background velocity. This image depends on an additional "subsurface-offset" parameter allowing to assess the quality of the background velocity model with a focusing criterion and to correct it. However classical migration techniques do not provide a sufficiently accurate reflectivity image, leading to inconsistent velocity updates. In particular they do not take into account multiple reflections, usually regarded as noise and removed from the data before processing. Multiple removal is however a difficult step, and additional information contained in multiples is discarded.

In this thesis, we propose to determine the reflectivity model by iterative migration before subsequent velocity analysis, leading to a nested optimisation procedure. Iterative migration yields accurate reflectivity image and extends naturally to the case of multiples. One of its disadvantages is the associated increased computational cost. To limit the number of iterations in the inner loop, a preconditioner based on a pseudoinverse of the modelling operator is introduced. Another difficulty is the instability of the velocity update obtained with very close successive reflectivity models. We propose a modified approach, valid in the presence of multiples, and discussed through applications on 2D synthetic data sets.

Figure 1

 1 Figure 1.1. -Acquisition geometries for di erent types of seismic surveys.

Figure 1 . 2 .

 12 Figure 1.2. -An example of a 2D seismic data recorded in marine acquisition (from Anandakrishnan et al., 1998).

Figure

  Figure 1.3.-Spatial frequencies that can be resolved from seismic data (fromClaerbout, 1985).

  (a) Velocity model c(x). (b) Background model c 0 (x). (c) Model perturbation δc(x).

Figure

  Figure 1.4. -Illustration of the usual scale separation for the Marmousi model (Brougois et al., 1990). The complete velocity model c(x) is decomposed into a smooth background part c 0 (x) and a perturbation δc(x) of the background model c 0 (adapted from Billette, 1998).

Figure 1 . 5 .

 15 Figure 1.5. -Relationship between the wavenumbers k s and k r and the opening angle θ at the image point x.

Figure 1 . 6 .

 16 Figure 1.6. -Example of CIG with the surface o set as redundant parameter (taken from Chauris et al., 2002).

Figure 1 . 7 .

 17 Figure 1.7. -De nition of the subsurface-o set spatial delay.

  (a) Common Image Gathers. (b) Gradient of the MVA objective function.

Figure 1

 1 Figure 1.8. -(a) CIGs and (b) associated gradients of the MVA objective function computed with classical migration for a too low (left), correct (middle), and too high (right) initial velocity model with a single horizontal re ector. Blue, white and red colours correspond to negative, null and positive values, respectively (from Lameloise et al., 2014). Homogeneous update above the re ector would be expected in a tomographic approach.

  (a) Common Image Gathers. (b) Gradient of the MVA objective function.

Figure 1

 1 Figure 1.9. -Same as gure 1.8 with a quantitative (ray-based) migration instead of a classical (adjoint) migration (from Lameloise et al., 2014).

( a )

 a Without horizontal contraction. (b) With horizontal contraction.

Figure 1 .

 1 Figure 1.10. -(a) Illustration of the "gradient artefacts" on a homogeneous model with a single horizontal re ector. Oscillations appear at the truncation of the re ectivity. (b) The "horizontal contraction" approach allows to remove these spurious oscillations (from Fei and Williamson, 2010).

  relative data mis t for successive values of δc (a) Objective function representing data mis t for the successive value of δc across iterations. An example of expected gradient of the MVA objective function. Blue, white and red colours correspond to negative, null and positive values, respectively. Actual MVA gradient computed after 7, 8, 9, 10 and 11 iterations on the re ectivity δc.

Figure 1 .

 1 Figure 1.11. -Iterative resolution of the migration inverse problem and computation of the associated MVA gradient, showing the sensitivity of the gradient to minor changes in the re ectivity model.

  Observed data (left), re ectivity section at h = 0 (middle) and MVA gradient (right) recovered with an increasing level of noise (from top to bottom). Blue, white and red colours correspond to negative, null and positive values, respectively. re ectivity error in the MVA gradient y = x1.13 (b) Error in the recovered re ectivity model and in the associated MVA gradient, function of the level of noise introduced in observed data.

Figure 1 .

 1 Figure 1.12. -In uence of noise in observed data in the re ectivity model recovered by migration and on the associated MVA gradient.

  Figure 1.13. -Principle of MVA in a simple 1D case. The left column describes the migration process. The source wave eld (c) is constructed by propagating the source (a) in the medium. Data observed at the surface (b) are backpropagated to construct the receiver wave eld (d). Correlation of the source and receiver wave eld produces the model perturbation image (e). Energy away from the position of the true re ector (400 m) is penalised by an annihilator (f) to produce a modi ed migrated image (g) called "image residual". The latter is used to construct scattered and backscattered wave elds ((j) and (i), respectively). The (opposite of the) velocity update is made of two contributions ((h) and (k)), obtained by correlating (c) and (i) on one side, and (d) and (j) on the other side.

  Figure 1.15.-Di erent kinds of multiple re ections (similar toVerschuur, 2013, pp. 8-9). The black point and the white triangle represent source and receiver positions, respectively.

  Figure1.16. -Long-period and short-period multiples (see alsoVerschuur, 2013, p. 10).

Figure 1 .

 1 Figure 1.17. -(a) A real marine data set containing both primaries and multiples is decomposed into (b) primaries only and (c) multiples only (from Trad et al., 2003).

Figure 1 .

 1 Figure 1.18. -Synthetic example of classic migration performed on data contaminated with multiples. WBM is a spurious events caused by a Water Bottom Multiple while M refers to surface related multiples repeating the bottom structure of the salt dome (from Sava and Guitton, 2005).

Figure 1 .

 1 Figure 1.19. -Example of conventional migration using primaries only (top) and using primaries and multiples (bottom) on the Sigsbee2B synthetic data set. Note the extended illumination provided by multiples below the salt dome in the areas marked with the white ellipses (from Liu et al., 2011).

Figure 1 .

 1 Figure 1.21. -Example of cross-talk artefacts. The exact velocity model is displayed on the left panel. The right panel shows the image reconstructed from the migration of multiple re ections. C and D correspond to the true re ectors. All other events including A and B are cross-talk artefacts (from Liu et al., 2011).

  Figure 1.22. -Sketch showing the algorithm of MVA with iterative migration. In the inner loop,we optimise the re ectivity δc for a given c 0 in order to minimise the objective function J 0 measuring the data mis t P[c 0 , δc] -P obs . In the outer loop, we update the velocity model to minimise the objective function J 1 measuring defocused energy in the CIGs of δc.

Figure 2

 2 Figure 2.1. -The Ricker wavelet with maximum frequency of 40 Hz.

  Position of the source (black dot) and the receivers (along the dashed lined). Traces extracted at the position indicated by the dashed lines in (c) Data recorded at receiver positions when the modelling is performed with nite di erences (left) and with the Born approximation (centre). The right panel display their di erence. The same colour scale is used for the three plots.

Figure 2 . 2 .

 22 Figure 2.2. -Accuracy of the second-order Born approximation with respect to the free-surface re ection.

  Data recorded at receiver positions when the modelling is performed with nite di erences (left) and with a free-surface condition and a second-order Born approximation (centre). The right panel display their di erence. The same colour scale is used for the three plots. Born: ξ(z)Born: c 0 (z) (b) 1D-models used for nite-di erences and Born modelling.

  Traces extracted at the positions indicated by the dashed lines in (a). The bottom row is a zoom of the top row allowing to compare multiples modelled with both approaches.

Figure 2 .

 2 Figure 2.3. -Comparison of nite-di erences modelling and free-surface modelling with a second-order Born approximation in the re ective case.

Figure 2 . 4 .

 24 Figure 2.4. -Same as gure 2.3, but with a stronger velocity contrast.

  Traces extracted at the positions indicated by the dashed lines in (a). The bottom row is a zoom of the top row allowing to compare multiples modelled with both approaches.

Figure 2 . 5 .

 25 Figure 2.5. -Comparison of nite-di erences modelling and free-surface modelling with a second-order Born approximation in the di ractive case.

Figure 2 . 6 .

 26 Figure 2.6. -Same as gure 2.5, but with a stronger velocity contrast.

  C ] -P[ξ A ] -2P[ξ B ]

Figure 2 . 7 .

 27 Figure 2.7. -Illustration of the linearity of primaries and non-linearity of multiples to the model perturbation. Two re ectivity models ξ A and ξ B and their combination ξ C are considered (top row). Data are modelled in the case of primary re ections only (middle row) and in the case of primaries and rst-order surface multiples (bottom row). The most right column shows the di erence between data modelled in ξ C and the combination of the data sets computed with ξ A and ξ B .

Figure 2

 2 Figure 2.8. -Modelling of primary re ections with the horizontal subsurface-o set h de ned as a spatial delay at depth.

2. 6

 6 .1. Primaries only in a homogeneous medium In the rst example, we consider a 450 m depth and 1620 m large model discretised along a 6 m × 6 m grid. The re ector is 300 m deep and the exact background velocity model is observed data P obs = P[ξ e ] data residuals P[ξ t ] -P obs

Figure 2 .

 2 Figure 2.10. -Model constructed to interpret the three parts of the gradients of J 0 as de ned in equation (2.42). (a) Exact re ectivity model ξ e ; (c) Corresponding observed data P obs ; (b) Trial re ectivity model ξ t ; (d) Residuals (P[ξ t ] -P obs ). P e and M e are the primary and multiple corresponding to the true re ector, P t and M t the primary and multiple corresponding to the trial re ectivity.

  g 3 = Q[P 2 , λ 3 ]

Figure 2 .

 2 Figure 2.11. -Section at h = 0 m of the three contributions of the gradient (equation 2.42) computed at the point ξ t as de ned in gure 2.10. (a) g 1 = Q[P 0 , λ 3 ] is the interpretation of the events in the residuals as primary re ections; (b) g 2 = Q[P 0 , λ 4 ] and (c) g 3 = Q[P 2 , λ 3 ] are the interpretation of the same events as multiple re ections. The labels indicate the corresponding events in the data residual ( gure 2.10d).

Figure 2 .Figure 2 .

 22 Figure 2.12. -Observed data obtained for a source at the middle point of the model (left).Corresponding calculated data (middle) and residuals (right) obtained after 1, 2, 5 and 10 iterations (from top to bottom), corresponding to the re ectivity models presented in gure 2.13.

2 Figure 2 .Figure 2 .

 222 Figure 2.14. -Value of the objective function J 0 and norm of its gradient corresponding to the iterative migration results presented in gure 2.13.

Figure 2 .

 2 Figure 2.16. -Values of the objective function J 1 for di erent homogeneous velocity models. The correct velocity is 3000 m/s.

Figure 2 .

 2 Figure 2.17. -Observed data obtained for a source at the middle point of the model (left). Corresponding calculated data (middle) and residuals (right) obtained after 1, 5 and 20 iterations (from top to bottom), corresponding to the re ectivity models presented in gure 2.18.

Figure 2 . 2 Figure 2 .Figure 2 .

 2222 Figure 2.18. -Re ectivity model obtained after 1, 5 and 20 iterations (from top to bottom) in a too slow background velocity model c initial = 2500 m/s.

  Figure 3.3. -Results of inversion (left, 2nd and 3rd row) obtained with observed data modelledin the exact model shown on the rst row (left). These results are presented in more details in chapter 6. For each background velocity model, three CIGs are displayed at the lateral positions 500 m, 1500 m and 2000 m as well as three traces of the velocity models at the same locations. The red curve corresponds to the exact model. The blue and green curves correspond to the nal models displayed in the 2nd and 3rd row, respectively. These results are obtained with two di erent initial models (dashed curves), a homogeneous macro-model and a model with values increasing with depth.

2 Figure 3

 23 Figure 3.4. -(a) Value and (b) norm of the gradient of the objective function J 0 for ten iterations of conjugate gradients with su cient regularisation. (c) Norm of penalised CIGs. All plots are normalised by their maximum value. Note that c 0 (x) is xed.

Figure 3 . 5 .

 35 Figure 3.5. -Gradients of J 1 obtained with method A (left) and method B (right) after (from top to bottom) 1, 2, 5 and 10 iterations of iterative migration with su cient regularisation. Blue, grey and red represent negative, null and positive values respectively. Each velocity update is plotted with its own colour scale.

Figure 3

 3 Figure 3.6.-Results of migration after 1, 2, 5 and 10 iterations (from top to bottom) when using a too slow velocity model (2500 m/s). We show a section at h = 0 (left), a CIG at x = 1350 m (middle), and the same CIG multiplied by |h| (right). Blue, grey and red represent negative, null and positive values respectively. Each image is represented with its own colour scale. The two main events are indicated by dashed lines: the downward curved event corresponds to the re ector, while the upward curved one is caused by the limited acquisition aperture. This gure is very similar to gure 2.13, except that a di erent acquisition has been used here.

Figure 3

 3 Figure 3.8. -Same as gure 3.4 with a smaller regularisation weight a φ .

Figure 3

 3 Figure 3.9. -Same as gure 3.6 with a smaller regularisation weight a φ .at large values of h and ensures the convergence of the value of the MVA objective function with iterations, as illustrated by the former example. Finally, we would like to put these results in perspective with the theoretical study of the gradient stability led in section 3.2.4. We have shown that the stability of the gradient obtained with both methods in the linear case was ensured by the validity of the Lipschitz condition (3.20a). The results shown in gures 3.5 and 3.7 suggest that equation (3.20a) is not valid unless su cient regularisation is introduced during iterative migration. This observation is complementary to the observations of Huang

  Figure 3.10. -(a) Gradient of J 1 computed using method A after 8 iterations. (b) Section of this image at z = 96 m (red, solid) compared to the gradient obtained with nite di erences (blue, dashed). Both are normalised with the same constant.

Figure 3

 3 Figure 3.11. -Illustration of the e ect of migration/demigration on γ (8) . (a) Value of variable γ at iteration 8 (section at h = 0). (b) result of the application of the Hessian on this image (section at h = 0).

Figure 3

 3 Figure 3.12. -(a) Contribution of the line search

Figure 3

 3 Figure 3.13. -Results of migration after 1 (top) and 15 (bottom) iterations of conjugate gradient when we consider primaries and rst-order surface-related multiples. Section at h = 0 (left), CIG at x = 1350 m (middle), and the same CIG penalised by |h| (right). Two event are visible: the true re ector (TR) and the multiple imprint caused by cross-talk (MI)

Figure 3

 3 Figure 3.14. -Velocity updates obtained with method B after 1 (top) and 15 (bottom) iterative migration of data containing both primaries and rst-order surface-related multiples. They correspond to the migrated images of gure 3.13. Blue, grey and red represent negative, null and positive values respectively. Each velocity update is plotted with its own colour scale.

Figure 3

 3 Figure 3.15. -Re ectivity sections at h = 0 m (left) and velocity updates (right) obtained for three increasing (from top to bottom) values of the regularisation parameter a φ =1×10 -3 , 2×10 -3 and 3×10 -3 . In this example, positive and negative values correspond to black and white colours, respectively.

  ) l'utilisation d'une version déconvoluée de l'ondelette de source pour la construction du champ source ; (3) l'application d'une intégration temporelle d'ordre un à la place d'une dérivée seconde avant corrélation des champs sources et récepteurs, et l'application d'une dérivée verticale dans le domaine image au résultat de la corrélation de ces deux champs.

Figure 4 .

 4 Figure 4.1. -Schematic view of the wave elds (left: from the source, right: from the receiver) contributing to the gradient in the presence of a single re ector at depth z 0 . Both S and µ s and R and µ r correlate between the surface and depth z 0 .

Figure 4 . 2 .

 42 Figure 4.2. -Left: re ectivity sections ξ mig (blue) and ξ inv (red) for h = 0. Right: observed (blue) and computed data after inversion and modelling (red).We use a classical nite-di erence modelling tool to solve the constant density acoustic wave equation. Perfectly Matched Layers (PMLs) are implemented to deal with model boundaries, with no free-surface condition. The re ectivity model consists of a single interface at 500 m depth. After migration and inversion using the exact model v 0 = v e = 3000 m/s, the re ector is localised around the correct depth ( gure 4.2, left). The deconvolution aspect in the inversion

Figure 4 .

 4 Figure 4.3. -Gradients for the migration approach. The blue, red and black curves (almost superimposed) are respectively computed with the adjoint, nite-di erence and analytic approaches. In all cases, α = 0 Left: v 0 = 2500 m/s and v e = 3000 m/s. Right: same for v 0 = 3000 m/s and v e = 2500 m/s.

Figure 4 . 4 .

 44 Figure 4.4. -Same as for Figure 4.3, but here for the inversion approach (top: β = 0 and middle: β = -3/2). Bottom: decomposition of the gradient G inv (dashed line) for β = -3/2 into the three contributions G 1 inv (blue), G 2 inv (red) and G 3 inv (black).

  In the rst model, we consider a single horizontal interface at depth z = 240 m in the exact model corresponding to a homogeneous model v e = 3000 m/s. Sources and receivers are located at the surface, with a maximum o set of ±600 m. The maximum frequency content of the data is 40 Hz.

Figure 4 . 5 .

 45 Figure 4.5. -Re ectivity sections ξ mig (a) and ξ inv (b) for h = 0 and for v e = 3000 m/s and v 0 = 2500 m/s, in the case of a single horizontal interface, and associated CIGs at position x = 1800 m for the migration (c) and inverse (d) cases.

Figure 4 . 6 .

 46 Figure 4.6. -Gradients for the migration (a) and inversion (b and c) approaches, for β = 0 (b) and β = -3/2 (c).

Figure 4 . 7 .

 47 Figure 4.7. -Same as for gure 4.5, but here with v e = 2500 m/s and v 0 = 3000 m/s.

Figure 4 .

 4 Figure 4.8. -Same as for Figure 4.6, but here with v e = 2500 m/s and v 0 = 3000 m/s.

Figure 4 .

 4 Figure 4.9. -Single di raction case, with the migration (left) and inversion (right) approaches. Top: re ectivity section, middle: velocity gradient, bottom: smoothed version of the velocity gradient.

Figure 4 .

 4 Figure 4.10. -Multi-layer case, with the migration (left) and inversion (right) approaches. Top: re ectivity section, middle: velocity gradient, bottom: smooth version of the velocity gradient.

Figure 4 .

 4 Figure 4.11. -Exact velocity model (a), migrated section in v 0 = 2500 m/s and for h = 0 m (b) and associated gradient (c).

Figure 4 .

 4 Figure 4.12. -Inversion results for h = 0 m (a) and associated gradient for β = -3/2, before (b) and after smoothing with (c) σ = λ data /2 and (d) σ = λ data .

Figure 4 .

 4 Figure 4.13. -Observed (a) and computed shot gather at s = 1750 m, after inversion and modelling in the incorrect model v 0 = 2500 m/s (b) and di erences at the same scale (c).

Figure 4 .

 4 Figure 4.14. -Traces extracted from gure 4.13, for surface o set -300 (a), 0 (b) and +300 m (c). The blue colour corresponds to the observed shot and the red to the shot after inversion and modelling in the incorrect model v 0 = 2500 m/s.

Figure 4 .

 4 Figure 4.15. -Exact velocity model (a), migrated section in v 0 = 3000 m/s and for h = 0 m (b)and associated gradient (c). The colorbar ranges from 2500 (blue) to 3000 m/s (red).

Figure 4 .

 4 Figure 4.16. -Inversion section in v 0 = 3000 m/s and for h = 0 m (a), rst gradient (b), nal inverted model after 10 iterations (c), and associated inverted re ectivity (d). The colorbar ranges from 2500 (blue) to 3000 m/s (red).

Figure 4 .

 4 Figure 4.17. -CIGs for x = 1750 m computed in (a) the initial homogeneous model and (b) the nal model ( gure 4.16c).

  θ r . The application of z and hz on the Green's function provides m θ r(equations 4.21 and 4.22): in other words, the model value m 1/2 0-in front of cos θ s is estimated at xh and not at x + h as it should be in the determinant expression. The same holds for the receiver side. The new term p is de ned as a linear combination of z and hz (equation 4.7) to fully remove m θ r in equation(4.23). Finally, the expressions for the amplitudes are given in[START_REF] Zhang | Theory of True-Amplitude One-Way Wave Equations and True-Amplitude Common-Shot Migration[END_REF], ten Kroode (

  )P obs (FF † ) T P obs

Figure 4 .

 4 Figure 4.18. -Result of the application of (FF † ) ( rst column) and (FF † ) T (third column) to observed data P obs . The corresponding residuals are plotted in the second and fourth column, respectively. All plots share the same colour scale. Observed data are computed in a homogeneous velocity model (c ex 0 = 3000 m/s) with a single at re ector. The velocity model used to compute F and F † is too low (top, c 0 = 2500 m/s), correct (middle, c 0 = 3000 m/s), and too high (bottom, c 0 = 3500 m/s). The last two columns show traces extracted at positions indicated by dashed lines on the fourth column.

Figure 4 .

 4 Figure 4.19. -Common Images Gathers obtained in a too low (2500 m/s, top), correct (3000 m/s, middle), and too high (3500 m/s center) background velocity model. CIGs are computed by of application of the pseudo-inverse F † to observed data (1st column), to observed data premultiplied by (FF † ) (2nd column) and to observed data premultiplied by

Figure 4 .

 4 Figure 4.20. -Value of the normalised objective function of MVA obtained in homogeneous velocity models for the approximate inverse and after one iteration with and without preconditioning. The correct velocity is 3000 m/s. The oscillation in the dashed blue curve is due to low frequency energy appearing above the re ector in CIGs especially for too high velocities (third column in gure 4.19).

Figure 4 .

 4 Figure 4.21. -CIGs obtained after one (top) and ten (bottom) iterations of preconditioned iterative migration in a too low (2500 m/s, left), correct (3000 m/s, centre) and too high (3500 m/s, right) velocity model.

Figure 4 .

 4 Figure 4.22. -Results of iterative migration in a too slow velocity model (2500 m/s) withprimaries only. We plot the relative data mis t (left) and the relative normal residual associated with the linear system (4.43a). Red and blue curves correspond to the case of a zero initial guess (ξ (1) = 0) with classical and preconditioned iterative migration, respectively. The green curve corresponds to non-preconditioned migration initiated with the approximate inverse ξ (1) = F † P obs . The dashed line corresponds to the value obtained for the approximate inverse F † P obs .

Figure 4 .

 4 Figure 4.23. -Value of the MVA objective function obtained in homogeneous background velocity models after several iterations with preconditioning. The correct velocity is 3000 m/s.

2 P

 2 Exact background velocity model (red, solid) and some of the initial models considered (black, dashed). The re ectivity model consists of a single re ector localised at z = 475 m. obs = primaries and multiples P obs = primaries only P obs = multiples only (b) Normalised MVA objective function. Each function has been mapped linearly to the interval [0, 1]. The correct model corresponds to a velocity of 3000 m/s at 1000 m depth. Central CIGs ξ inv = F † P obs obtained for di erent background velocity models (top) and application of the annihilator to these CIGs Aξ inv / ξ inv (bottom). The plots of the bottom row have the same colour scale.

Figure 4 .

 4 Figure 4.24. -Result of application of the approximate inverse F † to observed data containing both primaries and rst-order surface-related multiples with several initial background velocity model increasing with depth.

Figure 4 .

 4 Figure 4.25. -CIGs obtained after ten iterations of migration without (left) and with (centre) preconditioning starting with ξ (1) = 0, and ten iterations of migration without preconditioning starting with ξ (1) = F † P obs (right).

  ξ (1) = 0 with preconditioning, ξ (1) = 0 without preconditioning, ξ (1) = F † P obs

Figure 4 . 5 Figure 4 .

 454 Figure 4.26. -Decrease of the migration objective function using three optimisation strategies, corresponding to the CIGs gure 4.24.

Figure 4 .

 4 Figure 4.28. -Convergence of the adjoint problem in the case of small regularisation.

Figure 4 .

 4 Figure 4.29. -Residuals related to the linear adjoint problem (4.46) after ten iterations. We show the central CIG of the adjoint variable η (1st column) obtained after ten iterations without preconditioning (top) and with preconditioning (bottom). Then we compare Hη (2nd column) with the right-hand side of the adjoint problem (3rd column) and compare this two vectors. On each line, the three most right plots share the same colour scale.

Figure 4 .

 4 Figure 4.30.-Gradients obtained, in the case of small regularisation, after seven iteration of preconditioned iterative migration for the successive values of the adjoint variable (from top to bottom). The left and right columns correspond to the resolution of the adjoint problem without and with preconditioner, respectively. The gradient is not really stable from one iteration to another.

Figure 4 .

 4 Figure 4.31. -Central trace (x = 810 m) of the gradients shown in gure 4.30. The value of the gradient above the re ector oscillates from one iteration to another.

Figure 4 .

 4 Figure 4.32. -Gradients obtained by direct inversion (top), and after iterative inversion (middle without preconditioning and bottom with preconditioning) with small regularisation (N = 7 and M = 10, corresponding to the bottom row of gure 4.31). The right column shows a trace extracted at the middle position (corresponding to the dashed orange lines in gure 4.31 for the iterative cases).

Figure 4 .

 4 Figure 4.33. -Convergence of the adjoint problem in the case of stronger regularisation.

Figure 4 .

 4 Figure 4.34. -Same as gure 4.29 in the case of stronger regularisation.

Figure 4 .

 4 Figure 4.35. -Same as gure 4.28, but with a stronger regularisation.

Figure 4 .

 4 Figure 4.36. -Central trace (x = 810 m) of the gradients shown in gure 4.35.

Figure 4 .

 4 Figure 4.37. -Same as gure 4.37 with stronger regularisation.

Figure 5 Figure 5 . 2 .

 552 Figure 5.1. -Source wavelet Ω and its approximate inverse Ω.

  Adjoint of inverse operator (F † ) T : → obs . Approximate inverse operator F † : obs → .

Figure 5 .

 5 Figure 5.3. -Matrix representation of the four linear operators de ned in equations (5.9) in the initial velocity model c 0 = 2500 m/s. The matrices have zero values for z ≤ 100 m because of the taper introduced in the -space ( gure 5.2).

Figure 5 . 4 ,

 54 left) would yield a smooth re ectivity model. The non triviality of the null space of F is important for the behaviour of the iterative case studied in sections5.2.3 and 5.2.4.

Figure 5 . 4 .

 54 Figure 5.4. -Exact re ectivity model and result of the application of the adjoint F T and F †to observed data (left). Observed data and data modelled from the adjoint and inverse re ectivity section (right).

  Operator Q inv : → .

Figure 5 . 5 .

 55 Figure 5.5. -Matrix representation of the operators Q mig and Q inv de ned in equations (5.14a) and (5.14b).

  Relative data mis t Fξ (n) -P obs 2 obs / P obs 2 obs (left) and relative normal residual Hξ (n) -b / b (right) across iterations. Re ectivity model obtained after one and six iterations, without preconditioning (left), and with preconditioning (right).

Figure 5 . 7 .

 57 Figure 5.7. -Results of six iterations of iterative migration with and without preconditioning.

  101) without preconditionnig η (101,101) with preconditionnig Hη (101,101) without preconditionnig Hη (101,101) with preconditionnig C T A T ACξ (101) (b) Adjoint variables obtained after 100 iterations (left). Application of the Hessian matrix H to this result and comparison with the right-hand side term of the adjoint problem (right).

Figure 5 .

 5 Figure 5.8. -Results of 100 iterations for the adjoint problem. The direct problem has been solved with 100 iterations too. A small 2 -regularisation is applied.

  Normal operator F T F : → when no preconditioner is applied. Normal operator (FF † ) T FF † : obs → obs when F † is used as a preconditioner. Normalised right-hand side term of the direct (red) and adjoint (blue) problems. Eigenvalue distribution (black) and projection (absolute value) of the right-hand side term of the direct (red) and adjoint (blue) problems.

Figure 5 .

 5 Figure 5.9. -Analysis of the di erence of convergence speed between the direct and adjoint problems.

Figure 5 .

 5 Figure 5.10. -Final solution (top row) of the direct (left) and adjoint (right) problem obtained after 100 iterations. Application of the normal operator to these results and comparison with the right-hand side term (middle row). Comparison of observed data and nal calculated data and associated residuals (bottom row).

Figure 5 .

 5 Figure 5.11. -Gradient obtained with ξ (101) and η(101,101) (top left) and its decomposition (top right). The three remaining panels illustrate that all contributions but one cancel with one another (equations 5.25).

  Fξ(11) -P obs C T A T ACξ(11)

Figure 5 .

 5 Figure 5.13. -Normal residual across iterations for the adjoint problem (left). Eigenvalue distribution of (F † ) T (H + H φ )F † and projection on its eigenvectors of the right-hand side term of the adjoint problem in the preconditioned case.

Figure 5 .

 5 Figure 5.14. -Same as gure 5.11 with the Huber norm as regularisation, and after 10 iterations for both the direct and adjoint problems.

Figure 5 .Figure 5 .

 55  of the application of a lter K to the nal re ectivity ξ(11) (red) in the depth (top left) and wavenumber (top right) domain. We display the result of the application of the adjoint (blue) and inverse (green) operator for reference. The bottom row displays the same comparison after application of the annihilator A.

Figure 5 .

 5 Figure 5.18. -Same as gure 5.14 without regularisation and with the introduction of a lter K in the MVA objective function.

Figure 5 .Figure 5 .

 55 Figure 5.19. -Same as gure 5.15 for the lter K = F † F. The curves corresponding to ξ inv and Kξ (11) are almost superimposed.

Figure 5 .

 5 Figure 5.23. -Same as gure 5.22 except that 100 iterations (instead of 10) are performed to solve both the direct and adjoint problems.

Figure 5 .

 5 Figure 5.24. -Mean value of the gradient between z =0 m to 250 m (above the re ectorwhere it is supposed to be constant with depth) obtained without and with the lter K = F † F after performing N =1 to 100 iterations of iterative migration. The adjoint problem is solved with the same number of iterations as the direct problem (M = N). Small 2 regularisation is applied. For each case we plot the result obtained without (left) and with (right) preconditioner introduced in the resolution of both direct and adjoint problems. Each plot is normalised individually by the mean value obtained between iterations 50 to 100.The expected mean value is negative as this is the case of the blue curves. Locally the sign of the gradient is incorrect (positive values, red curves).

Figure 5 .

 5 Figure 5.25. -Central CIG (top) obtained after sever iteration of preconditioned iterative migration (left). After application of F † F, we obtain a CIG (middle) very close to the result of direct inversion (right). The bottom row display the same CIGs multiplied by the absolute value of the subsurface o set. The same colour scale is used in each row.

Figure 5 .

 5 Figure 5.26.-Central CIG of the last iterate η(8,11) obtained in the resolution of the adjoint problem (1st column) and application of the Hessian H to η (8,11) (2nd column). We also display the right-hand side term b (8) of the linear system (3rd column) and the corresponding residual (4th column). We consider the case where no lter is introduced in the MVA objective function (top, similar to gure 4.29), and the case with a lter K = F † F based on the approximate inverse (bottom). The three most right plots of each line share the same colour scale.

Figure 5 .

 5 Figure 5.27. -Convergence rate of the adjoint problem without (top) and with (bottom) application of the lter F † F. The left column shows the decrease of the norm associated with the linear conjugate gradient algorithm, the right column shows the norm of residuals of the linear system.

Figure 5 .

 5 Figure 5.28. -Gradient G (7,M) obtained with ξ (8) and with the successive adjoint variables η (8,M+1) without (left) and with (right) application of the lter K = F † F. The left column is close to the right column of gure 4.30 where a very similar model was considered.

Figure 5 .Figure 5 .Figure 5 .

 555 Figure 5.29. -Section of the gradients at the middle x-position obtained with the nal reectivity ξ (8) and the successive values of the adjoint variables η (8,M+1) , without (left) and with (right) the lter K = F † F.

Figure 5 .Figure 5 .Figure 5 .

 555 Figure 5.32. -Section of the re ectivity ξ (8) at h = 0 m (top) and gradient G (7,10) (bottom) obtained with x max = 1620 m (left, same gradient as gure 5.30 bottom left), and with x max = 2430 m. Extending the model laterally for a similar value of the maximum surface o set results in a larger homogeneous central part, the edge e ects remaining similar.

Figure 5 .

 5 Figure 5.35. -Gradient and CIGs obtained in the initial model shown in gure 5.33, after direct inversion with observed data containing (a) primaries only or (b) both primaries and multiples.

Figure 5 .

 5 Figure 5.36. -Gradient and CIGs obtained in the initial model shown in gure 5.33, after iterative inversion with observed data containing primaries only. The direct and adjoint problems are solved in N = 7 and M = 10 iterations respectively, and with small 2 regularisation.

Figure 5 .

 5 Figure 5.37. -Gradient and CIGs obtained after iterative inversion with observed data containing both primaries and multiples. Di erent optimisation strategies are considered, regarding regularisation and the number N and M of iterations used to solve the direct and adjoint problems: (a) small 2 regularisation, N = 1, M = 9; (b) small 2 regularisation, N = 10, M = 10; (c) small 2 regularisation, N = 30, M = 20; (d) stronger Huber regularisation, N = 10, M = 10.

Figure 5 .

 5 Figure 5.38. -Decomposition of the gradients of gure 5.37 into three parts, similar to the one presented in gure 5.31 in the case of primaries only. The rst contribution (left) can be interpreted as the gradient that would be obtained by applying the direct inversion strategy to recalculated primaries remodelled with the last re ectivity image of the iterative migration processing. The second contribution (middle) is directly related to data residuals and should be zero if iterative migration has converged. The remaining contributions (right) are di cult to interpret and are not necessarily zero if convergence is reached for both the direct and adjoint problems.

Figure 6

 6 Figure 6.1. -Exact and initial background velocity model (left) and dense exact re ectivity model (right).

Figure 6 . 2 .

 62 Figure 6.2. -Observed data computed in the model of gure 6.1 under the rst-order Born approximation (left) and a secondorder Born approximation (right).

Figure 6 .Figure 6 . 4 .

 664 Figure 6.3. -Gradient and CIGs obtained with the model of gure 6.1 with di erent inversion strategies with and without multiples: (a) direct inversion on primaries only; (b) direct inversion on primaries and multiples; (c) iterative migration on primaries only (N = 7 and M = 10); (d) iterative migration on primaries and multiples (N = 10 and M = 8).

Figure 6 . 5 .

 65 Figure 6.5. -Source wavelet used to generate observed data in test A. It is obtained by setting to zero low frequencies in the original Ricker.

Figure 6 . 6 .Figure 6 . 7 .

 6667 Figure 6.6. -Same as gure 6.3 when observed data are generated with the source wavelet shown in gure 6.5 (test A).

Figure 6 .

 6 Figure 6.8. -Source wavelet used to generate observed data in test B, obtained as the time derivative of the original Ricker wavelet.

Figure 6 .

 6 Figure 6.9. -Same as gure 6.3 when observed data are generated with the source wavelet shown in gure 6.8 (test B).

Figure 6 .

 6 Figure 6.10. -Stretching factor c 0 / cos θ, with θ the half opening angle at the image point, obtained for di erent depths and the maximum surface o set value h surface = 700 m in the initial velocity model of gure 6.1.

Figure 6 .

 6 Figure 6.11. -Same as gure 6.4 for the gradients presented in gure 6.9 (test B).

Figure 6 .

 6 Figure 6.14. -Norm associated with the conjugate gradient algorithm (top) and relative normal residuals (bottom) obtained across iterations for the resolution of the adjoint problem in the reference case as well as in the tests A, B, C and D both in the case of primaries only (left),and in the case of primaries and multiples (right). The ve case shown here correspond to di erent linear systems and cannot be directly compared. Note that the relative normal residuals are actually the norm of the gradient of the CG-objective and may not decrease from one iteration to the following.

Figure 6 .

 6 Figure 6.15. -Observed data obtained with a perturbation of the velocity model (1st column) and with a perturbation of the density model (2nd column). Both plots use the same colour scale. Traces extracted at the position indicated by the dashed lines are shown on the two most right panels.

Figure 6 .

 6 Figure 6.16. -Same as gure 6.3 when observed data are generated with a density perturbation instead of a velocity perturbation (test C).

Figure 6 .Figure 6 .

 66 Figure 6.17. -Observed data and calculated data obtained after ten iterations of iterative migration for two values of the surface oset, h = 0 m (top) and h = 417.6 m (bottom) (test C).

  Figure 6.19. -Same as gure 6.4 for the gradients presented in gure 6.16 (test C).

Figure 6 .

 6 Figure 6.20. -Exact discontinuous velocity and density 1D pro les used to model observed data (red, rst and third columns). The blue curves correspond to smoothed velocity and density models, and associated velocity and density perturbations used to model data with second-order Born approximation. The dashed green line is the initial velocity model used to compute the gradient (test D).

Figure 6 .Figure 6 .

 66 Figure 6.21. -Observed data computed with a nite-di erence modelling code and a freesurface condition in the multiple case (left column, corresponding to the red 1D pro les of gure 6.20). We compare this data set with data modelled under the second-order Born approximation (2nd column, corresponding to the blue 1D pro les of gure 6.20) (test D).

Figure 6 .

 6 Figure 6.23. -Same as gure 6.3 when observed data are generated with a nite-di erence modelling code without Born approximation (test D).

Figure 6 .

 6 Figure 6.24. -Same as gure 6.4 for the gradients presented in gure 6.23 (test D).

Figure 6 .

 6 Figure 6.26. -Value of the data mis t across iterations (test D).

Figure 6 .

 6 Figure 6.27. -Exact velocity and density model (top), and their respective smoothed versions (bottom) used to generate observed data with a second-order Born approximation.

Table 6 . 2 .Figure 6 .

 626 Figure 6.28. -Result of iterations on the background velocity model with the direct inversion strategy on data containing primaries only. The colour bar ranges from 1950 m/s to 2750 m/s and is similar to the one used in the left column of gure 6.27.

Figure 6 .

 6 Figure 6.29. -Re ectivity image obtained by direct inversion in the background velocity models shown in gure 6.28. The bottom row displays the results obtained in the exact velocity model. The same colour scale is used for all sections at h = 0 (left column). All CIGs are also plotted with the same colour scale, which has been truncated for a better representation of deeper re ectors.

Figure 6 .

 6 Figure 6.30. -Value of the MVA objective function (6.10) across iterations (blue, solid). The dashed black line represents the value obtained in the exact background velocity model.Three di erent inversions strategies are shown: direct inversion on primaries only (left, corresponding to gure 6.28), direct inversion on primaries and multiples (centre, corresponding to gure 6.31), and iterative inversion on primaries and multiples with application of the lter F † F and the "truncated gradient" as background velocity update (right, nal result shown in gure 6.33b, bottom right).

Figure 6 .

 6 Figure 6.31. -Same as gure 6.28 with both primaries and rst-order surface multiple in observed data.

Figure 6 .

 6 Figure 6.32. -Same as gure 6.29 in the case of multiples for the background velocity models of gure 6.31.

  a) Migrated images in the initial velocity model, using direct inversion on primaries only (top), direct inversion on primaries and multiple (middle), and iterative migration and ltering on primaries and multiples (bottom).

  First gradient of the inversion (left), corresponding to the migrated images of (a), and background velocity model recovered after 15 iterations.

Figure 6 .

 6 Figure 6.33.-Results of 20 iterations on the background velocity model using the iterative migration strategy with observed data containing both primaries and rst-order surface multiples, and comparison with the results obtained with direct inversion. In the iterative case, the "truncated gradient" strategy is used to compute the background velocity update.

Figure 6 .Figure 6 .

 66 Figure 6.34. -Laterally invariant velocity model considered in section 6.4. It is constructed by extending the central part of the 2D velocity model shown in gure 6.27. The horizontal black arrows indicate sources lateral position. There are no sources between x = 1500 m and x = 2250 m/s.

Figure 6 .

 6 Figure 6.36. -Relative data mis t across iterations when performing iterative migration in a homogeneous velocity model (2000 m/s) with primary only (left, corresponding to gure 6.35, middle) and primaries and rst-order surface multiples (left, corresponding to gure 6.35, right).

  Figure B.1. -The strong Wolfe conditions for a relatively complex scalar function ψ (blue, solid): (a) su cient decrease (or Armijo) condition; (b) strong curvature condition; (c) combination of the su cient decrease condition and the strong curvature condition. The coloured areas show the range of acceptable step length values.

  W r P(s, r, ω) s, x = (x, z), ω = , r, ω) if z = z r -1,+ 1 2∆z P(s, r, ω) if z = z r + 1∆zthe size of a grid point in the z direction and z r the position of the receivers. It can be interpreted as a multiplication by the cosine of the take-o angle at the receiver position ( gure C.1), iω c 0 (r) cos β r ; (C.6) • Finally, S is a modi ed source term, which involves a deconvolved version Ω of the source wavelet Ω and an other weighting operator W s de ned in the same way as W r for the source depth z s S = W s δ(x -s) Ω . (C.7) W s can be interpreted as a multiplication by the cosine of the take-o angle at the source position ( gure C.1), iω c 0 (s) cos β s . (C.8)

  Figure C.1. -Angles involved in the interpretation of the weights de ning the inverse operator F † .

  x -2h, ω)ξ(x -h, h) dh, x + 2h, ω)ξ(x + h, h) dh.

  is obtained by application of the approximate inverse F † to observed data (equation 5.7c), ξ inv (z) = F † P obs (z) we have derived the following expression for the gradient G inv of J inv 1 with respect to c 0 ,G inv ( y) = βc 2β-1 0 ( y)a 2 ( y)ξ 2 inv ( y) + z Q inv ( y, z)η inv (z) dz (D.3)where η inv is obtained as the partial derivative of J 1 with respect to ξ inv ,

2 0-

 2 s, y, ω)G 0 ( y, z, ω), ∂ G 0 (s, z, ω) respect to c 0 ( y) (D.6a) is given by the rst-order Born approximation, reviewed in section 2.3.1: the perturbation δG 0 (s, z, ω) of a reference Green's function G 0 (s, z, ω) due to a perturbation δc of the reference background velocity model c 0 obeys (equation 2.6b)(iω) 2 c ∆ δG 0 (s, z, ω) = 2δc(z) c 3 0 (z) (iω) 2 G 0 (s, z, ω), (D.7)and δG 0 (s, z, ω) can be expressed as (equation 2.10b), δG 0 (s, z, ω) 6a) is obtained as the derivative of this expression with respect to δc. We now detail the important steps in the derivation of Q inv . Starting from the expression of F † P obs given in equation (D.2), we compute the following quantity, explicitly separated the term ∂ η inv /∂ z as is done in the calculation with the adjoint-state method (equation 4.27g). Then, using equation (D.6b), z, ω)G * 0 (s, y, ω)G * 0 ( y, z, ω)P obs (ω) dω , (D.9d)where we have used equation (D.6a). Eventually we need to compute the derivative with respect to z of the product of Green's functions inside the integral, whose value depends on the relative position of y and z, c 0 (z) ω (iω) 4 Ω * (ω)G * 0 (s, z)G * 0 (s, y)G * 0 ( y, z) dω if y < z,

  2, we have assumed an homogeneous background velocity model (c 0 (z) = c 0 ) and replaced the value of the Green's function G 0 in equations (D.2) and (D.11) by its asymptotic expression in a 1D homogeneous medium (c 0 (z) = c 0 ), G 0 (s, z, ω) = c 0 2iω e -iωτ(s,z) . (D.12)leading to the expressions given in equations (5.9c) and (5.15d)ξ inv (z) = 2 c 0 ω Ω * (ω)e 2iωτ(s,z) P(ω) dω, Ω * (ω)e (iω) τ(s,z)+τ(s, y)+τ( y,z) P obs (ω) dω if y < z,

  Figure D.1. -Comparison of the re ectivity (left) and background velocity update (right)obtained with the analytic expressions (equations D.13 and D.14) (green dashed line) with the results obtained by solving wave-equations with a nite di erences code (red) as described in section 4.2.11. A third value for the gradient (blue) is obtained by computing the value of ξ inv with wave-equations propagation for slightly perturbed velocity model. For this example, we took β = 3/2 and use the same re ectivity and background velocity model as in section 5.2, that is homogeneous exact and initial velocity models (3000 m/s and 2500 m/s) and an exact re ectivity model made of a single re ector at 400 m depth.

Figure D. 2 .

 2 Figure D.2. -Perturbation of the Green's function G 0 (s, z, ω) giving the response in z to a source in s, when the perturbation is located above z (left) and below z (right).

  5) where C, Q, K -, K + , W mod , W * mod , W r and S are de ned in equations (3.19), (C.2), (C.3), (C.5), (C.7), (C.14a), (C.14b) and (C.15), and f mod : ( × × ) → is de ned as f mod [c 0 , η, ξ 0 ](x) = kαc α

  compute ξ = F † P obs and evaluate J 1 [c 0 ] 2 P 0 , λ 1 given ξ, compute ∂ J 1 ∂ c 0 [c 0 ] 4 P 0 , λ 1 , ν 1 , µ 0 given ξ and ∂ J 1 ∂ c 0 [c 0 ], compute ∂ J 1 ∂ c 0 [c 0 ] V 12 2 P 0 , λ 1 , M 0 , N 1 , ν 1 , µ 0 , L 1 , Π 0

Table 2 .

 2 1. -Data and model spaces involved in this study and size of their elements. n z , n x

  Contents 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Computing the gradient of J 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Method A: Exact gradient of an approximate objective function . .

3.2.2. Method B: approximate gradient of the ideal objective function . . 3.2.3. Computation of the second-order derivatives of J 0 . . . . . . . . . . 3.2.4. Stability of the gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Comparison in terms of implementation . . . . . . . . . . . . . . . . 3.3. Synthetic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Examples with primaries only . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Importance of regularisation . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Example with multiples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  .34) Using equations (3.20c) and (3.34), we can bound the third contribution in equation (3.32) by the error ξ ∞ξ (N+1) . In the linear case, equation (3.20c) is automatically veri ed and the validity of equation (3.20b) is implied by equation (3.20a), as observed above. We conclude that for both methods A and B, the validitity of equation (3.20a) is essential to the stability of the MVA gradient across inner-iterations. The numerical examples of section 3.3.1 will help us to determine under which conditions this hypothesis is veri ed.

Table 4 .

 4 ) The objective function may also exhibit spurious oscillations 1. -Dimensions of the data and space domains, with (s, r) being the source and receiver coordinates, t the time, (x, y, z) the spatial coordinates, and h the subsurface o set.

	Dimension 1D 2D 3D	Data domain t (s, r, t) (s x , s y , r x , r y , t)	Space domain Extended domain z z (x, z) (x, z, h) (x, y, z) (x, y, z, h x , h y )

Table 6 .

 6 1. -Series of tests performed in chapter 6. The coloured cells correspond to the modi cation introduced in the modelling code for observed data compared to the modelling code used during inversion.

	test section	gures	source wavelet	modelling tool for observed data
	ref.	6.2.1	6.3 and 6.4	Ricker wavelet	2nd-order Born approximation
	A	6.2.2	6.6 and 6.7	Ricker wavelet without low frequencies	2nd-order Born approximation
	B	6.2.3	6.9 and 6.11	time derivative of Ricker wavelet	2nd-order Born approximation
	C	6.2.4	6.16 and 6.19	Ricker wavelet	2nd-order Born approximation density perturbation instead of velocity perturbation
	D	6.2.5	6.23 and 6.24	Ricker wavelet	nite di erence and free-surface condition with variable density
	chapter 5,

Table 7 .

 7 1. -Summary of the main conclusions of each chapter.

  TableE.1. -Number of wave-equations to be solved to compute the value of the MVA objective function in the direct inversion case, its gradient and the product of its Hessian with a vector of V ∈ . The number indicated here should be multiplied by the number of source positions considered in the acquisition. Résumé Les expériences de sismique active sont couramment utilisées pour estimer la valeur d'un modèle de vitesse de propagation des ondes P dans le sous-sol. Les méthodes dites d'« analyse de vitesse par migration » ont pour but la détermination d'un macromodèle de vitesse, lisse, et responsable de la cinématique de propagation des ondes. Dans une première étape de « migration », une image de réflectivité est obtenue à partir des données enregistrées en utilisant une première estimation du macro-modèle. Cette image dépend d'un paramètre additionnel permettant dans un second temps d'estimer la qualité du macro-modèle puis de l'améliorer. Les images de réflectivité obtenues par les techniques de migration classiques sont cependant contaminées par des artefacts, altérant la qualité de la remise à jour du macro-modèle. En particulier, elles ne prennent pas en compte les réflexions multiples, habituellement retirées des données avant traitement. Cette étape reste cependant délicate et on se prive alors de l'information supplémentaire contenue dans les multiples. Nous proposons dans cette étude une stratégie d'optimisation imbriquée en itérant l'étape de migration avant de remettre à jour le macro-modèle. La migration itérative produit des images de réflectivité satisfaisantes pour l'analyse de vitesse et s'étend naturellement aux réflexions multiples. Un désavantage de la méthode est son coût de calcul. Un pseudo-inverse de l'opérateur de modélisation est alors utilisé comme préconditionneur pour limiter le nombre d'itérations dans la boucle interne. Une autre difficulté est l'instabilité de la remise à jour du modèle de vitesse calculée pour des modèles de réflectivité successifs proches les uns des autres. Une nouvelle approche plus robuste est proposée, valide aussi dans le cas de multiples. Son efficacité est testée sur des jeux de données synthétiques 2D.

another de nition for the re ectivity (e.g. ξ(z) =

2δc(z)/c 3 0 (z), as in equation 2.8) would modify the gradient expressions, but not the conclusions.

The optimal value is β = -1/2 if ξ is de ned as a velocity perturbation ξ = δc/(2c 0 ) and β =

3/2 if ξ is de ned as a squared slowness perturbation ξ = 2δc/c 3 0 .

Remerciements

of the following ten contributions,

(5.40b) (5.40c) (5.40d)

(5.40e)

(5.40f)

(5.40g)

(5.40h)

(5.40i) (5.40j) with P (N+1) 7

= Fξ (N+1) and P (N+1) 8

= (F † ) T C T ACKξ (N+1) . Assuming that both problems are perfectly solved, that the regularisation weight is small and that K is close to the identity, eight contributions cancel out,

(5.41b)

(5.41c) (5.41d) and the two remaining contributions G 7 and G β are close to the two contributions of the direct inversion gradient (equation 5.13b). Still using the same numerical example, we solve the direct and adjoint problems in 10 iterations with very small regularisation. Applying the lter yields a re ectivity section close to the direct inverse ξ inv ( gure 5.19). The convergence of the adjoint problem is not as good as with the previous "hand-made" lter, but is still better than in the reference case ( gure 5.20), and 10 iterations already provide a good solution ( gure 5.21). Eventually we obtain a gradient which is very close to the one obtained after direct inversion ( gure 5.22). Some unwanted oscillations are still visible. They are progressively attenuated if more iterations are performed both on the direct and adjoint problems ( gure 5.23). Contrary to the preceding examples, applying a gaussian blur to the gradient obtained after 10 iterations would yield a velocity Figure 6.12. -Observed data (containing both primaries and multiples) and calculated data obtained after ten iterations of iterative migration in the reference case (top, gure 6.3d), and in test B (bottom, gure 6.9d). (bottom). We also compute the data mis t of primaries alone and multiples alone (blue and green, respectively). Note that the red and blue curves in the reference case are almost superimposed.

Appendix A.

Modelling of free-surface reflection under the Born approximation

In this appendix related to section 2.3.2, we justify on a 1D case the expression for the operator M s de ned in equation ( 2.12) accounting for the re ection at the free-surface. Intuitively, we would like to use the Born formula and de ne a source term made of primaries recorded at the surface multiplied by a constant re ection coe cient R = (-1), as the free surface is known to act as a mirror [START_REF] Schuster | Basics of Seismic Wave Theory[END_REF]. Numerical comparison with a nite-di erence modelling with a free-surface condition suggests that an additional coe cient and a time derivative 2iω/c 0 have to be applied as well. The purpose of this appendix is to explain the reason for this additional term. Actually, the origin of the re ection at the free surface can be described by a constant velocity perturbation above the surface with value (-1). However, in practice, the model considered for simulation does not extend above the free surface and we consider a re ectivity localised at the surface. This requires a modi cation of the Born formula.

Let us consider a pure 1D case with a single source and a single receiver both buried and localised at depth z s . The surface is at a depth z 0 < z s . In the asymptotic approximation, the 1D Green's function reads

where τ(s, z) is the traveltime between the source and the depth z the surface. First we consider a constant re ectivity model above the surface ξ(z) = R • H(z 0 -z) where H stands for the Heaviside function. Using equation (2.10), the upgoing incident wave reads

and the downgoing re ected wave recorded at the source position reads

We linearise the value of the traveltime around the surface,

where τ 0 = τ(z s , z 0 ). This leads to

The downgoing wave eld has the same shape as the upgoing wave eld, except for the multiplication by the re ection coe cient R = (-1). The surface acts as a mirror on a upgoing wave elds.

In practice we consider a re ectivity localised at z 0 : ξ(z) = R • δ(z -z 0 ). Following the same steps, this leads to

Compared to, equation (A.5), a coe cient c 0 2iω is missing in the result. To model the surface re ection properly, the source term of the Born modelling equation (2.10b) has to be multiplied by

Non-linear local optimisation

In this appendix, we review standard non-linear optimisation methods. For an extensive review, the reader is referred to [START_REF] Nocedal | Numerical Optimization[END_REF]. Non-linear optimisation is needed in chapter 2 to determine a re ectivity model minimising the data mis t when multiples are included in the calculation of reconstructed data. The objective function of DSO is also minimised with non-linear optimisation to determine a background velocity model minimising defocused energy in CIGs.

B.1. General form of the algorithm

Without loss of generality, we use in this appendix the notations of iterative migration, that is we want to nd a vector ξ ∈ minimising a scalar objective function J 0 (ξ). With gradient-based methods, the re ectivity model at iteration (n + 1) is updated from the re ectivity model at iteration (n) following

where d ∈ is called descent direction and the positive scalar α is called step size. The process is initialised with an initial guess ξ (1) = ξ ini , and is stopped when a termination criterion is satis ed, for example when the value of the objective function or the norm of its gradient goes below a given threshold. In the numeric examples shown in this thesis, a maximum number of iterations N is set, so that the nal result is ξ (N+1) .

In local optimisation methods, the value of the objective function should decrease at each iteration, that is J 0 (ξ (n+1) ) < J 0 (ξ (n) ). Then d and α have to verify some conditions:

• to ensure that there exists a step length α allowing the decrease of J 0 , d should be a descent direction, meaning that it should verify

where g is the gradient of J 0 ,

3)

The most obvious choice for d is the opposite of the gradient. Alternative strategies providing faster convergence are presented in section B.3.

B.3. Descent direction

We review classic strategies used to determine a descent direction.

Steepest Descent

The steepest descent is the simplest strategy. The descent direction is simply the opposite of the gradient:

It obviously satis es the descent condition (B.2). It is a natural choice because it is the direction providing the maximum decrease of J 0 in the neighbourhood of ξ (n) . However other strategies provide faster decrease of the objective function.

Non-linear conjugate-gradient

The conjugate-gradient algorithm originally designed for linear problems can be extended to non-linear optimisation. The descent direction is based on the steepest descent but an additional term takes into account the value of the gradient obtained in the previous iterations,

Many formulas have been proposed for the de nition of β [START_REF] Hager | A Survey of Nonlinear Conjugate Gradient Methods[END_REF]. The most popular are

g (n-1) g (n-1) (Fletcher and Reeves, 1964) (B.9a) and

g (n-1) g (n-1) [START_REF] Polak | Note Sur La Convergence de Méthodes de Directions Conjuguées[END_REF]. (B.9b)

The Fletcher-Reeves formula is actually the one implemented in the linear version of the algorithm. In the linear case, successive gradients are orthogonal and the Polak-Ribière formula is equivalent to the Fletcher-Reeves one.

The descent condition (B.2) reads

Then an exact linesearch automatically ensures that d (n) is a descent direction (equation B.4). More generally a step length ensuring the strong Wolfe conditions is su cient (Nocedal and Wright, 2006, p. 122). The Polak-Ribière formula is the one providing the faster convergence in most cases (Nocedal and Wright, 2006, p. 131). It is the one used in this study.

Newton method

Newton strategies are based on a second-order Taylor expansion of the objective function

where we have noted H the Hessian matrix of J 0 ,

Deriving equation (B.11) with respect to d (n) leads to the Newton equation

The Newton direction d (n) is the solution of this equation, which reads, provided that the Hessian matrix is invertible

The descent condition equation (B.2) reads

therefore if H is positive de nite, the Newton direction is a descent direction. If it is not positive de nite, it may not be invertible and may not de ne a descent direction.

asi-Newton methods The use of a Newton descent requires the computation of the Hessian matrix and its inversion, which are both computer expensive procedures. Quasi-Newton methods are designed to alleviate this di culty. They use an approximation of the inverse of the Hessian B

, and the quasi-Newton direction reads

The steepest descent can actually be interpreted as a quasi-Newton method with the Hessian approximated with the identity operator. A popular choice for the de nition of B uses the BFGS formula (named after Broyden, Fletcher, Goldfarb and Shanno) (Nocedal and Wright, 2006, p. 136). Starting with the identity, the approximation of H -1 is improved iteratively at each iteration

A di culty is that the huge matrix B (n) has to be kept in memory. This di culty can be overcome with the limited-memory BFGS (l-BFGS) technique [START_REF] Nocedal | Updating Quasi-Newton Matrices with Limited Storage[END_REF], which computes the value of B (n) from the values of ξ (n) and g (n) at the m preceding iterations (where m typically ranges between 3 and 20). With this method, there is no need to compute nor to store H (n) .

An alternative strategy called the truncated Newton method (Nocedal and Wright, 2006, p. 168) aims at resolving the linear system (B.13) iteratively with the classical linear conjugate gradient algorithm. The method needs only an e cient way of computing the application of the Hessian to a vector of (matrix-vector product) and there is no storage requirements. See [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF][START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method: Quantitative Imaging of Complex Subsurface Structures[END_REF] for applications of the truncated Newton method in the framework of Full Waveform Inversion.

Appendix C.

Computation of the adjoint of the approximate inverse of the extended Born modelling operator

In section 4.2, we have proposed a formula for an approximate inverse of the extended Born modelling operator F : → obs . In section 4.3, this approximate inverse, noted F † : obs → , is used as a preconditioner to improve the convergence rate of the iterative migration process. The derivation involves the adjoint of the approximate inverse, noted here (F † ) T : → obs . In this appendix, we derive an expression for (F † ) T , using the adjoint-state technique. We begin by reviewing the implementation of the approximate inverse F † .

C.1. Equations for the forward map

The application of the approximate inverse F † to a vector P of obs produces a vector ξ inv of . The implementation is very similar to the case of the adjoint F T described in section 2.5.1. It consists of computing modi ed source and receiver wave elds, noted here P 0 and λ 1 respectively, by analogy with the notations of section 2.5.1. The application of the approximate inverse to P is obtained by solving the following equations,

where we de ned

• the modi ed extended cross-correlation operator Q[u, v] : × → , de ned for

It is similar to the usual operator Q[u, v] (equation 2.38), except that a rst order integration replaces the second order derivative;

Note that the integration operator is causal in equation (C.14a) and anti-causal in equation (C.14b) (Claerbout, 2014). They are similar to operators K -and K + de ned in equations (2.19) and (2.35), except that the time integration replaces the second order time derivative;

• the adjoint W * mod : → of W mod , de ned as

• and the adjoint W * r : → obs of W r , de ned as

Note that only the value of ν 1 is required and µ 0 does not need to be computed. If we remove the weighting operators, that is W mod = I , W * r = M, K -= K -and Ω = Ω, then equations (C.12) are equivalent to the usual rst-order extended Born modelling. It means that in terms of implementation, (F † ) T and F have the same structure.

Appendix E.

Product of the Hessian of the MVA objective function with a vector of the -space

In this appendix we consider the case of non-linear iterations on the background velocity model c 0 to determine a minimum of the MVA objective function J 1 [c 0 ]. In the following calculations, we consider only the case of direct inversion of primary re ections, the re ectivity image ξ being de ned by the application of operator F † to observed data P obs . We keep a very general notation for the expression of J 1 ,

so that further expressions are applicable to the standard de nition J 1 [c 0 ] = Aξ 2 , to a normalised version J 1 [c 0 ] = Aξ 2 / ξ 2 and to similar versions with the introduction of a power of the background velocity model c β 0 as proposed in section 4.2. In Newton optimisation methods (appendix B.3), the descent direction d ∈ is de ned as the solution of the following linear system, called Newton equation,

where c

(n) 0 is the value of the background velocity model at the nth non-linear iteration. Most application of MVA do not solve this equation and use a quasi-Newton such as l-BFGS. In Liu et al. (2014b) and Shen and Symes (2015), d is de ned as the opposite of the gradient elementwise divided by the product of the Hessian with the unit vector of the -space. One could also use the truncated-Newton method, already studied in the framework of Full Waveform Inversion by [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF]. It consists of iteratively solving equation (E.2) with the linear conjugate-gradient algorithm, without explicitly computing the full Hessian matrix. Then several products of the Hessian matrix with a vector V ∈ are required. An e cient procedure to compute this product should then be de ned.

We propose here a method based on the adjoint-state method [START_REF] Plessix | A Review of the Adjoint-State Method for Computing the Gradient of a Functional with Geophysical Applications[END_REF], very similar to the one used by [START_REF] Métivier | Full Waveform Inversion and the Truncated Newton Method[END_REF] for FWI and to the method presented in section 3.2.3 to compute the Hessian-vector product in the case of the migration objective function c 0 . Given a vector V of , we de ne the scalar objective function Γ[c 0 , V] as