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1. Velocity Model Building for Depth Imaging of Carbonate Reservoirs to the Deep Salt Incorporating Walkaway VSP and Gravity-Magnetic Data From OBC Survey Offshore Abu Dhabi

Author

Waqas, Muhammad, additional, Hou, Lian, additional, Prieux, Vincent, additional, Meffre, Adrien, additional, Sabetian, Raffaela, additional, Prigent, Hervé, additional, Alkaabi, Ahmed Saeed, additional, van Kleef, Franciscus, additional, Jan, Kamran Saeed, additional, Alkobaisi, Abdulla Saad, additional, and Aljaberi, Salem Mubarak, additional

Published
2023
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2. Revival of Legacy Land Seismic Surveys Using Advanced Processing Technologies: An Example from the Carpathian Foothills

Author

Meffre, Adrien, primary, Prieux, Vincent, additional, Retailleau, Matthieu, additional, Meur, David Le, additional, Monteiro, Abel Afonso, additional, Bouzouita, Zied, additional, Wang, Fang, additional, Mestiri, Sofia, additional, Markos, Tünde, additional, Vermeulen, Justin, additional, Orosz, Jozsef, additional, and Tyler, Emma, additional

Published
2022
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3. Modelling Seismic Wave Propagation for Geophysical Imaging

Author

Virieux, Jean, primary, Etienne, Vincent, additional, Cruz-Atienza, Victor, additional, Brossier, Romain, additional, Chaljub, Emmanuel, additional, Coutant, Olivier, additional, Garambois, Stphane, additional, Mercerat, Diego, additional, Prieux, Vincent, additional, Operto, Stphane, additional, Ribodetti, Alessandra, additional, and Tago, Josu, additional

Published
2012
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5. Imagerie sismique des milieux visco-acoustiques et visco-élastiques à deux dimensions par stéréotomographie et inversion des formes d'ondes: applications au champ pétrolier de Valhall

Author

Prieux, Vincent, Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Université Nice Sophia Antipolis, and Stéphane Operto et Jean Virieux(operto@geoazur.obs-vlfr.fr,Jean.Virieux@obs.ujf-grenoble.fr)

Subjects
Construction de macromodèles, [SDU.STU.GP]Sciences of the Universe [physics]/Earth Sciences/Geophysics [physics.geo-ph], High-resolution quantitative seismic imaging, Données multicomposantes, [SDE.MCG]Environmental Sciences/Global Changes, Multicomponent data, Imagerie sismique quantitative haute résolution, [PHYS.PHYS.PHYS-GEO-PH]Physics [physics]/Physics [physics]/Geophysics [physics.geo-ph], Tomographie multiparamètres, Local optimization, Stereotomography, Multiparameter tomography, Macromodel reconstruction, Valhall oilfield, Non-linear inverse problem, Optimisation locale, Champ pétrolier de Valhall, Stéréotomographie, Problème inverse non-linéaire
Abstract

Geophysics is a very insightful field to gain inferences on the internal structure of the Earth at different scales, and on the past events related to its slow evolution. It also constitutes a major issue in our society, as the resources are becoming increasingly rare, and because it is of great consideration for town and country planning. Full waveform inversion is an optimization process differs from other seismic imaging approaches by its ability to extract the full information content of the seismic data to construct a high-resolution quantitative imaging of one or more physical parameters. I discuss in my thesis several issues related to this method and I illustrate them with applications to multi-component Ocean Bottom Cable (OBC) data recorded at the Valhall oilfield in North Sea. I first discuss the footprint of anisotropy in vertical transverse isotropic media on the seismic Valhall dataset by comparing the subsurface models built by full waveform inversion when anisotropy is taken into account or not in the seismic modeling. I show some biases in the velocity reconstruction when the imaging is performed in the isotropic approximation, these biases leading to mispositioning of the reflectors at depth. %The meaning of the velocities is different in subsurface and in depth, where they are representative of horizontal and NMO velocities respectively. I then address the problem of constructing an initial velocity model for full waveform inversion from wide-aperture data. The selected approach is an extension of stereotomography, a slope tomographic method, where the traveltimes and the slopes of the reflected and refracted arrivals are sequentially introduced in the inversion process through a multiscale approach. The potential of the method is discussed based on the synthetic Valhall case study before the application to the real Valhall data-set. The final chapters are devoted to the reconstruction of several classes of parameters within the visco-acoustic and visco-elastic approximations. In order to reduce the nonlinearity of the inverse problem, I propose a methodology based on the hierarchical reconstruction of the parameter classes, and on the progressive introduction of the different data components in the inversion process. I first test different inversion strategies on the synthetic Valhall case to build the compressive velocity, the density and the attenuation parameters, before their application to the real data within the visco-acoustic approximation. I reconstruct in a second step the shear velocity from the three components of the sensor (one hydrophone and a vertical and horizontal geophones). Quality of the results is assessed with various tools, and a geological interpretation of the results is proposed.; La géophysique interne est une discipline riche en enseignements sur la structure de la Terre à différentes échelles, et sur les phénomènes passés liés à sa lente évolution. Elle constitue de plus un enjeu présent et d'avenir de première importance dans notre société, à l'heure où les ressources de toutes sortes se font de plus en plus rares, et dans le cadre d'une meilleure gestion de l'aménagement du territoire. La méthode d'inversion des formes d'ondes complètes, fondée sur un processus d'optimisation local, se distingue des autres méthodes d'imagerie sismique par sa vocation à extraire de manière aussi complète que possible l'intégralité de l'information sismique, afin de construire une image quantitative haute résolution d'un ou plusieurs paramètres physiques. Je discute dans ma thèse de plusieurs problématiques liées à cette méthode, en les illustrant par des applications à des données multicomposantes enregistrées par des câbles de fond de mer (OBC) sur le champ pétrolier de Valhall en mer du nord. Je discute tout d'abord de l'empreinte de l'anisotropie engendrée par des milieux transverses isotropes à axe de symétrie vertical sur les données du champ de Valhall. J'illustre cette empreinte sur les résultats de l'imagerie en comparant les modèles du sous-sol obtenus lorsque l'anisotropie est prise en compte ou pas dans la modélisation sismique. Je mets en évidence un biais dans les vitesses reconstruites par une méthode d'inversion isotrope, ce biais induisant un mauvais positionnement des réflecteurs en profondeur. %Les vitesses n'ont pas la même signification près de la surface et en profondeur, où elles sont représentatives des vitesses horizontales et NMO respectivement. J'aborde ensuite le problème de la construction du modèle initial nécessaire à l'inversion des formes d'ondes à partir de données à grands déports. L'approche sélectionnée est une extension de la stéréotomographie, une méthode de tomographie de pente, où les attributs temps de trajet+pente des ondes réfractées et réfléchies sont inversées au sein d'un algorithme hiérarchique multi-échelle. Le potentiel de la méthode est discuté à partir d'un cas synthétique représentatif du champ pétrolier de Valhall, avant l'application aux données réelles du champ de Valhall.

Published
2012

6. On the footprint of anisotropy on isotropic full waveform inversion: the Valhall case study

Author

Prieux, Vincent, Brossier, Romain, Gholami, Yaser, Operto, Stéphane, Virieux, Jean, Barkved, O.I., Kommedal, J.H., Géoazur (GEOAZUR 6526), Institut de Recherche pour le Développement (IRD)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de la Côte d'Azur, COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Risques (Risques), Laboratoire de Géophysique Interne et Tectonophysique (LGIT), Observatoire des Sciences de l'Univers de Grenoble (OSUG), Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Laboratoire Central des Ponts et Chaussées (LCPC)-Centre National de la Recherche Scientifique (CNRS)-Observatoire des Sciences de l'Univers de Grenoble (OSUG), Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Laboratoire Central des Ponts et Chaussées (LCPC)-Centre National de la Recherche Scientifique (CNRS), SEISCOPE consortium, Observatoire de la Côte d'Azur, Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut de Recherche pour le Développement (IRD)-Centre National de la Recherche Scientifique (CNRS), Risques, Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Laboratoire Central des Ponts et Chaussées (LCPC)-Institut des Sciences de la Terre (ISTerre), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-PRES Université de Grenoble-Institut de recherche pour le développement [IRD] : UR219-Institut national des sciences de l'Univers (INSU - CNRS)-Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS)-PRES Université de Grenoble-Institut de recherche pour le développement [IRD] : UR219-Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-Observatoire des Sciences de l'Univers de Grenoble (OSUG), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-PRES Université de Grenoble-Institut de recherche pour le développement [IRD] : UR219-Institut national des sciences de l'Univers (INSU - CNRS)-Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR)-Université Joseph Fourier - Grenoble 1 (UJF)-Centre National de la Recherche Scientifique (CNRS)-PRES Université de Grenoble-Institut de recherche pour le développement [IRD] : UR219-Institut Français des Sciences et Technologies des Transports, de l'Aménagement et des Réseaux (IFSTTAR), BP Norge, BP NORGE, Institut de Recherche pour le Développement (IRD)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire de la Côte d'Azur, Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Université Côte d'Azur (UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Central des Ponts et Chaussées (LCPC)-Observatoire des Sciences de l'Univers de Grenoble (OSUG), Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire Central des Ponts et Chaussées (LCPC)-Observatoire des Sciences de l'Univers de Grenoble (OSUG), and Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Computational seismology, Controlled source seismology, Seismic tomography, Wave propagation, [SDU.STU.GP]Sciences of the Universe [physics]/Earth Sciences/Geophysics [physics.geo-ph], [SDU.STU]Sciences of the Universe [physics]/Earth Sciences, Inverse theory, Seismic anisotropy
Abstract

International audience; The validity of isotropic approximation to perform acoustic full waveform inversion (FWI) of real wide-aperture anisotropic data can be questioned due to the intrinsic kinematic inconsistency between short- and large-aperture components of the data. This inconsistency is mainly related to the differences between the vertical and horizontal velocities in vertical-transverse isotropic (VTI) media. The footprint of VTI anisotropy on 2-D acoustic isotropic FWI is illustrated on a hydrophone data set of an ocean-bottom cable that was collected over the Valhall field in the North Sea. Multiscale FWI is implemented in the frequency domain by hierarchical inversions of increasing frequencies and decreasing aperture angles. The FWI models are appraised by local comparison with well information, seismic modelling, reverse-time migration (RTM) and source-wavelet estimation. A smooth initial VTI model parameterized by the vertical velocity V0 and the Thomsen parameters δ and ε were previously developed by anisotropic reflection traveltime tomography. The normal moveout (inline image) and horizontal (inline image) velocity models were inferred from the anisotropic models to perform isotropic FWI. The VNMO models allows for an accurate match of short-spread reflection traveltimes, whereas the Vh model, after updating by first-arrival traveltime tomography (FATT), allows for an accurate match of first-arrival traveltimes. Ray tracing in the velocity models shows that the first 1.5 km of the medium are sampled by both diving waves and reflections, whereas the deeper structure at the reservoir level is mainly controlled by short-spread reflections. Starting from the initial anisotropic model and keeping fixed δ and ε models, anisotropic FWI allows us to build a vertical velocity model that matches reasonably well the well-log velocities. Isotropic FWI is performed using either the NMO model or the FATT model as initial model. In both cases, horizontal velocities are mainly reconstructed in the first 1.5 km of the medium. This suggests that the wide-aperture components of the data have a dominant control on the velocity estimation at these depths. These high velocities in the upper structure lead to low values of velocity in the underlying gas layers (either equal or lower than vertical velocities of the well log), and/or a vertical stretching of the structure at the reservoir level below the gas. This bias in the gas velocities and the mispositioning in depth of the deep reflectors, also shown in the RTM images, are required to match the deep reflections in the isotropic approximation and highlight the footprint of anisotropy in the isotropic FWI of long-offset data. Despite the significant differences between the anisotropic and isotropic FWI models, each of these models produce a nearly-equivalent match of the data, which highlights the ill-posedness of acoustic anisotropic FWI. Hence, we conclude with the importance of considering anisotropy in FWI of wide-aperture data to avoid bias in the velocity reconstructions and mispositioning in depth of reflectors. Designing a suitable parameterization of the VTI acoustic FWI is a central issue to manage the ill-posedness of the FWI.

Published
2011
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13. Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation.

Author

Prieux, Vincent, Brossier, Romain, Operto, Stéphane, and Virieux, Jean

Subjects
*WAVE analysis, *OCEAN bottom, *DATA analysis, *ATTENUATION of seismic waves, *SEISMIC wave velocity, *IMAGE compression, *INVERSIONS (Geology)
Abstract

Multiparameter full waveform inversion (FWI) is a challenging quantitative seismic imaging method for lithological characterization and reservoir monitoring. The difficulties in multiparameter FWI arise from the variable influence of the different parameter classes on the phase and amplitude of the data, and the trade-off between these. In this framework, choosing a suitable parametrization of the subsurface and designing the suitable FWI workflow are two key methodological issues in non-linear waveform inversion. We assess frequency-domain visco-acoustic FWI to reconstruct the compressive velocity (VP), the density (ρ) or the impedance (IP) and the quality factor (QP), from the hydrophone component, using a synthetic data set that is representative of the Valhall oil field in the North Sea. We first assess which of the (VP, ρ) and (VP, IP) parametrizations provides the most reliable FWI results when dealing with wide-aperture data. Contrary to widely accepted ideas, we show that the (VP, ρ) parametrization allows a better reconstruction of both the VP, ρ and IP parameters, first because it favours the broad-band reconstruction of the dominant VP parameter, and secondly because the trade-off effects between velocity and density at short-to-intermediate scattering angles can be removed by multiplication, to build an impedance model. This allows for the matching of the reflection amplitudes, while the broad-band velocity model accurately describes the kinematic attributes of both the diving waves and reflections. Then, we assess different inversion strategies to recover the quality factor QP, in addition to parameters VP and ρ. A difficulty related to attenuation estimation arises because, on the one hand the values of QP are on average one order of magnitude smaller than those of VP and ρ, and on the other hands model perturbations relative to the starting models can be much higher for QP than for VP and ρ during FWI. In this framework, we show that an empirical tuning of the FWI regularization, which is adapted to each parameter class, is a key issue to correctly account for the attenuation in the inversion. We promote a hierarchical approach where the dominant parameter VP is reconstructed first from the full data set (i.e. without any data preconditioning) to build a velocity model as kinematically accurate as possible before performing the joint update of the three parameter classes during a second step. This hierarchical imaging of compressive wave speed, density and attenuation is applied to a real wide-aperture ocean-bottom-cable data set from the Valhall oil field. Several geological features, such as accumulation of gas below barriers of claystone and soft quaternary sediment are interpreted in the FWI models of density and attenuation. The models of VP, ρ and QP that have been developed by visco-acoustic FWI of the hydrophone data can be used as initial models to perform visco-elastic FWI of the geophone data for the joint update of the compressive and shear wave speeds. [ABSTRACT FROM AUTHOR]

Published
2013
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14. Hierarchical Approach of Seismic Full Waveform Inversion

Author

Institut des sciences de la Terre (ISTerre) ; INSU - OSUG - Université de Savoie - Université Joseph Fourier - Grenoble I - IFSTTAR - Institut de recherche pour le développement [IRD] : UR219 - CNRS, Géoazur (GEOAZUR) ; CNRS - Institut de recherche pour le développement [IRD] - INSU - Université Nice Sophia Antipolis (UNS) - Université Pierre et Marie Curie (UPMC) - Paris VI - Observatoire de la Côte d'Azur, GENCI- [CINES/IDRIS] (grant 2010-046091)., Asnaashari, A., Brossier, Romain, Castellanos, C., Dupuy, B., Etienne, V., Gholami, Y., Hu, G., Métivier, L., Operto, Stéphane, Pageot, D., Prieux, Vincent, Ribodetti, A., Roques, Aurélien, Virieux, Jean, Institut des sciences de la Terre (ISTerre) ; INSU - OSUG - Université de Savoie - Université Joseph Fourier - Grenoble I - IFSTTAR - Institut de recherche pour le développement [IRD] : UR219 - CNRS, Géoazur (GEOAZUR) ; CNRS - Institut de recherche pour le développement [IRD] - INSU - Université Nice Sophia Antipolis (UNS) - Université Pierre et Marie Curie (UPMC) - Paris VI - Observatoire de la Côte d'Azur, GENCI- [CINES/IDRIS] (grant 2010-046091)., Asnaashari, A., Brossier, Romain, Castellanos, C., Dupuy, B., Etienne, V., Gholami, Y., Hu, G., Métivier, L., Operto, Stéphane, Pageot, D., Prieux, Vincent, Ribodetti, A., Roques, Aurélien, and Virieux, Jean

Abstract

Full waveform inversion (FWI) of seismic traces recorded at the free surface allows the reconstruction of the physical parameters structure on the underlying medium. For such a reconstruction, an optimization problem is defined, where synthetic traces, obtained through numerical techniques as finite-difference or finite-element methods in a given model of the subsurface, should match the observed traces. The number of data samples is routinely around 1 billion for 2D problems and 1 trillion for 3D problems while the number of parameters ranges from 1 million to 10 million degrees of freedom. Moreover, if one defines the mismatch as the standard least-squares norm between values sampled in time/frequency and space, the misfit function has a significant number of secondary minima related to the ill-posedness and the nonlinearity of the inversion problem linked to the so-called cycle skipping. Taking into account the size of the problem, we consider a local linearized method where gradient is computed using the adjoint formulation of the seismic wave propagation problem. Starting for an initial model, we consider a quasi-Newtonian method, which allows us to formulate the reconstruction of various parameters such as P and S waves velocities or density or attenuation factors. A hierarchical strategy based on the incremental increase of the data complexity starting from low-frequency content to high-frequency content, from initial wavelets to later phases in the data space from narrow azimuths to wide azimuths and from simple observables to more complex ones. Different synthetic examples on realistic structures illustrate the efficiency of this strategy based on the data manipulation. This strategy related to the data space has to be inserted into a more global framework where we could improve significantly the probability to converge to the global minimum. When considering the model space, we may rely on the construction of the initial model or add constraints such as sm

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