Applying Gauss-Newton and Exact Newton method to Full Waveform Inversion

International audienceFull Waveform Inversion (FWI) applications classically rely on efficient first-order optimization schemes, as the steepest descent or the nonlinear conjugate gradient optimization. However, second-order information provided by the Hessian matrix is proven to give a useful help in the scaling of the FWI problem and in the speed-up of the optimization. In this study, we propose an efficient matrix-free Hessian-vector formalism, that should allow to tackle Gauss-Newton (GN) and Exact-Newton (EN) optimization for large and realistic FWI targets. Our method relies on general second order adjoint formulas, based on a Lagrangian formalism. These formulas yield the possibility of computing Hessian-vector products at the cost of 2 forward simulations per shot. In this context, the computational cost (per shot) of one GN or one EN nonlinear iteration amounts to the resolution of 2 forward simulations for the computation of the gradient plus 2 forward simulations per inner linear conjugate gradient iteration. A numerical test is provided, emphasizing the possible improvement of the resolution when accounting for the exact Hessian in the inversion algorithm

Time-angle ocean acoustic tomography using sensitivity kernels: Numerical and experimental inversion results

International audienceIn shallow water acoustic tomography, broadband mid-frequency acoustic waves (1 to 5 kHz) follow multiple ray-like paths to travel through the ocean. Travel-time (TT) variations associated to these raypaths are classically used to estimate sound speed perturbations of the water column using the ray theory. In this shallow water environment, source and receiver arrays, combined with adapted array processing, provide the measurement of directions-of-arrival (DOA) and directions-of-departure (DOD) of each acoustic path as new additional observables to perform ocean acoustic tomography. To this aim, the double-beamforming technique is used to extract the TT, DOA and DOD variations from the array-to-array acoustic records. Besides, based on the first order Born approximation, we introduce the time-angle sensitivity kernels to link sound speed perturbations to the three observable variations. This forward problem is then inverted with the maximum a posteriori method using both the extracted-observable variations and the proposed sensitivity kernels. Inversion results obtained on numerical data, simulated with a parabolic equation code, are presented. The inversion algorithm is performed with the three observables separately, namely TT, DOA and DOD. The three observables are then used jointly in the inversion process. The results are discussed in the context on ocean acoustic tomography

Mesure d'angles par double formation de voies appliquée à la tomographie acoustique sous-marine

National audienceCet article traite de la tomographie acoustique océanique utilisant les angles de départ et les angles d'arrivée des ondes sonores. Dans ce but, deux antennes (en émission et en réception) sont utilisées pour séparer les arrivées acoustiques et mesurer leur temps de trajet, direction de réception et direction d'émission. Ces mesures sont ensuite utilisées dans un processus d'inversion visant à retrouver la carte des variations de vitesse du son au sein du milieu de propagation (guide d'onde océanique peu profond). L'apport de ces travaux est double : (1) ils montrent qu'il est possible de faire de la tomographie acoustique océanique en utilisant les angles d'émission et de réception, ce qui n'avait jamais été fait à ce jour ; (2) ils permettent de se passer de synchronisation émetteur-récepteur haute précision, qui est un problème technique délicat à régler en pratique. Cet article présente des résultats de tomographie acoustique océanique sur des données simulées. Les résultats obtenus à partir des seuls angles d'émission et de réception sont équivalents à ceux obtenus à partir des temps de trajet des ondes acoustiques, nécessitant la synchronisation émetteur-récepteur

A review of the use of optimal transport distances for high resolution seismic imaging based on the full waveform

We consider the high-resolution seismic imaging method called full-waveform inversion (FWI). FWI is a data fitting method aimed at inverting for subsurface mechanical parameters. Despite the large adoption of FWI by the academic and industrial communities, and many successful results, FWI still suffers from severe limitations. From a mathematical standpoint, FWI is a large scale PDE-constrained optimization problem. The misfit function that is used, which measures the discrepancy between observed seismic data and data calculated through the solution of a wave propagation problem, is non-convex. After discretization, the size of the FWI problem requires the use of local optimization solvers, which are prone to converge towards local minima. Thus the success of FWI strongly depends on the choice of the initial model to ensure the convergence towards the global minimum of the misfit function. This limitation has been the motivation for a large variety of strategies. Among the different methods that have been investigated, the use of optimal transport (OT) distances-based misfit functions has been recently promoted. The leading idea is to benefit from the inherent convexity of OT distances with respect to dilation and translation to render the FWI problem more convex. However, the application of OT distances in the framework of FWI is not straightforward, as seismic data is signed, while OT has been developed for the comparison of probability measures. The purpose of this study is to review two methods that were developed to overcome this difficulty. Both have been successfully applied to field data in an industrial framework. Both make it possible to better exploit the seismic data, alleviating the sensitivity to the initial model and to various conventional workflow steps, and reducing the uncertainty attached to the subsurface mechanical parameters inversion.Comment: 18 figure

High resolution quantitative seismic imaging of a strike-slip fault with small vertical o set in clay-rocks from underground galleries. Experimental Platform of Tournemire, France.

Imaging tectonic faults with small vertical offsets in argilittes (clay-rock) using geophysical methods is challenging. In the context of deep radioactive waste disposals, the presence of such faults has to be assessed since they can modify the rock confining properties. In the Tournemire Experimental Platform (TEP, France), fault zones with small vertical offsets and complex shape have been identified from underground works. However, 3D high-resolution surface seismic methods have shown limitations in this context that led us to consider the detection and characterization of the faults directly from underground works. We investigate here the potential of seismic full waveform inversion (FWI) applied in a transmission configuration to image the clay-rock medium in a horizontal plane between galleries, and compare it with first-arrival traveltime tomography (FATT). Our objective is to characterize seismic velocities of a block of argilittes crossed by a subvertical fault zone with a small vertical offset. The specific measurement configuration allows us to neglect the influence of the galleries on the wave propagation and to simplify the problem by considering a 2D isotropic horizontal imaging domain. Our FWI scheme relies on a robust adaptation of early-arrival waveform tomography. The results obtained with FATT and FWI are in accordance and both correlate with the geological observations from the gallery walls and boreholes. We show that even though various simplifications are done in the inversion scheme and only a part of the data is used, FWI allows to get higher resolution images than FATT, and is especially less sensitive to the incomplete illumination as it uses also diffracted energy. The results provided in this study highlight the complexity of the fault zone, showing a complex interaction of the main fault system with a secondary system composed of decimetric fractures associated with the presence of water

Modelling Seismic Wave Propagation for Geophysical Imaging

International audienceThe Earth is an heterogeneous complex media from the mineral composition scale (10−6m) to the global scale ( 106m). The reconstruction of its structure is a quite challenging problem because sampling methodologies are mainly indirect as potential methods (Günther et al., 2006; Rücker et al., 2006), diffusive methods (Cognon, 1971; Druskin & Knizhnerman, 1988; Goldman & Stover, 1983; Hohmann, 1988; Kuo & Cho, 1980; Oristaglio & Hohmann, 1984) or propagation methods (Alterman & Karal, 1968; Bolt & Smith, 1976; Dablain, 1986; Kelly et al., 1976; Levander, 1988; Marfurt, 1984; Virieux, 1986). Seismic waves belong to the last category. We shall concentrate in this chapter on the forward problem which will be at the heart of any inverse problem for imaging the Earth. The forward problem is dedicated to the estimation of seismic wavefields when one knows the medium properties while the inverse problem is devoted to the estimation of medium properties from recorded seismic wavefields

Imagerie sismique à deux dimensions des milieux visco-élastiques par inversion des formes d'ondes : développements méthodologiques et applications

The knowledge of Earth intern structures at different scales is of major interest for economy, humans, environment and science. Several methods have been developed for Earth imaging using seismic wave information. The full waveform inversion attempts to build quantitative high-resolution images of the subsurface physical parameters using the full wavefield, solved as an optimization procedure. In this thesis, I present application of two-dimensional frequency-domain full waveform inversion for imaging visco-elastic parameters from large offsets data. In a first time, methodologies and algorithms are presented. The frequency-domain P-SV waves propagation modelling, the forward problem of the inversion process, is solved with a finite element discontinuous Galerkin method. This method allows a flexible choice of interpolation orders and the use of triangular unstructured meshes. The inverse problem is linearized in order to limit the number of forward problem simulations, and solved with the quasi-Newton L-BFGS algorithm in order to exploit information contained in the estimated Hessian matrix. The full imaging process is implemented in a massively-parallel algorithm for the distributed-memory architectures of modern computing centres. In a second time, algorithms are applied to several case studies. Applications are performed in realistic synthetic models, representative of onshore and offshore environments. These studies show the difficulties associated to elastic parameters imaging from complex data including converted waves, multiples and surfaces waves. Several multi-scale hierarchical procedures are proposed in order to mitigate non-linearity of the inverse problem and to improve convergence toward the global minimum. Finally, a sensitivity study is performed to analyse the behaviour of several minimization norms and criteria, when noisy data are inverted. A first application in realistic synthetic models is presented before an acoustic application to field data. These tests show some limits of the classic L2 norm in the data space, while the L1 norm appears to be a robust alternative for frequency-domain inversion of decimated data.La connaissance des structures internes de la Terre, à différentes échelles, présente des enjeux majeurs d'ordres économiques, humains, environnementaux et scientifiques. Diverses méthodes d'imagerie ont été développées en utilisant les informations contenues dans les ondes sismiques. La méthode d'inversion des formes d'ondes construit des images quantitative haute résolution des paramètres physiques du sous-sol, en exploitant le champ d'onde complet, sous la forme d'un problème d'optimisation. Dans ce travail de thèse, je présente l'application de l'inversion des formes d'ondes en domaine fréquentiel, pour imager les paramètres visco-élastiques dans des géometries à deux dimensions à grands offsets. Dans un premier temps les développements méthodologiques et algorithmiques sont présentés. La modélisation de la propagation des ondes P-SV en domaine fréquentiel, le problème direct du processus d'imagerie, est assurée par une méthode d'éléments finis Galerkin discontinus, assurant une grande flexibilité dans le choix des ordres d'interpolation et dans l'utilisation de maillages triangulaires non-structurés. Le problème inverse est résolu sous une forme linéarisée, afin de limiter le nombre de simulations directes, et utilise l'algorithme quasi-Newton L-BFGS permettant de tirer bénéfice de l'estimation "économique" du Hessien. Le processus global d'imagerie est implémenté sous la forme d'un algorithme massivement parallèle destiné aux calculateurs modernes à mémoire distribuée. Dans un deuxième temps, les algorithmes développés sont appliqués à des cas d'étude. Des applications sont menées dans des modèles synthétiques réalistes représentatifs d'environnements terrestres et marins. Ces études montrent les difficultés associées à la reconstruction des paramètres élastiques à partir de données mettant en jeu des phénomènes de propagations complexes (ondes converties, multiples, ondes de surfaces...). Des solutions sont proposées sous forme de processus hiérarchiques multi-échelles, afin de limiter les effets des non-linéarités du problème inverse et ainsi d'améliorer la convergence du processus vers le minimum global. Enfin, la sensibilité de différentes normes et critères de minimisation est analysée, à partir de données bruités issues de modèles synthétiques réalistes, ainsi que sous l'approximation acoustique pour un jeu de données réelles pétrolière. Ces tests montrent certaines limites du formalisme classique basé sur la norme L2 dans l'espace des données, tandis que la norme L1 apparaît comme alternative robuste pour l'inversion de données décimées en domaine fréquentiel

Two-dimensional frequency-domain visco-elastic full waveform inversion: Parallel algorithms, optimization and performance

International audienceFull waveform inversion (FWI) is an appealing seismic data-fitting procedure for the derivation of high-resolution quantitative models of the subsurface at various scales. Full modelling and inversion of visco-elastic waves from multiple seismic sources allow for the recovering of different physical parameters, although they remain computationally challenging tasks. An efficient massively parallel, frequency-domain FWI algorithm is implemented here on large-scale distributed-memory platforms for imaging two-dimensional visco-elastic media. The resolution of the elastodynamic equations, as the forward problem of the inversion, is performed in the frequency domain on unstructured triangular meshes, using a low-order finite element discontinuous Galerkin method. The linear system resulting from discretization of the forward problem is solved with a parallel direct solver. The inverse problem, which is presented as a non-linear local optimization problem, is solved in parallel with a quasi-Newton method, and this allows for reliable estimation of multiple classes of visco-elastic parameters. Two levels of parallelism are implemented in the algorithm, based on message passing interfaces and multi-threading, for optimal use of computational time and the core-memory resources available on modern distributed-memory multi-core computational platforms. The algorithm allows for imaging of realistic targets at various scales, ranging from near-surface geotechnic applications to crustal-scale exploration

New insights on the graph space optimal transport distance for full waveform inversion

International audienceNon-convexity issues in full waveform inversion is a topic still deserving significant research efforts. One direction relies on modifying the function measuring the distance between observed and synthetic data on which is based the full waveform inversion process. Recently, optimal transport distances have been considered to play this role. As optimal transport theory has been developed for the comparison of positive functions, adaptation needs to be brought to apply it to the comparison of seismic data which are oscillatory. Among different propositions, the graph space optimal transport distance consists in considering each seismic trace as a point cloud in a time/amplitude two-dimensional space. The method has shown interesting properties in application both to synthetic and three-dimensional field data. In this abstract, we present new insights on this misfit function. We first provide a theoretical comparison with the dynamic time-warping approach. We propose a novel formulation of the graph space optimal transport problem making its application more flexible. We demonstrate the simple form of the second-order derivatives of the corresponding misfit function, making it possible to use standard preconditioning method such as pseudo-Hessian which is illustrated on a synthetic experiment with the Marmousi model