True amplitude one-way propagation in heterogeneous media

This paper deals with the numerical analysis of two one-way systems derived from the general complete modeling proposed by M.V. De Hoop. The main goal of this work is to compare two different formulations in which a correcting term allows to improve the amplitude of the numerical solution. It comes out that even if the two systems are equivalent from a theoretical point of view, nothing of the kind is as far as the numerical simulation is concerned. Herein a numerical analysis is performed to show that as long as the propagation medium is smooth, both the models are equivalent but it is no more the case when the medium is associated to a quite strongly discontinuous velocity

Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter to be recovered is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing. We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables and denoising). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the Full Waveform Inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method is efficient for the challenging reconstruction of media with salt-domes. We employ the method in two and three-dimensional experiments and show that the eigenvector representation compensates for the lack of low frequency information, it eventually serves us to extract guidelines for the implementation of the method.Comment: 45 pages, 37 figure

Signal and noise in helioseismic holography

Helioseismic holography is an imaging technique used to study heterogeneities and flows in the solar interior from observations of solar oscillations at the surface. Holograms contain noise due to the stochastic nature of solar oscillations. We provide a theoretical framework for modeling signal and noise in Porter-Bojarski helioseismic holography. The wave equation may be recast into a Helmholtz-like equation, so as to connect with the acoustics literature and define the holography Green's function in a meaningful way. Sources of wave excitation are assumed to be stationary, horizontally homogeneous, and spatially uncorrelated. Using the first Born approximation we calculate holograms in the presence of perturbations in sound-speed, density, flows, and source covariance, as well as the noise level as a function of position. This work is a direct extension of the methods used in time-distance helioseismology to model signal and noise. To illustrate the theory, we compute the hologram intensity numerically for a buried sound-speed perturbation at different depths in the solar interior. The reference Green's function is obtained for a spherically-symmetric solar model using a finite-element solver in the frequency domain. Below the pupil area on the surface, we find that the spatial resolution of the hologram intensity is very close to half the local wavelength. For a sound-speed perturbation of size comparable to the local spatial resolution, the signal-to-noise ratio is approximately constant with depth. Averaging the hologram intensity over a number $N$ of frequencies above 3 mHz increases the signal-to-noise ratio by a factor nearly equal to the square root of $N$. This may not be the case at lower frequencies, where large variations in the holographic signal are due to the individual contributions of the long-lived modes of oscillation.Comment: Submitted to Astronomy and Astrophysic

Derivation of high order absorbing boundary conditions for the Helmholtz equation in 2D

We present high order absorbing boundary conditions (ABC)for the Helmholtz equation in 2D, that can adapt to any regular shapedsurfaces. The new ABCs are derived by using the technique ofmicro-diagonalisation to approximate the Dirichlet-to-Neumann map.Numerical results on different shapes illustrate the behavior of thenew ABCs along with high-order finite elements

L²(H¹γ) finite element convergence for degenerate isotropic Hamilton–Jacobi–Bellman equations

In this paper we study the convergence of monotone P1 finite element methods for fully nonlinear Hamilton–Jacobi–Bellman equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in a weighted Sobolev space L²(H¹γ(Ω)) to the viscosity solution without assuming uniform parabolicity of the HJB operator

Schémas numérique d'ordre élevé en temps et en espace pour l'équation des ondes du premier ordre. Application à la Reverse Time Migration.

L imagerie du sous-sol par équations d onde est une application de l ingénierie pétrolière qui mobilise des ressources de calcul très importantes. On dispose aujourd hui de calculateurs puissants qui rendent accessible l imagerie de régions complexes mais des progrès sont encore nécessaires pour réduire les coûts de calcul et améliorer la qualité des simulations. Les méthodes utilisées aujourd hui ne permettent toujours pas d imager correctement des régions très hétérogènes 3D parce qu elles sont trop coûteuses et /ou pas assez précises. Les méthodes d éléments finis sont reconnues pour leur efficacité à produire des simulations de qualité dans des milieux hétérogènes. Dans cette thèse, on a fait le choix d utiliser une méthode de Galerkine discontinue (DG) d ordre élevé à flux centrés pour résoudre l équation des ondes acoustiques et on développe un schéma d ordre élevé pour l intégration en temps qui peut se coupler avec la technique de discrétisation en espace, sans générer des coûts de calcul plus élevés qu avec le schéma d ordre deux Leap-Frog qui est le plus couramment employé. Le nouveau schéma est comparé au schéma d ordre élevé ADER qui s avère plus coûteux car il requiert un plus grand nombre d opérations pour un niveau de précision fixé. De plus, le schéma ADER utilise plus de mémoire, ce qui joue aussi en faveur du nouveau schéma car la production d images du sous-sol consomme beaucoup de mémoire et justifie de développer des méthodes numériques qui utilisent la mémoire au minimum. On analyse également la précision des deux schémas intégrés dans un code industriel et appliqués à des cas test réalistes. On met en évidence des phénomènes de pollution numériques liés à la mise en oeuvre d'une source ponctuelle dans le schéma DG et on montre qu'on peut éliminer ces ondes parasites en introduisant un terme de pénalisation non dissipatif dans la formulation DG. On finit cette thèse en discutant les difficultés engendrées par l'utilisation de schémas numériques dans un contexte industriel, et en particulier l'effet des calculs en simple précision.Oil engineering uses a wide variety of technologies including imaging wave equation which involves very large computing resources. Very powerful computers are now available that make imaging of complex areas possible, but further progress is needed both to reduce the computational cost and improve the simulation accuracy. The current methods still do not allow to image properly heterogeneous 3D regions because they are too expensive and / or not accurate enough. Finite element methods turn out to be efficient for producing good simulations in heterogeneous media. In this thesis, we thus chose to use a high order Discontinuous Galerkin (DG) method based upon centered fluxes to solve the acoustic wave equation and developed a high-order scheme for time integration which can be coupled with the space discretization technique, without generating higher computational cost than the second-order Leap Frog scheme which is the most widely used . The new scheme is compared to the high order ADER scheme which is more expensive because it requires a larger number of computations for a fixed level of accuracy. In addition, the ADER scheme uses more memory, which also works in favor of the new scheme since producing subsurface images consumes lots of memory and justifies the development of low-memory numerical methods. The accuracy of both schemes is then analyzed when they are included in an industrial code and applied to realistic problems. The comparison highlights the phenomena of numerical pollution that occur when injecting a point source in the DG scheme and shows that spurious waves can be eliminated by introducing a non-dissipative penalty term in the DG formulation. This work ends by discussing the difficulties induced by using numerical methods in an industrial framework, and in particular the effect of single precision calculations.PAU-BU Sciences (644452103) / SudocSudocFranceF

Analysis of PML formulations for the Maxwell system including pseudo-differential operators : existence and uniqueness results

12 pagesWe consider a formulation of the Maxwell system which involves pseudo-differential operators in time. This formulation has been analyzed in a previous work and it has been proved that it is a Perfectly Matched Layer model for electromagnetic waves propagating into the vacuum. We establish that the system admits a single solution by adapting the Hille-Yosida theory

Time Domain Full Waveform Inversion involving Discontinuous Galerkin approximation

International audienc

Long-Time Stability Analysis of Acoustic Absorbing Boundary Conditions for Regular-Shaped Surfaces

International audienceThis work deals with the stability analysis of a one-parameter family of Absorbing Boundary Conditions (ABC) that have been derived for the acoustic wave equation. We tackle the problem of long-term stability of the wave field both at the continuous and the numerical levels. We first define a function of energy and we show that it is decreasing in time. Its discrete form is also decreasing under a Courant-Friedrichs-Lewy (CFL) condition that does not depend on the ABC. Moreover, the decay rate of the continuous energy can be determined: it is exponential if the computational domain is star-shaped and this property can be illustrated numerically