PhD project

Subject : Eulerian approach of Hamilton-Jacobi equation with a Discontinuous Galerkin method in heterogeneous anisotropic media: Application to seismic imaging.

Supervisors : Jean Virieux (ISTerre), Ludovic Métivier (ISTerre - LJK)

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**Summary:

The aim of this project is to develop robust and accurate algorithms for wave propagation modeling in the high-frequency regime, mainly focusing on solving Eikonal and transport equations, yielding high-order approximations for traveltimes and amplitudes in heterogeneous anisotropic media, and eventually asymptotic Green’s functions reconstruction.

For this I combine state-of-the-art tools coming from two separate communities:

– computational techniques from geophysics that are related to TTI anisotropy, point-source factorization, fast-sweeping methods, mainly applied in finite-difference framework;
– discontinuous Galerkin finite-element schemes related to Hamilton—Jacobi equations in the field of applied mathematics.

*****This work can be extended to various scales as well as other kinds of waves such as electromagnetic waves for GPR (ground penetrating radar) data processing or medical imaging.

This combination yields highly accurate numerical estimation of traveltime and its spatial derivatives, leading to very good approximations of derived quantities like amplitudes, take-off angles, or adjoint-state variables. Moreover, the finite-element approach naturally handles problems with complex geometry, allowing for a nice accounting of topography.

In terms of applications, these numerical developments allow us to revisit two techniques related to seismology and seismics:

– first traveltime tomography based on adjoint computation with complex free surfaces (topographies) for velocity model building, prior or not to a full-waveform inversion process;
– Kirchhoff and Gaussian beam migration based on asymptotic Green’s functions for reflectivity model retrieving.

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