Seismic Imaging - Week 2

Laurent Demanet
Massachusetts Institute of Technology

With two guest lectures and one computer lab by Felix Herrmann (EOS, UBC).

The course will pick up where Cheney and Natterer left it. Multiple scattering series, backprojection in uniform media, and the Radon transform are assumed known from week 1, but may be reviewed if necessary. An effort will be made to harmonize notations and synchronize topics with the lecturers from week 1.

The course will start by a brief description of the experimental setups in global and exploration seismology.

It is then shown how to linearize the seismic inversion problem, and how to solve it in its linearized form. Reflection and transmission of waves at interfaces are explained. The adjoint-state method for seismic imaging, a.k.a. reverse-time migration, is introduced as an imaging operator. A constrained optimization formulation, which uses Lagrange multipliers, motivates the introduction of the adjoint field.

The frequeny formulation of the adjoint-state method is covered next, and establishes the link with filtered backprojection for radar imaging (seen with Cheney on week 1.)

Seismic imaging departs from radar in the sense that the medium in which waves propagate is inhomogeneous. In this context, geometrical optics is a beautiful high-frequency simplification that explains waves through rays and wavefronts. The eikonal equation is derived for the traveltime function.

The modeling and imaging operators are reformulated as generalized Radon transforms (a.k.a. Kirchhoff migration) in the scope of geometrical optics. Some microlocal analysis is performed to quantify how singularities are mapped by these operators. This material essentially settles the seismic imaging problem in the linearized, high-frequency regime.

One lecture will also be devoted to sparse regularization for radar and seismic imaging problems, including a short review of compressed sensing, and methods of $\ell_1$ optimization.