IGPP Seminar
Friday, February 17, 2006
Bill Symes, Rice University

Title: Inverse scattering for the wave equation in heterogenous media

Abstract: Active source ("reflection") seismology leads to a version of
the inverse scattering problem for the wave equation. The mechanical
properties of the earth are heterogeneous on all scales. However most
useful work to date depends on linearization of the coefficient - solution
relation about smooth (slowly varying on wavelength scale) reference
coefficients. The nature of this linearized relation is well understood,
up to a point.  The operators mapping perturbations in coefficients to
perturbations in solutions belong in some useful cases to well-behaved
classes of Fourier Integral Operators, and this fact underlies the
construction, and explains the behaviour, of operational seismic imaging
algorithms.

Reference coefficients must also be estimated for use in these algorithms,
however, as these are no more known {\em a priori} than are any other
aspects of the earth model. Practical algorithms for estimating reference
coefficients (velocity analysis) rely on {\em extensions} of the
coefficient - solution map, and may be viewed as infeasible point
formulations of the inverse problem. There exist at least two inequivalent
extensions, one of which has much better global properties than the other.
I will describe these extended coefficent - solution maps, explain how
they are used to estimate the reference coefficients in linearized
scattering, sketch a possible generalization to nonlinear inverse
scattering, and mention a few of the mathematical puzzles that abound in
this subject.