# Acoustic Wave Imaging with Discontinuities (Joint with IP Seminar)

Tuesday, January 21, 2020 - 2:30pm to 3:30pm

Abstract: Consider a medium (e.g. the human body, or the Earth) governed by the
acoustic wave equation $\partial_t^2 u = c(x)^2\Delta u$, where the
wave speed $c$ is unknown, but assumed piecewise smooth. Given
reflection data --- the response of the medium to incoming waves ---
the aim is to identify $c$ inside the medium. For the human body, $c$
is an ultrasound image; for Earth, $c$ is an image of underground
strata. Discontinuities in $c$ cause waves to scatter --- reflect and
refract --- providing the needed reflection data, but the fact that
waves may scatter multiple times greatly complicates the problem of
reconstructing $c$.

In 2002, Rose developed an iterative "single-sided autofocusing"
procedure in 1D, based on the simple physical idea of replaying
reflected data in reverse, and showed his procedure focuses waves
inside the medium without knowing anything about $c$. Starting with
Rose's autofocusing, we developed the scattering control series, which
produces a similar focused wave, free from multiple reflections, in a
3D acoustic medium. I will briefly introduce the scattering control
series and its interesting properties in both exact and microlocal
settings. Returning to the acoustic imaging problem, I will then show
how to use scattering control's microlocal properties to determine the
location of $c$'s discontinuities in boundary normal coordinates, and
how to use its exact properties to reconstruct $c$ itself.

This research was joint work at Rice University with Vitaly
Katsnelson, Maarten V. de Hoop, and Gunther Uhlmann.

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