Analysis of 'Wave-equation' Imaging of Reflection Seismic Data with Curvelets

Friday, October 21, 2005 - 3:50pm - 4:40pm
EE/CS 3-180
Maarten De Hoop (Purdue University)
in collaboration with Gunther Uhlmann and Hart Smith

In reflection seismology one places sources and receivers on the
Earth's surface. The source generates waves in the subsurface that are
reflected where the medium properties vary discontinuously; these
reflections are observed in all the receivers. The data thus obtained
are commonly modeled by a scattering operator in a single scattering
approximation: the linearization is carried out about a smooth
background medium, while the scattering operator maps the (singular)
medium contrast to the scattered field observation. In seismic
imaging, upon applying the adjoint of the scattering operator, the
data are mapped to an image of the medium contrast.

We discuss how multiresolution analysis can be exploited in
representing the process of `wave-equation' seismic imaging. The frame
that appears naturally in this context is the one formed by
curvelets. The implied multiresolution analysis yields a full-wave
description of the underlying seismic inverse scattering problem on
the one the hand but reveals the geometrical properties derived from
the propagation of singularities on the other hand. The analysis
presented here relies on the factorization of the seismic imaging
process into Fourier integral operators associated with canonical

The approach and analysis presented in this talk aids in the
understanding of the notion of scale in the data and how it is coupled
through imaging to scale in - and regularity of - the background
medium. In this framework, background media of limited smoothness can
be accounted for. From a computational perspective, the analysis
presented here suggests an approach that requires solving for the
geometry on the one hand and solving a matrix Volterra integral
equation on the other hand. The Volterra equation can be solved by
recursion - as in the computation of certain multiple scattering
series; this process reveals the curvelet-curvelet interaction in
seismic imaging. The extent of this interaction can be estimated, and
is dependent on the Hölder class of the background medium.
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