# 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

transformations.

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.

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

transformations.

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.

MSC Code:

35L05

Keywords: