Analysis of 'Wave-equation' Imaging of Reflection Seismic Data with Curvelets
October 21, 2005
- Wave equation
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.