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An \emph {explicit} algorithm for the extrapolation of one-way wavefields is proposed which combines recent developments in information theory and theoretical signal processing with the physics of wave propagation. Because of excessive memory requirements, explicit formulations for wave propagation have proven to be a challenge in {3-D}. By using ideas from ``\emph{compressed sensing}'', we are able to formulate the (inverse) wavefield extrapolation problem on small subsets of the data volume{,} thereby reducing the size of the operators. According {to} compressed sensing theory, signals can successfully be recovered from an imcomplete set of measurements when the measurement basis is \emph{incoherent} with the representation in which the wavefield is sparse. In this new approach, the eigenfunctions of the Helmholtz operator are recognized as a basis that is incoherent with curvelets that are known to compress seismic wavefields. By casting the wavefield extrapolation problem in this framework, wavefields can successfully be extrapolated in the modal domain via a computationally cheaper operatoion. A proof of principle for the ``compressed sensing'' method is given for wavefield extrapolation in {2-D}. The results show that our method is stable and produces identical results compared to the direct application of the full extrapolation operator.

Presented at Cyber-Enabled Discovery and Innovation: Knowledge Extraction as a success story lecture. See for more detail https://www.ipam.ucla.edu/programs/cdi2007/

Seismic imaging involves the solution of an inverse-scattering problem during which the energy of (extremely) large data volumes is collapsed onto the Earth's reflectors. We show how the ideas from 'compressive sampling' can alleviate this task by exploiting the curvelet transform's 'wavefront-set detection' capability and 'invariance' property under wave propagation. First, a wavelet-vaguellete technique is reviewed, where seismic amplitudes are recovered from complete data by diagonalizing the Gramm matrix of the linearized scattering problem. Next, we show how the recovery of seismic wavefields from incomplete data can be cast into a compressive sampling problem, followed by a proposal to compress wavefield extrapolation operators via compressive sampling in the modal domain. During the latter approach, we explicitly exploit the mutual incoherence between the eigenfunctions of the Helmholtz operator and the curvelet frame elements that compress the extrapolated wavefield. This is joint work with Gilles Hennenfent, Peyman Moghaddam, Tim Lin, Chris Stolk and Deli Wang.

An explicit algorithm for the extrapolation of one-way wavefields is proposed which combines recent developments in information theory and theoretical signal processing with the physics of wave propagation. Because of excessive memory requirements, explicit formulations for wave propagation have proven to be a challenge in {3-D}. By using ideas from ``compressed sensing'', we are able to formulate the (inverse) wavefield extrapolation problem on small subsets of the data volume, thereby reducing the size of the operators. According {to} compressed sensing theory, signals can successfully be recovered from an imcomplete set of measurements when the measurement basis is incoherent} with the representation in which the wavefield is sparse. In this new approach, the eigenfunctions of the Helmholtz operator are recognized as a basis that is incoherent with curvelets that are known to compress seismic wavefields. By casting the wavefield extrapolation problem in this framework, wavefields can successfully be extrapolated in the modal domain via a computationally cheaper operatoion. A proof of principle for the ``compressed sensing'' method is given for wavefield extrapolation in 2-D. The results show that our method is stable and produces identical results compared to the direct application of the full extrapolation operator.